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Full text of "Introduction To Radiochemistry"

DAMAGE BOOK 



OU_1 58537 >m 



Introduction to 
Radiochemistry 



Introduction to 
Radiochemistry 



BY 

Gerharf Friedlander 

Chemist/ Brookhaven National Laboratory 
(Visiting Lecturer, Washington University, St. Louis) 

AND 

Joseph W. Kennedy 

Professor of Chemistry 
Washington University, St. Louis 



1949 

JOHN WILEY & SONS, INC., NEW YORK 
CHAPMAN & HALL, LIMITED, LONDON 



COPYRIGHT, 1949 

BY 
JOHN WILEY & SONS, INC. 



All Rights Reserved 

This book or any part thereof must not 
be reproduced in any form without 
the written permission of the publisher. 



PRINTED IN THE UNITED STATES OP AMERICA 



To 
JOYCE CLENNAM STEARNS 



PREFACE 

An increasing number of universities are offering courses in 
radioactivity for chemists. Very likely many teachers and stu- 
dents in these courses feel as we do that there has been no suitable 
textbook for this purpose. There is the very excellent Manual of 
Radioactivity by G. Hevesy and F. A. Paneth; however, advances 
in the science since its last edition, in 1938, have been more than 
any authors should have to expect in one decade. Moreover, no 
recent book on the subject has been written specifically for chem- 
ists. We have tried to prepare a textbook for an introductory 
course in the broad field of radiochemistry, at the graduate or 
senior undergraduate level, taking into account the degree of pre- 
vious preparation in physics ordinarily possessed by chemistry 
students at that level. 

We would like to offer definitions of terms, including radio- 
chemistry, nuclear chemistry, tracer chemistry, and radiation 
chemistry that are heard increasingly today. Unfortunately, the 
meanings of some of these vary from laboratory to laboratory, and 
they are hardly used concisely at all. By one group nuclear chem- 
istry is used to mean all applications of chemistry and nuclear 
physics to each other (including stable-isotope applications) . How- 
ever, to our minds nuclear chemistry emphasizes the reactions of 
nuclei and the properties of resulting nuclear species, just as 
organic chemistry is concerned with reactions and properties of 
organic compounds. We think of tracer chemistry as the field of 
chemical studies made with the use of isotopic tracers, including 
studies of the essentially pure tracers at extremely low concen- 
trations. In the title of this book we have meant the term radio- 
chemistry to include all the fields just described, but to exclude 
stable-isotope tracer applications. Radiation chemistry, which is 
not discussed in this text, deals with the chemical effects produced 
by nuclear and other like radiations, and although it involves some 
of the phenomena of radiochemistry it is really closely related to 
photochemistry. 

Some comments on the order in which the subject matter is 
presented are perhaps appropriate. We believe that the sequence 



viii PREFACE 

of chapters after chapter VI is the logical one; the order of presen- 
tation of the material of the first five chapters is much more nearly 
a matter of individual choice. Our plan, which we have found quite 
teachable, is to use the historical background as a brief introduction 
to the concepts and terminology; this makes the going much easier 
in the succeeding topics. Chapter V actually follows logically 
after chapter I, and nothing in the arrangement of the material 
prevents its introduction there if preferred, but we feel that it 
is more effective first to present further descriptive information 
about atomic nuclei and nuclear reactions than to confront the 
student at this point with the quantitative treatment of growth 
and decay processes. 

The development of the subject matter in this book has grown 
out of an introductory course in radiochemistry, first given in the 
informal "Los Alamos University" in the latter part of 1945 by 
the authors (principally G. F.) with the help of Drs. R. W. Dodson 
and A. C. Wahl, and offered each year since in the Department of 
Chemistry at Washington University, St. Louis, by one of us 
(J. W. K.). The formal teaching of radiochemistry to graduate 
students in chemistry at Washington University is divided into 
two one-semester courses. In the first course the subject matter 
of this book is covered as an introduction to radiochemistry; the 
second semester is a seminar which treats in some detail the appli- 
cations of radioactivity to chemistry. The subject matter of this 
second course is covered in the new reference book Radioactivity 
Applied to Chemistry, edited by Prof. Wahl, and is reviewed only 
very briefly in chapters XI, XII and XIII of our text. The inclu- 
sion of these chapters in our book should not be taken as an 
effort to cover the field of tracer chemistry completely ; rather it is 
meant to provide an introduction to the subject which should prove 
particularly useful in shorter courses in radiochemistry. 

It is a pleasure to express our appreciation for assistance to a 
number of colleagues who have examined the manuscript and 
offered many valuable suggestions, including Professors A. C. 
Wahl, W. S. Koski, and M. D. Kamen; Mr. J. A. Miskel and Dr. 
N. Elliott, and particularly Prof. R. W. Dodson who in addition 
has contributed much to our understanding of the subject matter 
of chapter IX and to its formulation, through his lectures on the 
Statistical Nature of Radioactivity presented at the Los Alamos 
Laboratory. We are grateful both to the Brookhaven National 



PREFACE ix 

Laboratory and to Washington University for cooperation and 
hospitality during the writing of the text. We have also valued 
the assistance of Adrienne Kennedy and Gertrude Friedlander in 
the preparation of the manuscript and in proof-reading. Pro- 
fessors E. Feenberg, G. T. Seaborg, and E. Segr6 were kind 
enough to look over the proofs and to point out a number of 
errors. 

Throughout the book we have drawn upon the work and ideas 
of other investigators, but in keeping with its character as a text- 
book have not attempted to include more than a very few specific 
references. The references listed at the end of each chapter were 
selected to call attention to a large number of standard works and 
to introduce the student to some of the recent literature on specific 
topics. The exercises given at the end of each chapter are in- 
tended as an integral part of the course, and only with them docs 
the text contain the variety of specific examples which we consider 
necessary for an effective presentation. 

GERHART FRIEDLANDER 
JOSEPH W. KENNEDY 
Upton, N. Y. 
St. Louis, Mo. 
March 1949 



CONTENTS 

CHAPTER I 
NATURALLY OCCURRING RADIOACTIVE SUBSTANCES 

A. Discovery of Radioactivity 1 

B. Radioactive Decay and Growth 7 

C. Radioactive Series 10 

CHAPTER II 

ATOMIC NUCLEI 

. Atomic Structure 22 

B. Nuclear Structure 29 

C. Nuclear Properties 34 

D. Isotopy and Isotope Separations 45 

E. Nuclear Systeraatics 47 

CHAPTER III 

NUCLEAR REACTIONS 

A. The Nature of Nuclear Reactions 54 

B. Energetics of Nuclear Reactions 56 

C. Mechanisms and Types of Nuclear Reactions 61 

D. Cross Sections 71 

CHAPTER IV 
SOURCES OF BOMBARDING PARTICLES 

A. Heavy Charged Particles 79 

B. Electrons 90 

C. Gamma Rays and X Rays 96 

D. Neutrons 98 

CHAPTER V 
QUANTITATIVE TREATMENT OF RADIOACTIVE PROCESSES 

A. Exponential Decay 107 

B. Growth of a Radioactive Product 109 

C. More Complicated Cases 116 

D. Units of Radioactivity . 117 

E. Determination of Half-lives 119 



xii CONTENTS 

CHAPTER VI 

TYPES OP RADIOACTIVE DECAY 

Alpha Decay 124 

Beta Decay 130 

Gamma Decay and Isomerism 140 

Other Modes of Decay 144 

CHAPTER VII 

INTERACTION OF RADIATIONS WITH MATTER 

A. Alpha Particles 147 

B. Electrons 156 

C. Electromagnetic Radiation 166 

D. Neutrons * 171 

E. Biological Radiation Units 171 

CHAPTER VIII 

INSTRUMENTS FOR RADIATION DETECTION AND MEASUREMENT 

A. Methods not Based on Ion Collection 176 

B. Saturation Currerft Collection 179 

C. Multiplicative Ion Collection 186 

D. Auxiliary Instruments 194 

E. Health Physics Instruments 195 

CHAPTER IX 

STATISTICAL CONSIDERATIONS IN RADIOACTIVITY MEASUREMENTS 

A. Random Phenomena 199 

B. Probability and the Compounding of Probabilities 200 

C. Radioactivity as a Statistical Phenomenon 203 

D. Poisson and Gaussian Approximations of the Distribution Law 208 

E. Experimental Applications 209 

CHAPTER X 
TECHNIQUES FOR MEASUREMENT AND STUDY OF RADIATIONS 

A. General Techniques 218 

B. Critical Absorption of X Rays 221 

C. Beta-particle Sign Determination 223 

D. Absolute Disintegration Rates 224 

E. Preparation of Samples for Counting 228 



CONTENTS xiii 

CHAPTER XI 

IDENTIFICATION, CONCENTRATION, AND ISOLATION OF RADIOACTIVE SPECIES 

A. General Remarks 236 

B. Identification of Nuclear-reaction Products 237 

C. Chemical Isolation of Non-isotopic Species 244 

D. Chemical Concentration of Isotopic Species: The Szilard-Chalmers 
Process 252 

CHAPTER XII 

CHEMISTRY OF Low CONCENTRATIONS AND THE STUDY OF NEW ELEMENTS 

A. The Low-concentration Region 262 

B. Coprecipitation and Adsorption 263 

C. Other Chemical Properties 267 

D. Discoveries of New Elements by Tracer Methods 269 

CHAPTER XIII 

TRACERS IN CHEMICAL APPLICATIONS 

A. The Tracer Method; Diffusion Studies 278 

B. Exchange Reactions 282 

C. Applications to Analytical Chemistry 290 

D. Radiocarbon Tracer Studies 292 

APPENDIX 

A. Table of Radioactive and Stable Isotopes of the Elements 297 

B. Table of Isotopic Thermal Neutron Activation Cross Sections 390 

C. Table of Thick-target Yields for Some Nuclear Reactions Obtained 

with 14-Mev Deuterons 393 

D. TablS of Physical Constants and Conversion Factors 393 

E. Selected Examination Questions from an Introductory Course in 

Radiochemistry 394 

INDEX 399 



I. NATURALLY OCCURRING RADIOACTIVE SUBSTANCES 
A. DISCOVERY OF RADIOACTIVITY 

BecquerePs Discovery. The more or less accidental series of 
events which led to the discovery of radioactivity depended on 
two especially significant factors: (1) the mysterious X rays dis- 
covered about one year earlier by W. C. Roentgen produced 
fluorescence (the term phosphorescence was preferred at that 
.time) in the glass walls of X-ray tubes, and in some other mate- 
rials; and (2) Henri Becquerel had inherited an interest in phos- 
phorescence from both his father and grandfather. The father, 
Edmund Becquerel (1820-1891), had actually studied phosphores- 
cence of uranium salts, and about 1880 Henri Becquerel prepared 
potassium uranyl sulfate, K 2 UO2(SO 4 )2-2H 2 O, and noted its 
pronounced phosphorescence excited by ultraviolet light. Thus, 
in 1895 and 1896 when several scientists were seeking the connec- 
tion between X rays and phosphorescence and were looking for 
penetrating radiation from phosphorescent substances, it was 
natural for Becquerel to experiment along this line with the potas- 
sium uranyl sulfate. 

It was on February 24, 1896, that Henri Becquerel reported his 
first resists : after exposure to bright sunlight crystals of the uranyl 
double sulfate emitted a radiation which blackened a photographic 
plate after penetrating black paper, glass, and other substances. 
During the next few months he continued the experiments, obtain- 
ing more and more puzzling results. The effect was as strong 
with weak light as with bright sunlight; it was found in complete 
darkness and even for crystals prepared and always kept in the 
dark. The penetrating radiation was emitted by other uranyl 
and also uranous salts, by solutions of uranium salts, and even 
by what was believed to be metallic uranium, and in each case 
with an intensity proportional to the uranium content. Proceed- 
ing by analogy with a known property of X rays Becquerel ob- 
served that the penetrating rays from uranium would discharge 
an electroscope. These results were all obtained in the early part 

1 



2 NATURALLY OCCURRING RADIOACTIVE SUBSTANCES CH. I 

of 1896. Although Becquerel and others continued .investigations 
for several years, the knowledge gained in this phase of the new 
science was summarized in 1898 when Pierre and Marie Sklodowska 
Curie concluded that the uranium rays were an atomic phe- 
nomenon characteristic of the element and not related to its 
chemical or physical state, and they introduced the name "radio- 
activity" for the phenomenon. 



TABLE 1-1 
SOME URANIUM AND THORIUM MINERALS 



Name 
Uraninite 
(pitch- 
blende) 

Thorianite 



Carnotite 
Monazite 

Pilbarite 
Autunite 



Thorite or 
orangite 



Composition 

Uranium oxide, 
UO 2 to U 3 8 , 
with rare-earth 
and other oxides 

Thorium and 
uranium oxides, 
(Th, U)0 2 , with 
UOa and rare- 
earth oxides 

Potassium uranyl 
vanadate, 
K(U0 2 )VO 4 -nH 2 O 

Phosphates of the 
cerium earths 
and thorium, 
CeP0 4 + 

Th 3 (P04)4 

Thorium lead ura- 
nate and silicate 

Calcium uranyl 
phosphate, 
Ca(U02) 2 (PO 4 )r 
8H 2 O 

Thorium ortho- 
silicate, ThSiO 4 



Uranium 
Content 
60-80% 



-45% 



-25% 
-50% 



Thorium 

Content 

0-10% 



4-40% 30-82% 



Up to 
16% 



Up to 

70% 



Color and 
Form 



Grayish, greenish 
or brownish 
black; cubic, 
amorphous 

Gray, brownish, or 
greenish gray, 
black; cubic 



Yellow; 

hexagonal, 

rhombic 
Red, brown, 

yellowish 

brown; 

monoclinic 

Yellow 

Greenish yellow; 
rhombic 



Brown or black 
(thorite), 
orange-yellow 
(orangite); 
tetragonal 



The Curies. Much new information appeared during the year 
1898, mostly through the work of the Curies. Examination of 
other elements led to the discovery, independently by Mme. Curie 



EARLY CHARACTERIZATION OF THE RAYS 3 

and G. C. Schmidt, that compounds of thorium emitted rays 
similar to the uranium rays. A very important observation was 
that some natural uranium ores were even more radioactive than 
pure uranium, and more active than a chemically similar "ore" 
prepared synthetically. The chemical decomposition and frac- 
tionation of such ores was the first exercise in radiochemistry and 
led immediately to the discovery of polonium as a new substance 
observed only through its intense radioactivity and of radium, 
a highly radioactive substance recognized as a new element and 
soon identified spectroscopically. .The Curies and their coworkers 
had found radium in the barium fraction separated chemically 
from pitchblende (a dark almost black ore containing 60 to 90 
per cent U 3 8 ), and they learned that it could be concentrated 
from the barium by repeated fractional crystallization of the 
chlorides, the radium salt remaining preferentially in the mother 
liquor. By 1902 Mme. Curie reported the isolation of 100 mg of 
radium chloride spectroscopically free from barium and gave 
225 as the approximate atomic weight of the element. .(The 
work had started with about two tons of pitchblende, and the 
radium isolated represented about a 25 per cent yield.) Still 
later Mme. Curie redetermined the atomic weight to be 226.5 
(the 1942 value is 226.05) and also prepared radium metal by 
electrolysis of the fused salt. 

Becqucrel in his experiments had shown that uranium, in the 
dark and not supplied with energy in any known way, continued 
for years to emit rays in undiminished intensity. E. Rutherford 
had made some rough estimates of the energy associated with 
the radioactive rays; the source of this energy was quite unknown. 
With concentrated radium samples the Curies made measure- 
ments of the resulting heating effect, which they found to be 
about 100 cal per hr per g of radium. The evidence for so large 
a store of energy not only caused a controversy among the scien- 
tists of that time but also helped to create a great popular interest 
in radium and radioactivity. (An interesting article in the St. Louis 
Post-Dispatch of October 4, 1903, speculated on this inconceivable 
new power, its use in war and as an instrument for destruction of 
the world.) 

Early Characterization of the Rays. The effect of radioactive 
radiations in discharging an electroscope was soon understood in 
terms of the ionization of the air molecules, as J. J. Thomson and 



4 NATURALLY OCCURRING RADIOACTIVE SUBSTANCES CH. I 

others were developing a knowledge of this subject in their studies 
of X rays. The use of the amount of ionization in air as a measure 
of the intensity of radiations was developed into a more precise 
technique than the photographic blackening, and this technique 
was employed in the Curie laboratory, where ionization currents 
were measured with an electrometer. In 1899 Rutherford began 
a study of the properties of the rays themselves, using a similar 
instrument. Measurements of the absorption of the rays in 
metal foils showed that there were two components. One com- 
ponent was absorbed in the first few thousandths of a centimeter 
of aluminum and was named a. radiation; the other was absorbed 
considerably in roughly 100 times this thickness of aluminum 
and was named ft radiation. For the ft rays Rutherford found 
that the ionization effect was reduced to the fraction e' 1 ^ of its 
original value when d centimeters of absorber were interposed; 
the absorption coefficient p was about 15 cm""" 1 for aluminum and 
increased with atomic weight for other metal foils. 

Rutherford at that time believed that the absorption of the 
a radiation also followed an exponential law and gave for it 
H = 1600 cm"" 1 in aluminum. About a year later Mme. Curie 
found that ju was not constant for a rays but increased as the rays 
proceeded through the absorber. This was a very surprising fact, 
since one would have expected that any inhomogeneity of the 
radiation would result in early absorption of the less penetrating 
components with a corresponding decrease in absorption coeffi- 
cient with distance. In 1904 the concept of a definite range for 
the a particles (they were recognized as particles by that time) 
was proposed and demonstrated by W. H. Bragg, He found that 
several radioactive substances emitted a rays with different char- 
acteristic ranges. 

The recognition of the character of the a. and ft rays as streams 
of high-speed particles came largely as a result of magnetic and 
electrostatic deflection experiments. In this way the ft rays 
were seen to be electrons moving with almost the velocity of light. 
At first the a rays were thought to be undeviated by these fields. 
More refined experiments did show deflections; from these the 
ratio of charge to mass was calculated to be about half that of 
the hydrogen ion, with the charge positive, and the velocity was 
calculated to be about one-tenth that of light. The suggestion 
that the a particle was a helium ion immediately arose, and this 



RUTHERFORD AND SODDY TRANSFORMATION HYPOTHESIS 5 

was confirmed after much more study. The presence of helium 
in uranium and thorium ores had already been noticed and was 
seen to be significant in this connection. A striking demonstra- 
tion was later made, in which a rays were allowed to pass through 
a very thin glass wall into an evacuated glass vessel; within a 
few days sufficient helium gas appeared in the vessel to be detected 
spectroscopically. 

Before the completion of these studies of the a. and ft rays, an 
even more penetrating new radiation, not deviated by a magnetic 
field, was found in the rays from radioactive preparations. The 
recognition of this 7 radiation as electromagnetic waves, like 
X rays in character if not in energy, came rather soon. For a 
long time no distinction was made between the nuclear 7 rays 
and some extranuclear X rays which often accompany radio- 
active transformations.^ 

Rutherford and Soddy Transformation Hypothesis. In the 
course of measurements of thorium salt activities Rutherford 
observed that the electrometer readings were sometimes quite 
erratic. During 1899 it was determined that the cause of this 
effect was the diffusion through the ionization chamber of a radio- 
active substance emanating from the thorium compound. Similar 
effects were obtained with radium compounds. Subsequent 
studies, principally by Rutherford and F. Soddy, showed that 
these emanations were inert gases of high molecular weight, 
subject to condensation at about 150C. Another radioactive 
substance, actinium, had been separated from pitchblende in 
1899, and it too was found to give off an active emanation. 

The presence of the radioactive emanations from thorium, 
radium, and actinium preparations was a very fortunate circum- 
stance for advancement of knowledge of the real nature of radio- 
activity. Essentially the inert gaseous character of these sub- 
stances made radiochemical separations not only an easy process, 
but also one which forced itself on the attentions of these early 
investigators. Two very significant consequences of the early 
study of the emanations were: (1) the realization that the activity 
of radioactive substances did not continue forever but diminished 

1 In the nomenclature of this book concerning radioactive decay processes 
the term 7 rays will include only nuclear electromagnetic radiation; accom- 
panying X rays will be designated as such, even though this is not an entirely 
uniform practice in the current literature. 



8 NATURALLY OCCURRING RADIOACTIVE SUBSTANCES CH. I 

in intensity with a time scale characteristic of the substance; 
and (2) the knowledge that the radioactive processes were accom- 
panied by a change in chemical properties of the active atoms. 
The application of chemical separation procedures, especially by 
W. Crookes and by Rutherford and Soddy, in 1900 and the suc- 
ceeding years, revealed the existence of other activities with 
characteristic decay rates and radiations, notably uranium X 
which is separated from uranium by precipitation with excess 
ammonium carbonate (the uranyl carbonate redissolves in excess 
carbonate through formation of a complex ion), and thorium X 
which remains in solution when thorium is precipitated as the 
hydroxide with ammonium hydroxide. In each case it was found 
that the activity of the X body decayed appreciably in a matter 
of days, and that a new supply of the X body appeared in the 
parent substance in a similar time. It was also shown that both 
uranium and thorium, when effectively purified of the X bodies 
and other products, emitted only a rays, and that uranium X 
and thorium X emitted ft rays. 

By the spring of 1903 Rutherford and Soddy had reached an 
excellent understanding of the nature of radioactivity and pub- 
lished their conclusions that the radioactive elements were under- 
going spontaneous transformation from one chemical atom into 
another, that the radioactive radiations were an accompaniment 
of these changes, and that the radioactive process was a subatomic 
change within the atom. However, it should be remembered 
here that the idea of the atomic nucleus did not emerge until 
eight years later and that in 1904 Bragg was attempting to under- 
stand the a particle as a flying cluster of thousands of more or 
less independent electrons. 

Statistical Aspect of Radioactivity. In 1905 E. v. Schweidler 
used the foregoing conclusions as to the nature of radioactivity 
and formulated a new description of the process in terms of dis- 
integration probabilities. His fundamental assumptions were 
that the probability p for a particular atom of a radioactive ele- 
ment to disintegrate in a time interval A is independent of the 
past history and the present circumstances of the atom; it depends 
only on the length of the time interval At and for sufficiently 
short intervals is just proportional to A; thus p = XA, where 
X is the proportionality constant characteristic of that species of 
radioactive atoms. The probability of the given atom not disinte- 



RADIOACTIVE DECAY AND GROWTH 7 

grating during the short interval At is 1 p = 1 \At. If the 
atom has survived this interval, then its probability of not dis- 
integrating in a second like interval is again 1 \At. By the 
law for compounding such probabilities the probability for the 
given atom to survive the first interval and also the second is 
given by (1 XA2) 2 ; for n such intervals this survival probability 

/ t\ n 

is (1 XA2) n . Setting nkt , the total time, we have I 1 X - I . 

V n/ 

Now the probability that the atom will remain unchanged after 
time t is just the value of this quantity when At is made indefinitely 

/ t\ n 

small; that is, it is the limit of ( 1 X - ) as n approaches infinity. 

\ n/ 

( x \ n 

Recalling that e x = lim 1 1 H ) , we have e~ M for the limiting 

n ~* \ n/ 

value. If we consider not one atom, but a large initial number 
NO of the radioactive atoms, then the fraction remaining un- 
changed after time t we may take to be N/N Q = e~, where N 
is the number of unchanged atoms at time t. This exponential 
law of decay is just that which had already been found experi- 
mentally for the simple isolated radioactivities. 

Chapter IX will present a more detailed discussion of the statis- 
tical nature of radioactivity. 



B. RADIOACTIVE DECAY AND GROWTH 

In the preceding section we mentioned that the decay of a 
radioactive substance followed the exponential law N = Noe~, 
where N is the (large) number of unchanged atoms at time t, 
NQ is the number present when t = 0, and X is a constant charac- 
teristic of the particular radioactive species. This will be recog- 
nized as the rate law for any monomolecular reaction, and, of 
course, this should be expected in view of the nature of the radio- 
active process. It may be derived if the decay rate, dN/dt, 
is set proportional to the number of atoms present: dN/dt = \N. 
(This is to say that we expect twice as many disintegrations per 
unit time in a sample containing twice as many atoms, etc.) On 
integration the result is In N = \t + a, and the constant of 
integration a is evaluated from the limit N = NO when t = 0: 
a = In NQ. Combining these terms we have: ]nN/N Q = X, 
or N/No = e~ u . 



8 NATURALLY OCCURRING RADIOACTIVE SUBSTANCES CH. I 

The constant X is known as the decay constant for that radio- 
active species. As may be seen from the differential equation, 
it is the fraction of the number of atoms transformed per unit 
time, provided the time unit is chosen short enough so that only 
a small fraction of the atoms transform in that interval. In any 
case X has the dimensions of a reciprocal time and is most often 
expressed in sec" 1 . It is to be noticed that no attempt to alter 
X through variation of ordinary experimental conditions, such 
as temperature; chemical change; pressure; gravitational, mag- 
netic or electric fields; has ever given a detectable effect. (2) 

The characteristic rate of a radioactive decay may very con- 
veniently be given in terms of the half -life t^ which is the time 
required for an initial (large) number of atoms to be reduced to 
half that number by transformations. Thus, at the time t = ty 2 
N = No/2, and: 

, In 2 0.69315 

In f = -X^, or 1^ = = . 

A A 

In practical work with radioactive materials the number of 
atoms N is not directly evaluated, and even the rate of change 
dN/dt is usually not measured absolutely. The usual procedure 
is to determine, through its electric, photographic, or other effect, 
a quantity proportional to XN; we may term this quantity the 
activity A, with A = c\N = c(dN/df). The coefficient c, 
which we may term the detection coefficient, will depend on the 
nature of the detection instrument, the efficiency for the recording 
of the particular radiation in that particular instrument, and the 
geometrical arrangement of sample and detector; a usual feature 
of the experimentation is careful precaution to keep all these 
factors under control. We may now write the decay law as it is 
commonly observed, A = A e~ x '. 

The usual procedure for the treatment of data measuring A at 
successive times is to plot log A vs. t] for this purpose semilog 
paper (with a suitable number of decades) is most convenient. 
Now X could be found from the slope of the resulting straight line 
corresponding to the simple decay law; however, in this procedure 
there is a possibility for the confusion of units or of different 
logarithm bases. It is more convenient to read from the plot on 

8 A possible exception is mentioned in chapter VI, page 137, footnote 4. 



RADIOACTIVE DECAY AND GROWTH n 

semilog paper the time required for the activity to fall from any 
value to half that value; this is the half-life t w 

In this discussion we have considered only the radioactivity 
corresponding to the transformation of a single atomic species; 
however, the atom resulting from the transformation may itself 
be radioactive, with its own characteristic radiation and half- 
life, as well as its own chemical identity. Indeed among the 
naturally occurring radioactive substances this is the more com- 
mon situation, and eventually (in chapter V) we must treat quite 
complicated interrelated radioactive growths and decays. For 
the moment consider the decay of the substance uranium I, or 
Ui. This species of uranium is an o-particle emitter with 
^ = 4.51 X 10 9 years. The immediate product of its transfor- 
mation is the radioactive substance uranium Xi, or UXi, a 
emitter with half-life 24.6 days. For this pair of substances, the 
parent uranium may be separated from the daughter atoms by 
precipitation of the daughter with excess ammonium carbonate, 
as already mentioned. The daughter precipitate will show a 
characteristic activity, which will decay with the rate indicated; 
that is, it will be half gone in 24.6 days, three fourths gone in 49.2 
days, seven eighths gone in 73.8 days, etc. The parent fraction 
will, of course, continue its a activity as before, but will for the 
moment be free of the @ radiations associated with the daughter. 
However, in time new daughter atoms will be formed, and the 
daughter activity in the parent fraction will return to its initial 
value with a time scale corresponding to the rate of decay of the 
isolated daughter fraction. 

In an undisturbed sample containing NI atoms of Ui, a steady 
state is established in which the rate of formation of the daughter 
UXi atoms (number N 2 ) is just equal to their rate of decay. 
This means that dNi/dt = \2AT 2 in this situation, because the 
rate of formation of the daughter atoms is just the rate of decay 
of the parent atoms. Using the "earlier relation we have then, 
\iNi = X 2 #2> with \i and X 2 the respective disintegration con- 
stants, 'this is sometimes more convenient in terms of the two 
half-lives: N\/(t^)i = AT 2 /(^) 2 . This s ^ a ^ e f affairs is known 
as secular equilibrium. No account is taken of the decrease of 
NI with time, since the fraction of Ui atoms transformed even 
throughout the life of the experimenter is completely negligible. 
In general, wherever a short-lived daughter results from the decay 



10 NATURALLY OCCURRING RADIOACTIVE SUBSTANCES Cn.I 

of a very long-lived parent, this situation exists. The same rela- 
tion, \iNi = A 2 # 2 = >^3, etc., may be applied when several 
short-lived products arise from successive decays beginning with 
a long-lived parent, provided again that the material has been 
undisturbed (that is, no daughter substances removed or allowed 
to escape) for a long enough time for secular equilibrium to be 
established. 

The concept of secular equilibrium suggests a convenient way 
to handle experimental data concerned with the rate of growth 
of a short-lived daughter substance in a freshly separated long- 
lived parent fraction. Because all the rates of decay are entirely 
independent of the chemical manipulations in the separation (say 
of UXi from Ui), the sum of the amounts of daughter UXi in 
the two fractions always continues at the constant value given 
by \\NI = X 2 A^2' Thus by the time the isolated daughter prepara- 
tion is practically inactive the growth in the parent will have 
practically re-established the secular equilibrium condition. If 
measurements of the amount of daughter activity in the parent 
fraction are obtained as a function of the time 2, then these activity 
values may be subtracted from the final value approached as t 
becomes long compared to (< H ) 2 , and the differences plotted on 
semilog paper vs. t to give a straight line like a decay curve. In 
fact this curve describes the decay of the isolated daughter frac- 
tion. In this way the daughter half-life may be obtained from 
its rate of growth in a very long-lived parent. (It may be useful 
here to emphasize that a plot directly of the growing daughter 
activity in the parent fraction, on either a linear or semilog basis, 
cannot give a straight line and in either case gives a curve 
approaching the secular equilibrium value as an asymptote. See 
figure V-3, page 113.) 

C. RADIOACTIVE SERIES 

Uranium, Thorium, and Actinium Series. All elements found in 
natural sources with atomic number greater than 83 (bismuth) 
are radioactive. They belong to chains of successive decays, and 
all the species in one such chain constitute a radioactive family 
or series. Three of these families include all the natural activities 
in this region of the periodic chart. One has Ui (mass 238 on the 
atomic weight scale) as the parent substance, and after 14 trans- 



RADIOACTIVE SERIES 11 

formations (8 of them by a-particle emission and 6 by 0-particle 
emission) reaches a stable end product, radium G (lead with 
mass 206); this is known as the uranium series. (This series 
includes radium and its decay products; these are sometimes 
called the radium series.) Since the atomic mass is changed by 
four units in a decay and changed much less than one unit by /3 
decay, the various masses found in members of the family differ 
by multiples of 4, and a general formula for the approximate 
masses is 4n + 2, where n is an integer. Therefore, the uranium 
series is known also as the 4n + 2 series. Figure 1-1 shows the 
members and transformations of the uranium series. The exist- 
ence of branching decays should be noticed; it is almost certain 
that very many more branchings would be found if sufficiently 
sensitive means of recognizing them were available. 

Thorium (mass 232) is the parent substance of the 4?i series, 
or thorium series, with lead of weight 208 as the stable end product. 
This series is shown in figure 1-2. The 4n + 3, or actinium, 
series has actino-uranium, AclJ (uranium of mass 235), as the 
parent and lead of mass 207 as the stable end product. This 
series is shown in figure 1-3. 

The fairly close similarity of the three series to each other and 
in their relations to the periodic chart is interesting and helpful 
in remembering the decay schemes and nomenclature for the 
active bodies. Actually, these historical names may some day 
become obsolete, and the designations of chemical element and 
atomic mass become standard; already we are more familiar with 
U 238 , U 235 , and U 234 than with U r , AcU, and U n . (This trend 
is favored by the fact that names like UXi and RaD do not imme- 
diately suggest that these substances are chemically like thorium 
and lead, respectively; also, in some of the early literature the 
nomenclature is different from current usage, which leads to some 
confusion. On the other hand, many of the historical names 
like RaA, RaB indicate immediately positions in the decay chain; 
and, further, the name Pa 234 would not distinguish between UX 2 
and UZ.) 

The 4n + 1 Series and Synthetic Transuranium Elements ; Addi- 
tional Natural Activities. In past years much comment has been 
given to the failure to find in nature a 4n + 1 radioactive series; 
the most plausible presumption has been that no member of this 
series was sufficiently long-lived to survive the many years 



12 NATURALLY OCCURRING RADIOACTIVE SUBSTANCES CH. I 





u^.u, 


U 234 ,^ 




u 


uranium I) 


uranium II) 




92 


4.51 xlO 9 


2.33 xlO 5 






years 


/ 


years 






/ Pa 234 ,UX2 ft 


/ 




Pa 
91 


/l.Uminutes, 
a 1 


(99.85%) 
I.T. 
' (0.15%) 


a 


/ t>o234TT7 < 

//I ra ,U j 
j p 6.7 hours / 




Th 234 ,^ 


Th'Mo 




Th 
90 


uraniumX}) 


(ionium) 
8.3 xlO 4 






24.5 days 




years 




Ac 


a 


89 


Ra 

88 




(radium) 








1590 years 




Pr 






87 










Rn^.Rn 




Ri 










86 




(radon) 








3.825 days 






a 


At 218 




At 






85 


few 






, seconds 




Po 
84 




Po 218 RaA 

(radium A) 
minutes 


ft 
(0.03%) 
a 


Po 214 , RaC' Po 210 ,RaF 
(radiumC') (polonium) 
1.5X10- 4 140 days 
.seconds / 


Bi 
83 




a 

(99.97%) 


Bi 2M ,RaC 

(radium C) 
19.7 


3 >210 x ^3 

99.%%) Bi RaE HOO%) t 
a (radium E) 








,, minutes 


^5.0 days 






Pb^.RaB 


ft 


Pb m .EaD^ Pb^RaG 


Pb 
82 




(radium B) 
26.8 
minutes 


(0.04%) 


(radium D) (5x (stable lead 
22years 10^%) isotope) 


81 




(radium C*) 


(radium E') 




1.32 

minutes 


minutes 



FIGURE 1-1. The uranium series. 



RADIOACTIVE SERIES 



13 



Th 
90 


Th^Th 
(thorium) 

1.39 xlO 10 
years 




Th^.RdTh 

(radiothorium) 
j 1.90 years 






Ac 
89 


or 


Ac^MsTh/ 

(meso- 
thorium2) 

4 6.13 hours 





a 






Ha 
88 


Ra^MsThf 
(meso- 
thorium 1) 

6.7 years 



a 


Ra^.ThX 
(thorium X) 
3.64 days 






Fr 
87 








Of 






Rn 

86 






Rn^.Tn 
(thoron) 
54.5 seconds 






At 
85 








a 


At 218 

< 1 minute 
/ 




Po 

84 






Po 216 ,ThA / 
(thorium A) 
0.158 second 


0(0.013%) 
a 


Po^^ThC' 
(thorium C') 

3xlO' 7 second 

s 


Bi 

83 








a 
H00%) 


Bi 2l2 ,ThC ' 
(thorium C) 

60.5 minutes 

* 


0(66.3%) 
a 


Pb 
82 






Pb^ThB 
(thorium B) 
10.6 hours 


& 
(33.7%) 


Pb 808 ,) 

(stable 
lead isotope) 


Tl 
81 








Tf.ThC^ 

(thorium C') 
3J, minutes 






FIGURE 1-2. The thorium series. 



14 NATURALLY OCCURRING RADIOACTIVE SUBSTANCES CH. I 





U 236 , AcU 










u 


(actino 








92 


uranium) 
.07x10 years 








Pa 




Pa 231 , Pa 








91 


a 


protoactinium) 










3.2 xlO 4 years 






Th 


Th^.UY ' 





Th 227 ^ 


Ac 






90 


(uranium Y) 


a 


radioactinium) 








24.6 hours 




^ 18.9 days 






Ac 




Ac 227 . Ac ' 


0(98.8%) 








89 




(actinium) 




OL 










21 years 










Ra 




a. 


Ra M .AcX 






88 




(1.2%) 


(actinium X) 












11. 2 days 










223 A If ' 


ft 








Fr 




r r , AcK 










87 




(actinium K) 


a 












21 minutes 










Rn 






Rn 219 ,An 






86 






(actinon) 












3.92 seconds 






At 






or 


.215 










At 




85 
















Po 215 ,AcA x 


0(5xlO' 4 %) 


Po 211 ,AcC / 


84 






(actinium A) 
1.83 xlO' 3 


a 


(actinium C') 
5xlO' 3 








second 




s second 


Bi 






a 




Bi 211 ,AcC / 


0(0.32%) * 


83 






(-100% 


(actinium C) 


a. 












2.16 minutes 














/ 




Pb 






Pb 211 .AcB X 


ft 
ct 


Pb^.AcD 


82 






(actinium 8) 
36.1 minutes 


(99.68% 


(stable 
lead isotope) 


Tl 








Tl^.AcC" 


ft 


81 






(actinium C") 










4.76 minutes 





FIGURE 1-3. The actinium series. 



RADIOACTIVE SERIES 



15 



Cm 
96 






'Cm m ' 

55 days 








Am 
95 




Am 241 X 
(americum) 
,500 years 


K 








Pu 
94 


Pu*> ' 
~10 years 


ft 
a 


r Pu 217 ] 

~40 days 

/L J 








Np 
93 




Np 237 ' 
(neptunium) 
2.25 xlO 6 
/ years 


K 








U 
92 


r u 237 T 

6.8 days 


ft 
<x 


U 233 
1.63 xlO 5 
x years 








Pa 

91 




PP' 

, 27.4 days 



a 


[ Pa 229 ! 
x [ 1-4 days] 






Th 
90 


r Th 233 T 

23 
L minutes J 





Th 228 * 
7 x 10 s years 


K(?) 
a 






Ac 
89 






a 


Ac 225 
10.0 days 






Ra 
88 






Ra 225 
14.8 days 


& 
a. 






Fr 
87 








Fr m 
4.8 minutes 






Rn 
86 








a 






At 
85 








At 217 

0.020 
second 






Po 
84 








a 


Po 213 
4.2 x 
,10" 6 second 




Bi 
83 








Bi 213 
47 minutes 


/3(98%) 
a 


Bi 209 
(stable 
, bismuth) 


Pb 
82 








a (2%) 


Pb 209 

3.3 hours 

S 


ft 


Tl 
81 








-n* ' 

<1 hour 


ft 


~ 



FIGURE 1-4. The 4n -f- 1 series. 



16 NATURALLY OCCURRING RADIOACTIVE SUBSTANCES CH. I 

it might have been formed. In the same way there has been 
speculation as to heavier members of the known families, with 
the assumption that the half-lives of any transuranium elements 
were short compared to geologic time. By recent artificial trans- 
mutation techniques a rather well-developed 4n + 1 series includ- 
ing several transuranium species has been prepared and investi- 
gated. This series is displayed in figure 1^. The properties 
of several other transuranium species may be found in table A 
in the appendix. 

Several investigators at one time or another have examined 
essentially all the remaining known elements for evidences of 
naturally occurring radioactivity, and with some positive results, 
first in the case of potassium and later in a number of others. 
The properties of the known natural radioactivities other than 
those of the three radioactive families are collected in table 1-2. 

TABLE 1-2 
ADDITIONAL NATURALLY OCCURRING RADIOACTIVE SUBSTANCES 











Stable 


Active 


Type of 




Relative 


Disinte- 


Sub- 


Disinte- 




Isotopic 


gration 


stance 


gration 


Half-life 


Abundance 


Products 


R 40 


A-K 


4.5 X 10 8 years 


0.012% 


Ca 40 , A 40 


Rb 87 


ft 


6.0 X 10 10 years 


27.2% 


Sr 87 


Sm 162 


a 


2.5 X 10 11 years 


26.6% 


Nd 148 


Lu 176 


ft,K 


2.4 X 10 10 years 


2.5% 


Hf 176 , Yb 176 


Re 187 


ft 


4 X 10 12 years 


62% 


Os 187 



It may be seen that the samarium radioactivity is the only case 
of a disintegration; the type of decay listed as K is discussed in 
chapter VI, section B. In attempts to extend the search for 
new radioactivities to very low intensity levels difficulty arises 
from the general background of detectable radiations present in 
every laboratory. In part this general background is due to 
presence of traces of uranium, thorium, potassium, etc., and in 
large part to the cosmic radiation of unknown origin. The cosmic 
rays reach every portion of the earth's surface; their intensity 
is greater at high altitudes but persists measurably even in deep 
caves and mines. The magnitude of the background effect is 
indicated in the discussion of radiation-detection instruments 
in chapter VIII, page 193. 



AGE OF THE EARTH 17 

Age of the Earth. As already suggested, the existence of the 
radioactive substances uranium and thorium gives some informa- 
tion concerning the tune that may have passed since the genesis of 
these elements, and perhaps of all elements. Clearly conditions 
as we know them today cannot have existed for a tune very long 
compared to the half-lives, 4.51 X 10 9 years, 7.07 X 10 8 years, 
and 1.39 X 10 10 years for Ui, AclJ, and Th, respectively. Minerals 
have been studied with the object of determining their age in 
relation to radioactive constants by at least four methods; these 
are discussed in the following paragraphs. 

1. Intensity of Coloration of Pkochroic Haloes. Many types of 
radiation are capable of producing coloration or discoloration in 
glass, quartz, mica, and a number of similar materials. Intense 
sources of a particles are effective in producing colorations in a 
short time, and even exceedingly small amounts of uranium or 
thorium are capable of producing visible effects within geological 
time intervals when present as a minute inclusion in a mineral 
such as mica. The range of coloration from a particles is of the 
order of a few thousandths of a centimeter in mica, and the charac- 
teristic a-particle ranges for the various decay products cause the 
production of tiny concentric shells of varying coloration. Exam- 
ined in thin sections under a microscope, these appear as circular 
areas known as pleochroic haloes; in polarized light the colors 
change with the plane of polarization. The many observed radii 
of the color bands have been fairly well correlated with the known 
a-particle ranges in mica of decay products of uranium or thorium, 
and in this way the nature of the inclusion is established. The 
amount of inclusion may be judged roughly from its size in the 
microscope field. Attempts have been made to evaluate the 
amount of radiation required for a particular degree of coloration 
and thus to establish a geologic time scale for these mica samples. 
Effects such as reversal of the intensity of coloration caused by 
overexposure (analogous to photographic solarization) must be 
taken into account, and no accurate results can be claimed. 

2. Ratio of Uranium to Helium Content. Once an atom of Ui 
disintegrates, the chain of successive decays soon (say in less than 
about a million years) produces eight a particles. Because the 
ranges of these particles are very short in dense matter, most of 
the resulting helium atoms (the helium ions at rest are easily 
capable of acquiring two electrons by oxidizing almost any sub- 



18 NATURALLY OCCURRING RADIOACTIVE SUBSTANCES CH. I 

stance) may be trapped in the interior of the rock. In favorable 
cases, with very impervious fine-grained rocks and a low helium 
concentration (pressure) from small uranium contents, this helium 
has been retained throughout the geologic ages and now serves 
as an indicator of the fraction of uranium transformed since the 
formation of the ore. The thorium content of the rock also is a 
source of helium, and this must be taken into account. Very 
sensitive methods of assay for helium, uranium, and thorium are 
available and have permitted determinations on rocks with 
uranium and thorium contents below 1 part per million, and on 
metallic iron meteorites (where loss of helium in any process short 
of melting seems quite unlikely). The ages found, usually to be 
taken as lower limits, range up to a little over 2000 million years. 
For some meteorites larger values have been found; however, 
evidence has been presented recently which suggests that a con- 
siderable part of the helium in meteorites may have resulted from 
the action of cosmic radiation and that some meteorites may, 
therefore, be considerably younger than indicated. 

3. Ratio of Uranium or Thorium to RaG or ThD Content Lead 
(RaG and ThD) is a stable end product of the disintegration of 
uranium and thorium and provided there is no other source of 
lead in an ore may be used as a quantitative indicator of disinte- 
gration. This lead method might be expected to be more reliable 
than the helium method since lead is not so likely to have been 
lost by slow diffusion; however, it is still quite possible that a 
lead-uranium or lead-thorium ratio has been changed by leaching 
or some other process. The distinction between these lead decay 
products (Pb 206 from Uj and Pb 208 from Th) and ordinary lead 
is made in a satisfactory way by mass spectrographic analysis; 
it is usually presumed that absence of Pb 204 establishes the absence 
of ordinary lead. Age determinations from uranium-to-Pb 206 and 
from thorium-to-Pb 208 ratios do not always agree, but values 
ranging up to roughly 3000 million years are indicated. 

4. Ratio of Uranium Lead to Actinium Lead. Probably the 
best method so far devised involves determination of the ratio of 
Pb 206 to Pb 207 . This method should be free of many experimental 
errors and is less sensitive to chemical or mechanical loss of either 
uranium or lead than method 3. The Pb 206 /Pb 207 ratio is an 
indicator of age because Ui and AcU decay at different rates. 
From mass spectrographic measurements of this ratio a convinc- 



AGE OF THE EARTH 19 

ing age scale of minerals can be established. Samples of two 
ores from a region known to be geologically very old (Huron 
Claim monazite and uraninite) have ages close to 2000 million 
years. 

These times of about 2 X 10 9 years are of the same order as 
the "age of the universe" estimated from the red-shift phenomenon. 
(This disputed hypothesis assigns to each star system a velocity 
which would give the observed shift towards the red of all spectral 
lines as a Doppler effect; the result is an exploding universe model 
with the "age" as the time calculated for the systems to have 
achieved their present separation distances starting from the 
point of common origin.) 

It may be noted that studies of relative abundances of radio- 
active species yield some other information on ancient time scales. 
It has been carefully demonstrated that the ratio of AclJ to Ui is 
the same (1:139) for uranium samples from various sources; this 
is plausible if all our uranium had been formed at the same time; 
even if it had been formed in the same way at different times, the 
different rates of decay would leave in the younger samples more 
of the shorter-lived component. In the same way, the concentra- 
tion of the radioactive K 40 in potassium from various terrestrial 
and meteoric sources is constant, giving evidence for a common 
genesis of elements in our solar system. More careful attention 
to the distribution of isotopes, especially in meteorites, may give 
new knowledge of the time between genesis of the elements and 
formation of the discrete rock phases, since any comparatively 
short-lived element naturally included in one phase could give 
rise later by radioactive transformation to an "unnatural" inclu- 
sion in that phase. In considerations of this kind it may not be 
justified to neglect nuclear reactions of other types; for example 
it has been suggested that an era of high neutron flux may have 
at one time considerably upset the pattern of elementary and 
isotopic abundances. 

Quite recently it has been demonstrated that one action of 
neutrons associated with the cosmic radiation is the continuing 
production of radioactive carbon (C 14 ) ; mostly in the upper 
regions of the earth's atmosphere. The half-life of about 5000 
years for this substance is no doubt short compared to the duration 
of the cosmic irradiation; therefore, we may expect a steady-state 
concentration (analogous to the situation in secular equilibrium) 



20 NATURALLY OCCURRING RADIOACTIVE SUBSTANCES CH. I 

of this radioactivity in all carbon of the living carbon cycle. It 
may be possible to date archeological objects, bones, mummies 
and the like, through measurements of the level of carbon radio- 
activity, as decreased by radioactive decay since the specimen 
was removed from participation in the life cycle. 



EXERCISES 

1. One hundred milligrams of Ra would represent what percentage 
yield from exactly 2 tons of a pitchblende ore containing 75 per cent 

Answer: 26 per cent. 



2. Calculate the rate of energy liberation (in calories/hr) for 1.00 g 
of pure radium free of its decay products. What can you say about the 
actual heating effect of an old radium preparation? 

Answer to first part: 25 cal/hr. 

3. A certain active substance (which has no radioactive parent) has 
a half -life of 8.0 days. What fraction of the initial amount will be left 
after (a) 16 days, (b) 32 days, (c) 4 days, (d) 83 days? Answer: (a) 0.25. 

4. How long would a sample of radium have to be observed before the 
decay amounted to 1 per cent? (Neglect effects of radium A, B, C, etc., 
on the detector.) 

5. Find the number of disintegrations of uranium I atoms occurring 
per minute in 1 mg of ordinary uranium, from the half-life of U r , t^ = 
4.51 X 10 9 years. 

6. How many ft disintegrations occur per second in 1.00 g of pitch- 
blende containing 70 per cent uranium? You may assume that there has 
been no loss of radon from the ore, Answer: 53,000 per sec. 

7. Estimate the age of a rock which is found to contain 5 X 10 ~ 5 cc 
of helium at standard temperature and pressure and 3 X 10~ 7 g of ura- 
nium per gram. Answer: 1.2 X 10 9 years. 

8. By artificial transmutation techniques a number of new radioactive 
species have been produced in the region of the natural radioactive families. 
These may be classified into the four families according to mass and are 
known as collateral members of the families. Find all of these in table A 
in the appendix and place them in figures 1-1, 1-2, etc., so as to indicate 
their decay relationships to other members of the series. 

9. Calculate the age of a mineral which shows a Pb 206 -to-Pb 207 ratio 
of 14.0 and is essentially free of Pb 204 . Answer: 1 X 10 9 years. 



REFERENCES 21 

REFERENCES 

G. E. M. JAUNCEY, "The Early Years of Radioactivity," Am. J. Phys. 14, 

226 (1946). 
E. RUTHERFORD, J. CHADWICK, and C. D. ELLIS, Radiations from Radioactive 

Substances, Cambridge University Press, 1930. 
G. HEVESY and F. A. PANETH, A Manual of Radioactivity, Oxford University 

Press, 1938. 

M. S. CURIE, Traite de radioactivity, Paris, Gauthier-Villars, 1935. 
G. T. SEABORG, "The Neptunium (4n + 1) Radioactive Family," Chem. and 

Eng. News 26, 1902 (1948). 
A. O. NIER, R. W. THOMPSON, and B. F. MURPHY, "Isotopic Constitution of 

Lead and the Measurement of Geologic Time," Phys. Rev. 60, 112 (1941). 
R. D. EVANS, "Measurements of the Age of the Solar System," Field Museum 

of Nat. History, Geol. Ser. 7, No. 6, 79 (1943). 

G. C. McViTTiE, Cosmological Theory, London, Methuen and Co., 1937. 
C. A. BAUER, "Rate of Production of Helium in Meteorites by Cosmic 

Radiation," Phys. Rev. 74, 501 (1948). 

M. H. STUDIER and E. K. HYDE, "A New Radioactive Series The Protac- 
tinium Scries/' Phys. Rev. 74, 591 (1948). 
A. GHIORSO, W. W. MEINKE and G. T. SEABORG, "Artificial Collateral Chains 

to the Thorium and Actinium Families," Phys. Rev. 74, 695 (1948). 



H. ATOMIC NUCLEI 

A. ATOMIC STRUCTURE 

Rutherford's Nuclear Model of the Atom. At the time the phe- 
nomenon of radioactivity was discovered the chemical elements 
were regarded as unalterable; that is, they were thought to retain 
their characteristic properties throughout all chemical and physical 
processes. This view became untenable when it was recognized 
that radioactive processes involved the disintegration of elements 
and the formation of other elements. Thus it became clear about 
1900 that atoms, until then regarded as the indivisible building 
blocks of the elements, must have some structure and that it 
must be possible for the atoms of one element to be transformed 
into those of another, with the emission of radiations. However, 
it was not until 1911 that the nuclear model of the atom which is 
now generally accepted was proposed by Rutherford. 

Rutherford was led to propose this model by experimental 
results on the scattering of a particles in matter. He discovered 
the scattering phenomenon when he noticed that a collimated 
beam of a particles was spread out in passing through a thin layer 
of matter. A quantitative study by H. Geiger of the scattering 
in very thin foils showed that single encounters between a particles 
and atoms resulted in scattering of the a. particles through very 
small angles. On the basis of this information the probability of 
larger scattering angles resulting from multiple scattering processes 
in thicker foils could be calculated statistically, and this proba- 
bility falls off very rapidly with increasing angle. When experi- 
ments of Geiger and E. Marsden showed that scattering angles 
of 90 or more were much more frequent than could be accounted 
for by multiple scattering, Rutherford proposed that these large 
angles were due to a special type of single scattering process. 

To explain the scattering of a. particles through large angles in 
single processes Rutherford postulated an atomic model in which 
the positive charge resides in a small massive nucleus and negative 
charge of the same magnitude is distributed over a sphere of 

22 



RUTHERFORD'S NUCLEAR MODEL OF THE ATOM 23 

atomic dimensions. The large deflection of an a particle from 
its path was then supposed to result from Coulomb repulsion 
between the a particle and the positively charged nucleus of an 
atom. With this simple assumption and the additional restric- 
tion that the nucleus is so heavy as to be considered at rest during 
the impact Rutherford set up the conditions for conservation of 
momentum and energy and derived from these his famous scat- 
tering formula, 

Ndf Zc 

5" \M~ ava , 

sin 4 - 



<? \ 2 

) 

a^a / 



where rc(0) is the number of scattered a particles falling on a unit 
area at a distance r from the scattering point when the angle 
between the directions of the initial and scattered a particles is 
6] n is the incident number of particles, d the thickness of the 
scatterer, N the number of nuclei per unit volume of scatterer 
and Ze the charge per nucleus. M a and v a are the mass and initial 
velocity of the a particle. 

Rutherford's hypothesis thus predicted that the number of 
scattered particles per unit area was proportional to the thickness 
of the scatterer and to the square of the nuclear charge, arid in- 
versely proportional to the square of the a-particle energy and to 
the fourth power of the sine of half the scattering angle. The 
number of scattered particles has been carefully measured as a 
function of scattering angle, a-particle energy, and thickness of 
scatterer, and the results have been found to be in excellent agree- 
ment with the Rutherford formula provided heavy elements were 
used as scatterers. For light scatterers, that is, for the case where 
the nuclei of the scatterer cannot be considered at rest during the 
impact, a more complicated expression must be substituted for 
the Rutherford formula, and if this is done the agreement between 
theory and experiment is again satisfactory. 

This experimental verification of the scattering formula led to 
a general acceptance of Rutherford's picture of the atom as con- 
sisting of a small positively charged nucleus, containing practically 
the entire mass of the atom, and surrounded by a distribution of 
negatively charged electrons. In addition, the scattering law 
made it possible to study the magnitude of the nuclear charge in 
the atoms of a given element, because the scattering intensity 



24 ATOMIC NUCLEI CH. II 

depends on the square of the nuclear charge. It was by the 
method of a-particle scattering that nuclear charges were first 
determined, and this work led to the suggestion that atomic num- 
ber was identical with the nuclear charge (expressed in units of 
the electronic charge e). This suggestion was subsequently con- 
firmed by H. G. Moseley's work on the X-ray spectra of the 
elements. 

Bohr's Theory of Electron Orbits. In the preceding section we 
have seen that according to Rutherford's hypothesis an atom con- 
sists of a small positively charged nucleus and a "cloud" of elec- 
trons surrounding it. Since the charge on the nucleus is an inte- 
gral multiple Z of the electronic charge e, the number of electrons 
surrounding the nucleus of a neutral atom must also be equal to 
Z. This number Z is known as the atomic number. 

Classical mechanics and electrodynamics cannot account for 
the stability of a system consisting of a heavy positive nucleus 
surrounded by moving electrons. According to classical theory 
such an atom would lose energy continuously because the elec- 
trons, being accelerated in the Coulomb field of the nucleus, 
would emit electromagnetic radiation. In 1913 N. Bohr intro- 
duced a quantum theory of atomic structure. He postulated 
that an atom could exist only in certain discrete energy states 
corresponding to particular circular orbits of the electrons around 
the nucleus, and that it could lose or gain energy only in transi- 
tions from one of these quantum states to another. The mono- 
chromatic radiation absorbed or emitted in such a transition 
should then be related to the energy difference AE between the 
two energy states by the quantum relation AE = hv, where v is 
the frequency of the radiation and h is Planck 's constant. In 
considering the simplest atom, that of hydrogen, Bohr found he 
could obtain good agreement with observed frequencies in the 
hydrogen spectrum if he made the assumption that the electron 
was restricted to those orbits whose angular momenta were whole 
multiples of h/2ir. 

The angular momentum of an electron of mass m and velocity v 
traveling in a circular orbit of radius a is mva, and Bohr's quantum 
condition is, therefore, given by 

nh 
mva = , (II-l) 



BOHR'S THEORY OF ELECTRON ORBITS 25 

where n is an integer. An additional condition, which follows 
from classical mechanics, is that the centripetal force due to the 
Coulomb attraction between electron and nucleus must equal 
the centrifugal force due to the electron's motion in its orbit. 
This condition is expressed by 

^ = - (H-2) 

a 2 a 

By solving both equations II-l and II-2 for v and equating the 

expressions we get 

n 2 h 2 
a - -- (H-3) 



Thus, the radius of each Bohr orbit of the electron in the hydro- 
gen atom is characterized by the so-called principal quantum 
number n. In the lowest or normal energy state of the hydrogen 
atom, n = 1, or the radius of the electron orbit is h 2 /4ir 2 mZe 2 
5.3 X 10~ 9 cm. 

Bohr's original theory was very successful in accounting for 
the main features of the hydrogen spectrum. However, the 
splitting of many of the hydrogen lines into several components 
could not be explained on this basis, and no quantitative agree- 
ment was obtained for the spectra of more complex elements. In 
the years following 1913 Bohr's theory underwent a series of 
refinements. The motion of the nucleus was taken into account 
through replacement, in Bohr's equations, of the electron mass 

[Mm 

m by the reduced mass of the system - , where M is the 

M + m 

mass of the nucleus. A. Sommerfeld generalized the theory by 
introducing elliptical as well as circular orbits, with the nucleus 
at one focus. This introduced a second quantum condition restrict- 
ing the eccentricities of the ellipses to certain allowed values. A 
third quantum condition allowing only a limited number of orien- 
tations of an atom in an external magnetic field was found to 
explain the Zeeman effect, that is, the splitting of spectral lines 
into several closely spaced components under the influence of a 
magnetic field. Finally the concept of electron spin was intro- 
duced, and the condition that an electron can have its spin- 
momentum vector oriented either parallel or antiparallel to its 



26 ATOMIC NUCLEI CH. II 

orbital angular momentum vector represents a fourth quantum 
condition. 

Summarizing we can say that each electron orbit in an atom is 
characterized by four quantum numbers: 

1. The principal quantum number n (related to the average 
distance from the nucleus). 

2. The azimuthal quantum number I (related to the orbital 
angular momentum). 

3. The magnetic quantum number mi. 

4. The spin quantum number m a . 

It turns out that these quantum numbers are not independent of 
each other, but that for each value of n only certain values of the 
other quantum numbers are possible: for any n, I can have any 
integral value from to n 1 ; for any I, m t can have any integral 
value from I to +1 including 0; and m a can be either J^ or 



As an illustration the possible combinations of quantum num- 
bers f or n 1 and n = 2 are given in table II-l. Also included 

TABLE II-l 
POSSIBLE COMBINATIONS OF QUANTUM NUMBERS FOK n ** 1 AND n 2 

Energy Level 
n I mi m s Designation 

1 +Kor-M 1 
20 +Mor-^ 2s 
21-1 -hJ^or-H 2p 

2 1 +J^or-^ 2p 
21+1 +Hor-^ 2p 



are the designations of the orbits or energy states used in dis- 
cussions of atomic structure and spectroscopy. The states corre- 
sponding to Z = 0, 1, 2, 3 are referred to as s, p, d, f states, 
respectively, and the numerical value of n precedes the letter. 
Another terminology, taken over from X-ray spectroscopy, refers 
to the electron orbits characterized by n = 1, 2, 3, 4, 5, 6, etc., 
as the K, L, M y N, 0, P, etc., shells. 

Building Up of the Periodic Table. It follows from the exclusion 
principle formulated by W. Pauli in 1925 that in a system such 
as an atom no more than one electron can be in an energy state 
characterized by the same values of the four quantum numbers 



BUILDING UP OF THE PERIODIC TABLE 27 

n, Z, m/, and w a . This principle, together with the very plausible 
assumption that the normal state of an atom always corresponds 
to the electron configuration of lowest energy, determines the 
distribution of electrons in the quantum levels of any given atom. 
In hydrogen the electron is normally in the orbit characterized 
byn = l,Z = 0(als orbit); in helium the two electrons are both 
in the Is orbit but with opposite spin orientations. In lithium 
two electrons are again in the Is orbit, but the third one has to 
go to the next higher energy state, the 2s state (n = 2, Z = 0). 
In beryllium the Is and 2s orbits are filled, from boron to neon 
the six 2p electrons are being added, in sodium the eleventh elec- 
tron goes into the 3s orbit, etc. An irregularity occurs at potas- 
sium (Z = 19) where one might expect the nineteenth electron 
to go to the 3d orbit (Is, 2s, 2p, 3s, 3p being filled), but actually 
the nineteenth electron in the normal atom appears in the 4s 
state. Only after the second 4s electron is also added (in calcium), 
do the 3d orbits fill (from scandium to copper). A similar situa- 
tion occurs between rubidium and silver. These irregularities in 
the filling of the electron shells can be understood qualitatively 
in terms of the different eccentricities of the various orbits in a 
given shell. For any given n the orbits become increasingly 
eccentric with decreasing Z; the largest value of Z always corre- 
sponds to a circular orbit. Thus, although an electron in a 4s 
orbit is on the average further away from the nucleus than a 3d 
electron, its orbit penetrates inside some of the inner shells, while 
the 3d electron is rather well shielded from the positively charged 
nucleus by the inner electrons. Thus, it is plausible that in a 
particular atom an electron in a 4s orbit may actually have a 
somewhat lower potential energy than it would have in a 3d orbit. 
Such qualitative arguments as this are borne out by approximate 
quantitative calculations of the electron energies in the various 
orbits. The distribution of electrons among the quantum states 
can now be reasonably well calculated for the lowest state of most 
elements in the periodic table, and the results relate remarkably 
well to much of the chemical behavior of the elements. Thus, to 
name only a few examples, the similarity between elements in 
the same column of the periodic table can now be ascribed to the 
similarity in their electron structures in the outermost shells; 
valence can be correlated with the number of electrons in the 
outer shell; the striking similarity between the fourteen rare 



28 ATOMIC NUCLEI CH. II 

earths can be accounted for by the fact that they differ only in 
the number of electrons in the 4/ shell (which is well inside the 
atom) while the population of the 5s, 5p, 5d, and 6s shells is prac- 
tically the same for all of them. 

Wave Mechanics. So far we have discussed atomic structure in 
terms of the picture introduced by Bohr in 1913. Actually this 
has been superseded by another approach to the problem which 
may be harder to visualize, but which is of much wider applica- 
bility and gives results that are quantitatively in better accord 
with experimental observations. This is the wave-mechanical 
approach developed by E. Schroedinger. 

In 1923 L. de Broglie had pointed out that Bohr's quantum 
condition (equation II-l) followed directly from the assumptions 
that an electron of mass m and velocity v had associated with it a 
phase wave of wave length X = h/mv and that the orbit circum- 
ference must be an integral multiple of X. In pursuing further 
the idea of wave properties associated with material particles 
Schroedinger arrived at his wave equation for the representation 
of material systems. In solving this equation for the hydrogen 
atom one finds that proper solutions exist only for certain values 
of the energy of the system, and these so-called eigen values (or 
proper values) agree closely with the energy values expected from 
spectroscopic data, even in those cases where the older Bohr pic- 
ture does not give satisfactory agreement with experimental 
results. Furthermore, the quantum restrictions on the allowed 
energy states, which were somewhat artificially introduced into 
the Bohr theory, appear naturally in the solution of the wave 
equation. 

A physical interpretation has been given by M. Born to the 
wave function \l/ which is the solution of Schroedinger's equation 
for a given system such as an electron in an atom. He assumes 
that ^ is the amplitude of the wave associated with the electron 
and that the square of this amplitude at any point in space repre- 
sents the fraction of the total time that the electron spends there. 
Thus according to wave mechanics the notion of well-defined 
orbits for the electrons has to be abandoned, because only the 
time average of the positions of the electrons can be derived from 
the equations. This is also in accord with the Heisenberg uncer- 
tainty principle which states that the inherent uncertainty As 
in the position of a particle whose momentum is known within an 



PROTON-NEUTRON MODEL 29 

accuracy Ap is such that the product Ap-Ax is of the order of 
magnitude of h. 

However, it should be noted that it is still entirely legitimate to 
speak of the energy states or quantum states of bound electrons, 
and that everything said in the preceding section about the rela- 
tion of atomic structure to the building up of the periodic system 
is still essentially valid provided the well-defined electron orbits 
are replaced by smeared-out probability distributions. 

B. NUCLEAR STRUCTURE 

Proton-Neutron Model. Having discussed, at least qualita- 
tively, the arrangement of the external electrons in atoms we 
shall now return to a consideration of the atomic nuclei whose 
existence was revealed by the a-particle-scattering experiments. 
We have seen that a nucleus is a small positively charged particle 
whose mass accounts for almost the entire atomic mass and whose 
charge is equal in magnitude but opposite in sign to the sum of 
the charges of all the electrons in the neutral atom. The dimen- 
sions of nuclei are of the order of 10~~ 12 to 10~ 13 cm while atomic 
dimensions, as determined, for example, from gas kinetics, are 
about 10~ 8 cm. Nuclei, therefore, are very much more dense 
than ordinary matter; the density of nuclear matter is in the 
neighborhood of 10 14 g per cm 3 or 10 8 tons per cm 3 . 

According to present ideas nuclei consist of protons and neu- 
trons. The simplest nucleus is that of the common hydrogen 
atom; a single proton constitutes this nucleus. A proton, there- 
fore, carries a positive charge equal in magnitude to the charge 
on an electron, 4.8025 X 10~ 10 electrostatic unit (esu). The 
mass of a proton is approximately equal to that of a hydrogen 
atom and, therefore, nearly equal to one on the atomic weight 
scale. The neutron is an uncharged particle whose mass is very 
nearly equal to but slightly greater than the proton mass. 

The nature of the forces which hold neutrons and protons 
together in nuclei is still not well understood. It is clear that the 
force cannot be simple electric (Coulombic) attraction, because 
the neutron carries no charge. Gravitational forces are too weak 
by very many orders of magnitude to account for nuclear binding. 
Marry experimental facts indicate that nuclear forces have a very 



30 ATOMIC NUCLEI CH. II 

short range, in fact, a range somewhat smaller than nuclear dimen- 
sions. The type of force which is now rather generally believed 
to act between nucleons (a collective term for protons and neu- 
trons) is a so-called exchange force, that is, a force which holds 
the nucleons together through a continuous exchange of some 
constituent particles between them. These particles are referred 
to as mesons; this name was chosen because they are believed to 
be intermediate in mass between electrons and protons. (l) The 
exact properties ascribed to them vary in the different meson 
theories. According to the theory of exchange forces nuclear 
binding is to be pictured as a resonance phenomenon somewhat 
analogous to the binding of the two hydrogen atoms in a hydrogen 
molecule through the sharing or exchange of the pair of electrons 
between the two atoms. 

Atomic Number and Mass Number, The number of protons in 
a nucleus is called the atomic number Z and ordinarily deter- 
mines the chemical properties of the element. The atomic num- 
bers of the known elements range from 1 for hydrogen to 96 for 
curium. The number of neutrons in the nucleus is sometimes 
called the neutron number N. Nuclei are known with values of 
N from to 147. 

The total number of nucleons (neutrons plus protons) in the 
nucleus of a given atomic species is called its mass number A; 
this is the whole number nearest the atomic weight of that par- 
ticular atom. Mass numbers are known in the range 1 to 242. 
The difference N Z (or A 2Z) between the number of neu- 
trons and protons in a nucleus is referred to as the neutron excess 
or isotopic number. 

The symbol used to denote a nuclear species is the chemical 
symbol of the element with the atomic number as a left subscript 
and the mass number as a superscript, usually to the right, for 
example, 2He 4 , 2?Co 59 , 9 2U 235 . The atomic number is often 
omitted because it is uniquely determined by the chemical 
symbol. 

Isotopes, IsobarStlsotones^and Isomers. Atomic species of 
the same atomic number/that is, belonging to the same element, 

1 In some laboratories the name mesotron is preferred. Intermediate mass 
particles have been found in cosmic-ray studies, and there is evidence for the 
existence of more than one type. See also chapter VI, page 134. 



PROTON-ELECTRON HYPOTHESIS 31 

but having different mass numbers are called isotopes. (2) In the 
nuclei of the different isotopes of a given element the same number 
of protons is combined with different numbers of neutrons. For 
example, a irCl 35 nucleus contains 17 protons and 18 neutrons, 
whereas a irCl 37 nucleus contains 17 protons and 20 neutrons. 
Deuterium, a rare isotope of hydrogen, has a nucleus containing 
one proton and one neutron. 

Atomic species having the same mass number but different 
atomic numbers are called isobars. A few examples of isobars 
are 32 Ge 76 and 34 Se 76 ; 52 Te 130 , 5 4Xe 130 and seBa 130 ; 80 Hg 204 and 
82 Pb 204 . 

Atomic species having the same number of neutrons but dif- 
ferent mass numbers are sometimes referred to as isotones. For 
example, ^Si 30 , isP 31 , and i 6 S 32 are isotones because they all 
contain 16 neutrons per nucleus. 

Among the natural radioactive bodies discussed in chapter I 
there are two, UX 2 and UZ, which have the same mass number 
as well as the same atomic number, but differ in their radioactive 
properties. This is an example of nuclear isomerism. Although 
UX 2 and UZ had been known for several years the phenomenon 
of nuclear isomerism did not receive much attention until another 
pair of isomers, Br 80 , was discovered among artificially produced 
radioactive species in 1937. Some 70 pairs of nuclear isomers 
are now known. Nuclear isomers are regarded as different energy 
states of the same nucleus, each having different radioactive 
properties. The upper state is always radioactive; the lower 
state may be stable or radioactive. Isomerism is discussed more 
fully in chapter VI. 

Comparison of the Proton-Neutron with the Older Proton-Elec- 
tron Hypothesis. Before the discovery of the neutron there existed 
the idea that nuclei were composed of protons and electrons. In 
this model the nucleus contained enough protons to account for 
its approximate mass (one proton for each unit of atomic weight) 
and enough electrons to reduce the net positive charge to the 

2 For some years the word isotope has been used also in a broader sense to 
signify any particular nuclear species characterized by its A and Z values. 
The word nuclide has recently been suggested [T. P. Kohman, Am. J. Phys 
15, 356 (1947)], with the following definition: "a species of atom characterized 
by the constitution of its nucleus, in particular by the number of protons and 
neutrons in its nucleus." 



32 ATOMIC NUCLEI CH. II 

proper value; thus the neutral atom was to contain as many total 
electrons as protons, with some in the known atomic orbits or 
energy levels and the remainder within the nucleus. This model 
now not only is unfashionable but also presents difficulties which 
are not easy to resolve, when compared with the proton-neutron 
nuclear model. One difficulty is that the electron is "too large 
to be in the small space of the nucleus"; we will consider two 
aspects of the question of the size of the electron. 

With the assumption of unlimited validity for Coulomb's law 
(probably an improper assumption) it is clear that the electron 
cannot have zero radius since its potential energy would then be 
infinite. We may use the known mass of the electron to set an 
upper limit to this potential energy through the mass-energy 
relation E = mc 2 ? with the rest mass of the electron m = 9.11 
X 10~ 28 g and the velocity of light c = 3.00 X 10 10 cm sec"" 1 . 
Imagine the assembly of the electronic charge e (= 4.80 X 10~ 10 
esu) from infinitesimal units dq of charge; take q for the charge 
already collected within the radius R. The (repulsive) force on 
dq at a distance r from the center of the electron under construc- 
tion is given by Coulomb's law / = qdq/r 2 . The energy re- 
quired for assembly of the electron of radius R is then 

**e /*r~R qdq e 2 

_ ..Q J r=a r 2 2R 



From this, 

e 2 (4.80 X HT 10 ) 2 



2E 2 X 9.11 X 10~ 28 X (3.00 X 10 10 ) 



= 1 4 X 10" cm 



This figure is not to be believed as a literal radius of the electron; 
often the quantity e 2 /E (that is, twice the preceding value) is 
considered merely as a convenient unit of length and termed the 
"classical radius of the electron. J? 

The length just obtained is of the order of nuclear dimensions. 
However, if the electron is to be thought of as being within the 
nucleus its de Broglie wave length X = h/mv must be of the order 
of nuclear dimensions; this requires a very high momentum and 
consequently high kinetic energy. For example, to make 
X = 10~~ 12 cm the required total energy from the necessary rela- 
tivistic formula is 



PROTON-ELECTRON HYPOTHESIS 33 

he 6.62 X 10~ 27 X 3.00 X 10 10 

E = m<r mvc = - = 

X 10~ 12 

= 2.0 X 10~ 4 erg. 

This energy is 125 million electron volts (125 Mev) (3) and is far 
in excess of the energies known to be associated with nuclear 
changes involving one nucleon. On the other hand, a similar 
calculation for a neutron or proton (with mass M = ^1.66 
X 10~ 24 g) from a satisfactory nonrelativistic formula gives the 
kinetic energy, 

1 , h * 

E = -Mv 2 = = 0.13 X 10~ 4 erg = 8.3 Mev. 

2 2MX 2 

This value is just in the range of experimental energy changes 
for the addition or removal of one nucleon. On the basis of this 
argument the proton-neutron model of nuclear structure is more 
acceptable than the proton-electron model. 

As may be seen in table 1 1-3, the particles, proton, neutron, 
and electron, all have spin values of J^J that is, in each case the 
quantized angular momentum is just J^> of the unit h/2ir. For 
the deuterium nucleus (called the deuteron) the spin is 1 (in the 
same unit). Now it is easily imagined that the neutron and proton 
are disposed in the deuteron with spins parallel giving the observed 
one unit of spin; but no such result is possible with the proton- 
electron model, where the combined particles (two protons and 
one electron) would have resultant spin % or %. The same 
argument makes very questionable an earlier view that the neu- 
tron itself was composed of one proton and one electron. This 
argument may be extended to any nucleus whose spin has been 
determined, with the same result; all nuclei with even mass num- 
ber show integral spins, and all with odd mass number show half- 
integral spins. For many nuclei the spin value is not known, 
but often the system of statistics obeyed is known. Particles 
with Fermi statistics are considered to have half-integral spin, 
and those with Bose statistics, integral spin; all this evidence is 
compatible with the proton-neutron rather than the proton- 
electron model. 

Some additional qualitative remarks might be made about the 
structure of the deuteron, since especially in this simple case 

8 This unit is defined on page 36. 



34 ATOMIC NUCLEI CH. II 

progress has recently been made in achieving a working quantum- 
mechanical description. The evidence is that the ground state 
consists predominantly of overlying proton and neutron waves, 
with a region of very strong mutual attraction smaller in extent 
than the dimensions (wave lengths) of the particles, and with no 
orbital angular momentum. Thus, in spectroscopic notation this 
is a 3 Si (triplet Si) state; triplet because of the three permitted 
orientations of the nuclear spin of 1. The singlet S state ^So is 
known to be virtual (energetically unstable). The actual ground 
state must contain with 3 S\ an admixture of about 4 per cent 
3 Z>i, as evidenced by the electric quadrupole moment and the 
failure of a simple additivity relation between the proton and 
neutron magnetic moments and the deuteron moment. 



C. NUCLEAR PROPERTIES 

Mass and Energy. Masses of atomic nuclei are so small when 
expressed on ordinary mass scales (less than 10~ 21 g) that they 
are generally expressed on a different scale. The scale used is the 
so-called physical atomic-weight scale in which the mass of an 
atom of O 16 is taken as the standard and assigned a mass of exactly 
16.00000 units. It should be noted that this scale is not identical 
with the chemical atomic-weight scale (which is used for expressing 
atomic weights in chemical calculations); in the chemical scale 
the atomic weight of the natural isotopic mixture of oxygen (con- 
taining small amounts of O 17 and O 18 ) is assigned the value of 
exactly 16.00000. The unit used is, therefore, larger in the chem- 
ical than in the physical scale, and the numerical value of any 
atomic weight is smaller when expressed on the chemical scale. 
The conversion factor between the two scales is 1.000272 
0.000005, the uncertainty being due to the uncertainty in the 
isotopic composition of normal oxygen. 

The values of isotopic masses given in this book and in most 
of the literature on nuclear physics and chemistry are on the 
physical scale and are not nuclear but atomic masses; that is, 
they include the masses of the extranuclear electrons in the neu- 
tral atoms. This convention turns out to have some advantages 
in the treatment of nuclear reactions and energy relations. 

The experimental determination of exact atomic masses involves 
the use of a mass spectrograph. A number of different types of 



MASS AND ENERGY 35 

mass spectrographs have been devised. In all of these the charge- 
to-mass ratio of positive ions is determined from the amount of 
deflection in a combination of magnetic and electric fields; but 
they use different arrangements for bringing about either velocity 
focusing or directional focusing or both, for ions of a given e/M. 
Instruments which use photographic plates for recording the mass 
spectra are called mass spectrographs; those which make use of 
collection and measurement of ion currents are referred to as mass 
spectrometers. 

With modern techniques mass determinations can be made with 
a precision of 1 part in 10 5 for light atoms (up to about A = 40) 
and with somewhat lower precision (sometimes as low as 1 part 
in 10 4 ) for heavier atoms. For precision mass determinations the 
method generally used is the so-called doublet method. This 
substitutes the measurement of the difference between two almost 
identical masses for the direct measurement of absolute masses. 
All measurements must, of course, eventually be related to the 
standard O 16 . But for convenience the masses of H 1 , H 2 , and 
C 12 have been adopted as substandards and for this purpose have 
been carefully measured by determinations of the fundamental 
doublets: 

(C 12 Hl)+ and (0 16 )+ at mass-to-charge ratio 16, 
(Hi) 4 " and (C 12 ) 4 " 4 " at mass-to-charge ratio 6, 
(H 2 ) 4 " and (H^) 4 " at mass-to-charge ratio 2. 

On the physical scale the mass of a hydrogen atom (sometimes 
loosely called the proton mass) is 1.00812, the mass of the neutron 
1.00893, and that of an electron 0.0005486 mass units. One mass 
unit equals 1.661 X 10~ 24 g. 

One of the important consequences of Einstein's special theory 
of relativity is the equivalence of mass and energy. The total 
energy content E of a system of mass M is given by the relation 

S = Me 2 , 

where c is the velocity of light (2.99776 X 10 10 cm per sec). 
Therefore, the mass of a nucleus is a direct measure of its energy 
content. The measured mass of a nucleus is always smaller than 
the combined masses of its constituent nucleons, and the differ- 
ence between the two is called the binding energy of the nucleus. 



36 ATOMIC NUCLEI CH. II 

To find the energy equivalent to 1 mass unit we merely have 
to put M 1.661 X 10" 24 g and c = 2.998 X 10 10 cm sec- 1 , 
and we find E = Me 2 1.493 X 10~ 3 erg. However, energy 
units much more useful in nuclear work than the erg are the elec- 
tron volt (ev), the kiloelectron volt (kev; 1 kev = 1000 ev) and 
the million electron volt (Mev; 1 Mev = 10 6 ev). The electron 
volt is defined as the energy necessary to raise one electron through 
a potential difference of 1 volt. 

1 ev = 1.602 X 10" 12 erg; 1 Mev = 1.602 X lO"" 6 erg. 
Using these new units, we find 

1 mass unit = 931 Mev, 
and 

1 electron mass = 0.5107 Mev. 

As an example we shall calculate the binding energy of He 4 . 
The mass of He 4 is 4.00390; the combined mass of two hydrogen 
atoms (4) and two neutrons is 2 X 1.00812 + 2 X 1.00893 = 
4.03410. Thus the binding energy of He 4 is 4.03410 - 4.00390 
= 0.03020 mass unit or 0.03020 X 931 = 28.12 Mev. The 
binding energy per nucleon in He 4 is, therefore, approximately 
7.0 Mev. 

The binding energy of the deuteron calculated by the same 
method is found to be 2.18 Mev. Actually this value was deter- 
mined experimentally from the threshold for the photodisintegra- 
tion of the deuteron and combined with the mass-spectrograph- 
ically-measured masses of proton and deuteron to calculate the 
neutron mass. No accurate method for a direct measurement of 
the neutron mass is known. 

The average binding energy per nucleon is about 6 to 8 Mev 
throughout the table of elements except in a few of the lightest 
nuclei. The binding energy per nucleon is found to have a maxi- 
mum (near 8.7 Mev) for nuclei in the neighborhood of iron 
(mass ^ 55). Toward the heavy elements the dropping off is 
very gradual, and the average binding energy per nucleon reaches 
values near 7.5 Mev for the heaviest nuclei. On the light side of 
iron the values drop much faster, and among the lightest nuclei 
a number of irregularities occur; in particular the binding energies 

4 Since the mass of He 4 includes the mass of two electrons it is clear that it 
is the atomic mass of H 1 which must be used. 



MASS AND ENERGY 



37 



8 



a j I 

P-< 

8 7 



s 

8 




^01 x uojpejj 8un|08d 



38 ATOMIC NUCLEI CH. II 



of 2He 4 3 eC 12 , and 8 O 16 are abnormally high. These trends have 
some important consequences. The sun's radiant energy is be- 
lieved to result from a series of nuclear transformations whose 
net effect is the building up of helium atoms from hydrogen atoms 
which is a very exoergic process. Fission of the heaviest nuclei 
is energetically possible because nuclei near the middle of the 
periodic table have higher binding energies per nucleon. There 
is some evidence that the earth's core consists largely of iron and 
nickel, and this may well be connected with the maximum in the 
nuclear stability curve in the region of these elements. 

Quantities related to the binding ener _gy are the, mass jlefecj 
and the gacking fraction.^ These are, in fact, more frequently 
tabuIate3^"S[ian the binding energies. The mass defect A is the 
difference between the atomic mass M and the mass number 
A: A = M A. (Some authors call this the mass excess.) The 
packing fraction / is the mass defect divided by the mass number: 
/ = A/ A. (Sometimes / is defined as A/Af; the difference is negli- 
gible.) The packing fraction goes through a minimum in the 
region of iron. Since the atomic masses fall below the correspond- 
ing mass numbers between A 20 and A 180 the packing 
fractions are negative in that region. For convenience in tabula- 
tion packing fractions are often multiplied by 10 4 . A packing 
fraction curve is reproduced in figure II-l. 

Before leaving the subject of binding energies we should men- 
tion the binding energy of a nucleus for an additional nucleon, in 
particular an additional neutron; this is the energy that would be 
liberated if another neutron were added to the nucleus. This 
quantity is known for quite a number of nuclei, and values are 
shown in table 1 1-2. 

The masses of some radioactive nuclei can be determined from 
an accurate knowledge of the energy balance in nuclear reactions 
involving these nuclei and from their disintegration energies. 
This subject is discussed in chapter III. 

Charge and Radius. Nuclear charges were first determined in 
the a-particle-scattering experiments mentioned in section A. 
The most reliable way of determining nuclear charge is by 
Moseley's relation between Z and the energy characteristic of the 
K X rays emitted when the element is used as a target in an X-ray 
tube: 

Energy = 10.25( - I) 2 ev. 



CHARGE AND RADIUS 



TABLE II-2 
BINDING ENERGY FOB AN ADDITIONAL NEUTRON 





Binding 
Energy for 




Binding 
Energy for 




Binding 
Energy for 


Nu- Added Neutron 


Nu- Added Neutron 


Nu- Added Neutron 


cleus 


(in Mev) 


cleus 


(in Mev) 


cleus 


(in Mev) 


H 1 


2.19 


Ne 21 


9.42 


K 39 


7.1 


H 2 


6.16 


Ne 22 


4.88 


Ca 42 


7.2 


He 8 


20.51 


Na 23 


7.0 


Ti 46 


9.6 


He 4 


-0.8 


Mg 24 


7.1 


Ti 47 


9.8 


Li 6 


7.15 


Mg 26 


12.0 


Ti 48 


6.9 


Li 7 


1.98 


Mg 26 


5.8 


Ti 49 


10.6 


Be 9 


6.69 


Al 27 


7.5 


Ti 60 


11.5 


B io 


11.42 


Si 28 


8.3 


Cr 62 


8.9 


B 11 


3.2 


Si 29 


11.5 


Cr 63 


9.2 


C 12 


4.81 


Si 30 


5.5 


Fe 56 


6.9 


C 13 


8.23 


p31 


9.9 


Fe 67 


12.0 


N 14 


10.72 


S 32 


9.1 


Ni 60 


4.4 


Q16 


4.12 


S 33 


11.0 


Ni 61 


12.4 


O 17 


7.9 


S 34 


6.7 


Zn 66 


7.0 


Q18 


4.8 


C1 35 


8.2 


Zn 67 


7.7 


Ne 20 


7.51 


A 40 


7.0 







The data in this table are calculated from the isotopic masses given in table 
A in the appendix. 

A number of different methods have been employed to measure 
nuclear radii. The results are not all in agreement, and to under- 
stand the discrepancies we must consider the shape of the field 
of force around a nucleus. At distances outside the range of 
nuclear forces only Coulomb forces act, but closer in towards the 
center of the nucleus the nuclear attractive forces play the domi- 
nant role. This gives rise to a potential-energy curve for a nucleus 
and a positive particle separated by a distance r (measured 
between centers) somewhat as sketched in figure II-2. 

The first experiments that gave an indication of nuclear dimen- 
sions were those on a-particle scattering. In scattering experi- 
ments with elements heavier than copper no deviations from the 
Rutherford scattering formula were observed, because the avail- 
able a-particle energies did not permit approach to the surface of 
these nuclei. The minimum separation distances achieved, 
1.2 X 10~ 12 cm for copper and 3.2 X 10~~ 12 cm for gold, merely 
set upper limits to these nuclear radii. In lighter elements defi- 



40 



ATOMIC NUCLEI 



CH.H 



\ 



nite deviations from the Coulomb law were observed. Scattering 
experiments in aluminum, for example, show marked deviations 
from inverse-square forces at a distance of about 8 X 10~~ 13 cm. 
This value presumably corresponds roughly to R in. figure II-2. 

The radius R of the potential well inside which only the short- 
range nuclear forces are of importance can be determined in sev- 
eral other ways. The cross- 

\ sectional "area" which a nu- 

cleus presents to a beam of fast 
neutrons can be determined 
from experiments on fast neu- 
tron absorption and scattering. 
This method yields values of 
R from about 6 X KT 13 for 
A 50 to about 1 X 10~ 12 
for A ~ 200. A quantum-me- 
chanical treatment of a-particle 
decay yields a formula con- 
necting half-life and nuclear 
radius (see chapter VI); from 
the known half-lives of the 
naturally occurring a emitters 
nuclear radii between 8.4 and 
9.8 X 10~ 13 cm are obtained 
for these heavy elements. This 
method involves a calculation 
of the probability of penetra- 
tion of the potential barrier 
around the nucleus by the a particle. By computation of the 
barrier-penetration probabilities for charged particles from the 
yields of nuclear reactions between charged particles and nuclei, 
radii of lighter nonradioactive nuclei can be determined. 

The nuclear radii determined by the methods described in the 
last paragraph can be represented approximately by the empirical 
formula R = 1.5 X 10~ 13 A^ cm. We see from this that the 
nuclear volume is roughly proportional to the nuclear mass and 
that the volume per nucleon is about constant. 

Spin and Magnetic Moment. The intrinsic angular momentum 
of a nucleus is an integral or half-integral multiple / of h/2ir 
(often written as ft). This spin / is zero or integral for nuclei of 



FIGURE II-2. Potential energy in the 

neighborhood of a nucleus. R is the 

radius of the potential well. 



SPIN AND MAGNETIC MOMENT 41 

even A and half-integral for nuclei of odd A. Nuclei of even A 
and even Z seem to have zero spin. 

Since the rotation of a charged particle produces a magnetic 
moment, nuclei with spin also have magnetic moments associated 
with them. For a particle of mass M , charge Ze, and angular mo- 
mentum P the magnetic moment is, according to classical theory, 

/ 1 h\ 
ZeP/2Mc. For the electron I with P = 1 this formula gives 

\ t tT\ / 

one half of what is called the Bohr magneton (1 Bohr mag- 
neton = 0.927 X 10~~ 20 erg per gauss). Since proton and elec- 
tron have the same magnitude of charge and the same spin the 
proton magnetic moment calculated from the classical formula 
is 1/1835 of the electron moment; 1/1835 Bohr magneton is called 
a nuclear magneton and is used as the unit of nuclear magnetic 
moments. Actually neither the proton's nor the electron's mag- 
netic moment agrees with this too simple theory. The electron 
has a magnetic moment of one (instead of one-half) Bohr magne- 
ton, and the proton has 2.79 (instead of one-half) nuclear mag- 
netons. The magnetic moments of other nuclei also differ from 
the classical value (7 nuclear magnetons). These moments are 
sometimes expressed in terms of nuclear g factors; the magnetic 
moment is then g-I nuclear magnetons. If the magnetic moment 
is in the direction of the spin, g is taken as positive; if magnetic 
moment and spin are opposed, g is negative. A negative g factor 
results from the presence of some neutrons arranged with unpaired 
spins. The negative magnetic moment of the neutron presumably 
results from a charge distribution with some negative charge 
(perhaps due to negative mesons) concentrated near the periphery 
and overbalancing the effect of an equal positive charge nearer the 
center. 

Nuclear spins and magnetic moments can sometimes be deter- 
mined from hyperfine structure in atomic spectra. Plyperfine 
structure arises from the fact that the energy of an atom is slightly 
different for different (quantized) orientations between nuclear 
spin and angular momentum of the electrons because of the inter- 
action between the nuclear magnetic moment and the magnetic 
field of the electrons. From the number of lines in a spectroscopic 
"hypermultiplet" the nuclear spin 7 can be determined under 
suitable conditions. By this method many nuclear spins such as 
those of Bi 209 (7 = %) and Pr 141 (7 = %) have been measured. 



42 ATOMIC NUCLEI CH. II 

Although the number of components in a hypermultiplet is 
determined by the nuclear spin, the amount of the splitting (which 
is usually of the order of 1 angstrom or less) depends on the value 
of the nuclear magnetic moment. This dependence is well enough 
understood to allow a calculation of the magnetic moment from 
the magnitude of the splitting provided the nuclear spin is known. 
Thus the nuclear magnetic moments of Bi 209 and Na 23 have been 
determined by this method to be +3.8 and +2.215 nuclear mag- 
netons, respectively. 

Another method of determining nuclear spins is based on the 
alternating intensities found in rotational spectra of homonuclear 
diatomic molecules. Molecules in which the two nuclear spins 
are parallel and antiparallel, respectively, give rise to two sets of 
alternate lines in the rotational spectrum. The intensity ratio 
between successive lines is a measure of the abundance ratio for 

the two types of molecules, which can be shown to be 

at equilibrium except at very low temperatures. For the case of 
hydrogen the intensity ratio is 3:1, thus confirming the assign- 
ment of spin / = J/2 to the proton. The two kinds of hydrogen, 
orthohydrogen (with spins parallel) and parahydrogen (with 
spins antiparallel) can actually be isolated. In normal hydrogen 
at moderate and high temperatures they are present in the ratio 
3:1. 

A second method for the determination of nuclear magnetic 
moments (and for the measurement of nuclear spin) is the atomic 
beam method of I. I. Rabi and coworkers. In this method (which 
is an extension of the Stern-Gerlach experiment for the determina- 
tion of magnetic moments of atoms) a beam of atoms is sent 
through an inhomogeneous magnetic field. The nuclear spin /, 
uncoupled from the electron angular momentum J by the external 
field, orients itself with respect to the field. This orientation is 
governed by the usual quantum conditions and the beam is, 
therefore, split into 21 + 1 components whose separations are 
dependent on the nuclear magnetic moment. The energies of 
these splittings may be found in terms of characteristic alternating 
magnetic-field frequencies which induce transitions between com- 
ponents. Various modifications of this method, notably the addi- 
tion of focusing devices and the adaptation to molecular rather 
than atomic beams, have greatly improved the accuracy of the 



STATISTICS AND OTHER PROPERTIES 43 

results obtainable. The magnetic moment of the neutron has 
been directly determined by a suitable (and rather drastic) modifi- 
cation of this principle; it was found to be 1.91 nuclear mag- 
netons. 

A recent technique for the study of nuclear spins, and especially 
of magnetic moments, uses the so-called nuclear-induction method. 
The magnetic dipoles of nuclei of spin / can align themselves with 
a strong external magnetic field in 27 + 1 different orientations. 
The energy differences between the resulting 27+1 energy states 
correspond to the radio-frequency region, and their magnitude 
depends on the so-called gyromagnetic ratio, that is, the ratio of 
magnetic moment to spin. Resonance absorption of radio-fre- 
quency radiation will, therefore, take place at a frequency corre- 
sponding to these transitions, and the resonance frequency is a 
measure of the gyromagnetic ratio and, if / is known, of the mag- 
netic moment. In some cases / can be determined separately 
because the intensity of absorption is a function of / (and of 
other factors). 

Statistics and Other Properties. All nuclei and elementary par- 
ticles are known to obey one of two kinds of statistics: Bose sta- 
tistics or Fermi statistics. If the coordinates of two' identical 
particles in a system can be interchanged without change in the 
sign of the wave function representing the system, Bose statistics 
applies. If the sign of the wave function does change with the 
interchange of the coordinates, the particles obey Fermi statistics. 
In Fermi statistics each completely specified quantum state can 
be occupied by only one particle; that is, the Pauli exclusion 
principle applies to all particles obeying Fermi statistics. For 
particles obeying Bose statistics no such restriction exists. Pro- 
tons, neutrons, electrons (and some other "elementary" particles 
such as positrons, neutrinos, and perhaps some types of mesons) 
all obey Fermi statistics. A nucleus will obey Bose or Fermi 
statistics, depending on whether it contains an even or odd num- 
ber of nucleons. 

The statistics of nuclei can be deduced from the alternating 
intensities in rotational bands of the spectra of diatomic homo- 
nuclear molecules. With Bose statistics the even-rotational states 
and with Fermi statistics the odd-rotational states are more popu- 
lated. This can be illustrated by the cases of hydrogen and 
deuterium molecules. In normal hydrogen H2, the ratio of the 



44 



ATOMIC NUCLEI 



CH. II 



populations in the states of odd- and of even-rotational quantum 
numbers is 3:1 corresponding to spin ^ and Fermi statistics; in 
deuterium D 2 , the ratio is 1:2 corresponding to spin 1 and Bose 
statistics. 

Another nuclear property connected with symmetry properties 
of wave functions is parity. According to whether or not the wave 
function of a nucleus changes sign with an inversion of signs of 
all space coordinates, the nucleus is said to have odd or even 
parity. States of even and odd parity do not resonate, and it is 
for this reason that the deuteron ground state, a mixture of 3 /Si 
and 3 Z>i, cannot contain a P-state component. 

Finally mention should be made of the electric quadrupole 
moment of nuclei. This property may be thought of as arising 
from an elliptic charge distribution in the nucleus. The quad- 
rupole moment q is given by the equation q = %Z(a 2 6 2 ), 
where a is the semiaxis of rotation of the ellipsoid, and b is the 
semiaxis perpendicular to a; q has the dimensions of an area. For 
the deuteron q = +2.73 X 10"~ 27 cm 2 , and the charge distribu- 

TABLE II-3 

PROPERTIES OF SOME ELEMENTARY AND OTHER SIMPLE PARTICLES 
Particle or Nucleus 

Charge * 
i 

4-1 




+1, -1 
+1, -i, (0) 

+1 
+1 
+1 
+2 
+2 

* In units of e - 4.8025 X KT 10 esu. 

f For the proton and other nuclei the mass of the neutral atom is listed. 
The unit is the physical atomic weight unit. 

J In units of /i/2r. 

In units of the nuclear magneton (eh/irMc), where M is the proton mass. 
Positive values indicate moment orientations with respect to spin orientations 
that would result from spinning positive charges. 



Symbol 


Name 


e~ or 0~~ 


electron 


e+ or 0+ 


positron 


y 


photon 


V 


neutrino 


n 


neutron 


M 


n meson 


IT 


TT meson 


H l orp 


proton 


H 2 , d, or D 


deuteron 


H 3 , t or T 


triton 


He 3 




He 4 or a 


a particle 



Rest 




Magnetic 


Mass f 


Spin J 


Moment 


0.0005486 


1 A 


-1835 


0.0005486 


1 A 


4-1835 





i 





<0. 00002 


1 A 


<0.3 


1.00893 


1 A 


-1.913 


0.118 


or H? 




0.156 


or 1 




1.008123 


1 A 


4-2.793 


2.01471 


i 


4-0.857 


3.01702 


1 A 


4-2.976 


3.01700 


(K) 


( )2.13 


4.00390 





(0) 



OCCURRENCE OF ISOTOPES IN NATURE 45 

tion is cigar-shaped. Quadrupole moments, including both posi- 
tive and negative values, are known for a number of other nuclei 
with / > J^. They may be determined from abnormal hyperfine 
splittings in atomic spectra. Some attempts have been made to 
obtain information on quadrupole moments from a study of transi- 
tions between hyperfine components by means of microwave 
spectroscopy. 



D. ISOTOPY AND ISOTOPE SEPARATIONS 



Occurrence of Isotopes in Nature. The phenomenon of isotopy 
was discovered when different radioactive bodies in the natural 
decay series, for example, RaB, AcB, and ThB, were found to 
exhibit identical chemical properties (in the case mentioned the 
properties are those of lead). This led to a search for the existence 
of isotopes in nonradioactive elements. In early experiments 
with ion. deflections in magnetic and electric fields J. 3. Thomson 
showed in 1913 that neon consisted of two isotopes of mass num- 
bers 20 and 22 (now a third isotope Ne 21 is known). Since that 
time the development of improved instruments, notably 
F. W. Aston's mass spectrograph and its various modifications, 
has made very careful searches for isotopes possible in all the ele- 
ments existing in stable form. As a result of these mass-spectro- 
graphic investigations we now know that the elements with atomic 
numbers between 1 and 83 have on the average more than three 
stable isotopes each. Some elements such as beryllium, phos- 
phorus, arsenic, and bismuth have a single stable nuclear species 
each, whereas tin, for example, has as many as 10 stable isotopes. 
In addition each element from Z = 1 to Z = 96 has at least one 
known radioactive isotope, and in some cases there are as many 
as 12 or 15. The total number of radioactive species now known 
is about 700. 

The stable isotopes of a given element generally occur together 
in constant proportions. This accounts for the fact that atomic 
weight determinations on samples of a given element from widely 
different sources generally agree within experimental errors. 
However, there are some notable exceptions to this rule of con- 
stant isotopic composition. One is the variation in the abundances 
of lead isotopes, especially in ores containing uranium and thorium, 
which has already been discussed in chapter I in connection with 



46 ATOMIC NUCLEI CH. II 

the determination of the age of the earth. Similarly the isotope 
Sr 87 has been found to have an abnormally high abundance in 
rocks which contain rubidium; this is explained by the fact that 
Rb 87 is a naturally occurring ft emitter and decays to Sr 87 . Helium 
from gas wells probably has its origin in radioactive processes 
(a disintegrations) and contains a much smaller proportion of the 
rare isotope He 3 than does atmospheric helium. Water from 
various sources shows slight variations in the H 1 /!! 2 ratio. This 
is in some cases due to the fact that heavy water has a slightly 
lower vapor pressure than ordinary water and is, therefore, con- 
centrated by evaporation. The enrichment of H 2 in the water of 
the Dead Sea and in certain vegetables is ascribed to this cause. 
The waters which show abnormally high H 2 concentrations usu- 
ally also have slightly higher 18 /O 16 ratios than normal. Another 
cause for small variations in isotopic composition is the fact that 
chemical equilibria are slightly dependent on the molecular weights 
of the reactants, and this may lead to isotopic enrichments in the 
course of reactions occurring in nature. For example, the slight 
enrichment of C 13 in limestones relative to some other sources of 
carbon is explained by the fact that the equilibrium in the reaction 
between CO2 and water to form carbonic acid lies somewhat further 
towards the side of carbonic acid for C 13 O 2 than for C 12 O 2 . 

Isotope Separations. Some of the principles involved in isotope 
fractionutions in nature have been exploited for the artificial con- 
centration and separation of isotopes. We stated earlier that 
chemical properties are determined by the nuclear charge, and it 
would follow from this that isotopes of a given element are com- 
pletely identical in their chemical behavior. However, the iso- 
topic mass does have a very slight effect on chemical equilibria. 
In fact, for light elements such as carbon and nitrogen multistage 
exchange reactions have been used to produce separated isotopes 
on a commercial scale. These chemical effects become vanishingly 
small for isotopes of heavier elements because they depend essen- 
tially on percentage differences in mass. 

Other methods for isotope separations should be mentioned 
briefly. Diffusion of gases or liquids through porous membranes 
results in separation, the lighter isotopes diffusing more rapidly. 
This method has been successfully applied to the large-scale 
separation of the uranium isotopes. The thermal diffusion tech- 
nique of K. Clusius and G. Dickel makes use of the fact that in a 



BINDING ENERGIES 47 

thermal gradient the heavy isotopic component concentrates at 
the cold end and, by means of vertical adjacent hot and cold walls, 
provides a very ingenious arrangement for obtaining multistage 
separations in simple apparatus through the combined actions of 
convection and thermal diffusion. Separations have also been 
effected by use of high-speed centrifugal ion. Differences in vapor 
pressure between compounds containing different isotopes lead 
to concentration of the heavy constituents in the residues from 
slow evaporations; this method has been used for the concentra- 
tion of heavy water (H^O or D2O). Most heavy water is now 
produced by electrolysis, which also enriches the heavy com- 
ponent in the residues. Finally the electromagnetic methods of 
separation must be mentioned. Mass spectrographs have long 
been used to separate small quantities of isotopes, but during 
World War II the electromagnetic method was developed from a 
microgram to a kilogram scale for the purpose of separating U 235 
in quantity. The large electromagnetic separators (calutrons) at 
Oak Ridge, Tenn., are now used also for the separation of isotopes 
of a large number of elements throughout the periodic table. 

E. NUCLEAR SYSTEMATICS 

Binding Energies. ljumeroufi attempts have been made to ac- 
count for the shape of the packing-fraction or binding-energy 
curve. On a semiempirical basis an expression for the binding 
energy of a nucleus as a function of its proton and neutron com- 
position, that is as a function of Z and N = A Z, can be ob- 
tained. Several different expressions are given in the literature; 
one that gives rather good agreement with measured binding 
energies at least for A > 80 is 

E B =* 14.1 A - 13.1 A^ - 0.585 Z(Z - 1) A~* 

- 18.1 (JV - Z) 2 A" 1 + d A- 1 , (II-4) 

where 5 = 132 for Z even and N even, 5 = 132 for Z odd and 
N odd, and 5 = for A odd. The binding energy EB is here ex- 
pressed in million electron volts (Mev). 

The first term in equation II-4, and this is the most important 
term, is proportional to the number of nucleons A. This observa- 
tion can be interpreted to mean that the nuclear forces have short 



ATOMIC NUCLEI 



CH. II 



ranges and act between a small number of nucleons only. The 
saturation of these forces is apparently almost (but certainly not 
entirely) complete when four particles, two protons and two 
neutrons, interact, as is indicated by the large observed binding 
energies of He 4 , C 12 , O 16 . 

Those nucleons at the surface of a nucleus can be expected to 
have unsaturated forces, and, consequently, a reduction in the 
binding energy proportional to the nuclear surface should be 
taken into account. This is the second term, containing A^ 
which is a measure of the surface since A is proportional to the 
volume. 

The coulombic repulsive force between protons is, of course, 
not of the saturation type and is of sufficient range to be effective 
for all the protons in a nucleus. Therefore, each of the Z protons 
interacts with the other Z 1 protons to reduce the binding 
energy as shown in the third term. The factor A~^ enters this 
term because it measures the average separation distance for 
protons distributed in a volume proportional to A. 



100 

> 90 

80 

70 

60 

N50 
40 
30 
20 
10 



10 20 30 40 50 60 70 80 90 100 110 120 130 

N 

FIGURE II-3. The known stable nuclei on a plot of Z versus N. Note the 
gradual increase in the neutron-proton ratio; the 45 line indicates a neutron- 
proton ratio of 1. 



TYPES OF NUCLEI 49 

Other factors being equal, maximum stability, that is, minimum 
binding energy per nucleon, is found for nuclei with equal num- 
bers of protons and neutrons. This would indicate that (except 
for the coulombic energy effect) the protons and neutrons have 
similar energy-level spacings in nuclei. In this case the levels of 
lowest energy are occupied when Z = N; additional nucleons, 
most often neutrons, are by the Pauli exclusion principle required 
to go into levels of greater energy. This effect is expressed in 
quantitative (empirical) fashion by the fourth term. 

Closed Shells in Nuclear Structure. There is some evidence in 
binding energies and radii of nuclei that there is a shell structure 
somewhat analogous to the electron-shell structure in atoms, 
although less pronounced and much less understood. Attempts 
have been made to derive the quantum numbers for the succes- 
sive shells from assumptions about the nuclear forces, and the 
order of the first few levels for protons and neutrons appears likely 
to be Is, 2p, 2s, 3d. Remembering that s, p, d stand for angular- 
momentum quantum numbers I = 0, 1, 2, and that for each I 
there are 21+1 different states (corresponding to m values from 
+lto I including 0) each of which may now bo occupied by two 
protons and two neutrons, we get for the numbers of nucleons in 
nuclei with these shells successively completed: 

^4=4; A - 4 + 12 = 16; 

A = 4 + 12 + 4 = 20; A = 4 + 12 + 4 + 20 = 40. 



These completed shell structures correspond to 2He 4 , 8 O 10 ; ioNe 20 , 
and 2oCa 40 . Experimental observations on the stabilities (binding 
energies) of these nuclei are not in disagreement with this con- 
clusion. Above 2oCa 40 there are no stable nuclei with Z = N as a 
consequence of the coulombic repulsion, but there is evidence 
that energy shells fill with protons or neutrons to produce par- 
ticularly stable structures when Z or N equals 20 (as in Ca 40 ), 
50, 82, or 126. 

Types of Nuclei. Nuclei can be classified according to whether 
they contain even or odd numbers of protons and neutrons. There 
are then four types of nuclei as shown in the following tabulation. 
The distribution of the known stable nuclear species among these 
four types and the corresponding values of 6 in equation II-4 are 
as follows: 



50 ATOMIC NUCLEI CH. IJ 

162 even-even (6) (Z even, N even) 5 = +132 < 6 > 

56 even-odd (Z even, N odd) 5 = 

52 odd-even (Z odd, N even) 6 = 

4 odd-odd (Z odd, # odd) 6 = - 132 <> 

A relation between the frequency of occurrence and the stability 
as measured by the binding energy is apparent. The striking 
preponderance of even-wen nuclei and complete absence of odd- 
odd nuclei outside the region of the lightest elements (the four 
odd-odd nuclei are ill 2 , aLi 6 , sB 10 , and yN 14 ) can be explained in 
terms of a tendency of two like particles to complete an energy 
level by pairing opposite spins. 

The greater stability of nuclei with filled energy states is appar- 
ent not only in the larger number of even-even nuclei, but also 
in their greater abundance relative to the other types of nuclei. 
On the average, elements of even Z are much more abundant than 
those of odd Z (by a factor of about 10). For elements of even 
Z the isotopes of even mass (even N) account in general for about 
70 to 100 per cent of the element (beryllium, xenon, and dyspro- 
sium being exceptions). 

Stabttity Ruj,_ lielajbjgd to the foregoing is the observation that 
foFlmyodd Z there are never more than two stable isotopes, and 
if there are two their mass numbers differ by two units. Analo- 
gously, for any odd N there are no more than two stable isotones, 
and if there are two their mass numbers differ by two units. The 
only exceptions to these two rules are the four light odd-odd 
nuclei. Beyond Z 7, the only odd values of TV for which two 
stable isotones exist are 55, 05, and 85. 

Another empirical rule about the stable isotopes of elements of 
odd Z is that their mass numbers do not have values other than 
A 3, A 1, or ^1 + 1, where A is the mass number of the 
heaviest stable isotope of the preceding element. This rule applies 
without exception above oxygen. 

For any even Z there exist no more than two stable isotopes of 
odd mass number. The one exception to this rule is soSn with 

6 Sometimes these are called g-g, g-u, etc., from the German gerade even 
and ungerade * odd. 

* The quantity 6 is actually not a constant but varies rather irregularly 
with A. The value given is an average for A > 80. For A < 60 a lower 
value, perhaps 5 ^ 65, should be used, but in that region equation II-4 does 
not give reliable results anyway. 



EXERCISES 51 

three odd isotopes; but there is reason to suspect that Sn 118 is 
actually not stable (see below). The analogous rule can again be 
stated for isotones with even N; in fact, even N with two stable 
isotones of odd mass number occurs only at N = 20 and 
N = 82. 

Between oxygen and bismuth the following two rules also apply : 
(1) for every stable nuclide with odd Z there are two stable iso- 
tones, one w r ith Z 1 and one with Z + 1 protons; (2) for every 
stable nuclide with odd N there are two stable isotopes, one with 
N 1 and one with N + 1 neutrons. 

Finally, the following important rule is stated here, although 
its discussion must be deferred to chapter VI, section B. No 
pairs of stable isobars exist with Z differing by one unit. Appar- 
ent exceptions to this rule aro 4 8 Od 113 -4 9 In 113 , 49 In 115 -5oSn 115 , 
51 Sb 123 - 52 Te 123 , and 66 Ba 138 - 57 La 138 - 58 Ce 138 . 

EXERCISES 

1. Show that h ( 6.624 X 10~ 27 erg sec) has the dimensions of angular 
momentum. 

2. Derive the expressions for the kinetic and potential energies of an 
electron in a Bohr orbit. 

3. Show that the circumference of a stable Bohr orbit is always an 
integral multiple of the de Broglie wave length of the electron in the orbit. 

4. From the definitions of binding energy and packing fraction of a 
nucleus, derive a relation between the two quantities. 

5. Calculate the binding energy per nucleon in Li 6 , P 31 , Ti 50 , Ni 58 , 
Ag 107 , La 139 , (id 168 , An 197 , U 238 . 

Answers f&r first three: 5.30, 8.42, 8.69 Mev. 

6. Calculate from the masses given in table A in the appendix the 
binding energy for an additional neutron in O 16 , P 31 , Ti 60 . 

7. Using the natural abundances of the oxygen isotopes from table A 
in the appendix, calculate the atomic weight of ordinary oxygen on the 
physical scale and the conversion factor between the physical and chemical 
scales of atomic weights. 

8. What is the energy of an electron whose de Broglie wave length is 
1.5 X 10- 13 cm? Answer: 830 Mev. 



52 ATOMIC NUCLEI CH. II 

9. Without reference to tables, calculate the approximate charge and 
mass of the proton in familiar units (coulombs and grams, respectively). 

10. The three fundamental mass doublets have been found to have the 
following separations: 

(C 12 H 4 )+ - (0 16 )+ = 36.30 millimass units 
H 2 + - D+ = 1.538 millimass units 
D 3 + - (C 12 )++ = 42.22 millimass units 
Calculate the atomic masses of II, D, and C 12 . 

11. The radiation from the sun at normal incidence at the earth (93 mil- 
lion miles away) amounts to 0.135 joule per cm 2 per sec. The source of 
this energy is believed to be the conversion of hydrogen into helium. At 
what rate is hydrogen being consumed, in grain atoms per second? 

Answer: 5.9 X 10 14 . 

12. Using what information you have on nuclear radii, estimate the 
minimum a-particle energy necessary to observe deviations from Ruther- 
ford scattering in silver. Answer: ~14 Mev. 

13. With the aid of equation II-4 estimate: (a) the energy liberated 
when one additional neutron is added to U 235 , (b) the energy liberated when 
one additional neutron is added to U 238 , (c) the amount of energy by which 
ji29 j s uns table with respect to ft decay to Xe 129 . Answer: (a) 7.3 Mev. 

REFERENCES 

E. RUTHERFORD, J. CHADWICK, and C. D. ELLIS, Radiations from Radioactive 

Substances, Cambridge University Press, 1930. 

F. RASETTI, Elements of Nuclear Physics, New York, Prentice-Hall, 1936. 
M.I.T. Seminar Notes (C. GOODMAN, Editor), The Science and Engineering of 

Nuclear Power, Cambridge, Mass., Addison- Wesley Press, 1947 (Chapter 

1, "Fundamentals of Nuclear Physics"). 
II. E. LAPP and II. L. ANDREWS, Nuclear Radiation Physics, New York, 

Prentice-Hall, 1948. 
F. K. RICHTMYER and E. H. KENNARD, Introduction to Modern Physics, 4th 

ed., New York, McGraw-Hill Book Co., 1947. 
H. A. BETHE and R. F. BACHER, "Nuclear Physics, A. Stationary States of 

Nuclei," Rev. Mod. Phys. 8, 83-105 (193G). 
S. FLUEGQE, An Introduction to Nuclear Physics, New York, Interscience 

Publishers, 1946. 
Lecture Series in Nuclear Physics (Document MDDC-1175), obtainable from 

Superintendent of Documents, Washington 25, D. C. 
H. A. BETHE, Elementary Nuclear Theory, New York, John Wiley & Sons, 1947. 



REFERENCES 53 

F. W. ASTON, Mass Spectra and Isotopes, 2d ed., New York, Longmans, Green, 

1942. 
A. ROBERTS, "Radio-Frequency Spectroscopy in Nuclear Studies," Nucleonics 

1 no. 2, 10 (Oct. 1947). 
H. D. SMYTH, Atomic Energy for Military Purposes, Princeton University Press, 

1945. 

D. W. STEWART, "Separation of Stable Isotopes," Nucleonics 1 no. 2, 18 (Oct. 

1947). 

E. FEENBERG, "Semi-Empirical Theory of the Nuclear Energy Surface " 

Rev. Mod. Phys. 19, 239 (1947). 
R. R. WILLIAMS, "Nuclear Energetics," J. Chem. Ed. 23, 508 (1946). 



HI. NUCLEAR REACTIONS 

A. THE NATURE OF NUCLEAR REACTIONS 

A nuclear reaction is a process in which a nucleus reacts with 
another nucleus, an elementary particle, or a photon, to produce 
in a time of the order of 10 ~ 12 sec or less one or more other nuclei 
(and possibly neutrons or photons). Most of the nuclear reac- 
tions studied to date are of the type in which a nucleus reacts 
with a light particle (neutron, proton, deuteron, alpha particle, 
electron, photon) and the products are a nucleus of a different 
species and again one or more light particles. The chief excep- 
tion to this description is the fission reaction. 

Notation. As an example of a nuclear reaction we may cite the 
first such process discovered (in 1919), the disintegration of nitro- 
gen by a particles. When Rutherford bombarded nitrogen with 
a particles from RaC' he could observe scintillations on a zinc 
sulfide screen even when enough material was interposed between 
the nitrogen and the screen to absorb all the a. particles. Further 
experiments proved the long-range particles causing the scintilla- 
tions to be protons, and the results were interpreted in terms of 
a nuclear reaction between nitrogen and a particles to give oxygen 
and protons, or, in the usual notation: 

7 N 14 + 2 He 4 -> 8 17 + jH 1 . 

Since 1919 well over 1000 different nuclear reactions have been 
studied. The recognition of reaction products is greatly facilitated 
when these are unstable because characteristic radioactive radia- 
tions can then be observed. Artificial radioactivity was discovered 
in 1933 when I. Curie and F. Joliot were studying the emission of 
positrons by light elements under a-particle bombardment and 
found that some of the elements such as B, Mg, and Al continued 
to emit positrons after the removal of the a-particle source. For 
example, in the reaction between aluminum and a particles the 
product is P 30 which decays by positron emission to Si 30 with a 

54 



COMPARISON OF NUCLEAR AND CHEMICAL REACTIONS 65 
2.5-min half-life. The reaction is 

13 AF + 2 He 4 -> 15 P 30 + on 1 . 

The notation used for nuclear reactions is analogous to that in 
chemical reactions, with the reactants on the left- and the reaction 
products on the right-hand side of the equation. In all reactions 
so far observed the total number of protons and the total number 
of neutrons (or total Z and total A) are conserved, just as in 
chemical reactions the number of atoms of each element is con- 
served. In addition other properties such as energy, momentum, 
angular momentum, statistics, and parity are conserved in nuclear 
reactions. 

A short-hand notation is often used for the representation of 
nuclear reactions. The light bombarding particle and the light 
fragments (in that order) are written in parentheses between the 
initial and final nucleus; in this notation the two afore-mentioned 

reactions would read 

f 

N 14 (a, p) O 17 and Al 27 (a, n) P 30 . 

The symbols n, p, d, a, e, 7, x are used in this notation to represent 
neutron, proton, deuteron, alpha particle, electron, gamma ray 
and X ray. 

Comparison of Nuclear and Chemical Reactions. Nuclear reac- 
tions, like chemical reactions, are always accompanied by a release 
or absorption of energy, and this is expressed by adding the term 
Q to the right-hand side of the equation^ Thus a more_complete 
statement of Rutherford's first transmutation reaction reads 

7 N 14 + 2 He 4 -> 8 17 + iH 1 + Q. 

The quantity Q is called the energy of the reaction or more fre- 
quently just "the Q of the reaction. " Positive Q corresponds to 
energy release (exoergic reaction) ; negative Q to energy absorption 
(endoergic reaction). 

Here an important difference between chemical and nuclear 
reactions must be pointed out. In treating chemical reactions we 
always consider macroscopic amounts of material undergoing 
reactions, and, consequently, heats of reaction are usually given 
per mole or occasionally per gram of one of the reactants. In the 
case of nuclear reactions we usually consider single processes, 
and the Q values are therefore given per nucleus transformed. If 



56 NUCLEAR REACTIONS CH. Ill 

the two are calculated on the same basis, the energy release in a 
representative nuclear reaction is found to be many orders of 
magnitude larger than that in any chemical reaction. For exam- 
ple, the reaction N 14 (a, p) O 17 has a Q value of 1.13 Mev or 
-1.13 X 1.602 X HT 6 erg or -1.13 X 1.602 X 10" 6 X 2.390 
X 10~ n kg cal = 4.33 X 10~ 17 kg cal for a single process. 
Thus to convert 1 g atom of N 14 to O 17 , the energy required would 
be 6.02 X 10 23 X 4.33 X 10~ 17 kg cal = 2.61 X 10 7 kg cal. 
This is about 10 5 times as large as the largest values observed for 
heats of chemical reactions. On the other hand, one must keep 
in mind the fact that nuclear reactions are very rare events com- 
pared with chemical reactions; one reason is that the small sizes 
of nuclei make effective nuclear collisions quite improbable. 

B. ENERGETICS OF NUCLEAR REACTIONS 

/"The Q of a Reaction. It is clear from the foregoing discussion 
that the energy changes involved in nuclear^ ^reactions are of such 
magnitude that the corresponding mass changes in nuclear par- 
ticles must be observable. (The mass changes accompanying 
chemical reactions are too small to be observable with the most 
sensitive balances available.) If the masses of all the particles 
participating in a nuclear reaction are known from mass spectro- 
graphic data, as is the case for the N 14 (a, p) O 17 reaction, the 
Q of the reaction can be calculated. The sum of the N 14 and 
Pie 4 masses is 18.01141 mass units, and the sum of the O 17 and 
II 1 masses is 18.01262 mass units; thus an amount of energy 
equivalent to 0.00121 mass unit has to be supplied to make the 
reaction energetically possible, or Q = 0.00121 X 931 Mev 
= 1.13 Mev. In cases where the Q value is known experimen- 
tally (from the kinetic energies of the bombarding particle and 
the reaction products), it is sometimes possible to compute the 
unknown mass of one of the participating nuclei. By this method 
the masses of a number of radioactive nuclei have been deter- 
mined (see exercise 4). 

It is often possible to calculate the Q value of a reaction even 
if the masses of the nuclei involved are not known if the product 
nucleus is radioactive and decays back to the initial nucleus with 
known decay energy. Consider, for example, the reaction 
Pd 106 (n, p) Rh 106 . The product Rh 106 decays with a 30-sec 



THE Q OF A REACTION 57 

half-life and the emission of 3.55-Mev particles to the ground 
state of Pd 106 . We can write this sequence of events as follows: 

46 Pd 106 + on 1 - 45 Rh 106 + 1 H 1 + Q; 

4 5 Rh 106 -> 46 Pd 106 + r + 3.55 Mev. 

Adding the two equations we see that the net change is just the 
transformation of a neutron into a proton and an electron with 
accompanying energy change; or, symbolically, 

on 1 -> jH 1 + p- + Q + 3.55 Mev. 

Note that the symbol iH 1 must here stand for a bare proton 
(evident from the charge conservation) whereas the listed "proton 
mass" includes the mass of one orbital electron. (1) For energy 
balance we therefore write 

M n = /1/H 1 + Q + 3.55 Mev 

where M n = 1.00893 and M H l = 1.00812 mass units. Then 
Q = (1.00893 - 1.00812) X 931 - 3.55 

= 0.75 - 3.55 = -2.80 Mev. 

In the first example calculated we found the Q value of the reac- 
tion N 14 (a, p) O 17 to be -1.13 Mev. Does that mean that this 
reaction can actually be produced by a particles whose kinetic 
energies are just over 1.13 Mev? The answer is no, for two 
reasons. First, in the collision between the a particle and the 
N 14 nucleus conservation of momentum requires that at least 
4/18 of the kinetic energy of the a particle must be retained by 
the products as kinetic energy; thus, only 14/18 of the a. particle's 
kinetic energy is available for the reaction. The threshold energy 
of a particles for the N 14 (a, p) O 17 reaction, that is, the kinetic 
energy of a particles just capable of making the reaction ener- 
getically possible, is 18/14 X 1.13 Mev = 1.45 Mev. The frac- 
tion of the bombarding particle's kinetic energy which is retained 

1 In general, for negative 0-particle emission and electron-capture processes, 
the masses of electrons never have to be included in calculations when atomic 
masses are used. However, whenever a positron is involved in a reaction, two 
electron masses have to be taken into account: one for the positron and one for 
the extra electron that has to leave the electron shells to preserve electrical 
neutrality. 



58 NUCLEAR REACTIONS CH. Ill 

as kinetic energy of the products becomes smaller with increasing 
mass of the target nucleus (see exercise 5). 

Barriers for Charged Particles. The second reason why the a 
particles must have higher energies than is evident from the Q 
value to produce the reaction N 14 (a, p) O 17 in good yield is the 
Coulomb repulsion between the a particle and the N 14 nucleus. 
The repulsion increases with decreasing distance of separation 
until the a. particle comes within the range of the nuclear forces 
of the N 14 nucleus. This Coulomb repulsion gives rise to the 
potential barrier already mentioned in chapter II. The height 
V of the potential barrier around a nucleus of charge Ze and 
radius R\ for a particle of positive charge Z 2 e and radius #2 may 
be estimated as the energy of Coulomb repulsion when the two 



particles are just in contact: V = - . Obtaining the 



nuclear radii from the formula (2) R = 1.5 X 10~" 13 A^ f we get 
for the barrier height between N 14 and an a particle a value of 
about 3.4 Mev. Thus, at least according to the classical theory 
an a particle of energy less than 3.4 Mev cannot enter the N 14 
nucleus. Rutherford actually used a. particles of over 7 Mev in 
his experiments. In the quantum mechanical treatment of the 
problem there exists a finite probability for "tunnelling through 
the barrier " by lower-energy particles, but this probability drops 
rapidly as the energy of the particle decreases. The penetration 
of potential barriers is discussed in connection with a decay (chap- 
ter VI, section A). 

The potential barrier around a given nucleus for protons and 
for deuterons is about half as high as for a particles. The height 
of the potential barrier increases with increasing Z of the target 
nucleus; it is roughly proportional to Z^ (not to Z, because the 
nuclear radius R increases approximately as Z M ). For the heaviest 
elements the potential barriers are about 15 Mev for protons and 
deuterons and about 30 Mev for a particles. In order to study 
nuclear reactions induced by charged particles, especially reac- 
tions involving heavy elements, it was therefore necessary to 
develop machines capable of accelerating charged particles to 
energies of many millions of electron volts. 

* For the lightest nuclei this formula for nuclear radii is actually a poor 
approximation; but for an estimate of barrier heights it is adequate. 



SLOW NEUTRONS 59 

It must be emphasized that potential barriers have an effect 
not only for particles entering, but also for particles leaving nuclei. 
For this reason a charged particle has to be excited to a rather 
high energy inside the nucleus before it can either go over the top 
of the barrier or, according to the quantum mechanical picture, 
leak through the barrier with appreciable probability. Therefore, 
charged particles are usually emitted from nuclei with consider- 
able energies (more than 1 Mev)./ 

Slow Neutrons. It is evident from the foregoing discussion that, 
in general, it should be much easier for neutrons to enter and leave 
nuclei than it is for charged particles. This is indeed the case; 
even neutrons of very low energy can enter most nuclei with 
comparative ease. In fact, the so-called thermal neutrons, that 
is, neutrons whose energy distribution is approximately that of 
gas molecules in thermal agitation at ordinary temperatures, 
have particularly high probabilities for entering nuclei. The 
fact that a neutron of thermal energy (about 0.035 ev at 0C) 
has a de Broglie wave length of 1.5 X 10 ~ 8 cm may be considered 
as responsible for the unusually high reaction probabilities of slow 
neutrons with some nuclei. This subject is discussed further in 
section D. 

Our only sources of neutrons are nuclear reactions, in which 
neutrons are emitted from highly excited nuclei and, therefore, 
usually have initially rather high kinetic energies. Because of 
the great importance of slow or thermal neutrons in bringing about 
nuclear reactions, processes for slowing down fast neutrons to 
thermal energies have received much attention, both theoretically 
aud experimentally. Although fast neutrons may lose energy in 
inelastic collisions with nuclei, most slowing down is accomplished 
through a process of many successive elastic collisions with nuclei. 
The lighter the nucleus with which a neutron collides, the greater 
is the fraction of the neutron's kinetic energy that can be trans- 
ferred in the elastic collision. For this reason hydrogen-containing 
substances such as paraffin or water are the best slowing-down 
media for neutrons. Heavy water, helium, and carbon are also 
used. The mean free path between collisions in water or paraffin 
is several centimeters for a neutron of a few million electron volts 
energy, and a few millimeters for a thermal neutron. In each 
collision between a neutron and a proton the neutron's energy is 
on the average distributed equally between the two particles. 



60 NUCLEAR REACTIONS CH. Ill 

The average percentage energy loss of the neutrons is constant 
in each collision, and, as a result of this logarithmic averaging, 
the average neutron energy after n collisions with protons is 
e~~ n times the initial neutron energy. Approximately 20 collisions 
are therefore necessary to reduce neutrons from a few million 
electron volts to thermal energies. Eight or ten inches of paraffin 
surrounding a neutron source are adequate for reducing most 
neutrons to the thermal energy distribution. The whole slowing- 
down process requires less than 0.001 sec. The probable eventual 
fate of a thermal neutron in a hydrogenous medium like water or 
paraffin is capture by a proton to form a deuteron; but, since the 
probability of this reaction is quite small compared with the 
probability for scattering, a neutron after reaching thermal 
energies makes about 150 further collisions before being captured. 
Paraffin and water are good substances to use for the slowing 
down of neutrons because the capture probabilities in oxygen 
and carbon are even much smaller than in hydrogen. Heavy 
water is better than ordinary water because of the low probability 
of neutron capture by deuterium. Carbon (graphite) is also useful 
as a slowing-down medium because of its extremely low capture 
probability for neutrons; many more collisions are, of course, 
necessary to reduce neutrons to thermal energies in carbon than 
in hydrogen, but after reaching thermal energies the neutrons 
can exist longer in carbon. In either substance the lifetime of a 
neutron is only a fraction of a second. Even if neutrons could 
be kept in a medium where they would not eventually be captured, 
they might not be able to exist very long; they are believed to be 
unstable with respect to decay into protons and electrons, and 
theory predicts a half-life of about 20 min for this process. This 
decay has not yet been definitely observed. 

It should be evident from the foregoing discussion that thermal 
neutrons do not all have the same energy. After neutrons are 
slowed to energies comparable to thermal agitation energies 
(about 0.035 ev) they may either lose or gain energy in collisions, 
and the result is a Maxwellian distribution of velocities, in which 
the fraction (8) of the total number of neutrons with velocity 

8 The complete expression for this fraction, denoted by N(v)dv, is 



THE BOHR PICTURE OF NUCLEAR REACTIONS 61 

between v and v + dv at any given temperature is proportional to 

Mo 2 

v 2 e 2kT dv, where M is the neutron mass, and k is the Boltzmann 
constant. The average energy of the neutrons can be varied by 
varying the temperature of the slowing-down medium. At very 
low temperatures the Maxwellian distribution function becomes a 
poor approximation because of the discrete energy levels of the 
bound atoms of the slowing-down medium. 

C. MECHANISMS AND TYPES OF NUCLEAR REACTIONS 

The Bohr Picture of Nuclear Reactions. We have talked about 
nuclear reactions without considering in any detail the mecha- 
nisms by which such reactions may take place. In 1936 Bohr 
developed a theory of nuclear reactions which has been very suc- 
cessful in explaining many features of reactions induced by parti- 
cles of moderate energies (at least up to 30 or 40 Mev). This 
theory Avas based on a concept of the nucleus as a densely packed 
system with distances between nucleons of the same order of 
magnitude as the range of the nuclear forces and interaction 
energies between nucleons of the same order of magnitude as the 
kinetic energies of the incident particles. Bohr argued that an 
incident particle hitting such a system would lose much of its 
kinetic energy in the first few collisions with the nucleons and 
would then be held by the nuclear forces. /Thus he postulated as 
the first step in any nuclear reaction the amalgamation of target 
nucleus and incident particle into a compound nucleus. In this 
compound nucleus the kinetic energy of the incident particle and 
the additional binding energy contributed by it are rapidly 
distributed among all the nucleons. The second step of the reac- 
tion, the breaking up of the compound nucleus into the reaction 
products, can take place only after a relatively long time because 
a large number of collisions is required before enough energy is 
likely to be "acciden tally " concentrated on one nucleon to allow 
it to escape from the nuclear binding forces. The lifetimes of 
compound nuclei are of the order of 10~~ 12 to 10~~ 14 sec (4) which 

4 These times are too short to have been measured directly. By use of the 
uncertainty relation between time and energy (AE-At * h/2w) the lifetime 
of a compound nucleus in a particular energy state can be deduced from the 
width in energy of this state. This in turn can be determined experimentally 
from the spread in the energies of the incident particle which can be used to 
reach the state in question. 



62 NUCLEAR REACTIONS CH. Ill 

is very long compared to the time required for a fast particle to 
traverse a distance equal to a nuclear diameter. For example, 
a 0.5-Mev neutron (velocity 10 cm per sec) would traverse a 
medium heavy nucleus (diameter 10~ 12 cm) in 10~~ 12 /10 9 = 10~" 21 
sec. 

An essential feature of the Bohr picture is that the two steps 
of a nuclear reaction, the formation and the breaking up of a 
compound nucleus, are independent of each other. A given com- 
pound nucleus may be formed in different \vays and may also 
be able to disintegrate in different ways (for example, by emission 
of a proton, a neutron, or an a particle). According to this model 
each mode of disintegration has a certain probability which is 
independent of the mode of formation of the compound nucleus. 

The Bohr picture is definitely statistical in nature and can 
therefore be expected to be valid only for nuclei containing a 
large number of nucleons and for excitation to high energies where 
the nuclear energy levels are closely spaced. Although there is 
at present very little detailed knowledge of nuclear energy levels 
we can say that, in general, the level density increases with in- 
creasing mass number (that is, with increasing complexity of the 
nuclear system) and with increasing excitation energy. Average 
level spacings (in electron volts) appear to be about as follows: 



Near Ground At 


At 




State 


8 Mev 


15 Mev 


Light nuclei (A ^ 


' 10) ~10 6 


10 4 -10 6 


10 3 


Heavy nuclei (A <- 


- 150) ~10 6 


10-100 


10~ 2 -1 



There is evidence that the individual level widths increase with 
increasing excitation energy, so that there is a great deal of over- 
lapping of levels in medium and heavy nuclei at excitation energies 
of 12 or 15 Mev. The experimental evidence on level spacings 
and level widths comes from 7-ray spectra for the region near 
the ground state and from resonance capture processes (see sec- 
tion D below) for excitation energies in the neighborhood of 
8 Mev. 

Competition among Different Reactions. The Bohr theory sug- 
gests that a given compound nucleus may break up in several 
different ways. This is in agreement with the experimental obser- 
vation that the bombardment of a given nuclide with one type 
of nuclear particle of one energy usually leads to a variety of 




COMPETITION AMONG DIFFERENT REACTIONS 63 

products. For example, the bombardment of Al 27 with fast neu- 
trons (say 10 Mev) produces the radioactive products isAI 28 , 
13 Al 28 , i 2 Mg 27 , and nNa 24 ; according to Bohr's picture this means 
that the compound nucleus ^Al 28 * is formed and that it may 
break up in any one of the following ways (the asterisk indicates 
a high state of excitation of the compound nucleus) : 



A1 27 

isAl 



The Bohr picture permits some tentative predictions about 
the relative probability of various competing reaction types and 
their energy dependence, and, within the limitations to be dis- 
cussed later, these predictions are in accord with experimental 
evidence. For example, the theory predicts that for a compound 
nucleus which has been formed by the entry of a thermal neutron 
into the target nucleus the probability of emission of any nuclear 
particle should be extremely low; for neutron emission the entire 
binding energy contributed by the incident neutron would have 
to be concentrated on one neutron again, and proton emission 
would require the concentration on one proton of a comparable 
binding energy plus the barrier energy. The compound nucleus 
from thermal-neutron capture generally gives up its excess energy 
in electromagnetic radiation (7 rays). Indeed almost the only 
reaction observed with slow neutrons is the n, 7 process, often 
referred to as radiative capture. A few exceptions are found 
among reactions with the light nuclei in cases where the binding 
energy of a proton or a particle is appreciably lower than that 
of a neutron: the reactions B 10 (n, p) Be 10 , N 14 (n, p) C 14 , 
Cl 35 (n, p) S 35 , B 10 (n, a) Li 7 , and Li 6 (n, a) H 3 occur with thermal 
neutrons. 

As the excitation energy of the compound nucleus increases 
above the binding energy of the most loosely bound particle the 
probability for 7-ray emission becomes small relative to the 
probability for heavy-particle emission. For this reason radiative 
capture is important only for neutrons (which can enter a nucleus 



64 NUCLEAR REACTIONS CH. Ill 

without kinetic energy) and is rarely observed for charged parti- 
cles. The type of heavy particle emitted depends not only on 
the binding energy but also on the height of the potential barrier 
relative to the excitation energy. For moderate excitation 
energies the emission of neutrons is favored over the emission of 
protons, and thgrfin turn is more probable than a-particle emis- 
sion. For example, at moderate bombarding energies a, n reac- 
tions have muchVfarger probabilities than a, p reactions. With 
increasing excitatpsqi energy the effect of the barrier becomes 
less pronounced. 

At excitation energies above about 15 Mev the competition 
among different reactions becomes even more complex because 
the emission of a single particle (say a neutron) may leave the 
nucleus still in a sufficiently high state of excitation to "boil off" 
a second particle. Thus reactions of the types (n, 2n), (d, 2n), 
(a, 2n), (n, up), (d, up) will appear and compete with the simpler 
reactions (d, n), (a, n), (n, p), etc. At sufficiently high excitation 
energies one would expect the probability of single-particle emis- 
sion to drop owing to the competition of the two-particle emis- 
sions, and this again agrees with experimental results. 

Excitation Functions. A plot^of reaction yield versus energy of 
the incident particle is called an excitation function. To illus- 
trate the effect of competition, figure III-l shows the excitation 
functions of some reactions produced by a-particle bombardment 
of silver. (6) The reaction yield determined from the amount of 
radioactive product formed is plotted (in arbitrary units) against 
the a-particle energy in million electron volts. The production 
of In 110 has an observed threshold at about 11 Mev, presumably 
due to the reaction Ag 107 (a, n) In 110 . The In 110 yield goes 
through a maximum at 17.5 Mev and drops off rapidly after that; 
this decrease in the a, n yield coincides with a rise in the yield of 
the competing reaction Ag 107 (a, 2n) In 109 which has its observed 
threshold at 13.5 Mev. However, after going through a minimum 
at about 24 Mev, the In 110 yield increases again, and this second 
rise can be interpreted as caused by the a, 3n reaction on the 
other silver isotope Ag 109 . The excitation function for the Ag 109 
(a, 2n)In in reaction is almost identical with that for Ag 107 (a, 2n) 
In 109 until it reaches a peak at 27 Mev; it drops off when the 

6 S. N. Ghoshal, Phys. Rev. 73, 417 (1948). We are grateful to Mr. Ghoshal 
for permission to reproduce the figure. 



REACTIONS AT HIGHER ENERGIES 65 

reaction Ag 109 (a, 3n) In 110 begins to compete. The fact that the 
In 109 yield does not follow this pattern beyond about 25 Mev 
has been interpreted to mean that the "In 109 activity" actually 
includes In 108 activity which is thought to have a very similar 
half-life; thus the "In 109 curve" may really be the sum of the 
excitation functions for Ag 107 (, 2n) In 109 and Ag 107 (a, 3n) In 108 . 



10 




o In 111 
+ In 110 
In 109 (4ln 108 ?) 



10 



20 



30 
Mev 



40 



50 



FIGURE III-l. Excitation functions for (a, n), (, 2n), and (a, 3n) reactions 

in silver. Energy of the incident a particles in million electron volts is plotted 

on the abscissa; yield in arbitrary units on the ordinate. The three curves 

represent the yields of the various indium isotopes as indicated. 

Deviations from the Bohr Mechanism with High-energy Pro- 
jectiles. The Bohr picture leads one to believe that with higher 
and higher bombarding energies more and more complex reactions 
may be expected to replace the simpler ones; this expectation is 
again borne out by experiment, at least with heavy-particle pro- 
jectiles up to 40 or 50 Mev. Reactions such as (d, 4n), (d, pZn\ 
and (a, 6n), (a, p5ri) have been obtained in the heaviest elements 
with the 20-Mev deuterons and 40-Mev a. particles available 
from the Berkeley 60-inch cyclotron; similarly complex reactions 
have been produced by high-energy photons from the Schenectady 
100-Mev betatron. More recently the 190-Mev deuteroas and 
380-Mev a. particles from the new 184-inch cyclotron in Berkeley 
have been used for transmutation experiments, and it appears 
that with the excitation energies thus available the mechanism of 
the reactions is no longer entirely the one described by the Bohr 



66 NUCLEAR REACTIONS CH. Ill 

theory. Actually reactions of the type predicted by the Bohr 
theory, with the emission of a large number of light fragments 
because of the high excitation energies, are observed. (These 
have been termed "spallation reactions." As produced with 
190-Mev deuterons these processes have been reported to result 
in decreases of as many as 30 units of mass and 14 units of charge.) 
However, at the same time the whole spectrum of simpler reac- 
tions also occurs, with probabilities very much larger than the 
Bohr theory would permit. In fact the excitation function (6) 
of the reaction C 12 (p, pri) C 11 shows a rise up to a proton energy 
of about 60 Mev and then stays constant up to 140 Mev. Similar 
curves were found (7) for the reactions C 12 (d, dri) C 11 with deu- 
terons up to 190 Mov and C 12 (a, an) C 11 with a particles up to 
380 Mev. 

The following explanation for the observed phenomena in the 
high-energy region under discussion has been given by R. Serber. 
A nucleon with an energy of about 100 Mev or more will have a 
mean free path in nuclei comparable to nuclear dimensions, and, 
therefore, nuclei will not be entirely opaque for such particles, 
and the compound nucleus picture becomes inapplicable. Further- 
more, the impinging nucleon will transfer only a small part of its 
kinetic energy to a nucleon struck by it and will therefore, in 
general, have more than enough energy left to escape from the 
nucleus, even after one or two collisions. Thus the particular 
mechanism depends on whether the impinging particle strikes the 
nucleus near the periphery (in which case it may escape after 
a single collision and transfer only about 25 Mev to the nucleus) 
or near the center (in which case it may be stopped in the nuclear 
matter by several collisions and transfer its entire kinetic energy 
to the nucleus). Clearly there are intermediate possibilities, and 
according to this picture it is reasonable to expect a variety of 
modes of disintegration corresponding to the different possible 
excitation energies of the compound nucleus, all of which may 
have comparable probabilities. The reactions (p, pn), (d, dri), 
and (a, ari) previously mentioned are then thought to be brought 
about by collisions in which the incident particle takes away 
most of its energy, leaving the nucleus sufficiently excited to boil 
off a neutron. The probability of this type of reaction depends on 

W. W. Chupp and E. M. McMillan, Phys. Rev. 72, 873 (1947). 
7 L. R. Thornton and R. W. Senseman, Phys. Rev. 72, 872 (1947). 



INELASTIC SCATTERING 67 

the mean free path of the incident particle in the nucleus, which 
varies only slowly with energy at the high energies under discus- 
sion; this- accounts for the observed flatness of the excitation 
functions. Deuterons and a particles may be considered in terms 
of their individual nucleons because their kinetic energies are 
large compared with their binding energies. 

An additional observation made in connection with the high- 
energy reactions is that the incident nucleon may undergo an 
exchange collision: a proton may emerge as a neutron and vice 
versa. This result is apparent in the angular distribution of 
emitted particles and represents one of the most direct pieces of 
evidence for a charge exchange nature of nuclear forces. Other 
recent experiments in Berkeley have demonstrated the produc- 
tion of mesons in nuclear reactions at these very high energies. 

Types of Reactions. Returning now to reactions in the more 
usual energy range (8) we shall discuss a few particular t} r pes of 
reactions because of their special interest. Bombarding particles 
which have been used to effect nuclear reactions are neutrons, 
protons, deuterons, II 3 nuclei, a. particles, y rays (and X rays), 
and electrons, and the same particles except electrons and deu- 
terons are commonly observed among the fragments produced in 
nuclear reactions. 

The n, y reaction has already been mentioned as the only type 
commonly occurring with slow neutrons. The reaction is always 
exoergic and occurs with very nearly every target. In about 150 
cases it is known to lead to radioactive products. This reaction 
type is particularly important for the production of radioactive 
isotopes, because of the relatively high reaction yields and because 
of the enormous neutron fluxes now available in the chain-reacting 
piles. 

Inelastic Scattering. Reactions in which the incident and emit- 
ted particles are of the same type (n, n), (p, p), (y, 7), etc. 

8 The overwhelming majority of the artificially produced radioactive species 
now known (and listed in table A in the appendix) have been produced by 
relatively simple reactions involving the emission of one or two or at most 
three light fragments. Furthermore, it seems almost certain that for the 
production of practical amounts of radioactive tracers these simple reactions 
will continue to be used. However, the number of known radioactive species 
will undoubtedly increase considerably with the use of bombarding particles 
of higher and higher energies. In general, these will be further away from 
the region of stable nuclides and will tend to have short half-lives. 



68 NUCLEAR REACTIONS CH. Ill 

lead to an excited state of the initial nucleus which usually reverts 
to the ground state by y emission. The outgoing particle has less 
energy than the incident one, and, therefore, such a -process is 
referred to as inelastic scattering. In general, this type of reaction 
can be detected only by a measurement of the energy of the 
emitted (inelastically scattered) particle, but occasionally the 
residual nucleus is left in a metastable state of measurable life- 
time; that is, an isomer of a stable nucleus is formed. More than 
20 isomers of stable nuclei (9) are now known, among them 
Sr 87 * H -2.7 hr), In 115 * (4.5 hr), Ta 181 * (22 X 10^ sec), 
Pb 204 * (68 min). The n, n reaction is the most probable process 
with neutrons between a few hundred kev and a few Mev because 
in this region the n, 7 reaction is no longer so important, and the 
?i, p reaction cannot compete favorably because of the potential 
barrior. The p, p reaction is not very prevalent at any energy 
because of competition from the p, n reaction. For the same 
reason d, d and a, a reactions are rarely observed. Excitation to 
isomeric levels by 7, 7 reactions and even by e~~, e~~ reactions has 
been observed; in the latter case the excitation is not caused by 
an amalgamation of the electron with the nucleus, but by inter- 
action between the nucleus and the electromagnetic field of the 
high-speed electron. The e~~ t ne~~ reactions recently observed (10) 
in exceedingly small yields in Cu 63 , Ag 107 , and Ag 109 probably 
occur by a similar mechanism. 

Oppenheimer-Phillips Process. The rf, p reaction deserves spe- 
cial mention because it occurs very commonly and sets in at much 
lower energies than one would expect from the Bohr theory. In 
fact, the observed d, p thresholds are usually even lower than the 
corresponding d, n thresholds. This apparent anomaly has been 
explained by Oppenheimer and Phillips (11) as being due to the 
polarization of the deuteron by the Coulomb field of the nucleus. 
As the deuteron approaches the nucleus, its "neutron end" is 
thought to be turned toward the nucleus, the "proton end" being 
repelled by the Coulomb force. Because 'of the relatively large 
neutron-proton distance in the deuteron (several times 10~~ 13 cm), 

9 An asterisk after the mass number is often used to indicate an excited 
isomeric state. 

10 L. S. Skaggs, J. S. Laughlin, A. O. Hanson, and J. J. Orlin, Phys. Rev. 73, 
420 (1948). 

11 J. R. Oppenheimer and M. Phillips, Phys. Rev. 48, 500 (1935). 



FISSION 69 

the neutron reaches the surface of the nucleus while the proton is 
still outside most of the potential barrier. Since the binding 
energy of the deuteron is only 2.18 Mev, the action of the nuclear 
forces on the neutron tends to break up the deuteron, leaving the 
proton outside the potential barrier. The process just described 
is now generally called an Oppenheimer Phillips (or O-P) process. 
An analogous mechanism appears to be responsible for the 7/ 3 , p 
reaction recently reported. (12) An interesting feature of the OP 
process is that the emergent protons have a spread of energies 
which includes values in excess of the incident deuteron energy, 
so that in a small fraction of the cases the excitation of the com- 
pound nucleus is that which would result from the capture of a 
neutron of negative kinetic energy. 

Fission. Finally we mention a reaction of special practical sig- 
nificance, the fission process. By fission is meant the breakup 
of a heavy nucleus into two or more medium-heavy fragments. 
The process is usually accompanied by the emission of neutrons 
and much more rarely by the emission of a. particles and possibly 
other light fragments. Fission has been produced in some nuclides 
(notably U 235 , U 238 , and Th 232 ) by neutrons, protons, deuterons, 
a particles, and y rays, and more recently such elements as tanta- 
lum, platinum, thallium, lead, and bismuth have been found to 
undergo fission with 200-Mev deuterons and 400-Mev a. particles. 
By far the most important of these reactions is neutron-produced 
fission. The species 9 2 U 235 and 94 Pu 239 undergo fission with 
thermal or with fast neutrons, whereas fission of 90 Th 232 , 91 Pa 231 , 
and Q2U 238 requires fast neutrons. The fission process may occur 
in many different modes, and a very large number of fission 
products are known, ranging from Z = 30 (zinc) to Z = 63 
(europium), and from A = 72 to A = 158 for the case of U 235 
neutron fission. Because of the fact that the neutron excess 
required for stability is much greater in the region of the heaviest 
elements than in the fission-product region, the primary fission 
products have neutron excesses far greater than the stable isotopes 
of the same elements. These primary fission products achieve 
stability through successive /3~~ decays; some chains with as many 
as six successive &~~ decays are known. The yields of fission 

11 D. N. Kundu and M. L. Pool, Phys. Rev. 73, 22 (1948). 



70 



NUCLEAR REACTIONS 



CH. in 



products plotted against mass number (figure III-2) show two 
peaks separated by a very pronounced minimum; for slow-neutron 
fission of U 235 the maxima in the yield curve occur at A = 95 




110 120 130 140 150 160 170 

Mass Number 

FIGURE III-2. Yields of fission-product chains as a function of mass number 

(for the slow-neutron fission of U 236 ). O Mass assignment certain. D Mass 

assignment uncertain. 



DEFINITIONS 71 

and A 139, and the fission yield at mass 117 (at the minimum) 
is about 600 times smaller than at the peaks. In other words, 
fission is preferably asymmetric. The asymmetry appears to 
become less pronounced with higher excitation energy. The fission 
process is accompanied by the very large energy release of about 
200 Mev. The unique importance of the fission reaction is mainly 
due to the release of more than one neutron in each neutron- 
produced process, which makes a divergent chain reaction possible. 

The analogy between a nucleus and a liquid drop which Bohr 
had used when he proposed the idea of the compound nucleus 
can be extended to explain fission at least in a qualitative way. 
A heavy nucleus is held together by nuclear forces analogous to 
the cohesive forces holding together a liquid drop. Just as a 
liquid drop tends to assume spherical shape under the action of 
surface tension, the unsaturation of the nuclear forces at the 
surface makes a sphere (which has the smallest surface for a given 
volume) the most stable configuration for a heavy nucleus, Bohr 
and J. A, Wheeler have shown that there will be a certain critical 
size for nuclei, depending on Z 2 /A, above which the force of elec- 
trostatic repulsion will be greater than the surface forces holding 
the nucleus together. This critical size has been calculated to 
occur for Z somewhere near 100, and it is therefore reasonable 
that for a nucleus only slightly below this limit of stability a small 
excitation should be sufficient to induce breakup into two frag- 
ments. Bohr and Wheeler have calculated the energetic condi- 
tions for fission of various heavy nuclear species on the basis of 
this model, and their theory is in fair agreement with the facts. 
They were able to predict the fission of Pa 231 and to estimate its 
threshold energy before the reaction Jmd been discovered. 

Fission is not the only reaction that can occur with the heaviest 
elements. Rather, it competes with other reactions such as 
radiative capture and inelastic scattering, and the compound 
nucleus picture appears to account adequately for the relative 
probabilities of the competing processes at moderate excitation 
energies. 

^/D. CROSS SECTIONS 

Definitions. We shall now turn to a more quantitative consid- 
eration of reaction probabilities. The probability of a nuclear 
process is generally expressed in terms of a cross section v which 
has the dimensions of an area. This originates from the simple 



72 NUCLEAR REACTIONS CH. Ill 

picture that the probability for the reaction betw^n__ajiucleus 
and an impinging paHIde is^roportipnal to thecross-sectional 
target area presented by' the nucleus. Although this picture cer- 
taiSIydoes not hold for reactions with charged particles which 
have to overcome Coulomb barriers or for slow neutrons (it does 
hold fairly well for the total probability of a fast neutron inter- 
acting with a nucleus), tjje cross section is a very useful measure 
of the probability for any nuclear reaction. ./For a beam of par- 
ticles sinking a thin target, that is, a target in which the beam is 
attenuated only infinitesimally, the cross section for a particular 
process is defined by the equation, 

N = ITKTX, (III-l) 

where N is the number of processes of the type under considera- 
tion occuring in the target, 

I is the number of incident particles, 

n is the number of target nuclei per cubic centimeter of 
target, 

a is the cross section for the specified process, expressed 
in square centimeters, and 

x is the target thickness in centimeters. 

The total cross section for collision with ajfest particle is never 
greater than the geometrical cross-sectional area of the nucleus, 
and therefore fast-particle cross sections are rarely as large as 
10~ 24 cm 2 (radii of the heaviest nuclei are about 10 ~ 12 cm). 
Therefore, a cross section of 10~ 24 cm 2 is considered "as big as a 
barn" and l()~ 24 cm 2 has been named the barn, a unit often used 
in expressing cross sections. 

If instead of a thin target we consider a thick target, that is, 
one in which the intensity of the inciden ^particle beam is atten- 
uated, then the attenuation dl in the infinitesimal thickness dx 
is given by the equation, 

dl = Ina dx, 

where a must be the total cross section. If we are able to neglect 
the variation in a as the incident particles traverse the target, 
which is often the case for neutron reactions, we may obtain, 
by integration, 

I = lo*-"*, 
Io - I - I (l ~ *-""), (III-2) 



PARTIAL AND TOTAL CROSS SECTIONS 73 

where I is the intensity of the beam after traversing a target 
thickness #, IQ is the incident intensity, and IQ I is the number 
of reactions occurring. 

As an illustration we shall calculate the number of radioactive 
Au 198 nuclei produced per second in a sheet of gold 0.3 mm thick 
and 5 cm 2 in area exposed to a thermal neutron flux of 10 7 neu- 
trons per cm 2 per sec. The capture cross section of Au 197 for 
thermal neutrons is 95 barns, and we neglect any other reactions 
of neutrons with gold. The density of gold is 19.3 g per cm 3 , 
and its atomic weight is 197.2; therefore, 

19 3 

n = X 6.02 X 10 23 = 5.89 X 10 22 Au 167 nuclei per cm 3 ; 
197.2 

x = 0.03 cm; 

Io = 5 X 10 7 incident neutrons per sec. 
Therefore, according to equation 1 1 1-2: 
I - I = 5 X 10 7 (1 - e-a.wxio-xwxio-^xac*) 

= 5 X 10 7 (1 - t'- ' 108 ) 
= 7.8 X 10 Au 198 nuclei formed per see. 

Partial and Total Cross Sections. A cross section may bo given 
for any particular nuclear process. For example, the total cross 
section for absorption of 10-Mev neutrons by a particular nuclear 
species may be measured. This corresponds to the cross section 
for formation of the compound nucleus; but that compound 
nucleus may break up in various ways, and one may therefore 
wish to define partial cross sections for particular processes Buch 
as (n, n), (n, 7), (n, /?), and (n, a) reactions. The sum of all the 
partial cross sections equals the tot aj^ cross sec tion . We have 
already indicated that in equation lfi--2 total cross sections should 
be used, and only the total number of reactions may be obtained 
directly. This may be multiplied by the ratio of a partial cross 
section to the total cross section to obtain the number of reactions 
of a particular kind. The partial cross section might be for a 
single isotope, but the total cross section mustpBe that Tor thcT 
Target substance. ""* 



74 / NUCLEAR REACTIONS CH. Ill 

^The total cross section for fast-neutron reactions is approxi- 
mately equal to the geometrical cross section of the nucleus, 
n72 2 , and nuclear radii R have been determined by measurements 
of total fast-neutron cross sections. For charged particles the 
total cross section is generally less than irR 2 because of the poten- 
tial barrier. Charged particles with energies near the barrier 
height often have <r's in the neighborhood of lO" 1 barn. Gamma- 
ray cross sections are commonly observed in the range of 10"" 4 
to 10" 1 barn. 

Slow-neutron Cross Sections. The cross section for capture of 
slow neutrons is in many cases much greater than the nuclear 
geometrical cross section irR 2 . f This is explained by the fact that 
the de Broglie wave length X of a thermal neutron is much larger 
than nuclear dimensions. One would expect the theoretical upper 
limit for slow-neutron cross sections to be something less than X 2 
or about 10~~ 16 cm 2 (10 8 barns). Some of the largest thermal- 
neutron cross sections observed are those of gadolinium (30,000 
barns), samarium (6500 barns), and cadmium (2900 barns). 
These values are average cross sections for the natural mixtures 
of isotopes; since in each case one or two particular isotopes seem 
to be responsible for the large cross section the isotopic cross sec- 
tions are even bigger (for Cd 113 o- thermal 23,000 barns). Cad- 
mium is commonly used as an absorber of slow neutrons. The 
slow-neutron cross section varies in a quite irregular way from 
element to element. Data on neutron cross sections of most ele- 
ments are now available in the literature. A number of thermal- 
neutron activation cross sections are listed in table B in the appen- 
dix. 

One-over-v Law and Resonance Processes. For neutron ener- 
gies in the thermal region, and somewhat above, the cross sections 
for many substances are found to obey a 1/v law; that is, they 
vary inversely with the neutron velocity. For some light elements 
the 1/v law is valid over a wide energy range. Neutron capture 
in B 10 , for example, shows a 1/v dependence up to about 50 kev; 
neutron energies below that limit can therefore be measured by 
absorption in boron. 

..In practically all elements there are deviations from the 1/v 
law in one or more energy regions due to the existence of resonance 
levels. If the neutron entering a nucleus has an energy which 



ONE-OVER-F LAW AND RESONANCE PROCESSES 



75 



will just bring the compound nucleus into one of its energy levels, 
the cross section will be particularly high. Reactions occurring 
in this way are called resonance processes. If a cross section is 
measured as a function of neutron energy a curve such as the one 
in figure III-3 may be obtained, with resonance peaks superim- 
posed on the 1/v curve. It is from the spacings and widths of 
these resonance peaks that conclusions can be drawn about the 




FIGURE III-3. Typical curve of neutron cross section versus neutron energy. 



level densities and level widths in nuclei with excitation energies 
in the neighborhood of 6 to 8 Mev (the binding energy of a neu- 
tron). For light elements resonances have also been observed 
with protons and a particles. Some typical neutron resonances 
are at 1.44 ev in indium (cr res 26,000 barns), at 5.5 ev in silver 
foes ~ 7200 barns) and at about 100 kev in lithium; the large 
thermal-neutron cross section of cadmium is due to a resonance 
at 0.18 ev (<r res 7250 barns). The energy widths of resonance 
levels vary over wide limits; in general, the levels are narrower 
at low than at high energies. In many elements (including indium 
and rhodium) resonance capture leads to the formation of a radio- 
active isotope, and such substances are useful as detectors fgr 
neutrons of particular energies. Resonances occur not only for 
capture but also for scattering processes. 



70 NUCLEAR REACTIONS CH. Ill 

EXERCISES 

1. Compute, from masses in table A in the appendix, the Q values 
for the following reactions: 

(a) Mg 24 (d, p) Mg 25 
(6) B 10 (n, a) Li 7 

2. Would the Ca 43 (n, a) reaction go with thermal neutrons? Justify 
your answer. Answer: No. 

3. By what nuclear reactions may I 129 be made? What are some 
important considerations that would determine the best practical choice? 

4. In the reaction N 14 (n, p) C 14 0.60 Mev is liberated. What is the 
mass of the nucleus C 14 ? 

5. Show from conservation of momentum that in any nuclear reaction 
A (a, b)B the fraction of the kinetic energy of the bombarding particle a 
which goes into kinetic energy of the products is M a /(M a + MA), where 
M a and MA are the masses of a and A, respectively. 

6. Calculate the approximate heights of the potential barriers around 
18 A1 27 , 26 Fe 56 , 4 7 Ag 107 , 73 Ta 181 , and 92 U 238 for protons. 

Answers to first two: 3.1, 5.2 Mev. 

7. Calculate the de Broglie wave length of a neutron of (a) 1 ev, (6) 
1 kev, (c) 1 Mev energy. Answer: (b) 0.9 X 10~ 10 cm. 

8. What are the average energies (in electron volts) of thermal neutrons 
at 25C and at -196C (liquid nitrogen temperature)? 

9. Estimate (from Coulomb barrier considerations) the minimum total 
kinetic energy (in Mev) of the two fission fragments obtained when a 
uranium nucleus splits into (a) krypton and barium (6) rhodium and silver. 

Answer: (a) 198 Mev. 

10. Using the table of nuclear species in the appendix, decide what 
reactions would be practical for the production of (a) Mn 63 , (b) Gd 159 , (c) 
Tc 100 , if you had at your disposal 12-Mev deuterons, 6-Mev protons, 
24-Mev a particles, and neutrons up to 16 Mev. 

11. Approximately what thickness of cadmium (o- t h = 2900 barns) is 
necessary to reduce a beam of thermal neutrons to 0.1 per cent of its 
intensity? 

12. At room temperature (say 27 C) a beam of neutrons is brought to 
thermal equilibrium in graphite. This beam then falls on a thin boron 
absorber, 0- t h for boron = 703 barns. The beam is reduced to 90 per cent 



EXERCISES 77 

of its original intensity by the absorber. What would have been the 
intensity reduction if the entire experiment had been performed at 327 C? 

Answer: To 93 per cent. 

13. The nuclide Cl 33 can be produced by the reaction S 33 (p, n) Cl 33 . 
The Cl 33 emits positrons with an upper energy limit of 4.13 Mev. What 
is the Q value of the reaction? What is the height of the potential barrier 
around the S 33 nucleus for the proton? Estimate the minimum proton 
energy required to produce the reaction. 

Answer: Minimum proton energy 6.1 Mev. 

14. With the very crude assumption that in the highly excited com- 
pound nucleus the energy partition between nucleons is the same as in an 
ideal gas, estimate the temperature of a heavy nucleus (say A = 150) 
after the capture of a 10-Mev neutron. Answer: About 10 9 K. 

15. isA 35 decays to the ground state of nCl 35 with the emission of posi- 
trons of 4.4 Mev maximum energy. Without using the masses of A 35 
and Cl 35 , calculate the Q value of the reaction Cl 36 (d, 2n) A 36 . 

Answer: 8.37 Mev. 

16. (a) The measured cross section for absorption of thermal neutrons 
in B 10 is 3500 barns at 300K. Assuming the l/v law to hold, find the 
discrete velocity v' at which a = 3500 barns. 

(b) Using the answer to (a) estimate the mean life of a thermal neutron 
in 0.1 M borax (Na2B 4 O7-10H20) solution. You may assume B 10 to be 
the only capturing material. 

Answer: (a) 1.97 X 10 6 cm per sec; (b) 3.2 X 10~ 6 sec. 

17. With deuteron energies up to 100 Mev available, how would you 
propose to make each of the following nuclides with deuteron-induced 
reactions: Ti 43 , Gd 150 , Pb 200 , Eu 168 , La 143 ? Give specific answers, in terms 
of targets and deuteron energies. State your reasoning. 

18. Ordinary microbalances have a capacity of 20 g and a sensitivity 
of about 1 tig. How much better would the sensitivity have to be to 
permit detection of the mass change accompanying the reaction Na + 
y 2 Cl 2 = NaCl? 

19. The energy liberated in the sun is known to be the result of a cycle 
of nuclear reactions. We may think of the first step as the radiative 
capture of a high-speed proton (a thermal proton in the very hot interior 
of the sun) by a C 12 nucleus. The resultant N 13 is a positron emitter, 
half -life 10 min, and so transforms into C 13 , which in turn undergoes a 
p, 7 reaction. With the help of table A in the appendix, can you complete 
this mechanism to obtain the net reaction, that is, the transformation of 
four hydrogen atoms into one helium atom? You may want to refer to 
H. A. Bethe, "Energy Production in Stars," Phys. Rev. 56, 103 and 434 
(1939). 



78 NUCLEAR REACTIONS CH. Ill 

REFERENCES 

P. MORRISON, "Introduction to the Theory of Nuclear Reactions," Am. J. 

Phys. 9, 135 (1941). 
R. E. LAPP and H. L. ANDREWS, Nudear Radiation Physics, New York, 

Prentice-Hall, 1948. 

F. RASETTI, Elements of Nuclear Physics, New York, Prentice-Hall, 1936. 
J. MATTAUCH and S. FLUEGGE, Nudear Physics Tables and An Introduction 

to Nuclear Physics, New York, Interscience Publishers, 1946. 
H. A. BETHE, "Nuclear Physics, B. Nuclear Dynamics, Theoretical," Rev. 

Mod. Phys. 9, 69 (1937). 
M. S. LIVINGSTON and H. A. BETHE, "Nuclear Physics, C. Nuclear Dynamics, 

Experimental," Rev. Mod. Phys. 9, 245 (1937). 
Lecture Series in Nuclear Physics (Document MDDC-1175), obtainable from 

Superintendent of Documents, Washington 25, D. C. 
M.I.T. Seminar Notes (C. GOODMAN, Editor), The Science and Engineering of 

Nuclear Power, Cambridge, Mass., Addison- Wesley Press, 1947. 
E. POLLARD and W. L. DAVIDSON, Applied Nuclear Physics, New York, John 

Wiley <fe Sons, 1942. 

N. BOHR, "Transmutation of Atomic Nuclei," Science 86, 161 (1937). 
R. SERBER, "Nuclear Reactions at High Energies," Phys. Rev. 72, 1114 (1947). 
N. BOHR and J. A. WHEELER, "The Mechanism of Nuclear Fission," Phys. 

Rev. 56, 426 (1939). 
V. F. WEISSKOPP and D. H. EWING, "On the Yield of Nuclear Reactions with 

Heavy Elements," Phys. Rev. 67, 472 (1940). 
H. H. GOLDSMITH, H. W. IBSER and B. T. FELD, "Neutron Cross Sections of 

the Elements," Rev. Mod. Phys. 19, 259 (1947). 

L. SEREN, H. N. FRIEDLANDER and S. H. TURKEL, "Thermal Neutron Activa- 
tion Cross Sections," Phys. Rev. 72, 888 (1947). 

PLUTONIUM PROJECT, "Nuclei Formed in Fission: Decay Characteristics, Fis- 
sion Yields, and Chain Relationships," J.A.C.S. 68, 2411 (1946). 
R. H. GOECKERMANN and I. PERLMAN, "Characteristics of Bismuth Fission 

with High-energy Particles," Phys. Rev. 73, 1127 (1948). 



IV. SOURCES OF BOMBARDING PARTICLES 
A. HEAVY CHARGED PARTICLES 

Natural Sources of a Particles. From the discovery of nuclear 
transmutations in 1919 until 1932 the only known sources of 
particles which would induce nuclear reactions were the natural 
a emitters. In fact the only type of nuclear reaction known dur- 
ing that period of 13 years was the a, p reaction. The natural 
a-particle sources most frequently used in transmutation experi- 
ments were Po 210 (5.30 Mev, t^ = 140 days) and RaC' (7.68 
Mev, tft = 1.5 X 10"" 4 sec) used in equilibrium with its 0-emitting 
parent RaC. Today natural a-particle sources for nuclear reac- 
tions are chiefly of historical interest because of the much higher 
intensities and higher energies now available from man-made 
accelerators for charged heavy particles. 

Voltage Multiplier. The first transmutation by artificially accel- 
erated particles was achieved in 1932 when J, D. Cockroft and 
E. T. Walton bombarded lithium with protons from their voltage- 
multiplying rectifier set at the Cavendish Laboratory. The reac- 
tions were Li 6 (p, a) He 3 and Li 7 (p, a) He 4 ; the protons used in 
the initial studies had energies of 100 to 500 kev. The Cockroft- 
Walton type of machine consists essentially of a moderately high- 
voltage a-c transformer with an arrangement of vacuum-tube 
rectifiers and filter condensers so that the d-c output voltage is 
several times greater than the peak alternating voltage. With 
this type of equipment to supply the high voltage, protons have 
been accelerated up to about 1 Mev, and currents as high as 
200 pa, have been, obtained. 

Cascade Transformer. C. C. Lauritsen and his coworkers at the 
California Institute of Technology adapted multistage trans- 
formers to the acceleration of positive ions. In this arrangement 
a number of high-voltage transformers are placed in cascade (to 
reduce transformer-winding insulation requirements) by having 
each primary winding excited by a portion of the preceding second- 
ary winding. Units with four or five stages and with 200 or 250 kv 

79 



80 



SOURCES OF BOMBARDING PARTICLES 



CH. IV 



Spherical conductor 



per stage have been built, giving a maximum potential difference 
of about 1 million volts. 

The cascade transformer is entirely an a-c device, and, there- 
fore, positive ions can be accelerated only during half a cycle; in 

fact only during a small frac- 
tion of the cycle can they be 
accelerated to the maximum 
energy. To obtain nearly 
monoenergetic beams an ar- 
rangement is used whereby the 
ions are admitted to the accel- 
erating tube during the appro- 
priate fraction of the positive 
half-cycle. Ion currents of 
about 30 /ia have been obtained 
with cascade transformers. 

Electrostatic (Van de Graaff ) 
Generator. The adaptation of 
the electrostatic machine to 
the production of high poten- 
tials for the acceleration of 
positive ions was pioneered by 
R. J. Van de Graaff of the 
Massachusetts Institute of 
Technology, beginning in 1931. 
In the Van de Graaff machine 
a high potential is built up and 
maintained on a conducting 
sphere by the continuous trans- 
fer of static charges from a 
moving belt to the sphere. 

This is illustrated in figure IV 1. The belt, made of silk, rubber, 
paper, or some other suitable insulator, is driven by a motor and 
pulley system. It passes through the gap AB which is connected to 
a high-voltage source (10,000 to 30,000 volts d-c, usually from a vac- 
uum-tube rectifier arrangement) and adjusted so that a continuous 
discharge is maintained from the sharp point B. Thus positive (or 
negative) charges are sprayed from B onto the belt which carries 
them to the interior of the insulated metal sphere; there another 
sharp point or sharp-toothed comb C connected to the sphere 




FIGURE IV- 1. Schematic representa- 
tion of the charging mechanism of a 
Van de Graaff generator. 



ELECTROSTATIC (VAN DE GRAAFF) GENERATOR 81 

takes off the charges and distributes them to the outside surface 
of the sphere. (Because of their coulombic repulsion the charges 
always go to the outside of a spherical conductor.) The sphere 
will continue to charge up until the loss of charge from the surface 
by corona discharge and by leakage along its insulating support 
balances the rate of charge transfer from the belt. The continuous 
current that can be maintained with an electrostatic generator 
depends on the rate with which charge can be supplied to the 
sphere. The use of wide belts (2 to 4 feet) driven at high speeds 
is therefore indicated. 

Van de Graaff' s first installation consisted of two spheres, each 
2 feet in diameter; one charged to a positive potential of about 
750,000 volts above ground and the other to an equal negative 
potential. All recent electrostatic generators use a single elec- 
trode, with acceleration of the ions between that electrode and 
ground potential, because considerable practical advantages are 
gained if much of the auxiliary equipment can be operated at 
ground potential. 

Since the voltage of an electrostatic generator is limited by the 
breakdown potential of the gas surrounding the charged electrode 
it is desirable to use conditions under which this breakdown poten- 
tial is as high as possible. The breakdown potential is a function 
of pressure and goes through a minimum at a rather low pressure 
(small fraction of an atmosphere). It is therefore advantageous 
to operate an electrostatic generator either in a high vacuum, 
which presents formidable difficulties, or in a high-pressure atmos- 
phere. R. G. Herb and his associates at the University of Wis- 
consin have developed electrostatic generators completely enclosed 
in steel tanks in which pressures of several atmospheres are main- 
tained. This has considerably increased the potentials attainable. 
A further improvement is the use of gases which have higher 
breakdown potentials than air. Admixtures of carbon tetra- 
chloride and Freon to air or nitrogen, and more recently the use 
of sulfur hexafluoride, have been successful in increasing voltages 
of existing generators. A number of pressure-type electrostatic 
generators capable of accelerating protons (or other positive ions) 
to 2 to 5 Mev are in operation, and it appears likely that energies 
of 10 to 12 Mev may be reached with larger machines of similar 
design. Proton currents of 5 to 100 pa, are common. The chief 
application of machines of this type is in nuclear physics work 



82 



SOURCES OF BOMBARDING PARTICLES 



CH. IV 



requiring high precision because unlike other machines such as 
cyclotrons they supply ions of precisely controllable energies 
(constant to about 0.1 per cent). 

Accelerating Tubes. We have so far discussed methods of ob- 
taining high voltages for the acceleration of positive ions without 
going into any detail about the accelerating tube across which 
the potential is applied. The need for such a tube is common to 
all the types of machines described so far. A source of ions near 







s 












^ v 





Ion beam 



FIGURE IV- 2. Schematic diagram of a portion of an accelerating tube. 

the high-voltage end, a system of accelerating electrodes, and a 
target at the low-voltage end must be provided and enclosed in a 
vacuum tube connected to the necessary pumping system. The 
ion source is essentially an arrangement for ionizing the proper 
gas (hydrogen, deuterium, helium) in an arc or electron beam; 
the ions are drawn through an opening into the accelerating sys- 
tem. A typical accelerating tube (figure IV-2) is built of glass or 
porcelain sections S. Inside this tube, sections of metal tube T 
define the path of the ion beam. Each metal section is supported 
on a disk which passes between two sections of insulator out into 
the gas-filled space to a corona ring R which is equipped with 
corona points P. The purpose of the corona rings and points is 
to carry the corona discharge from the high- to the low-voltage 
end of the tube and to distribute the voltage drop uniformly along 
the tube. Depending on the number of sections used a potential 
difference somewhere between ten and several hundred kilovolts 



LINEAR ACCELERATOR 83 

exists between successive sections. Each gap between successive 
sections has both a focusing and a defocusing action on the ions 
traveling down the tube. The ions tend to travel along the elec- 
tric lines of force (see figure IV-2 for the pattern of these lines 
between a pair of sections). In entering the gap the ions are there- 
fore focused, and in leaving it they are defocused; but because the 
ions move more slowly on entering the gap than on leaving it the 
focusing effect is stronger than the subsequent defocusing. Well- 
focused beams (cross-sectional area less than 0.1 cm 2 ) can be 
obtained. It should be mentioned that from hydrogen gas in an 
ion source not only protons but also hydrogen molecule ions 
(H 2 + ), and H 3 + ions are obtained; these are also accelerated in 
the tube but can be separated from the protons before striking 
the target by means of a magnetic analyzer. 

In a pressure electrostatic generator the charging system, high- 
voltage electrode, and accelerating tube are all enclosed in the 
steel tank containing the high-pressure gas. The high-potential 
electrode is often more like a cylinder than a sphere. 

Linear Accelerator. In all the devices for accelerating ions dis- 
cussed so far the full high potential corresponding to the final 
energy of the ions must be provided, and the limitations of this 
type of device are introduced by the insulation problems. These 
problems are very much reduced in machines which employ re- 
peated acceleration of ions through relatively small potential 
differences. The most successful scheme of this type is the cyclo- 
tron. But the first device proposed for achieving high-voltage 
acceleration of ions by multiple application of a much lower volt- 
age was the linear accelerator suggested by R. Wider 6e in 1929 
and built in the United States by D. H. Sloan and E. O. Lawrence 
(1931). 

In this machine a beam of ions from an ion source is injected 
into an accelerating tube containing a number of coaxial cylin- 
drical sections (see figure IV-3 for a schematic diagram). Alter- 
nate sections are connected together, and a high-frequency alter- 
nating voltage from an oscillator is applied between the two 
groups of electrodes. An ion traveling down the tube will be 
accelerated at a gap between electrodes if the voltage is in the 
proper phase. By choosing the frequency and the lengths of suc- 
cessive sections correctly one can arrange the system so that the 
ions arrive at each gap at the proper phase for acceleration across 



84 SOURCES OF BOMBARDING PARTICLES CH. IV 

the gap. The successive electrode lengths have to be such that 
the ions spend just one half-cycle in each electrode. Acceleration 
takes place at each gap, and the focusing action is the same as in 
the accelerating tubes of high- voltage sets. 

In early models of the linear accelerator, mercury ions were 
accelerated to 2.85 Mev with an input voltage of about 80 kv. 
However, because the cyclotron was developed almost simul- 
taneously and had obviously great advantages, the linear accelera- 



J 



Ion bearn^ 



j_ 



i 

FIGURE IV-3. Schematic diagram of the accelerating tube of a linear accel- 

erator. 

tor did not receive much further attention after about 1934. Very 
recently, interest in modified linear accelerators has been renewed 
because the development of high-power microwave oscillators 
apparently offers the possibility of extending the energy range of 
linear accelerators to the billion-electron-volt (Bev) region. Pre- 
liminary work in this field indicates that energy increments of 
about 1 Mev per linear foot can be achieved. But it appears 
that for the acceleration, at least of positive ions, to ultrahigh 
energies other devices are more practical than the linear accel- 
erator. 

JBy far the most important and successful device to 



date for accelerating positive ions to millions of electron volts is 
the cyclotron proposed by Lawrence in 1929. A remarkable 
development has taken place from the first working model which 
produced 80-kev protons in 1930 to the giant machine recently 
put in operation in Berkeley which accelerates a particles to 
380 Mev and deuterons to 190 Mev. 

In the cyclotron, as in the linear accelerator, multiple accelera- 
tion by a radio-frequency potential is used. But the ions, instead 
of traveling along a straight tube, are constrained by a magnetic 
field to move in a spiral path consisting of a series of semicircles 
with increasing radii. The principle of operation is illustrated in 



CYCLOTRON 



85 



figure IV-4. Ions are produced in an arc ion source I near the 
center of the gap between two hollow semicircular electrode boxes 
called "dees" (figure IV-5). The dees are enclosed in a vacuum 
tank, which is located between the circular pole faces of an electro- 
magnet and is connected to the necessary vacuum pumping system. 
A high-frequency potential supplied by an oscillator is applied 
between the dees. A positive ion starting from the ion source is 
accelerated toward the dee which is at negative potential at the 



deflector 





FIGURE IV-4. Ion path in the 
cyclotron. The magnetic field 
is perpendicular to the plane of 
the paper. Shaded area repre- 
sents the "decs." 



FIGURE IV-5. Perspec- 
tive view of the dees. 



time. As soon as it reaches the field-free interior of the dee, the 
ion is no longer acted on by electric forces, but the magnetic field 
perpendicular to the plane of the dees constrains the ion to a semi- 
circular path. If the frequency of the alternating potential is 
such that the field has reversed its direction just at the time the 
ion again reaches the gap between dees, the ion again is acceler- 
ated, this time toward the other dee. Now its velocity is greater 
than before, and it therefore describes a semicircle of larger radius; 
however, as we shall see from the equations of motion, the time 
of transit for each semicircle is independent of radius. Therefore, 
although the ion describes larger and larger semicircles it continues 
to arrive at the gap when the oscillating voltage is at the right 
phase for acceleration. At each crossing of the jjapthe ion acquires 
an amount of kinetic energy equal to 'the proouxjtTof the voltage 
difference between the dees and the ion charge. Finally, as the 



86 SOURCES OF BOMBARDING PARTICLES CH. IV 

ion reaches the periphery of the dee system it is removed from its 
circular path by a negatively charged deflector plate and is allowed 
to strike a target. "^ 

The equation of motion of an ion of mass M, charge e, and 
velocity v in a magnetic field H is given by the necessary equality 
of the centripetal magnetic force Hev and the centrifugal force 
MiP/r where r is the radius of the ion's orbit: 

Mv 2 

Hev = - , 
r 

and, therefore, 

Mv 
r - _. (IV-1) 

Remembering that the angular velocity = v/r, we see that 

He 



From this equation it is evident that the angular velocity is inde- 
pendent of radius and ion velocity and that the time required for 
half a revolution is constant for ions of the same e/M provided 
the magnetic-field strength is constant. In practice, the mag- 
netic field is kept constant, e/M is a characteristic of the type 
of ion used, and therefore o> is constant; the radio frequency has 
to be chosen such that its period equals the time it takes for 
the ions to make one revolution. For H = 15,000 gauss, (1) and 
e/M for a proton, the revolution frequency co/27r, and therefore 
the necessary oscillator frequency turns out (from equation 
IV-2) to be about 23 X 10 6 cycles per sec. For deuterons or 
helium ions (He+ + ) at the same H the frequency is half that 
value. Most cyclotrons are operated as deuteron sources, and 
they often use r-f oscillators tuned to about 11 or 12 megacycles. 
It is clear from equation IV-2 that on a given cyclotron both 
the magnetic field and the oscillator frequency can be left un- 
changed when different ions of the same e/M, such as deuterons 
and a particles, are accelerated. Equation IV-1 shows that the 
velocity reached at a given radius is the same for ions of the same 

1 Actually to make the equations dimensionally correct, not the field strength 
H, but the magnetic induction B (maxwells per square centimeter in emu) 
should be used. 



CYCLOTRON 87 

e/M; therefore, a particles are accelerated to the same velocity 
and, hence, twice the energy as deuterons. To accelerate protons 
in a cyclotron designed for deuterons either the frequency must 
be approximately doubled (which is very impractical) or H must 
be about halved. Although the latter makes inefficient use of 
the magnet, it is occasionally done, and the final velocity is again 
the same as for deuterons (see equation IV-1); therefore, protons 
are accelerated to half the energy available for deuterons. 
By squaring equation IV-1 one gets 

M 2 v 2 1 H 2 e* 

or -M*2 = - r 2 . 



H 2 e 2 2 2M 

Thus the final energy attainable for a given ion varies with the 
square of the radius of the cyclotron. With H = 15,000 gauss the 
deuteron energy E = 0.035r 2 for E in million electron volts and 
r in inches. 

From the equations of motion it is clear that an ion can reach 
the dee gap at any phase of the dee potential and still be in reso- 
nance with the radio frequency. As we have just derived, the 
final energy acquired by an ion is entirely independent of the 
energy increment the ion receives at each crossing of the dee gap. 
However, in practice it is found that ions making too many revo- 
lutions generally do not reach the exit slit because they may be 
deflected in collisions with residual gas molecules and may strike 
the surface of the dees. Therefore, only ions which enter the 
first gap in a favorable phase of the radio frequency (perhaps 
during about one third of the cycle) contribute to the beam cur- 
rent. To avoid difficulties due to excessively long paths for the 
ions, rather high dee voltages (20,000 to 50,000 volts) are generally 
used. 

A very important feature of the cyclotron is the automatic 
focusing action it provides for the ion beam. The electrostatic 
focusing at the dee gap is entirely analogous to that in the high- 
voltage accelerating tubes. However, as the energy of the ions 
increases, this effect becomes almost negligible. Fortunately a 
magnetic focusing effect becomes more and more pronounced as 
the ions travel towards the periphery. This can be seen from the 
shape of the magnetic field as shown in figure IV-6. Near the 
edge of the pole faces the magnetic lines of force are curved, and, 
therefore, the field has a horizontal component which provides a 



88 



SOURCES OF BOMBARDING PARTICLES 



CH. IV 



restoring force toward the median plane to an ion either below or 
above that plane. The focusing is so good that a cyclotron beam 
generally covers less than 1 cm 2 at the target. 

One difficulty we have so far neglected is presented by the rela- 
tivistic mass increase of the ions as they reach high energies. This 
increase is about J^ per cent for a 10-Mev deuteron and about 
5 per cent for a 100-Mev deuteron. It is clear from equation IV-2 
that if the revolution frequency is to be kept constant the increase 
in mass must be compensated by a proportional increase in field 




FIGURE IV-6. Shape of magnetic 
field in the gap of a cyclotron mag- 
net. Curvature of lines of force 
near the edge of the magnet gives 
rise to focusing action. 



FIGURE IV-7. Exaggerated repre- 
sentation of defocusing effect of 
cyclotron magnet shimming. 



strength. When the relativity effects are small this increase of 
the magnetic field toward the periphery can be readily achieved 
by slight radial shaping or shimming of the pole faces. (2) Notice, 
however, that this shaping of the field creates regions of magnetic 
defocusing (exaggerated in figure IV-7). For moderate relativistic 
mass increases this difficulty has been overcome, mainly by the 
use of higher dee voltages and correspondingly shorter ion paths. 
The 60-inch cyclotron in Berkeley operates excellently for 20-Mev 
deuterons or 40-Mev a particles, producing currents of about 
100 M& and several microamperes, respectively. But it is clear 
that the energies available with the conventional cyclotron are 
limited by the relativity effects. 

* In a given cyclotron the field should be shaped slightly differently for 
protons and for deuterons because of the different relativity effects. For this 
reason deuteron cyclotrons do not give very good proton beams without major 
readjustments. 



FM CYCLOTRON 



89 



FM Cyclotron. The relativity limitation can be overcome if the 
oscillator frequency can be modulated corresponding to the mass 
increase of the ions. The 184-inch cyclotron in Berkeley actually 
operates as a frequency-modulated cyclotron (sometimes also 




FIGURE IV-8. A general view of the giant (184-inch) Berkeley cyclotron, now 
producing 200-Mev deuterons and 400-Mev a particles operating as a synchro- 
cyclotron, taken before the installation of the concrete shielding. In the right 
foreground, on the truck which moves on rails in the floor, are mounted the 
round vacuum housing for the rotating condenser, its associated vacuum 
pump, and behind these the oscillator housing. At the left can be seen the 
target probe, a shaft entering a port in the main vacuum chamber. (Photo- 
graph courtesy of Radiation Laboratory, University of California) 

called synchrocyclotron). Many other large accelerators of this 
type are now under construction or design. In the FM cyclotron 
the decrease in rotation frequency due to the increasing mass is 
compensated by a decrease in the frequency of the r-f voltage 
applied to the dees; this frequency modulation may be brought 
about by means of a rotating condenser in the oscillator circuit. 
Obviously, for successful acceleration ions have to start their 
spiral path at or near the time of maximum frequency. Because 



90 SOURCES OF BOMBARDING PARTICLES CH. IV 

ions are accepted into stable orbits only during about 1 per cent 
of the FM cycle the ion currents are appreciably lower than in 
ordinary cyclotrons. On the other hand, the magnetic field can 
actually be shaped so as to increase the magnetic focusing effects. 
Frequency-modulated cyclotrons can be operated with relatively 
low dee voltages. The operating dee voltage for the 184-inch 
Berkeley cyclotron is 15 kv, which means that the deuterons 
make about 10 4 revolutions for a total path length of the order of 
10 5 meters, and take about 1 millisec to reach the target. 

Proton Synchrotron. Even with frequency modulation, cyclo- 
trons probably become impractical for still higher energy ranges 
(above 1 Bev), mainly because of prohibitive magnet costs. (The 
184-inch Berkeley cyclotron has about 4000 tons of iron in its 
magnet.) On the other hand, it appears that the synchrotron 
principle which is discussed in the next section can be extended to 
the acceleration of heavy particles in the billion-electron-volt 
range. A machine of this type for the acceleration of protons to 
1.3 Bev is under construction in England, and work has begun 
on 3-Bev and 6-Bev proton synchrotrons at the Brookhaven 
National Laboratory and at the Berkeley Radiation Laboratory, 
respectively. 

B. ELECTRONS 

Very little work has been done with electron-induced nuclear 
reactions. There are several reasons for this: the cross sections 
for such reactions appear to be extremely low (10~ 5 barn and 
less), natural and artificially produced /? emitters do not give out 
electrons of sufficiently high energy to be useful as electron sources 
for nuclear reactions, and the cyclotron cannot be used for the 
acceleration of electrons because of the high frequencies that 
would be required and because of the enormous relativistic mass 
increase of electrons even at moderate energies (at 1 Mev the 
total mass is three times the rest mass). The direct high-voltage 
machines discussed in section A can be applied to the acceleration 
of electrons as well as of positive ions, and especially the electro- 
static generator has been extensively used for this purpose; e~, e~~ 
reactions and the Be 9 (e~", ne~~) reaction have been observed this 
way. However, in recent years, methods for accelerating elec- 
trons to much higher energies have been developed, and these will 
be briefly described. 



BETATRON &1 

Betatron. The use of a varying magnetic field for the accelera- 



tion of electrons had been suggested by a number of investigators, 
but the first successful device using this idea was the induction 
accelerator or betatron developed by D. W. Kerst in 1940. 

The betatron may be thought of as a transformer in which the 
secondary winding is replaced by a stream of electrons in a vacuum 
"doughnut." To understand the principle of operation consider 
the circular orbit described by an electron in a magnetic field. 
Suppose that the magnetic flux <p perpendicular to the plane of 
the orbit and inside it increases at the rate d<p/dt. As long as this 
increase in central flux continues, an induced emf will persist at 
the position of the electron orbit, steadily increasing the momen- 
tum of the electrons. In order for the electrons to continue to 
move in a fixed orbit it is necessary that the field at the orbit 
change proportionally with the momentum of the electrons. This 
can be seen from the condition of equality of centripetal magnetic 
force Hev and centrifugal force mv 2 /r, which can be written 
mv = Her. Now to keep r constant, the rate of change of momen- 
tum d(mv)/dt must be proportional to the rate of change of field 
strength dH/dt: 

d(mv) dH 

= er -- 

dt dt 

On the other hand, the rate of change of the momentum of an 
electron in an orbit of radius r due to a rate of change of flux 
d<f)/dt inside the orbit is given by 

d(mv) e d(p 
- L = -- 1. 

dt 2*r dt 

Combining equations IV-3 and IV-4, we see that the condition 
dH 1 1 d<? 



must be fulfilled. This means that the field at the orbit must 
increase at exactly half the rate at which the average flux inside 
the orbit increases. This condition can be achieved by the proper 
tapering of the pole faces as indicated schematically in figure 



92 



SOURCES OF BOMBARDING PARTICLES 



Cn.IV 




IV-9. The radial variation of the field in the region of the elec- 
tron orbit is important in the focusing problem. It turns out 
that, if the field falls off as the inverse nth power of the radius in 
the region of the orbit and if y% < n < 1, the electrons will describe 
damped oscillations about the equilibrium orbit. Betatrons are 
generally designed with n = %, and the focusing is excellent, 
permitting the electrons to make hundreds of thousands of revolu- 
tions. 

Electrons for acceleration are pulse-injected into the doughnut 
at 20 to 50 kev from an electron gun. The injection takes place 

'//////////////QA^ 'rt^*'/////////////^* 

rVacuum 
: doughnut 

Stable 

electron 

orbit 

FIGURE IV-9. Cross section through central region of a betatron (schematic), 
with the magnetic lines of force indicated. 

for 1 to 2 jusec when the field at the orbit is zero. The rapid varia- 
tion of the magnetic field necessary for betatron operation is 
achieved with a magnet constructed of low-loss laminated iron 
and energized by an alternating current. The magnetizing coils 
are resonated with a capacitor bank. Since electrons are injected 
only once during each field cycle, the time average electron cur- 
rent obtainable is proportional to the frequency. Various fre- 
quencies between 60 and 1900 cycles have been used; the cooling 
of the magnet becomes more difficult with increasing frequency. 

The energy obtainable with a given betatron is limited by the 
saturation of the iron in the pole pieces. As the central flux in- 
creases to the point where the iron in the pole pieces begins to 
saturate, the orbit begins to shrink. The electron beam can then 
be allowed to strike a target (usually a tungsten wire) mounted 
in the doughnut inside the equilibrium orbit. The maximum 
momentum and, therefore, the maximum energy attainable for a 
given orbit radius can be estimated by integration of equation 
IV-4, provided the maximum flux that can be reached without 
saturation is known. By orbit shift pulses from auxiliary magnet 
coils the orbit can be expanded, raised, lowered, or contracted 
toward a suitably placed target at any desired phase and, there- 



BETATRON 93 

fore, at any desired electron energy. Recently Kerst and his 
associates at the University of Illinois have succeeded in bringing 
the electron beam out of the vacuum doughnut into the air for 
more convenient experimentation. 

A number of 20-Mev betatrons are in operation in the United 
States. Most of these are used as X-ray sources (see section C) 




FIGURE IV-10. 100-Mev betatron, Research Laboratory, General Electric 

Company, Schenectady, N. Y. (The authors are indebted to Dr. E. E. 

Charlton who kindly supplied this photograph.) 

for radiographic work. The largest betatron now in operation 
is the 100-Mev model at the General Electric Company in Schenec- 
tady which has a magnet weighing about 130 tons and an equi- 
librium orbit of 1 meter radius. A 250-Mev betatron is under 
design at the University of Illinois. It appears that for energies 
much in excess of this value the betatron becomes impractical, 
because with magnets of reasonably small size the energy loss by 
radiation as the electrons travel in circular paths becomes very 
serious for electrons of several hundred million electron volts. 



94 SOURCES OF BOMBARDING PARTICLES CH. IV 

Synchrotron* Another device for electron acceleration which has 
some advantages over the betatron, particularly from the point 
of view of size and cost, was proposed independently by V. I. Vek- 
sler and by E. M. McMillan and has come to be known as the 
synchrotron. In the synchrotron as in the betatron the radius of 
the electron orbit is kept approximately constant by a magnetic 
field which at the orbit increases proportionally with the momen- 
tum of the electrons. However, the acceleration (or rather the 
increase in energy since the velocity must remain essentially con- 
stant at v c) is provided not by a changing central flux but 
(more nearly as in the cyclotron) by a r-f oscillator which supplies 
an energy increment every time the electrons cross a gap in a 
resonator which forms part of the vacuum .dou^mut. It turns 
out that with this arrangement phase stability isT obtained at the 
orbit, because the electrons tend to cross the gap at a time when 
the field changes from accelerating to decelerating. Electrons 
arriving at the gap too early gain energy, and, therefore, their 
mass and period of revolution increase, and they will arrive rela- 
tively later at the next revolution. Those electrons which arrive 
too late have their mass and period of revolution decreased and 
are more nearly "on time" the next time around. As the mag- 
netic field is steadily increased, a slight phase difference is main- 
tained between the electron orbits and the resonator voltage, and 
on the average the electrons gain some energy at each passage of 
the gap. 

With a constant radio frequency the stable orbit in a synchro- 
tron has a constant radius only after the electrons have reached 
constant velocity. Therefore, if the electrons are injected from 
an electron gun at relatively low energies (say 50 kev, with velocity 
= 0.41c), the synchrotron orbit expands until the velocity of 
light is practically reached. However, this expansion of the 
orbit, which requires a large vacuum tube and magnet, can be 
avoided if the electrons are initially accelerated in the same ma- 
chine by its operation as a betatron. The changeover from 
the betatron to the synchrotron principle in each cycle is achieved 
by saturation of small central flux bars in the magnet when the 
electrons reach an energy of about 2 Mev (velocity = 0.98c). 
At this time the r-f circuit is turned on, and thereafter the elec- 
trons receive their additional energy almost entirely from the 
synchrotron operation. 



SYNCHROTRON 



95 




FIGURE IV-11. 70-Mev synchrotron, Research Laboratory, General Electric 
Company, Schenectady, N. Y. Notice the light spot near the left edge of 
the magnet gap. This light emerging tangentially from the doughnut is due 
to the energy loss by the accelerating electrons in their circular path. (The 
authors are indebted to Dr. E. E. Charlton who kindly supplied this photo- 
graph.) 

In the synchrotron as in the betatron the rapidly varying mag- 
netic field is provided by an a-c magnet, but because no large 
central flux is necessary the magnet size and cost are substantially 
lower. The maximum energy available for an orbit of given 
radius is determined by the peak value of the magnetic field at 



96 SOURCES OF BOMBARDING PARTICLES CH. IV 

the orbit (see equation IV-3). If radio frequency is turned off 
before the magnetic guide field has reached its peak, the electron 
beam spirals inward, and electrons of any desired energy below 
the maximum can thus be allowed to hit a target. If the r-f field 
is turned off after the magnetic field has passed its peak, the elec- 
trons will spiral out as the field decreases. 

A number of synchrotrons designed for electron energies be- 
tween 30 and 300 Mev are under construction. The largest unit 
in operation now is the 80-Mev synchrotron at the General Elec- 
tric Company in Schenectady with an orbit radius of about 
12 inches and a magnet weighing about 8 tons. 

The synchrotron principle appears to be applicable to positive 
ions as well as to electrons. In this case, however, the velocity of 
light is not approached until very high energies are reached (at 
200 Mev for a proton v = 0.57c). Therefore, in order to achieve 
a constant orbit radius a wide-band modulation of the radio fre- 
quency is proposed (much wider than in the FM cyclotron which 
uses spiraling orbits). This presents a serious but probably not 
insurmountable problem. An advantage of a synchrotron over 
an FM cyclotron is the fact that the synchrotron requires a mag- 
netic field only at the circular orbit; thus a ring-shaped magnet 
can be used, which represents an enormous saving. Proton syn- 
chrotrons for the l-to-10-Bev range are in various stages of design 
and construction. 

Linear Accelerator. The principle of the linear accelerator is 
applicable to electrons as well as to positive ions, and a number 
of linear electron accelerators, using series of cavity resonators 
excited by microwave oscillators, are under construction. Some 
of these units are designed to accelerate electrons to energies of 
several hundred million electron volts. 



C. GAMMA RAYS AND X RAYS 

Radioactive Sources. Since nucleons are bound in most nuclei 
with binding energies of 6 to 8 Mev, photons having energies less 
than 6 Mev cannot be expected to induce many nuclear reactions. 
No 7 rays known to be emitted in radioactive processes (from either 
natural or artificially produced radioactive substances) have 
energies that high; nor do the X rays from conventional X-ray 



NUCLEAR REACTIONS AS SOURCES 97 

tubes. The only 7-ray- or X-ray-induced (8> nuclear reactions 
produced with such sources are, therefore, excitations of nuclei 
to isomeric levels and the "photodisintegrations" of the deuteron 
(threshold 2.18 Mev) and of Be 9 (threshold 1.63 Mev). Some 
of the radioactive 7-ray sources that have been used are listed 
in table IV-1. 

TABLE IV-1 

SOME TYPICAL RADIOACTIVE T-RAY SOURCES 

Number of 

Energy Quanta per 

Source Half-life (Mev) Disintegration 

ThC" 3.1m 2.62 1 

Na 24 14. 8h 2.76 1 

1.38 1 

Y 88 105d 0.9 1 

1.87 1 

Sb 124 60d 1.70 0.5 

0.60 1 

Nuclear Reactions as Sources. In a number of nuclear reactions, 
especially with the lightest elements (which have large nuclear 
level spacings), very energetic y rays are emitted. These in turn 
have been used to induce other nuclear reactions. Such sources 
are of particular value if the y rays are monoenergetic. The most 
important sources of this type are listed in table IV-2. With 
these reactions relatively high intensities of y rays (of the order 
of 10 6 quanta per sec) can be obtained from rather moderate high- 
voltage sets operating at 500 to 1000 kev. 

TABLE IV-2 
HIGH-ENERGY y RAYS FROM NUCLEAR REACTIONS 

7-Ray Energy 

Reaction (Mev) Remarks 

Li 7 (p, y) Be 8 17.2 Resonance at 440 kev proton energy. 

Other components, at ~15 Mev and 
at lower energies, also found. 

B 11 (p, y) C 12 11.8, 16.6 11.8 Mev has 7 times intensity of 16.6 

Mev; also low-energy component (^ 
4 Mev) found. 

8 For convenience we shall speak of 7-ray-induced reactions even when the 
electromagnetic radiation used is not of nuclear origin but is produced by the 
deceleration of electrons in a target in the manner of continuous X rays. 



98 SOURCES OF BOMBARDING PARTICLES CH. IV 

Bremsstrahlung. The continuous X rays produced when elec- 
trons are decelerated in the Coulomb fields of atomic nuclei are 
called bremsstrahlung (German for "slowing-down radiation")- 
This type of radiation is produced whenever fast electrons pass 
through matter, and the efficiency of the conversion of kinetic 
energy into bremsstrahlung goes up with increasing electron 
energy and with increasing atomic number of the material. In 
tungsten, for example, a 10-Mev electron loses about 50 per cent 
of its energy by radiation, whereas a 100-Mev electron loses over 
90 per cent of its energy by that mechanism (see chapter VII, 
section B). 

The spectrum of bremsstrahlung from a monoenergetic elec- 
tron source extends from the electron energy down to zero, with 
approximately equal amounts of energy in equal energy intervals; 
in other words, the number of quanta in a narrow energy interval 
is about inversely proportional to the mean energy of the interval. 

The stopping of fast electrons in matter thus produces a con- 
tinuous spectrum of X rays, and any electron accelerator also 
serves as an X-ray source. Van de Graaff machines, betatrons, 
and synchrotrons have all been used as sources of X rays for pro- 
ducing nuclear reactions. In fact, unless special devices are used 
to bring the electron beam out of betatron or synchrotron dough- 
nuts, the X rays are the only radiation available outside the 
vacuum systems of these machines. The higher the energy of 
an electron producing bremsstrahlung, the more the X-ray emis- 
sion is concentrated in the forward direction; with the 100-Mev 
betatron, for example, about half the intensity of the X-ray beam 
is contained in a 2 cone. The chief disadvantage of brems- 
strahlung sources for nuclear work is their spectral distribution. 
However, they are capable of producing electromagnetic radiation 
in energy and intensity ranges not accessible by other means. 

D. NEUTRONS 

Radioactive Sources. Our only sources of neutrons are nuclear 
reactions. There are several naturally occurring and several 
artificially produced a and y emitters which can be combined 
with a suitable light element to make useful neutron sources. 
Because of the short ranges of the a particles, a emitters must be 
intimately mixed with the light element (usually beryllium, be- 







Approximate 




Energy of 


Yield 




Neutrons 


(neutrons 


Reaction 


(Mev) 


sec" 1 nT 1 ) 


Be 9 (, n) C 12 


Up to 13 


460 


Be 9 (a, n) C 12 


Up to 11 


400 


Be 9 (a, n) C 12 


Up to ~11 


80 




(avg. ~4) 




B 11 (a, n) N 14 


Up to 8 


180 


Be 9 (7, n) Be 8 


0.12, 0.51 


0.8 * 


H 2 (7, n) H 1 


0.27 


7.8* 


Be 9 (7, n) Be 8 


0.029 


5.1 * 


Be 9 (7, n) Be 8 


1.00 


3.8* 


Be 9 (7, n) Be 8 


0.75 


0.062* 



NEUTRON-PRODUCING REACTIONS WITH ACCELERATORS 99 

cause it gives the highest yield). Aj^jemitter may be enclosed 
in a capsule surrounded by the light-element target. Some of 
the most commonly used sources are listed in table IV-3. 

TABLE IV-3 f 



Source 

-f Be (mixed) 
Rn + Be (mixed) 
Po '+ Be (mixed) 

Ra + B (mixed) 
Ra + Be (separated) 
Na 24 + D 2 
Sb 124 + Be 
Na 24 -f Be 
La 140 + Be 

* The photoneutron yields are given for 1 g of target (D 2 or Be) at 1 cm 

from the 7-ray source. 

t Data for this table were obtained from: 

H. L. ANDERSON, "Neutrons from Alpha Emitters," Preliminary Report 
No. 3 in Nuclear Science Series published by the National Research 
Council; 

B. RUSSELL, D. SACHS, A. WATTENBERG, and R. FIELDS, "Yields of Neutrons 
from Photo-Neutron Sources/' Phys. Rev. 73, 545 (1948); 

A. WATTENBERG, "Photo-Neutron Sources and the Energy of the Photo- 
Neutrons," Phys. Rev. 71, 497 (1947). 

Neutron-producing Reactions with Accelerators. Much more 
copious sources of neutrons than can be obtained with radioactive 
a and j emitters are available with ion accelerators. The reaction 
H 2 (d, n) He 3 (often called a D, D reaction) is exoergic (Q = +3.25 
Mev), and because the potential barrier is very low good neutron 
yields can be obtained with deuteron energies as low as 100 to 
200 kev. With thick targets of solid D 2 0, the yields are about 
0.7, 3, and 80 neutrons per 10 7 deuterons at 100 kev, 200 kev, 
and 1 Mev deuteron energy, respectively. High-voltage sets and 
electrostatic generators are often used to produce the D, D reac- 
tion. The neutrons are monoenergetic if monoenergetic deuterons 
of moderate energies (up to a few million electron volts) fall on a 
sufficiently thin target. 



100 SOURCES OF BOMBARDING PARTICLES CH. IV 

For a controlled source of monoenergetic neutrons of very low 
energy (down to about 30 kev) the Li 7 (p, n) Be 7 reaction is suit- 
able, especially when produced with the protons of well-defined 
energy available from electrostatic generators. The reaction is 
endoergic (Q = 1.63 Mev) and has a threshold of 1.86 Mev. 
Advantage may be taken of the differences in neutron energy in 
the forward and backward (and intermediate) directions. 

With X rays from electrostatic generators, betatrons, and the 
like, neutrons can be produced by means of the Be 9 (7, n) or 
H 2 (r, ri) reactions. (4) The yields of these reactions go up quite 
sharply with energy. With an electrostatic generator operating 
at 2.5 Mev with 100 /ua electron current the neutron yield per 
gram of beryllium is equivalent to that from about 4 g of radium 
mixed with beryllium; at 3.2 Mev energy the corresponding figure 
is 26 g of radium. 

Where high-energy deuterons are available, the most prolific 
neutron source is obtained by bombarding beryllium with deu- 
terons. The reaction Be e (d, ri) B 10 has a positive Q value of 
about 4.3 Mev, but the neutrons are far from monoenergetic. 
When a beryllium target is bombarded with deuterons of E Mev 
energy, neutrons with a distribution of energies up to about 
(E + 4) Mev are emitted. The neutron yield goes up rapidly 
with deuteron energy; it is about 10 8 neutrons per sec per ^a for 
1-Mev deuterons, about 10 10 neutrons per sec per jua for 8-Mev 
deuterons, and about 3 X 10 10 neutrons per sec per jua for 
14-Mev deuterons. 

With a given deuteron source a lithium target gives the highest 
neutron energies because the Li 7 (d^ n) reaction is exoergic by 
14.6 Mev. The neutron yield is only about one-third that for the 
Be 9 (d, n) B 10 reaction. Neutrons are also obtained in the bom- 
bardment of practically any element with fast protons, deuterons, 
or a. particles. The yields and energies vary from reaction to 
reaction, but if a neutron bombardment is needed for the activa- 
tion of some substance it is often sufficient to place the sample 
near a cyclotron target which is being bombarded by deuterons, 
even if the target is not beryllium or lithium. 

In the bombardment of targets with deuterons of very high 
energy (the 200-Mev deuterons of the 184-inch FM cyclotron in 

4 The product Be 8 is unstable and decomposes into two a particles. The 
threshold for the Be 9 (r,n)Be 8 reaction has been measured as 1.63 Mev, so 
thatQ - -1.63 Mev. 



NUCLEAR CHAIN REACTORS (PILES) 10, 

Berkeley) it has been found (5) that high-energy neutrons are 
emitted in a rather narrow cone in the forward direction; the 
energy distribution of these neutrons is approximately Gaussian, 
with the maxiinum at half the deuteron energy. On striking the 
edge of a nucleus the high-energy deuteron may have its proton 
or neutron stripped off; thus the cone of neutrons and a similar 
cone of protons are produced. (See chapter III, section C.) 

Neutrons from nuclear reactions initially are fast neutrons. 
The slowing down of neutrons and some properties of thermal 
neutrons have already been discussed in chapter III, section D. 

Nuclear Chain Reactors (Piles). By far the most prolific sources 
of neutrons known are the nuclear chain reactors or piles^. A 
nuclear reactor is an assembly of fissionable material (such as 
uranium, enriched U 235 , Pu 239 , or U 233 ) arranged in such a way 
that a self-sustaining chain reaction is maintained. In each fission 
process a number of neutrons (somewhere between one and three) 
are emitted. The requirement common to all reactors is that at 
least one of these neutrons must be available to produce another 
fission rather than escape from the assembly or be used up in 
some other type of nuclear reaction. Therefore, for a given type 
of reactor there is a minimum (or critical) size, below which the 
chain reaction cannot be self-sustaining. It is also necessary to 
avoid as much as possible the presence in the reactor of materials 
which consume neutrons in processes other than the fission reac- 
tion. This imposes a severe restriction on structural materials, 
coolants, and moderators. 

If a reactor has exactly the critical size, the neutrons produced 
in each fission will, on the average, give rise to exactly one new 
fission, and the neutron density will remain constant. The ratio 
of the number of neutrons in one generation to that in the pre- 
ceding generation is called the multiplication constant or repro- 
duction factor k, and, for an exactly critical assembly, k = 1. In 
practice, reactors are designed in such a way that k can be made 
slightly larger than one. The reactor can then be made super- 
critical (k > 1), just critical (k = 1), or subcritical (k < 1) by 
means of so-called control rods. These are rods made of material 
with large neutron-absorption cross section (in the case of slow- 
neutron piles usually cadmium or boron steel) which can be intro- 

8 R. Serber, Phys. Rev. 72, 1008 (1947). A. C. Helmholz, E. M. McMillan, 
and D. C. Sewell, Phys. Rev. 72, 1003 (1947). 



i02 SOURCES OF BOMBARDING PARTICLES CH. IV 

duced into the pile to accurately adjustable depths. The farther 
the control rods are introduced, the smaller becomes k. In order 
to increase the power level and the neutron density of a reactor it 
is necessary to allow k to become greater than 1. It is possible 
to do this without letting the reactor get out of control because a 
certain fraction (about 0.6 per cent) of the neutrons emitted in 
fission are not emitted instantaneously, but with half-lives rang- 
ing from about 0.4 sec to about 1 min (see chapter VI, section D). 
Therefore, as long as k 1 is smaller than this fraction, the 
exponential increase in neutron density is relatively slow, and 
reactors can then be controlled very easily by means of the con- 
trol rods. Since no elements with very high capture cross sections 
for fast neutrons are known, control rods seem hardly feasible 
for reactors operating on fast neutrons; in such reactors one might 
be able to control the reproduction factor by adjusting the posi- 
tion of some of the fissionable material itself or of a surrounding 
neutron reflector* In addition to the accurately adjustable oper- 
ating controls, reactors are equipped with safety devices which 
allow rapid shutdown. These may consist of neutron-absorbing 
rods which can be rapidly introduced into the reactor and whose 
introduction insures a decrease in k to well below 1. Such safety 
devices may, of course, be triggered automatically, for example, 
when a certain neutron density is reached. 

In most of the nuclear reactors now in operation thermal neu- 
trons are used for the propagation of the chain reaction. Since 
the neutrons produced in fission are emitted with kinetic energies 
of the order of 1 Mev, such reactors contain so-called moderators, 
that is, materials whose function it is to slow down the neutrons 
to thermal energies. Moderatojg sh^uj^ojf^yrse, be^of^lQ\Yjna^ 
number^ and^ Jiaye low_ cross^ sections for neutron absorption. 
Graphite, neavy water, and ordinary water have been used as 
moderators. Moderators may be either homogeneously mixed 
with the fissionable material or separated from it in a heterogene- 
ous arrangement. Actually all the reactors which have been con- 
structed with ordinary uranium as the "fuel" are of the hetero- 
geneous type. The reason is that the resonance absorption in 
U 238 which accounts for much of the loss of neutrons (6) in thermal 

6 This is a "loss" in terms of the chain-reaction propagation; but the neutron 
capture in U 288 is the process responsible for the production of plutonium in 
piles by the set of reactions: U 238 (n, y) U 239 ; U 239 2?J Np 239 + 0~; Np 239 ii d 



NUCLEAR CHAIN REACTORS (PILES) 103 

uranium reactors is reduced considerably when the uranium is 
arranged in aggregates such as lumps or rods. In a homogeneous 
mixture of uranium and moderator the probability that a neutron 
during the slowing-down process is absorbed by U 238 in the reso- 
nance region is quite large. If, however, the uranium is arranged 
in aggregates, the probability for resonance capture will be large 
only in a relatively thin layer at the surface of the aggregate, 
whereas this layer effectively shields the interior from neutrons 
of the resonance energy. In most of the existing reactors uranium 
cylinders (canned in aluminum jackets for corrosion protection) 
are imbedded in a graphite lattice. Such reactors are now in 
operation at Oak Ridge, Tenn.; Hanford, Wash.; and Harwell, 
England; and one is under construction at Brookhaven National 
Laboratory. Heavy-water moderators are used in heterogeneous 
Uranium reactors at Argonne National Laboratory, at Chalk 
River, Canada, and in a pile recently put into operation at Fort de 
Chatillon, France. 

A homogeneous thermal reactor (called the "water boiler") 
using uranium enriched in U 235 has been in operation at 
Los Alamos, N. Mex. The enriched uranium is in the form of a 
uranyl salt, and this is in solution in ordinary water which serves 
as the moderator. The solution is in a spherical container sur- 
rounded by a neutron reflector (in this case beryllium oxide). 
The critical size of a reactor may often be significantly reduced 
by a neutron reflector surrounding the reactor; ts^bej^good^jfi^ec- 
tor a substance must have a low capture cross section^ and a high 
scattering cross section for neutrons^. 

Although many schemes for reactors operating in the resonance 
(or epithermal) and in the high-energy regions have been discussed, 
only one such reactor seems to have been put into operation so 
far: the so-called "Fast Reactor" at Los Alamos. In this device 
plutonium metal is used as the nuclear fuel, and no moderator is 
present. A circulating liquid coolant is employed. 

The total energy release in each fission process is about 200 Mev 
or 3.2 X 10" 4 erg or 3.2 X 10~ n watt-sec. (Over 80 per cent of 
this energy appears as kinetic energy of the fission fragments; 
most of the remainder is released in subsequent radioactive 
processes.) Therefore, in a reactor operating at a power of 1 watt, 
about 3 X 10 10 fissions take place every second. The power 
ratings of most of the operating piles are measured in kilowatts 
OT* even megawatts. For example, the air-cooled graphite-uranium 



104 SOURCES OF BOMBARDING PARTICLES CH. IV 

piles at Oak Ridge and Harwell, England (BEPO, the larger of 
the two reactors there), have been reported to operate at greater 
than 2000 kw and about 6000 kw, respectively. The power rating 
of the water-cooled Hanf ord piles (whose chief function is plu- 
tonium production) has not been released but is certainly much 
larger than that of the Oak Ridge pile. The first nuclear reactor 
built in 1942 in Chicago and later reconstructed at the Argonne 
Laboratory near Chicago consists of uranium and uranium oxide 
lumps in a graphite lattice with no cooling provisions and has 
been reported to operate at a few kilowatts. The water-cooled 
Los Alamos water boiler has operated at about 10 kw. The 
heavy-water reactor at Argonne (in which the heavy water also 
serves as a coolant) has been reported to run at a power level of 
300 kw. 

Since we are here considering nuclear reactors chiefly as neutron 
sources, not as plutonium production plants or as power sources, 
we are concerned more with available neutron fluxes than with 
power ratings. Although for a given reactor the two quantities 
are proportional to each other, the proportionality constant 
depends on the design details of the reactor. Neutron fluxes of 
about 10 12 neutrons per cm 2 per sec have been reported both for 
the Oak Ridge graphite pile and the Los Alamos Fast Reactor. 
A maximum flux of about 5 X 10 12 neutrons per cm 2 per sec is 
expected for the graphite-uranium pile under construction at 
Brookhaven National Laboratory. The neutron flux generally 
falls off from the center towards the outside of a reactor. In most 
reactors, especially in those designed primarily for research pur- 
poses (such as the Brookhaven and Harwell reactors), facilities 
are available for exposure of samples in holes or channels in the 
interior. The neutron energy distribution in these spots depends, 
of course, on the type of reactor; in graphite-uranium piles thermal 
neutrons predominate, but the flux of neutrons even up to several 
million electron volts energy is not entirely negligible. To pro- 
vide pure thermal-neutron sources so-called thermal columns are 
often attached to reactors. A thermal column is a column of 
graphite (or some other moderator) of sufficient length to insure a 
thermal-energy distribution for the neutrons which have passed 
through it. The neutron flux at the end of a thermal column is, 
of course, several orders of magnitude smaller than that available 
inside the associated reactor. 



EXERCISES 106 

EXERCISES 

1. Show that the magnetic field near the edge of a cyclotron magnet 
provides a focusing action for positive ions (see figure IV-6). 

2. See p. 100. Explain the difference between the Q value and the 
threshold of the Li 7 (p, n) Be 7 reaction. 

3. Estimate (a) the percentage frequency modulation and (6) the pole 
diameter required for an FM cyclotron designed to accelerate protons to 
350 Mev. Assume H = 16,000 gauss. 

4. What would be the minimum r?, 7 cross section detectable by means 
of the product activity in a sample of 10 cm 2 area containing 1 mg-equiva- 
lent of target isotope, with a mixed Ha-Be source containing 1 g radium? 
Assume that the bombardment is continued to saturation and that 1 per 
cent of the neutrons emitted by the source strike each square centimeter 
of the target sample as slow neutrons. Consider 30 disintegrations per 
min as the minimum detectable activity. 

5. Suppose you want to prepare some 5.3-year Co 60 with a cyclotron 
and have the choice of bombarding a cobalt sample directly with 14-Mev 
deuterons for 2 hr or of surrounding it with paraffin and placing it near a 
beryllium target bombarded with 14-Mev deuterons for a total of 100 hr. 
Which is more advantageous from the point of view of total activity ob- 
tained? Use data in tables B and C in the appendix, and make reasonable 
assumptions about the solid angle subtended by the neutron-irradiated 
sample. 

6. (a) What would be the approximate neutron energy from a beryllium 
target bombarded with As 76 7 rays? (6) What would be the energy 
difference between neutrons in the forward and in the reverse directions? 

Answer: (a) 107 kev; (6) 5.8 kev. 

7. Estimate (a) the equilibrium quantity of Ba 140 present in a ura- 
nium-graphite reactor operating at 1000 kw, and (6) the total amount of 
Ce 140 accumulated in the same reactor after 1 year's operation followed 
by 2 months' shutdown. Answer: (a) 0.70 g. 

8. What time would be required to convert 2 per cent of the Cd 118 in a 
thin cadmium foil to Cd 114 if the foil were placed in a reactor with a ther- 
mal neutron flux of 5 X 10 11 neutrons per cm 2 per sec? 



106 SOURCES OF BOMBARDING PARTICLES CH. IV 



REFERENCES 

E. POLLARD and W. L. DAVIDSON, Applied Nudear Physics, New York, John 

Wiley & Sons, 1942, 
R. E. LAPP and H. L. ANDREWS, Nuclear Radiation Physics, New York, 

Prentice-Hall, 1948. 
W. W. SALISBURY, "Accelerators for Heavy Particles," Nucleonics 1 no. 3, 

34 (Nov. 1947). 
L. I. SCHIFP, "Production of Particle Energies beyond 200 Mev," Rev. Sci. 

Inst. 17, 6 (1946). 
R. G. HERB, D. B. PARKINSON, and D. W. KERST, "The Development and 

Performance of an Electrostatic Generator Operating under High Air 

Pressure," Phys. Rev. 61, 75 (1937). 

M. S. LIVINGSTON, "The Cyclotron," J. App. Phys. 16, 2 and 128 (1944). 
W. M. BROBECK, E. O. LAWRENCE, K. R. MACKENZIE, E. M. McMiLLAN, 

R. SERBER, D. C. SEWELL, K. M. SIMPSON, and R. L. THORNTON, "Initial 

Performance of the 184-inch Cyclotron at the University of California," 

Phys. Rev. 71, 449 (1947). 

D. W. KERST, "The Betatron," Radiology 40, 115 (1943). 
W. F. WESTENDORP and E. E. CHARLTON, "A 100 Million Volt Induction 

Electron Accelerator," /. App. Phys. 16, 581 (1945). 
R. F. ELDER, A. M. GUREWITSCH, R. V. LANGMUIR, and H. C. POLLOCK, "A 

70-Mev Synchrotron," J. App. Phys. 18, 810 (1947). 
M.I.T. Seminar Notes (C. GOODMAN, Editor), The Science and Engineering of 

Nuclear Power, Cambridge, Mass., Addison- Wesley Press, 1947. 
C. GOODMAN, "Nuclear Principles of Nuclear Reactors," Nucleonics 1 no. 3, 

23 (Nov. 1947) and 1 no. 4, 22 (Dec. 1947). 

F. L. FRIEDMAN, "Nuclear Reactors," Electrical Engineering 67, 685 (1948). 



V. QUANTITATIVE TREATMENT OF RADIOACTIVE 
PROCESSES 

A. EXPONENTIAL DECAY 

^JJalf-ltfe._jye have seen (in chapter I) that a given radio- 
active species decays according to an exponential law: N = JV e~ xt 
or A = Aoe~ x *, where N and A represent the number of atoms and 
the measured activity, respectively, at time t, and N and AO the 
corresponding quantities when t = 0, and X is the characteristic 
decay constant for the species. The half-life ^ is the time interval 
required for N or A to fall from any particular value to one-half 
that value. The half-life is conveniently determined from a plot 
of log A vs. t when the necessary data are available and is related 

, , In 2 0.69315 
+o the decay constant: t\^ = = 

'"Average Life. We may determine the average life expectancy 
of the atoms of a radioactive species. This average life is found 
from the sum of the times of existence of all the atoms divided by 
the initial number; if we consider AT to be a very large number 
we may approximate this sum by an equivalent integral, finding 
for the average life T: 



I /" 

= ^7 I 

NQ JO 

r ru + i i w i 

= XI te-*dt=-\ - e~M = - 
Jo L X Jo X 



"7T tdN 

NQ Jt = 



We see that the average life is greater than the half-life by the 
factor 1/0.693; the difference arises because of the weight given 
in the averaging process to the fraction of atoms that by chance 
survive for a long time. It may be seen that during the time 
1/X an activity will be reduced to just l/e of its initial value. 

Mixtures of Independently Decaying Activities. If two radio- 
active species, denoted by subscripts 1 and 2, are mixed together, 

107 



108 



QUANTITATIVE TREATMENT 



CH.V 



then the observed total activity is the sum of the two separate 
activities: A = AI + A 2 - ci\iNi + c 2 X 2 AT 2 . The detection co- 
efficients ci and c 2 are by no means necessarily the same and often 



ouu 
200 

100 
80 

60 
50 
40 

30 
20 

10 


\ 






























\\ 

\ 
\ 

\ 


\ 


a 


























\ 




V 






























s 


























*"* *^* 


V- 




^ 


























\ ' 


**^ 


^ 


























\ 


b* 




* 


**^ 






















\ 










"^^, 


^ 




















\ 












"*^^ 


-. 


. 














\ 

\ 


















^^**, 


, 


"- 


^^ 






\ 
\ 


V 
-V 
























8 

6 

5 

4 
3 

2 

>, 








. v 






























\ 






























































% 






























\ 

\ 






























\ 

t 
































\ 






























\ 

t 

\ 

\ 




















) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 



Time (hr) 
FIGUKE V-l. Analysis of composite decay curve. 

(a) Composite decay curve. 

(b) Longer-lived component (14 ** 8.0 hr). 

(c) Shorter-lived component (t^ 0.8 hr). 

are very different in magnitude. In general, A = A! + A 2 H A* 

for mixtures of n species. 

For a mixture of several independent activities the result of 
plotting log A vs. t is always a curve concave upward (convex 
toward the origin). This curvature results because the shorter- 
lived components become relatively less significant as time passes. 



GENERAL EQUATION 109 

In fact, after sufficient time the longest-lived activity will entirely 
predominate, and its half-life may be read from this late portion 
of the decay curve. Now, if this last portion, which is a straight 
line, is extrapolated back to t = and the extrapolated line sub- 
tracted from the original curve, the residual curve represents the 
decay of all components except the longest-lived. This curve may 
be treated again in the same way, and in principle any complex 
decay curve may be analyzed into its components. In actual 
practice experimental uncertainties in the observed data may be 
expected to make it most difficult to handle systems of more than 
three components, and even two-component curves may not be 
satisfactorily resolved if the two half-lives differ by less than 
about a factor of two. The curve shown in figure V-l is for 
two components with half-lives differing by a factor of 10. 

B. GROWTH OF A RADIOACTIVE PRODUCT 

General Equation. In chapter I we considered briefly a special 
case in which a radioactive daughter substance was formed in the 
decay of the parent. Let us take up the general case for the decay 
of a radioactive species, denoted by subscript 1, to produce another 
radioactive species, denoted by subscript 2. The behavior of 
NI is just as has been derived; that is, dNi/dt = \iNi, and 
NI = JVie"~ Xl ' ; where we use the symbol JVi to represent the 
value of NI at t = 0. Now the second species is formed at the 
rate at which the first decays, \iNi, and itself decays at the rate 
X 2 AT 2 . Thus, 

dN 2 



at 
or 

- + X 2 #2 - \ 1 N l e^ lt = 0. (V-l) 

at 

For this linear differential equation of the first order we assume a 
solution of the form N% = uv t where u and v are functions of t. 
Differentiating, we obtain 



dv du 
u -- h v 
dt dt dt 



110 QUANTITATIVE TREATMENT CH. V 

Substituting in equation V-l, we have 

dv du . . . 

u + v + \ 2 uv - Xi#ie~ Xl ' = 0, 
dt dt 

which may be rearranged to give 

du n 



n . t 

V Xl < = 0. (V-2) 

dt 

We may choose the arbitrary function v so that the term in paren- 
thesis is zero, 

dv 

- + \ 2 v = 0, 

at 

dv 

7t = ^ 

v = e-^. 

By substitution of this result in equation V-2 a differential equa- 
tion in u is obtained: 



u jy O e (X2-XO< i f>. 

X2 ~ \i 

and 

AT 2 = uv = - ^ JV!" M + Ce~ X2i . (V-3) 

X 2 Xi 

The constant C is evaluated from the condition N 2 = N 2 at ^ = 0: 



X2 Xi 

Substituting in equation V-3 and rearranging we obtain the final 
solution for N% as a function of time: 

N 2 = - NS^ - e~) + N 2 e~*. (V-4) 
X 2 Xi 



TRANSIENT EQUILIBRIUM 



111 



Notice that the first group of terms shows the growth of daughter 
from the parent and the decay of these daughter atoms; the last 
term gives the contribution at any time from the daughter atoms 
present initially. 

Transient Equilibrium. In applying equation V-4 to consider- 
ations of radioactive (parent and daughter) pairs one can dis- 
tinguish two general cases, depending on which of the two sub- 




^ t Slopes correspond 

to *!/ 2 =:8.0houn 



789 
Time (hr) 



10 11 12 13 14 15 



FIGURE V-2. Transient equilibrium, 
(a) Total activity of an initially pure parent fraction. 
(6) Activity due to parent (t\^ = 8.0 hr). 

(c) Decay of freshly isolated daughter fraction (^ 0.80 hr). 

(d) Daughter activity growing in freshly purified parent fraction. 

(e) Total daughter activity in parent-plus-daughter fractions. 



112 QUANTITATIVE TREATMENT CH. V 

stances has the longer half-life. If the parent is longer-lived than 
the daughter (\i < X 2 ) a state of so-called radioactive equilibrium 
is reached; that is, after a certain time the ratio of the numbers 
of atoms and, consequently, the ratio of the disintegration rates 
of parent and daughter become constant. This can be readily 
seen from equation V-4; after t becomes sufficiently large e~* 2t 
is negligible compared with e~* lt , and N<e~^ also becomes 
negligible; then 



X 2 ~~ Xi 
and, since NI = Nie~ Xl ', 

Ni X 2 - 



(V-5) 

The relation of the two measured activities is found, from 
i, A 2 = c 2 X 2 A^2> to be 



(v_ 6) 



Notice that the right-hand sides of equations V-5 and V-6 are 
not the same. Injbj^e special case of equal detection coefficients 
(GI = c 2 ) the ratio^of the two activities, Ai/A 2 = 1 (X 1 /X 2 ), 
may have anv^ value bet\ioeen an^ 1, depending on the ratio of 
Xi to X 2 ; thaj is, in equilibrium the daughter activity will be 

"* X 2 " 

greater than that of the E&rent by the factor -- In equi- 

X 2 Xi 
librium both activities decay with the parent's half-life. -^ 

Secular Equilibrium. A limiting case of radioactive equilibrium 
in which Xx X 2 and in which the parent activity does not de- 
crease measurably during many daughter half-lives is known as 
secular equilibrium. We illustrated this situation in chapter I 
and now may derive the equation presented there, as a useful 
approximation of equation V-5: 

NI X 2 

= , or XiATi = X 2 AT 2 . 

N 2 \i 



SECULAR EQUILIBRIUM 113 

In the same way equation V-6 reduces to 



1, or 



and the measured activities are equal if c\ = C2. 

The production of a radioactive substance (daughter) by any 
steady source, for example, steadily operating nuclear chain 



^ 

<0 

K 
S 
| 





JUU 
200 
































100 


/ 

c 


^ 

b 


a 


























RO, 






x" 




























\ 


X 




























fin 




/ 




























en 
































Af\ 




k 








r 




















^n 


/ 


\ 




























on 




% 

\ 




























10 


Id 




\ 
\ 


































































































\ 
























ft 

































5l 








% 
























! 








\ 
% 
































\ 
























; 










\ % 
\ 






















1 










\ 
% 


i 




















c 


) 1 


t. 


) 2 


\ 4 


\ 5 


6 


7 





t S 


) 1 


3 1 


1 1 


2 1 


3 1 


4 1! 



Time (hr) 

FIGURE V-3. Secular equilibrium, 
(a) Total activity of an initially pure parent fraction. 
(6) Activity due to parent (^ *>) ; this is also the total daughter activity 
in parent-plus-daughter fractions. 

(c) Decay of freshly isolated daughter fraction (fa - 0.80 hr). 

(d) Daughter activity growing in freshly purified parent fraction. 



114 QUANTITATIVE TREATMENT CH. V 

reactors (piles) or cyclotrons, presents a situation analogous to 
the approach to secular equilibrium. We may obtain the growth 
formula for N 2 as a function of time from equation V-4, setting 
A/2 = at t = 0, using AI \2 and e~~ Xl ' = 1, and replacing 
AiNi by the rate R of production of the active atoms: 

N 2 = - (1 - e~ X2 '). (V-7) 



As t becomes long compared to the half-life of the activity then 
N 2 approaches R/X 2 as a maximum limiting value, and we may 
rewrite equation V-7 in this way: 

N 2 = (# 2 ) max (l - e- x '<). (V-8) 

Thus if an activity with a 20-min half-life (say Ga 70 ) is being 
steadily produced, one-half the maximum attainable yield is 
reached after 20 min, three-fourths after 40 min, seven-eighths 
after 60 min, fifteen-sixteenths after 80 min, and so on. If you 
must pay by the hour for cyclotron running time, you would want 
to irradiate for not more than about two half-lives of the desired 
product. 

Figure V-2 presents an example of the transient equilibrium 
with Xi < \2 (actually with Ai/A 2 = Mo); the curves represent 
variations with time of the parent activity and the activity of a 
freshly isolated daughter fraction, the growth of daughter activity 
in a freshly purified parent fraction, and other relations; in pre- 
paring the figure we have taken GI = c 2 . Figure V-3 is a similar 
plot for secular equilibrium; it is apparent that as AI becomes 
smaller compared to A 2 the curves for transient equilibrium shift 
to approach more and more closely the limiting case shown in 
figure V-3. 

The Case of no Equilibrium. If the parent is shorter-lived than 
the daughter (A t > A 2 ), it is evident that no equilibrium is attained 
at any time. If the parent is made initially free of the daughter, 
then as the parent decays the amount of daughter will rise, pass 
through a maximum, and eventually decay with the characteristic 
half-life of the daughter. This is illustrated in figure V-4; for this 
plot we have taken A!/A 2 = 10, and GI = c 2 . In the figure the 
final exponential decay of the daughter is extrapolated back to 



THE CASE OF NO EQUILIBRIUM 



115 



t = 0; this method of analysis is useful if Xi X 2 for then this 
intercept measures the activity c 2 X 2 7Vi, the NI atoms giving 
rise to JV 2 atoms early enough to take NI equal to the extra- 
polated value of N 2 at t = 0. The ratio of the initial activity, 
ci\iNi Q , to this extrapolated activity gives the ratios of the half- 
lives if the relation between c\ and c 2 is known: 



s 



.& 



200 

100 
80 

60 
50 
40 

30 
20 

10 


\ 






























\ 


k 






























~^\ 






























\ 






























































\ \ 


a 




























V 
\~ 


\ 


























*^^, 






V 






















dJ 


' 




V 




"*^ 


^, 





"^ 


fSlo 
I to 


peco 
\,\/ = 

^, 


rrespc 
8.0 hr 

^, 


>nds 






6 


-f 








\ 




















5^5* 


/ 








\ 






















/ 








\ 






























\ 
\ 




















































I 










\ 
\ 




















1 












\ 


\ 


















"123456789 10 11 121 11 



Time (hr) 

FIGURE V-4. The case of no equilibrium. 
(a) Total activity. 
(6) Activity due to parent (^ = 0.80 hr). 

(c) Extrapolation of final decay curve to tune zero. 

(d) Daughter activity in initially pure parent. 



116 QUANTITATIVE TREATMENT CH. V 

If X 2 is not negligible compared to Xi, it can be shown that the 

Xi ~~" X2 

ratio Xi/X 2 in this equation should be replaced by - , and 

X 2 
the expression involving the half-lives changed accordingly, 

Both the transient-equilibrium and the no-equilibrium cases 
are sometimes analyzed in terms of the time tm for the daughter 
to reach its maximum activity when growing in a freshly sepa- 
rated parent fraction. This time we find from the general equa- 
tion V-4 by differentiating, 



dt X 2 AI A 2 ~ 

and setting dN 2 /dt = when t = t m : 



2.303 X 2 
or , - -log -?. 

A 2 AX AI 



At this time the daughter decay rate, \ 2 N 2 , is just equal to the 
rate of formation, \iNi (this is obvious from equation V-l); and 
in figures V-2, 3, 4, where we assumed c\ = c 2 , we have the parent 
activity AI intersecting the daughter growth curve A 2 at the time 
t m . (The time t m is infinite for secular equilibrium.) 

C. MORE COMPLICATED CASES 

Bateman Solution. If we consider a chain of three or more 
radioactive products it is clear that the equations already derived 
for N\ and N 2 as functions of time are valid, and N% may be 
found by solving the new differential equation* 

dN 3 

= X 2 ^V 2 XsJVs. (V-9) 

dt 

This is entirely analogous to the equation for dN 2 /dt, but the 
solution calls for more labor since N 2 is a much more complicated 
function than NI. The next solution, for N, is still more tedious. 
H. Bateman has given the solution for a chain of n members with 
the special assumption that at t = only the parent substance is 



UNITS OF RADIOACTIVITY 117 

present, that is, that N 2 = JV 3 = -N n - 0. This solution is 

N n - 

Cl = 



'X n -l _ TQ 

C 2 = - - NI 9 etc. 
(Xi -X 2 )(X 3 -X 2 )---(X n -X 2 ) 

If we do require a solution to the more general case with JV 2 , 
N 3 - -N n i* 0, we may construct it by adding to the Bateman 
solution for JV n in an remembered chain, a Bateman solution 
for N n in an (n l)-membered chain with substance 2 as the 
parent, and, therefore, N% = N 2 at t = 0, and a Bateman solu- 
tion for N n in an (n 2)-membered chain, etc. 

Branching Decay. Another variant that is met in general decay 
schemes is the branching decay, illustrated by 




Here both X& and X c must be considered when the general rela- 
tions in either branch are studied because, for example, the sub- 
stance B is formed at the rate X&A^? but A is consumed at the 
rate (X& + X C )A^. Notice that A can have but one half-life, given 

f\ AO*!J 

in this case by t^ = By definition the half-life is related 

X& + X c 

to the total rate of disappearance of a substance regardless of the 
mechanism by which it disappears. 

D. UNITS OF RADIOACTIVITY 

A familiar unit of radioactivity is the curie. Originally the 
term referred to the quantity of radon in equilibrium with one 
gram of radium. Later it came to be used as a unit of disintegrar 
tion rate for any radioactive preparation, defined as that quantity 
of the preparation which undergoes the same number of disinte- 
grations per second as one gram of pure radium. This use for 



118 QUANTITATIVE TREATMENT CH. V 

substances other than members of the radium series has never 
received the sanction of the appropriate international committees. 
With this definition the value of the curie should vary with suc- 
cessive refinements in the measurement of the decay constant or 
atomic weight of radium; the International Radium Standard 
Commission has recommended the use of the fixed value 3.7 X 10 10 
disintegrations per sec. The millicurie and the microcurie are 
practical units also in common use. 

A new absolute unit of radioactive disintegration rate has 
recently been recommended by the National Bureau of Standards. 
This unit, the rutherford (abbreviated rd), is defined as that 
amount of a radioactive substance which undergoes 10 6 disinte- 
grations per sec. The abbreviations mrd for millirutherford 
(10 3 dis per sec) and jurd for microrutherford (1 dis per sec) should 
be noted. The kilorutherford and the megarutherford are likely 
to be useful for very active materials. 

As an illustration we may calculate the weight in grams W of 
1 rd of C 14 from its half-life of very nearly 5000 years: 

0.693 _ t 0.693 

X = year = sec 

5000 5000 X 365 X 24 X 60 X 60 

= 4.4 X 10" 12 sec" 1 

dN W 

= AAT = X X 6.02 X 10 23 

dt 14 

= 1.9TF X 10 disintegrations per sec. 

dN 

With = 10 6 disintegrations per sec (1 rd), 

dt 

10 6 

W = = 5.3 X 10~ 6 g. 

1.9 X 10 11 

The lack of rigor in previous usage of units of radioactivity is 
no doubt largely a result of the experimental difficulty associated 
with determinations of absolute disintegration rates. These diffi- 
culties remain, and are discussed with the experimental techniques 
in chapter X. For cases in which the radioactivity is observed 
as a 7 radiation the problem is particularly awkward. A common 
practice in the absence of information on the disintegration rate 
has been the comparison of the -/-radiation intensity with the 7 
intensity from a unit amount (curie) of radium in equilibrium 



HALF-LIVES FROM DECAY CURVES 119 

with its decay products, with some absorber interposed between 
sample and detector in each measurement. This procedure is 
uncertain and arbitrary for a number of obvious reasons. The 
Bureau of Standards proposes that 7-ray intensities be measured 
as such, without reference to absolute disintegration rates, and 
recommends as the unit one roentgen per hour at one meter 
(abbreviated rhm or r.h.m. and pronounced "rum")' The y 
radiations from 1 curie of radium with decay products give a 
little less than 1 rhm (0.97 rhm bare, and 0.84 rhm through 0.5 mm 
of platinum absorber; the latter is standard practice). The 
roentgen as a unit of radiation intensity is defined in chapter VII, 
section E. 

E. DETERMINATION OF HALF-LIVES 

From Decay Curves. Half-lives in the range from several sec- 
onds to several years are usually determined experimentally by 
measuring the activity with an appropriate instrument at a num- 
ber of suitable successive times. Then log A is plotted versus 
time, and the half-life found by inspection, provided that the 
activity is sufficiently free of other radioactivities that a straight 
line (exponential decay) is found, preferably extending over 
several half-life intervals. As we have already discussed, the 
decay curve resulting from a mixture of independent activities 
may often be analyzed to yield the half-lives of the various com- 
ponents. When difficulties arise in this analysis it is often ade- 
quate to measure separately decay curves through several different 
thicknesses of absorbing material to obtain curves with some 
components relatively suppressed. Our treatments of the more 
general equations have already suggested methods of finding half- 
lives from more complicated growth and decay curves. 

The manipulations necessary for activity measurements become 
difficult as the time scale to be investigated becomes short. The 
use of electronic and photographic recording devices can extend 
the working region to half-lives well below 0.1 sec. With short- 
lived gaseous products, or products in solution, a method that has 
been particularly useful for fission products with half-lives of the 
order of a few seconds is to measure the activity at different points 
along a tube through which the fluid flows at a measured rate. 
The ordinary decay curve is then found on a plot of log A vs. dis- 
tance along the tube. A method based on a similar principle 



120 QUANTITATIVE TREATMENT CH. V 

using a rapidly rotating wheel has been employed for solid samples; 
the half-life (0.022 sec) of B 12 was determined in this way. In 
these procedures the limitation usually arises not in the activity 
measurements but in the rapid preparation, and possibly isola- 
tion, of the short-lived sample. The possible use of a modulated 
source, such as a betatron or synchrotron, or a cyclotron modified 
to produce periodic pulses of accelerated ions, may be mentioned. 
Appropriate electric circuits will divide the time between pulses 
into an arbitrary number of intervals and measure the average 
resulting activity in each interval. 

From Variable-delay Coincidences. When a body of very short 
half-life results from a radioactive decay with moderate or long 
half-life the method of variable-delay coincidences can be used. 
In one form of apparatus the electric pulse produced in a detection 
instrument by a ray from the parent is electrically delayed by a 
time d and then recorded in coincidence with any ray from the 
daughter that may produce a pulse in a detector at that time, or 
more correctly at that time within the limits of the resolving time 
T of the coincidence equipment. Now as d is varied by electrical 
means the coincidence counting rate coincidences per unit T 
will vary; the effect is essentially to record disintegrations over 
the period T: at a time d after formation of the short-lived nucleus. 
The very short half-life is determined from the typical decay 
curve with the logarithm of the coincidence rate plotted against 
d. The Ta 181 isomer of 22 /xsec half-life was found in this way in 
the ft decay of Hf 181 . In an earlier form of apparatus no pro- 
vision was made for the introduction of a delay time d, and coinci- 
dences C were recorded as T was varied. Clearly, with T very 
short (compared to the half-life sought) no coincidences would 
result; with T long (compared to ty) a maximum number of 
coincidences C max would be observed. The expected relation, 
C = C max (1 0~ XT ), is analogous to the formula for radioactive 
growth, equation V-8, and the half-life is obtained from the 
semilog plot of (C max - C) vs. T. The half-lives of the C 1 bodies 
in the three natural decay series were measured by this method. 

From Specific Radioactivity. If the half-life, or disintegration 
constant, is to be determined for a substance of very long half- 
life (very small X), the activity A = c\N may not change meas- 
urably in the time available for observation. In such cases X may 
be found from the relation \N = dN/dt = A/c, provided that 



GEIGER-NUTTALL RULE AND SARGENT RELATION 121 

N is known and dN/dt may be determined in an absolute way 
(through knowledge of the detection coefficient c). This method 
is most accurate for a emitters, and the absolute rates of emission 
of a particles from uranium samples have been investigated with 
great care to measure the half-life of Uj. In an accurate deter- 
mination of the half-life of Pu 239 the value of dN/dt was estab- 
lished in a calorimetric measurement of the heating effect, with 
the a-particle energy known from the a-particle range. 

In some instances the disintegration rate is better obtained 
from a measurement of the equal disintegration rate of a daughter 
in secular equilibrium. Early determinations of the half-life of 
U 235 were based on the a-particle counting rate of Pa 231 obtained 
in known yield from old uranium ores; the U 236 a particles were 
not measurable in a direct way because of the much larger number 
of a disintegrations occurring in the U 238 and U 234 . 

From Oei^-Nu^ll^Rule QJ gqrgpnt Relation. The half-lives 
for a emitters in the radioactive families may be estimated from 
measurements of the range R (or the energy) of the emitted 
a particles from an empirical relationship known as the Geiger- 
Nuttall rule: log R = A log X + J5, where A is a general constant, 
and B is a constant characteristic of the radioactive series^ There 
may be large uncertainties in half-lives determined by this'iormula, 
as may be seen in exercise 5 at the end of this chapter. There 
exists an even less precise relation (the Sargent relation) between 
the energy (or range) and disintegration constant for ft emitters; 
very roughly, X = KE 5 , where K is very different for different 
classes of transitions (allowed, forbidden, doubly forbidden, etc.). 
This relation is considered in chapter VI, but we may note here 
that it is sometimes useful in revealing discrepancies in one direc- 
tion half-lives too short for the measured energies in ft decay. 



122 QUANTITATIVE TREATMENT CH. V 

EXERCISES 

1. The following experimental data were obtained when the activity 
of a certain beta-active sample was measured at frequent intervals: 

Time Activity Time Activity 

(in hours) (in counts/mm) (in hours) (in counts/min) 

7300 4.0 481 

0.5 4680 5.0 371 

1.0 2982 6.0 317 

1.5 1958 7.0 280 

2.0 1341 8.0 254 

2.5 965 10.0 214 

3.0 729 12.0 181 

3.5 580 14.0 153 

Plot the decay curve on semilog paper and analyze it into its components. 
What are the half -lives and the initial activities of the component activi- 
ties? 

2. Calculate the weight of (a) 1 curie of radon; (6) 1 "curie" of P 32 
(see table A in the appendix for the half-life); (c) 1 rd of P 32 ; (d) 10,000 rd 
of H 3 . Answer: (a) 6.47 jug; (c) 9.43 X 1Q- 11 g. 

3. What was the rate of production, in atoms per second, of I 128 during 
a 1-hr cyclotron (neutron) irradiation of an iodine sample, if the sample is 
found to contain 3.5 rd of I 128 activity at 15 min after the end of the 
irradiation? 

4. From data from table A in the appendix, calculate the total rate of 
emission of a particles from 1 mg of ordinary uranium. Calculate this 
answer also for the case of 1 mg of very old uranium, in secular equilibrium 
with all its decay products. Answer to first part: 25.0 per sec. 

5. Given the half -lives of Ra and RaA equal 1590 years and 3.05 min, 
and their a ranges equal 3.39 and 4.72 cm, respectively, estimate the half- 
lives of U 238 and RaC'; the ranges for these a emitters are 2.63 and 6.96 cm. 
Compare the results with the half-lives given in table A of the appendix 
(obtained for RaC' from coincidence counting with variable resolving 
time). 

6. A 0.100 mg sample of pure 9 4?u 239 (an a-particle emitter) was found 
to undergo 1.40 X 10 7 disintegrations per min. Calculate the half-life 
of this isotope. Pu 239 is formed by the decay of Np 239 . How many 
rutherfords of Np 239 would be required to produce a 0.100 mg sample of 
Pu 239 ? 



REFERENCES 123 

7. A sample of 1.00 X 10~ 10 g of RaE is freshly purified at time t 0. 
(a) If this sample is left without further treatment, when will the amount 
of Po 210 in it be a maximum? (6) At that time of maximum growth, what 
will be the weight of Po 210 present, the a. activity in disintegrations per 
second, the beta activity in disintegrations per second, the number of 
microcuries of Po 210 present? (c) Sketch on semilog paper a graph of 
a activity and /} activity versus time. 

8. In the slow-neutron activation of a sample of separated Mo 100 
isotope some 14.6-min Mo 101 is produced; this decays to 14.0-min Tc 101 . 
A sample of Mo 101 is chemically freed of technetium and then immediately 
placed under a counter. Sketch the activity as a function of time, assum- 
ing the detection coefficient to be the same for the Tc 101 as for the Mo 101 
radiation. 

9. Carry out the solution of the differential equation V-9. Compare 
your result with the Bateman solution for this case with N and AV 
not equal to 0. 

10. A sample of an activity whose half-life is known to be 30 min was 
measured from 10:03 to 10:13. The total number of counts recorded in 
this' 10-min interval was 34,650. What was the activity of the sample 
(in counts per minute) at 10:00? Answer: 4159. 

REFERENCES 

G. HEVESY and F. A. PANETH, A Manual of Radioactivity, Oxford University 

Press, 1938. 
E. RUTHERFORD, J. CHADWICK, and C. D. ELLIS, Radiations from Radioactive 

Substances, Cambridge University Press, 1930. 
H. BATEMAN, "Solution of a System of Differential Equations Occurring in 

the Theory of Radio-active Transformations," Proc. Camb. Phil Soc. 

15, 423 (1910). 
R. D. EVANS, "Radioactivity Units and Standards/ 7 Nucleonics 1 no. 2, 32 

(Oct. 1947). 
W. RUBINSON, "The Equations of Radioactive Transformation in a Neutron 

Flux," J. Chem. Phys., in press (1949). 

S. ROWLANDS, "Methods of Measuring Very Long and Very Short Half- 
lives," Nuckonics 3 no. 3, 2 (Sept. 1948). 



r l. TYPES OF RADIOACTIVE DECAY 
A. ALPHA DECAY 

n<T 

As we have seen in chapter I, <x particles soon after their dis- 
covery were identified as doubly charged helium ions, first by the 
measurement of their charge-to-mass ratio and then by the spec- 
troscopic evidence for the accumulation of helium in a tube sur- 
rounding an a emitter. Alpha-particle emission is therefore 
always accompanied by a decrease of two in atomic number and 
a decrease of four in mass number. 

The a particles from a given isotope either all have the same 
energy or are distributed among a few monoenergetic groups. 
Where a single a-particle energy occurs, for example, in the 
decays of AcA (Po 215 ), Rn 222 , and probably U 238 , the transition 
evidently takes place between a single energy level of the a-emit- 
ting nucleus and a single energy level (generally the ground state) 
of the product nucleus. The emission of a. particles of several 
different energies by one nuclear species may be due to the exist- 
ence of several energy levels either preceding or following the 
a emission (or both, although this has not been observed). 

Alpha-particle Groups. In the majority of cases, for example, in 
Ra 226 , AcX (Ra 223 ), ThC (Bi 212 ), the emission of several a-particle 
groups of different energies from a given substance is due to the 
fact that the product nucleus can be left in different states of 
excitation which subsequently transform to the ground state by 
y emission. Each r-ray energy observed is then as a rule equal 
to the energy difference between the disintegration energies asso- 
ciated with two of the a-particle groups. (1) From a complete 

1 The total disintegration energy associated with an a-particle emission is 
larger than the a-particle energy by the recoil energy of the nucleus, which 
for the heavy a emitters is of the order of magnitude of 0.1 Mev. Since the 
momentum p and energy E of a particle of mass M are related by the equa- 
tion p 2 2ME, it follows from the conservation of momentum that 
M a E a -Mnucieus^nucieus- For example, in the case of the 6.083-Mev a 
particles of ThC the recoil energy is 6.083 X 4/208 - 0.117 Mev. 

124 



ALPHA-PARTICLE GROUPS 



125 



knowledge of the a and y energies, an energy-level diagram of 
the product nucleus can often be constructed (see figure VI-1). 
Usually not all the possible 7 transitions between the levels of 



Energy above ground 
state of ThC"(Mev) 
6.200 



ThC" 



ThC 




V.TJ A X / 




0.472 


/ 
















y 










y 


0.451 




y 




y 


0.43? 


Mev 




0.471 
Mev 


y 

0.327 


0.287 
Mev 


Mev 


7 






Mev 






// 






\y 0.040 Mev / 



FIGURE VI-1. Energy-level diagram for the a-particle decay of ThC to 
ThC'". The a-particle energies given are the kinetic energies, not total dis- 
integration energies. 

the product nucleus are actually observed; in the ThC > ThC" 
decay, for example, only six of the ten theoretically possible 
y rays are found (figure VI-1). The reason is that in nuclear 
as in atomic or molecular transitions selection rules are operative 
(see section C). 

In most of the cases where more than one a-particle energy is 
observed from a single a emitter, the groups of highest energy 



126 TYPES OF RADIOACTIVE DECAY CH. VI 

also have the largest abundance. This can be understood from 
the fact that the probability for the penetration of the potential 
barrier increases with increasing energy. However, there are 
other factors (such as the angular momenta of the initial and 
final states) which also play a part in determining the relative 
probabilities for the emission of a particles of different energies. 

Long-range a Particles. The disintegration of each of the 
short-lived a emitters RaC' and ThC' takes place predominantly 
with the emission of a particles of a single energy (7.683 Mev for 
RaC' and 8.778 Mev for ThC'). However, in both cases small 
numbers of higher-energy a particles have been observed. Twelve 
higher-energy groups (with energies up to 10.51 Mev) are known 
in RaC' and two (with energies up to 10.54 Mev) in ThC'; only 
0.003 per cent of all the RaC' a particles and 0.02 per cent of all 
the ThC 7 a particles belong to these high-energy groups. The 
high-energy a particles are due to transitions from excited states 
of RaC' and ThC', as can be seen from a comparison with the 
different RaC and ThC /3-particle transitions leading to these 
various states, The energy-level diagram for the ThC ThC' 
> ThD transitions is shown in figure VI-2. 

Alpha emission from the excited levels in ThC' and RaC' must 
compete with 7 transitions from these same levels to the respec- 
tive ground states. The decay constants for a emission can be 
calculated approximately for the excited states from the Geiger- 
Nuttall rule which relates decay constant and a-particle energy 
(chapter V, section E). If the relative numbers of a particles and 
7 quanta from a given excited state are known experimentally, 
the decay constant of that excited state for y emission can be 
estimated. This is actually the only experimental method of 
determining the "lifetimes" for the fast 7-ray transitions; it leads 
to values in the neighborhood of 10~ 13 sec. Gamma-ray transi- 
tions are. discussed further in section C. 

Penetration of Potential Barriers. The 10.54-Mev a particles of 
ThC' are the most energetic a. particles known from radioactive 
sources. At the other extreme are the 2.0-Mev a particles of 
samarium. Most of the naturally occurring a particles are emitted 
with energies between 4 and 8 Mev. Before the advent of quantum 
mechanics, the fact that a particles could be emitted with such 
low energies from nuclei which were known to have much higher 
potential barriers was very puzzling. For example, scattering 



PENETRATION OF POTENTIAL BARRIERS 



127 



experiments with 8.78-Mev a. particles (from ThC') on uranium 
show that around a uranium nucleus no deviation from the Cou- 



Energy above 

ground state of 

ThD(Mev) 

11.195- 



ThC 



.0 1.52 (14%) 
0.62 (4%) 
'00.45(2%) 




ThD 



FIGURE VI-2. Energy-level diagram for the ThC-ThC'-ThD decays showing 

the origin of the long-range a. particles of ThC'. The energies given do not 

include the rest energies of the a and /3 particles. 

lomb law exists at least up to 8.78 Mev; yet U 238 emits 4.5-Mev 
a particles. How do these a particles get outside the potential 
barrier which is at least 8.78 Mev high (and actually quite a bit 
higher)? 



180 INSTRUMENTS FOR RADIATION DETECTION CH. VIII 

increases again, and, finally, the gap breaks down into a glowing 
discharge or arc, with a very sharp rise in the current. In the 
measurement of gas ionization it is obviously of some advantage 
to measure the saturation current: the current is easily interpreted 
in terms of the rate of gas ionization, and the measured current 
does not depend critically on the applied voltage or other like 
factors. The range of voltage over which the saturation current 
is obtained depends on the geometry of the electrodes and their 
spacing, the nature and pressure of the gas, and the general and 
local density and spatial distribution of the ionization produced 
in the gas. In air, for many practical cases, this region may be 
taken to extend from ~10 2 to ~10 4 volts per cm of distance 
between the electrodes. 

We may classify detection systems (of the ion-collection type) 
according to whether saturation collection is employed or whether 
the multiplicative collection region is used. In the multiplicative 
region, where V is above the maximum value for saturation collec- 
tion, the additional current is due to secondary ionization proc- 
esses which result from the high velocities reached by the ions 
(particularly electrons) moving in the high field gradient. The 
use of this current amplification makes multiplicative collection 
methods inherently sensitive but unfortunately also inherently 
critical to many experimental variables. 

Lauritsen-type Electroscope. We will call the gas-filled elec- 
trode systems designed for saturation collection ionization cham- 
bers. Saturation current instruments consist of the ionization 
chamber, in which ions produced are collected with as little recom- 
bination or multiplication as possible, and an electric system for 
measuring the very small currents obtained. The essential differ- 
ences between the various instruments of this sort are in the 
nature of the current-measuring systems. In one common and 
relatively inexpensive instrument, the Lauritsen-type electroscope, 
a sensitive quartz-fiber electrometer measures the change in volt- 
age produced on the fiber and its support by collection of the 
ionization charge. An external battery or rectifier is used to 
provide the initial voltage V (by means of a temporary connection 
to the fiber support) ; then, the fiber position is observed through 
a small telescope to measure AF as a function of time. For a 
collected charge q, the resulting A 7 = g/C, where C is the approx- 
imately constant capacitance of the fiber and electrode system. 



PENETRATION OF POTENTIAL BARRIERS 129 

It is clear from this expression that the probability for barrier 
penetration decreases with increasing value of the integral in the 
exponent, that is, with increasing barrier height and width. (The 
higher the barrier, the larger is the difference U(r) E; and the 
wider the barrier, the greater is the difference between the limits 
of integration RQ and RI.) 

The decay constant X may be considered as the product of 
P and the frequency / with which an a. particle strikes the poten- 
tial barrier; the order of magnitude of / may be estimated as 
follows. The de Broglie wave length h/Mv of the a. particle of 
velocity v and momentum Mv inside the nucleus is taken com- 
parable to RQ, thus 

h . h 

RQ, or # - 
Mv MR Q 

If the a. particle is considered as bouncing back and forth between 
the potential walls, 

or f 



Therefore, the decay constant 



By a more elaborate treatment more accurate expressions for / 
and X are obtained. If a simple shape is assumed for U(r) (for 
example a Coulomb law up to #0 a s indicated by a dotted line in 
figure VI-3), it is possible to evaluate the integral in the exponent 
and thus to obtain an explicit expression for X in terms of RQ, Z, 
and E. The values of X calculated for the known a emitters are 
in fair agreement with experimental values. Actually the known 
values of X, Z, and E were used to calculate nuclear radii from 
this relation, and these were found to be in good agreement with 
values obtained by other methods (see chapter II, section C). 

The Geiger-Nuttall rule relates X and E by an empirical equa- 
tion which may be written logX = a-logl? + 5; this is different 
in form from the Gamow theory, log X = f(Z)E~^ + F(Z, R ) 
where /(Z) and F(Z, RQ) are complicated functions of the variables 
indicated. However, owing to some compensating factors and 
mainly because log-log plots tend to minimize small irregularities, 
a Geiger-Nuttall plot fits experimental data fairly well. 



130 TYPES OF RADIOACTIVE DECAY CH. VI 

It may be appropriate to say a few words about the fact that 
proton decay has not been observed. Protons are emitted by 
nuclei in less than about 10~ 12 sec or not at all. The explanation 
of the difference in behavior between a. particles and protons is 
as follows. The proton barrier is half as high as the a-particle 
barrier and therefore also narrower, and for high excitation 
energies the life times for proton emission are thus much smaller 
than for a emission. For lower excitation energies a emission is 
more probable because the energy balance is much more favorable 
than for proton emission: the nucleons in the a particle are still 
about as tightly bound as in the original nucleus, whereas a proton 
to be emitted must be supplied with an energy of several million 
electron volts (its binding energy). 

^^ B. BETA DECAY 

Electrons and Positrons. Electrons had been recognized as the 
ultimate units of negative electricity, and their properties had 
begun to be investigated, by the time radioactivity was discovered. 
Not long after their discovery rays were characterized as elec- 
trons. Much later the existence and properties of positrons had 
been both predicted theoretically and established in cosmic-ray 
studies before positron emission from artificially produced radio- 
active bodies was discovered. The existence of positrons or posi- 
tive electrons was first postulated by P. A. M. Dirac on the basis 
of his relativistic quantum theory of the electron. Solutions of 
the relativistic wave equations reveal possible states of electrons 
with energies always larger than me 2 (where m is the electron 
mass), but with either positive or negative signs. As to the 
physical meaning of the undeserved negative energy states of 
electrons Dirac suggested that normally all the negative energy 
states are filled. The raising of an electron from a negative- to a 
positive-energy state (by the addition of an amount of energy 
necessarily greater than 2w 2 ) should then be observable not 
only through the appearance of an ordinary electron, but also 
through the simultaneous appearance of a "hole" hi the infinite 
"sea" of electrons of negative energy. This hole would have the 
properties of a positively charged particle, otherwise identical 
with an ordinary electron. The subsequent discovery of posi- 
trons, first in cosmic rays and then hi radioactive disintegrations, 



BETA-RAY SPECTRUM AND THE NEUTRINO 131 

was soon followed by discoveries of the processes of pair produc- 
tion and positron-electron annihilation, which may be regarded 
as experimental verifications of Dirac's theory. 

Pair production is the name for a process which involves the 
creation of a positron-electron pair by a photon of at least 1.02 
Mev (2 me 2 ). It can be shown (see exercise 4 at the end of this 
chapter) that in this process momentum and energy cannot both 
be conserved in empty space; however, the pair production may 
take place in the field of a nucleus which can then carry off some 
momentum and energy. The cross section for pair production 
goes up with increasing Z and with increasing photon energy. 
Pair production may be thought of as the lifting of an electron 
from a negative- to a positive-energy state. The reverse process, 
the. falling of an ordinary electron into a hole in the sea of elec- 
trons of negative energy, with the simultaneous emission of the 
corresponding amount of energy in the form of radiation, is ob- 
served in the so-called positron-electron annihilation process. 
This process accounts for the very short lifetime of positrons: 
whenever a hole in the sea of electrons is created it is quickly 
filled again by an electron. The energy corresponding to the 
annihilation of a positron and electron is released either in the 
form of two 7 quanta emitted in nearly opposite directions (to 
conserve momentum) or, much more rarely, in the form of a 
single quantum if the electron involved in the annihilation is 
strongly bound (say, in an inner shell of an atom) so that a nucleus 
is available to carry off the excess momentum. (The latter proc- 
ess, although theoretically possible, has not been definitely estab- 
lished experimentally.) The two-quantum annihilation occurs 
principally with very slow positrons, that is, positrons which have 
almost come to rest by ionization processes; it is then accom- 
panied by the emission of two 7 quanta, each of energy equal to 
me 2 (0.51 Mev); this radiation is often referred to as annihilation 
radiation. 

Beta-ray Spectrum and the Neutrino. In contrast to a particles, 
particles (2) from a given radioactive species do not belong to a 

2 By ft particles we shall mean any electrons, positive or negative, emitted 
from nuclei. Whenever necessary, negative ft particles or negatrons (/3~) 
and positive ft particles or positrons (ft + ) will be distinguished. Electrons 
originating in the extranuclear shells should not be referred to as ft particles; 
they are often represented by the symbol e~. In the early literature any 
electrons emitted in radioactive processes are usually called ft particles. 



132 



TYPES OF RADIOACTIVE DECAY 



CH.VI 



limited number of energy groups but are emitted with a con- 
tinuous energy distribution extending from zero up to a maximum 
value. The shapes of -ray spectra have been studied in some 
detail by magnetic-deflection methods. Some typical shapes of 
ft spectra are shown in figure VI-4. The average energy is usually 
about one-third the maximum energy. Maximum energies rang- 
ing from 15 kev to 15 Mev occur among known ft emitters. Ever 
since its discovery, by J. Chad wick in 1914, the continuous spec- 
trum of ft rays has presented a very puzzling problem. Studies 




Energy 
FIGURE VI-4. Typical shapes of jS-ray spectra. 

of the or and 7-ray spectra have revealed that nuclei exist in 
definite energy states. Yet in every known 0-decay process the 
transition from one such definite energy state to another takes 
place with the emission of ft particles of variable kinetic energy. 
It was proved by calorimetric measurements that when all the 
ft particles are absorbed in a calorimeter the measured energy per 
ft particle is the average and not the maximum energy of the 
ft spectrum. Thus the law of the conservation of energy might 
appear to be violated in ft decay. 

Furthermore, the observations show discrepancies with other 
conservation laws also. As we have seen in chapter II, all nuclei 
of even mass number have integral spins and obey Bose statistics; 
all nuclei of odd mass number have half-integral spins and obey 
Fermi statistics. Since the mass number remains unchanged in 
ft decay the spins of initial and final nuclei should belong to the 
same class, either integral or half -integral, and the statistics should 



THEORY OF BETA DECAY 133 

remain the same. Yet electrons (and positrons) have one-half 
unit of spin and obey Fermi statistics. Thus angular momentum 
and statistics appear not to be conserved in ft decay. Finally, 
recent experiments in which the recoil momenta of nuclei as well 
as the corresponding 0-particle momenta were measured seem to 
indicate that linear momentum conservation is also violated in 
ft decay. 

To avoid the necessity of abandoning all these conservation 
laws for the case of /3-decay processes, W. Pauli postulated that 
in each ft disintegration an additional unobserved particle is 
emitted. The properties attributed to this hypothetical particle 
which has come to be known as the neutrino are such that the 
conservation difficulties are eliminated. The neutrino is sup- 
posed to have zero charge, spin Y^ and Fermi statistics, and it is 
thought to carry away the appropriate amount of energy and 
momentum in each ft process to conserve these quantities. To 
account for the fact that neutrinos have never been detected, it 
is in addition necessary to assume that they have a very small or 
zero rest mass and a very small or zero magnetic moment. By 
careful measurements of the maximum energy of a ft spectrum 
and determination of the masses of the corresponding ft emitter 
and product nucleus, experimenters have set an upper limit to 
the rest mass of the neutrino at about 0.05 electron mass (or 25 
kev). 

The difficulties regarding conservation laws are entirely analo- 
gous in positron decay. In this case the hypothetical particle 
emitted is sometimes called the antineutrino. Since the properties 
of neutrino and antineutrino are supposed to be the same we shall 
speak only of neutrinos. 

Theory of ft Decay. Using the neutrino hypothesis, E. Fermi in 
1934 constructed a theory of ft decay, somewhat analogous to the 
theory of emission of radiation from atoms. In this theory 
ft emission is treated as the transition of a nucleon from the neu- 
tron state to the proton state (or the reverse) with the simul- 
taneous creation of an electron (or positron) and a neutrino. 
Choosing a form for the interaction between nucleons and the 
electron-neutrino field analogous to that between atoms and the 
radiation field, Fermi arrived at an equation relating the decay 
constant with the maximum momentum of the ft particles and 



134 TYPES OF RADIOACTIVE DECAY CH. VI 

with the shape of the ft spectrum. (3) Although the Fermi theory 
accounts qualitatively for the observed shapes of ft spectra it 
appears that the theory may not in all cases give quantitatively 
correct results. The best fit at the upper-energy limit is obtained 
when the neutrino mass is taken to be zero or negligibly small. 

Selection Rules. According to the Fermi theory the disintegra- 
tion probability of a ft emitter depends on, among other factors, 
the spin difference Al between initial nucleus and final nucleus. 
In first approximation only transitions with A7 = and no change 
in parity are allowed. In higher approximations it turns out 
that transitions with A/ = 1 should be roughly 100 times less 
probable than those with A/ = 0; those with A/ = 2 another 
factor of 100 less probable, etc. These are called the Fermi selec- 
tion rules for ft decay. If the possibility that a nucleon may 
reverse its spin orientation during ft decay is taken into account, 
transitions with A/ = or 1 and no change in parity become 
allowed; those with A/ = 1 or 2 and no change in parity 
' 'singly forbidden," etc. These latter rules, the so-called Gamow- 
Teller selection rules, appear to agree much better with experi- 
mental evidence than do the Fermi rules. 

For maximum ft energies considerably greater than 0.5 Mev 
the Fermi equation for the decay constant reduces to the approx- 
imate form X = k(E m&x ) 5 y where E max is the maximum energy of 
the ft spectrum, and the value of the constant k depends on the 
"degree of forbiddenness" of the transition. Thus for a given 

3 Once an interaction between nucleons and electron-neutrino field is chosen, 
one might hope this same interaction would account for the nuclear binding 
forces in terms of an exchange of electron-neutrino pairs between protons and 
neutrons (and perhaps an exchange of electron-positron or neutrino-anti- 
neutrino pairs between like nucleons). The Fermi theory leads to exchange 
forces much too small to account for nuclear binding. Exchange of a single, 
positive, negative, or neutral particle (rather than a pair of particles) to account 
for nuclear forces was first proposed by H. Yukawa; in his theory decay was 
thought of as the emission of such a particle followed by immediate breakup 
into electron (or positron) and neutrino. This theory gives better agreement 
with known nuclear binding energies than the Fermi theory if the particle 
responsible for the exchange forces has a mass approximately 200 to 300 
times the electron mass. Attempts have been made to identify this particle 
with some of the types of mesons found in cosmic rays. It is now thought that 
the TT mesons (table 1 1-3) may be responsible for nuclear binding. However, 
none of the meson theories of nuclear forces proposed so far appears to be 
entirely satisfactory. 



COMPLEX BETA SPECTRA 135 

degree of f orbiddenness a linear relation would be expected between 
log X and log S max , and on a plot of log X vs. log max the points 
representing the various ft emitters should fall on a series of 
parallel lines, the top line representing the allowed transitions, 
the next lower one the first forbidden transitions, etc. Prior to 
the development of the Fermi theory B. W. Sargent called atten- 
tion to the fact that the natural ft emitters could be represented 
on two such lines, and these lines are called Sargent curves. For 
the quantitative correlation of decay constants, maximum energies, 
and spin changes for the large number of artificially produced 
ft emitters, Sargent diagrams have proved of little value. The 
data scatter over a large area in a Sargent plot, with little evi- 
dence for definite lines. However, it may be noted that few if 
any points fall above the line of allowed transitions; thus for a 
given max an approximate lower limit for the half-life can be 
determined, and for a given half-life an approximate lower limit 
for E m&x can be found. 

Complex ft Spectra. In connection with the relation between 
decay constant and E m&K it should be noted that a given ft emitter 
may decay to several different quantum states of the product 
nucleus, with corresponding 0-particle spectral distributions of 
different maximum energies. The relative probability of the 
transition to a particular state will then, of course, depend on the 
selection rules governing that transition. However, for the dif- 
ferent transitions to be observed their relative probabilities must 
not differ by many orders of magnitude. 

Whenever a ft transition leads to an excited state of the product 
nucleus, it is followed by the emission of one or more y quanta 
(or the excitation energy is lost in some other way discussed in 
section C). A complete diagram of the energies and genetic rela- 
tionships of the ft and 7 rays is called a decay scheme. A few 
typical decay schemes are shown in figure VI-5. 

The analysis of complex ft spectra into components is very 
difficult. The existence of more than one component can be 
rather easily established from 0-ray spectrograph data (and some- 
times even from absorption data) if the intensities are not too 
different and the energies not too similar; but it is hard to deter- 
mine the maximum energy of the lower-energy components. 
Sometimes this can be accomplished by the coincidence methods 
mentioned in chapter VIII, section D. 



136 



TYPES OF RADIOACTIVE DECAY 



CH.VI 






(a) 



(b) 






(<*) (e) (/) 

FIGURE VI-5. A few typical /3-decay schemes. 

(a) Simple ft spectrum. 

(b) Simple ft spectrum followed by 7 quantum. 

(c) Two ft spectra; the 7 energy equals the difference between the maximum 
ft energies. 

(d) Two ft spectra; the lower-energy one being followed by two 7 quanta in 
cascade. 

(e) Three ft spectra with four 7 rays. 
(/) ft + ~ft~~ branching. 

X'-electron Capture. As we have seen, positron emission may be 
considered as the transformation of a proton into a neutron with 
the simultaneous emission of a positron (and a neutrino). An 
alternative way for a proton to transform into a neutron (and 
thus for a nucleus to decrease its charge by one unit) is by the 
capture of an electron, and this process is also observed. As the 
K electrons in an atom are, on the average, closest to the nucleus 
(or quan turn-mechanically the wave functions of the K electrons 
have larger amplitudes at the nucleus than those of the L, M , 



STABILITY CONSIDERATIONS 137 

etc., electrons) the capture probability is greatest for the K elec- 
trons; the process is therefore called K -electron capture, or K 
capture, although L, M, etc., electrons may also be captured, but 
with smaller probabilities. (4) (In the sense of the Dirac theory, 
positron emission is the capture of an electron from the con- 
tinuum of negative-energy states.) In ./^-electron capture, as in 
other /8 processes, momentum, angular momentum, and statistics 
cannot be conserved unless a neutrino is simultaneously emitted. 
However, since the electron is captured from a definite energy 
state, the neutrinos emitted in this process are presumably mono- 
energetic. 

The electron-capture process may be difficult to observe because 
it is not necessarily accompanied by the emission of any detectable 
nuclear radiation, except in cases where the product nuclei are 
left in excited states and 7 rays are emitted. The most charac- 
teristic radiations accompanying electron capture are the X rays 
emitted as a consequence of the vacancy created in the K (or L, 
etc.) shell. These X rays ordinarily include the entire X-ray 
spectrum of the product element although the K lines are usually 
by far the most prominent ones. Emission of Auger electrons 
from the excited extranuclear structure accompanies K capture. 
Auger electrons result from what may be described as an internal 
photoelectric effect; for example, the emission of a K X ray may 
be replaced by the ejection of an L electron with a kinetic energy 
equal to the difference between the K X-ray energy and the L 
binding energy. 

Stability Considerations. In general, nuclei with neutron-proton 
ratios greater than that corresponding to the stability region 
decay by p~ emission, and most nuclei with neutron-proton 
ratios less than that required for stability decay by /?"*" emission 
or ^-electron capture. Occasionally a nuclide with two adjacent 
stable isobars may decay both by fT~ emission and by one or both 

4 It is clear that a nucleus stripped of all extranuclear electrons cannot 
undergo K capture. By the same reasoning it is evident that it should be 
possible to change the half-life for a K-capture process by altering the electron 
density near the nucleus. Attempts to detect differences in the half-life of 
X-capturing Be 7 in different states of chemical combination have recently 
been reported [see E. Segrfc and C. E. Wiegand, Phys. Rev. 75, 39 (1949)]. 
Although this is a very favorable case for testing the effect, the reported 
differences in half-life are surprisingly small and may not be real. 



138 TYPES OF RADIOACTIVE DECAY CH. VI 

of the other modes (for example, Cu 64 and As 74 ). Electron cap- 
ture (6) by a nucleus Z A is energetically possible if 

M Z A > M (Z _DA + pi (VI-1) 

where the AFs are the atomic masses, (6) and (A is the mass of the 
neutrino. Positron emission by the nucleus Z A requires that 

M Z A > M (Z ^A + 2m e + [A, (VI-2) 

where m e is the electron mass (5.486 X 10~~ 4 atomic weight unit). 
In cases intermediate between conditions VI-1 and VI-2, that 
is, when 

M ( z-u* + 2m e + (JL, 

protons can transform into neutrons only by electron capture; 
for nuclei to which relation VI-2 applies, both electron capture 
and positron emission are possible. The ratio of the probability 
of /?"*" emission to that of K capture increases with increasing 
disintegration energy and decreases with increasing Z. No case 
of p + emission has been definitely established for any element 
heavier than the rare earths. 

The condition for instability of a nucleus (Z 1) A with respect 
to /3~ emission can be written 



Considering now two isobars of atomic numbers Z and 
Z 1 we see from relations VI-1 and VI-3 that, if (JL is zero, 

5 Relation VI-1 is the condition for capture of a free electron. If the avail- 
able energy is very little, it may not permit the capture of an electron bound 
in the K shell; in such cases either a free electron or one from a shell of suffi- 
ciently low binding energy would have to be captured, and this probability 
might be quite small. 

8 Conditions VI-1 ,2,3 can perhaps be more readily derived by considering 
the masses of the bare nuclei; denoting these by small ra's, we can write down 
the following conditions: 

For electron capture by Z A : mz A -\- rn e > m(z-i) A 4- ^; 

For 0+ emission by Z A : m z A > ni(z-D A + m e + ^; 

For /3~ emission by (Z l) A : rn (Z -i) A > m z A + m e + \L. 



STABILITY CONSIDERATIONS 139 



one of the two isobars must be unstable: if MZ* > M^-i) A t 
the nucleus Z A is unstable with respect to electron capture; 
if M z * < M (Z _i)A, the nucleus (Z 1) A is unstable with respect 
to 0~~ emission. If the neutrino has a small but finite rest mass 
it may be possible for a pair of neighboring isobars to be ener- 
getically stable. (7) 

Three pairs of apparently stable neighboring isobars are known, 
and are listed in table VI-1. Whether both members of each 

TABLE VI-1 
PAIRS OP APPARENTLY STABLE NEIGHBORING ISOBARS 

48 Cd 113 4 9 In 113 49 In 115 50 Sn 115 Bl Sb 128 52 Te 123 
Relative abun- 

dance (%) 12.3 4.2 95.8 0.4 42.8 0.89 

Spin (units of h) % % % % Yz 

pair are really stable or whether the half-lives are so long or the 
radiations so soft that no activity has yet been detected cannot 
be stated with certainty; indeed long half-lives could be ration- 
alized in terms of the large spin differences. The fact that the 
member of each pair which has the higher Z has a relatively low 
abundance has sometimes been taken as an indication that these 
substances may have been disappearing by ^-capture activity. 
Another pair of neighboring isobars until recently regarded as 
stable is TsRe 187 and 76 0s 187 ; however Re 187 (62.9 per cent abun- 
dant) has been shown to decay to Os 187 (1.64 per cent abundant) 
by 0~ decay with a half-life of (4 =t 1) X 10 12 years. The recently 
discovered naturally occurring La 138 (abundance 0.089%) has 
two stable neighboring isobars (Ba 138 and Ce 138 ) and is very 

7 Conditions similar to equations VI-1 to VI-3 can be set up for isobars 
differing by two units of charge. For example the nucleus (Z 2) A is unstable 
with respect to decay into Z A with the simultaneous emission of two /3~~ 
particles if M(z-2) A > MZ*. It is interesting to note that conservation 
laws do not make it necessary to postulate neutrino emission in such a double /3 
decay process. The expected half -lives for simultaneous emission of two par- 
ticles (or simultaneous capture of two orbital electrons) are exceedingly long, 
and it is therefore not surprising that many pairs of apparently stable isobars 
differing by two units of charge are known. Recently some experimental 
evidence has been presented for the double p~ decay of Sn 124 with a half-life 
of about 10 16 years. [E. L. Fireman, Phys. Rev. 75, 323 (1949)]. 



140 TYPES OF RADIOACTIVE DECAY CH. VI 

probably radioactive although its activity has not yet been 
found. Three other naturally occurring nuclides with stable 
neighboring isobars have long been known to be radioactive. 
These are K 40 , Rb 87 , and Lu 176 ; their properties may be found 
in table 1-2, or in table A in the appendix. 

An interesting consequence of the prohibition (or near pro- 
hibition) of stable neighboring isobars is the absence of stable 
isotopes of the elements of atomic numbers 43 (technetium) and 
61. All mass numbers from 94 to 102 are occupied by isotopes 
of 42Mo or 44 Ru or both, and all mass numbers from 142 to 150 
are occupied by isotopes of eoNd or 62 Sm or both. Yet it is just 
in these mass regions that stable isotopes of technetium and 
element 61, respectively, would be expected. 

C. GAMMA DECAY AND ISOMERISM 

Gamma Radiation and Internal Conversion. In the discussion 
of the high-energy a particles from RaC' and ThC' (on page 126), 
we mentioned that a study of these a particles and the 7 rays 
emitted from the same nuclear levels permits an estimate of the 
"lifetimes" of these levels with respect to 7 emission. These 
lifetimes are of the order of 10~ 13 sec. In general, theoretical 
calculations indicate that ordinarily the half-lives for 7 emission 
are unobservably short (<10~ 9 sec) if dipole or quadrupole 
radiation is emitted. This is the case for the vast majority of 
7 rays following a- or /3-decay processes as well as neutron capture 
and other nuclear reactions. Gamma-ray energies between about 
10 kev and about 6 Mev have been observed in radioactive decay. 

Gamma-ray emission may be accompanied, or even replaced, 
by another process, the emission of internal-conversion electrons. 
Internal conversion has been pictured as a photoelectric effect 
produced by a 7 ray in the electron shell surrounding the 7-emit- 
ting nucleus. An extranuclear electron is emitted with a kinetic 
energy equal to the difference between the 7 energy and the bind- 
ing energy of the electron in the atom. Actually the emission 
of internal-conversion electrons may be regarded as an additional 
alternative process for the de-excitation of a nucleus. Internal- 
conversion electrons have line spectra, and for a ft transition 
followed by internal conversion the conversion electron lines are 
superimposed on the /3 spectrum. 



GAMMA RADIATION AND INTERNAL CONVERSION 141 

We define for a given transition between two quantum states 
an internal conversion coefficient <8) 

number of internal-conversion electrons emitted 

a = 

number of 7 quanta emitted 

This coefficient a may have any value between and oo. The 
internal-conversion coefficient generally decreases with increasing 
energy of the transition, and 7 rays of 0.5 Mev or greater energy 
are usually not accompanied by appreciable numbers of conver- 
sion electrons. The magnitude of the internal-conversion coeffi- 
cient increases with increasing multipole order of the transition 
(see below). 

Internal conversion involving emission of electrons from the 
K shell is most probable (if K conversion is energetically possible), 
and the probabilities decrease for successive shells. Some cases 
are known in which the energy of the transition is less than the 
binding energy of the K electrons; then only L, M, etc., conversion 
can take place (for example, the 1.5-min Ir 192 isomer). Partial 
conversion coefficients for the various shells are often used; for 
example the .fiT-conversion coefficient 

number of conversion electrons emitted from the K shell 

OLK = 

number of 7 quanta emitted 

The total conversion coefficient equals the sum of the partial 
coefficients. The energy differences between the conversion elec- 
trons from different shells but due to transitions between the 
same pair of nuclear levels are characteristic of the element; this 
fact is often used either to identify the element or to decide which 
electron lines (in an electron spectrogram) result from the same 
nuclear transition. 

Since internal conversion leaves a vacancy in one of the inner 
electron shells, the emission of conversion electrons is always 
accompanied by the emission of characteristic X rays and Auger 
electrons. The mode of decay of an active X-ray-emitting isotope 

8 Caution should be exercised in the use of data on internal-conversion co- 
efficients from the literature. Some authors define the internal-conversion 
coefficient as the ratio of the number of internal-conversion electrons to the 
total number of transitions (number of quanta -f- number of conversion elec- 
trons). The coefficient defined in that way is restricted to values between 
and 1. 



142 TYPES OF RADIOACTIVE DECAY CH. VI 

of element Z can often be determined from a study of the charac- 
teristic X-ray spectrum by critical-absorption methods as is dis- 
cussed in chapter X, section B. 

Isomerism. The phenomenon of nuclear isomerism, that is, the 
existence of a nucleus of given Z and A in more than one energy 
state of measurable lifetime, (9) is generally explained in terms of 
highly forbidden 7-ray transitions making the excited states 
metastable. Attempts have been made to formulate on a theo- 
retical basis quantitative relations between the lifetime of an 
excited state, the energy of the transition, the spin change, and 
the internal-conversion coefficient. The predicted correlations 
appear to be in fairly good agreement with experimental observa- 
tions. Some of the theoretical results are briefly summarized in 
the next paragraph. 

Electric dipole radiation is emitted when the initial and final 
states of the nucleus have opposite parity and a spin difference 
A/ of or dbl. Electric quadrupole radiation corresponds to 
A/ = 1 or 2 with no parity change. Lifetimes of excited 
states increase with increasing multipole order and with decreasing 
energy of the transition. For electric octopole radiation (A/ = 2 
or 3, parity change) half-lives of about 10~ 5 sec to minutes may 
be expected for energies between 200 kev and 20 kev; 2 4 -pole 
radiation corresponds to half-lives of seconds to days for that 
energy range, etc. When, for a given A/ = Z, electric 2^-pole 
radiation is forbidden by the parity selection rule, magnetic 
2'-pole radiation may compete with electric 2 /+1 -pole radiation, 
with comparable lifetimes. (10) It has also been shown that a transi- 
tion between two states of zero spin is highly forbidden if the two 
states have the same parity. A few cases of excited states of 
measurable lifetimes have been tentatively ascribed to this effect, 
for example, a 5 X 10~~ 7 -sec isomeric state of Ge 72 . For low 
transition energies and fairly large spin differences, emission of 
conversion electrons is a more probable process than y emission, 
and lifetimes calculated from y-emission probabilities must be 

9 With present techniques the lower limit for a measurable lifetime is about 
1(T 8 sec. 

10 M. L. Wiedenbeck [Phys. Rev. 69, 567 (1946)] has shown that most known 
isomeric transitions fit quite well the simple relations X * 42J 3 - 67 (for electric 
2 4 - or magnetic 2 3 -pole radiation) and X 45.8 X 10~ 6 # 3 - 67 (for electric 2 6 - or 
magnetic 2 4 -pole radiation), where X is in sec" 1 and E in Mev. 



ISOMERISM 



143 



In 11 



-105m 



IT 0.39 




IT 0.05% 
e-0.39 



y 


0.73 


y 


0.94 y 



18*99.95% 
2.66 



(a) 



Co 60 Ni 81 

10.7m ^ rpf 90%,<T0.056 
5.3y- 



1.46 



(6) 




Sb 124 Te 124 



f IT 0.006 
Plm / /ITQ.012 
1.3m v\V -FlTO.018 



71.5 




JT0.7 



71.3 



./T2.4 
\ 7 


1.7 




70.6 



(c) 



(d) 



FIGURE VI 6. Some typical decay schemes of nuclear isomers. Knergies are 

given in million electron volts. 
(a) 105-min excited state of In 113 decays entirely to ground state by isomeric 

transition. 
(6) 21-min upper state of Mn 52 decays 99.95 per cent by p + emission to Cr 52 . 

(c) 10.7-min upper state of Co 60 decays 90 per cent by isomeric transition 
to 5.3-year lower state, 10 per cent by /3~ emission to Ni 60 . 

(d) The first case of triple isomerism reported is in Sb 124 . The 21-min and 
1.3-min excited states decay by partially L-con verted isomeric transi- 
tions as well as by 0~ emission. The decay scheme shown is incomplete; 
for example several other /3~ and 7 energies are known to be associated 
with the 60-day activity. 



144 TYPES OF RADIOACTIVE DECAY CH. VI 

correspondingly reduced ( multiplied by the factor ) when 

\ 1 + / 

the conversion coefficients are high. Experimentally, very high 
conversion coefficients are common in low-energy isomeric transi- 
tions. 

Excited isomeric states with half -lives ranging from about 10~~ 7 
sec to several months are known. Depending on the relative 
decay probabilities an excited isomeric state may transform pre- 
dominantly into its associated lower isomeric state by 7-ray or 
conversion-electron emission or predominantly to a neighboring 
isobar by a ft or ./^-capture process. For example, the 105-min 
In 113 transforms into stable In 113 by emission of conversion elec- 
trons, and the 24.5-min Ag 106 emits positrons to become Pd 106 ; 
in 21-min Mn 52 both the transition to the lower isomeric state 
and /3 + emission occur to an observable extent. Transition to a 
lower isomeric state is called isomeric transition, abbreviated IT. 
Some typical decay schemes for isomers are shown in figure VI-6. 
At least one case of triple isomerism has been established; this is 
in siSb 124 , with 21-min, 1.3-min, and 60-day periods (figure VI-6). 

D. OTHER MODES OF DECAY 

Spontaneous Fission. The explanation of the fission process in 
terms of the liquid drop model naturally* leads one to inquire 
whether a nucleus that can be split by the addition of a relatively 
small amount of excitation energy may not have a finite proba- 
bility of undergoing fission spontaneously. This process of spon- 
taneous fission has indeed been discovered in uranium. The 
isotope U 238 accounts for most of the spontaneous fission processes 
observed; its disintegration constant for spontaneous fission is 
about 2 X 10~ 24 sec"" 1 . If spontaneous fission were the only 
process by which U 238 decayed, this would correspond to a "half- 
life" of 10 16 years. Also Pu 239 has been investigated, and its 
"half-life" for spontaneous fission reported as >10 14 years. 
Obviously spontaneous fission contributes only a negligible frac- 
tion of the decay of these nuclear species, and their half-lives are 
practically entirely determined by their a decay. However, it is 
quite possible that still heavier elements may have much higher 
disintegration constants for spontaneous fission and that this 



EXERCISES 146 

may at least in part account for the absence of transuranium 
elements in nature. 

"Delayed" Neutrons. As has been pointed out before, nucleon 
emission with measurable lifetimes has not been observed and is 
not expected on theoretical grounds. The emission of so-called 
"delayed" neutrons, with half-lives of from fractions of a second 
to about 1 min, following fission, has been shown to be due to 
practically instantaneous emission of neutrons from highly excited 
states of fission products following /9~~ decays. This process can 
occur when 0""" decay leaves the product nucleus in a state of such 
high excitation (at least several million electron volts) that neu- 
tron emission can compete successfully with 7 emission. The 
neutron activity then persists with the half-life of the preceding 
ft" decay. Some of the delayed neutron periods have been identi- 
fied with particular ^""-emitting fission products by chemical 
analysis; the 56-sec ft" decay of Br 87 and the 22-sec ft~~ decay of 
I 137 , for example, are followed by neutron emission. Recently 
the 4.1-sec ft~~ decay of N 17 (formed in various nuclear reactions) 
has been shown to be followed by neutron emission. 

EXERCISES 

1. In the decay of AcC(Bi 211 ) to AcC^T! 207 ) the following a- and 7-ray 
energies have been observed: a. 6.611, 6.262 Mev; 7 0.354 Mev. Con- 
struct a reasonable decay scheme for this disintegration. Indicate the 
total energy difference between each energy state involved and the ground 
state of AcC". 

2. Estimate the decay constant for the 10.54-Mev a-particle emission 
in ThC', and the decay constant for 7 emission from the same ThC' level. 

Answer: X a 5 X 10 11 sec~ l ; X T - 100X a . 

3. Represent on a Sargent diagram the ft" decays of H 8 , C 14 , Na 24 , 
P 32 , S 35 , K 40 , Co 60 (5.3 y), Zn 69 (57 m), Br 82 , Sr 89 , Y 91 (57 d), Cb 95 (35 d), 
Te 127 (9.3 h), Te 129 (70 m), I 131 . What conclusions, if any, can you draw? 

4. Show that the production of a positron-electron pair by a photon 
in vacuum is impossible. (Note: Set up the conditions for momentum 
and energy conservation, using relativistic expressions, and show that they 
lead to a contradiction, for example to the inequality cos 6 > 1, where 
is the angle between the directions of motion of positron and electron.) 

5. Estimate the muitipole order of the isomeric transition in Te 125 . 



146 TYPES OF RADIOACTIVE DECAY CH. VI 

6. Collect from the literature as complete information as you can on 
the decay schemes of V 48 , Na 22 , I 131 . (Note. The Physical Review is a 
good source of information.) 



REFERENCES 

E. RUTHERFORD, J. CHADWICK, and C. D. ELLIS, Radiations from Radioactive 

Substances, Cambridge University Press, 1930. 

F. RASETTI, Elements of Nuclear Physics, New York, Prentice-Hall, 1936. 
M. S. CURIE, Traitt de radioactivite, Paris, Gauthier-Villars, 1935. 

G. HEVESY and F. A. PANETH, A Manual of Radioactivity, 2d ed., Oxford 

University Press, 1938. 
R. E. LAPP and H. L. ANDREWS, Nuclear Radiation Physics, New York, 

Prentice-Hall, 1948. 
G. GAMOW, Structure of Atomic Nuclei and Nuclear Transformations, New 

York, Clarendon Press, 1937 (particularly pp. 87-107). 
H. A. BETHE, "Nuclear Physics, B. Nuclear Dynamics, Theoretical," Rev. 

Mod. Phys. 9, 161-171 (1937). 

E. J. KONOPINSKI, "Beta Decay," Rev. Mod. Phys. 16, 209 (1943). 
H. A. BETHE, Elementary Nuclear Theory, New York, John Wiley & Sons, 1947. 
H. PRIMAKOPP, "Introduction to Meson Theory," Nucleonics 2 no. 1, 2 (Jan. 

1948). 
N. R. CRANE, "The Energy and Momentum Relations in the Beta Decay, 

and the Search for the Neutrino," Rev. Mod. Phys. 20, 278 (1948). 

C. W. SHERWIN, "The Neutrino," Nucleonics 2 no. 5, 16 (May 1948). 

S. N. NALDRETT and W. F. LIBBY, "Natural Radioactivity of Rhenium," Phys. 

Rev. 73, 487, 929 (1948). 

T. P. KOHMAN, "Limits of Beta-Stability," Phys. Rev. 73, 16 (1948). 
S. M. DANCOFF and P. MORRISON, "The Calculation of Internal Conversion 

Coefficients," Phys. Rev. 66, 122 (1939). 
A. C. HELMHOLZ, "Energy and Multipole Order of Nuclear Gamma Rays," 

Phys. Rev. 60, 415 (1941). 

R. G. SACHS, "A Note on Nuclear Isomerism," Phys. Rev. 67, 194 (1940). 
A. BERTHELOT, "Contribution a l'6tude de 1'isome'rie nucle"aire" (Review 

Article), Ann. Phys. 19, 117 (1944). 

D. J. HUGHES, J. DABBS, A. CAHN, and D. HALL, "Delayed Neutrons from 

Fission of U 235 ," Phys. Rev. 73, 111 (1948). 

M. L. WIEDENBECK, "Note on Lifetime of Metastable States," Phys. Rev. 69, 
567 (1946). 



.OF RASIAIIQNS WTTHJilATTER 

A. ALPHA PARTICLES 

Processes Responsible for Energy Loss. Nuclear radiations, 
both corpuscular and electromagnetic, are detectable only through 
their interactions with matter. If this interaction is sufficiently 
small, as in the case of the neutrino, the radiation remains unde- 
tected. For an understanding of the methods and instruments 
used for the detection, measurement, and characterization of 
nuclear radiations it is necessary to consider the manner in which 
these radiations interact with matter. 

In passing through matter a particles lose energy chiefly by 
interaction with electrons. (1) This interaction may lead to the 
dissociation of molecules or to the excitation or ionization of 
atoms and molecules. The effect which is most' easily measured 
and most often used for the detection of a particles is ionization. 
The details of the ionization processes and other effects associated 
with a-particle passage are more readily investigated in gases 
than in liquids or solids, although the processes are presumably 
about the same. We shall therefore speak mostly of phenomena 
observed in the passage of a particles through gases. 

Because a particles have relatively short ranges a known num- 
ber of a particles of known initial energy can be made to spend 
their entire energy inside an ionization chamber, and thus the 
total ionization produced per a particle is readily measured. 
These experiments show that on the average about 35 ev (35 elec- 
tron volts of energy) are dissipated for each ion pair formed in 
air. At least in the case of air this value is quite independent of 
the initial energy of the a particles. The energy required to form 
an ion pair is listed for a number of other gases in table VII-1, 
together with the first ionization potentials of these gases. In 
the noble gases a larger fraction of the a-particle energy is spent 

1 The interactions of a particles with nuclei (scattering and nuclear reactions) 
have been discussed in previous chapters. The contribution of these processes 
to energy loss of a particles in passing through matter is entirely negligible. 

147 



INTERACTION OF RADIATIONS WITH MATTER CH. VII 

i ionization processes than in the diatomic and polyatomic gases 
where dissociation of molecules is also possible. 

Part of the energy loss of a. particles is accounted for by the 
kinetic energy given to the electrons removed from atoms or 
molecules in close collisions with the a. particle. It can easily be 
shown from conservation of momentum that the maximum veloc- 
ity which an a. particle of velocity v can impart to an electron is 
about 2t>; therefore, the maximum energy which an electron can 

TABLE VII-1 

AVERAGE ENERGY LOST BY a PARTICLES IN PRODUCING ONE ION PAIR IN 

VARIOUS GASES 



H 2 

He 

N 2 

2 

Ne 

A 

Kr 

Xe 

CH4 

NH 3 



receive from the impact of a 6-Mev a particle, for example, is 
about 3000 ev. The average energy imparted to electrons by 
a particles in their passage through matter is of the order of 100 
to 200 ev. Many of these secondary electrons or d rays are fast 
enough to ionize other atoms. In fact about 60 to 80 per cent of 
the ionization produced by a particles is due to secondary ioniza- 
tion; the exact ratio of primary to secondary ionization is very 
difficult to determine. Delta-ray tracks are often seen in cloud- 
chamber pictures of a-particle tracks. 

Range. Because an a particle loses only a very small fraction of 
its energy in a single collision with an electron and is not appre- 
ciably deflected in the collision, a-particle paths are very nearly 
straight lines. Furthermore, because of the very large number 
of collisions (of the order of 10 5 ) necessary to bring an a particle 
of a few million electron volts initial energy to rest, the ranges 



Energy per 


First Ioniza- 




Ion Pair 


tion Potential 


Difference 


(K) (ev) 


(I) (volte) 


(K-D 


33.0 


15.4 


17.6 


27.8 


24.5 


3.3 


35.0 


15.5 


19.5 


32.3 


12.5 


19.8 


27.4 


21.5 


5.9 


25.4 


15.7 


9.7 


22.8 


13.9 


8.9 


20.8 


12.1 


8.7 


30.0 


14.5 


15.5 


38.9 


11.2 


27.7 


25.6 


10.4 


15.2 



RANGE 



149 



of all a particles of the same initial energy are the same within 
narrow limits. In figure VII-1 the number of a particles found 
in a gas at a distance r from the source is plotted against r for 
the case of a source which emits a particles of a single energy. 
It is seen that the ranges of all the particles in a given medium 
are not exactly the same but show a small spread of about 3 or 4 
per cent. This phenomenon, called the straggling of a-particle 



r (distance from source) 




Rmax 



FIGURE VII-1. Number of a particles from a point source as a function of 
the distance from the source (full curve). The derivative of that function is 
also shown (dotted curve); the latter represents the distribution in ranges. 

ranges, is caused by the statistical fluctuations in the number of 
collisions and in the energy loss per collision. The dotted curve 
in figure VII-1 is obtained by differentiating the other (integral) 
curve and represents the distribution of ranges or the amount of 
straggling; it is approximately a Gaussian curve. The distance 
r corresponding to the maximum of the differential curve (point 
of inflection of the integral curve) is called the mean range U of 
the a particles. The distance r obtained by extrapolating to the 
abscissa the approximately straight portion of the integral curve 
is the extrapolated (or practical) range J? ex . The largest observed 
range is called the maximum (or true) range # max . Thus, although 
the range of an a particle is a measure of its energy and can be 
rather readily determined experimentally, it is necessary to specify 



150 INTERACTION OF RADIATIONS WITH MATTER CH. VII 

which range is expressed. The mean range is least dependent on 
the particular experimental setup and is now generally used in 
range tables and in range-energy relations. Extrapolated ranges 
were often given in the older literature and are more easily deter- 
mined experimentally. 

Alpha-particle ranges are usually given for air at 15C and 
760 mm pressure (although 0C is used as this standard tempera- 
ture by some authors). The range is inversely proportional to 
the density of air. Range-energy relations based on calculations 
and on energy determinations by magnetic-deflection experiments 
coupled with range measurements have been published by many 
authors. (2) For mean a-particle ranges between 3 and 7 cm the 
empirical equation R = 0.318 E**, with E in centimeters of air 
at 15C and 760 mm pressure and E in millions of electron volts, 
holds fairly well. Ranges of a. particles are generally determined 
by absorption methods, either with solid absorbers or, more accu- 
rately, by varying the pressure of an absorbing gas. Cloud- 
chamber photographs of a-particle tracks are also used. Ranges 
may be determined with a precision of about one part in 5000^- 

Stopping Power. Comparisons of the ranges of a particles in 
different substances introduce the concept of the stopping power 
of a particular substance. Formally the stopping power F is 
defined as the space rate of energy loss of an a particle in the 
substance, F = dE/dx, and thus has the dimensions of a retard- 
ing force. Because F is generally not independent of energy E 

we write dx = dE/F(E), and the range R = - | dE/F(E) 

JE Q 

for a particle of initial energy E . Experimentally one can deter- 
mine F(E) by measuring the range as a function of energy and 
then taking the derivative of that function: dR/dE = l/F(E). 
In common practice the energy dependence of stopping power is 
often neglected and an average value for the whole range used in 
approximate calculations. 

Bragg's empirical rule for stopping-power variations among 
different substances is that the stopping effect per atom, called 
the atomic stopping power s, is about proportional to A^ y that 

* See, for example, M. S. Livingston and H. A. Bethe, Rev. Mod. Phys. 9, 
261-276 (1937). The same paper (p. 285) also gives the difference between 
extrapolated and mean range as a function of mean range. 



STOPPING POWER 151 

is to the square root of the atomic weight. Because the number 
of atoms per cubic centimeter is proportional to the ratio of the 
density p to the atomic weight, another statement of this rule is 
that the stopping power per unit weight, called the mass stopping 
power, is about proportional to A~~^. The similarity of these 
two statements is just enough to be confusing; it is to be remem- 
bered that for stopping <x particles a heavier atom is more effec- 
tive, according to A ^, but is less effective per unit weight, accord- 
ing to A~~ l//2 . 

Another empirical rule for the atomic stopping power, resem- 
bling Bragg's rule but giving a better result in most instances, is 
stated in terms of Z, the atomic charge, rather than A : 

Z 

8 = 0.563 . (VII-1) 

Vz + io ' 

The proportionality constant 0.563 is chosen to give the best 
fit with experimental data in terms of a value of s = 1 for 
air. 

In most actual cases, indeed for air, the stopping substance is 
not a single element but rather is a compound or mixture of ele- 
ments. For practical purposes in using the approximate rules 
just stated we make the further approximation that the stopping 
power of a molecule or of a mixture of atoms or molecules is given 
by the sum of the stopping powers of all the component atoms. 
In view of the fact that a considerable fraction of the a-particle 
energy is expended in molecular excitation and dissociation 
processes, this simple additivity relation is somewhat surprising. 
The stopping power of water vapor has been measured to be 
about 3 per cent less than that of the equivalent mixture of hydro- 
gen and oxygen; measurements for a number of organic isomers 
show that their molecular stopping powers are the same within 
less than 1 per cent. For systems containing deuterium equation 
VII-1 should be used rather than Bragg's rule, because H 1 and 
H 2 have very nearly the same atomic stopping power. 

Probably the best procedure for making calculations of ranges 
in different substances compared to air is to apply to the given 
range appropriate correction factors for density, atomic or molecu- 
lar stopping power, and so on, just as the pressure-volume-tem- 
perature relations are commonly handled in gas-law problems. 



152 INTERACTION OF RADIATIONS WITH MATTER CH. VII 

For example, a range in aluminum fl A1 is given in terms of the 
range in air: 



tt _ y pair v \r A1 i? v Q- 001226 v 

MAI = ^air X X \ = /l air X X 

P M A^ 2.70 

flAl == 0.000622 fl a ir. 

The value 14.4 for the "atomic weight of air" is the square of the 
weighted average square-root value: (0.8VTJ+ 0.2VT(3) 2 = 
14.4. Other mixtures and compounds are handled in the same 
way when Bragg's rule is used. For the same calculation but 
using equation VII-1, we obtain a slightly different result: 

no. atoms air per cm 3 . 

flAl = flair X ~ X [ 0.563 

no. atoms Al per cnr 
0.001226/14.4 
* A1 = * air X 2.7/27.0 

flAl = 0.000558 flair. 

As another illustration we calculate the range in S0 2 compared 
to the range in air, taking for the density of SO 2 at 15C the 
ideal-gas-law value, (64.1/22,400) X (273/288) = 0.00272; we 
have: 

no. atoms air per cm 3 

flsO 2 == flair X 




X 



no. molecules SO 2 per cm 3 

1 
molecular stopping power of SO 2 ' 



0.001226/14.4 1 

= flair X X 

0.00272/64.1 

16 
= s s + 2s = 0.563 




= 0.515fl a ij.. 



SPECIFIC IONIZATION 



153 



Often it is convenient to express absorber thickness not in centi- 
meters but in milligrams per square centimeter. As an illustra- 
tion of the use of this unit, a foil of gold of thickness 12.0 mg per 
cm 2 would be equivalent to about 12.0 X Vl4.4/197 = 3.24 
mg per cm 2 of air, or 3.24/1.226 = 2.65 cm of air at 15C. 
jf Specific lonization. The energy and velocity of an a particle 
decrease in each interaction. If the velocity of an a particle in 
an absorbing medium is plotted against distance from the source, 
a curve such as the one sketched in figure VII-2 is obtained. The 



2.0 






I 



1.0 2.0 3.0 4.0 5.0 
Distance from the source (cm) 



6.0 R 



FIGURE VII-2. Velocity of a typical a particle versus distance from the source. 

velocity of an a particle of E million electron volts is about 
6.9 X lO 8 ^ cm sec"" 1 (in the nonrelativistic region). 

The specific ionization the number of ion pairs formed per 
millimeter of path varies with the velocity of the a. particle and 
is approximately proportional to l/v at energies in excess of about 
1 Mev (v > 7 X 10 8 cm sec" 1 ). The maximum specific ionization 
is reached by an individual a particle about 3.0 mm from the end 
of its range where it has a velocity of 4.2 X 10 8 cm sec"^ 1 , or an 
energy of about 370 kev. After that the specific ionization falls 
rapidly to zero. For a beam of initially homogeneous a particles 
the maximum of the specific ionization occurs at about 4.7 mm 
from the extrapolated range. A plot of specific ionization versus 
distance from the source is called a Bragg curve (figure VII-3). 
For a particles of different energies the Bragg curves are almost 
identical over corresponding regions measured from the end of 



154 



INTERACTION OF RADIATIONS WITH MATTER CH. VII 
6000 

5000 
| 4000 
I 3000 
| 2000 

J- 1000 




1 



ueL) 



765432 

Residual range (cm) 

FIGURE VII-3. Bragg curve for initially homogeneous a. particles. 



the range. In table VII-2 specific ionization values are given 
for a number of residual ranges, that is for various distances from 
the end of the extrapolated range. _ 

TABLE VII-2 

SPECIFIC lONIZATIONS AND ENERGIES OF a PARTICLES AT VARIOUS 
RESIDUAL RANGES * 



Residual Range 

(in cm of air) Energy 
(15C and 760 mm pressure) (Mev) 

0.21 

0.47 

1.0 1.9 

1.5 2.7 

2.0 3.4 

4.0 5.5 

7.0 7.8 



Ion Pairs 

per mm of air 

(15C and 760 mm pressure) 

4500 
6000 
4800 
3960 
3440 
2680 
2440 



* The data in columns one and three are taken from E. Rutherford, J. Chad- 
wick, and C. D. Ellis, Radiations from Radioactive Substances, Cambridge 
University Press, England, and The Macmillan Co., New York. 

It should be noted that for a given a-particle source the extra- 
polated range obtained from a Bragg curve is not exactly the 
same as that obtained from a number-versus-distance curve, the 
latter being a few tenths of 1 per cent larger. 



OTHER HEAVY CHARGED PARTICLES 155 

The straggling in specific ionization is complicated by the fact 
that near the end of their ranges the a. particles frequently pick 
up electrons (becoming He + or even He) and subsequently lose 
them in later collisions. Several thousand such fluctuations in 
charge occur for each a. particle, but they are almost completely 
confined to the last few millimeters of the range where the velocity 
of the a particle becomes comparable to the orbital-electron 
velocities in helium atoms. An a particle spends over 90 per cent 
of its entire path as He" 1 " 1 ". The relative abundances of He" 1 "" 1 ", 
He + , and He at various energies were studied by magnetic- 
deflection measurements with varying thicknesses of absorbers 
between source and measuring device. 

Other Heavy Charged Particles. Any ion moving at high speed 
through matter loses its energy by essentially the same mechanism. 
Rather elementary calculations show that the rate of energy loss 
at a given velocity may be expected to vary approximately as 
Z 2 , where Z is the net charge of the ion. This is found to be true 
experimentally. Alpha particles and protons of the same velocity 
have equal ranges because the a. particle, with Z 2 four times as 
large, loses energy four times as fast and has just four times as 
much energy to lose. Deuterons and protons of the same velocity 
lose energy at the same rate, with Z 2 = 1 for both; then because 
the deuteron has twice the energy it has just twice the range of 
the proton. Range-energy relations for protons and deuterons 
are found in the literature. (3) 

Fission fragments are 20 to 40 times heavier than a particles 
and have initial energies 10 to 20 times greater; thus their initial 
velocities are close to those of a particles. If a fission fragment 
were completely stripped of electrons the large value of Z 2 , several 
hundred times greater than for an a particle, would lead to a 
range of the order of 1 mm in air. Actually the particle probably 
will be stripped only of electrons with orbital velocities smaller 
than its own velocity and so will hold all electrons with binding 
energies greater than about 1 kev. (4) Therefore, the fragment 

8 For example, Livingston and Be the, Rev. Mod. Phys. 9, 268 (1937). 

4 The kinetic energy of a bound electron (which is equal to its binding 
energy; see chapter II, page 25) with velocity equal to that of a fission frag- 
ment is in a nonrelativistic approximation the energy of the fission fragment 
multiplied by the ratio of the electron-to-fragment masses. For a 100-Mev 
fragment with mass 100, the binding energy of the corresponding electron is 
100 X 10 6 X (0.00055/100) - 550 ev. 



156 INTERACTION OF RADIATIONS WITH MATTER CH. VII 



starts with net Z 25 and gains electrons as its velocity decreases 
until Z at about 1 Mev (approximately 3 mm before the end 
of the range). Measured fission fragment ranges are about 1.9 
to 2.9 cm in air; a curve of specific ionization versus residual range 
looks about as shown in figure VI 1-4. If particles like fission 
fragments but of very much higher energy were available, the 
curve in figure VII-4 could be extended to larger residual ranges, 
and, no doubt, the same general features as in the Bragg curve 




I 



2.0 



1.0 



Residual range (cm of air) 
FIGTOE VII-4. "Bragg curve" for a fission fragment. 

for a. particles would be seen. The relative stopping powers of 
various substances are very nearly the same for fission fragments 
as for a particles. 

B. ELECTRONS 

Comparison with a-particle Behavior. The interaction of elec- 
trons with matter is in many ways fundamentally similar to that 
of a particles. The processes which are responsible for the energy 
loss are the same in both cases. In fact, the average energy loss 
per ion pair formed is almost the same for electrons as for a par- 
ticles (about 32.5 ev for electrons in air). The primary ionization 
by electrons accounts for only about 20 to 30 per cent of the total 
ionization; the remainder is due to secondary ionization. 

We must consider a number of differences between the interac- 
tions of the two types of particles with matter. First, for a given 
energy the velocity of an electron is much larger than that of an 
a particle (see table VII-3), and, therefore, the specific ionization 



COMPARISON WITH ALPHA-PARTICLE BEHAVIOR 157 



is less for electrons. In table VII-3 the specific ionization in air 
is given for electrons of a few different energies. In the region 
from about 10 kev to 2 Mev the specific ionization has been shown 
to be nearly proportional to the inverse square of the velocity. 
The largest specific ionization, 770 ion pairs per millimeter, occurs 

TABLE VII-3 



VELOCITY AND SPECIFIC IONIZATION FOR ELECTRONS OF 
VARIOUS ENERGIES 




Velocity (in units 






of the velocity 


Ion Pairs 


E (Mev) 


of light, c) 


per mm of air 


10-* 


0.001979 





10~ 6 


0.006257 





1(T 4 


0.01978 




1.46 X 10~ 4 


0.0240 


770 (maximum) 


10~ 3 


0.06248 




icr 2 


0.1950 


~110 


0.05 


0.4127 


20 


0.10 


0.5483 


15 


0.50 


0.8629 


6 


1.0 


0.9411 


5 


2.0 


0.9791 


4.5 


3.0 


0.9893 




4.0 


0.9934 




5.0 


0.9957 




10 


0.9988 




20 


0.99969 





at 146 ev (velocity = 0.024c), which is a much lower energy but 
somewhat higher velocity than corresponds to the peak in the 
Bragg curve for a particles. In air, ionization stops when the 
electron energy has been reduced to 12.5 ev (the ionization poten- 
tial of oxygen molecules). 

An electron may lose a large fraction of its energy in one col- 
lision; therefore, a statistical treatment of the energy-loss processes 
is much less justified than for a particles, and straggling is much 
more pronounced. In the passage of an initially homogeneous 
beam of electrons through matter the apparent straggling is 
further increased by the pronounced scattering of the electrons 
into different directions, which makes possible widely different 
path lengths for electrons traversing the same thickness of ab- 



158 INTERACTION OF RADIATIONS WITH MATTER CH. VII 

sorber. Nuclear scattering as well as scattering by electrons is 
important. 

For electrons of high energy an additional mechanism for losing 
energy must be taken into account: the emission of radiation 
(bremsstrahlung) when an electron is accelerated in the electric 
field of a nucleus. The ratio of energy loss by this radiation to 
energy loss by ionization in an element of atomic number Z is 

E-Z 

approximately equal to , where E is the electron energy in 

800 

millions of electron volts. Thus, in heavy materials such as lead 
the radiation loss becomes appreciable even at 1 Mev, whereas 
in light materials (air, aluminum) it is unimportant for the energies 
available from ft emitters. 

Finally, the additional fact that ft particles are emitted with a 
continuous energy spectrum makes their absorption in matter a 
phenomenon too complicated for theoretical analysis. 
^Absorption of ft Particles. The combined effects of continuous 
spectrum and scattering lead quite fortuitously to an approx- 
imately exponential absorption law for ft particles of a given 
maximum energy. Absorption curves, that is, curves of activity 
versus thickness of absorber traversed, are for this reason usually 
plotted on semilogarithmic paper. The nearly exponential de- 
crease applies both to numbers and specific ionizations of ft par- 
ticles, although absorption curves taken with counters and ioniza- 
tion chambers cannot be expected to be completely identical. 
The exact shape of an absorption curve depends also on the shape 
of the /3-ray spectrum and, because of scattering effects, on the 
geometrical arrangement of active sample, absorber, and detector. 
If sample and absorber are as close as possible to the detector, 
the semilog absorption curve becomes most nearly a straight line; 
otherwise, some curvature toward the axes is generally found. 
When ft particles belonging to two spectra of widely different 
maximum energies are present in a source, this is apparent from 
the change of slope in the absorption curve; such an absorption 
curve is roughly analogous to the semilog decay curve of an 
activity containing two different half-life periods. 

If the absorption of ft rays is represented by an exponential law 
Ad = AO^""^, where AQ is the measured activity without absorber 
and Arf the activity observed through absorber of thickness d y 
then /x is known as the absorption coefficient. The ratio of the 



DETERMINATION OF BETA-PARTICLE RANGES 



159 



absorption coefficient to the density p, known as the mass absorp- 
tion coefficient, is nearly independent of the nature of the absorber. 
More accurately it varies about as Z/A\ that is, the number of 
electrons per unit mass determines the mass stopping power of a 
substance for ft particles. The thickness required to reduce the 
activity to one half of its initial value is called the half-thickness 
di, = 0.693/V; more frequently the half -thickness is expressed in 
grams per square centimeter, 
and then is equal to 0.693p//* 
and varies about as A/Z. Ab- 
sorption coefficients and half- ^ 
thickness values given in the |j 
literature usually refer to the > 
initial portions of absorption ~~ 
curves. These values cannot 
be relied on as accurate meas- 
ures of /5-particle energies. 

Determination of ^-particle 
Ranges. It is generally the 
purpose of absorption measure- 
ments to determine the upper 
energy limit of a 0-ray spec- 




. 0.01 - 



0.001 - 



d|/2 Range 

Absorber Thickness (mg/cm 2 ) 
(.linear scalej 

FIGURE VII-5. Idealized -ray ab- 
sorption, curve (semilog plot). 



trum. We should say at the 
outset that precision determina- 
tions of upper energy limits can be made only with electron 
spectrographs; yet for most purposes absorption measurements 
are much more convenient. 

To get a measure of the upper energy limit of a 0-ray spectrum 
one must find the range in the absorber of the most energetic 
ft particles. The fact that a range exists for a given 0-ray spec- 
trum means that the absorption curve cannot continue as an 
approximate exponential but must eventually turn downward 
toward oo on a semilog plot (see figure VII-5). The ratio of 
range to initial half-thickness is generally between 5 and 10. In 
practice a /9-ray absorption curve is never found to reach oo 
on a semilog plot and may not even turn in that direction, because 
of the presence of more penetrating radiation beyond the range 
of the ft rays. Even if neither nuclear y radiation nor character- 
istic X rays are present, there is always some background of 
bremsstrahlung from the deceleration of the ft particles in the 



160 INTERACTION OF RADIATIONS WITH MATTER CH. VII 



sample itself and in the absorbers. If elements of low Z are used 
as absorbers, the difference in slopes between /3-ray and 7- or 
X-ray absorption curves is particularly marked; the absorption 
curve then exhibits a fairly sharp break where the /3-ray com- 
ponent turns over into the photon "tail" (see figure VII-6). For 
this reason -ray absorption curves are always taken with absorbers 



1000 




Absorber Thickness (mg/cm 2 ) 
(linear scale) 

FIGURE VII-6. Typical /3-ray absorption curve in aluminum (semilog plot). 
A 7-ray component is present. 

of low atomic number; aluminum or plastic absorbers are most 
commonly used, and for differentiating ft particles from soft 
X rays beryllium absorbers are particularly useful. (Beta-ray 
ranges expressed in milligrams per square centimeter vary only 
about as A/Z, whereas the absorption of electromagnetic radiation 
increases rapidly with Z as is discussed in section C.) 

The relative efficiencies of measuring instruments for ft and 7 
rays vary with the energies of the radiations and with different 
instruments; most Geiger-Mtiller counters and air-filled ioniza- 
tion chambers are about 100 times as efficient for ft particles as 
for 1-Mev 7 quanta, and the efficiency for 7 rays is roughly pro- 



DETERMINATION OF BETA-PARTICLE RANGES 161 

portional to the -y-ray energy from a few hundred kiloelectron 
volts to a few million electron volts. Thus a typical /3-ray absorp- 
tion curve for a ft spectrum accompanied by y rays will have a 
y-ray "tail" with intensity of the order of 1 per cent of the initial 
ft activity. A pure bremsstrahlung tail is usually at least an order 
of magnitude smaller, about 0.05 per cent or less of the initial 
ft activity. In ft~*~ absorption curves there is, in addition to other 
electromagnetic radiation which may be present, always a back- 
ground of annihilation radiation, about I per cent of the initial 
ft + intensity in a typical detector. 

The maximum range for a ^-particle spectrum may be obtained 
from an experimental absorption curve in various ways. Visual 
inspection gives a rough value (usually too small) for the point 
at which the ft activity ceases to be detectable above the 7- or 
X-ray background; the lower the 7- or X-ray background, the 
better is the visual method. Better results can sometimes be 
obtained by subtraction of the penetrating background radiation 
from the total absorption curve, which should result in a curve 
similar to the one in figure VII-5. 

The best method for the determination of /3-ray ranges from 
absorption curves is the comparison method suggested by 
N. Feather. Here the absorption curve to be analyzed is com- 
pared with the absorption curve (measured under identical con- 
ditions) of a standard ft emitter, usually RaE (# max =1.17 Mev, 
range = 476 mg per cm 2 in aluminum) or UX 2 (# max = 2.32 Mev, 
range = 1105 mg per cm 2 in aluminum). (5) The net fi-ray absorp- 
tion curves after subtraction of all backgrounds due to electro- 
magnetic radiation are used. If the same percentage reduction 
of initial activity would correspond to the same fraction of the 
range for each ft emitter, the absorber thickness corresponding to 
a certain fraction (say 0.5) of the range could be readily determined 
for the unknown by comparison with the standard of known 
range. Actually this procedure gives a somewhat different appar- 
ent range for each fraction of the range at which the comparison 
is made, because of the different shapes of the absorption curves. 

6 The ranges given for the standard substances are those corresponding to 
the spectrographically determined E m&K values according to the best range- 
energy relations (see p. 164). These, rather than the visual ranges determined 
for the standard substances in a particular experimental arrangement should 
be used in the Feather analysis. 



162 



INTERACTION OF RADIATIONS WITH MATTER CH. VII 



But, if these apparent ranges are plotted against the fractions of 
the range at which they were determined, a smooth curve results 
which can be extrapolated to fraction 1.0 of the range to find the 
true range. This curve is called the Feather plot; one is shown 
in figure VII-7. If the spectra of the unknown and the standard 
have the same shape, the Feather plot is a straight line parallel 
to the abscissa. When RaE, which has an unusually large per- 
centage of low-energy electrons in its spectrum, is used as a stand- 



I I 




I 



J I 



J I 



_L 



Range 



0.1 0.2 



0.3 0.4 0.5 0.6 0.7 
Fraction of Range 



0.8 0.9 1.0 



FIGURE VII-7. A typical Feather plot. 

ard, the Feather plot often has a shape similar to that shown. 
The UX2 spectrum is more nearly normal, as is also that of P 32 
which has been suggested as a standard. A more complicated 
comparison method has recently been proposed; (6) it uses as 
standards a number of absorption curves and takes into account 
the fact that the shapes of ft spectra vary with the atomic number 
of the ft emitter and with max . 

When two /3-spectral components are emitted from a sample it 
is usually very difficult to obtain a reliable end point for the 
softer component; extrapolation of the line representing the harder 
component to zero absorber thickness and subtraction of this 
extrapolated curve from the total absorption curve is the best 
that can be done. In this case too the Feather method gives 
much better results than does visual determination. 

Absorption of Monoenergetic Electrons. When soft conversion 
electrons are emitted in addition to ft particles, the absorption 
curve (on semilog paper) usually has an initial portion which is 



8 E. Bleuler and W. Zttnti, "On the Absorption Method for the Determina- 



RANGE-ENERGY RELATIONS 



163 



concave toward the origin (see figure VII-8a). The range of the 
conversion electrons cannot be obtained reliably from such a 
graph. If conversion electrons are emitted without accompanying 
13 rays, for example following K capture or in an isomeric transi- 
tion, their range can often be determined better from an extra- 
polation to zero activity on a linear plot, because the absorption 



1.0 



0.1 



0.01 



0.4 



Absorber Thickness 



Absorber Thickness 



(a) (6) 

FIGURE VII-8. Typical absorption curves of conversion electrons, 
(a) Semilog absorption curve of soft conversion electrons superimposed on a 

(3 spectrum. 
(6) Linear absorption curve of pure conversion electrons. 

of monoenergetic electrons turns out to be more nearly linear 
than exponential (see figure VII-86). The only really good method 
for measuring energies of conversion electrons uses the electron 
spectrograph. 

It should be noted that in plotting absorption curves the plotted 
absorber thickness must include not only the added absorbers 
but also the window or wall of the measuring instrument, the air 
between sample and instrument, and any material covering the 
sample. The sample itself should be thin compared to the half- 
thickness value for the radiation. 

Range-energy Relations. Once the range of particles or con- 
version electrons is known, a range-energy relation can be used 
to deduce the maximum energy. (7) Many empirical relations have 

7 Monoenergetic electrons of a given energy and /3 particles of the same max- 
imum energy should, of course, have the same ranges, and at low energies 
this is verified experimentally. Above 0.5 Mev, apparent 0-particle ranges 
drop off a few per cent compared to the observed ranges of corresponding 
monoenergetic electrons; this is probably due to the relatively small number 
of electrons with energies near the upper energy limit in a -ray spectrum. 



164 INTERACTION OF RADIATIONS WITH MATTER CH. VII 

































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BACK-SCATTERING AND SELF-ABSORPTION 165 

been proposed. One given by Feather for energies above 0.6 Mev 
has been most widely used: R = 0.5432? 0.160, where E is 
the maximum ft energy in million electron volts and R the range 
in aluminum in grams per square centimeter. Another relation 
proposed by C. D. Coryell and L. E. Glendenin for energies above 
0.8 Mev appears to give a somewhat better fit at higher energies 
(2-3 Mev): R = OM2E - 0.133. In the lower-energy region 
(below about 0.7 Mev) it is best to use a range-energy curve such 
as the one plotted in figure VII-9. 

Back-scattering and Self-absorption. As already mentioned, 
scattering of electrons, both by nuclei and by electrons, is much 
more pronounced than scattering of heavy particles. A very 
significant fraction of the number of electrons falling on a piece 
of material may be reflected (that is, scattered through more 
than 90) as a result of single and multiple scattering processes. 
The reflected intensity increases with increasing thickness of 
reflector until the thickness equals about one-fifth of the range of 
the electrons; further increase in thickness does not add to the 
reflected intensity. The amount of back-scattering increases also 
with atomic number of the reflector. Table VII-4 lists the rela- 

TABLE VII-4 * 

Amount of 
Reflector Z of Reflector lonization 

None Io 

C G 1.17I 

Al 13 1.30I 

S 16 1.32I 

Cu 29 1.45Io 

Ag 47 1.57I 

AU 79 1.68I 

Pb 82 1.70Io 

* The data for this table were taken from G. Hevesy and F. A. Paneth, 
A Manual of Radioactivity, Oxford University Press, 1938, p. 46. 

tive amounts of ionization produced in an electroscope by a given 
sample of RaE (E m& ^ =1.17 Mev) with various reflectors of effec- 
tively infinite thickness behind it. Numerical values will depend, 
of course, on the particular geometrical arrangement. It is clear 
that in the measurement of radioactive materials care must be 
taken to mount all samples whose activities are to be compared 



166 INTERACTION OF RADIATIONS WITH MATTER CH. VII 

on the same thickness of the same backing material; otherwise, 
cumbersome and uncertain corrections for back-scattering are 
necessary. It is sometimes convenient to increase measured 
0-ray activities by placing heavy reflectors (such as lead) immedi- 
ately behind the samples. 

A very complex effect compounded of absorption and back- 
scattering occurs in all but the thinnest /3-particle sources. This 
so-called self-absorption effect, for a given sample thickness in 
milligrams per square centimeter, appears to decrease with in- 
creasing Z, perhaps about as Z~~^. Some other empirical facts 
about self-absorption will be given in chapter X, section E. 

C. ELECTROMAGNETIC RADIATION 

Processes Responsible for Energy Loss. The specific ionization 
caused by a 7 ray is about Koo of that caused by an electron of 
the same energy, at least for energies greater than 100 kev. The 
practical ranges of y rays are very much greater than those of 
ft particles. The ionization observed for y rays is almost entirely 
secondary in nature as we shall see from a discussion of the three 
processes by which y rays (and X rays) lose their energy. The 
average energy loss per ion pair formed is the same as for ft rays, 
namely, 32.5 ev. 

At low energies (and, therefore, of particular significance for 
characteristic X rays) the most important process is the photo- 
electric effect. In this process the electromagnetic quantum of 
energy hv ejects a bound electron from an atom or molecule and 
imparts to it an energy hv E y where E is the energy with which 
the electron was bound. The quantum of radiation completely 
disappears in this process, and momentum conservation is pos- 
sible only because the remainder of the atom can receive some 
momentum. 

In the energy region of characteristic X rays the probability 
for photoelectric absorption has sharp discontinuities at energies 
equal to the binding energies of the K, L, etc., electrons. For 
hv greater than the X-binding energy the photoelectric absorption 
first falls off rapidly (about as E y ~ 7/i ), then more slowly (even- 
tually as E y ~~ l ) with increasing energy. It is also approximately 
proportional to Z 5 . Except in the heaviest elements photoelec- 
tric absorption is relatively unimportant for energies above 1 Mev. 



PROCESSES RESPONSIBLE FOR ENERGY LOSS 167 

The ionization produced by photoelectrons accounts largely for 
the ionization effect of low-energy photons. The photoelectric 
effect is frequently used to determine 7-ray energies. This is 
accomplished by measurements in an electron spectrograph of 
the energies of photoelectrons ejected from a thin foil, called the 
"radiator" or "converter," placed over the 7-active sample; an 
element of high atomic number such as gold is used as the radiator. 

Instead of giving up its entire energy to a bound electron a 
photon may transfer only a part of its energy to an electron, which 
in this case may be either bound or free; the photon is not only 
degraded in energy but also deflected from its original path. This 
process is called the Compton effect or Compton scattering. The 
relation between energy loss and scattering angle can be derived 
from the conditions for conservation of momentum and energy. 
The Compton scattering per electron is independent of Z, and, 
.therefore, the scattering coefficient per atom is proportional to Z. 
For energies in excess of 0.5 Mev it is also approximately (8) pro- 
portional to E y ~~ l . Thus Compton scattering falls off much more 
slowly with increasing energy than photoelectric absorption, at 
least at moderate energies (up to 1 or 2 Mev), and even in heavy 
elements it is the predominant process in the energy region from 
about 0.6 to 2.5 Mev. Photon energies can be determined from 
the upper energy limits of Compton electrons. For this purpose 
a radiator of relatively low Z, often copper, is used in the electron 
spectrograph, so that the Compton effect predominates over the 
photoelectric effect. 

The third mechanism by which electromagnetic radiation can 
be absorbed is the pair-production process (discussed in chapter 
VI, section B). Pair production cannot occur when E y < 1.02 
Mev. Above this energy the atomic cross section for pair produc- 
tion first increases slowly with increasing energy and above about 
4 Mev becomes nearly proportional to logJEy. It is also pro- 
portional to Z 2 . The energy dependence of pair production is 
satisfactorily predicted by a theory due to H. A. Bethe and 
W. Heitler. At high energies, where pair production is the pre- 
dominant process, 7-ray energies can best be determined by 
measurements of the total energies of positron-electron pairs. 

8 A formula for the scattering coefficient which contains a very complicated 
function of E y has been derived from relativistic quantum mechanics by 
O. Klein and Y. Nishina. 



168 INTERACTION OF RADIATIONS WITH MATTER CH. VII 

Pair production is always followed by annihilation of the posi- 
tron, usually with the simultaneous emission of two 0.51-Mev 
photons. The absorption of quanta by the pair-production 
process is, therefore, always complicated by the appearance of 
this low-energy secondary radiation. 

The atomic cross sections for all three processes discussed in- 
crease with increasing Z, except for the photoelectric effect at very 
low energies. For this reason heavy elements, atom for atom, 
are much more effective absorbers for electromagnetic radiation 
than light elements, and lead is most commonly used as an 
absorber. Because photoelectric effect and Compton effect de- 
crease and pair production increases with increasing energy, the 
total absorption in any one element has a minimum at some 
energy. For lead this minimum absorption, or maximum trans- 
parency, occurs at about 2.7 Mev; for copper at about 10 Mev; 
and for aluminum at about 22 Mev. 

Determination of Photon Energies by Absorption. The only 
mechanism for absorption of quanta which gives rise to a true 
exponential absorption is the photoelectric effect. The produc- 
tion of degraded electromagnetic radiation in the Compton- 



. Aluminum absorber to cut out 
/3 particles and secondary electrons 



Ior2 (say Igpercm2) 

inches 




- Lead absorbers 

Active source 

FIGURE VII-10. Recommended arrangement for 7-ray absorption measure- 
ments. 

scattering and pair-production processes tends to distort the 
exponential absorption. However, if an appropriate experi- 
mental arrangement is used, these distortions can be minimized 
to such an extent that exponential absorption curves can be 
obtained in practice. One such arrangement is shown schemati- 
cally in figure VII-10. Active source and absorbers are as far as 
practicable from the detector to prevent most of the scattered 
quanta and secondary electrons from falling on the detector. The 
additional absorber of low Z near the detector stops a large frac- 



DETERMINATION OF PHOTON ENERGIES BY ABSORPTION 169 



tion of the secondary electrons (as well as any particles emitted 
by the source which otherwise might enter the detector when 
little or no lead absorber is used). In an ideal arrangement, source 



18 



16 



14 



12 



3 
^jf* 



i 



t 



0.5 



1.0 



1.5 2.0 

Energy (Mev) 



2.5 



3.0 



35 



FIGURE VII-11. Half-thickness values in lead and aluminum for photons of 

various energies. (Reproduced from a paper by L. E. Glendenin, Nucleonics 

2, no. 1, 19 (1948), by permission of the McGraw-Hill Publishing Co.) 

and detector would be far apart, with the lead absorbers midway 
between them. 

Absorption curves for electromagnetic radiation are plotted 
on semilog paper. For a single energy a line results which is 
straight over a factor of 10 or 20 in intensity if the afore-mentioned 
experimental arrangement is used. When two components dif- 
fering by at least a factor of two in energy are present, the absorp- 
tion curve can often be resolved into two straight lines, in the 



170 INTERACTION OF RADIATIONS WITH MATTER CH. VII 



same manner as a decay curve is resolved. Resolution into more 
than two components with any precision is generally not possible. 
For an exponential absorption law the intensity Id measured 
through an absorber thickness d is given by Id = Io^~^ where 
I is the intensity without absorber and /* is called the absorption 
coefficient. The half- thickness d\^ is defined as the thickness 
which makes Id = %!$', dy 2 = 0.693//Z. Absorber thicknesses are 

1000 



100 



10 



0.1 



1.0 



1000 



10,000 



10 100 

Half-thickness in Al (mg/cm 2 ) 

FIGURE VII-12. Half-thickness values in aluminum for low-energy photons. 

(Reproduced from a paper by L. E. Glendenin, Nucleonics 2, no. 1, 21 (1948), 

by permission of the McGraw-Hill Publishing Co.) 

more frequently given in terms of surface density (pc?, expressed 
in grams per square centimeter). Then Id = I e~^ /p)pd ' f n/p is 
called the mass absorption coefficient. In the energy region 
where the Compton effect predominates the mass absorption 
coefficient varies only slowly with Z (as in /3-ray absorption), but 
in the low- and high-energy ranges it increases rapidly with in- 
creasing Z. This dependence of the absorption on Z is sometimes 
used to distinguish low-energy X rays from electrons. 

The energy of y or X rays is usually deduced from measured 
half-thickness values. Curves of half-thickness versus energy 
are reproduced in figures VII-11 and VII-12 for different energy 
regions and for lead and aluminum absorbers. It should be 
remembered that a given half-thickness may correspond to two 

different energies henAllSA nf t.ViA minimum in fh^ nnrvfk nf aVvanrn- 



BIOLOGICAL RADIATION UNITS 17] 

tion versus energy. For example, a half-thickness of 13.5 g pe] 
cm 2 in lead may correspond either to an energy of 1.5 Mev or t( 
an energy of about 5.5 Mev. One can always eliminate this 
ambiguity by taking absorption curves in two different materials 



D. NEUTRONS 



Becausp neutrons carry no charge, their interaction with elec- 
trons is exceedingly small, and primary ionization by neutronsls 
"a^comjpletely negligible effect^ The interaction of neutrons with 
to nuclear effects; these include elastic and 



inelastic scattering and nuclear reactions sunh M fy r y\ (n t p). 
^Torjj fa. 2n) fL _aji(j fission^ All these subjects have been discussed 
in chapter III, and here we shall merely indicate how each of 
these types of interaction may be applied to the detection and 
measurement of neutrons. 

The recoil protons produced by fast neutrons in hydrogenous 
materiaiare olten used for the detection of such neutrons. About' 
7 'protons' leave a thick paraffin layer per 10 4 incident neutrons 
of 1 Mev energy , and for other energies the ratio of JPotonsto 
neutrons is roughly proportional to neutron energy The energy 
ffi ihjTTastest recoil protons equals the neutron energy. 

The ionization produced by protons or a particles created iq 
n, p or n, a reactions can also be used for neutron detection! 
lomzatioh chambers or proportional counters (described in chap- 
ter VIII) may be lined with boron or filled with gaseous BF 3 , 
and the particles from the B 10 (n, a) Li 7 reaction may be detected. 
The separated isotope B 10 is particularly effective. Fission frag- 
ments may be detected in an ionization chamber lined with fis- 
sionable material and exposed to a neutron source. 

Neutronrcapture reactions leading to radioactive yroducts are 
frequently used for detection of neutrons hy rneagff of the* inrhipgd 
activity.^ This technique is especially useful in the resonance 
region, where the number of neutrons of a particular energy cai^ 
often be determined by the amount of resonance capture observed, 

E. BIOLOGICAL RADIATION UNITS 

The subject of interaction of radiations with matter is of great 
importance in the study of the biological effects of radiation. 
These effects are generally assumed to be almost entirely due to 



172 INTERACTION OF RADIATIONS WITH MATTER CH. VII 

ionization processes. Actually the disruption of molecules by 
recoiling atoms is undoubtedly also a factor, especially in the case 
of neutron irradiation, but the effects of such processes are not 
easily isolated for study. 

In determining radiation effects on living organisms, whether 
from external radiation or from ingested or inhaled radioactive 
material, one has to take into consideration not only the total 
dosages of ionization produced in the organism but also such 
factors as the density of the ionization, the dosage rate, the local- 
ization of the effect, and the rates of administration and elimina- 
tion of radioactive material. 

The units which are used for biological radiation dosage are 
derived from the roentgen unit. One roentgen unit, or r unit, is 
"that quantity of X or 7 radiation such that the associated cor- 
puscular emission per 0.001293 g (9) of air produces, in air, ions 
carrying 1 esu of quantity of electricity of either sign." This 
means that 1 r produces 1.61 X 10 12 ion pairs per g of air which 
corresponds to the absorption of 83.8 ergs of energy per g of air. 

The r is a unit of the total quantity of ionization produced by 
7 or X rays, and dosage rates for these radiations are therefore 
expressed in terms of roentgens per unit time. The maximum 
allowable daily dose for humans exposed to X or 7 radiation is 
usually taken (in the United States) as 0.1 r, or 100 mr. (It is 
believed by some that this value should be lowered to 50 mr.) 

Because of its definition, the r unit should not be used for radia- 
tions other than X or 7 rays. Another unit, the roentgen-equiva- 
lent-physical, rep, has therefore been proposed to express ioniza- 
tion in tissues caused by other radiations (electrons, protons, 
a particles, neutrons). One roentgen-equivalent-physical is the 
quantity of ionization produced when 83 ergs are dissipated by 
the radiation per gram of tissue. 

The same amount of energy dissipation per gram of tissue may 
cause different amounts of biological damage when brought about 
by different radiations. For this reason still another unit, the 
roentgen-equivalent-man, rem, has been introduced. One rem 
unit corresponds to an energy dissipation in tissue which is biolog- 
ically equivalent in man to 1 r of 7 or X rays. For example, since 
the secondary ionization due to recoil protons from fast neutrons 
has been found to be about 5 times as effective biologically as the 

9 This is the weight of 1 cc of dry air at 0C and 760 mm pressure. 



EXERCISES 173 

same quantity of ionization due to y rays, for fast neutrons 
1 rem = 83/5 ergs per g of tissue = 0.2 rep. Therefore, the 
maximum allowable daily dose is 0.02 rep for fast neutrons. 
Similarly the maximum allowable daily doses for other types of 
radiation are estimated; these are usually taken as 0.1 rep for ft 
radiation, 0.01 rep for a radiation, 0.02 rep for protons, and 0.05 
rep for thermal neutrons. 



EXERCISES 

1. Show that the maximum velocity an electron can receive in an 
impact with an a. particle of velocity v is approximately 2v. 

2. Estimate the ranges in air of (a) 10-Mev H 3 ions, (b) doubly charged 
10-Mev He 3 ions. Answer: (a) 51.3 cm. 

3. What is the velocity of a 20-Mev a particle (a) in a nonrelativistic 
approximation and (b) calculated with the relativistic correction? 

4. An absorption curve of a sample emitting /3 and y rays was taken, 
using a Lauritsen electroscope, with aluminum absorbers. The data 
obtained were: 

Absorber Thickness Activity 

(g/cm 2 ) (diviaions/min) 

5.8 

0.070 3.5 

0.130 2.2 

0.200 1.3 

0.300 0.60 

0.400 0.28 

0.500 0.12 

0.600 0.11 

0.700 0.11 

0.800 0.10 

1.00 0.10 

2.00 0.092 

4.00 0.080 

7.00 0.065 

10.00 0.053 

14.00 0.040 

(a) Find the maximum energy of the ft spectrum (in million electron 
volts). 

(b) Find the energy of the y ray. 

(c) What would be the absorption coefficient of that y ray in lead? 



174 INTERACTION OF RADIATIONS WITH MATTER CH. VII 

5. What are the approximate initial charges of the following fission 
fragments: (a) Kr 97 of 100-Mev initial energy, (6) Xe 140 of 70-Mev initial 
energy? Answer: (a) 26. 

6. What are the mean and the extrapolated ranges of a particles from 
Th 232 in 15C air? Estimate their mean ranges (in milligrams per square 
centimeter) in argon, uranium hexafluoride, and gold. 

Answer: R in argon = 5.5 mg/cm 2 by Bragg's rule or 4.8 mg/cm 2 
by equation VI I- 1. 

7. In a certain measuring arrangement the /3 rays of 13.7-day Cs 136 
are absorbed as follows (7-ray background has been subtracted). 

Absorber Thickness Relative 

(mg Al/cm 2 ) Intensity 

100 

12 47 

27 17 

41 7.3 

53 2.7 

72 0.30 

85 0.037 

For comparison the absorption of P 82 ft rays is measured with the same ar- 
rangement; the results are 

Absorber Thickness Relative 

(mg Al/cm 2 ) Intensity 

250 

160 100 

245 50 

360 20 
420 10 

480 5.0 

530 2.30 

580 0.95 

620 0.45 

680 0.15 

725 0.05 

780 (range) 0.0 

Determine the maximum ]8 energy of 13.7-day Cs 136 by means of a 
Feather plot, using the P 32 as a standard. 

8. At 1.00 meter from 1.00 g radium (in equilibrium with its decay 
products and enclosed in 0.5 mm of platinum) the 7-ray dosage rate is 
0.84 r per hr. What is the minimum safe working distance from a 1 mg 
radium source for an 8-hr day? Answer: 26 cm. 



REFERENCES 175 

9. Estimate the linear absorption coefficient in air (at 15C and 760 
mm pressure) for 1-Mev 7 rays. 

10. (a) Estimate the r-dosage rate at a distance of 30 cm from a 
2 X 10 3 rd source of 1.5-Mev 7 rays, (b) What is the minimum thickness 
of lead that must be placed around this source to allow an experimenter 
to work at 30 cm from it for 2 hr every day? 

11. At sea level the cosmic radiation produces about 2 ion pairs per 
sec per cm 3 of air. At higher altitudes the intensity depends on the 
latitude but for much of the United States is about 10 ion pairs sec" 1 cm"" 8 
at 10,000 feet and about 200 ion pairs sec" 1 cm~ 3 at 40,000 feet above sea 
level. Estimate the radiation dosage received per 24 hr in r units (probably 
we should say in rep units) from this source at (a) sea level, (b) 10,000 feet, 
(c) 40,000 feet. 

REFERENCES 

G. HEVESY and F. A. PANETH, A Manual of Radioactivity, Oxford University 
Press, 1938. 

E. RUTHERFORD, J. CHADWICK, and C. D. ELLIS, Radiations from Radioactive 

Substances, Cambridge University Press, 1930. 

F. RASETTI, Elements of Nuclear Physics, New York, Prentice-Hall, 1936. 

M. S. LIVINGSTON and H. A. BETHE, "Nuclear Physics, C. Nuclear Dynamics, 

Experimental," Rev. Mod. Phys. 9, 245 (1937). 
J. KNIPP and E. TELLER, "On the Energy Loss of Heavy Ions," Phys. Rev. 69, 

659 (1941). 
L. E. GLENDENIN," Determination of the Energy of Beta Particles and Photons 

by Absorption," Nucleonics 2 no. 1, 12 (Jan. 1948). 

N. FEATHER, "Further Possibilities for the Absorption Method of Investi- 
gating the Primary /3 Particles from Radioactive Substances," Proc. 

Camb. Phil Soc. 34, 599 (1938). 
R. D. EVANS, "Radioactivity Units and Standards," Nucleonics 1 no. 2, 32 

(Oct. 1947). 
K. Z. MORGAN, "Tolerance Concentrations of Radioactive Substances," 

/. Phys. Colloid Chem. 61, 984 (1947). 



INSTRUMENTS FOR RADIATION DETECTION 
AND MEASUREMENT 

A. METHODS NOT BASED ON ION COLLECTION 

In the preceding chapter we saw that the principal interactions 
of the radioactive radiations with matter result in the production 
of ions with a reduction in energy of the radiation of about 33 ev 
per ion pair formed. All methods for detection of radioactivity 
are based on interactions of the charged particles or electromag- 
netic rays with matter traversed. The uncharged neutron is 
detected only indirectly, through recoil protons (from fast neu- 
trons) or through nuclear transmutations or induced radio- 
activities (from fast or slow neutrons). Neutrinos have no charge 
and do not seem to interact measurably with matter to produce 
either ions or recoil particles, and, therefore, are not detectable 
by any of these methods. (It may be presumed that neutrinos 
should be capable of causing nuclear transmutations, but the 
cross section for the process is estimated from the principle of 
microscopic reversibility to be less than 10 ~ 40 cm 2 being hard 
to emit they must be hard to absorb.) 

Photographic Film. The historical method for the detection of 
radioactivity was the general blackening or fogging of photo- 
graphic negatives, apparent on chemical development in the usual 
way. This method was soon supplanted by ionization measure- 
ments but has reappeared recently in the "film badge" for per- 
sonnel exposure control (see section E) and in the y raying (anal- 
ogous to X raying) of castings and other heavy metal parts for 
hidden flaws. Also, in the radioautograph technique the distri- 
bution of a radioactive tracer (preferably an a or soft-/? emitter) 
is revealed when a thin section, perhaps of biological material, is 
kept in contact with a photographic plate. An improvement 
over this "contact radioautograph" technique may eventually 
be achieved through perfection of an "enlargement" method, in 
which soft ft rays from the specimen are accelerated electro- 
statically and then focused on the photographic plate as in an 



CLOUD CHAMBER 177 

electron microscope; however, the range of useful magnification 
is not likely to be very great. 

Photographic emulsions exposed to densely ionizing radiations 
such as a. rays, protons, and mesons, on development show black- 
ened grains along the path of each particle; since the range of 
such rays is small, these tracks are observed under a microscope. 
The direction and range of each particle are indicated, and nuclear 
transmutations may be studied. The number of developed grains 
along a track is smaller by several orders of magnitude than the 
number of ion pairs produced. The technique is particularly 
useful for the recording of very rare events, such as are of interest 
in cosmic-ray studies. 

Cloud Chamber. A pictorial representation of the paths of ion- 
izing particles similar to the photographic track but capable of 
much finer detail is given by the cloud chamber (Wilson chamber). 
In this instrument the particle track through a gas is made visible 
by the condensation of water droplets on the ions produced. To 
accomplish this, an enclosed gas saturated with vapor (water, 
alcohol, and the like) is suddenly cooled by adiabatic expansion 
to produce supersaturation. Ordinarily a fog would be formed, 
but, if conditions are right and the gas is free of dust, scattered 
ions, and so on, the supersaturation is maintained except for local 
condensation along the track where the ions serve as condensa- 
tion centers. The piston or diaphragm causing the expansion is 
operated in a cyclic way, and a small electrostatic gradient is 
provided to sweep out ions between expansions. There is usually 
an arrangement of lights, camera, and mirrors to make stereo- 
scopic photographs of the fresh tracks at each expansion. 

The a tracks appear as straight lines of dense fog droplets, with 
thousands of droplets per centimeter. The ft tracks are much 
less dense, with discrete droplets visible, several per centimeter 
along the path. In both cases 5 rays are visible, and scattering 
and straggling may be studied. Electron energies may be deter- 
mined from track curvature in a magnetic field, and positrons are 
distinguished from negative electrons by the curvature if beginning 
and end of the tracks can be recognized. Cosmic-ray experts 
have learned to tell much about a particle's charge, mass, and 
energy from the relation of magnetic curvature and track density. 
Gamma rays in the cloud chamber produce scattered droplets 
and 5-ray tracks, with no obvious indication of a particular photon 



178 INSTRUMENTS FOR RADIATION DETECTION CH. VIII 

path; the Compton-recoil electrons and photoelectric-conversion 
electrons may be studied. 

Scintillation Counting. When a particles strike a prepared fluo- 
rescent screen of zinc sulfide, discrete flashes of light may be seen 
by the dark-accustomed eye. The counting of a. rays by this 
scintillation method was of great value in the early studies of 
radioactivity. Although it is no longer used in this way, there 
is a modern adaptation of scintillation counting for & and 7 rays. 
The rays produce light in a naphthalene or anthracene crystal 
(or polycrystalline mass); the light can produce photoelectrons 
from the first, photosensitive electrode of a photomultiplier tube 
such as the RCA 931 A, 1P21, or 1P28, and the output pulse may 
be recorded. Other fluorescent materials recommended for this 
purpose include calcium tungstate and corundum (synthetic 
sapphire). Insofar as the substance is translucent, the effective 
detector thickness may be quite great (more than 1 cm in models 
already working); consequently, high 7-ray efficiencies are pos- 
sible in principle and have been indicated in current reports. 
Also, because the time for light emission is probably very short, 
the instrument should be capable of high counting rates and short 
coincidence resolving times. A special type of electron multiplier 
tube (not yet commercially available) has been used for counting 
in a more direct way also: the rays are caused to fall on its first 
electrode, and the secondary electrons produced there initiate 
pulses. 

Other Methods. A few other detection methods not based on 
ion collection have historical and occasional current interest. 
The heating effect of the radiations can be measured with pre- 
cision for very active preparations to the extent that the radia- 
tions are stopped in the calorimeter; with a emitters this condi- 
tion is readily met. One of the more curious (and certainly not 
practical) detection methods described involves a calibration of 
the error in weighing when a strong radium sample (presumably 
many curies) is placed under one pan of an analytical balance. 
(This should not be confused with the quite ordinary use of much 
smaller 7-active sources near balance cases to prevent the accumu- 
lation of disturbing static electric charges.) The partial precipi- 
tation of certain colloids by radioactive rays has been used as a 
simple indicator of radiation dosage. 



THE IONIZATION CHAMBER 



179 



B. SATURATION ION CURRENT COLLECTION 

The lonization Chamber; Relation of Current to Voltage and 
lonization Intensity. Many common radiation detectors make use 
of the electric conductivity of a gas resulting from the ionization 
produced in it. This conductivity is somewhat analogous to the 
electric conductivity of solutions caused by the presence of elec- 
trolyte ions. In gas conduction as produced by radiation the ion 
current first increases with applied voltage (as in the electrolyte 



I i 

Current-/) 



/^- 

Gas-filled 
ionization 
chamber 




.=_ (Battery) 



(Zero potential) 



(with radiation 
source) 

-Saturation current (number of ion pairs 
per sec x 1.60xlO" 19 a 



1 



_L 



1 



1 



Applied voltage (V] 
FIGURE VIII-1. lonization current. 

case); with increasing voltage the current eventually reaches a 
constant value which is a direct measure of the rate of production 
of charged ions in the gas volume. This constant value of the 
current is called the saturation current. A schematic representa- 
tion of a gas volume and collecting electrodes, with potential dif- 
ference y and meter to measure the ionization current I, is shown 
in figure VIII-1, along with a plot of I vs. V that might be ob- 
tained. 

In the region of applied voltage below that necessary for the 
saturation current, recombination of positive and negative ions 
reduces the current collected. As the applied voltage is increased 
beyond the upper limit for saturation collection, the current 



180 INSTRUMENTS FOR RADIATION DETECTION CH. VIII 

increases again, and, finally, the gap breaks down into a glowing 
discharge or arc, with a very sharp rise in the current. In the 
measurement of gas ionization it is obviously of some advantage 
to measure the saturation current: the current is easily interpreted 
in terms of the rate of gas ionization, and the measured current 
does not depend critically on the applied voltage or other like 
factors. The range of voltage over which the saturation current 
is obtained depends on the geometry of the electrodes and their 
spacing, the nature and pressure of the gas, and the general and 
local density and spatial distribution of the ionization produced 
in the gas. In air, for many practical cases, this region may be 
taken to extend from ~10 2 to ~10 4 volts per cm of distance 
between the electrodes. 

We may classify detection systems (of the ion-collection type) 
according to whether saturation collection is employed or whether 
the multiplicative collection region is used. In the multiplicative 
region, where V is above the maximum value for saturation collec- 
tion, the additional current is due to secondary ionization proc- 
esses which result from the high velocities reached by the ions 
(particularly electrons) moving in the high field gradient. The 
use of this current amplification makes multiplicative collection 
methods inherently sensitive but unfortunately also inherently 
critical to many experimental variables. 

Lauritsen-type Electroscope. We will call the gas-filled elec- 
trode systems designed for saturation collection ionization cham- 
bers. Saturation current instruments consist of the ionization 
chamber, in which ions produced are collected with as little recom- 
bination or multiplication as possible, and an electric system for 
measuring the very small currents obtained. The essential differ- 
ences between the various instruments of this sort are in the 
nature of the current-measuring systems. In one common and 
relatively inexpensive instrument, the Lauritsen-type electroscope, 
a sensitive quartz-fiber electrometer measures the change in volt- 
age produced on the fiber and its support by collection of the 
ionization charge. An external battery or rectifier is used to 
provide the initial voltage V (by means of a temporary connection 
to the fiber support) ; then, the fiber position is observed through 
a small telescope to measure AF as a function of time. For a 
collected charge g, the resulting AF = g/C, where C is the approx- 
imately constant capacitance of the fiber and electrode system. 



D-C AMPLIFIERS 



181 



(The order of magnitude of C may be guessed from the dimensions; 
the fiber and support arrangement are about 1 cm long, and C is 
of the order of 1 cm, which is the electrostatic unit of capacitance 
equal to about 10~ 12 farad or ^1 /zjuf.) The Lauritsen electroscope 
is simple and rugged and can be used to detect a activity of 
about 2000 disintegrations per min (so arranged that about 50 
per cent of the rays enter the ionization chamber). With the same 
arrangement samples up to about 1000 times this activity may be 
measured accurately. To set corresponding limits of usefulness 
for 7-ray measurements, account must be taken of the roughly 
100-fold smaller specific ionization produced by these rays. Care 
must be used in measurement because the instrument is not 
strictly linear over different portions of the eyepiece scale and 
may be somewhat erratic in behavior when first charged. Most 
workers find it best to use only one chosen portion of the scale 
and to have the fiber charged for several hours before use. 

D-c Amplifiers. Instruments of another type use ionization 
chambers with electronic d-c amplifiers. The ionization current 
I is caused to flow through a very high resistance R, and the volt- 




wwvw ilijili 



FIGURE VIII-2. Ionization chamber with balanced d-c amplifier. 

age developed, V = IR, is applied to the control grid of a vacuum 
tube and measured in terms of the plate current of the tube by a 
galvanometer G. (See the schematic circuit in figure VIII-2.) 
To measure the smallest currents the vacuum tube must be chosen 
fbr low inherent grid current. A common amplifier has been one 
using the General Electric FP54 or Western Electric D-96475 



182 INSTRUMENTS FOR RADIATION DETECTION CH. VIII 

vacuum tube; the Victoreen VX-41A is also used, especially in 
portable equipment. High stability of the circuit, particularly 
against variations in the battery voltage, is essential; balanced 
circuits are in use in which a single battery supplies plate, filament, 
and bias voltages, and through proper adjustment of the voltage- 
dividing networks the plate current is made insensitive to the 
battery voltage over a limited range, in the balanced condition. 
An instrument of this type with II = 10 11 ohms is easily sensitive 
to 1000 /5 disintegrations per min (with the same assumption as 
in the last paragraph that about 50 per cent of the rays enter the 
ionization chamber). With R = oo the time rate of drift of the 
plate current may be measured; by this means the sensitivity 
may be extended to about 200 disintegrations per min. Ordinarily 
a switch is provided so that R may be selected from values such 
as oo, 10 11 , 10 10 , 10 9 , and 10 8 ohms. On this last position a sample 
of ~20,000,000 disintegrations per min may be measured in the 
50-per cent geometry. 

The vacuum tube, grid resistors R, selector switch, and so on, 
for the FP54-type instrument are enclosed in an evacuated or at 
least sealed and desiccated can to minimize stray leakage currents, 
and insulators of the highest quality are needed in the grid circuit. 
The ionization chamber may be connected through a screw-in 
fitting so that different chambers can be used as desired. A cham- 
ber containing air at atmospheric pressure and closed with an 
exceedingly thin aluminum leaf window is common for particles 
of low penetration; 7 and X rays are detected with considerably 
improved sensitivity in a closed chamber filled to a pressure of 
2 or 3 atmospheres with Freon (a chlorofluoromethane) or methyl 
bromide. 

As an example we might estimate the ionization current I and 
the voltage drop IR for an ionization chamber under these assump- 
tions: R = 10 11 ohms; the sample is an emitter of moderately 
energetic ft rays with 1000 disintegrations per min; the geometry is 
such that 50 per cent of the ft particles enter and spend an average 
8-cm path length in the effective volume of an air-filled cham- 
ber. The number of ion pairs to be expected (1) is about 
1000 X 0.50 X 80 X 10 = 4 X 10 5 per min, or 6.7 X 10 3 per 
sec. The current I will be the corresponding charge per second: 

1 The estimate of 10 ion pairs per mm over the 8Q-mm path is taken from 
the information on -ray ionization in chapter VII. 



LINEAR PULSE AMPLIFIER 183 

I = 6.7 X 10 3 X 1.6 X 10- 19 

= 1.1 X 10~~ 15 coulomb per sec, or ampere. 
IR = 1.1 X 10~ 15 X 10 11 

= 1.1 X 10- 4 volt, or 0.11 mv. 

If the use of the number of ion pairs rather than the total number 
of ions in this calculation is not entirely clear, remember that 
only half of the ions those with the proper sign of charge are 
collected at either electrode. (Also, although the current in the 
gas space consists of moving ions of both signs, for each ion pair 
formed neither ion traverses the whole path length between the 
electrodes, but rather the sum of the two ion paths is that 
distance.) 

Vibrating-reed Electrometer. Even though special tubes and 
balanced circuits may be used with ionization chambers, d-c 
amplifiers are more susceptible to disturbance and drift and more 
difficult to arrange with several successive stages of amplification 
than those designed for amplifying alternating currents. A recent 
development is the use of a continuously vibrating reed which, 
through its oscillating electrostatic capacitance to a fixed elec- 
trode, converts the IR voltage to an approximately sinusoidal 
alternating potential; the a-c signal is then amplified in a highly 
stable audio-frequency amplifier. This instrument has a sensi- 
tivity comparable to the FP54 unit and can be unusually free 
from troublesome zero drift and external disturbances. The out- 
put signal level may be used to operate a continuously recording 
milliammeter to give a permanent record of the ionization-chamber 
current. The vibrating reeds are constructed with great care, 
and the entire set of equipment is rather expensive. 

Linear Pulse Amplifier. An ionization chamber with directly 
connected a-c amplifier, as in the schematic representation of 
figure VIII-3, will, of course, give no response to any steady 
ionization current. A short burst of intense ionization, such as 
results from the passage of an a. particle through the chamber, 
will give a sudden change of voltage on the first grid; this grid 
voltage will return to normal in a time of the order of RC, where 
C represents the distributed capacitance of the grid and collecting- 



184 



INSTRUMENTS FOR RADIATION DETECTION CH. VIII 



electrode system and R is the effective resistance to ground. (2) 
With sufficient amplification a large pulse will appear at the 
amplifier output terminal; the shape in time of this voltage pulse 
will depend on several factors, including the value of RC and the 
frequency-response characteristics of the amplifier. It is ordi- 
narily desirable to have the height of the output pulse propor- 
tional to the amount of ionization produced by the particle in the 



-HOQO volts 
ionization 
chamber 

about 
4 mm 



- Ion pairs along a-particle track; 
Q coulombs of each sign. 




(distributed 
capacitance 
to ground=C)< 10 ohms 



+45 volts +300 volts +300 volts 

FIGURE VIII-3. Schematic representation of ionization chamber with linear 

pulse amplifier. 

chamber; thus, the name linear amplifier or linear pulse amplifier 
is often applied to this instrument. 

Since the instrument is used for counting single a particles we 
may estimate the voltage amplification factor (gain) needed. A 
fast a particle traveling 1 cm in the chamber would give in 
air about 25,000 ion pairs, and the collected ion charge, 
q = ~25,000 X 1.6 X 10" 19 = ~4 X 10"~ 15 coulomb; and guess- 
ing C = ~10 /zjuf we have then V = q/C = ~4 X 10"" 4 volt. If 
an output pulse of 100 volts is wanted (convenient for oscillo- 
graphic observation and photographic recording) the required 
gain is 100/(4 X 10" 4 ) = 2.5 X 10 5 . Four amplifier stages 
might be used, each with gain of roughly 22. 

2 The charge in a capacitor of capacitance C shortr-circuited with a high 
resistance R will be dissipated exponentially; the half-time for the process is 
given by 0.693 RC; RC is known as the time constant of the circuit and is the 
time required for the charge to be reduced to l/e of its value. 



LINEAR PULSE AMPLIFIER 185 

A practical ionization chamber may have a background rate of 
the order of 0.1 to 1 a per min; the lower limit of sample strength 
easily detectable we may take as ~1 a. disintegration per min 
(with 50 per cent geometry). With appropriate amplifier and 
recording equipment the maximum usable rate is limited by the 
duration (~RC) of the voltage pulse, because, if the average rate 
of arrival of pulses is such that there is an appreciable chance of 
one following another within the time RC, appreciable counting 
error results. With R = 10 8 ohms, RC = ~10~~ 3 sec, and a 
few thousand counts per minute would be the useful upper limit. 
Of course R is easily made smaller, but the full voltage q/C is 
achieved only if RC is long compared to the time of collection of 
ions in the chamber. The velocity v of ions in air under a voltage 
gradient E volts per centimeter is about (perhaps 1.5 times) E 
centimeters per second; with 1000 volts applied to a 0.4 cm cham- 
ber, v = ^4000 cm per sec, and the ion collection time is 
^0.4/4000 = ~10~* sec. In practice RC is usually made some- 
what longer than this time; to waste much of the voltage pulse 
is not advisable because with higher amplifier gains much trouble 
would be caused by tube "noise" and by microphonic effects 
(sensitivity of the chamber and amplifier to vibration). However, 
in a closed ionization chamber filled with pure argon or nitrogen 
the negative ions will be principally free electrons, which may be 
collected very much faster than heavy gas ions; then with an 
appropriate amplifier much higher counting rates may be used. 

If an ionization chamber is large enough to contain the entire 
range of the most energetic a rays, then the ionization produced 
will be an accurate measure of the a-particle energy, which is 
characteristic of the particular a emitter. Instruments which 
record in many separate channels the counting rates of a particles 
of various energies have been constructed, and they are very 
useful for the analysis of complex mixtures of the heavy a-active 
nuclides. Linear amplifiers may be used to count fissions; because 
fission fragments have roughly ten times the specific ionization 
of a particles they are easily distinguished. 

It has recently been shown that certain crystals, including 
selected diamonds, and AgCl crystals at low temperatures, when 
fitted with electrodes and connected to a source of high voltage 
and to a fast amplifier, give pulses under the action of radioactive 
rays. The crystal presumably acts as an ionization chamber, 



186 INSTRUMENTS FOR RADIATION DETECTION CH. VIII 

with electrons and vacant electron sites moving through the crys- 
tal in the potential gradient. That this detector can respond to 
ft and 7 rays in addition to a. particles, is due to the much greater 
specific ionization produced in a solid crystal than in a gas. The 
special requirements on the nature of the crystal may be related 
to the random production of free electrons by thermal processes 
and to the existence of many electron traps in ordinary imperfect 
crystals. This crystal counting technique may be developed to 
have a favorable sensitivity for y rays; it is also of interest in 
coincidence counting because of its short characteristic time 
delay. 

C. MULTIPLICATIVE ION COLLECTION 

In the preceding section we discussed detection techniques 
utilizing saturation collection of ions in ionization chambers. The 
arrangements of electrodes for the multiplicative collection of 
ions as described in the following we will call counters. Usually 
the currents collected in counters, even from as little as one initial 
ion pair, may be large enough so that no very sensitive amplifiers 
or extremely low-capacitance or extremely high-resistance circuits 
are required. Practical difficulties are found rather in the con- 
struction and operation of the counters themselves. To obtain 
multiplicative collection of the type desired one might at first 
think of simply increasing the voltage applied to an ordinary 
parallel-plate ionization chamber. This is ordinarily (3) not prac- 
tical, for several reasons which are suggested by the following 
discussion. 

Voltage Gradients and Electrode Shapes. Figure VIII-4 shows 
the electrostatic lines of force between parallel-plate electrodes. 
The density of the lines of force is a measure of the voltage gradient 
E in any region. The voltage gradient is the same everywhere 
between the plates, except for effects near the edge, and is given 
by the applied voltage difference AF divided by the plate separa- 
tion. Lines of force converge on a curved electrode such as a 
sphere or wire or point, and indeed unless the parallel plates are 

8 This qualification is made because some success has been obtained with 
parallel-plate counters. They are poor counters in most respects but do seem 
to have shorter time lags between ionization and counting action, and this 
characteristic can be very valuable in some coincidence counting work. 



VOLTAGE GRADIENTS AND ELECTRODE SHAPES 



187 



perfectly smooth high local gradients will exist at surface irregu- 
larities. 

In counters one electrode is usually a cylinder, the other an 
axial wire. Figure VIII-5 shows a cross-sectional view, with the 
wire radius exaggerated; the lines of force are sketched in. It is 
readily seen that the density of these lines is inversely proportional 
to the radial distance r; that is, E = k/r. Now E is by definition 




FIGURE VIII-4. Electrostatic lines 
of force between parallel-plate elec- 
trodes. 



FIGURE VIII-5, Electro- 
static lines of force be- 
tween coaxial cylindrical 
electrodes. 



dV/dr y and we may represent the voltage difference between the 
electrodes of radii a and 6: 

-6 /%& fib h /fc\ 

dV = I Edr = I -dr = fcln(-)- 
a Ja Ja T \d/ 

In a practical case we might have 6=1 cm, a = 4 X 10~" 3 cm, 
AV = 1000 volts. Then: 



1000 



= k In ( J - 5.5/c; * = 180. 

\4 X 10~ 3 / 



The voltage gradients at wall and wire are 
E b = 180 volts per cm; 
180 



4 X 10~ 3 



= 4.5 X 10 4 volts per cm. 



The gradient at the wire and for a small space around it is above 
the maximum value for saturation collection (say ~10 3 volts 



188 INSTRUMENTS FOR RADIATION DETECTION CH. VIII 

per cm in a practical counter gas). The voltage difference AF 
is always applied with the wall (cathode) negative with respect 
to the wire (anode) ; in this way free electrons and heavy negative 
ions move to the wire. 

The Geiger-Muller (G-M) Counter. If an electrode system 
like that just described is filled with a suitable gas such as 90 
per cent argon and 10 per cent ethyl alcohol (total pressure about 
10 cm) and connected to a high-gain amplifier, and the pulses 
produced are studied as a function of the applied voltage with 
different types of ionizing particles, the following voltage regions 
are observed: 

1. At relatively low voltages (of the order of 100 volts) there 
is no multiplication of the ionization current; the system operates 
as an ordinary ionization chamber, and only the pulses produced 
by a particles are seen and these only at very high amplification. 

2. At moderate voltages (several hundred volts) there is ampli- 
fication (10- to 100-fold or more) of the pulse heights; a-particle 
pulses are seen with moderate external amplification, and even 
ft particles are detectable with high external amplification. The 
pulse height at fixed voltage is approximately proportional to the 
amount of ionization caused by the particle ; when operated in this 
region the device is known as a proportional counter. (At least 
one commercial instrument uses this principle to count /3 
particles.) 

3. As the voltage is increased further (to about 1000 volts) the 
pulse heights increase, and their dependence on the initial ioniza- 
tion intensity disappears; this is the beginning of the Geiger 
counting region, where a single ion pair or the intense ionization 
from an a particle produces the same large pulse (perhaps about 
10 volts on the counter wire and requiring little or no amplifica- 
tion for observation or recording). 

To investigate the extent of the Geiger counting region we often 
arrange the counter with a fixed source of radiation and determine 
the counting rate produced as a function of the applied voltage. 
Figure VIII-6 shows this curve for a good counter. The region 
EC in which the rate is very nearly independent of the voltage is 
the "plateau" region; its length may be as much as several hun- 
dred volts; the voltage is always set in this region for counting. 
At voltages below B pulses exist, but are not uniform in size, and 
only some trip the recording circuits. We call A the starting 



THE GEIGER-MttLLER (G-M) COUNTER 189 

voltage, where the largest of these pulses just begin to be counted. 
The voltage difference between A and B obviously will depend 
on the circuit characteristics; if it exceeds 30 or 40 volts for a 
counter connected directly to the amplifier, a reduction of the 
stray capacitance to ground of the anode wire connections or an 
increase in the amplifier gain will probably lengthen the plateau 
(move B nearer to A). 



Starting voltage s 

J I I 



Plateau 



JL 



A 

Applied Voltage >- 

FIGURE VIII-6. Plateau curve for a good Geiger-Muller counter. 

To understand the rise in counting rate beyond C we must 
consider (very briefly) some of the things that happen when the 
counting action occurs: 

(a) The negative ion of the original ion pair moves toward the 
wire, traveling most or all the way as a free electron and thus at 
high speed, and it very quickly reaches the region of pronounced 
multiplicative processes. 

(b) The intense region of secondary, tertiary, etc., ions and elec- 
trons formed in the high field gradient immediately around the 
wire spreads along the wire over all its effective length; spreading 
occurs at least partly through the photoelectric ionization of the 
gas by photons of high absorption coefficient (short mean path). 
There may also be some effect of photoelectrons from the cathode 
wall. 

(c) The negative ions formed, mostly free electrons, very quickly 
reach the wire, and the intensely ionized region is left as a sheath 
of positive ions surrounding the wire. The effect of this positive 
charge is to reduce the voltage gradient below the value necessary 
for ion multiplication. All this has occurred in less than about 



190 INSTRUMENTS FOR RADIATION DETECTION CH. VIII 

0.5 microseconds (0.5 ^ sec), and now the counting action is com- 
plete except that the counter is left insensitive and must recover 
before another event can be counted. 

(d) Recovery is effected through migration of the positive gas 
ions away from the wire. From the rough formula already given 
for ion mobilities, correcting for a roughly linear dependence of 
velocity on reciprocal pressure and taking account of the variable 
voltage gradient, we can estimate that migration of an ion from 
wire to wall will require ~200 /* sec, and this is about the dead 
time found experimentally. 

(e) When positive ions reach the cathode secondary electrons 
might be emitted from the surface; this would produce a new 
counter discharge just about 200 /* sec after the first, and quite 
independent of the source of radiation that the counter is intended 
to measure. Double, triple, and other multiple pulses with about 
this time spacing are observed with counters operating above 
the upper voltage limit of the plateau. 

The various recipes for counter construction contain pro- 
visions designed to repress the emission of secondary electrons 
from the cathode. The argon-alcohol filling mixture seems to be 
effective because the positive ions are by electron transfers all 
converted to alcohol ions while moving to the cathode, and the 
polyatomic alcohol ions may dissipate energy by predissociation 
and so reduce enormously the probability of secondary-electron 
emission. Also the alcohol may serve to quench metastable 
states of the argon atoms. It is significant that the alcohol is 
considerably consumed after 10 8 or 10 9 counts and that a poly- 
atomic filling gas, tetramethyl lead, requires no additive such as 
alcohol. The special cathode treatments recommended by some 
are probably related to changes in surface work function. The 
fact that various counter recipes differ widely probably means 
that the nature of the counter discharge action may be quantita- 
tively different in different styles of counter tubes. The references 
at the end of this chapter will supply details on a number of con- 
struction techniques. Our feeling is that, although it is easy to 
make a useful counter, it is difficult to make one that does not 
sacrifice at least one of the following "ideal" features, chosen for 
their general usefulness and with the understanding that any 
given feature may readily be had at the expense of others: 



THE GEIGER-MttLLER (G-M) COUNTER 191 

1. The counter should operate without quench circuit, with a 
series resistor of ~0.25 megohm. 

2. The plateau should be at least 50 volts long, preferably 
longer, and have a slope of not more than about 3 per cent per 
hundred volts. 

3. The counter should be indefinitely stable against aging, 
though it may require refilling after the usual 10 8 counts, or 
thereabouts. 

4. For the measurement of soft ft particles a window not more 
than about 3 mg/cm 2 in thickness and with an area of several 
square centimeters should be provided close to the sensitive region. 

5. The counter should be free of the troublesome hysteresis 
effects, in which counting rates and plateau curves may be dif- 
ferent by a few per cent before and after the counting of active 
samples. 

6. The dead time should not be more than a few hundred micro- 
seconds. 

As suggested in item 4 provision for the effective introduction 
of low-energy ft rays into the closed counter gas volume offers 
some problems. The window specified can be made of mica with- 
out a supporting grid and is thin enough to be usable with serious 
though not prohibitive loss of sensitivity for C 14 and S 36 , but 
not H 3 . For very soft rays, there are several counter modifica- 
tions that may be used : 

1. With helium-plus-alcohol-vapor filling, the counters can be 
made to work at atmospheric pressure and used with very thin 
windows or even no window at all. Other gases can be used 
similarly. 

2. In the screen wall counter the cathode is an open screen, 
and the sample is placed between it and an outer tube which is 
the counter gas container; with the sample on the inner surface 
of a cylindrical holder, it can be moved into counting position 
and moved away for background determination, by gravity, 
without opening the counter seal. 

3. The counter may be made as part of a larger gas space which 
encloses a rotating wheel with positions for assorted samples or 
absorbers, and many measurements can be made before the system 
must be opened. 

4. If the sample can be converted into gaseous form it may be 
introduced directly in the filling mixture; because strange gases 



192 INSTRUMENTS FOR RADIATION DETECTION CH. VIII 

may upset the counter properties this is a tricky technique. 
However, with ionization chambers rather than counters it is a 
more useful method. In both cases considerable precautions are 
necessary to minimize errors caused by adsorbed radioactive gas. 
Counter Quench Circuits. Before self-quenching counters were 
made it was common procedure to avoid spurious counts from 
double, triple, etc., pulses by connecting the counter with a very 
high series resistor (R = 10 8 to 10 9 ohms). See figure VIII-7. 
With the discharge a large negative voltage appears on the anode 
wire and leaks away with a time constant RC = ^lO"" 3 sec = 
^1000 M sec. In this way, the voltage across the tube may be 
kept below the starting voltage until after the positive ion sheath 



Counter tube-^. 




Negative pulse to amplifier grid circuit 
(distributed capacitance to ground =C) 



-1000 volts 

>/?-10 8 tol0 9 ohms 



FIGURE VIII-7. Resistor-quenched counter circuit. With self-quenching 
counter tube the value of R is reduced to about 10 5 to 10 6 ohms. 

is discharged. Use of this resistor quench seriously reduces the 
maximum useful counting rate. A number of electronic quenching 
circuits have been devised that are faster or more effective or both. 
Essentially these circuits use the amplification factor of a vacuum 
tube to provide a more positive and better-timed voltage reduc- 
tion pulse applied to the wire or wall of the counter tube. 

Practical Counting Instruments. In addition to the G-M coun- 
ter tube and the optional quenching circuit, some auxiliary circuits 
are always required. Today excellent complete circuits are avail- 
able from a number of manufacturers. The high-voltage supply 
(~1000 volts) might be a bank of batteries but is usually a com- 
bination of transformer, rectifier, and filter. Regulation of the 
output voltage is essential; a good commercial unit should provide 
a stabilized high voltage which will vary not more than 1 or 2 
volts (~0.1 per cent) for a 10-volt (^10 per cent) change in line 
voltage. In almost all sets scaling circuits are used to reduce the 
rate for easier recording; with few exceptions these employ multi- 



PRACTICAL COUNTING INSTRUMENTS 193 

vibrator (4) scaling pairs, each pair reducing the rate by a factor 
of 2; common scaling factors include 2 3 = 8; 2 6 = 64; 2 8 = 25(3. 
The scaled impulses are recorded mechanically, with electric 
power supplied by a suitable output circuit. Although with a 
fast mechanical recorder a scaling factor of about 8 can be ade- 
quate for most purposes, the present trend is toward higher scaling 
factors with slow mechanical recorders; the costs of register plus 
sealer are about the same in both cases. 

There are some modifications of the basic circuit arrangements 
useful with Geiger counters and other types of counting instru- 
ments. In one design the time required for a fixed number of 
input pulses is automatically measured; this unit has no mechan- 
ical register, but rather a very high scaling factor, and the first 
output pulse turns off the counter and stops an electric timer, 
In other variations of the same principle the time for each output 
pulse is printed by a triggered time clock, or the output pulses 
are recorded by a recording galvanometer on paper tape moving 
at a constant known speed; these modifications are particularly 
useful for determinations of decay curves. In the counting rate 
meter the pulses are integrated electrically, and their average 
rate of arrival (averaged over a suitably long time interval) is 
indicated directly by a meter deflection. 

The limit of sensitivity of a G-M counter is set by the back- 
ground counting rate. Even in a laboratory not contaminated by 
radiochemical work small amounts of activity are present as im- 
purities in construction materials. Also the air contains an appre- 
ciable and variable amount of radon and thoron and their decay 
products. It has been estimated that in free air at the earth's 
surface most of the ionization is from these two causes, with the 
cosmic radiation contributing a smaller part. However, because 
the counter is itself closed, and enclosed in a building, it is not 
accessible to most of the radioactive a, 0, and even 7 radiation, 
and the cosmic-ray effect is the most significant. A counter with 
a radius of 1 cm and length 10 cm may have a background rate 

4 In this application both tubes of the basic multivibrator pair (square 
wave-form oscillator) are biased to prevent oscillation; each incoming pulse 
triggers the pair through a half-cycle; completion of each cycle provides an 
output pulse to trigger the next pair. One of the most dependable circuits is 
that employing direct rather than capacitor coupling, devised by W. A. 
Higinbotham. 



194 INSTRUMENTS FOR RADIATION DETECTION CH. VIII 

of about 50 counts per min; this may be reduced to about 25 
counts per min by the usual lead shield of a few centimeters 
thickness. We may take about 10 ft disintegrations per min as 
the minimum sample strength easily detected by a counter (with 
the 50 per cent geometry estimate as before). Most counter sets 
are not dependable at counting rates greater than that from a 
sample of about 3000 times this activity placed in the same posi- 
tion; however, with rather special precautions some workers use 
counters up to rates higher by another order of magnitude. For 
7 rays samples must have roughly 100-fold higher disintegration 
rates to give comparable counting rates. 

D. AUXILIARY INSTRUMENTS 

Electron Spectrograph ; Coincidence Spectrometer. A number 
of other special instruments, designed more for the study of radia- 
tions than for intensity measurements, are likely to be found in 
laboratories working in many phases of radiochemistry. The 
/3-ray spectrograph, or electron spectrograph, permits a much 
more accurate study of /3-ray spectra than is possible by absorp- 
tion measurements. In the various forms of this instrument the 
ft particles are deflected by a magnetic (or possibly electrostatic) 
field, with some provision for angular focusing, and in this way are 
resolved according to energy. They may be recorded by a photo- 
graphic plate or by a counter; in the latter case the instrument is 
called a /3-ray spectrometer. Internal-conversion electrons are 
best studied in the same instrument, and the energies of 7 rays 
and X rays may be deduced from measured energies of photo- 
electrons (external-conversion electrons). In studies of complex 
ft spectra for the elucidation of disintegration schemes a very 
valuable technique is the coincidence spectrometer, in which a 
spectrum of ft rays coincident with characteristic 7 rays is dis- 
tinguished by coincidence counting. 

Curved-crystal Spectrograph. An instrument for the precision 
measurement of X rays and low-energy 7 rays is the curved-crystal 
spectrograph. This is analogous to an optical spectrograph of 
the grating type, with the atomic planes of a bent crystal replacing 
the ruled lines of the curved grating. The detector may be a 
photographic plate, or a counter in the curved-crystal spectrom- 
eter. 



PORTABLE COUNTERS AND D-C AMPLIFIERS 195 

E. HEALTH PHYSICS INSTRUMENTS 

By the term health physics instruments we refer to detection 
and measuring instruments designed for the monitoring of per- 
sonnel radiation exposures, and for the surveying of laboratories, 
equipment, clothing, hands, and the like, for biologically harmful 
radioactive contaminations. Very many types of such instru- 
ments have been made and used, especially within the wartime 
Manhattan Project laboratories and the Atomic Energy Com- 
mission laboratories and affiliated installations. Those which 
are generally available commercially may be divided into a few 
categories, and are all derived from the instrumentation principles 
already discussed in this chapter. 

Pocket Ion Chambers; Film Badges. Perhaps the most widely 
used radiation monitor is the pocket ionization chamber. This 
is an ordinary ionization chamber in most respects, made small 
enough to be worn clipped in the pocket like a fountain pen. The 
charging potential is applied through a temporary connection, 
and at the end of the day or end of exposure the residual charge 
is read on an electrometer. A pocket ionization chamber with a 
built-in electrometer and scale is more convenient, but also more 
expensive; this style of chamber is initially charged on an external 
device and then may be read directly at any time without auxiliary 
apparatus and without effect on the indication. 

These pocket meters are calibrated in roentgen units, with full 
scale corresponding most commonly to 0.1 or 0.2 r, so that they 
easily detect general radiation dosage below tolerance levels. 
They may not give a measure of local exposure (say, of the hands, 
while other parts of the body are shielded by a lead screen) and, 
of course, are not sensitive to soft radiations that do not penetrate 
the chamber wall. These limitations are also found in the photo- 
graphic-film badge, which may be worn for determination of 
exposures integrated over longer times usually over several days 
or weeks. 

Portable Counters and D-c Amplifiers. More sensitive detection 
instruments are used to determine the rate at which exposure is 
being received in a given radiation field. These may be larger 
ionization chambers with compact d-c amplifiers operated from 
self-contained batteries, so that a readily portable survey meter 
weighing perhaps 10 pounds is achieved. Models of this type 



196 INSTRUMENTS FOR RADIATION DETECTION CH. VIII 

ordinarily have several calibrated scale ranges, from about 0-20 
mr per hr to about 0-3000 mr per hr. Battery-operated porta- 
ble Geiger-counter sets of about the same size and weight are 
available and can be used for the same purpose. They are usually 
arranged as counting-rate meters, with full-scale readings cali- 
brated at about 0.2 to about 20 mr per hr. Although the counter 
type of meter is much more sensitive than the ionization chamber, 
the latter is usually sensitive enough and may be expected to give 
a response more nearly proportional to the biological effects of 
the radiation. Both types are ordinarily provided with a movable 
shield to permit a distinction between hard and soft radiations. 
These two types of instruments are also useful for surveying the 
laboratory and its apparatus for radioactive contamination. The 
G-M counter instrument with its higher sensitivity, especially 
when used with earphones so that each count may be heard, is 
more convenient in rapid surveys for small amounts of activity 
but in its usual form is not useful for very soft ft rays such as those 
from C 14 . The ionizaiton-chamber instrument is more easily 
fitted with a window than enough for this purpose (not more than 
a few milligrams per squire centimeter), and some available models 
have very thin windows (or simply open screens) that will pass 
even a particles. 

Other Procedures. A number of other more specialized instru- 
ments have been devised. Geiger counters and atmospheric- 
pressure proportional counters may be arranged particularly to 
detect ft and a. contaminations on the hands. The monitoring 
of air-borne contamination requires special instruments, and may 
be particularly important in laboratories handling long-lived 
a activities. One method for air-borne dusts is to filter a large 
volume of the air and to assay the activity left on the filter paper 
with a standard counter or linear-amplifier instrument. (The 
radon decay products ordinarily present in air can be detected ir 
this way.) A very simple and widely applicable semiquantitativf 
method of contamination monitoring wilich requires no specia 
instrumentation is worth mentioning here; this is the so-callec 
"swipe" method. A small piece of clean filter paper of a standarc 
size is wiped over a roughly uniform path length on the suspectec 
desk top, floor, wall, laboratory ware, or almost anywhere, anc 
then measured for a, ft, or 7 activity on a standard instrument 
Even air-borne contamination may be checked in a rough way ty 



EXERCISES 197 

swipe samples of accumulated dust from an electric-light fixture 
or some such place which is exposed only to contamination from 
the air. 

EXERCISES 

1. Estimate roughly the voltage (IR) applied to the grid of a d-c 
amplifier tube; use these assumptions: R = 10 11 ohms; the sample emits 
1-Mev 7 rays at the rate of 10 6 per min; the geometry is such that 30 per 
cent of the y's spend an average 8-cm path length in the ionization chamber 
which is filled with CF2C12 at 2 atmospheres total pressure. 

2. (a) What is the thickest mica window that will pass some particles 
from C 14 ? Give the answer in milligrams per square centimeter and in 
inches of thickness. 

(6) How thin must a window be to pass some of the H 3 ft particles? 
(c) What is the energy of the softest ft ray that could enter a counter 
tube with a 0.10-mm Pyrex-glass wall? 

3. A ft particle of 2 Mev enters a G-M counter and spends a 1.0-cm 
path length in the gas, which is Pb(CH 3 ) 4 at a pressure of 1.0 cm. What 
is the (average) number of ion pairs that should be expected to result from 
ionization in the gas? Answer: 5 

4. Calculate the time required for a positive ion to move from the wire 
to the wall of a Geiger counter; take 0.005 inch for the wire diameter, 
1> inch for the cathode diameter, 1000 volts as the applied voltage, 10 cm 
as the gas pressure, and 1.5 cm per sec for the mobility of the ion at 1 volt 
per cm gradient and at 76 cm pressure. Answer: 490 jus 

5. A Lauritsen electroscope is to be used for work with Ci 38 . Its fiber 
system has a capacity of 0.3 esu. Its chamber has a diameter of 5 cm. 
In order to get a sufficiently accurate reading, its discharge rate must be 
at least 0.10 volt per min (corresponding to about 0.1 division per min). 
Making appropriate assumptions, estimate the minimum sample strength 
that should be used. 

6. What type of instrument would you use for each of the following: 
(a) detection of 10~ 5 rd of H 3 ; (b) detection of 10~ 3 rd of Mn 66 ; (c) detection 
of 10~ 6 rd of At 211 ; (d) following the decay of a sample of Cu 64 (initially 
0.3 rd) over a period of 8 days. Briefly state the reason for each choice. 

7. The radiations emitted by a certain radioactive species were studied 
in a jS-ray spectrometer. The ft~ spectrum was resolved into two com- 
ponents of 0.61 rb 0.01 Mev and 1.438 0.007 Mev maximum energies. 
The higher-energy component was about four times as abundant as the 
lower-energy one. When 7 rays were allowed to strike a thin silver radiator 



198 INSTRUMENTS FOR RADIATION DETECTION CH. VIII 

placed in the source position of the spectrometer, the following five 
photoelectron energies were measured : 

Energy in Mev Intensity 

0.216 0.002 Strong 

0.237 0.002 Weak 

0.801 0.003 Weak 

. 823 . 003 Very weak 

1 . 046 . 005 Very weak 

The K and L binding energies in silver are 25 and 4 kev, respectively. 
Draw a plausible decay scheme for the radioactive species under investi- 
gation. 

REFERENCES 

R. E. LAPP and H. L. ANDREWS, Nuclear Radiation Physics, New York, 

Prentice-Hall, 1948. 
J. STRONG, Procedures in Experimental Physics, New York, Prentice-Hall, 

1938. 
M. D. KAMEN, Radioactive Tracers in Biology, New York, Academic Press, 

1947. 
S. A. KORFF, Electron and Nuclear Counters, New York, D. Van Nostrand Co., 

1946, 
D. R. CORSON and R. R. WILSON, "Particle and Quantum Counters," Rev. Sci. 

Inst. 19, 207 (1948). 
S. C. BROWN, "Theory and Operation of Geiger-Mtiller Counters," Nucleonics 

2 no. 6, 10 (June 1948). 
,). D. CRAGOS and A. A. JAFFE, "Discharge Spread in Geiger Counters," Phys. 

Rev. 72, 784 (1947). 
S. H. LIEBSON, "The Discharge Mechanism of Self-Quenching Geiger-Mueller 

Counters," Phys. Rev. 72, 602 (1947). 
W. GOOD, A. KIP, and S. BROWN, "Design of Beta-Ray and Gamma-Ray 

Geiger-Muller Counters," Rev. Sci. Inst. 17, 262 (1946). 
D. H. COPP and D. M. GREENBERG, "A Mica Window Geiger Counter Tube 

for Measuring Soft Radiations," Rev. Sci. Inst. 14, 205 (1943). 
S. H. LIEBSON and H. FRIEDMAN, "Self-Quenching Halogen Filled Counters," 

Rev. Sci. Inst. 19, 303 (1948). 
K. K. D ARROW, Electrical Phenomena in Gases, Baltimore, Williams & Wilkins 

Co., 1932. 
L. B. LOEB, Fundamental Processes of Ekctrical Discharge in Gases, New York, 

John Wiley & Sons, 1939. 

J. D. COBINE, Gaseous Conductors, New York, McGraw-Hill Book Co., 1941. 
D. B. PENNICK, "Direct-Current Amplifier Circuits for Use with the Elec- 
trometer Tube," Rev. Sci. Inst. 6, 115 (1935). 
M. DEUTSCH, "Naphthalene Counters for Beta and Gamma Rays," Nucleonics 

2 no. 3, 58 (March 1948). 
J. S. ALLEN, "An Improved Electron Multiplier Particle Counter," Rev. Sci. 

hist. 18, 739 (1947). 



IX. STATISTICAL CONSIDERATIONS IN RADIOACTIVITY 
MEASUREMENTS 

A. RANDOM PHENOMENA 

The occurrence of nuclear disintegrations is a random phe- 
nomenon subject to established methods of statistical analysis. 
We must study in some detail the application of these methods 
and the nature of the statistical laws. First consider the set of 
data actually obtained with a Geiger counter measuring a ' 'steady" 
source, as given in table IX 1. The number of counts recorded 
per minute (the counting rate) is clearly not uniform. Which 
minute gave the most accurate result? The best thing we can 
do is to compute the arithmetic mean (the average value) and 
consider this to represent the proper counting rate. 

Average Value. If the determinations, minute by minute, arc 
denoted by x iy x 2 , X{ for the 1st, 2d, zth minute, then the 
arithmetic mean value x is, by definition, 



X == 



where NO is the number of values of x to be averaged. For the 
counting rates in the table x = 990/10 = 99.0. 

Standard Deviation. After the experiment which gave these 
data we might have repeated the measurements and so obtained 
another average rate. The best figure would then have been the 
average of all the results. Knowing merely the average value 
we do not know anything about the statistical dependability of 
the data from which the average was computed, that is, the degree 
of agreement between the individual results. We want to define 
a quantitative measure of the closeness of agreement. Consider 
for a moment the deviation A; of each number x^ defined as the 
difference between Xi and x: A; = Xi x. These deviations are 
tabulated in the third column of the table. The average value 
of A t - cannot be taken as a measure of internal agreement because 
it is just zero: 

199 



200 STATISTICAL CONSIDERATIONS CH. IX 

1 t '-^o i J^o i JVo 

^ t - = y^ xi y^ ^ = x - z = o. 



The average of the squares of the individual deviations, called the 
dispersion and denoted by 0- x 2 , gives a useful measure of the degree 
of agreement among the results. 



tfo 

The standard deviation <r x , which is just the square root of the 
dispersion, is commonly used. For the data of table IX-1 we 



Minute Counts A,- A 2 

1 89 -10 100 

2 120 +21 441 

3 94 -5 25 

4 110 +11 121 

5 105 +6 36 

6 108 +9 81 

7 85 -14 196 

8 83 -16 256 

9 101 +2 4 
10 95 -4 16 

Totals 990 1276 

compute a x 2 = 1270/10 = 127.6^ <r x = 11.3. A useful relation 
that may be derived is: c x 2 = x 2 x 2 ; that is, the dispersion is 
given by the difference between the average of the squares of the 
x values and the square of the average value. 

B. PROBABILITY AND THE COMPOUNDING OF PROBABILITIES 

The ideas and definitions just presented may be applied, with 
varying degrees of usefulness, to any set of data, whether or not 
strictly random phenomena are involved. Before proceeding 
further, we must consider very carefully the concept of probability. 
As illustrations we will investigate the answers to questions such 
as: 



ADDITION THEOREM 201 

(a) What is the probability that a card drawn from a deck be 
an ace? 

(6) If a coin is flipped twice, what is the probability of it falling 
"heads up" both times? 

(c) Given a sample of a radioactive material what is the proba- 
bility that exactly 100 disintegrations occur during the next 
minute? We shall define probability in this way: Given a set of 
A^o objects (or events, or results, etc.) containing n\ objects of 
the 1st kind, n 2 objects of the 2d kind, and n t - objects of the zth 
kind; the probability pi that an object specified only as belonging 
to the set is of the tth kind is given by: pi = n t /ZV . By applying 
this definition we find that the probability that one card drawn 
from a full deck be an ace is just 4/52. 

We may now rewrite the definition of the average value x of a 
set of quantities Xi taking into account the possibility that any 
particular value may appear several, say n iy times. Then: 



Q 

This may be generalized, and the expression for the average value 
of any function of x is 

/(*) = :wfe). (ix-i) 

In experimental measurements we may make a large number 
K of observations and find the ^th result ki times. Now the ratio 
ki/K is not the probability pi of the zth result as we have defined 
it, but for our purposes we assume that ki/K approaches arbi- 
trarily closely to Pi as K becomes very large: 

h 

lim = p^ 
K->K * 

This assumption is not subject to mathematical proof, because a 
limit may not be evaluated for a series with no law of sequence 
of terms. 

Addition Theorem. We turn now to the compounding of several 
probabilities, and consider first the addition theorem. Given a 
set of NO objects (or events, or results, etc.) containing U{ objects 
of the kind a t , and given that the kinds a\, a^ y a* have no 
members in common, then the probability that one of the NO 



202 STATISTICAL CONSIDERATIONS CH. IX 

objects belongs to a combined group 0,1 + a 2 H ---- ay is just 
f pi. Thus for two mutually exclusive events with probabilities 

Pi and p 2 the probability of one or the other occurring is just 
Pi + p 2 . When one card is drawn from a full deck the chance of 
its being either a five or a ten is 4/52 + 4/52 = 2/13. (When 
one draws one card while already holding, say, four cards none 
of which is a five or ten, the probability then of getting either a 
five or a ten is slightly greater, 4/48 + 4/48 = 1/6, provided 
there is available no information as to the identity of other cards 
that may already have been withdrawn.) When a coin is tossed 
the probability of either "heads" or "tails" is % + % = 1. 

Multiplication Theorem. Another type of compounding of prob- 
abilities is described by the multiplication theorem. If the prob- 
ability of an event i is p^ and if after i has happened the probabil- 
ity of another event j is py , then the probability that first i happens 
and then j happens is pi X py. If a coin is tossed twice the prob- 
ability of getting "heads" twice is % X 1 A = Y. If two cards 
are drawn from an initially full deck the probability of two aces 
is 4/52 X 3/51. The probability of four aces in four cards drawn 
is 4/52 X 3/51 X 2/50 X 1/49. (The probability of drawing 
five aces in five cards is 4/52 X 3/51 X 2/50 X 1/49 X 0/48 
= 0.) 

Binomial Distribution. The binomial distribution law treats one 
fairly general case of compounding of probabilities. Given a 
very large set of objects in which the probability of occurrence of 
an object of a particular kind w is p, then, if n objects are with- 
drawn from the set, the probability W(r) that exactly r of the 
objects are of the kind w is given by 

n\ 



- 
(n r)!r! 

To see how this combination of terms actually represents the 
probability in question, think for a moment of just r of the n 
objects. That the first of these is of the kind w has the probability 
p; that the first and second are of the kind w has the probability 
p 2 , etc., and the probability that all r objects are of the kind w 
is p r . But, if exactly r of the n objects are to be of this kind, the 
remaining n r objects must be of some other kind; this proba- 



BINOMIAL DISTRIBUTION 203 

bility is (1 p) w ~ r . Thus we see that, for a particular choice of 
r objects out of the n objects, the probability of exactly r of kind 
w is p r (l p) n ~~ r 'j this particular choice is not the only one. The 
first of the r objects might be chosen (from the n objects) in n 
different ways; the second in n 1 ways; the third in n 2 
ways, and the rth in n r + 1 ways. The product of these terms, 

n\ 
n(n l)(n - 2) - (n - r + 1), is - , and this coeffi- 

(n-r)I 

cient must be used to multiply the probability just found. But 
this coefficient is actually too large in that it not only gives the 
total number of possible arrangements of the objects in the way 
required but also includes the number of arrangements which 
differ only in the order of selection of the r objects. So we must 
divide by the number of permutations of r objects which is r!. 
Thus, the final coefficient is n\/(n r)!r!, which is that in equa- 
tion IX-2. The law (equation IX-2) is known as the binomial 
distribution law because this coefficient is just the coefficient of 
the x r term in the binomial expansion of (1 + x) n . 

C. RADIOACTIVITY AS A STATISTICAL PHENOMENON 

Binomial Distribution for Radioactive Disintegrations. We 

may apply the binomial distribution law to find the probability 
TF(ra) of obtaining just m disintegrations in time t from NQ orig- 
inal radioactive atoms. We think of NQ as the number n of objects 
chosen for observation (in our derivation of the binomial law) 
and we think of m as the number r that are to have a certain 
property (namely, that of disintegrating in time t), so that for 
this case the binomial law becomes 



M I - 

m)!m! 

Now the probability of an atom not decaying in time t, I p in 
the equation above, is given (1) by the ratio of the number N that 
survive the time interval t to the initial number NQ, 

N 

1 See chapter I, p. 7. 



204 STATISTICAL CONSIDERATIONS CH. IX 

and p is then 1 e~ Xf . We now have 



, . . 

m)lml 

Time Intervals Between Disintegrations. Since the time of 
Schweidler's derivation of the exponential decay law from prob- 
ability considerations a number of experiments have been made 
to test the applicability of these statistical laws to the phenomena 
of radioactivity. As an example of the positive evidence obtained 
we consider the distribution of time intervals between disintegra- 
tions. The probability of this time interval having a value be- 
tween t and t + dt y which we write as P(t) dt, is given by the 
product of the probability of no disintegration between and t 
and the probability of a disintegration between t and t + dt. 
The first of these two probabilities is given by equation IX-3 
with m = 0: 



(Notice that 0! = 1.) The probability of one of the N atoms 
disintegrating in the time dt is clearly NQ\ dt. (See chapter I, 
page 6, or obtain this result as W(l) from equation IX-3 with 
m = 1, t replaced by dt, and all terms in (dt) 2 and higher neglected.) 
Then, 

P(t) dt = N Q \e- N xt dt. 

Experiments designed to test this result usually measure a large 
number s of time intervals between disintegrations and classify 
them into intervals differing by the short but finite length At-> 
then the probability for intervals between t and t + At should 
be NQ\e~ N u At, and the number of measured intervals between 
t and t + At should be sN G \e~ NQ ^ At. For example, Feather has 
found experimentally that the logarithm of the number of intervals 
between t and t + At is proportional to t, as required by this 
formula. 

Average Disintegration Rate. Another application of the bi- 
nomial law to radioactive disintegrations may be seen if we calcu- 
late the expected average value for a set of numbers obeying the 
binomial distribution law. We will for the moment revert to the 



AVERAGE DISINTEGRATION RATE 205 

notation of equation IX-2 and for further convenience will repre- 
sent 1 p by q. 



. 

(n r)\rl 

The average value to be expected for r is obtained from equation 
IX-1: 



To evaluate this awkward-appearing summation consider the 
binomial expansion of (px + q) n : 



Differentiating with respect to x we obtain 
np(px + q) n ~ l = 



Now letting x = 1 and using q = 1 p, we have the desired 
expression : 



np 



r = n 

= f ^ rW(r) 



This result should not be surprising; it means that the average 
number f of the n objects which are of the kind w is just n times 
the probability for any given one of the objects to be of the kind 
w. ^ 

The foregoing result may be interpreted for radioactive disinte- 
gration if n is set equal to NO and p = 1 e~ H , as before. Then 
the expected average number M of atoms disintegrating in the 
time t is: M = N Q (l e~ x % For small values of X, that is, for 
times of observation short compared to the half-life, we may use 
the approximation e~ u = 1 \t, and then M = N Q \t. The 
disintegration rate R to be expected is: R = M/t = N Q \. (This 
corresponds to the familiar equation dN/dt = \N.) 



206 STATISTICAL CONSIDERATIONS CH. IX 

Expected Standard Deviation. A more interesting question is: 
What may we expect for the standard deviation o> for this ex- 
pected average value f (or M)? If we differentiate equation 
IX-4 again with respect to x we get 



n(n - I)p 2 (pz 
Again letting x = 1 and using p + q = 1 we have 

r=-r? 

n(n - l)p 2 = 



r-0 



n(n l)p 2 = r 2 f. 
Recall that the dispersion o> 2 is given by 

<r r 2 = 7 2 - f 2 ; 
now combining we have 

ff r 2 = n(n l)p 2 + f f 2 , 
and with f = up: 

<r r 2 = n 2 p 2 np 2 + np n 2 p 2 = np(l p) = 



For the case of radioactive disintegration this becomes 



In counting practice X^ is usually small; that is, the observation 
time t is short compared to the half-life, and when this is so 
<r = V M. If a reasonably large number m of counts has been 
obtained that number m may be used in the place of M for the 
purpose of evaluating <r. Thus if 100 counts are recorded in 1 min 
the expected standard deviation is <7 VlOO = 10, and the 
counting rate might be written 100 10 counts per min. If 
1000 counts are recorded in 10 min the standard deviation of this 

, . . _ . 1000 32 

number is a = v 1000 = 32; the counting rate is 



10 
100 3.2 counts per min. Thus we see that for a given counting 



EXPECTED STANDARD DEVIATION 207 

rate R the a for the rate is inversely proportional to the square 
root of the time of measurement: 



m 



\/m 

ffrate = - = - = Y~ (IX-5) 

tit 

What is the result in an experiment in which the counting time 
is long compared to the half-life? As \t > oo ; e~ > 0, and in 
this limit o- = V Me~ u = 0. The explanation is clear; if we start 
with NO atoms and wait for all to disintegrate, then the number 
that disintegrate is exactly N Q . However, in actual practice we 
observe not the number of disintegrations but that number times 
a coefficient c which denotes the probability of a disintegra- 
tion resulting in an observed count. Taking this into account 
we see that^ in this limiting case the proper representation 
of or = Vnpq is <r = VN O C(! - c). If c 1 then <r = 



= v no. of counts as before. For cases where \t ~ 1 and 
c is neither unity nor very small a more exact analysis 
based on a = Vnpq should be made, with the result that <r = 
VMc(l - c + ce~ xt ). 

The introduction of the detection coefficient c in the preceding 
paragraph may raise the question as to why it is not necessary 
to take account of this coefficient in the more familiar case with 
Xt small, where we have written o- = \/m. If we do consider c 
in this case, we have for the probability of one atom producing a 
count in time t, p = (1 - e~ M )c; and ^=1 p=l-c + ce~ M . 
Then, 



- c ~ c + c, 
and for \t small and the same approximations as before: 

o- = v NQ\IC = vMc = v no. of counts recorded. 

This is just the conclusion we had reached without bothering 
about the detection efficiency. However, it should be emphasized 
that actual counts and not scaled counts from a scaling circuit 
must be used in these equations. 



208 STATISTICAL CONSIDERATIONS CH. IX 

D. POISSON AND GAUSSIAN APPROXIMATIONS OF THE 
DISTRIBUTION LAW 

Poisson Distribution. The binomial distribution law may be put 
into a more convenient approximate form if we impose the restric- 
tions \t 1, NO 1, m< N 0) that is, if we consider a large 
number of active atoms observed for a time short compared to 
their half-life. It is also necessary to make use of these mathe- 
matical approximations: 

(1) 6 M = 1 + \t, neglecting subsequent terms. 

(2) xl = v 2irx x x e~ x (Stirling's approximation). 

(771, \ ATo / r fYi \NQ 
1 ) = lim ( 1 ) = e~ m , since NO 1- 
NQ/ tf -\ NQ/ 

With these restrictions and approximations and with M = No\t, 
equation IX-3 may in a straightforward way be put into the form 
known as the Poisson distribution: 

M m e~ M 

W(m) = - 

ml 

In words, the probability of obtaining the particular number of 
counts m, where the average to be expected is M, is M m e~ M /m\. 
This approximation is very good even for N as small as 100 and 
\t as large as 0.01. Two features of this distribution might be 
noticed in particular. The probability of obtaining m = M 1 
is equal to the probability of obtaining m = M, or W(M) = 
W(M 1). For large M the distribution is very nearly sym- 
metrical about m M if values of m very far from M be excluded. 
Gaussian Distribution. A further approximation of the distribu- 
tion law may be made for large m (say >100) and for 
| M m | <<C M. With these additional restrictions and with 
the approximate expansion, 

M m\ M m (M m) 2 



In 



\ 

U 

/ 



m / m 2m 



neglecting subsequent terms, we may modify the Poisson distri- 
bution to obtain the Gaussian distribution: 



W(m) = 







RATIOS AND PRODUCTS OF COUNTING RATES 209 

It will be noticed that this distribution is symmetrical about 
m = M. For both the Poisson and Gaussian distributions we 
may derive <r = VM, or o- = \/m for large m. 



E. EXPERIMENTAL APPLICATIONS 

Addition and Subtraction of Counting Rates. An important 
practical consideration is the addition and subtraction of counting 
results or counting rates. The Poisson distribution expression is 
suitable for the treatment of these cases, but the derivations are 
much too tedious to be included here. The very significant results 
are these: 

1. The sum of two Poisson distributions is itself a Poisson 
distribution. Hence, the dispersion <r s 2 and standard deviation 
<T 8 of a sum are given by: <? 8 2 = v<? + <r 2 2 + and <r a = 
V<r 1 2 + * 2 2 +.... 

2. The difference of two Poisson distributions is not a Poisson 
distribution; the dispersion <r d 2 and standard deviation <r<i of the 
difference are given by: <r d 2 = <?i 2 + <r 2 2 and <r d = Wi 2 + o- 2 2 . 

As an example suppose that the background counting rate of a 
counter is measured, and 600 counts are recorded in 15 min. 
Then with a sample in place the total counting rate is measured, 
and 1000 counts are recorded in 10 min. We wish to know the 
net counting rate due to the sample and the standard deviation 
of this net rate. First the background rate R& is 

600 =fc V600 600 =fc 24 

Rb = - - = - = 40 1.6 counts per min. 
15 15 

The total rate R* is 

1000 db \/1000 1000 =fc 32 






= 100 3.2 counts per min. 



The net rate Rn = 100 40 = 60 counts per min, and its stand- 
ard deviation is <r n = Vl.6 2 + 3.2 2 = 3.6; and R n = 60 db 3.6 
counts per min. 

Ratios and Products of Counting Rates. In many types of ex- 
periments the ratio of two counting rates is wanted. What is the 
standard deviation of this ratio Q = Ri/R 2 , if the two standard 
deviations <TI and <r 2 are known? It may be shown by straightfor- 



210 STATISTICAL CONSIDERATIONS CH. IX 

ward algebraic operations that for small deviations the ratio 
<TQ*/Q is given by 



ffQ * ff\ (72 



where for the moment we mean by <TQ* the particular deviation in 
Q resulting from possible combinations of the deviations cr\ and 
ff 2 in RI and R 2 . To obtain instead the standard deviation <TQ 
we must assume that the two fractional deviations of the rates 
are not simply additive but on the average combine to give a root- 
mean-square deviation: 




This is the formula to be used for evaluation of the standard 
deviation (TQ of the ratio Q of two quantities. A similar expression 
is applicable for the product P of two or more rates: 



(See exercise 10 at the end of this chapter.) 

Gaussian Error Curve. Knowledge of the distribution law per- 
mits a quantitive evaluation of the probability of a given devia- 
tion of a measured result m from the proper average M to be ex- 
pected. The Gaussian distribution is convenient for this purpose. 
With the absolute error | M m \ = e, and with the assumption 
that the integral numbers are so large that the distribution may be 
treated as continuous, the probability W (t)de of an error between 
e and c + dc is given by 

-^ 

2M 



6 2M de. 



The factor 2 arises from the existence of positive and negative 
errors with equal probability within the limits of validity of this 
approximation. Recalling that = v M we have 

5 -4 



COMPARISON WITH EXPERIMENT 



211 



The probability of an error greater than far is obtained by integra- 
tion from e = fccr to c = oo. Numerical values of this integral as 
a function of k may be found in handbooks. For example, we 
have taken for table IX-2 some representative values from the 
table, " Probability of Occurrence of Deviations " in the Chemical 
Rubber Publishing Company Handbook of Chemistry and Physics. 

TABLE IX-2 



k 





0.674 


1 


2 


3 


4 


Probability of e > far 


1.00 


0.50 


0.32 


0.046 


0.0027 


0.00006 



Notice that errors greater than 0.674o- and errors smaller than 
0.674a are equally probable; 0.674<r is called the "probable error " 
and is sometimes given rather than the standard deviation when 
counting data are reported. In plots of experimental curves it 
can be very convenient to indicate the probable error of each 
point (by a mark of the proper length); then on the average the 
smooth curve drawn should be expected to pass through about 
as many "points" as it misses. 

Comparison with Experiment. We may now return to a con- 
sideration of the typical counting data in table IX-1. We have 
already found from the deviations between the ten measurements 



= \j 

NQ 



x) 2 = 11.3. Now, if the counting rate meas- 



ured there represents a random phenomenon, as we expect it 
should, we may evaluate the expected a for the result in any 
minute as the square root of the number of counts. For a typical 
minute, the 9th, we find <r = A/101 = 10, and for other minutes 
other values not much different. Because these agree reasonably 
with the 11.3 there is evidence for the random nature of the 
observed counting rate. This test should occasionally be made 
on the data from a counting instrument. 

In addition to estimating a for any minute's counting in table 
IX-1, we may now estimate the a for the average of the ten obser- 
vations (which we could not do directly from the definition of 



212 STATISTICAL CONSIDERATIONS CH. IX 

<r). The average counting rate with its standard deviation is 

990 A/990 

$ 99 o 3.1 counts per min. This means 

that the probability that the true average is between 95.9 and 
102.1 is from table IX-2 just 1 - 0.32 = 0.68. Actually, when 
the counting data given in table IX-1 were obtained the average 
rate was measured much more accurately in a 100-min count, 



10,042 V 10,042 

and the result was = 100.4 1.0 counts per 

100 F 

nun. 

Case of Very Few Counts. A question sometimes met in count- 
ing experiments is what can be done with results which show very 
few total counts (not net counts), or even no counts at all. A 
treatment of this problem has been given by R. W. Dodson. He 
shows that for the Poisson distribution the best value to assume 
for the true average number of counts M is not the measured 
value m, but rather m + 1. (This is clearly related to one of the 
features already pointed out for the Poisson distribution.) The 
best guess for the standard deviation is a = V m + 1. Thus, 
if the observed count is m = 0, we take for the answer M = 1 1 ; 
if the observed m = 1 we take M = 2 1.4; etc. 

Counter Efficiencies. As another application of the methods of 
this chapter to counting techniques we may estimate the efficiency 
of a Geiger counter for rays of a given ionizing power, with the 
assumptions that any ray which produces at least one ion pair 
in the counter gas is counted and that effects at the counter walls 
are negligible. Knowledge of the nature of the radiation and the 
information given in chapter VII permit an estimate of the aver- 
age number of ion pairs a to be expected within the path length 
of the radiation in the counter filling gas. The problem then is 
to find the probability that a ray pass through the counter leaving 
no ion pairs and thus not be counted. We think of the path of 
the ray in the counter as divided into n segments of equal length; 
if n is very large, each segment will be so small that we may 
neglect the possibility of having two ion pairs in any segment. 
Then just a of the n segments will contain ion pairs, and by defini- 
tion the probability of having an ion pair in a given segment is 
p = a/n. Now by equation IX-2 for the binomial distribution 



COINCIDENCE CORRECTION 213 

we have the probability for no ion pairs in n segments; that is, 
for r = 0: 

nl / /AW 

TF(0) = p(l p) n = (1 p) n - 



Since the probability (2) is evaluated correctly only as n becomes 
very large, 



W(0) - lim 1 



Y 

- - 1 
n) 



The probability of counting the ray, which is the efficiency to be 
determined, is then 1 W (0) = 1 ""*. As a particular 
example consider a fast ft particle with the relatively low specific 
ionization of 5 ion pairs per mm in air and a path length of 10 mm 
in a counter gas which is almost pure argon at 7.6 cm pressure. 
We estimate a from these assumptions, correcting for the relative 
densities of air and the argon : 

7.6 40 

a = 5 X 10 X X = 7. 
76 29 

The corresponding estimated counter efficiency is 1 e~~~ 7 ~ 99.9 
per cent. It should not be expected that an efficiency calculated 
in this way is very precise; wall effects may be important, and the 
assumption of random distribution of ion pairs along the 0-ray 
path is not entirely consistent with the mechanism of energy loss 
by ionization presented in chapter VII. 

Coincidence Correction. If a counter has a recovery time (or 
dead time or resolving time) T after each recorded count during 
which it is completely insensitive, the total insensitive time per 
unit time is RT, where R is the observed counting rate. If R* is 
the rate that would be recorded if there were no coincidence losses 
then the number of lost counts per unit time is R* R, and is 
given by the product of the rate R* and the fraction of insensitive 
time RT: 

R* - R = R*Rt, 

R* = (IX-6) 

1 K.T 

1 We might have evaluated this probability more easily from the Poisson 
distribution expression: W(Q) ae~ a /0! e~. 



214 STATISTICAL CONSIDERATIONS CH. IX 

A number of variants of this formula are also in use. One expres- 
sion (the Schiff formula) is R* = Re R * T ; this is derived from a 
calculation of the probability TF(0) of having had no event during 
the time T immediately preceding any event, but it neglects the 
possibility that any preceding event itself may not have led to a 
count through coincidence loss. (3) Another approximate expres- 
sion is derived from the first two terms in the binomial expansion 
of (1 Rr)"" 1 appearing in equation IX-6: 

R* = R(l + RT) = R + R 2 T. 

This form is especially convenient for the interpretation of an 
experiment designed to measure T by measuring the rates RI 
and R 2 produced by two separate sources and the rate R; pro- 
duced by the two sources together, all these rates including the 
background effect R&. Obviously, 



where we have neglected the coincidence loss in the measurement 
of the low background rate. Replacing by R x * = Rj + Ri 2 T, 
etc., and rearranging we have, 



Integrating Measuring Instruments. For measurements with 
counting instruments we have seen the convenience of the simple 
expression a = V no. of counts. In the counting rate meter, 
where a steady meter deflection is observed, what value may be 
assigned to the standard deviation? In this instrument a com- 
bination of resistance R and capacitance C effectively averages 
the rate of arrival of pulses over an interval of the order of magni- 
tude of the time constant RC; a quantitative analysis shows that 
the effective interval is 2RC. Representing the counting rate in 
counts per minute by R x we obtain the standard deviation of this 

8 It may be noticed that the Schiff formula might be expected to correspond 
more closely to the conditions of coincidence loss in a mechanical register, 
where a new pulse within a dead time could initiate a new dead-time period 
although it would not be recorded. There exists also the opportunity for 
coincidence losses in the electric circuits. 



EXERCISES 215 

rate from equation IX-5, where t = 2RC/60 min for R in ohms 
and C in farads: 

ORl 



For an instrument using an ionization chamber with a d-c ampli- 
fier the same expression may be used, provided the activity can 
be evaluated approximately as a rate of arrival of ionizing particles 
and the value of the longest time constant is known (ordinarily 
that of the collecting electrode and first grid circuit unless a very 
slow galvanometer is employed). If necessary the time constant 
may be approximated as the time for a deflection to be reduced 
to l/e of its value after removal of an active sample. 

In a rate-of-drift-measuring method, as in the d-c amplifier with 
infinite grid resistance or in the Lauritsen electroscope, the num- 
ber ra of ionizing particles arriving during the time of measure- 
ment is estimated, and the standard deviation of the activity A 

\/m A 

is given by a A = . These considerations are not 

m v m 

easily applied to 7-ray measurements in ionization chambers, but 
the statistical uncertainties would be at least as great as indicated 
by these formulas. 

EXERCISES 

1. Derive the relation given on p. 200: <r x 2 = x* 2 . 

2. Mr. Jones' automobile license carries a six-digit number. What is 
the probability that it has (a) exactly one 4, (b) at least one 4? 

Answer: (b) 0.46856. 

3. Given an atom of a radioactive substance with decay constant X, 
what is 

(a) The probability of it decaying between and dt? 

(b) The probability of it decaying between and J? 

4. The following numbers were obtained in the measurement of a 
physical quantity x. 

Set 1:90,110,100. 
Set 2: 99,101,100. 

What is the average value obtained in each set? In which set would you 
consider the measurements more reliable? What is the standard devia- 
tion for each set? 



216 STATISTICAL CONSIDERATIONS CH. IX 

5. A given Geiger counter has a measured background rate of 900 
counts in 30 min. With a sample of a long-lived activity in place, the 
total measured rate was 1100 counts in 20 min. What is the net sample 
counting rate and its standard deviation? 

Answer: 25.0 1.9 counts per min. 

6. Denote by R* and R& the total and background counting rates for a 
long-lived sample, and calculate the optimum division of available count- 
ing time between sample and background for minimum a on the net 

counting rate. t t /R* 

Answer: = \l~ 
it, * R& 

7. Refer to exercise 3, chapter VIII. What would be the detection 
efficiency for that counter and that /3 ray? 

8. (a) Sample A, sample B, and background alone were each counted 
for 10 min; the observed total rates were 110, 205, and 44 counts per min, 
respectively. Find the ratio of the activity of sample A to that of sample 
B and the standard deviation of this ratio, (b) Sample C was counted on 
the same counter for 2 min, and the observed total rate was 155 counts 
per min. Find the ratio, and its standard deviation, of the activity of C 
to that of A. Answer: (a) 0.41 db 0.027. 

9. Show that equation IX-6 may be derived by summing up the proba- 
bilities of losing one count, two counts, three counts, etc., during each dead 
time interval. (The Poisson distribution is convenient for this.) 

10. The expressions given for the standard deviations of sums, differ- 
ences, quotients, and products, and indeed for any function F of the indi- 
vidual rates, F = F(Ri, R2 )> may be found by the following approxi- 
mate method, provided ail deviations are small and all rates are large 
numbers : replace the usual differential expression, 



by a form showing the root-mean-square differential deviation, 



Use this equation to derive for the sum or difference of two rates RI and 
R 2 , r that is, for F = RI + R 2 and F = RI - R 2 , the standard deviation dF 
corresponding to the individual standard deviations dRi and dR 2 . Find 
the formulas for standard deviations of quotients and products in the same 
way, with F = Ri/R a and F = RiR 2 . 



REFERENCES 217 

11. What is the probability of a penny turning heads up at least once 
in n throws? 

12. Use the data given in exercise V-l to find the half-lives by the 
method of least squares rather than by graphical solution. Assume that 
there is no error in the measured time intervals and that all the measured 
counting rates have standard deviations of =tl per cent. 



REFERENCES 

L. J. RAINWATER and C. S. Wu, "Applications of Probability Theory to 

Nuclear Particle Detection," Nucleonics 1 no. 2, 60 (Oct. 1947) and 

2 no. 1, 42 (Jan. 1948). 
R. A. FISHER, Statistical Methods for Research Workers, London, Oliver and 

Boyd, 1936. 
L. J. SCHIFP and R. D. EVANS, "Statistical Analysis of the Counting Rate 

Meter," Rev. Sci. Inst. 7, 456 (1936). 
N. FEATHER, "On the Distribution in Time of the Scintillations Produced by 

the Particles from a Weak Source," Phys. Rev. 35, 705 (1930). 
R. PEIERLS, "Statistical Error in Counting Experiments," Proc. Roy. Soc., 

Series A 149, 467 (1935). 



X. TECHNIQUES FOR MEASUREMENT AND STUDY 
OF RADIATIONS 

A. GENERAL TECHNIQUES 

The principal types of instruments used for detection and meas- 
urement of radiations from radioactive substances have been dis- 
cussed in chapter VI I L Here we wish to emphasize the important 
role of proper techniques in the course of measurements of this 
kind. The choice of instruments and techniques will be deter- 
mined in large part by the kinds of information desired. In a 
simple tracer application, employing one active isotope of favor- 
able properties and available with ample activity and known 
purity, a single instrument (counter, electrometer, or electroscope) 
is likely to be sufficient, and measuring techniques may offer no 
problems. The opposite extreme would be a large radiochemical 
laboratory devoted to the identification and characterization of 
new radioactive species as made in a nuclear chain reactor or a new 
high-energy accelerator; here a large number and wide variety of 
instruments, including highly specialized types, must be available, 
and the associated techniques and manipulations will be complex 
and ingenious. Most radiochemical laboratories represent inter- 
mediate situations. A number of tracers are likely to be in use, 
calling for different choices of detectors and sample handling 
procedures. Frequently the desired radioactivity must be iso- 
lated and identified and checked for purity with regard to con- 
tamination by other radioactive substances. Occasions are 
recalled when undiscovered isotopes have been sought for use as 
chemical tracers (I 131 and C 14 were found in this way). The 
typical laboratory may have G-M counters arranged for y count- 
ing, thin-window counters for most (3 counting, perhaps a Lauritsen 
electroscope, and possibly an ionization chamber with d-c ampli- 
fier, useful because of its convenience and its linear response over 
a very wide range of sample strengths. It is very convenient to 
have a standard arrangement for holding standard-size samples 

218 



GENERAL TECHNIQUES 219 

in various positions, the same arrangement being used for as 
many of the instruments as possible. 

In one of our laboratories, that at the Department of Chemistry, 
Washington University, St. Louis, all instruments, G-M counter 
sets, ionization chamber, and Lauritsen electroscopes, except 
special-purpose instruments for C 14 counting and for 7 counting 




FIGURE X-l. A side-window Geiger-Miiller tube with sample holder. A 
radioactive sample mounted on a standard card is in place in the third step. 
The first step is being used to hold a thin aluminum absorber, and space is 
available for insertion of additional absorbers between the sample and the 
counter window (not shown). 

with samples in solution, are fitted with a standardized sample- 
holder arrangement with milled slots to hold samples at various 
distances below the detector; the top step is 0.50 cm below the 
instrument window, and three additional steps are spaced below 
at successive 1.50-cm intervals. The samples are mounted on 
pieces of cardboard 2 inches by 2.5 inches, supplied precut by a 
stationer; for the highest reproducibility the cardboard may be 
replaced by pieces of aluminum or plastic of the same size. These 
sample cards slide broadwise into the milled slots, the 2-inch edges 
being held by the slots, and stops are provided at the back of the 
slots for convenience in locating the samples in the proper position. 



220 MEASUREMENT AND STUDY OF RADIATIONS CH. X 

If the detector is a G-M counter, it and the sample-holding 
arrangement are usually enclosed in a lead shield 1 to 1^ inch 
thick to reduce background, including background effects from 
strong samples in the laboratory. 

The spaces between the milled sample slots are also milled out 
to take absorbers cut to the same 2 X 2.5-inch size as the sample 
cards. Sets of aluminum absorbers are the most useful. They 
should be available in an assortment of thicknesses (nearly inte- 
gral values are convenient), so that absorption curves may include 
points from about a milligram per square centimeter up to several 
grams per square centimeter. Our thicker absorbers are cut from 
thick aluminum stock, with a small tab left on one of the long 
sides which is marked with the thickness in milligrams per square 
centimeter. The thinnest absorbers are compounded of thin 
aluminum foils, held in rectangular 2 x 2.5-inch aluminum frames, 
again provided with tabs. These tabs are spaced along the side 
like index tabs and serve as convenient handles as well as labels. 
A set of lead absorbers is more convenient for determining 7-ray 
absorption curves, but it need not include very thin pieces. A 
few beryllium, paraffin, or polyethylene, (CH 2 ) n , plastic absorbers 
of the same shape can be useful in that they are effective in stop- 
ping ft rays but are almost completely transparent to 7 rays. 

All the measuring instruments should be checked occasionally 
Geiger counters daily with standard samples; it is best if a stand- 
ard is chosen to have a radiation similar to the activity to be 
measured. An intercalibration of all the instruments for activities 
of interest can be useful but ordinarily should be depended on 
only for semiquantitative results. Each instrument must have 
its response to samples of different strengths determined; outside 
the linear-response region it should be used cautiously, with cali- 
brated corrections. This calibration can be made in several 
ways: (1) with samples of different activity carefully prepared 
from aliquot portions of an active solution; (2) by comparison of 
the decay curve of a very pure short-lived activity of known half- 
life with the exponential decay to be expected; and (3) by meas- 
urements of the separate and combined effects of samples located 
in reproducible assigned positions (see chapter IX, page 214). 
With counters the failure of linearity at high counting rates is 
attributed to coincidence losses; the correction is known as a 



CRITICAL ABSORPTION OF X RAYS 221 

coincidence correction. (1) Ordinarily the necessity for corrections 
amounting to more than a few per cent should be avoided. 



B. CRITICAL ABSORPTION OF X RAYS 

Absorption curves in aluminum and lead absorbers are used to 
determine the energies of 0, 7, and X rays, as discussed in chapter 
VII, but the values obtained are rarely as accurate as those deter- 
mined by the use of the electron spectrograph and the curved- 
crystal spectrograph mentioned in chapter VIII, section D. In 
radiochemical work accurate measurements of X-ray energies are 
often wanted to establish the identity of the chemical element. 
The X radiation is characteristic of the particular value of Z at 
the time of emission of the ray; X rays following /3~~ decay of a 
nucleus of charge Z correspond to atomic number Z + 1 ; those 
following 0" 1 " or Jf-capture decay correspond to atomic number 
Z 1; those following converted isomeric transitions correspond 
to atomic number Z. The ordinary absorption study does not 
establish the X-ray energy with sufficient accuracy for this pur- 
pose, and if the special instruments are not available a technique 
based on critical absorption edges may be employed. 

To understand this method we have to recall that the emission 
of an X ray from an atom is due to the transition of an electron 
from one of the outer shells to a vacancy in a shell further in, say, 
from the L to the K shell. In X-ray terminology, X rays due to 
transitions from the L to the K shell are called K a X rays (K a i 
and K a2 corresponding to the electron originating in different 
sublevels of the L shell); X rays due to transitions from the M 
to the K shell are called Kp, etc. Similarly there are L a , L^ 
etc., X rays. 

An X ray can be absorbed by an atom only if its energy corre- 
sponds at least to the energy difference between one of the filled 
shells and the lowest lying shell with a vacancy (the latter being 
very close in energy to the continuum). It is therefore found 
that the absorption coefficient (or the half-thickness) in a given 
element has discontinuities called absorption edges at those 

1 If the dead time of the counter tube is known to be much larger than the 
time constants of any of the other components the coincidence correction for 
any counting rate can be calculated from the dead time; see chapter IX, page 
213. 



222 



MEASUREMENT AND STUDY OF RADIATIONS CH. X 



X-ray energies corresponding to the binding energies of the K, 
L, etc., electrons. Thus X rays of energy insufficient for the 
removal of a K electron (essentially to the continuum) are poorly 
absorbed and have large half-thickness values, whereas those of 
energy just exceeding that of the K absorption edge are absorbed 
much more strongly (perhaps ten times more strongly). 



1.0 



0.1 



$0.05 



0.02 








5 10 15 

Absorber Thickness (mg/cm 2 ) 



20 



FIGURE X-2. Absorption of zinc K a X rays in zinc, copper, and nickel. 

(These absorption curves were calculated from data given in A. H. Compton 

and S. K. Allison, X-rays in Theory and Experiment, New York, D. Van 

Nostrand Co., 1935.) 

It is clear from this that an element is a poor absorber for its 
own characteristic X rays. The K a X rays of an element have 
an energy corresponding to the difference between the K and L 
shells and can, therefore, not lift a K electron to one of the outer 
vacant shells in the same element. However, the binding energy 
of electrons decreases with decreasing Z] therefore, the K a emis- 
sion line of an element Z has an energy rather close to but slightly 
greater than the K absorption edge of an element of somewhat 
lower Z and is, therefore, strongly absorbed by that element, but 
not by the next higher one. 



MAGNETIC DEFLECTION 223 

For example, the K a X rays of zinc (Z = 30) have a wave 
length of 1.43 A (energy 8.7 kev). The K absorption edges of 
2gCu and 2 gNi are at 1.38 A (9.0 kev) and 1.48 A (8.4 kev), respec- 
tively. Therefore, nickel is a good absorber for zinc K a X rays, 
and copper is not (figure X-2). The K a X rays of gallium 
(Z = 31), on the other hand, are strongly absorbed both in nickel 
and copper because their wave length is 1.34 A (9.3 kev), but 
they are not absorbed well in zinc whose K absorption edge is at 
1.28 A (9.7 kev). 

Both the X-ray emission lines and the absorption edges of the 
elements can be found in tables, and suitable elements can be 
chosen as absorbers to decide the origin of a set of X rays accom- 
panying a nuclear decay process. The K a X rays are usually the 
most prominent lines, but occasionally, especially with very heavy 
elements, the absorption of other lines (Kp and L) also must be 
taken into account. Absorption curves are taken with each of 
two or three neighboring elements, and their comparison usually 
brackets the energy of the emission line sufficiently to determine 
the corresponding atomic number. It should be emphasized that 
it is entirely unnecessary to use pure elements as absorbers (this 
would be difficult or impossible for some elements). Compounds 
of the desired element can be used, provided other elements hi 
the compounds do not appreciably absorb the X rays under inves- 
tigation. Light elements such as hydrogen, oxygen, nitrogen, 
carbon are very poor absorbers for energetic X rays, and oxides, 
hydroxides, or carbonates are, therefore, usually suitable as 
absorbers. A very convenient method of preparing such absorbers 
is to suspend weighed amounts of the compound in water (or in 
an organic liquid) and to filter the suspension through filter paper 
on a sintered glass or Buchner funnel; the addition of a small 
amount of binder to the suspension is sometimes useful. Even 
solutions in shallow plastic dishes are used. 

C. BETA-PARTICLE SIGN DETERMINATION 

Magnetic Deflection. It is occasionally necessary to distinguish 
between 0"~ and 0+ particles. In the electron spectrograph this 
distinction is made easily according to the polarity of the magnetic 
or electrostatic field. Without such an instrument a crude deter- 
mination may be made to distinguish a predominantly positron 



224 MEASUREMENT AND STUDY OF RADIATIONS CH. X 

from a predominantly negatron emitter. If the sample and detec- 
tor are separated by a few inches and shielded from each other 
a magnetic field from an electromagnet or sizable permanent 
magnet (2) may be arranged to bend particles of one sign around 
the shield toward the detector. 

Annihilation Radiation. Another method for the identification 
of positrons is based on the presence of the annihilation radiation 
(two 7 rays of 0.51 Mev for each positron annihilated at the end 
of its range). For this procedure the sample on its card is placed 
several centimeters from the detector and facing away from the 
detector. Then enough beryllium, paraffin, or plastic absorber is 
inserted between sample and detector to stop all ft particles. In 
this arrangement the detector should give a response to 7 rays 
plus possible annihilation radiation from about 50 per cent of the 
positrons. The other 50 per cent of the positrons leave the sample 
in the direction opposite to the detector and end their ranges far 
away from it. Now with the addition of beryllium, paraffin, or 
plastic absorber on the face of the sample away from the detector, 
an increase in detector response would indicate positrons stopping 
in this new absorber and giving rise there to annihilation radiation. 
For a sample which emits positrons and no nuclear 7 rays, and 
with the absorber-sample-absorber sandwich located at a large 
distance (compared to its own thickness) from the detector, the 
increase in response should amount to almost a factor of two. 

D. ABSOLUTE DISINTEGRATION RATES 

Alpha-disintegration Rates. The determination of absolute dis- 
integration rates with precision is difficult with a. emitters, more 
difficult with ft emitters, and still more difficult in cases in which 
only 7 rays may be measured. It is fortunate that absolute 
disintegration rates are not often needed; for most radiochemical 
work relative rates for various samples of the same substance or 
for the same sample at various times are sufficient. The methods 
for determination of absolute rates for long-lived a emitters (for 
the purpose of half-life determination) have already been men- 
tioned in chapter V, section E. The absolute counting of a par- 
ticles requires: (1) a detector of known efficiency, usually an 

2 We have used a permanent magnet designed for radar magnetrons: it 
provides a field of 1300 gauss over a volume of about 1 cubic inch. 



KNOWN GEOMETRY AND DETECTOR EFFICIENCY 225 

ionization chamber with linear amplifier adjusted to count 100 
per cent of the particles crossing the chamber; (2) a sample of 
known weight spread so uniformly thin that corrections for 
a. particles stopped in the sample are small; (3) a known geo- 
metrical factor, usually about 50 per cent, including the correc- 
tion for back-scattering of a. rays from the sample support (which 
may amount to several per cent). With reasonable care values 
reliable to about 10 per cent can be obtained, and with great care 
the errors may be reduced to about 1 per cent. The calorimetric 
method mentioned in chapter V, section E, for the measurement 
of absolute a-disintegration rates is capable of at least comparable 
accuracy. It requires a larger sample activity but makes no 
demands on the sample's geometrical arrangement, thinness, 
and the like. It does require a knowledge of the a-particle energy; 
this is conveniently obtained from the range. 

Known Geometry and Detector Efficiency. The remainder of 
this section will deal with the problem of determining absolute 
disintegration rates for samples of ft emitters. (Because y rays 
usually are found accompanying other nuclear changes deter- 
mination of their absolute intensities ordinarily may be avoided; 
if not, comparison with a similar radiation is probably the best 
that can be done.) We will consider three methods; the first 
method involves estimates of the geometrical factor and detector 
efficiency as in the a-particle case. For this procedure the detec- 
tor is almost always a Geiger counter because it counts individual 
ft rays with an efficiency that may be close to 100 per cent. How- 
ever, it is not easy to arrange the sample and counter so that the 
geometrical factor is known, and so that absorption of the ft par- 
ticles in the counter wall or window is known or negligible. The 
mica end window counter minimizes window absorption (windows 
as thin as 1.5 mg per cm 2 are available), but even with the sample 
made small and placed close to the window the geometrical factor 
will not approach 50 per cent so closely as would be desired, and 
a factor nearer 30 per cent is common. Also, the back-scattering 
for ft particles is more important than for a particles (see table 
VII^4). Some of these difficulties are minimized in a ' 'low- 
geometry" arrangement, with the sample placed several centi- 
meters from the counter window and with a shield and aperture 
to define a cone of rays directed toward the sensitive region of 
the counter. The sample may be mounted on a very thin support 



226 MEASUREMENT AND STUDY OF RADIATIONS CH. X 

(mica is often convenient) to reduce back-scattering. Inversion 
of the sample will eliminate this back-scattering effect (except 
for the very small effect of the air backing), but then the effect 
of the support as an absorber must be taken into account. 

Although the Geiger-counter efficiency for ft particles at rates 
below those involving appreciable coincidence error is ordinarily 
taken very close to 100 per cent, the ideal situation in which even 
a single ion pair produces a count may not be realized for some 
counters or for some gas-filling mixtures. The efficiency of a 
counter for a given /3-ray spectrum can be estimated by a coinci- 
dence counting experiment. The counter must have very thin 
windows on opposite sides or at opposite ends, or, better, be 
designed to require no windows at all, so that a collimated beam 
of the ft rays may be passed through the counter and into a second 
counter without appreciable loss of lower-energy rays. Then each 
count in the second counter (rate = R 2 ) shows the presence of a 
ft particle that must have passed through the first counter (count- 
ing rate = RI), and the efficiency of the first counter in recording 
these particles is given by R 12 /R2, where Ri 2 is the real net coinci- 
dence counting rate between the two counters. This determina- 
tion can be made with three counters in a row, and then the effi- 
ciency of the second counter is given by Ri23/Ri3, the ratio of 
triple coincidences to double coincidences between the first and 
third counters. (Coincidence methods have been used also for 
approximate efficiencies for y rays; in this case beryllium, paraffin, 
or plastic absorbers thick enough to stop any recoil electrons 
should be placed between counters.) 

Calibrated Detector. In the second method for absolute disinte- 
gration rates a calibrated detector is used; here the absorption of 
ft rays in the window must be small, but the only requirements on 
the geometry and counter efficiency are that these be reproducible 
and relatively insensitive to the /3-ray spectrum. The ft source 
used in calibrating these factors must have a known disintegra- 
tion rate; ordinarily a ft emitter in secular equilibrium with an 
a activity is used, and the absolute disintegration rate evaluated 
for the a emitter. Also, the energy spectrum of the ft rays should 
be as nearly as possible the same for the standard and for the 
unknown activity. The rather energetic ft particles from UX 2 in 
secular equilibrium with Ui, or possibly in transient equilibrium 
with freshly isolated UXi, are often used as standard, with a thin 



COINCIDENCE METHOD 227 

absorber to exclude Ui a particles and soft UXi (and UY) ft par- 
ticles. (Absorption of UX 2 rays in this absorber and the counter 
window is corrected for by extrapolation of a measured absorption 
curve back to zero absorber.) (3) The calibration of such a standard 
could be made by a-particle counting, but since the disintegration 
constant for Ui is already known it is necessary only to weigh 
the uranium. The National Bureau of Standards supplies (among 
other radioactivity standards) carefully standardized RaD-plus- 
RaE samples; the RaD ft particles and RaF a. particles may be 
absorbed in a 0.001-inch aluminum foil, and the medium-energy 
RaE ft particles used as a standard for calibration. As supplied 
these preparations are deposited electrolytically (in PbC^) on 
palladium-clad silver disks, and considerable back-scattering is 
to be expected. Identical blank disks are available so that the 
activity to be measured may be mounted on the same type of 
surface. 

Coincidence Method. A different method for absolute disinte- 
gration rates of ft sources whose spectra include 7 rays has recently 
come into use. This method is easily understood for a simple case 
in which one 7 quantum follows each ft decay, and the spectrum 
is simple; in this case the method is free of all the sources of 
error already discussed. Consider two counters arranged to count 
ft rays and 7 rays, respectively, with measured counting rates 
R0 and Ry, and with ft - y coincidences also measured with rate 

dN dN 

R0 7 ; then Rg = c$, Ry = c y , where the coefficients 

dt dt 

c$ and c y may be thought of as defined by these equations, and 

dN H0R, dN t t 

R*., = caCy. Now = , and the absolute count- 

w dt R0 r dt 

ing rate is determined with few assumptions. If the ft spectrum 

3 For such an extrapolation to zero absorber the relative positions of sample, 
absorber, and sensitive region of the counter are very important. If the 
sample is placed at some distance, say 2 or 3 cm, from the counter, the measured 
counting rate with an absorber directly above the sample will be appreciably 
(perhaps as much as 10 or 15 per cent) higher than with the same absorber 
placed directly under the window. In fact, the addition of a thin absorber near 
the sample may cause an increase in the counting rate. This effect results 
from the scattering of ft particles into the counter by the absorbers and is 
clearly related to the "self-scattering" effect discussed below (on p. 231). 
When a straight-line extrapolation to zero absorber thickness is wanted the 
absorbers should be placed close to the counter window. 



228 MEASUREMENT AND STUDY OF RADIATIONS CH. X 

is complex this ratio will give the correct disintegration rate only 
if the ft counter window is sufficiently thin, and that counter does 
not discriminate appreciably against any component group of 
the spectrum. However, other sources of error must be con- 
sidered. It is possible, although not likely, that strong angular 
correlation between the directions of emission of coincident ft 
and 7 rays might exist; this effect could be detected through varia- 
tion of the angle between the two counters. A source of error of 
uncertain magnitude results from detection in the 7 counter of 
bremsstrahlung produced by particles stopping in the absorber 
used to shield the 7 counter from ft rays. This effect will increase 
Ry without a corresponding increase in R0 r because the particular 
ft particles giving rise to this bremsstrahlung have little chance of 
being counted in the ft counter. The magnitude of this error is 
not easy to estimate but should be fairly small; it will be in such 
a direction as to make values found for dN/dt too large. In 
most cases it would be reduced by a lead filter interposed between 
the aluminum absorber and the 7 counter. A very appreciable 
error of the same sort would be expected for any sample emitting 
positrons because of the annihilation radiation. Also, any delayed 
7 rays which might trip the 7 counter too late to be recorded in 
coincidence would lead to too large values of dN/dt. 

The National Bureau of Standards is currently encouraging 
the intercomparison of absolute ^-disintegration rates between 
various radiochemical laboratories. The intercomparisons of 
I 131 samples which have been reported show that uncertainties of 
the order of at least 20 per cent exist in most of the measure- 
ments. This may be seen in determinations made by all three 
methods in our laboratory (at St. Louis) on one of the I 131 prepara- 
tions: 

(1) Calculated low-geometry counter 
(average of 3 samples each measured in 

2 geometries) 108 db 9 mrd/ml 

(2) Comparison with RaD + RaE standard 125 db 7 mrd/ml 

(3) Coincidence counting: 

(a) with Al shield in front of <y counter 168 zb 9 mrd/ml 
(6) with Al shield plus KG in. Pb filter 131 db 11 mrd/ml 

E. PREPARATION OF SAMPLES FOR COUNTING 

Gamma versus Beta Counting. The determination of relative 
disintegration rates for several samples of the same kind, or for 



SELF-ABSORPTION CORRECTION 229 

the same sample at several times, is the measuring problem most 
often met in radiochemical work. This problem is much simpler 
than the determination of absolute disintegration rates, but some 
points of technique arise even here. If an active sample emits 
and 7 rays the relative advantages of arranging the detector 
to respond principally to ft rays or principally to 7 rays must be 
considered; ordinarily it is not advisable to have the detector 
response divided almost equally between the two radiations. For 
a given thin sample the 0-ray effect will offer a greater sensitivity 
by a factor of very roughly 100, and this may be decisive. How- 
ever, lack of reproducibility of absorption and self-absorption 
effects can be troublesome. Because the 7 rays are almost always 
much less strongly absorbed, these effects are not so large when 
7 rays are counted, even for thicker counter walls and for much 
thicker samples; in fact, usually no uncertainties need be intro- 
duced if the various samples are prepared and mounted in some 
simple reproducible way for example, in similar solutions of the 
same volume in test tubes placed in a fixed holder near the detector. 
In experiments where the specific activity (activity per gram or 
per milligram) rather than the total activity is a limiting factor, 
the sensitivity for 7 counting may approach that for counting 
because in 7-ray counting the sample can be larger. 

Self-absorption Correction. For the measurement of soft radi- 
ation from an appreciably thick sample it is theoretically possible 
to calculate the effect of absorption of the radiation in the sample 
(self-absorption); however, no rigorous calculation is practical 
because it would require that the absorption curve for the radia- 
tion, the thickness of the sample, the solid angle subtended by 
the counter and the back-scattering effect be taken into account. 
If possible the samples should be made thin compared to the half- 
thickness value for the radiation. When thicker samples must be 
used it is advisable either to standardize the thickness at a fixed 
value or to prepare an empirical calibration curve for different 
thicknesses; in either case careful attention must be given to a 
reproducible mechanical form for the sample, and reproducibility 
should be tested by experiment. 

Work with appreciably thick samples, in this sense, is most 
frequently necessary for the low-energy ft emitters, especially 
C 14 and S 35 . Some approximate equations have been proposed 
for normal ft-r&y spectra with upper limits below 200 kev. The 
range R in milligrams per square centimeter is given very closely 



230 



MEASUREMENT AND STUDY OF RADIATIONS CH. X 



by R = jF*Vl50, where E is the upper limit in kiloelectron volts. 
The absorption coefficient AC is roughly 5//J. The half-thickness 
is then d^ = In 2/n 0.14/2. The detection coefficient c for a 
sample of thickness d is given by the approximate relation: 



600 f 



500 



400 



300 



200 



100 





5 10 15 20 25 

Thickness of BaC0 3 (mg/cm 2 ) 

FIGURE X-3. Self-absorption of C 14 /3 rays in BaCOa. [Data are taken from 
A. K. Solomon, R. G. Gould, and C. B. Anfinsen, Phys. Rev. 72, 1097 (1947).! 

O Experimental points. The solid curve is calculated from c = CQ , 

with ft * 0.29 cnAng"" 1 . 



CQ 



where CQ is an arbitrary coefficient. These equa- 



tions are intended for the geometry which is customary with end 
window counters in which a flat sample is placed close to the 
window. 

When thicker and thicker samples are prepared from an active 
material, say, for example, BaCO 3 containing C 14 , the measured 
counting rate at first increases because of the greater total activity 
in the sample and then approaches a constant value. This "satu- 



USEFUL SAMPLE-MOUNTING TECHNIQUES 231 

ration value" is clearly not a measure of the total activity of the 
sample, but rather is related to the activity of the amount of 
sample material in an upper layer no thicker than the particle 
range /2, and is thus a measure of the specific activity of the sample 
material. This fact is sometimes used to advantage in the meas- 
urement of the low-energy tracers; no correction for self-absorp- 
tion is applied, and it is only necessary to measure the activities 
of thick samples of the same uniform area and the same chemical 
composition. Indeed in many tracer experiments the specific 
activity is more directly significant than the total activity. The 
minimum sample thickness required for this type of measurement 
is clearly not more than the range 72, and for most practical pur- 
poses 0.75/2 is adequate because of the very small relative contri- 
bution of the lowest layers and the additional absorption due to 
the counter window. 

In measurements of very thin samples an effect known as 
"self-scattering" has recently been noticed. A slightly thicker 
sample may give a counting efficiency higher by several per cent 
than the infinitely thin sample; this has been attributed to scat- 
tering of rays into the detector by the material of the sample, 
although the observed effects may in part have other causes, 
such as mechanical loss of sample. Figure X-3 shows the appar- 
ent specific activity of BaCO 3 precipitates, containing C 14 , as a 
function of the sample thickness. The experimental values fit 

1 - e-* 
the relation c = c rather well, except that at very low 

Md 
thicknesses the measured activities fall off (perhaps because the 

samples become too thin to exhibit self-scattering). 

Useful Sample-mounting Techniques. If thin samples are to be 
used, so that the self-absorption correction is small, it is usually 
necessary to arrange that they be spread uniformly over the 
sample-mounting area. If a solution is merely evaporated to 
dryness in a very shallow cup (4) of the right size the deposit left 
will probably show very obvious lack of uniformity; most of the 

4 Some workers use shallow flat-bottom porcelain ashing dishes. Very shal- 
low cups stamped from a suitably inert metal foil are better for some purposes. 
Actually cups are not necessary since a flat disk with a smooth (and possibly 
greased) edge will hold water. Most often we use microscope cover glasses, 
support them on smaller bits of cardboard, and carry out the evaporations 
under an ordinary infrared lamp. 



232 MEASUREMENT AND STUDY OF RADIATIONS CH. X 

residue will ordinarily be found near the edge. If the active 
substance is first precipitated and the slurry evaporated, pref- 
erably with stirring, a much improved deposit usually results. 
In another procedure the slurry is added and dried a little at a 
time. It is sometimes helpful to place a circular piece of cigarette 
paper, slightly smaller than the dish or disk, on the flat surface; 
the solution or slurry is allowed to spread over the paper and then 
to dry; when dry this type of paper weighs about 1 mg per cm 2 . 

Other methods of preparing samples are sometimes more con- 
venient, especially if the bulk of the deposit or the volume of 
solution or suspension is large. Filtration on a small Buchner or 
Gooch filter can give reasonably uniform and very nearly quanti- 
tative deposits of precipitates on the filter paper. If the precipi- 
tate and filter are washed finally with alcohol (6) the bits of pre- 
cipitate creeping up the sides of the filter are likely to be washed 
down; also the paper is then more easily dried, either with or 
without an ether wash. A glass chimney held tightly against the 
paper in a Buchner funnel with perforated surface ground flat 
will help to confine the precipitate to a definite area. Sedimenta- 
tion of a precipitate followed by washing and drying can be used 
to give uniform deposits. Special cells with demountable bottoms 
are employed for this purpose so that the sample may be removed 
on its mounting for measurement. Sedimentation cells of this 
type fitted into laboratory centrifuge cups may be used to give 
harder deposits in very much less time. 

Samples prepared in any of these ways should be thoroughly 
dry before measurement, otherwise the observed activity will 
grow with time as water evaporates with a consequent reduction 
in self-absorption of the softer rays. Samples may be found 
subject to loss of precipitate through powdering when dry; this 
can be especially troublesome if the active dust should contaminate 
a measuring instrument. To avoid this effect a few drops of a 

5 The use of alcohol (presumably to lower surface tension) has other applica- 
tions in the manipulation of samples. For example, when a precipitate is 
centrifuged down in a semimicro cone some is almost always trapped in the 
liquid surface (meniscus); addition of a few drops of alcohol on top of the 
solution followed by another centrifugation will usually bring down most of 
this precipitate. In the mounting of slurries as described previously a few 
drops of alcohol as a wash will often clean the residual slurry from the transfer 
micropipet (a glass tube drawn down to have a long tip less than 1 mm in 
diameter and fitted with a rubber bulb, like an "eye dropper"). 



USEFUL SAMPLE-MOUNTING TECHNIQUES 233 

solution of Duco cement in acetone may be used to wet the sample 
when it is first dried; the concentration of the cement in the ace- 
tone solution should be so small that only about 0.1 mg per cm 2 
of its dry residue will be left on the sample. 

Dry powders may be compacted into disks with a piston-and- 
cylinder arrangement, or simply pressed smoothly into shallow 
depressions cut in aluminum or other suitable sample-mounting 
cards. These techniques cannot be recommended for the prepara- 
tion of very thin samples for measurements of soft radiations. 
If the radiation is penetrating, solutions of active materials may 
be counted directly in a thin-walled glass jacket slipped over a 
tubular counter, or better in an outer jacket built as a part of a 
glass counter tube. A dipping counter is arranged to permit 
immersion of the whole counter (except an end for electric con- 
nections) in the solution to be measured. Radiations from rela- 
tively thick layers of solutions may be counted reproducibly only 
if the solution densities are comparable, and if the relative com- 
position of the solutions in terms of elements of various atomic 
numbers is kept approximately constant, especially for 7-ray 
measurements (see chapter VII, section C). 

If the radioactive element is a metal like copper excellent sam- 
ples for measurement may be prepared by electrodeposition. 
Other types of ions may often be deposited by different electrode 
reactions under suitable conditions. For example, lead may be 
deposited on an anode as Pb0 2 from alkali plumbite solutions. 
Insoluble hydroxides may be deposited from neutral solutions on 
cathodes because of the liberation of hydroxide ions there: 
H 2 + e~ = 3/2H 2 + OH~~. Insoluble ferrocyanides may form 
on a cathode from the reduction of some metal ferricyanide solu- 
tions. Metal fluorides may be precipitated in adherent form for 
some metals that can be oxidized or reduced at an electrode from 
a fluoride-soluble to a fluoride-insoluble state; for example, UF 4 
may be deposited in this way and then ignited in air to U 3 8 . 

Some specialized sample-spreading techniques have been devel- 
oped to a high state of the art for the preparation of standard 
foils, especially of uranium. In one of these a wetting agent, 
tetraethylene glycol (TEG), is used to improve spreading on 
evaporation. In this procedure a platinum foil of the right dimen- 
sion is prepared with a Zapon lacquer border, then a few micro- 
liters of TEG per square centimeter are applied, and the uranium 



234 MEASUREMENT AND STUDY OF RADIATIONS CH. X 

as chloride in dilute hydrochloric acid solution is added. Evapora- 
tion is carried out under an infrared lamp, with some rotation of 
the foil to insure mixing. Finally, the plate is ignited to red heat 
for a few minutes to convert the deposit to an adherent highly 
colored layer of UaOg. This is rubbed gently with lens paper 
and then may be weighed to establish the uranium content. This 
procedure is suitable for the preparation of uranium foils of the 
order of 0.1 mg per cm 2 and with larger amounts of TEG can be 
used up to about 1 mg per cm 2 . Another technique uses uranyl 
nitrate dissolved in alcohol (about 50 mg per ml), with about 
1 per cent of Zapon lacquer added; this mixture is painted on the 
platinum or aluminum foil with a brush; each coat is dried, ignited 
to U 3 O 8 , and rubbed with lens paper. Thick foils, weighable for 
uranium content, may be built up by this method. Other ele- 
ments with alcohol-soluble salts may give satisfactory plates 
with the same technique. Thin standard boron films are some- 
times wanted. These can be prepared by the decomposition of 
B 2 H 6 on hot tungsten at about 900C under carefully controlled 
conditions. 

EXERCISES 

1. The 7-ray activities of two portions of a solution of Co 60 were 
found under identical conditions to be: 350 counts per min for a 0.10-cc 
portion, and 9900 counts per min for a 3.00-cc portion. 

(a) What would you expect for the counting rate of a 2.00-cc por- 
tion? 

(b) What is the approximate dead time of the Geiger counter? 
Assume all other resolving times are negligibly small. Neglect self- 
absorption effects. Answer: (b) 346 jus. 

2. From data on X-ray spectra and X-ray absorption coefficients (in 
the Chemical Rubber Company Handbook, for example) locate the critical 
absorbers for the identification of the X rays following K capture (a) in 
A 37 , (6) in Pd 101 . 

3. Calculate the field strength necessary to give a 2.00-cm radius of 
curvature for the path of a /3 particle of 0.40 Mev. 

4. (a) Estimate the half -thickness d\/^ for a 1-Mev 7 ray in (1) lead, 
(2) aluminum, (3) beryllium, (b) Make the same estimates for a 50-kev 
7 ray in aluminum and beryllium. 

5. A sample of I 131 , contained in a 6.0-mg precipitate of Agl uniformly 
distributed over a circular area 1.00 cm in diameter, was placed on the 



REFERENCES 235 

fourth (lowest) step of the sample holder described on page 219 and counted 
with an end-window (mica 3 mg per cm 2 ) Geiger counter with cathode 
inner diameter \Y% inch. It gave a net counting rate of 2000 counts per 
min. Addition of a few milligrams per square centimeter of aluminum 
absorber gave a reduction in measured activity corresponding to a mass 
absorption coefficient /z/p = 25 cm 2 g -1 . Estimate as well as you can the 
absolute strength of this sample (in rutherfords). 

Answer: We estimate 2.4 X 10 ~ 8 rd. 

6. A sample of 6.6-min Cb 94 is counted in identical geometrical posi- 
tions, first with a neon-filled thin-window counter, then with an otherwise 
identical krypton-filled counter. When corrected for the decay between 
measurements, the counting rate in the second counter is about 30 per cent 
higher than that in the first. How can you explain this fact? 

7. If you wished to count the Rb 88 /3 rays without interference from 
the Rb 86 or Rb 87 /J rays you might interpose an aluminum absorber. 
(a) What thickness (in grams per square centimeter or milligrams per 
square centimeter) of aluminum would be just sufficient to stop the 
Rb 86 /3's? (b) By what factor would this absorber reduce the intensity 
of the Rb 88 /3's? (The mass absorption coefficient for this radiation you 
may estimate from the very rough rule that the range is about seven 
times the half-thickness.) 

REFERENCES 

M. D. KAMEN, Radioactive Tracers in Biology, New York, Academic Press, 
1947. 

D. W. WILSON, A. 0. NIER, and S. P. REIMANN, Preparation and Measurement 
of Isotopic Tracers, J. W. Edwards Co., 1946. 

W. F. LIBBY, "Measurement of Radioactive Tracers/' Ind. and Eng, Chem., 
Anal Ed. (Analytical Chemistry) 19, 2 (1947). 

M. L. WIEDENBECK, "The Absolute Strength of Radioactive Sources/' Phys. 
Rev. 72, 974 (1947). 

P. E. YANKWICH and J. W. WEIGL, "The Relation of Backscattering to Self- 
Absorption in Routine Beta-Ray Measurements/ 7 Science 107, 651 (1948). 

L. R. ZUMWALT, "Absolute Beta Counting Using End-Window Geiger-Mueller 
Counter Tubes," U. S. Atomic Energy Commission Declassified Document 
MDDC-1346 (available for 10 cents from Document Sales Agency, 
P. 0. Box 62, Oak Ridge, Tenn.). 

M. CALVIN, C. HEIDELBERGER, J. C. REID, B. M. TOLBERT, and P. F. YANK- 
WICH, Isotopic Carbon, New York, John Wiley & Sons, 1949. 



XI. IDENTIFICATION, CONCENTRATION, AND ISO- 
LATION OF RADIOACTIVE SPECIES 

A. GENERAL REMARKS 

In previous chapters we have discussed the production of radio- 
active species, the radiations emitted by them, and the instru- 
ments and techniques for their detection and measurement. 
Before we can proceed to consider some of the chemical applica- 
tions of radioactive materials we should give some attention to 
the problem of identifying, purifying, and isolating a given radio- 
active species following its production in a nuclear reaction. 

In this connection the radiochemist may be confronted with 
one of two tasks. He may need to prepare a known reaction 
product free from other radioactive contaminants and sometimes 
free from certain inactive impurities and in a specified chemical 
form for use in subsequent experimentation. Or he may wish to 
identify a hitherto unknown or unidentified radioactive species 
as to its atomic number, mass number, half-life, and radiation 
characteristics. In both cases chemical separations are usually 
required for two reasons: (1) In almost any nuclear bombardment 
more than one type of reaction occurs, and, therefore, the reac- 
tion products have to be separated; (2) impurities present in the 
target material usually give rise to radioactive products. Apart 
from the ordinary chemical impurities, further contamination is 
often introduced in the bombardment procedure; particularly in 
cyclotron bombardments, where for cooling purposes the target 
material is usually soldered, pressed, or bolted onto a copper or 
brass backing plate, transmutation products of the sample con- 
tainer, backing material, solder, and fluxes must be considered. 

In a slow-neutron bombardment the only type of reaction pro- 
duced in almost any target element is the n, -y reaction. There- 
fore, if an element of sufficient purity is bombarded with slow 
neutrons, chemical separations are often not required. However, 
in this case the radioactive product is isotopic with the target, 
and it is sometimes desirable to free the product isotope from the 

236 



CROSS BOMBARDMENTS 237 

bulk of the target material in order to obtain high specific activities. 
Special techniques developed for this purpose are discussed in 
section D. 

The chemical techniques used for the separation and isolation 
of radioactive species are essentially the same whether an unknown 
nuclide is to be identified or a known product to be prepared. In 
the former case chemical identification is usually not sufficient, 
and we shall defer a discussion of the chemical separation tech- 
niques to section C. First we take up the question of how to 
obtain the additional information necessary for the unambiguous 
identification of radioactive species. 

B. IDENTIFICATION OF NUCLEAR-REACTION PRODUCTS 

Cross Bombardments. For the purposes of this section we 
assume that the atomic number of a given radioactive species 
can be determined by chemical analysis and turn our attention 
to methods for the assignment of the correct mass number. The 
only case in which the mass number of a reaction product is prac- 
tically uniquely determined in a single bombardment is the slow- 
neutron activation of a single nuclear species. For example the 
slow-neutron bombardment of arsenic (with the single stable 
species As 75 ) produces a 26.8-hr ^-emitting arsenic isotope which 
is, therefore, readily assigned to As 76 . 

One can frequently make a mass assignment by investigating 
whether or not the radioactive species being studied is formed 
in a number of different types of bombardments; this is called 
the method of cross bombardments. For this purpose target 
elements as well as projectiles may be varied. In each bombard- 
ment the possible products are limited by the stable isotopes of 
the target element and by the types of reactions possible with 
the projectile and energy used. As an illustration consider some 
radioactive isotopes of strontium; figure XI-1 displays the stable 
nuclides in the region of strontium. Slow-neutron activation of 
strontium produces strontium isotopes with 2.7-hr and 54-day 
half-lives; either or both of these activities might be assigned to 
any of the isotopes Sr 85 , Sr 87 , Sr 88 , Sr 89 , since they are produced 
by neutron capture from the stable strontium isotopes. The 
fact that the same two activities are produced by fast neutrons 
(say 15 Mev) in zirconium, presumably by n, a reactions, elimi- 



238 IDENTIFICATION, CONCENTRATION, ISOLATION CH. XI 

nates Sr 85 . The fast-neutron bombardment of yttrium (say with 
10- or 15-Mev neutrons) produces only the 54-day strontium, 
presumably by n, p reaction, and this activity is therefore assigned 
to Sr 89 . The 2.7-hr activity is then to be assigned to an isomeric 
state of Sr 87 or Sr 88 . The fact that 2.7-hr strontium is also pro- 
duced by proton bombardment of rubidium by p, n reaction leads 
to its assignment to Sr 87 *. 

In the interpretation of the results of cross bombardments the 
relative abundances of isotopes in the target elements often have 
to be considered. For example, failure to observe a certain reac- 



40 
39 
38 
37 
36 


























Zr 90 


Zr" 


Zr" 2 




Zr" 




Zr 96 
























Y 89 




























Sr 84 




Sr" 


Sr" 


Sr 88 
































86 

Rb 




87 

Rb 




















78 

Kr 




80 

Kr 




82 

Kr 


83 

Kr 


84 

Kr 




86 

Kr 























78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 

A 
FIGURE XI-1. Naturally occurring nuclear species in the region of strontium. 

tion product may simply be due to low abundance of the isotope 
from which that product could have been made. 

Effects of Bombarding Energies and Bombardment Times. The 
energies of the bombarding particles used in cross bombardments 
are very important in the analysis of the results. An excitation 
function may help very much to determine the type of reaction 
involved, through comparison with similar excitation functions 
for known reactions (see, for example, the a-particle reactions of 
silver discussed in chapter III, page 64). Even a knowledge of 
the energy of the bombarding particle together with an approxi- 
mate yield of the product can usually be of value for the mass 
assignment. For example, if a certain activity of element Z + 1 
is produced in good yield in the bombardment of element Z with 
5-Mev deuterons, the reaction can hardly be anything but a d, n 
reaction; if the deuteron energy had been 20 Mev, a d, 2n reaction 
or perhaps a d, 3n reaction might more likely have been responsi- 
ble. Referring again to figure XI-1, we see that the 2.7-hr 
strontium cannot be expected to result from the bombardment 
of rubidium with 5-Mev deuterons if it is Sr 87 , but it could be 
formed in such a bombardment if it were Sr 88 . 



BEAM CONTAMINATION AND SECONDARY REACTIONS 239 

In the case of neutron-induced reactions the effect of neutron 
energy on the possible reactions was discussed in chapter III, 
section C. Thermal neutrons produce, in general, only n, 7 reac- 
tions. For neutrons of 1 or 2 Mev, capture cross sections are 
usually much smaller than for thermal neutrons, but formation 
of excited states by n, n reactions becomes more important at 
those energies. At 8 or 10 Mev n, 2n reactions may set in. Thus 
in our example the yield of 54-day strontium in neutron bombard- 
ments of strontium would be expected to drop off more with an 
increase in neutron energy from say 1 ev to 15 Mev than that of 
the 2.7-hr Sr 87 , because the former is made by n, y reaction only, 
whereas the latter is produced in the reactions Sr 86 (n, y), 
Sr 87 (n, n), and Sr 88 (n, 2n). (1) It is evident that with increasing 
bombardment energies the mass assignment problems become 
increasingly difficult because of the greater variety of possible 
reactions. 

If several radioactive isotopes of the same element are pro- 
duced in one bombardment the study of the properties of one 
is often greatly hindered by the presence of others. When the 
half-lives are sufficiently different, this difficulty can be mini- 
mized by proper choice of the bombardment and "cooling" times. 
In our example of the strontium isotopes it may be desirable to 
bombard one sample with slow neutrons for many weeks to build 
up sufficient 54-day activity; the 2.7-hr activity can then be 
allowed to decay practically completely in a day or two before 
the 54-day activity is studied. Another sample may be bom- 
barded for about 2 or 3 hr which builds up the 2.7-hr activity to 
about 50 per cent of its saturation value while producing only a 
very small fraction of the saturation amount of the long-lived 
isotope (see chapter V, section B, for the quantitative treatment). 

Beam Contamination and Secondary Reactions. In connection 
with the subject of cross bombardments it should be noted that 
extraneous bombarding particles or secondary particles produced 

1 It should be noted that no fast-neutron irradiation can be completely free 
of slow neutrons because in traversing any matter some neutrons are always 
slowed down. Cadmium shielding is usually employed to prevent thermal 
neutrons from reaching a sample. The difference between the activities pro- 
duced with and without cadmium is often taken as an approximate measure 
of the thermal-neutron effect. The cadmium shield does not completely 
eliminate n, 7 reactions because they may occur with appreciable cross sections 
at energies above the cadmium resonance. 



240 IDENTIFICATION, CONCENTRATION, ISOLATION CH. XI 

by bombardment of the targets may sometimes give rise to nuclear 
reactions to an extent which can interfere with the recognition of 
the primary-reaction products. For example, a-particle beams 
in cyclotrons are commonly contaminated with small amounts of 
deuterons (from residual deuterium in the ion source); because 
deuteron cross sections are often much higher than a-particle 
cross sections the deuteron-produced activities may actually 
exceed in intensity those from a particles. 

The neutrons produced in the bombardment of targets with 
charged particles or 7 rays usually give rise to nuclear reactions 
in the target material. The products of these reactions may be 
confused with those that are found or expected to be produced by 
the primary bombarding particle. For example, in the deuteron 
bombardment of a thick sodium target to make Na 24 by the 
d, p reaction, the Na 24 activity is found at depths beyond the 
range of the deuterons, because it can be produced there by rc, 7 
reaction. Even protons and probably a particles of sufficient 
energy to cause secondary reactions may be produced in bombard- 
ments with high-energy projectiles. 

Use of Isotope Separations. In many cases the mass number of 
a radioactive species is uncertain because it is not known from 
which isotope of the target element the activity has been produced. 
The bombardment of separated isotopes, or of isotopic mixtures 
sufficiently enriched in some isotope, is, therefore, of great advan- 
tage. Enrichment or impoverishment of an isotope by a factor of 
two or even less may be sufficient for this purpose; by comparing 
the yield of the activity of interest from samples of normal and 
altered isotopic composition one can deduce the origin of the 
activity. For a long time the assignment of the 37-min chlorine 
activity produced by slow neutrons or by deuterons in chlorine 
(stable isotopes Cl 35 and Cl 37 ) was uncertain and could not be 
readily determined by cross bombardments; but when it was 
shown that this activity was not produced by slow-neutron bom- 
bardment of a sample of almost pure Cl 35 the 37-min period could 
be assigned to Cl 38 rather than Cl 36 . Enriched lead isotopes have 
recently been used to aid in the assignments of a number of radio- 
active isotopes of lead, bismuth, and polonium. 

Until recently, separated or enriched stable isotopes of most 
'elements were not available in appreciable quantities, and they 
are only now coming into extensive use for the identification of 



TARGET MATERIAL AND EFFECT OF IMPURITIES 241 

radioactive products. Some fifty different enriched stable iso- 
topes have now been made available in milligram or gram quan- 
tities through the United States Atomic Energy Commission; 
some of these have already been used in assignment studies, and 
the technique will undoubtedly be employed much more in the 
future. 

Isotope separation can be used in an even more direct way for 
the identification of radioactive species: the reaction products 
themselves may be subjected to an isotope separation process. 
This procedure was used after the bombardment of separated 
nickel isotopes to establish the assignment of a 1.75-hr period to 
Co 61 by calutron analysis. The mass numbers of quite a number 
of long-lived activities, particularly fission products and rare 
earths, have been established by an interesting modification of 
mass-spectrographic analysis. The material containing the active 
isotope of interest is sent through an ordinary mass-spectrograph, 
and the ions are collected on a photographic plate. After develop- 
ment that plate is placed face to face with another unexposed 
plate called the transfer plate; the particles from a radioactive 
isotope deposited along a narrow line on the first plate will blacken 
the transfer plate, and from the position of the exposed line on 
the transfer plate the mass number of the active isotope can be 
deduced. By exposing successive transfer plates to the first plate 
at various times and in each case finding the exposure time neces- 
sary to get the same degree of blackening, one can obtain a rough 
check on the half-life of the activity. To establish the half-life 
more accurately the area of the original plate on which the activity 
is located can be counted over a period of time. The mass-spectro- 
graphic technique is limited to half-lives longer than about 1 hr. 

Target Material and Effect of Impurities. When the product of 
a particular nuclear reaction is to be studied, it is important that 
the target not contain other elements from which the same product 
might result. As an obvious example, in a study of the produc- 
tion of 34-hr bromine (Br 82 ) by neutron bombardment of rubidium 
one would not wish to use rubidium bromide as a target. For the 
same reason impurities of neighboring elements, such as iridium 
impurity in an osmium target, may lead to misinterpretation of 
results. In general, it is advisable to use free elements as targets; 
however, sometimes the bombardment of a compound is indi- 
cated, for example, if the element is too reactive, or in a physical 



242 IDENTIFICATION, CONCENTRATION, ISOLATION CH. XI 

state unsuitable for bombardment, or if the dissolving of the 
element would make a slow step in the subsequent chemical- 
separation procedure. 

As was pointed out at the beginning of this chapter, the presence 
of some impurities in a target is in most cases unavoidable. If 
the impurities present are known, then the chemical separation 
procedure can be designed to separate the products of interest 
from the products likely to result from the bombardment of the 
impurities. The purity requirements for the target material may 
become very exacting if a reaction of low cross section is studied, 
for then the expected product yield is small, and care must be 
taken that comparable amounts of the same species are not pro- 
duced from an impurity by a much more probable reaction. For 
example, the formation of Mg 23 from Al 27 by 7, p3n reaction 
could be studied only with aluminum very free of magnesium, 
because the Mg 24 (7, n) Mg 23 reaction has a much larger cross 
section. 

Importance of Decay Relationships and Radiations. Both in the 
study of new species and in the isolation of known tracer materials 
the half-life is generally considered the chief characteristic of a 
radioactive isotope. Complete resolution of the experimentally 
found decay curve into its components is, therefore, usually desir- 
able, and this may require reproducible activity measurements 
over long periods of time (months or even years). Some special 
techniques used in the determination of half -lives have already 
been discussed in chapter V, section E. 

If a radioactive product of a nuclear reaction decays into 
another radioactive product, the genetic relationship may be 
investigated by studies of the growth and decay curves of frac- 
tions chemically separated at successive times. An understand- 
ing of genetic relationships often helps in the assignment of the 
activities. This is particularly important in the fission-product 
decay chains. For example the fission product chain of mass 89 
was identified with that mass number because its last active mem- 
ber was shown to be the 54-day Sr 89 already discussed. The 
decay chain containing 275-day cerium was assigned the mass 
number 144 by mass-spectrographic determination of the mass 
of that long-lived cerium. 

Half-life and atomic number are frequently not sufficient to 
characterize a radioactive species. It happens rather often that 



IMPORTANCE OF DECAY RELATIONSHIPS AND RADIATIONS 243 

two isotopes of the same element have not very different half- 
lives. In that case the isotopes can usually be distinguished by 
the types and energies of the radiations they emit. We may use 
again our example of the strontium isotopes; an investigation 
of the radiations emitted by the slow-neutron-bombarded stron- 
tium sample after the 2.7-hr period has decayed reveals, in addi- 
tion to the 1.5-Mev /3~ particles of the 54-day Sr 89 , some 7 rays 
and characteristic rubidium X rays. On closer examination these 
last two radiations are found to follow a 65-day half-life. The 
54-day and 65-day periods could certainly not be resolved in a 
gross decay curve of the neutron-bombarded sample. The pres- 
ence of rubidium X rays shows that the 65-day isotope decays 
to rubidium by p + emission or K capture; no positrons are ob- 
served, and the process must be an electron capture. From the 
evidence the assignment is probably to Sr 85 or Sr 87 ; the latter is 
ruled out by the fact that the 65-day species is not formed by 
n, a reaction from Zr 90 . Proton bombardment of rubidium pro- 
duces the 65-day period (but not the 54-day Sr 89 ) as well as the 
2.7-hr Sr 87 * already discussed, and in addition a 70-min strontium 
which also fails to appear in the fast-neutron bombardment of 
zirconium. 

Using these four strontium activities as examples we shall illus- 
trate how a study of the mode of decay of an isotope helps in its 
assignment. The fact that the 54-day isotope emits fi~ particles 
rules out its assignment to any mass number less than 89 : a stron- 
tium nucleus of mass 88 or less would by /?~ decay move away 
from rather than toward the stability region. (2) A study of the 
radiations from the 70-min and 2.7-hr activities reveals that both 
emit strontium X rays, indicating that these periods are asso- 
ciated with isomcric transitions. The fact that the 65-day isotope 
decays by electron capture does not allow its assignment to any 
mass number greater than 87. From the facts listed in this and 
in the previous paragraph the 65-day and 70-min periods can 
both be assigned to Sr 85 ; the 70-min period is associated with an 
isomeric transition to the lower state, which in turn decays by 
K capture to Rb 85 with a 65-day half-life. Figure XI-2 is iden- 
tical with figure XI-1 except that the strontium activities dis- 
cussed are now entered at their proper places. 

2 This type of restriction is without known exception; however, it is not 
absolute a nucleus in an excited state could transform to an unstable nucleus 
which then returns to the ground state of the first nucleus. 



244 IDENTIFICATION, CONCENTRATION, ISOLATION CH. XI 

On the basis of arguments similar to the ones just presented 
the modes of decay of active isotopes can generally be used as an 
aid in their assignment. Some attempts have been made to 
correlate not only modes of decay but also half-lives and decay 
energies with the positions of isotopes with respect to the stability 
region. Semiquantitative predictions based on binding-energy 
formulas like equation II-4 can be useful in the interpretation of 
experimental data. 



40 
39 

N38 

37 
36 


























90 

Zr 


91 

Zr 


92 

Zr 




94 

Zr 




96 

Zr 
























89 

Y 




























84 

Sr 


/Urn 
IT 


86 

Sr 


2./ 
h 
IT 

Sr 87 


88 

Sr 


54d 

/T 
















65d 
K 
















Rb 




87 

K? 




















78 

Kr 




80 

Kr 




82 

Kr 


83 

Kr 


84 

Kr 




86 

Kr 























78 79 80 81 82 83 84 85 86 87 08 89 90 91 92 93 94 95 96 

A 

FIGURE XI-2. The four strontium activities discussed in the text are snown 

at their proper places (with half-lives and modes of decay) in the chart of 

naturally occurring nuclear species. 



C. CHEMICAL ISOLATION OF NONISOTOPIC SPECIES 

Comparison with Ordinary Analytical Practice. In many re- 
spects the chemical separations which the radiochemist carries 
out on irradiated targets are very similar to ordinary analytical 
procedures. However, there are a number of important differ- 
ences. One of these is the time factor which is often introduced 
by the short half-lives of the species involved. An otherwise very 
simple procedure such as the separation of two common cations 
may become quite difficult when it is to be performed, and the 
final precipitates are to be dried and mounted, in a few minutes. 
Where the usual procedures involve long digestions, slow filtra- 
tions, or other slow steps, completely different separation pro- 
cedures must be worked out for use with short-lived activities. 
Ingenious chemical isolation procedures taking as little as 30 sec 
have been developed. 

In radiochemical separations, at least those subsequent to 
bombardments with projectiles of moderate energies, we are usu- 



COMPARISON WITH ORDINARY ANALYTICAL PRACTICE 245 

ally concerned with several elements of neighboring atomic num- 
bers. Thus the procedures given in complete schemes of 
qualitative analysis can often be modified and shortened. On 
the other hand, the separation of neighboring elements often pre- 
sents considerable difficulties as can readily be seen by considering 
such groups as Ru, Rh, Pd or Hf, Ta, W or any sequence of 
neighboring rare earths. In the cases of very high-energy reac- 
tions and of fission the products are spread over a wide range of 
atomic numbers. In these cases the separation procedures either 
become more akin to general schemes of analysis or, more fre- 
quently, are designed for the isolation of a single element away 
from all the others; the latter type of procedure is required par- 
ticularly when a short-lived substance is to be isolated, and for 
such cases many specialized techniques have been developed. 

High yields in radiochemical separations are not always of great 
importance, provided the yields can be evaluated. It may be 
more valuable to get 50 per cent (or perhaps even 10 per cent) 
yield of a radioactive element separated in 10 min than to get 
99 per cent yield in 1 hr (this is certainly so if the activity has 
a half-life of 10 or 20 min). High chemical purity may or may not 
be required for radioactive preparations, depending on their use; 
for identification and study of radioactive species and for many 
chemical tracer applications it is not important; for most biolog- 
ical work it is. On the other hand radioactive purity is usually 
required and often has to be extremely good. 

Some specific effects of the radiations from radioactive sub- 
stances on the separation procedures may be noted. In case of 
very high activity levels (say 10 5 rd or more of ft particles per 
milliliter of solution) the chemical effects of the radiations, such 
as decomposition of water and other solvents, and heat effects, 
may affect the procedures. However, this is generally not so 
important as the fact that even at much lower activity levels, 
especially in the case of y-ray emitters, the person carrying out 
the separation receives dangerous doses of radiation unless pro- 
tected by shielding or distance. (3) For this reason it is often 
necessary to carry out separations behind lead shields and to 

3 Remember that radiation dosage falls off as the inverse square of the dis- 
tance from the source. The following are the dosages in milliroentgens per 
hour received at 10 cm distance from 1 rd of each of several typical y emitters: 
Na 24 5.2; Mn 64 1.3; Fe 59 1.8; Co 60 (5.3 year) 3.5; Br 82 4.1; I 128 0.05; I 130 3.4. 



246 IDENTIFICATION, CONCENTRATION, ISOLATION CH. XI 

perform operations with the use of tongs and other tools; for 
very high activity levels (say in excess of 10 5 or 10 6 rd of y activity) 
more elaborate remote-control methods are necessary. It is obvi- 
ous that separation procedures are more difficult under these 
conditions and in many cases have to be modified considerably 
to adapt them for remote-control operation. References to dis- 
cussions of the safe handling of radioactive materials and of 
appropriate health-protection measures may be found at the end 
of this chapter. 

Carriers. The mass of radioactive material produced in a nuclear 
reaction is generally very small. Notice, for example, that 1 rd 
of 37-min Cl 38 weighs about 2 X 10~ 13 g, 1 rd of 54-day Sr 89 weighs 
about 10~ 9 g, and 1 rd of 6000-year C 14 weighs about 6 X 10~ 6 g. 
Thus the substance to be isolated in a radiochemical separation 
may often be present in a completely impalpable quantity. (4) It 
is clear that ordinary analytical procedures involving precipita- 
tion and filtration or centrifugation may fail for such minute 
quantities. In fact, solutions containing the very minute concen- 
trations of solutes which can be investigated with radioactive 
tracers behave in many ways quite differently from solutions in 
ordinarily accessible concentration ranges; this subject is treated 
in the next chapter. Usually some inactive material isotopic with 
the radioactive transmutation product is deliberately added to 
act as a carrier for the active material in all subsequent chemical 
reactions. Most often it is not sufficient to add carrier only for 
the particular transmutation product to be isolated; frequently 
it is necessary to add carriers also for other activities which are 
known or assumed to be formed, including those which derive 
from target impurities. It is worth noting also that certain con- 
taminating activities such as P 32 (14.3 days) and Na 24 (14.8 hr) 
appear almost inevitably on all targets in cyclotrons which are 
often used for phosphorus and sodium bombardments. A special 
step for the removal of P 32 such as the precipitation of BiP0 4 in 
1 N nitric acid is therefore often necessary. 

It is in many cases not necessary to add carriers for all active 
species present, because several elements may behave sufficiently 
alike under given conditions so that traces of one will be carried 

4 Actually the mass of an element formed in a nuclear reaction is often 
exceeded by that of the inactive isotopes of the same element present as an 
impurity in the target and in the reagents used in the separation procedure. 



CARRIERS 247 

by macroscopic quantities of another. For example, an acid- 
insoluble sulfide such as CuS can usually be counted on to carry 
traces of ions such as Hg 4 "^, Bi" 1 "*" 1 ", Pb 4 "^ which also form acid- 
insoluble sulfides. On the other hand, since many precipitates 
(such as BaS04 or Fe(OH) 3 ) tend to occlude or adsorb many 
foreign substances, it is usually necessary to add carriers not only 
for ions to be precipitated but also for ions to be held in solution 
when other ions are precipitated. For example, if a zinc activity 
is to be separated from a ferric solution by ferric hydroxide pre- 
cipitation with excess ammonia, all the zinc will not be left in 
solution unless zinc carrier is present. The carrier in such cases 
is sometimes referred to as hold-back carrier. We shall later 
discuss cases where carriers are unnecessary. 

We have mentioned before that extreme radioactive purity is 
often very important. Frequently the desired product has an 
activity that constitutes only a very small fraction of the total 
target activity; yet this product may be required completely free 
of the other activities. Such extreme purification is usually quite 
readily attained by repeated removal of the impurities with suc- 
cessive fresh portions of carrier, until the fractions removed are 
sufficiently inactive. This so-called "washing-out" principle may 
be illustrated by the separation of a weak cobalt activity from 
radioactive copper contamination. Cobalt and copper carriers 
are added to a 0.3 N HC1 solution of the activities, CuS is pre- 
cipitated, and filtered or centrifuged off, excess H 2 S removed by 
boiling, then fresh copper carrier added to the filtrate and the 
procedure repeated until a final CuS precipitate no longer shows 
an objectionable amount of activity. The same principle can be 
applied to other than precipitation reactions. Radioactive iron 
impurity might be removed by repeated extraction of ferric chlo- 
ride from 6 N HC1 into ether, with fresh portions of FeCl 3 carrier 
added after each extraction. In applying the washing-out method 
one must, of course, make sure that the desired product is not 
partially removed along with the impurity in each cycle. If the 
washing out works properly, the activities of successive impurity 
fractions should decrease by large and approximately constant 
factors, provided the conditions in each step are about the same. 

In order that an added inactive material serve as a carrier for 
an active substance, the two must generally be in the same chem- 
ical form. For example, inactive iodide can hardly be expected 



248 IDENTIFICATION, CONCENTRATION, ISOLATION CH. XI 

to be a carrier for active iodine in the form of iodate ion; sodium 
phosphate would not carry radioactive phosphorus in elementary 
form. The chemical form in which a transmutation product 
emerges from a nuclear reaction is usually very hard to predict 
and has been investigated so far only in a very small number of 
cases. However, it is often possible to treat a target in such a 
way that the active material of interest is transformed to a certain 
chemical form. For example, if a zinc target is dissolved in a 
strongly oxidizing medium (say, HNO 3 , or HC1 + H 2 O 2 ), any 
copper present as a transmutation product is found afterwards 
in the Cu ++ form. If there is any uncertainty about the chemical 
form of the transmutation product as to its oxidation state or 
presence in some complex or undissociated compound, for exam- 
ple the only method which can be relied on to avoid difficulties 
is the addition of carrier in the various possible forms and a subse- 
quent procedure for the conversion of all of these into one form. 
To go through such a procedure prior to the addition of carrier 
may not be adequate. In fact, it appears that it may not always 
be sufficient to add the carrier element (say iodine) in its highest 
oxidation state (IO4~~) and carry through a reduction to a low 
oxidation state (12); in the case of the iodine compounds this 
procedure does not seem to reduce all the active atoms originally 
present in intermediate oxidation states. 

So far we have not spoken of the amounts of carriers used. For 
manipulative reasons it is often convenient to use about 10 or 
20 mg of each carrier, and less than about 1 mg is used only 
rarely; in separations from large bulks of target material larger 
amounts of carrier (perhaps 100 to 500 mg) are sometimes use- 
ful. (6) The amount of carrier frequently has to be measured at 
various stages of a chemical procedure to determine the chemical 
yield in the different steps, or at least in the over-all process. In 
such cases very small amounts of carrier are inconvenient. On 
the other hand, it is necessary to keep the quantities of carrier 
small in the preparation of sources of high specific activity. The 
specific activity of a sample of an element is sometimes expressed 
as the ratio of the number of radioactive atoms to the total num- 
ber of atoms of the element in the sample; more conveniently it is 
often expressed in terms of the disintegration rate (say in ruther- 

5 In a radiochemical laboratory it is convenient to have carrier solutions for 
a large number of elements on hand. These may, for example, be made up to 
contain 1 or 10 mg of carrier element per milliliter, or possibly 1 mg per drop. 



CHEMICAL SEPARATION TECHNIQUES 249 

fords) per unit weight (gram or milligram). High specific activities 
are essential particularly in many biological and medical applica- 
tions of radioactive isotopes, and also are often very desirable in 
samples to be used in physical measurements or chemical tracer 
studies, to insure small self-absorption or to permit high dilution 
factors. 

Sometimes it is possible to prepare samples of very high specific 
activities by the use of a nonisotopic carrier in the first stages of 
the separation; this may later be separated from the active mate- 
rial. In the isolation of radioyttrium (105-day Y 88 ) from deuteron- 
bombarded strontium targets, ferric ion can be used as a carrier 
for the active Y" 1 " 1 " 1 "; ferric hydroxide is then precipitated, cen- 
trifuged, washed, redissolved and, after the addition of more 
strontium as hold-back carrier, it is reprecipitated several more 
times to free it of strontium activity. Finally the ferric hydroxide 
which carries the yttrium activity is dissolved in 6 TV HC1, and 
ferric chloride is extracted into ether, leaving the active yttrium 
in the aqueous phase almost carrier-free. The use of nonisotopic 
carriers is particularly important in cases where no stable isotopes 
of the active material have been found in nature; these cases are 
treated in some detail in the next chapter. Criteria for the choice 
of nonisotopic carriers are also discussed there in connection with 
the behavior of substances at very small concentrations. 

Not all chemical procedures require the use of carriers. Par- 
ticularly procedures which do not involve solid phases may some- 
times be carried out at tracer concentrations without the addition 
of carriers. Because of the great importance of high specific 
activities, considerable work has been done on the preparation of 
carrier-free sources of many radioactive species. In the course 
of the following brief discussion of the various types of separation 
techniques we shall, therefore, point out those which lend them- 
selves to the production of carrier-free preparations. 

Chemical Separation Techniques. In most radiochemical sep- 
arations, as in conventional analytical schemes, precipitation 
reactions play a predominant role. The chief difficulties with 
precipitations arise from the carrying down of other materials. 
Some precipitates such as manganese dioxide and ferric hydroxide 
are so effective as "scavengers" that they are sometimes used 
deliberately to carry down foreign substances in trace amounts. 
Some other precipitates, such as rare-earth fluorides precipitated 
in acid solution, cupric sulfide precipitated in acid solution, or 



250 IDENTIFICATION, CONCENTRATION, ISOLATION CH. XI 

elementary tellurium brought down by reduction with S0 2 , have 
little tendency to carry substances not actually insoluble under 
the same conditions, and, therefore, can sometimes be brought 
down without the addition of hold-back carriers for activities 
that are to be left in solution. Most precipitates have an inter- 
mediate behavior in this regard. 

As mentioned before, adsorption on precipitates has been used 
to effect separations of tracer quantities. Adsorptions on the walls 
of glass vessels and on filter paper, which are sometimes bother- 
some, have also been put to successful use in special cases. Carrier- 
free yttrium activity has been quantitatively adsorbed on filter 
paper from an alkaline strontium solution at yttrium concentra- 
tions at which the solubility product of yttrium hydroxide could 
not have been exceeded. 

Ion-exchange Separations. An exceedingly useful separation 
technique closely related to adsorption chromatography has 
recently been developed for use both with and without carriers. 
This technique involves the adsorption of a mixture of ions on an 
ion-exchange resin followed by selective elution from the resin. 
The resin (such as the Amberlite IR-1 or Dowex-50) is usually a 
synthetic polymer containing free sulfonic acid groups; cations 
are adsorbed on it by replacing the hydrogen ions of these acid 
groups. The strength of the resin-ion bond increases with increas- 
ing ionic charge and decreasing (hydrated!) ionic radius. In 
practice a solution containing the ions to be separated is run 
through a column of the finely divided resin, and conditions (col- 
umn dimensions, solution volume, concentration, and flow rate) 
are chosen such that the ions are adsorbed in a narrow region near 
the top of the column. Then an eluting solution, usually con- 
taining a completing agent (such as oxalic acid or citric acid) 
which forms complexes of different stability with the various ions, 
is run through the column. There exists then a competition 
between the resin and the complexing agent for each ion, and if 
the column is run close to equilibrium conditions each ion will be 
exchanged between resin and complex form many times as it 
moves down the column. (6) The number of times an ion is ad- 

8 Slow flow rate, high resin-to-ion ratio, and fine resin particle size favor 
close approach to equilibrium. In practice, a compromise has to be made 
between high separation efficiency on the one hand and good yield and speed 
on the other. 



VOLATILIZATION 251 

sorbed and desorbed on the resin in such a column is analogous to 
the number of theoretical plates in a distillation column. The 
rates with which different ionic species move down the column 
under identical conditions are different, because the stabilities 
of both the resin compounds and the complexes vary from ion 
to ion; separations are particularly efficient if both these factors 
work in the same direction, that is, if the complex stability in- 
creases as the metal-resin bond strength decreases. As the various 
adsorption bands move down the column their separations increase, 
until finally the ion from the lowest band appears in the effluent. 
The various ions can then be collected separately in successive 
fractions of the effluent. 

The ion-exchange column technique \vas developed and has 
proved particularly useful for fission-product separations. Many 
of the carrier-free fission-product activities sold by the United 
States Atomic Energy Commission are isolated by this method. 
The most striking application of ion-exchange columns is in the 
separation of rare earths from each other, both on a tracer scale 
and in gram or hundred-gram lots. The eluting solution in this 
case may be 5 per cent citric acid solution buffered with ammonia 
to a pll somewhere between 2.5 and 3.0. The rare earths are 
eluted in reverse order of their atomic numbers, and yttrium falls 
between dysprosium and holmium. Very clean separations can 
be obtained, with impurities in some cases reduced to less than 
one part per million. By continuously recording the specific 
activity of the effluent solution as a function of time one obtains 
separate sharp peaks for the activities of the various rare earths 
when a mixture of rare-earth radioactivities is run through the 
column. This method led to the definite assignment of several 
decay periods to isotopes of element 61. 

Volatilization. Other separation methods avoiding the difficul- 
ties inherent in precipitations have frequently been used in radio- 
chemical work. Among these are volatilization, solvent extraction, 
elect rodeposition, and leaching. In special cases all these tech- 
niques lend themselves to the preparation of carrier-free tracers 
(as illustrated in chapter XII). Many volatile substances have 
been separated from less volatile ones by distillation. For exam- 
ple, after the solution of deuteron-bombarded tellurium in nitric 
acid, active iodine can be distilled out of the nitric acid solution 
at the boiling point. Radioactive noble gases can be swept out 



252 IDENTIFICATION, CONCENTRATION, ISOLATION CH. XI 

of aqueous solutions with some inert gas. The volatility of such 
compounds as GeCl 4 , AsCl 3 , SeCl 4 can be used to effect separa- 
tions from other chlorides by distillation from HC1 solutions. 
Similarly OsO 4 can be distilled from concentrated HNO 3 , and 
Ru0 4 from HC1O 4 solutions. 

Solvent Extraction. Distribution between two immiscible sol- 
vents has been used to effect a number of radiochemical separa- 
tions. Ferric chloride and gallium chloride, for example, are 
extracted quite efficiently (>95 per cent) into ether out of 
6 N HC1, whereas most other chlorides stay almost quantitatively 
in the aqueous phase. Similarly gold nitrate and mercuric nitrate 
can be extracted into ethyl acetate from nitric acid solutions. 
The extraction of uranyl nitrate into ether often serves as a con- 
venient method for the separation of fission products from the 
bulk of uranium. The basic acetate of beryllium is very soluble 
in chloroform, and a chloroform solution of this compound may 
be shaken with water to separate beryllium from many impurities. 

Electrodeposition. Electrolysis or electrochemical deposition 
may be used either to plate out the active material of interest or 
to plate out other substances leaving the active material in solu- 
tion. For example, it is possible to separate radioactive copper 
from a dissolved zinc target by an electroplating process. Carrier- 
free radioactive zinc may be obtained from a deuteron-bombarded 
copper target by solution of the target and electrolysis to remove 
all the copper. Sometimes care must be taken to avoid loss of 
active material due to electrochemical displacement; in the exam- 
ple of a deuteron-bombarded zinc target from which copper is 
to be separated after solution in HC1, it is essential that all the 
zinc be dissolved; as long as solid zinc is present the copper de- 
posits on the zinc surface. 

Occasionally it may be possible to leach an active product out 
of the target material. This has been done successfully in the 
case of neutron- and deuteron-bombarded magnesium oxide 
targets; radioactive sodium is separated rather efficiently from 
the bulk of such a target by leaching with hot water. 


D. CHEMICAL CONCENTRATION OF ISOTOPIC SPECIES: 
THE SZILARD-CHALMERS PROCESS 

Principle of the Method. The methods already discussed for 
preparation of sources of high specific activity apply only if the 



PRINCIPLE OF THE METHOD 253 

radioactive product is not isotopic with the target material. If 
target and product are isotopic the problem is much more diffi- 
cult; isotope separation by one of the usual physical methods is 
possible in principle but at present usually not practical for the 
preparation of tracer activities. However, a chemical separation 
is sometimes possible. In 1934 L. Szilard and T. A. Chalmers 
showed that after the neutron irradiation of ethyl iodide most 
of the iodine activity formed could be extracted from the ethyl 
iodide with water; they used a small amount of iodine carrier, 
reduced it to I ~ and finally precipitated Agl. Evidently the 
iodine-carbon bond was broken when an I 127 nucleus was trans- 
formed by neutron capture to I 128 . This type of process has 
since been used to concentrate the products of a number of n, 7 
reactions, and of some 7, n and n, 2n reactions, and is referred to 
as the Szilard-Chalmers process. Three conditions have to be 
fulfilled to make a Szilard-Chalmers separation possible. The 
radioactive atom in the process of its formation must be broken 
loose from its molecule; it must not recombine with the molecular 
fragment from which it separated, nor rapidly interchange with 
inactive atoms in other target molecules; and a chemical method 
for the separation of the target compound from the radioactive 
material in its new chemical form must be available. 

Most chemical bond energies are in the range of 1 to 5 ev (20,000 
to 100,000 cal per mole). In any nuclear reaction involving heavy 
particles either entering or leaving the nucleus with energies in 
excess of 10 or 100 kev the kinetic energy imparted to the residual 
nucleus far exceeds the magnitude of bond energies. (7) Injb 



of thermal-neutroiL^apture. where the Szilard-Chalmers method 
has its most important applications, the incident neutron does 
not impart nearly erigugh energy to the nucleus to cause anyjbond 
rugture. But jneutron capture is., always followed by 7;iay emis- 
sion, and thgjuicleus receives some ^er.oil energjrJa this process. 
A 7 ray of energy E y has a momentum p- = E y /c; to cqriserve 
momentum the_recoiling^Iom must have an identical momentujn, 
and, therefore, the recoil energy E r ~= p y 2 /2M = E y 2 /pMc 2 , 
where %L is the mass of the atom. For Af_in atomic mass units 



7 For reactions other than n, y, particularly for d, p reactions, the Szilard- 
Chalmers technique is not very useful because the energy dissipated by the 
incident radiation in the target is so great that many inactive molecules are 
also disrupted. The chemical effect of radiation the field of radiation 
chemistry is not treated in this book. 



254 IDENTIFICATION, CONCENTRATION, ISOLATION CH. XI 
and the energies in millions of electron volts we have 



1862 M 



(XI-1) 



Table XI-1 shows values of E r for a few values of E y and M. 
Neutron capture usually excites a nucleus to about 6 or 8 Mev, 

TABLE XI-1 

RECOIL ENERGIES IN ELECTRON VOLTS IMPARTED TO NUCLEI BY y RAYS OP 
VARIOUS ENERGIES 



M 


Ey = 

2 Mev 


E y = 
4 Mev 


E,~ 
6 Mev 


20 


107 


430 


967 


50 


43 


172 


387 


100 


21 


86 


193 


150 


14 


57 


129 


200 


11 


43 


97 



and a large fraction of this excitation energy is dissipated by the 
emission of one or more 7 rays; unless all the successive 7 rays 
emitted in a given capture process have low energies (say below 1 
or 2 Mev), which is a relatively rare occurrence, the recoiling 
nucleus receives more than sufficient energy for the rupture of 
one or more bonds. Of course, it is not the entire recoil energy 
but something more like its component in the direction of a bond 
that should be compared with the bond energy; furthermore, 
the momenta of several 7 rays emitted in cascade and in different 
directions may partially cancel each other. There is no evidence 
that two capture 7 rays in cascade are preferentially emitted in 
opposite directions, and momentum cancelation is, therefore, 
hardly expected to reduce the probability of bond rupture by a 
very large factor. In most n, 7 processes the probability of 
rupture is certainly very high. 

The second condition for the operation of the Szilard-Chalmers 
method requires at least that thermal exchange be slow between 
the radioactive atoms in their new chemical state and the inactive 



ILLUSTRATIONS 255 

atoms in the target compound. The energetic recoil atoms may 
undergo exchange more readily than atoms of ordinary thermal 
energies. These exchange reactions and other reactions of the 
high-energy recoil atoms (often called "hot atoms") determine 
to a large extent the separation efficiencies obtainable in Szilard- 
Chalmers processes. Hot-atom reactions are considered briefly 
after a discussion of some examples of Szilard-Chalmers separa- 
tions. 

Illustrations. The largest amount of work in the field of Szilard- 
Chalmers separations has been done on halogen compounds. 
Many different organic halides (including CC1 4 , C 2 H 4 C1 2 , C 2 H 5 Br, 
C 2 H 2 Br 2 , C 6 H 5 Br, CH 3 I) have been irradiated, and the products 
of neutron capture reactions (Cl 38 , Br 80 , Br 82 , I 128 ) removed by 
various techniques. Extraction with water, either with or without 
added halogen or halide carrier, results in rather efficient separa- 
tions. Yields are often improved, especially in the case of iodine, 
by extraction with an aqueous solution of a reducing agent such 
as HSO 3 ~~. Nearly complete extraction of activity has been 
reported in some cases when carrier was present. In the absence 
of carrier, yields of 50 per cent and large concentration factors 
have been found. Other methods which have been used for the 
removal of the active halogen from irradiated organic halides 
include adsorption on activated charcoal (with 30 or 40 per cent 
yields of halogen without added carrier) and collection on charged 
plates (with up to 70 per cent yields of halogen without added 
carrier). 

Szilard-Chalmers separations of halogens with 70 to 100 per cent 
yields have also been obtained in neutron irradiations of solid or 
dissolved chlorates, bromates, iodates, perchlorates, and perio- 
dates; from these the active halogen can be removed as silver 
halide after addition of halide ion carrier. Szilard-Chalmers 
separations based on differences in oxidation state before and 
after the neutron capture have been successful for a number of 
other elements. About half the P 32 activity formed in neutron 
irradiation of phosphates (solid or in solution) is found in +3 
phosphorus. Most of the Mn 56 activity can be removed from 
neutron-irradiated neutral or acid permanganate solutions in the 
form of Mn0 2 . Tellurium and selenium activities can be concen- 
trated through the separation of tellurite or selenite carrier from 
irradiated tellurate or selenate solution by reduction of the lower 



256 IDENTIFICATION, CONCENTRATION, ISOLATION CH. XI 

oxidation state to the element with S0 2 (reduction of the +6 
state proceeds much more slowly than of the +4 state). Similarly 
a Szilard-Chalmers separation for arsenic has been reported by 
the addition of arsenite carrier to irradiated arsenate solution and 
precipitation of As 2 S 3 . Reduction to the metal has been found to 
occur in the neutron capture of gold in various compounds. 
Whether or not an active element can be successfully isolated in a 
different oxidation state from the bombarded compound depends 
not only on the relative stabilities of the two oxidation states but 
also on the speed of exchange between them under the conditions 
of the experiment. (Exchange reactions are discussed in chapter 
XIII, section B.) 

Collection of charged fragments on electrodes has been used 
successfully for a number of Szilard-Chalmers separations. 
Arsenic activity has been separated by this method from arsine 
gas with yields up to 34 per cent. Deposition occurs on both 
positive and negative electrodes in this case. 

The bombardment of metal-organic compounds and complex 
salts is often useful for Szilard-Chalmers separations if the free 
metal ion does not exchange with the compound, and if the two 
are separable. Some of the compounds which have been used 
successfully are: cacodylic acid, (CH 3 ) 2 AsOOH, from which 
As 76 can be separated as silver arsenite in 95 per cent yield; copper 
salicylaldehyde o-phenylene diamine, from which as much as 97 
per cent of the Cu 64 activity can be removed as Cu ++ ion; uranyl 
benzoylacetonate, U0 2 (C 6 H 5 COCHCOCH 3 ) 2 , from which U 239 
activity has been extracted in about 10 per cent yield. It has 
been suggested that metal ion complexes which exist in optically 
active forms and do not racemize rapidly may be generally suitable 
for Szilard-Chalmers processes because the metal ion in such a 
complex is not expected to exchange rapidly with free metal ion 
in solution. Some complexes of this type have been used success- 
fully, for example the triethylenediamine nitrates of iridium, 
platinum, rhodium, and cobalt, Ir(NH 2 CH 2 CH 2 NH 2 ) 3 (N0 3 ) 3 
etc. 

Hot-atom Chemistry. The highly excited recoil atoms resulting 
from neutron-capture reactions have been shown to undergo 
various types of chemical reactions. One of these is recombina- 
tion with the fragment from which the "hot" atom had broken 
away. Insofar as it occurs, such recombination increases the 



HOT-ATOM CHEMISTRY 257 

retention of the activity, retention being defined as the fraction 
of the active atoms not separable from the target compound. By 
means of retention studies recombination reactions have been 
shown to be more probable in the liquid than in the gas phase, 
and often more probable in the solid than in the liquid phase; for 
example, in liquid ethyl bromide the retention of bromine activity 
was found to be 75 per cent, and in ethyl bromide vapor (390 mm 
partial pressure with 370 mm of air) only 4.5 per cent. Retention 
has been shown to decrease markedly when the target substance 
is diluted. For example, the retention of bromine activity is 
about 60 per cent in solid carbon tetrabromide, about 28 per cent 
in an alcohol solution containing 1.15 mol per cent CBr 4 , and 
db 2 per cent in an alcohol solution containing 0.064 mol per cent 
CBr 4 . These results have been interpreted as indicating that 
retention cannot be caused entirely by recombination of fragments 
(in a so-called reaction cage) but is at least in part brought about 
through replacement by recoil atoms of isotopic atoms in other 
molecules of the target substance. 

It has been shown that in reacting with a molecule a "hot" 
atom may replace another atom or group. For example, after 
the slow-neutron irradiation of CH 3 I, 11 per cent of the I 128 
activity was found in the form of CH 2 I 2 ; furthermore, this result 
was shown to be temperature-independent between 195C and 
15C, which proves that the substitution is not an ordinary 
thermal reaction. The formation of labeled CH 2 Br 2 in the irra- 
diation of CH 2 BrCOOH, of labeled CH 3 I and C 2 H 5 I in the ir- 
radiation of iodine dissolved in ethyl alcohol, and of labeled 
C6H 5 Br in the irradiation of aniline hydrobromide show that the 
excited halogen atoms can replace such groups as COOH, 
OH, CH 2 OH, NH 2 , and probably many others. The yield 
of active atoms in one of these substitution products is usualty 
less than about 10 per cent. Reactions of this type might con- 
ceivably be used to synthesize labeled compounds of high specific 
activity. 

Some hot-atom reactions have been studied in inorganic sys- 
tems. The retention of activity in permanganate, phosphate, 
and arsenate in thermal-neutron irradiations has been determined 
under varying conditions of pH and concentration. The results 
have been interpreted by W. F. Libby on the basis of competition 
between hydration reactions and oxidation-reduction reactions 



258 IDENTIFICATION, CONCENTRATION, ISOLATION CH. XI 

for the hot atoms. In the case of permanganate, for example, 
Libby found the retention to be practically independent of per- 
manganate concentration in neutral and alkaline solution and 
concluded from this that the manganese in the primary recoil 
fragment is in the +7 oxidation state (MnOa 4 ", Mn0 2 +3 , MnO* 5 , 
or Mn +7 ). These species are then considered to undergo either 
hydration reactions, such as MnOs^ + 20H~ = Mn04~~ + H 2 O, 
or reduction by water, such as 4Mn03 + + 2H20 = 4 MnC>2 + 
30 2 + 4H + . At pR > 12 the retention is nearly 100 per cent, 
presumably because the hydration reactions strongly predominate. 
Below pH 12 the retention falls rapidly with decreasing pH and 
reaches a constant value of about 7 per cent for pH values between 
8 and 2. According to Libby's interpretation the reduction by 
water is faster than hydration in neutral and acid solutions. At 
still higher acid concentrations the retention again rises slightly, 
perhaps because exchange reactions between the active MnOa" 1 " 
and inactive MnO 4 ~~ can then compete with reduction by water; 
in support of this Libby showed that in acid solutions the reten- 
tion increases with permanganate concentration. It is interesting 
to note that in arsenate bombardments the retention is nearly 
100 per cent over a wide pH range; here the hydration reactions 
apparently far outweigh the oxidation-reduction reactions with 
water. The phosphate experiments showed retentions of about 
50 per cent under all conditions tried, which might possibly be 
taken to indicate that only about 50 per cent of the primary recoil 
fragments contain phosphorus in the +5 state. 

Isomer Separations. "Hot" atoms may result not only from 
nuclear reactions but also from radioactive-decay processes. The 
chemistry of hot atoms formed as a result of /3-decay processes 
has been studied in a number of cases; for example, reactions such 
as 

TeOa" -> IO 3 ~ + /3~ and Mn0 4 "~ -> CrC^ + p + 

can occur in addition to molecular disruption leading to other 
forms. Of course, for these studies the nucleus resulting from 
the decay must itself be radioactive if its fate is to be investi- 
gated. 

Up to the present time most of the published studies of the 
chemistry of recoil atoms following radioactive decay have been 



ISOMER SEPARATIONS 259 

confined to one type of decay process, isomeric transition. It is 
perhaps not immediately clear why isomeric transitions may lead 
to bond rupture. The 7-ray energies in isomeric transitions are 
much lower than in neutron-capture processes, often below 100 kev 
and rarely above 500 kev. According to equation XI-1, a 100- 
kev 7 ray would give a nucleus of mass 100 a recoil energy of only 
about 0.05 ev, which is not nearly enough to break a chemical 
bond. Although internal-conversion electron emission gives rise 
to roughly 10 times greater recoil energy than 7 emission at the 
same energy ; (8) even this is not sufficient for bond rupture in most 
cases. However, the vacancy left in an inner electron shell by 
the internal conversion leads to electronic rearrangements and 
emission of Auger electrons; the atom is, therefore, in a highly 
excited state (and positively charged), and molecular dissociation 
may take place if the atom is bound in a molecule. 

Separations of nuclear isomers analogous to Szilard-Chalmers 
separations have been performed in a number of cases where the 
isomeric transition proceeds largely by conversion-electron emis- 
sion. The 18-min Br 80 has been separated from its parent, the 
4.4-hr Br 80 , by a number of different methods analogous to the 
Szilard-Chalmers methods used for bromine. The lower states 
of Te 121 (17 days), Te 127 (9.3 hr), Te 129 (72 min), and Te 131 (25 
min) have been separated as tellurite in good yield from tellurate 
solutions containing the corresponding upper isomeric states. 
Isomer separations are sometimes useful for the assignment of 
isomer activities and for the elucidation of genetic relationships. 
That the possibility of obtaining isomer separations depends on 
internal conversion was shown in experiments using the gaseous 
compounds Te(C 2 H 5 ) 2 containing 90-day Te 127 and 32-day Te 129 , 
and Zn(C 2 H 5 ) 2 containing the 13.8-hr upper isomeric state of 
Zn 69 . The lower isomeric states of the tellurium isomers could 
be separated on the walls of the vessels or on charged plates, but 
under identical conditions no separation of the zinc isomers was 
obtained; the isomeric transition in Zn 69 proceeds by an uncon- 
verted 440-kev 7 ray, whereas the tellurium transitions have 
energies of only about 100 kev but are almost completely con- 
verted. 

8 In nonrelativistic approximation an atom of mass M receives a recoil 
energy E r = E e (m/M) from a conversion electron of energy E e . 



260 IDENTIFICATION, CONCENTRATION, ISOLATION CH. XI 

EXERCISES 

1. Suggest hopeful easily prepared compounds for use in Szilard- 
Chalmers processes of (a) iron, (6) mercury, (c) technetium. 

2. A certain activity chemically proved to be associated with tech- 
netium (element 43) is produced in the bombardment of molybdenum with 
12-Mev deuterons and in the bombardment of ruthenium with 12-Mev 
deuterons, but not in the bombardment of ruthenium with fast (up to 
15 Mev) neutrons. To what isotope of technetium should the activity 
be assigned? What mode of decay would you expect? 

3. A sample of sodium iodide is irradiated with fast neutrons to produce 
90-day Te 127 . Suggest a chemical procedure for the isolation of the tel- 
lurium. How would you modify this procedure if you knew that the 
sodium iodide contained some sodium bromide impurity? 

4. What is the recoil energy imparted to a Te 129 atom by the emission 
of a 70-kev conversion electron? (Use the relativistic expression for the 
electron energy.) Answer: 0.318 ev. 

5. Element Z has a single stable nuclide of mass number A. In the 
bombardment of element Z with 28-Mev deuterons the following activities 
chemically identified with element Z + 1 were found: 

a moderately strong 3-hr positron emitter, 

a strong 2.6-day activity emitting mostly y and X rays, 

a weak 30-min positron emitter. 

The last activity was not produced when the deuteron energy was lowered 
to 20 Mev. 

Critical-absorption measurements showed the X rays of the 2.6-day 
activity to be those of element Z. An 11-day isotope of element Z was 
shown to grow from the 2.6-day activity, whereas the 3-hr activity decayed 
to a 50-min X-ray emitter chemically identified with element Z. The 
X rays from this latter activity were characteristic of element Z. 

One-million-electron-volt neutrons produced in a target of element Z 
the 50-min X-ray emitter, in addition to a 14-hr "-emitting isotope of Z 
which had also been identified in the deuteron bombarded Z samples. 
With 15-Mev neutrons the 11-day isotope of Z was also produced. 

Make mass assignments for the various radioactive isotopes of elements 
Z and Z + 1, and indicate the most likely mode of decay for each. The 
stable nu elides of Z + 1 have mass numbers A + 1, A + 2, and A + 3; 
those of Z 1 have mass numbers A 3, A 2, and A I. 



REFERENCES 261 

6. Suggest methods for the chemical identification of 

(a) V 62 produced in the fast-neutron bombardment of a chromate 
solution, 

(6) Mn 52 produced in the deuteron bombardment of iron, 
(c) O 14 produced in the proton bombardment of nitrogen gas. 



REFERENCES 

G. T. SEABORO, "Artificial Radioactivity," Chem. Rev. 27, 199 (1940). 

A. C. WAHL (Editor), Radioactivity Applied to Chemistry, New York, John 
Wiley & Sons, to be published. 

Chemical Institute of Canada, Proceedings of the Conference on Nuclear Chem- 
istry, May 15-17, 1947. 

G. T. SEABORQ, J. J. LIVINGOOD, and J. W. KENNEDY, "Radioactive Isotopes 
of Tellurium" (typical example of identification of radioactive species), 
Phys. Rev. 67, 363 (1940). 

D. H. TEMPLETON, J. J. ROWLAND, and I. PERLMAN, "Artificial Radioactive 
Isotopes of Polonium, Bismuth, and Lead," Phys. Rev. 72, 758, and 766 
(1947). 

R. J. HAYDEN, "Mass Spectrographic Mass Assignment of Radioactive 
Isotopes," Phys. Rev. 74, 650 (1948). 

W. E. COHN, "Radioactive Contaminants in Tracers," Anal Chem. 20, 498 
(1948). 

A. A. NOYES and W. C. BRAY, A System of Qualitative Analysis for the Rare 
Elements, New York, Macmillan Co., 1927. 

Various papers on the Ion Exchange Method, JACS 69, 2769-2881 (1947). 

R. R. EDWARDS and T. H. DA VIES, "Chemical Effects of Nuclear Transforma- 
tions," Nucleonics 2 no. 6, 44 (1948). 

W. F. LIBBY, "Chemistry of Energetic Atoms Produced by Nuclear Reac- 
tions," JACS 69, 2523 (1947). 

S. C. LIND, et al., "Symposium on Radiation Chemistry and Photochemistry/' 
J. Phys. and Colloid Chem. 62, 437-611 (1948). 

P. C. TOMPKINS, "Laboratory Handling of Radioactive Material," U. S. 
Atomic Energy Commission Declassified Document MDDC-1414, obtain- 
able from Document Sales Agency, P. 0. Box 62, Oak Ridge, Tenn., 
15 cents. 

K. Z. MORGAN, "Tolerance Concentrations of Radioactive Substances," 
/. Phys. Cottoid Chem. 51, 984 (1947). 

H. A. LEVY, "Some Aspects of the Design of Radiochcmical Laboratories," 
Chem. Eng. News 24, 3168 (1946). 

G. W. MORGAN, "Gamma and Beta Radiation Shielding," Circular B-3 (Jan. 
1948), obtainable from Isotopes Division, U. S. Atomic Energy Commis- 
sion, P. 0. Box E, Oak Ridge, Tenn. 



XII. CHEMISTRY OF LOW CONCENTRATIONS AND 
THE STUDY OF NEW ELEMENTS 

A. THE LOW-CONCENTRATION REGION 

Limits of Detection. The working region of concentrations in 
ordinary chemical studies is limited by the sensitivity of available 
analytical methods. The lower detection limits for different sub- 
stances vary widely. Gravimetric procedures rarely are useful 
for concentrations as low as one part per million (1 ppm); spectro- 
scopic elementary analysis in favorable cases offers about the best 
sensitivity, and detection of a number of elements at 0.01 ppm 
and less may be practical. It is true that some compounds of very 
pronounced odor may be noticed at much lower concentrations; 
for example, at 0.01 ppm in air the odor of mcrcaptan is very 
strong, and it may be recognized at 0.00001 ppm. The number 
of mercaptan molecules sufficient for recognition in this way is 
estimated to be about 3 X 10 l , corresponding to 2 X 10~ 12 g 
in about 100 ml of air. Other detection methods, especially biolog- 
ical assays, may approach and exceed even this sensitivity. But 
for the most part these methods have not offered practical means 
for extending knowledge of chemical behavior to such extremely 
low concentrations. The property of radioactivity does offer a 
rather convenient analytical method for concentrations so low 
and even much lower. As few as several thousand radioactive 
molecules may be detected, even when contained in sizable sam- 
ples. The practical working limits are fixed by the half-life and 
by the nature of the radiation. A polonium solution at 10~~ 12 
mole per liter has an easily detectable activity of 35 urd per ml 
(35 microrutherfords, or 35 disintegrations per sec, per milliliter). 
The shorter-lived La 140 (40-hr half-life) may be studied in 10~ 14 
molar solution. A new concentration region is opened for study 
by the technique of radioactive tracers. 

Radioc olio ids. It was observed many years ago that radioactive 
elements in some solutions where they existed at extremely low 
concentrations showed unusual physical properties in that they 

262 



FAJANS' PRECIPITATION RULE 263 

behaved more like colloids than true solutes. That the active 
atoms or molecules were clustered together in these solutions was 
shown in suitable cases by photographic registration of the spotty 
distribution of disintegration a rays. The size of the colloidal 
particles has been estimated from observed sedimentation rates 
on centrifugation, and it has been established that in some cases 
the phenomenon is one of adsorption of many of the active solute 
molecules or ions on particles of dust, silica, or the like, inevitably 
present even in "pure" water. The general adsorption phenomena 
are discussed in the next section. It should be remembered that 
effects of this sort, including adsorptions on container surfaces, 
filter paper, and so on, may be quite important in work at these 
very low "tracer" concentrations. The same effects, no doubt, 
occur in work at ordinary concentrations, but then the amount of 
material involved represents such a small fraction of the total 
amount that these effects are not noticed. 



B. COPRECIPITATION AND ADSORPTION 

Fajans* Precipitation Rule. Many of the manipulations of ordi- 
nary chemistry and also of radiochemistry require precipitation 
reactions. How are substances at tracer concentrations to be 
separated by precipitation, when often the concentration is too 
small to exceed the solubility-product condition, and when the 
amount of precipitate even if formed would be quite imperceptible? 
Of course, if the radioactivity is isotopic with an element available 
in quantity, more of the element may be added in a suitable 
chemical form as a carrier, and then the chemical problems become 
ordinary ones. If necessary or if desirable, nonisotopic carriers 
may be tried. Microscopic amounts of radium in solution are 
brought down with barium in the precipitation of BaS0 4 . Stron- 
tium ions have been carried by calcium salts, iodide ions by chlo- 
ride precipitates, and so on. In some cases precipitates carry 
down active substances where the chemical similarities are not 
so obvious; for example, tracer lead (ThB) is well carried by 
ammonium dichromate crystals, and many active bodies are 
carried by Fe(OH) 3 precipitates. On the basis of a number of 
such observations K. Fajans in 1913 formulated this principle: 
the lower the solubility of the compound formed by the radio- 
element (as cation) with the anion of the precipitate, the greater 



264 CHEMISTRY OF LOW CONCENTRATIONS CH. XII 

the amount of radioelement carried with the precipitate. As an 
illustration, bismuth tracer is carried by BaC0 3 and Fe(OH) 3 
but not by BaSO 4 or PbSO 4 from acid solutions. Lead tracer 
may be carried by all these, but is carried less well by AgCl, and 
is not carried by a nitron nitrate precipitate. Exceptions to this 
rule are not uncommon; ThB (a lead isotope) is not precipitated 
with Hgl2 or with cupric fumarate although both PbLj and lead 
fumarate are rather insoluble. The occurrence of carry ing not 
predicted by this rule is widely observed even on the macroscale 
in analytical chemistry; for example KNO 3 is appreciably copre- 
cipitated with BaS0 4 . Clearly factors in addition to that expressed 
in the Fajans rule are important in these phenomena. 

It should be noticed that precipitates previously formed in the 
absence of tracer may take up a tracer when added in suspension 
to the tracer solution. This is the method of preformed precipi- 
tates. This procedure and an intermediate case between it and 
ordinary coprecipitation find convenient uses in radiochemical 
separations. For example, consider the carrying of radioactive 
yttrium by lanthanum fluoride; after a single precipitation of 
LaF 3 with excess HF a small fraction of the yttrium activity may 
remain in solution. Now, when more lanthanum is added to 
excess, a new precipitate of LaF 3 forms immediately, probably 
before the La ++ + and Y+++ ions are well mixed. This precipi- 
tate may carry yttrium with it, but probably not so well as the 
first precipitate, or the third precipitate which may now be formed 
by addition of excess HF. 

Hahn's Classification. O. Halm in 1926 proposed a classification 
of carrying phenomena, distinguishing cases of true coprecipita- 
tion from cases of surface adsorption. The four principal types 
of carrying which he described are (1) isomorphous replacement, 
(2) surface adsorption, (3) anomalous mixed crystals, and (4) inter- 
nal adsorption. In discussing these we will give most emphasis 
to the first two; the others are not so well understood. 

1. Isomorphous Replacement. If the carrying ion and the ion 
carried form with the precipitating ion isomorphous crystalline 
compounds, coprecipitation of this type is to be expected. The 
radioelement is distributed throughout the precipitate crystals 
as may be shown by a radioautograph technique, and the mech- 
anism is simply one of replacement at normal ion sites in the 
crystal lattice. This true coprecipitation is not much affected 



HAHN'S CLASSIFICATION 265 

by conditions during precipitation such as acidity, order of addi- 
tion of reagents, rate of crystallization, and temperature; and 
repeated washing of the precipitate cannot remove the coprecipi- 
tated substance. The precipitation of radium with barium salts 
is an example of this class of carrying. 

2. Surface Adsorption. Freshly formed precipitate crystals with 
large surface areas may be capable of adsorbing radioelements 
effectively. For this type of carrying the Fajans rule is significant, 
but also another important factor is the surface charge on the 
precipitate relative to the ionic charge of the tracer substance. 
Because important adsorption occurs only when these charges 
are of opposite sign, experimental factors affecting the surface 
charge of the precipitate strongly influence the carrying, and this 
type is recognized by sensitivity to such factors as acidity, order 
of addition of reagents, and physical state of subdivision of the 
precipitate. In many instances an appreciable part of the adsorbed 
activity may be washed off, or displaced by another ion of similar 
charge. The carrying of ThB (a lead isotope) at tracer concen- 
trations by CaS0 4 or AgBr, and of Ra by Ag 2 Cr0 4 are examples. 
Table XII-1 shows the importance of excess of the anion, neces- 

TABLE XII-1 
CARRYING OF ThB BY CaSC>4 

Excess Excess ThB 

Ca++ SO 4 " carried 

600% 1.7% 

10% 5.2% 

5% 88 % 

900% 98 % 

sary to produce a negative surface charge on the precipitate 
crystals, for the carrying of cations. 

3. Anomalous Mixed Crystals. A type of carrying which at 
least superficially closely resembles isomorphous replacement is 
observed in a number of cases where true isomorphism is unexpect- 
ed and even unlikely. An example is the carrying, in a manner 
hardly affected by precipitate surface charges, of RaB or RaD 
(lead isotopes) by BaCl 2 -2H 2 O; these crystals are monoclinic, 
but PbCl2 in macroscopic amounts crystallizes in the rhombic 
system. The capacity of the barium chloride for lead ions has 



266 CHEMISTRY OF LOW CONCENTRATIONS CH. XII 

been found to be limited to about 0.1 mol per cent lead in the 
crystals. 

4. Internal Adsorption. There are some cases of carrying that 
do not fit into any of the three types already discussed and are 
characterized by a spotty distribution of tracer within the pre- 
cipitate crystals as shown by radioautographs. These cases may 
not be very numerous and seem to be associated with very poor 
carrying. For example, although lead tracer is carried in an 
anomalous mixed crystal by barium chloride, it is carried only 
very slightly by barium bromide; the small fraction of ThB that 
is carried is distributed in spots and patches scattered through 
the barium bromide crystals. 

In addition to these four types Hahn considers the mechanical 
inclusion in precipitate crystal masses of radiocolloids that might 
exist in the solution and inclusion of portions of the mother liquor 
itself. 

Doerner-Hoskins and Berthelot-Nernst Distributions. For 
true coprecipitation of the isomorphous-replacement type, and 
apparently also of the anomalous-mixed-crystal type, the progress 
of crystal separation and the detailed distribution of the active 
tracer within the precipitate crystals may tend to approach eithei 
of two limiting laws. For the assumption that the precipitate 
crystals grow progressively, with equilibrium conditions main- 
tained between the solution and the crystallizing layer, and witt 
both re-solution and solid-diffusion effects negligible, a quantita- 
tive treatment is easily made. Let x and y be the amounts of the 
two substances X and Y precipitated before a given instant, anc 
let a and b be the total amounts of X and 7; then: 

dx a x 
= \ ^ 

dy b - y 

where X is a constant characteristic of the system. In words, thi 
ratio of X to Y in the forming surface layer is proportional to th< 
ratio of the respective concentrations still remaining in the solu 
tion. On integration, 

a b 

log ~ \ fog _ j 

a x b y 

this is the logarithmic distribution law derived by H. A. Doerne 
and W. M. Hoskins. There is evidence that it is closel; 



PARTITION BETWEEN SOLVENTS 267 

approached in many actual coprecipitations; it is found especially 
for precipitations produced by gradual evaporation with care to 
avoid any supersaturation, and for precipitations from super- 
saturated solutions that are vigorously stirred and quickly sepa- 
rated (filtered). 

If for any reason the entire crystal rather than just the surface 
layer is brought into equilibrium with the solution, then the 
differential equation just given should be replaced by a similar 
nondifferential expression : 

x a x 

- = D 

y b - y 

Here the ratio of X to F in the crystals is proportional to the ratio 
of X to Y left in solution; this is the Berthelot-Nernst homo- 
geneous distribution law applicable to partition of a solute be- 
tween liquid phases. Coprecipitations made from strongly super- 
saturated solutions and coprecipitations in which a finely divided 
precipitate is left standing in contact with the solution are likely 
to approach this distribution law. In the first case failure to 
maintain equilibrium between the growing crystals, the imme- 
diately surrounding solution, and the bulk of the solution seems 
to be involved; in the other re-solution and recrystallization 
probably play an important role. 

C. OTHER CHEMICAL PROPERTIES 

Partition between Solvents. The partition of active solutes at 
tracer concentrations between two immiscible liquid phases one 
would suppose might be simpler than the distributions in pre- 
cipitations. Actually not much information on this subject has 
been reported, but such results as have been obtained do not give 
evidence of any abnormal effects at very low concentrations. 
The distribution of GaCl 3 between aqueous HC1 and ether phases 
has been studied at about 10~~ 12 molar, and the distribution coeffi- 
cient has been found to be the same as at a higher concentration 
(0.0016 molar). However, this cannot be expected to be true for 
all substances; for example, if the molecule extracted should be a 
dimeric form, then at sufficiently low concentration the distribu- 
tion would surely change. 



268 CHEMISTRY OF LOW CONCENTRATIONS CH. XII 

Volatility. The volatility of unweighable amounts of tracer sub- 
stances has been often investigated, although not for a wide 
variety of elements. This volatility is probably related to the 
macroscopic volatility of the substances but is, no doubt, modified 
by the nature of the surface attachment to the necessary sup- 
porting foil. Bismuth isotopes of the active deposit from emana- 
tion are volatilized from platinum in air at 800 to 900C, and 
the active lead isotopes are volatilized at 700 to 800C. These 
temperatures may be reduced somewhat by previous exposure of 
the substance to a free halogen, presumably because bismuth 
and lead halides are formed. Radioactive element 85 (astatine) 
formed in metallic bismuth by a-particle bombardment vola- 
tilizes almost completely in vacuum from liquid bismuth at about 
300C. Cadmium formed by transmutation from silver is sepa- 
rated in the same way at 900 C. Information is not available to 
permit quantitative comparisons of macrovolatilities with tracer 
volatilities from the interiors of solid substances. Volatility 
properties of unknown substances available only as tracers have 
been sought in experiments in which a solid containing the tracer 
(possibly by coprecipitation) is volatilized and the tracer either 
volatilized or left behind; these experiments too can be difficult 
to interpret because the tracer may be swept away with the carrier 
somewhat below its normal volatilization temperature. 

Electrochemistry. The electrochemistry of ions at tracer con- 
centrations has been the subject of many investigations, and a 
number of convenient radiochemical separation procedures are 
based on electrochemical methods. For example, RaF (polonium) 
is separated from RaD (lead) and RaE (bismuth) by its spon- 
taneous deposition from dilute HC1 solution on a silver foil; RaD 
and RaE are more electropositive and are not displaced from the 
solution by silver. Replacement by nickel with deposition on a 
nickel foil can be used to remove RaE and leave RaD in solution. 
The deposition potentials of some tracers have been measured 
. for the extremely low concentrations; the cathode potential 
(measured with respect to a suitable auxiliary reference electrode) 
necessary for the deposition of the tracer is interpreted as analo- 
gous to the decomposition potential. All these experiments give 
means of locating the trace substance in the electromotive series 
and permit approximate evaluation of the standard electrode 
potential, provided proper account is taken of the large shifts in 
emf caused by the extremely low ion concentrations. 



TECHNETIUM 269 

Tracer ions in a solution between electrodes move in a direction 
determined by their charge, and the rough average sign of charge 
is revealed by observation of the net average transport of the 
tracer. In simple cases some information on the magnitude of 
the charge can be obtained by a careful quantitative study. 

D. DISCOVERIES OF NEW ELEMENTS BY TRACER METHODS 

The Role of New Physical Methods. More than half a century 
ago the methods of chemistry conventional at that time had 
already reached a limit in the search for new and missing ele- 
ments; discoveries since that time have depended on the introduc- 
tion of new physical methods. Through studies of optical spectra 
the elements rubidium, cesium, indium, helium, and gallium were 
found. The first evidence for hafnium and rhenium came from 
X-ray spectra. Early investigations of the natural radioelements 
revealed the existence (often in extremely small amounts) of 
polonium (number 84), radon (86), radium (88), actinium (89), 
and protactinium (91), and recently the missing element number 
87 has been found in the same way. Through studies of nuclear 
reactions and artificially induced radioactivities the elements 43, 
61, and 85 have been identified, and more recently elements 93, 
94, 95, and 96 have been added to the periodic chart. In this 
section we will discuss briefly the discoveries of the last eight 
elements mentioned. 

Technetium. In 1925 W. Noddack, I. Tacke, and O. Berg, who 
had previously found the new element rhenium through its X-ray 
spectrum, reported observation of a faint X-ray line in a concen- 
trated rhenium sample that would correspond to the lighter homo- 
log of rhenium, the missing element 43; they proposed for it the 
name masurium, symbol Ma. The element could not be concen- 
trated, and the work has not been verified; present knowledge of 
the isotopes in this region of atomic weights (~100) makes it 
appear unlikely that this element exists in nature in stable form. 
(See chapter VI, page 140.) C. Perrier and E. Segre working in 
Italy isolated and studied radioactive isotopes of element 43 from 
an old molybdenum deflector plate of the Berkeley 37-inch cyclo- 
tron. These were long-lived X-capture activities, produced 
through d, n reactions by the deuteron beam. Perrier and Segrfe 
have recently offered for this element the name technetium, 
symbol Tc, derived from the Greek word meaning "artificial." 



270 CHEMISTRY OF LOW CONCENTRATIONS CH. XII 

In their early studies, Perrier and Segr& compared the chemical 
behavior of the tracer activity with that of several carrier sub- 
stances which might be guessed to be chemically similar, par- 
ticularly manganese and rhenium because technetium falls be- 
tween these in subgroup VII of the periodic table. They found 
the stability of the +7 oxidation state to be greater than for 
manganese and less than for rhenium, as expected. Technetium 
was carried by a precipitate of Re 2 S 7 on addition of H 2 S to HC1 
solutions up to 6 normal; the technetium seemed to concentrate 
somewhat in the solution on partial precipitation, which suggests 
a greater solubility for Tc 2 S 7 , although this evidence taken alone 
is subject to other interpretations. Precipitates of MnO 2 and 
Mn(OH) 2 did not carry technetium unless strong reducing agents 
were present. KReO 4 and CsReO 4 did carry the tracer, and the 
insolubility of the technetium compound seemed to be less than 
that of the perrhenate for the potassium salts and greater for the 
cesium salts. Nitron perrhenate was found to carry technetium 
quantitatively. The technetium tracer volatilized completely in 
air at about 300 C, probably as an oxide; TcCl 7 from Tc 2 S 7 plus 
C1 2 volatilized at 100C. Technetium may be separated from 
molybdenum by precipitation of the latter with 8-hydroxyquino- 
line. It may be separated from ruthenium by volatility of TcCl 7 
or by distillation of RuO 4 from perchloric acid solution. It may 
be freed of rhenium by volatilization of the rhenium at about 
180C in dry hydrogen chloride, by fractional crystallization of 
KReO 4 , or by precipitation of Rc 2 S 7 from 10 N HC1. It may 
be volatilized from columbium in oxygen at a high temper- 
ature. 

At this time twenty-one activities have been reported for tech- 
netium isotopes, and most of them are well established. Perhaps 
most interesting is the lower isomeric state of Tc"; it has a half- 
life of about 10 6 years and may be produced in quantity in a chain 
reactor. Its chemical properties have now been investigated at 
macroconcentrations and found to agree insofar as reported to 
date with the indications given by the tracer experiments. The 
sulfide Tc 2 S 7 precipitated from 4 M H 2 SO 4 is dark brown and 
highly insoluble. The pertechnetate ion Tc0 4 ~ is probably pink. 
Technetium metal has been prepared and found by X-ray diffrac- 
tion to be in the hexagonal close-packed arrangement, isomorphous 
with rhenium, with density 11.49 g per cc. 



ASTATINE 271 

Astatine. Isotopes of element 85 have been claimed and fairly 
well identified by their radiations as very short-lived branch 
products in the radium and thorium series, formed from RaA 
and ThA by ft decay in a very small fraction of the disintegrations 
(the normal mode is a decay for both). F. Allison claimed in 1931 
that this element had been observed as a natural substance by 
the magneto-optic technique. However, the first quite definite 
demonstration of element 85 was given by D. R. Corson, K. R. 
Mackenzie, and Segr&; they produced a radioactive isotope 85 211 
by the a, 2n reaction on Bi 209 using 30-Mev helium ions from 
the then newly completed 60-inch cyclotron in Berkeley. The 
half -life is 7.5 hr, and the decay is 40 per cent by a. emission and 
60 per cent by K capture. Several other isotopes are now known, 
but the half-lives are all short. Corson, Mackenzie, and Segrfe 
have recently proposed the name astatine, symbol At, derived 
from the Greek word meaning "unstable." 

Astatine is the halogen just heavier than iodine, and its chemical 
properties make a very interesting study. Because of the short 
half-life macroscopic quantities may not be accumulated, but tracer 
studies by G. L. Johnson, R. F. Leininger, and Segr& have given 
information which we may summarize. The element is separated 
from bismuth, from which it is formed, and from simultaneously 
produced radioactive polonium by its volatility from molten bis- 
muth in vacuum. The free element is quite volatile, particularly 
from a glass surface, even at room temperature. It has a rather 
specific affinity for metallic silver surfaces even at 325C. The 
free element exists in aqueous solution (presumably as At 2 or 
possibly At) and usually is partly lost on evaporation of acidic 
solutions. The free element is readily extracted from water solu- 
tions into benzene or carbon tetrachloride very much like iodine. 
It may not be extracted from alkaline solution, again like 
iodine. 

Astatine may be reduced by 862 or by zinc, but not by ferrous 
ion, almost certainly to the 1 oxidation state. This ion is pre- 
cipitated with Agl or Til from acidic or basic solutions. Cold 
concentrated nitric acid seems to oxidize elemental astatine only 
slowly. It is oxidized by bromine, and to some extent by ferric 
ion, to some positive oxidation state (possibly AtO~). This state 
is shown by migration experiments to be an anion; it is hardly 
AtO 3 ~ since it is not carried by a precipitate of AgIO 3 . With 



272 CHEMISTRY OF LOW CONCENTRATIONS CH. XII 



HC10 or hot S208 =a as oxidizing agent a different anion results; 
this is carried by AglOa and may be AtOa""". 

Francium. In 1914 F. Paneth observed some a-particle emission 
from 89 Ac 227 , which decays mostly by emission to RdAc. In 
1939 M. Perey observed the daughter produced in the 1 per cent 
a. branch and called the isotope of element 87 so produced AcK. 
This body decays by ft emission (to AcX); the half -life is 21 min. 
Mile. Perey has proposed the name francium, symbol Fr, for the 
new element. Its chemical properties appear to be as expected 
from its position in the periodic table in group I below cesium. 
It is carried along with CsC10 4 or Cs 2 PtCl 6 and also by the anal- 
ogous rubidium salts. Although the corresponding sodium and 
potassium salts apparently crystallize in the same crystal systems 
(rhombic for the perchlorates and cubic for the chloroplatinates), 
these do not carry francium effectively, presumably because of 
the great differences in ionic radii. Another isotope of francium 
may be formed in a rare a-particle branching of MsTh 2 (Ac 228 ), 
but this product has not been observed. Recently Fr 221 has been 
described as an a. emitter with a 4.8-min half-life in the 4n + 1 
series, and other very short-lived a-emitting francium isotopes 
have been produced artificially. 

Element 61. The history of element 61 has been and remains 
confused. In 1926 several groups of researchers reported evidence 
based on optical and X-ray spectral lines for the existence of the 
element in various minerals and rare-earth concentrates; the 
names of these workers included J. A. Harris, B. S. Hopkins, and 
L. F. Yntema; L. Rolla and L. Fernandes; and J. M. Cork, 
C. James, and H. C. Fogg. Names for the element proposed in 
this period were illinium, II, by Hopkins, and florentium by 
Rolla. If the element actually exists in nature in stable form and 
is detectable by methods then used, it is rather surprising that 
higher concentrations have not been prepared. About 1941 and 
shortly thereafter workers at Ohio State University including 
H. B. Law, M. L. Pool, J. D. Kurbatov, and L. L. Quill and later 
C. S. Wu and Segre in Berkeley obtained from cyclotron bombard- 
ments several activities which were attributed to isotopes of the 
missing element; however, the certainty of this interpretation 
was not positively established. Pool and Quill have recently 
proposed for the element the name cyclonium, symbol Cy. The 
fission of uranium produces several radioactive isotopes of ele- 



TRANSURANIUM ELEMENTS 273 

ment 61, and these have been investigated and definitely charac- 
terized by .workers at Oak Ridge including C. D. Coryell, J. A. 
Marinsky, and L. E. Glendenin. They were able to concentrate 
the tracer activities by the ion-exchange resin adsorption and 
elution technique. Their proposal for the name of the element 
is promethium, symbol Pm. Recently, visible amounts of 61 147 
have been exhibited. 

Transuranium Elements. When Fermi and his group in Rome 
first exposed uranium to slow neutrons they observed a number 
of activities, and in the following few years many more active 
species were found to be produced; most of these were at that 
time assigned to transuranium elements. The assignments were 
made because the substances were transformed by successive ft 
emissions which led to higher Z values, and because they could 
be shown by chemical tests to be different from all the known 
elements in the neighborhood of uranium in the periodic chart. 
This situation was resolved in the discovery by Hahn and F. Strass- 
man that these activities could be identified with known elements 
much lighter than uranium and that, therefore, the neutrons pro- 
duce fission of the uranium nuclei. Further investigation of the 
fission process and products led to the proof by E. M. McMillan 
and P. Abelson that one of the activities, the one with 2.3 days 
half-life, could not be a product of fission and was actually the 
daughter of the 23-min /3-particle-emitting U 239 which resulted 
from U 238 (n, 7) U 239 . Also, they devised a procedure for sepa- 
rating chemically the element 93 tracer from all known elements 
through an oxidation-reduction cycle, with bromate as the oxidiz- 
ing agent in acid solution, and with a rare-earth fluoride precipi- 
tate as carrier for the reduced state. They gave the name 
neptunium, symbol Np, to the new element, taking the name 
from Neptune, the planet next beyond Uranus in the solar system. 

At the present time seven isotopes of neptunium are known, with 
half-lives from 53 min for K- and a-active Np 231 to 2.25 X 10 6 
years for a-active Np 237 . Both Np 239 discovered by McMillan and 
Abelson and Np 238 from a, p3n or d, 2n reactions on U 238 emit 
ft particles and lead to known isotopes of element 94, very naturally 
named plutonium after Pluto (a planet beyond Neptune), with 
symbol Pu. These isotopes, Pu 238 and Pu 239 , are moderately 
long-lived a emitters first studied by McMillan, G. T. Seaborg, 
Segre, A. C. Wahl, and Kennedy; Pu 239 is distinguished for its 



274 CHEMISTRY OF LOW CONCENTRATIONS CH. XII 

practical usefulness in slow- and fast-neutron fission. Another 
isotope Pu 241 has been reported; it decays with a half -life of about 
10 years by 0~ emission to produce Am 241 , an isotope of the 
element 95, named americium with symbol Am. This americium 
isotope has a half-life of 500 years and emits a particles. By an 
n, 7 reaction Am 241 is converted to Am 242 , a ^-emitting 17-hr 
isotope; this decays to the isotope of curium, Cm 242 . This last 
substance emits a particles and has a half -life of 5 months; it is 
formed also by the reaction Pu 239 (a, n) Cm 242 . In addition 
Cm 240 , an a emitter with 1 month half -life, has been prepared by 
Pu 239 (a, 3n). Other isotopes of these elements are listed in 
table A in the appendix. 

The chemical properties of all these transuranium elements 
neptunium, plutonium, americium, and curium have been studied 
first as tracers and later by ultramicrochemical techniques. At 
the present time Pu 239 exists in some quantity, Np 237 has been 
isolated to the extent of hundreds of milligrams, Am 241 has been 
isolated on a microscale, and a very small amount of Cm 242 has 
been isolated. Most of the work has been done in Seaborg's 
laboratory, at the University of Chicago Metallurgical Project 
and at the University of California Radiation Laboratory; the 
elements americium and curium were discovered by Seaborg, 
R. A. James, L. 0. Morgan, and A. Ghiorso. The transuranium 
elements and uranium and thorium all have similar precipitation 
properties when in the same oxidation state; they differ principally 
in the ease of formation and in the existence of the various oxida- 
tion states. Seaborg has advanced the hypothesis, and with con- 
siderable evidence, that a new rare-earth series begins with actin- 
ium (number 89), with the 5/ electron orbitals being filled in 
subsequent elements. Thi$ would be analogous to the lanthanide 
rare-earth series beginning with lanthanum (number 57), with 
the 4/ orbitals filling in the next fourteen elements. Some of the 
evidence for this actinide series may be seen in these facts: (1) lan- 
thanum is chemically similar to actinium; (2) thorium is similar 
to cerium in the +4 state; (3) the ease of removal of more than 
three electrons decreases from uranium to curium. (Approximate 
oxidation potentials for uranium, neptunium, and plutonium are 
given in table XII-2.) There is additional evidence for the second 
rare-earth series from spectroscopic and crystal-structure data. 



TRANSURANIUM ELEMENTS 



275 



The interpretation of such magnetic data as are available is not 
at all clear. 

It does seem evident that this new series differs from the familiar 
rare-earth series in that the resemblance of successive elements 
is less than for the lanthanide series. The lanthanide earths are 
for the most part separable only by multiple fractionation proc- 
esses, or better by adsorption and elution from ion-exchange 
resins; the elements from 89 to 95 are separable by oxidation- 

TABLE XII-2 
OXIDATION POTENTIALS OF URANIUM, NEPTUNIUM, AND PLUTONIUM IONS 

TT ~i.7^ T +++ -0.5 TT++++ ? TT(V) ? TTO.++ 



1.4 



-0.35 



0.82 



+++ -- 1 



Np_L_Np 



-- 74 



-0.94 



- 1 - 2 Pu0 2 + ~- 93 PuQ 2 ++ 

-1.05 



reduction processes, but the separation of 95 from 96, both of 
which are stable in the +3 state, may require an ion-exchange 
column or a fractionation method. On the actinide hypothesis, 
curium, by analogy to gadolinium, would be expected to resist 
oxidation or reduction in the +3 state, because the 5/ 7 and 4/ 7 
structures, with one electron in each of the seven / orbitals, are 
particularly stable. Then americium, by analogy to europium, 
might possibly be reducible to a +2 state. (The oxide AmO has 
been reported, also Am0 2 .) Actually it is on the basis of these 
analogies that the names for 95 and 96 were suggested; curium 
after Pierre and Marie Curie and americium after North and 
South America. On the basis of an actinide series we might 
speculate on the properties of element 97, which likely will be 
found in time; by analogy to terbium it might be capable of oxida- 
tion to a +4 state. 



276 CHEMISTRY OF LOW CONCENTRATIONS CH. XII 

Some of the difficulties in work with substances like Cm 242 may 
be mentioned here, difficulties in addition to those naturally asso- 
ciated with work on the ultramicrochemical scale. The heavy 
short-lived a. emitters are extremely dangerous as radioactive 
poisons, and amounts of the order of a few micrograms taken 
into the body may produce harmful effects. Also, the high level 
of a radiation in concentrated samples can be expected to have 
some effect on chemical reactions; notice that a curium prepara- 
tion glows in the dark. In fact the rate of energy release is so 
great that if cooling effects are neglected it may be estimated that 
a 0.1 molar Cm 242 solution would begin to boil in about 15 sec 
and reach dryness in about 2 min. The question of possible exist- 
ence of longer-lived isotopes of curium, of course, arises. The 
stability curve in this region would suggest that less a-active iso- 
topes- might occur at slightly higher masses, nearer Cm 244 or 
Cm 245 . No very practical way of preparing these isotopes comes 
easily to mind. 

EXERCISES 

1. Calculate the electrode potentials corresponding to these half- 
reactions at the specified concentrations: 

(a) Ag = Ag+ + e~ with (Ag+) = 10~ 13 molar; 

(6) Al = A1+++ + 3e~, with (A1+++) = 10~ 15 molar; 

(c) 2Hg = Hg 2 ++ + 2e~, with (Hg 2 ++) = 10~ 8 molar. 

(d) Would the shift in emf in (c) continue to be proportional to the 
logarithm of the concentration of mercury ions in solution as that con- 
centration was indefinitely reduced? Answer: (a) 0.031 v. 

2. In analytical chemistry Fe 4 " 1 " 1 " is used as an oxidizing agent to 
convert I" to I 2 . Would Fe++ + be suitable to oxidize a solution of 
(pure) I 128 , concentration 150 /-ird per ml? 

3. On the basis of data given in table XII-2, estimate the concentra- 
tions of Pu ++ +, Pu02 + , and Pu02 ++ at equilibrium in a solution which 
has (Pu++ ++) = 1 molar and (H+) = 1 molar. 

Answer: (Pu +++ ) = about 0.1 molar. 

4. What is the electrode potential for Cu = Cu++ + 2e~ at a Cu+ + 
concentration equal to exactly zero? What do you suppose would be the 
result if a pure copper electrode were immersed in absolutely pure water? 

5. In an experiment on the crystallization of mixed radium-barium 
chlorides from supersaturated solutions the following data were obtained : 



REFERENCES 277 

Percentage of Radium Percentage of Barium 
Remaining in Solution Remaining in Solution 

87.41 97.48 

60.30 89.21 

59.01 88.45 

54.72 86.58 

47.61 83.53 

43.15 80.24 

Do these fractional crystallizations obey the Doerner-Hoskins or the 
Berthelot-Nernst equations? Find X or D. 

6. Take the partition coefficient for the distribution of astatine between 
carbon tetrachloride and water as 100 at 10 ~ 10 molar for the nonaqueous 
phase. Further, assume that the substance in carbon tetrachloride has 
the molecular formula At2. What might you expect for this partition 
coefficient at 10 ~ 11 molar (nonaqueous phase) if the zero state in water 
at these concentrations is predominantly (a) At2, (b) At, (c) At~ + HAtO, 
(d) Air + HAtO, (e) 3 Air + At0 2 ~. Answer: (b) 32. 

REFERENCES 

A. C. WAHL (Editor), Radioactivity Applied to Chemistry, New York, John 

Wiley & Sons, to be published. 

0. HAHN, Applied Radiochemistry, Cornell University Press, 1936. 
G. T. SEABORG, "Artificial Radioactivity," Chem. Reviews 27, 199 (1940). 
M.I.T. Seminar Notes (C. GOODMAN, Editor), The Science and Engineering of 

Nuclear Power, Vol. I, chapter 11, Addison- Wesley Press, Cambridge 

(Mass.), 1947. 
The Chemical Institute of Canada, Proceedings of the Conference on Nuclear 

Chemistry, May 15-17, 1947. 
C. PERKIER and E. SEGRE, "Some Chemical Properties of Element 43, " J. 

Chem. Phys. 5, 715 (1937), and 7, 155 (1939). 
G. L. JOHNSON, R. F. LEININGER, and E. SEGRE, "Chemical Properties of 

Astatine. I.", /. Chem. Phys. 17, 1 (1949). 

M. PEREY, "Chemical Properties of Element 87," /. chim. phys. 43, 262 (1946). 
J. A. MARINSKY, L. E. GLENDENIN, and C. D. CORYELL, "The Chemical 

Identification of Radioiso topes of Neodymium and of Element 61," JACS 

69, 2781 (1947). 
G. T. SEABORG, "Plutonium and Other Transuranium Elements," Chem. Eng. 

News 25, 358 (1947). 
L PERLMAN, "The Transuranium Elements and Nuclear Chemistry," J. Chem. 

Ed. 25, 273 (1948). 
F. A. PANETH, "The Making of the Missing Chemical Elements," Nature 159, 

8 (1947). 



Xm. TRACERS IN CHEMICAL APPLICATIONS 

A. THE TRACER METHOD; DIFFUSION STUDIES 

Iso topic Tracers. Most of the ordinary chemical elements are 
composed of mixtures of isotopes, and each mixture remains essen- 
tially invariant in composition through the course of physical, 
chemical, and biological processes. That this is so is shown by 
the constant isotopic ratios found for elements from widely scat- 
tered sources (1) and by the fact that atomic weights reliable to 
many significant figures may be determined by chemical means. 
It is true that isotopic fractionation may be appreciable for the 
lightest elements where the percentage mass difference between 
isotopes is greatest, and this effect must always be considered in 
the use of hydrogen tracer isotopes. However, apart from H 3 
(tritium) there is no practical radioactive tracer lighter than Be 7 , 
which differs in mass from stable Be 9 by only about 25 per cent, 
and the next heavier tracer is in carbon where already the specific 
isotope effect may be neglected in most tracer work of ordinary 
precision. 

The fact that a given nuclide may be radioactive does not in 
any way affect its chemical (or biological) properties; at least 
this is true for each atom until its nucleus actually undergoes the 
spontaneous radioactive change. Because the tracer-isotope 
atoms are detected by their radioactivity, this means that they 
behave normally up to the moment of detection; after that moment 
they are not detected, and their fate is of no consequence. Of 
course, if the resulting atoms after the nuclear transformation 
also should be radioactive and capable of a further nuclear change, 
the detection method must be arranged to give a response which 
measures the proper (in this case the first) radioactive species 
only. For example, if RaE (Bi 210 ) is used as a tracer for bismuth 
the a. particles from its daughter Po 210 should not be allowed to 

1 Some exceptions to the constancy of isotopic ratios were mentioned in 
chapter I, section C, and chapter II, section B. 

278 



ISOTOPIC TRACERS 279 

enter the detection instrument but should be absorbed by a suit- 
able absorber or by the counter wall. As a tracer for thorium 
TJX! is suitable in spite of the fact that most of the detectable 
radiation will be from its daughter UX 2 ; the reason is that the 
short half-life, 1.14 min, of UX 2 insures that it will be in transient 
equilibrium with the UXi by the time the sample is mounted and 
ready for counting, so that the total activity will be proportional 
to the UXi content. Many multiple decays are found in the 
fission-product activities. If Ba 140 (ty z = 12.8 days) is to be 
used as a tracer for barium, the isolated samples either should be 
freed chemically of the daughter La 140 (t^ = 40 hr) or should be 
kept before counting for a week or two until the transient equi- 
librium is achieved. The isomeric transition activities present 
interesting cases; if the 4.4-hr Br 80 is chosen as a tracer for bro- 
mine the 18-min lower isomeric state of Br 80 will always be pres- 
ent, and, because of the hot-atom effects accompanying the iso- 
meric transition as discussed in chapter XI, the two isomers may 
be present in different chemical forms. However one may use 
the 4.4-hr Br 80 with confidence, provided only that one measures 
isolated samples for the 4.4-hr period by analysis of the decay 
curves or simply by holding the prepared samples for a time long 
compared to 18 min before counting. These special cases do not 
arise in the use of the great majority of popular tracer isotopes. 
Another source of interference with the tracer principle is a possi- 
ble chemical (or biological) effect produced by the ionizing rays; 
this radiation chemistry effect is not often encountered at the usual 
tracer activity levels and may always be checked by experiments 
with a much higher or lower level of radioactivity. 

A definite limitation of the radioactive tracer method is the 
absence of known active isotopes of suitable half-life for a few 
elements, especially oxygen and nitrogen. There are radioactive 
oxygen isotopes, O 15 and O 19 , but these have half-lives of 126 sec 
and 31 sec, respectively. The 7-sec N 16 and 4-sec N 17 are useless 
as tracers, but some applications of the 10-min N 13 p + activity 
have been made. Also, helium, lithium, and boron do not have 
periods longer than 1 sec. The use of separated stable isotopes 
as tracers is a very valuable technique, and availability of the 
necessary concentrated isotopes is increasing. Enriched O 18 and 
N 15 are essential for many interesting and important purposes, 
and C 13 offers significant advantages for some carbon tracer 



280 TRACERS IN CHEMICAL APPLICATIONS CH. XIII 

experiments. Deuterium has found many applications as a hydro- 
gen tracer, and the use of tritium (H 3 ) is not entirely equivalent 
because its properties are even more different from those of pro- 
tium (H 1 ). 
The uses of isotopic tracers may be classified into two groups: 

(1) applications in which the tracer is necessary in principle, and 

(2) applications in which a tracer not necessary in principle may 
be a great practical convenience. Those applications which 
depend uniquely on the tracer principle although the tracers 
may be either radioactive or separated stable isotopes may be 
illustrated by studies of self-diffusion of an element or other sub- 
stance into itself; no other investigative techniques can give 
information on such matters. On the other hand, the studies on 
coprecipitation already mentioned in chapter XII might be done, 
at least at the higher concentrations, by careful application of 
conventional chemical methods, or perhaps by spectroscopic or 
other means of analysis. In some of the more involved applica- 
tions, particularly in biology, both of these aspects of tracer 
usefulness appear together. 

Self-diffusion. To illustrate the unique tracer method we will 
discuss first studies that have been made of self-diffusion. By 
the use of sensitive spectroscopic analyses the rates of diffusion 
of various metals (including gold, silver, bismuth, thallium, and 
tin) in solid lead at elevated temperatures have been investigated, 
but the first attempt (by G. Hevesy and his collaborators) to 
observe the diffusion of radioactive lead into ordinary lead failed, 
showing that the diffusion rate must be at least one hundred times 
smaller than that for gold in lead (which is the fastest of those just 
named, the others showing decreasing rates in the order listed). 
The method first used was a rather gross mechanical one, and the 
workers evolved a much more sensitive method based on the 
short range of the a particles from ThC in transient equilibrium 
with ThB. The lead containing ThB isotopic tracer was pressed 
into contact with a thin foil of inactive lead which was chosen just 
thick enough to stop all the a. rays, and then as diffusion pro- 
gressed an a activity appeared and increased as measured through 
this foil. The diffusion coefficient D is obtained through a suitable 

dc 3 2 c 

integration of Tick's diffusion law, = D - , where c is con- 

dt dx 

centration of the diffusing tracer, t is time, and x is the coordinate 



OTHER MIGRATION PROBLEMS 281 

along which the diffusion is measured; some typical values for D 
were 0.6 X 10" 6 cm 2 per day at 260C, 2.5 X 10~ 6 at 300C, 
and 47 X 10~ 6 at 320C. A similar but even more sensitive 
technique than the a-range method was based on the very much 
shorter ranges (a few millionths of a centimeter in lead) of the 
nuclei recoiling from a emission, with the radioactivity of the 
resulting ThC" as an indicator of the emergence of recoil nuclei 
from the very thin lead foils. At 200C the diffusion of lead in 
lead is about ten times slower than that of tin in lead, and roughly 
10 5 times slower than that of gold in lead. 

The diffusion of bismuth in bismuth has been studied with ThC 
as tracer; an interesting result is that the diffusion parallel to the 
c axis of the crystal is very different in magnitude (^-dO 5 times 
slower at 250C) and in temperature dependence from that per- 
pendicular to the c axis. Self-diffusion of gold was studied by 
H. A. C. McKay; he induced activity principally on one surface 
only of a thin gold disk by an ingenious activation with resonance 
neutrons, for which the capture cross section in gold is so great 
that few neutrons could penetrate far below the exposed surface. 
Self-diffusion of copper has been studied by several investigators, 
with the use of copper disks either activated on one surface by 
deuteron irradiation or plated on one side with active copper. 
Self-diffusions of zinc and of silver have been measured from elec- 
troplated surfaces. Diffusion of some ions in crystals has been 
investigated, for example, of Pb ++ in PbCl 2 and in PbI 2 . 

The rates of diffusion of ions through aqueous salt solutions of 
uniform composition may be determined with radioactive tracers, 
and this information may be of special significance since each 
rate is a property of the particular ion in that system, whereas 
salt-diffusion rates under a concentration gradient as ordinarily 
observed must necessarily depend on the diffusion tendencies of 
ions of both charges. Diffusion coefficients have been determined 
for ions such as Na + , Cl~~, and I~~ in salt solutions. 

Other Migration Problems. Radioactive tracers are useful 
in the study of numerous migration problems other than self- 
diffusion, particularly where movements of very small amounts 
of material are involved. In most such applications the tracer is 
serving only as a very sensitive and relatively convenient analytical 
tool. Erosion and corrosion of surfaces may be measured with 
great sensitivity if the surface to be tested can be made intensely 



282 TRACERS IN CHEMICAL APPLICATIONS CH. XIII 

radioactive. Transfer of very minute amounts of bearing-surface 
materials during friction has been studied in this way. Radio- 
active gases or vapors may be detected in small concentrations, 
and leakage, flow, and diffusion rates may therefore be studied by 
the tracer method. 



B. EXCHANGE REACTIONS 

Qualitative Observations. In a very early exchange experiment 
in 1920 Hevesy demonstrated by the use of ThB (Pb 212 ) the rapid 
interchange of lead atoms between Pb(N0 3 )2 and PbC^ in water 
solution. The experiment was performed by the addition of an 
active Pb(N0 3 )2 solution to an inactive PbCl2 solution and by 
the subsequent crystallization of PbCl2 from the mixture. The 
result is not at all surprising because the well-known process of 
ionization for these salts leads to chemically identical lead ions, 
Pb ++ . Hevesy also showed a rapid exchange of the lead atoms 
between Pb(C 2 H 3 O 2 ) 2 and Pb(C 2 H3O 2 )4 in acetic acid; we may 
conclude that the plumbous and plumbic forms enter a reversible 
oxidation-reduction reaction of some sort. Very many exchange 
systems have been examined since that time, and for the majority 
of cases where exchange is rapid at ordinary temperature there 
are known reversible reactions which would lead to interchange. 
For the other observed exchanges there either exist such reversible 
reactions, which are possibly unknown, or the exchange occurs 
by a simple collision mechanism (which may amount to about the 
same thing). 

It was soon shown when artificially radioactive isotopes became 
available that aqueous Cl~~ and C1 2 , Br~ and Br 2 , and I~ and I 2 
exchanged at room temperature so quickly that the rates could 
" not be measured by ordinary methods. These exchanges are 
interpreted as occurring through the reactions illustrated by 
I~~ + 12^ Is"". It has been found that Br 2 and HBr, either in 
the gas phase or in solution in dry carbon tetrachloride, exchange 
rapidly at room temperature, probably through reversible forma- 
tion of a complex HBr 3 , although the life of this intermediate may 
be very short and thus its concentration very low. Also I 2 and 
SbI 3 in dry pentane exchange within 20 min at 37 C, very possibly 
through Sbls. Rapid exchanges at room temperature in carbon 
tetrachloride are found between Br 2 and AsBr 3 and between Br 2 



QUALITATIVE OBSERVATIONS 283 

and SnBr 4 ; we may imagine that these proceed through inter- 
mediates like AsBr 5 and SnBr 2 , or bromides of other oxidation 
states. In aqueous solution PtBr 4 = or PtBr 6 = rapidly exchanges 
all bromine atoms with Br~~ ion; the four iodine atoms in Hgl^ 
exchange with I"" ion. 

In dilute-acid solution at room temperature there is no rapid 
exchange of halogen atoms between C1 2 and C10 3 ~" or C10 4 ~~, 
Br 2 and Br0 3 ~, I 2 and IO-T, C10 3 ~ and C10 4 ~ I0 3 ~ and I0 4 ""; 
however, some of these do exchange at measurable rates. No 
exchange was found in alkaline solution between Cl~ and C10 4 ~~, 
Br~" and Br0 3 ~, I"" and I0 3 ~. 

Interesting exchange studies have been made with the tracer 
S 35 . Sulfur and sulfide ions exchange in polysulfide solution. 
Even at 100C S = and SO*", S0 3 = and S0 4 ~ H 2 S0 3 and HS0 4 ~ 
do not exchange appreciably. If active sulfur is reacted with 
inactive SO^ to form S 2 3 == , and then the sulfur removed with 
acid, the H 2 SO 3 is regenerated inactive; therefore, the two sulfur 
atoms in thiosulfate are not equivalent. The ions S 2 O 3 X= and SOs* 3 * 
exchange only very slowly at room temperature, but exchange 
one sulfur fairly rapidly at 100 C. (Notice that this result can 
be found only by labeling the proper sulfur atom, the one attached 
directly to the oxygen atoms.) Sulfide ion and S 2 3 ==s slowly 
exchange (probably only one sulfur) at 100C. Sulfur does not 
exchange with CS 2 at 100 C; SO 2 and S0 3 do not exchange appre- 
ciably even at 280C. The short-lived C 11 has been used to show 
that CO and CO 2 do not exchange in 1 hr at 200C. 

Phosphoric and phosphorous acids, H 3 P0 4 and H 2 (HPO 3 ), 
and also phosphoric and hypophosphorous acids, H 3 P0 4 and 
H(H 2 P0 2 ), do not exchange phosphorus atoms even at 100C, 
although the first of these exchanges might be expected to proceed 
(at some unknown rate) through the formation of hypophosphoric 
acid, H 4 P 2 O 6 . Arsenate and arsenite ions, and H 3 AsO 4 and 
HAsO 2 do not exchange appreciably even at 100 C. 

The exchange reactions of various manganese compounds have 
been surveyed. The following do not exchange readily: MnO 4 ~ 
and Mn+ +, MnO 4 ~ and Mn(C 2 O 4 ) 3 s , Mn0 4 ~ and MnO 2 , Mn++ 
and MnO 2 . Rapid exchanges at room temperature were found 
for the pairs Mn^ 4 " and Mn(C 2 4 ) 3 s and Mn0 4 ~~ and MnO 4 "; 
in these cases oxidation-reduction by electron transfer only is 
involved. Ferrous and ferric ions transfer an electron (that is, 



284 TRACERS IN CHEMICAL APPLICATIONS CH. XIII 

they exchange) readily in 6 normal HC1, but apparently according 
to one recent report (la) they exchange only very slowly in HC104; 
this may mean that Fe^ and Fe +++ ions have difficulty in 
approaching sufficiently close for the oxidation-reduction reaction 
because of coulombic repulsion, but in HC1 the neutral FeCls 
may react. The two ions, Fe(CN) 6 s and Fe(CN) 6 =, probably 
exchange rapidly, although the method (precipitation) which has 
been used for separation might possibly be responsible for the 
exchange. As an example of this effect, Tl + and Tl 4 "*" 1 " ions 
were first reported to exchange rapidly, but recent investigations 
show that the exchange is a moderately slow bimolecular process 
inhibited by high concentrations of strong acids; a correction is 
applied to the experimental data for a partial exchange induced 
by the precipitation used as a separation method. Again in this 
case the slowness of the exchange may be due to coulombic repul- 
sion, and the effect of strong acid may be due to the reduction in 
number of the less-charged partially hydrolyzed forms of Tl(III); 
the induced exchange during precipitation may proceed through 
neutralized molecules of transient existence. The mercury ions 
Hg" 4 "*" and Hg 2 " f " f have been reported to exchange readily. 

Some metal surfaces exchange rather well with the correspond- 
ing metal ions in solution, as observed in the case of silver, zinc, 
and lead. With silver the exchange reaches to a depth of 10 to 
100 atomic layers in about 1 hr at room temperature. In part 
these effects may be due to local electrolysis caused by imperfec- 
tions in the metal surface. Precipitates such as AgBr exchange 
rather effectively with component ions such as Br~ when freshly 
formed, but only much more slowly when aged. An exchange of 
over 100 per cent completion is simulated in some cases; that is, 
initially inactive crystals may reach a higher specific activity 
than that of the final halide solution, because the inner and outer 
parts of grown crystals may not be in equilibrium. 

Alkyl halides of all types ordinarily do not exchange readily 
at room temperature with either free halogens or halide ions. 
However, the nature of the solvent (particularly ethyl alcohol, 
acetone, and amyl alcohol in some reported experiments) can 

10 Note added in proof: We now believe that this exchange is fast even in 
HC1O4. Moreover it appears that Ce(III) and Ce(IV) also exchange rapidly 
in HC1O4, so that the role of coulombic repulsion in electron-transfer exchanges 
is not clear. 



QUANTITATIVE EXCHANGE LAW 285 

exert a marked influence in producing exchange in these systems. 
In the solvents named, I"" exchanges with ethyl, n-propyl, iso- 
propyl, and methylene iodides and iodoform in about 15 min or 
less at 100C. Most remarkable, in these solvents I"~ and CH 3 I 
are reported to exchange in 1 min at room temperature; the ex- 
change of CH 3 I with I 2 is much less rapid. Iodide ion exchanges 
rapidly with iodoacetic acid, but rapidly only at elevated tempera- 
tures with /3-iodopropionic acid. The phenyl halides (including 
p-nitro and p-amino derivatives) exchange less readily than the 
alkyl halides. Halogen exchanges with alkyl halides have been 
produced photochemically. On the other hand, gaseous HBr 
and C 2 H 5 Br did not exchange photochemically but did exchange 
thermally at 300C. 

The behavior of aluminum bromide in exchange reactions is 
remarkable, paralleling its extraordinary character as a catalyst. 
It exchanges bromine atoms readily at room temperature with 
many alkyl bromides, with benzyl bromide, and with many 
aliphatic polybromides; it exchanges also, but more slowly, with 
aryl bromides. Aluminum iodide appears to behave in a similar 
way. Because these aluminum halides also exchange readily 
with gaseous halogen or hydrogen halide, a convenient synthesis 
of labeled organic halides is provided. Obviously the presence of 
aluminum bromide will catalyze an exchange between two organic 
bromides. 

Knowledge of the occurrence or nonoccurrence of rapid exchange 
has been used in the study of bond character. It is obvious that 
exchange data give information on the degree of stability of par- 
ticular bonds, but the relation of this information to the bond 
type and to other aspects of bond character is not established at 
the present time. 

Quantitative Exchange Law. Consider a schematic exchange- 
producing reaction, 

AX + BX = AX + BX, 

where X represents a radioactive atom of X. The radioactive 
decay of this species will be neglected; in practice if the decay is 
appreciable correction of all measured activities to some common 
time must be used to avoid error from this condition. The rate 
of the reaction between AX and BX in the dynamic equilibrium 
we call R, in units of moles liter"" 1 sec""" 1 ; notice that R is quite 



286 TRACERS IN CHEMICAL APPLICATIONS CH. XIII 

independent of the concentration and even of the existence of the 
active tracer X. We indicate mole-per-liter concentrations as 
follows: (AX) + (AX) = a, (BX) + (BX) = 6, (AX) = x 9 

(BX G ) = y and x + y = z. The rate of increase ( ) of (AX) 

\dt / 

is given by the rate of its formation minus the rate of its destruc- 
tion. The rate of formation of AX is given by R times the factor 
y/b, which is the fraction of reactions that occur with an active 
molecule BX, and times the factor (a x)/a, which is the frac- 
tion of reactions with the molecule AX initially inactive. The 
rate of destruction of AX is given by R times the factor x/a, 
which is the fraction of reactions in the reverse direction that 
occur with an active molecule AX Q , and times the factor (b y)/b, 
which is the fraction of reverse reactions with the molecule BX 
initially inactive. The differential equation is then 

dx _ y (a - x) x (b - y) 
j . j^ 

dt b a a b 

R R 

= (ay bx) = (az ax bx); 
ab ab 

dx a + b z 

= -- Rx + - R. 
dt ab b 

The solution of this first-order linear equation is found by stand- 
ard methods: 

-^R< a 

z. 



a + b 

After a very long time, that is at t = QO , let x = x^ and y = y 
substituting these values in the previous solution we have then 



= ~~T^> 
a + b 

and, since y^ = z x^, 

b 

2/oo = z. 
a + b 



QUANTITATIVE EXCHANGE LAW 287 

These two relations constitute an algebraic expression for the 
reasonable and well-known rule that when exchange is complete 
the specific activity (activity per mole or per gram of X) is the 
same in both fractions; at that time the specific activity of AX is 

= , and the specific activity of BX is = 

The solution may now be rewritten: 

- a + b Rt 

x = Ce ft + x*. 

If at t = we have x = 0, that is, if AX is inactive at the start, 
we find the constant C = # and some useful (final) forms may 
be obtained: 



x 



ab 

x e , 

^00 

2.303 log ( 1 - ) - R*; (XIII-1) 

\ XQO/ ab 

and, by differentiation with respect to t, 

ab d ( x\ 
R = - 2.303 log I 1 ) 

a + bdt \ xj 

The last result shows that R may be evaluated from the slope of a 
plot of log [1 (X/XK)} vs. t. Probably the most convenient 
procedure is to plot [1 (x/x^)] on semilog paper against /, read 
off the half-time Ty^ at which the fraction exchanged, x/x^ is J/2, 
and find R from an equation derived immediately from equation 
(XIII-1): 

ab 0.69315 

o ___ t t 

~~ a + b Ty 2 

It is important to notice that if a or b or both should be varied 
the variation in half-time for the exchange would not directly 

ab 

reflect the variation in R, because of the factor 

a + b 



288 TRACERS IN CHEMICAL APPLICATIONS CH. XIII 

For a number of practical exchange studies the simple formulas 
AX and BX may not represent the reacting molecules; for exam- 
ple, A X 2 or BX n might be involved. So long as the several atoms 
of X are entirely equivalent (or at least indistinguishable in ex- 
change experiments) in each of these molecules, the equations 
just derived may be applied without modification, provided only 
that we redefine all the concentrations in gram atoms of X per 
liter rather than moles of AX or AX 2 , etc., per liter. This is 
equivalent to considering (for this purpose only) one molecule of 
AX 2 as replaceable by two molecules of Ai A X, etc., in the deriva- 
tion. If in a molecule like A X 2 the two X atoms are not equiva- 
lent, and if they exchange through two different reactions with 
rates RI and R2, it may be seen that the resulting semilog plot 
will be not a straight line but a complex curve. The differential 
equations for the exchanges to the several positions may be set 
up and solved simultaneously, so that the curve may, at least 
in principle, be resolved to give values for the several R's; how- 
ever, this becomes very difficult for more than about two rates. 
A simplification may be made if a 6, with the several nonequiva- 
lent positions in the molecule AX n \ here the value of y is very 
nearly a constant, and in this limit the complex semilog curve is 
resolvable in the same way as a radioactive decay curve into 
straight lines measuring RI, R 2 , etc. No example of a complex 
exchange curve has been reported except the limiting case with 
RX measurable, but R 2 = within experimental accuracy (like 
the sulfur exchange between 8203 s and SO3 = ). In this limiting 
case no unusual feature appears if x^ is used in an experimental 
sense, although after a much longer time x^ may be expected to 
reach a higher value. 

Reaction Kinetics and Mechanisms. Radioactive tracers are 
sure to find an important place in the investigations of reaction 
kinetics and mechanisms. We will discuss several examples to 
illustrate the kinds of information in this field that can be obtained 
with tracers but hardly in any other way. Consider the reversible 
reaction: 

HAs0 2 + I 3 ~ + 2H 2 ^ H 3 AsO 4 + 31" + 2H+. 

The familiar theory of dynamic equilibrium takes K = fc//fc r , 
where K is the equilibrium constant and fc/ and k r are the specific 



REACTION KINETICS AND MECHANISMS 289 

rate constants of the forward and reverse reactions. Ordinarily 
K may be measured only at equilibrium and k/ or k r far from 
equilibrium. Using radioactive arsenic to measure the rate of 
exchange between arsenious and arsenic acids induced by an 
iodine catalyst in accordance with the foregoing equilibrium reac- 
tion, J. N. Wilson and R. G. Dickinson were able to find the rate 
law and specific rate constant at equilibrium. For the reverse 
direction as written they found R = k r (H 3 As0 4 )(H + )(I~), with 
k r = 0.057 liter 2 mole" 2 min"" 1 , which is in satisfactory agreement 
with the information from ordinary rate studies made far from 
equilibrium, R = 0.071 (H 3 As0 4 )(H- f )(I""). 

A theory of the Walden inversion calls for inversion at each 
substitution by the schematic mechanism : 



C I = IC + I 

/I l\ 



As shown here the substitution is by a like group, and if the initial 
molecules are optically active the final product will be the racemic 
mixture. It has been shown that for sec-octyl iodide (or for 
a-phenyl ethyl bromide) the rate of exchange with radioactive 
iodide ion (or bromide ion) is identical with the rate of racemiza- 
tion, which is a verification of the mechanism. 

A different type of racemization is that of chromioxalate ion, 
Cr(C 2 O 4 ) 3 ^, which may be optically active through different 
linkings of the six octahedral bonds of the chromium with the 
carbon-oxygen chains. This racemization in aqueous solution is 
fairly rapid and apparently first order; it had been proposed that 
the mechanism involved an ionization as the rate-determining 
step: 

4 ) 3 " = Cr(C 2 4 ) 2 - + C 2 O 4 =. 



Another theory favored an intramolecular rearrangement instead. 
The racemization has been allowed to proceed in a solution con- 
taining radioactive C 2 4 = (prepared with C 11 ). The activity did 
not enter the chromium complex compound; therefore, the ioniza- 
tion is disproved, and the intramolecular rearrangement hypothe- 
sis supported. 



290 TRACERS IN CHEMICAL APPLICATIONS CH. XIII 

C. APPLICATIONS TO ANALYTICAL CHEMISTRY 

Test of Separations. Radioactive tracers can be very conveni- 
iently used to follow the progress and test the completeness of 
chemical separation procedures. If one component of a mixture 
is radioactive, frequently it can be followed satisfactorily through 
successive operations if beakers containing filtrates, funnels with 
precipitates, and so on, are merely held near a counter or ioniza- 
tion chamber. We have seen good chemical isolations made by 
these methods in the almost complete absence of knowledge of 
specific chemical properties. The crude qualitative procedure 
may be refined as far as desired, and valuable tests of analytical 
separation methods have been made with tracers. As the first 
of a few examples \ve consider a procedure that has been used for 
the determination of gold-platinum-iridium compositions. The 
three elements in solution are precipitated by reduction with hot 
alkaline sodium formate; the residue after ignition is treated with 
aqua regia to dissolve gold and platinum but leave iridium; the 
resulting solution on treatment with hydrogen peroxide gives 
the gold as a precipitate of the metal, and the platinum is finally 
precipitated from the filtrate with sodium formate. By simple 
gravimetric studies on known compositions the iridium fraction 
was found to be too heavy, the gold fraction too heavy, and the 
platinum fraction too light. With radioactive tracer it was shown 
that the gold fraction actually contained only 97 per cent of the 
original gold, but with more than enough platinum to mask 
this; the remainder of the gold was mostly in the platinum 
fraction. 

The coprecipitation of cobalt with SnS 2 has been investigated 
in relation to various experimental conditions, with active cobalt 
as a tracer. The amount of coprecipitation was smaller at higher 
hydrogen ion concentrations; the cobalt contamination could be 
made negligible provided acrolein was present as a flocculating 
agent. Radioactive beryllium has been used to find that aluminum 
precipitated by 8-hydroxyquinoline carries some beryllium at pH. 
values greater than 6; below pB. 6 the coprecipitation is absent. 
The carrying down of tellurium by antimony oxide precipitated 
from boiling concentrated nitric acid has been studied with tracers. 
Also a number of solvent extraction procedures have been tested 
in this way for interference effects. 



ANALYSIS BY ACTIVATION 291 

The solubility of quite insoluble precipitates may be judged by 
the use of radioactivity. This has been done in the precipitation 
of tin as Sn 3 [Fe(CN) 6 ]2. The approach to equilibrium between 
solid phase and solution can be followed very conveniently by 
repeated measurements of the specific activity of the supernatant 
solution. A somewhat analogous tracer method is applicable to 
the determination of small vapor pressures. 

Analysis by Isotope Dilution. It may frequently occur that 
quantitative analysis for a component of a mixture is wanted 
where no quantitative isolation procedure is known. Particularly 
for complex organic mixtures it may be possible to isolate from 
the unknown the desired compound with satisfactory purityTut 
only in low and uncertain yield. In such a case the analysis may 
be made by the technique of isotope dilution. To the unknown 
mixture is added a known weight of the compound to be deter- 
mined containing a known amount (activity) of radioactively 
tagged molecules. Then the specific activity of that pure com- 
pound isolated from the mixture is determined and compared 
with that of the added material; the extent of dilution of the tracer 
shows the amount of inactive compound present in the original 
unknown. (You may think of the tracer as serving to measure 
the chemical yield of the isolation procedure. Obviously exchange 
reactions which would reduce the specific activity of the com- 
pound must be absent.) To date this powerful method has found 
uses principally in biochemistry and biology. 

Analysis by Activation. Throughout most of the tracer work 
discussed radioactive isotopes are assayed by measurement of 
their activities; this is actually an analytical procedure, but we 
have not emphasized that aspect because the samples are subject 
to analysis only if the tracer was provided earlier in the experi- 
ment. Of course, the naturally radioactive elements, and this 
includes besides uranium, thorium, radium, other elements, par- 
ticularly potassium and rubidium, may be assayed by radioactive 
measurement; a very practical although not very sensitive pro- 
cedure for potassium assay by means of its radioactivity has been 
reported. 

A somewhat different technique can be useful, in which an un- 
known sample is subjected to activation by neutrons, deuterons, 
or other irradiation for appropriately chosen lengths of time, and 
chemical elements are identified and assayed by analysis of the 



292 TRACERS IN CHEMICAL APPLICATIONS CH. XIII 

probably complex decay curve resulting. Some practical uses 
of this technique of analysis by activation have been made. By 
deuteron irradiation very small impurities of gallium in iron, 
copper in nickel, and iron in cobalt oxide have been found. With 
neutron activation small amounts of phosphorus and sulfur have 
been detected, and also hafnium in zirconium, europium in gado- 
linium, and dysprosium in yttrium. Analyses for sodium by 
neutron activation can be very convenient when there are not 
many other elements present which give strong activities of com- 
parable half-life. In all these procedures the method may be 
standardized by the use of known samples, and in some cases 
where the cross sections are known or may be guessed with confi- 
dence absolute orders of magnitude may be obtained without 
standardization. 

Radiometric Analysis. Analytical procedures by tracer methods 
for elements which are not themselves radioactive have been 
introduced and given the name radiometric analysis. For exam- 
ple, silver ion may be determined by the precipitation of Agl 
with radioactive iodide ion. A recent report describes the collec- 
tion by adsorption on Fe(OH) 3 of very small amounts of Agl 
formed, so that as little as 10 ppm of silver could be detected. In 
another procedure almost invisible amounts of Til are centrifuged 
onto a plate for counting; either Tl" 1 " or I~ might be determined 
by the use of the other in concentrated radioactive form. 



D. RADIOCARBON TRACER STUDIES 

The Special Importance of Tracer Carbon. The special impor- 
tance of carbon compounds in chemistry and biochemistry give a 
special importance to radioactive carbon tracer. When only the 
20-min C 11 was available, serious limitation was obviously im- 
posed on the location and duration of tracer studies with this 
element. However, in spite of this limitation some remarkable 
experiments were carried out; for example, S. Ruben, M. D. 
Kamen, and W. Z. Hassid studied pRotosynthesis by preparing 
C 11 in a Berkeley cyclotron, carrying the boron target to another 
building, removing the active carbon and forming C U 2 , allow- 
ing growing plants to assimilate this and photosynthesize com- 
pounds from it, then isolating dozens of compounds in attempts 



SOME TYPICAL RESULTS 293 

to identify the active intermediates, and even taking the active 
plant extracts to Palo Alto for determinations of the molecular 
weight range (~1000) of the active substances in a Stanford 
University ultracentrifuge. 

In 1940 Ruben and Kamen discovered the long-lived C 14 from 
d, p reaction on the rare isotope C 13 ; however, the very small 
activities obtainable even at great cost prevented any widespread 
use of the material at that time. Today this important isotope is 
made from nitrogen by N 14 (n, p) C 14 in the Oak Ridge, Tenn., 
chain reactor, and is available for purchase from the U. S. Atomic 
Energy Commission at the price of $50 per millicurie. One milli- 
curie, or 37 rd, of pure C 14 would weigh very nearly 0.25 mg, if 
the half-life is taken as 6400 years. As ordinarily supplied thi^ 
amount of active carbon is mixed with several milligrams of ordi- 
nary carbon. In many ways C 14 is a very excellent tracer and 
because it emits no 7 rays and only soft ft rays is easily shielded 
during chemical operations; however, the small penetration of 
the radiation does necessitate some special techniques of activity 
measurement. 

Some Typical Results. P. Nahinsky, C. N. Rice, Ruben, and 
Kamen studied the oxidation by alkaline permanganate of pro- 
pionate to the products carbonate and oxalate (1 mole of each 
from 1 mole of propionate). It might have been a plausible guess 
that the COs^ is formed from the carboxyl group; however, with 
the carboxyl carbon labeled they found that only about 25 per cent 
of the COa"" was from that part of the molecule. In acid solution 
in the oxidation of propionic acid by dichromate they found that 
all the C0 2 did originate from the COOH, demonstrating differ- 
ent mechanisms in the two instances. The oxidation of fumaric 
acid/ 2 > HOOC*CHCHC*OOH, by acid permanganate has been 

2 By common usage these asterisks indicate labeled positions (the two 
carboxyl carbons in this case) in the molecules; of course, because the active 
tracers are almost always very highly diluted with ordinary atoms it is very 
improbable for any given molecule actually to contain two radioactive atoms. 
Thus the asterisk denotes not a radioactive atom, but an atom taken at random 
from a sample containing some active atoms; in other words, the position 
marked with the asterisk will in some of the molecules be labeled with a radio- 
active atom. Notice that on page 285 we avoided the asterisk and chose a 
different superscript because there we wished to indicate an actual radioactive 
atom. 



294 TRACERS IN CHEMICAL APPLICATIONS CH. XIII 

investigated; the product HCOOH (1 mole per mole of fumaric 
acid) is formed always from one of the secondary carbon atoms, 
and the CO 2 (3 moles per mole of fumaric acid) is from the carboxyl 
carbons and the other secondary carbon. 

W. G. Dauben, J. C. Reid, P. E. Yankwich, and M. Calvin 
followed the mechanism of the Willgerodt reaction on acetophe- 
none, and obtained this result: 



C 6 H 6 C* CH 3 



S + NH 4 OH 



at 170C for 6 hr 



C 6 H 5 C*H 



\~10% 
\ 



C 6 H 6 CH 2 C* 

X)H 



Notice the positions of the tracer in the two products; clearly the 
two are not formed through the same mechanism, and one does 
not result entirely from the other by hydrolysis as had previously 
been supposed. 

At the present moment C 14 is finding extensive use in the 
preparation or study of amino acids, carcinogens, sugars, anti- 
biotics, hormones, vitamins, fuel hydrocarbons, and very many 
other important classes of organic compounds. 

EXERCISES 

1. A mixture is to be assayed for penicillin. You add 10.0 mg of 
penicillin of specific activity 15.0 mrd per mg (possibly prepared by 
biosynthesis) . From this mixture you are able to isolate only 0.35 mg of 
pure crystalline penicillin, and you determine its specific activity to be 
1.3 mrd per mg. What was the penicillin content of the original sample? 

2. Refer to the information on fumaric acid oxidation on page 293. 
The authors of that report (M. B. Allen and S. Ruben) in the experiment 
measured the specific activities of the original fumaric acid, the CO 2 
evolved, and the formic acid remaining. If the C 11 activity of the original 
acid had been 20,000 counts per min per mg of carbon at 6:00 P.M., what 
would they have found for the specific activity of the C0 2 if measured at 
7:22 P.M., and of the HCOOH if measured at 8:30 P.M.? What must have 
been the activity per milligram of carbon of the KC*N used to synthesize 
the fumaric acid from dichloroacetyiene, corrected to 6:00 P.M.? 

Answer to last part: 40,000 counts min" 1 mg"" 1 . 



EXERCISES 295 

3. Approximately what fraction of a gram of copper may be detected 
by the tracer method if the tracer copper (Cu 64 ) has been prepared by 
irradiation of copper with 14-Mev deuterons? See table C in the appendix 
and chapters IV, VII, and VIII. 

Answer: We estimate <~10~ 12 g for a thick copper target, less for a 
thin one. 

4. Would you expect almost complete exchange, and by what mecha- 
nism, in 1 hr at room temperature, between the members of each of the 
following pairs: 

(a) Ag+, Ag ++ (6) Hg+ +, very finely divided Hg 

(c) Cr*0 4 % Cr0 2 - (d) Cr*0 4 ssa , Cr 2 7 ~ 

(e) CH 3 I*, C 6 H 6 I (/) Hg*(CH 3 ) 2 ,Hg(C 2 H 6 )2 

5. A series of 10-mg sodium fluoride samples containing 0.1, 0.01, 
0.001, and 0.0001 per cent sodium iodide, respectively, have been mixed 
up before they were labeled. No fraction of the samples is available 
for chemical analysis, and it has therefore been suggested that they be 
identified by slow-neutron activation. Is this feasible with a thermal- 
neutron flux of 10 11 neutrons per cm 2 per sec available? 

6. The completeness of the separation of 10 mg of Ni from 100 mg 
of Co by dimethylglyoxime precipitation is to be tested with the use of a 
sample of 5-year Co 60 whose specific activity is 2 rd per g of cobalt. 
Estimate the minimum amount of coprecipitated cobalt that can be 
detected. 

7. The exchange between I 2 * and I0s~ has been studied under these 
conditions: (I 2 ) = 0.00050 mole per liter (formal), (HI0 3 ) = 0.00100 
formal, (HC104) = 1.00 formal, at 50C. At specified times samples 
were taken and measured for total (I 2 plus I0 3 ~) radioactivity by y count- 
ing; these counting rates corrected to the time t on the basis of the 
8.0-day half -life of I 131 are given below in the column "Corrected Total 
Activity." The I 2 fractions were removed by extraction with CCU and 
the residual (I0 3 ~) radioactivity measured and corrected in the same way; 
these rates are in the column "Corrected I0 3 ~~ Activity." 

Time Corrected Corrected 

(hours) Total Activity IO 3 ~ Activity 

0.9 1680 9.9 3.0 

19.1 1672 107 4.1 

47.3 1620 246 =t 6.6 

92.8 1653 438 9.4 

169.2 1683 610 13 

"oo" 1640 819 =t 9.8 



296 TRACERS IN CHEMICAL APPLICATIONS CH. XIII 

Find the half-time T^ for the exchange and the rate R of the exchange 
reaction. Answer: T^ = 90.6 hr; R = 3.83 X 10~ 6 moles liter" 1 hr" 1 . 

8. The experiment described in exercise 7 was repeated but with this 
difference, (12) = 0.00100 formal. The results are tabulated as before. 

Time Corrected Corrected 

(hours) Total Activity IO 3 ~ Activity 

0.9 1717 7. 6 2. 7 

19.1 1483 70.1 3. 6 

47.3 1548 178 5.2 

92.8 1612 305 7.3 

169.2 1587 413 9.8 

"oo" 1592 534 5.7 

For these conditions find T^ and R. What is the apparent order of the 
exchange reaction with respect to I 2 ? Note: Do not be surprised if the 
order is not an integer; according to O. E. Meyers and Kennedy, this order 
is consistent with this rate law for the exchange-producing reaction: 
R = &(I-)(H+) 3 (I0 3 -) 2 . 

REFERENCES 

A. C. WAHL (Editor), Radioactivity Applied to Chemistry, New York, John 

Wiley & Sons, to be published. 

G. T. SEABORG, ''Artificial Radioactivity," Chem. Reviews 27, 199-285 (1940). 
O. HAHN, Applied Radiochemistry, Cornell University Press, 1936. 
M. D. KAMEN, Radioactive Tracers in Biology, New York, Academic Press, 

1947. 

"Conference on Applied Nuclear Physics," J. Appl. Phys. 12, 259 (1941). 
J. T. BURWELL, JR., "Radioactive Tracers in Friction Studies," Nucleonics 1 

no. 4, 38 (Dec. 1947). 
H. GEST, M. D. KAMEN, and J. M. REINER, "The Theory of Isotope Dilution," 

Arch. Biochem. 12, 273 (1947). 
W. W. MILLER and T. D. PRICE, "Research with C 14 ," Nucleonics 1 no. 3, 4 

(Nov. 1947), and 1 no. 4, 11 (Dec. 1947). 
M. CALVIN, C. HEIDELBERGER, J. C. REID, B. M. TOLBERT, and P. F. YANK- 

WICH, Isotopic Carbon, New York, John Wiley & Sons, 1949. 
J. BIGELEISEN, "The Validity of the Use of Tracers to Follow Chemical Re- 
actions," Science, in press (1949). 
"Symposium on Nucleonics and Analytical Chemistry," Analytical Chemistry 

21, 318-368 (1949). 



APPENDIX 

TABLE A 
RADIOACTIVE AND STABLE ISOTOPES or THE ELEMENTS 

In this table are listed all the nuclear species (nuclides) whose existence has 
been fairly reliably established, together with the best available published 
information on some of their characteristics. Publications received before 
February 1, 1949, have been covered. Data from the following earlier compila- 
tions have been used extensively in the preparation of this table: 

G. T. SEABORG, "Table of Isotopes/' Rev. Modern Phys. 16, 1 (1944).W . 
"Nuclei Formed in Fission: Decay Characteristics, Fission Yields, and Chain 

Relationships" (issued by the Plutonium Project), JA CS 68, 2411 (1946). 
H. A. BETHE, "Table of Nuclear Species" in Elementary Nuclear Theory, 

John Wiley & Sons, New York, 1947. 
"1947 Summary of Nuclear Data" compiled by Nuclear Data Committee of 

Clinton National Laboratory, Nucleonics 2 no. 5, 81 (May 1948). 

Column 1 gives the atomic number and column 2 the chemical symbol of the 
element with the mass number as a superscript. If two or three mass numbers 
are listed for one species, this indicates that the activity in question may be 
assigned to one of these mass numbers. A question mark in the superscript 
after the mass number(s) indicates uncertainty in the mass assignment. A 
question mark after the symbol and mass number indicates uncertainty in the 
chemical identification of the activity. Whenever a nuclide is known to be 
an upper isomeric state this fact is indicated by underlining of the mass num- 
ber(s). The common designations for the members of the natural radioactive 
families (such as RaE for Bi 210 ) are given in parentheses after the designation 
by element and mass number. 

Column 3 lists per cent abundances for the stable and long-lived naturally 
occurring radioactive nuclides. Column 4 gives isotopic masses, that is atomic 
masses on the physical atomic weight scale; most of these values are taken 
from Bethe's table quoted previously. 

Half-lives are listed in column 5. In a few cases several different values 
reported in the literature are listed for a given activity. The symbols used 

1 A revised "Table of the Isotopes," by G. T. Seaborg and I. Perlman, is 
scheduled for publication in Reviews of Modern Physics for October, 1948. 
A copy of this table was kindly made available to us by Professor Seaborg 
prior to publication, and some previously unpublished data from that source 
are included here. 

297 



298 APPENDIX 

in this column are y = years, d = days, h = hours, m = minutes, s = seconds, 
fjLS = microseconds. 

Column 6 lists the modes of decay and the energies of the radiations emitted. 
The symbols used are: 

a alpha particles, 

/3~ negative beta particles (negatrons), 

/3+ positive beta particles (positrons), 

-y gamma rays, 

K electron capture (not necessarily restricted 

to K-clectron capture), 

IT isomeric transition, 

e~ internal-conversion electrons, 

n neutrons. 

x X rays 

The numbers following the symbols in this column indicate the measured 
energies in million electron volts; in case of /3~ and /8 + the energies listed are 
the maximum beta-particle energies for each transition. An attempt has 
been 'made to list different modes of decay or different energies in order of 
decreasing abundance where the relative abundances are known. Radiations 
found in less than 1 per cent of the disintegrations are given in parentheses. 
Column 7 lists some suggested modes of formation. In most cases not 
all the nuclear reactions by which the given nuclide has been produced are 
listed. On the other hand, some methods of production whose use has not 
been reported are sometimes included if they appear obviously suitable for 
the preparation of the nuclide in question. Different methods listed for the 
preparation of one radionuclide may be advantageous for different purposes, 
for example, to give good yields, or high specific activities, or minimum inter- 
ference from other radionuclides. In the choice of suggested modes of forma- 
tion, irradiation of natural iso topic mixtures only has been considered; the use 
of ultrahigh-energy projectiles (say > 20 or 30 Mev) has been disregarded except 
where it offers the only good method of producing the nuclide in question. 
When one radionuclide can be produced by the radioactive decay of another 
this has been indicated by such symbols as Zn 73 -/3~~, which means "produced 
by /3~ decay of Zn 73 ." The uranium fission reaction is represented by U (n, /), 
but this symbol is used only for the first (known) members of chains of fission 
products. For activities existing in nature by virtue of their own long lives 
the word "natural" appears in this column. 



II 

S PH 



APPENDIX 



299 



[ 



!'& 



I 

* 

w 



o 



11 



S 


I 
^ 

rH 

O 


1 

O 


1 

q 




TH 


<N 


CO 


CO 










O 


T 
^2 ^ 


1 


O 




X 

CO 


I;I 


Oi 


d 




rH 


rH 



'2 W W W 
I - 1 



8 



s 



0? 0) 

w w 







$ 


s 


12 
1 






O 


s 


O 

rH 


1 


1 


<N 


i 


rH 
05 


O 


O 


O 





8 


CO 


t>l 


06 


1>I 


od 



2 & 



a a a 



300 



APPENDIX 



V O 

I! 



& 



?: <N <N 1? 8 



O O ~ O 

PQ pQ PQ pQ PQ 



b 



ir 



s - 

f ,0 



CO 

14 



G5 

?7 f : 

ox <- ^ 



s 






Half-life 



X 




S 




1 


l> 




o 


& 




<N 




d 


rH 




CO 


00 

r- 1 

CO 


i 


O 


1 


o 


q 


o q 





q 


c5 


o 


rH (N 


d 


rH 



a 

6 






1 1 
-I 



b o 






APPENDIX 



301 



C<J 

o 

Cvf 

d 

oo~ 
-* *: 

T* 

5 <N 



r 



CO 
00. ^ 



s 

t* ^~^ 

co 5 

I Q 

ca^ 



eo 

(N* 
<^ 

ocT r>- 

-T -T 

CQ. CQ. 



r^ 
-H 



(N 

+ 



q 
16 






^H eo 
10 o 



o 
o 




CO l> 00 



06 oi 



g 

05 ^ 
OS O 



8 

^ 6 d 






bb 



00 



302 



APPENDIX 






PC, Ps 



52; 2 



v3 ^ V* 

^ *2 5! >5? 

jLt & $ tZ <5 f-\ -*. 



=3 



C 



co <xT ^ 

r2 rs 

CQ. ^ QQ. ^ 



H CO 
(N <N 



a 



.2 



'i I o 

55^^ 



APPENDIX 



303 



J 



> p < 

03. CQ. QQ_ 



00. 



CD (N 



iO 00 00 O 

(N 00 05 <N 

C5 O5 C30 O5 

Ci O5 C5 C5 

CO ^' lO CD 

<M (N <N (N 



CD t^> 

<N (N 




8 





OS 



10 

p 

CO 



I 



co 



CO 



304 



APPENDIX 



II 

a o 

eg* 



SS^S ~S 
-* - c^ si"' 



UO&fti 



1-2 

OS .4^ 



05 
H3 bfl co 

5 ^3 <o *o 



*^ <>- 

oa. ^ 



O3 

CO 






(53 

W 



g 00 
C? CO 



CO CO CO 






11 

cc I g 

i< PH 



CO O2 CQ 



APPENDIX 



305 



& 

o 



8 8 

O o 



00 M 



ss^ 

II& 



<N ^^"^ 

S^,S 

HfcS 



s 
S" 



1 <N 

CQ. <*- 



10 ^ 

+ CO 

QQ. ^ 



!S 



| 

oa. 



00 

C5 



CD 
00 



00 

O5 

J3 



CO 

rH 

q 
d 



o 



O 



O O O < 



5, 



306 



APPENDIX 



Some Mod< 
Formatk 



1 



O O 






T3 d 

d o 

s ' 



g 



00" 



S 

c4 

4- < 



10 

co 



o 
o"^ 



2 x 
x<3 






1 



u 
*p, 05 ^ 

Isg 

H-t CO 


i ^ 

t> 4 s - 

Oi Oi 


o 
O5 
4 s - 
o^ 

8 fe 


38.9747 


39.9760 


! 


J| 






00 


c* 




*S 


i 




i 


T^ 

o 
d 


s 

o 


al 

g 














WWW 



APPENDIX 307 



3^* ^ ^^^ ^ -a? 

tTft!^ S/ v? sf 1 ^^ s "e" $? 

sS" vT? ^ " " ' ' ' 



%w 



3 - y$ m % y s & i to 



w S 53 , 

I *-* I p I 4- 4^ 

CQ. ^ CQ. > QX <X1 <Q. 



o S 

i Is 






s a gj 

r* !- t>- 

O^ O) O^ 



o . '"I q q *"! 

35 o o ci d d 



9 S SI 5 3 * 3 5 r 9 S 

o oooooo ooj3o 



s 



308 



APPENDIX 



8 

IS !r 



s 



\n 



s 



l 



>>.5 



1 


5 > 

ivf <x> 


10 




Q 




2 




*o 

CO 


8=S j. 


23 oo t ! 




1 


'I a" 

fe CO CO > 

r ^T* ^ HI* 
QQ. ? S 


cocT ^06"^ ^cfoo^ 
poo o^^pw^-, 

To H P. I L *i ' Jz 


e* 

T-4 

-f 




a 








i 








w 1 3 


S <g 5 d i 


JR ?S 

OO QO 

10 o 




CO W 


06 c5 co ^ >o 


d co 



3| 
- S | 



i 



S 



APPENDIX 309 



S S * S S -j 



uj r^ 

^ <N <0 05 1-- c O 

P' q ^ ^ p co fl 

i-l I + + iH K? 



C T3 | 

I 3 8 

O 00 1-1 CO CO 



CD t < b- 

00 O5 O O 

^ ^1 JO lij 



Tt< i-H T*1 



10 



fSSS 8 



. . o - , S- 

5 5 S o ">! r- o ^- 04 

^ H M r^ * f y 1? f ^P 

HHHtH H >> >>> 



310 



APPENDIX 



"S' 



* 



! 
*.l 



10 ^ 
MJ 00 



o 
a of 



g 



1, 



lO iO 3 










. 

CO 
00 



i i & i i i 1 



s 



APPENDIX 



311 



-** 

_- 



II 111 S i>i 



s 



Eg 

o ci 




00 

r 

CQ. 



S 



CQ. 00. 



3 



,3 

S 



00 



s 



S? 



s 



g 2 S 53 

0) QJ OJ <U 



312 



APPENDIX 



itS 



1i? 



& 



tf 



s ' 
g 



O "^ "?- * > 

*H o O "* tf * "~* 

*-yr -fZ. jrX h-^ >-^ t-jr 

4-4 O U /H ^H /H 



li 



91 

>>4 



i 

o 



p 

<N 



." e 

O ^ rtj 

'^ o 

!>.,_( H VO 

sg t,s 

> ^"^ *" ' ^c 






^ 10 ^ 

^ S cq I-H 10 o 

o ^ ^ -^ ^ 

EH P I I. L +. 



*P 
P-i ' 



rH i-H CO 



a I 

08 | 

|l 

| gj s 

*IS 



s a 



111 



APPENDIX 313 



^ *N s~\ ^_ ^_ ** 

*- * ^ ^ & ^ i 


r '*-? 


^ 


$ & $ is ^ ^ & ^ 


3 "^ JX 


"^ 


SS JS'XSJJS Sooos 
%H ^ id %H t-t a> >H "?H 


s g 


8 


brr-\ r^b^^rfH ^^^ 
*^4 w CJ<;<i<i <5<5J2 


; kg 




O Oi 

P^H H 
CQ. ' T. ' + + 


CO 
CO 

X <*L 1^ 


<iX 



O5 CO CT> l>- 

* it} *$ ^ 

10 S 3 S 



8 



?^ ?^ ? M e, ? M 8^, s^ s^ a 8 a 

ggggg go OO o 



314 



APPENDIX 



"S 



8 *-' 

a fr tr, 



3 3 ! 
J 3 d 
) O S3 



8 3 g 

6^ 



i s d 
i O S3 



a -3 
*sl5 



II 
sJ 



CO 
9^ 



w -L <?. 



I i 

CQ. J- 



s 
f 






CO ff> 

I*! < 



d d 

3| 
s| 

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I 



:S 



I 



1 

3 



11 
2l 



g s 

o 



'3 I 



S 3 I 



S3 S3 



APPENDIX 315 



d j ej d 3 
NO N N O 



I N 



FH CO" 

f p 

CQ. c*- 00. 



-IS < ^ ^ 

r* co . os \/ 

o - w -^ v 



* 






316 



APPENDIX 



s 



:l 



i 



II 






SSSS SSSi 



II 

*.J 



o a > o 

^ m "1 o 

5s S 
* $3 



8 



r-4 r-J 

St,^ 

"!> 

jt>. i>T ^ b- 

>J-: ( 

s s ^r s < 



I 

M 



o 

1, 



i 



3 
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CO 

1-4 

s 



ll 



I 



O O 



I 



1 ! at 



x 



APPENDIX 317 



pow o o ffi rj 

d ^ V d oJ S fl "w 1 * *rt 

C5 N O O tsj O is OOOO 



2 s 



$$$$$ 



318 


APPENDIX 


8 




Si '5J-! 

j*ij 

cS<3 


-^ *? 'g ^ ^ <N 'e' ^ <N ^ <N 

j^ v S ^ vs?5s s^ ^^J> 
5$p & o ooS S S^S 


*H 1 

08 '^3 


H 
1 


s^l 

I- 8 / 

o 5 o 

s l f* 

<o. ; 


c^. r^ S <dT 

*. + TH ^ 

? ? + ^ ^i k "-i rll 

?* CQ. CQ. ^- CQ. ^ fc^ T- *^ *?> ^Q. ^ <? 


1 




T-4 


ai fjj Q S ^3 ^ 5 
<S ^ N P cp cO t 
iO <N <N CN b- tH 


O 

II 

HH 




*, 8 

Jl 




&l 




Symbol and 
Mass Number 

Ge 77 


* 

=4 \ ^ B. S 5 R 5 S 5 

c3 u <<< < < 



8 



APPENDIX 319 



p. S Sv -i -o" si e" 

* ^ %* ^ s v ^^ 

& oo m&o t= $o $o o 4^tJ5 



3s? 






So 8? 



S 4 I s ^ I 



320 



APPENDIX 



I I 



n? 

*5i g 

^s-a 

3-!~ i 

2 S3 c^ jff 

^ iv] uT o 

w M T- 



6 
HP <? 

HH > 



o 

5; 



St. 
2 A 
V 



3 



PI 

^ 



(V) O 

OQ OQ 



JJ- 8 



APPENDIX 



921 



o c$ <1 PQ 



o> o g o> o g g 

PQ PQ PQ PQ PQ c^ ^ 



PQ PQ 



CD 

fH 



f 



O 
O 



o oo 



,00.?- HH 



So 



CQ. 



I I 



(N 



S 



s 



1 *L 

PQ ffl 



322 



APPENDIX 



I 

*C *T3 

1| 
I" 2 



J5 **; *v 

c^ PQ PQ 



3-J 



Sa 
f a 



1| 



f 

oa. 



9 



o 

|>. CCf 

r t 

H 



s 



CO 

+ ^ : 

00, ^ 

5f 



O 
co" 



ta.^. c^ 

5f 2 



a 



O 

1 



a I 



1 



PQ 




S 



tt PQ 



PQ 



'i| 

s| 

^ K 



APPENDIX 



<- 

S^" 

t> M 



^-S"? 

^sv 

^l^s 
Mc 






I 

d 



o 

I* 

M 8 
hf 



OS N. 

PS 

CQ.T 



r 



I I I 

CQ. CQ. QQ. 



I 



10 



fc 



324 



APPENDIX 



II 

S JS 



e ^ 

PQ PQ 



a 1 



OH 02 02 



9-3 
>>.$ 



- 






1 

CQ. 



00 



q 

T-l 

of 

G> 



8" 

r 



P^O 
^ 

o 



f 



*p t OQ 

I S 



10 



X 

o 



rH CC 




4 4 % 45. 
g g g rt 



APPENDIX 



325 




41 



oq 

r 



I 



* , 

-r* VI VI VI VI 



VI 



3 

6 



{2 



S o 8 o S! 

3 "^ ** 
P4 PH P4 



1 



326 



APPENDIX 



8 

S 



16 sf 






7- 



Op. QQ. Qjl. 



111 1 a 



\, ja ^J 

N rt 






s" 

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o -^ - < 

s, ?: a. 



I 



1, 






|. 



S 



APPENDIX 327 



/ s / ^- N 

" 



5^tr 9S 95 <* 93 00 - Q A A $ r-l- 1 r-t-H 

09 00 00 00 00 gg 00 00 go . 00 S .Q > t_. * L. 

L* M M MM ^T^ M M Kl. M IT. K. ?^N M ,*^ M -7J M .'3 ^t 

OQ OQ OQ 02 OQ r^ 02 02 r* OQ r^ r^ ^^ 02 ^N 02 ^N 02 ^N PS 



8 



^ 



8? 



fc!^^ "S. ^ ? S.I C^l M ? n?- 



I 

H J-] 

lO l^" ^* 4 

CO iQ ^O CO 



328 



APPENDIX 






.2 



s s 



CSS 



T3 S 

d o 

03 



Is - * 

s w r ^ s^ ^ 



O f, 

-8 



CO CO 

p ^ 

I p 

00. > 



4 



?s s 



tS] tSJ 



H. 1 I 1 

N N N N N 



APPENDIX 



329 



CS3 



a 



i 1 



if. s 



> 

5?tf 

II 



*? ^ 7 ^ if ^ ^8 

/ _J / . _ J CO JO 

ii i + * ^ i o i 2 



a I < S ? 2 

CO T-H t>. T- C C<l 



S I 

A S 



8 



a a a 



8. 8. 



I g I 



330 



APPENDIX 



ilu 

A 2 



ts) S3 N 



OO 



1 
*3 



lu O V-* 

O fl <N 

02 3 Oi ^^ 

% % P P 



a" 



1 9. 
ox T- 



t 



I I 



^ I 

CO 1-1 



s I 

^^ 



8 

I 2 



I 5 a 

-16 



e-. e^. *.. 



g a_ 



O 



S o S o 



10 CO t- 







SQO 
52 ^ 

CO ^ CO 
rH O5 <N 



APPENDIX 



331 



~ 

> o o 



O <N 






If 



8,_l 
o 

*o b b b TO TO 



I 

^H CC 



* 



I 



332 



APPENDIX 



i d 

!.S 



ii 




ig 



2 

o S 

IM'O 

g 
1% 



iUK, 



00 

o 



^ *: 

O 



oq o 

P c<T i i c^ 

^ ^ d 



CO 

O rH 



1 
1 



3. 



O 

5. 







II- 



APPENDIX 333 



- * - 

1 1 I li II 5 ! 1 1 li 



00?? 



V 
A 




2 ^ 

t e 1 1 



g g 



a 



334 



"8 

I! 



is 



s's 



1-i 

03 ^> 

HU 

*i 

U* 

^ & 



o o 

^O 1C S CO 





tr 7 



q j*. 



1 



Ch C5 

8 S 



d c 

^1 

J 
^^ 



r-l rH rH CO 



06 



S 

fH ^ r- 



S 8 



S I 



APPENDIX 335 



-- S- aS - 
S'SS l3 3'8SS 8SS S'SS 8* T 

SS 933 33S 3S 



S 



8 






wL fc'l h i? 

oa.Qac* M^ffi oo.> 



336 



APPENDIX 



I - 



& i 



. 

P 



2 



g.2 

08 +a 



' 






eo 

H 00 H ~ 



S H 
V 



$: 



i 
s 



1 



oq 
d 



si 






T3 ^ 7 ? 
P^ p4 PH ^ 



3 8 1 
T3 ^5 -d T3 
g fc PU AH 



APPENDIX 337 



; 

i-J^ ^* & 

vS' s " / ^"' 

^' r T3 r O 

P S S 



Q <N 

CQ r 

2 1 

- O P 

12 ^' H 

^ " S" 

I 



s ^ 

^ ^ 

co w 



g 8 



I i 

O ^J CJ g" " 



ss 



338 



APPENDIX 



II" 

o8 fe ? 



STJTJ Ifl'bO bObO 77^ ? r3 <=l 

O < < < ^ P^P^O P-tOt-H 



T3 P 
p Q 
08 VM 

>> j 

s ^ 'p 

5^ 

W -S lO 

CO ffi ^ CO ,. 

-S'& s ^ M 

o 5 ;! o> oQ 

S P P <?* 

"* W H I <5 T 

HH v r* ^ 



^^ ^ v *^ 

EH I P L 9. I* 

HH > CQ. ^ V 



CO 

k^ 



I 



3 I 



p 

s 
^ 



APPENDIX 339 



^s < 3 3 

<J < O U U CM S 



f~ 

i L M! 



i/5 c^ co co 'v d 



(N 



32 SB 88 S3 SI SSBI-S 

s sss ssss 



ss s 

ss 



340 APPENDIX 



"8 '! 

a I 






QCL 



o 

II 



1. 

% d 

5 ^ 2 < 



n 8=2- 3 P 

'I 2"e, t-^. 4 a 

s l r3 r^ n & 



*4 8 

Ij 

1 5 a 31 S3 "3 

^| 3 3 33 "A "A A 



APPENDIX 841 



^-al-s 1 -? 



saigas r s S s a a 3 3 a33aa33aa 

333 >33 33 33 333 333 3333 a* 






I 



1 



r < 

CQ. 



a ai s J3i g si s 31 S| 

3 3 33 3331 A A 



342 



APPENDIX 



s 



si - ~ ^ - 

S S I -rf ff'S' 'a' s 

J ^ &' T s^^^ 

SB S3 3 3 



S 3 



sl 



"3 S 

08 '^ 



s 2 a 

00 ** 






Ci 



u u. 



O 

! Q 



I 

W 



T* 

^J CO 

CO 1-1 



1 



*&* 







N. t^ 

CO CO 

o^ o^ 

CD t>^ 



00 



i| 



CO 

d 



S3 



a 



v v 

d d d 

CC Cfi 03 



vi 

d 

CQ 



d d 

02 CQ 






02 



APPENDIX 



343 





"d d "i 

02 02 02 



d 
02 



d d 
02 QQ 



d d 
OQ 02 



d d ,_. 

02 02 M 



00 

p 



Cl 



f 

(0. 



a a 

d d 

CO CQ 



s 

VI 





vi' 



a g 



S 9 
d a 

GQ 02 



d 

CO 



344 



APPENDIX 



O "3 ^ O ^ g^X-stLg/^N 

sjl <?i? i?<i? s-s; J *J | 


-H 


3 **-. 'L : i_ ^ "% ID ^H "^D 'la "JD ^ d *Ib d d 

A W& A & 02 H HH H 02 020202 O2 O2 O2 


S S 


a 5 

08 '43 




5^ a ^ 

^ a .9 ^ , 


-i 1 , 


nrtQ)iO ^TrH,^ ^S ~^ 

|s2^S H ^ ri i-" r 1 ^ 

tx^ ^"^ ^' ^ ^ ^^ 


C^ 


1 




w . _, d pd cj ^ 

Cj -M rM O5 M O 

GO T-J 10 CO J^ 2 


1 


^2 

tS 
09 

o i^i 

s 








11 


1 


If 

3* 

||s 5 2 S 5 S H, 


8 I 

02 02 



02 



APPENDIX 



845 



-t H 

i i 



SB g 

' ' 



OQ OQ 



*. o 

o 



f? 8 

M O 



15 <N 



o X 

^ O O 



CO 



a a 



5 9 



8 



a a 

02 52 $ 



346 



APPENDIX 



J o 

1! 



-e 



1= EG 



*~U 'I-. '"J-s "J-v '^i' 
JD JD ^O ^O rO 
OQ OQ O2 C/i OQ 



eg 



i! 



82.9 

c 



1C 

d 



V 



s s s 

(M* CO ^ 



5 



s 

d 



6' 





d 



Ttj oq <^> q 
c$ d ^ t^ 



i 



sii si a s ai 
^ ^^5^^^^^ 



' 

^ sf 



APPENDIX 347 



Sjf 



r. <o o <o <o t-. 1 oo oo o>i oo oo o>' 1 2 2 S S cT 22 JS 

SSsssS s 2 sl s s S si s s j3 s vS 8 " 

^^^ HH^ HHH HH^ HHH H H P & 



a o 






O _ p .jco'H^-j ?P P ^ 

:&v OQ. M t ^^H 



S SI S S SI S - 



543 APPENDIX 

"8 

$ p P H oaHB HH$ wSnH^ 



II 

>>. 



1*8 



2 S 



8 g.g 

I -a^ 

O QJ 

c3 

oa 



1 U 



?- 



5 i: 



S 



a 3 

<s| 

^ I 



ri 

J!5 



ll 



APPENDIX 



349 



O jg CO O jj 

d) 0} GO ^ dj 

HHO HH 



p p 







OCO 
. 



8 



o 



CO 



-* . A 1FH 1 . . es 

I Q I A I ,H I <N I 

<n>* <a.fr- x^ ^^ <o.e 



1 



. 

(N 



e 

cd 



10 



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2 



350 APPENDIX 

8 



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APPENDIX 



351 



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352 



APPENDIX 



3 
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APPENDIX 353 



? 8 ^ s 

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do g ^ 

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III vJ* Q P 

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354 



APPENDIX 



o 
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355 



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APPENDIX 



o I 

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25C <^A?X <"**, ^ ^ ^ --r*TcT 

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g fe c^^S^ .T-rr 3 1 S .r - s 9 5S5JgS 



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APPENDIX 357 






r^ T *- a *rS $ 

393 3 3 5553 3|j^j 

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co co ^2 

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rH M 00 iH 



55 5? 3 g s 



PH 55 ^5 ^5 fc 



358 



APPENDIX 



1 1 "? 'a *? *- % 7: ^ ^ 

<2 i,ii is ^ i m 

s 

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APPENDIX 



359 



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360 



APPENDIX 



: d 

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s s s s 

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ns 3 

d O S 


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APPENDIX 



361 



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362 APPENDIX 



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APPENDIX 363 



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S S S S 8 S S S| ^83 8 

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364 



APPENDIX 



Some Modes 
Formation 
(P, n) 



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APPENDIX 365 



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366 APPENDIX 

8 

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APPENDIX 367 



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368 



APPENDIX 



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APPENDIX 



369 



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370 



APPENDIX 



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APPENDIX 



371 



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372 



APPENDIX 



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APPENDIX 373 



s^; $ $ s $ s 

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374 



APPENDIX 







'I + 1 
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APPENDIX 



375 



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APPENDIX 



II 

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APPENDIX 



377 



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378 



APPENDIX 



11 ~ 

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379 



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APPENDIX 



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APPENDIX 



381 



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APPENDIX 333 



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384 



APPENDIX 



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APPENDIX 



385 



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- 



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387 



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388 



APPENDIX 




si 
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APPENDIX 



389 



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390 



APPENDIX 



TABLE B 
ISOTOPIC THERMAL NEUTRON ACTIVATION CROSS SECTIONS f 



Target 
Isotope 

Q18 

Na 23 
Mg 26 
Al 27 
Si 30 

p31 

g34 

Cl 37 
K 41 

Ca 44 
Ca 48 
Ca 48 
Sc 45 

Ti 50 
Ti 5o 

y51 
Cr 50 

Cr 54 

Mn 55 

Fe 58 

Co 59 

Ni 64 

Cu 63 

Cu 66 

Zn 64 

Zn 68 

Zn 68 

Ga 69 

Ga 71 

Ge 70 

Ge 74 

Ge 76 

As 75 

Se 74 

Se 80 

Se 80 

Se 82 



Half-life 
of Product 
Isotope 
27s 
12s 
14.8h 
10.2m 
2.4m 
170m 
14.3d 
87. Id 
38m 
12.4h 
180d 
30m 
2.5h 
85d 
6m 
72d 
3.74m 
26.5d 
1.3h 
2.59h 
47d 
5.3y 
2.6h 
12.8h 
5m 
250d 
57m 
13.8h 
20m 
14.1h 
lid 
89m 
12h 
26.8h 
127d 
18m 
58m 
25m 



Isotopic Cross 
Section for (n, 7) 
Reaction (Barns) 

0.00022 

0.0094 

0.63 

0.048 

0.21 

0.159 * 

0.23 

0.26 

0.56 

1.0 

0.63 

0.55 

0.205 

22 

0.0075 

0.039 

4.50 

(H) 

(0.0061) 

10.7 

0.36 

21.7 

2.31 * 

2.88* 

1.83 

0.53 

1.02 

0.29 

1.40 

3.36 

(~0.45) 

0.38 

0.085 

4.2 

(22) 

0.48* 

0.035 * 

0.060 



t From L. Seren, H. N. Friedlander, and S. H. Turkel, Phys. Rev. 72, 888 
(1947). 



APPENDIX 



391 



TABLE B (Continued) 
ISOTOPIC THERMAL NEUTRON ACTIVATION CROSS SECTIONS 



Target 
Isotope 

Br 79 

Br 79 

Br 81 

Rb 86 

Rb 87 

Sr 86 

Sr 88 

y89 

Zr 94 

Zr 96 

Cb 93 

Mo 92 

Mo 98 

Mo 100 

Ru 102 

Ru 104 

Rh 103 

Rh 103 

p d !08 

Pd iio 

Ag 107 

Ag 109 

Ag 109 

Cd 112 

Cd 114 

Cd 114 

Cd 116 

In 113 

In 115 

In 115 

Sn 112 

Sn 120 

Sn 1227 

Sn 124 

Sb 121 

Sb 123 
Te i26 

Te i26 

Te 128 
Te 128 
Te 130 



Half-life 
of Product 
Isotope 
18m 
4.4h 
34h 
19.5d 
17.8m 
2.7h 
54d 
65h 
65d 
17.0h 
6.6m 
6.7h 
67h 
14.6m 
42d 
4.5h 
42s 
4.3m 
13.4h 
26m 
2.3m 
22s 
225d 
2.3m 
2.33d 
44d 
3.75h 
48d 
13s 
54m 
105d 
28h 
lOd 
9m 
2.8d 
60d 
9.3h 
90d 
70m 
32d 
25m 



Isotopic Cross 
Section for (n, -y) 
Reaction (Barns) 

8.1 

2.76 

2.25 

0.72 

0.122 

1.29 

0.0050 

1.24 

(0.43) * 

0.60 * 

(-1.0) 

0.007) 

0.42 * 

0.46 * 

1.2 * 

0.67 

[137] 

[11.6] 

11.2 

0.39 

44 

97 

2.3 

0.21 * 

1.0 * 

0.14 * 

1.3* 

[56.0] 

[51.8] 

[145] 

(-1.1) 

0.25* 

0.16* 

0.57 

6.8 

2.5 

0.80* 

0.075 * 

0.137 * 

0.016 * 

0.21 * 



392 



APPENDIX 



TABLE B (Continued) 
ISOTOPIC THEKMAL, NEUTKON ACTIVATION CROSS SECTIONS 



Target 
Isotope 

Te 130 

J127 

Cs 133 
Cs 133 



La 139 

p r !4l 

Sm 162 
Sm 154 
Eu 151 
Gd 160 

Tb l59 

Dy 164 

Ho 165 
Tm 169 
Lu 175 

Lu 176 
Hf iso 

Ta 181 
Ta 181 

W 184 
W 186 

Re 186 
Ite 187 

Q S 190? 

Os 1927 

Ir 191 
Ir 191 
Ir 193 
p t !96 
p t !96 
p t !98 

Au 197 

Hg 202 

Hg 204 
^203 



Half-life 
of Product 
Isotope 
30h 
25m 
3.1h 
2.0y 
85m 
40h 
19.3h 
47h 
25m 
9.2h 
18h 
3.9h 
2.4h 
27.5h 
105d 
3.4h 
6.6d 
46d 
16.2m 
120d 
73.2d 
24. Ih 
92.8h 
18.9h 
32h 
15d 
1.5m 
70d 
19h 
18h 
3.3d 
31m 
2.7d 
51.5d 
5.5m 
3.5y 
4.23m 
5.0d 



Isotopic Cross 
Section for (n, 7) 
Reaction (Barns) 

0.008) 

6.25 

0.016 

25.6 

0.51 

8.4 

10.1 

135* 

4.9* 

1420* 

(~4.3) * 

10.7 

[2660] * 

59.6 

106 

16.3* 

3640 

10.0 

0.034 

20.6 

2.08* 

35* 

101 

75 

2.50 

5.34 

(260) 

1000 

128 

1.1 

(4.5) 

3.9 

[96.4] 

2.44* 

0.34 

7.5* 

0.11 * 
. 0.015 



The probable errors of the cross-section values are estimated as 20 per 
cent, except where the cross section is given in parentheses (40 per cent) 
or in brackets [10 per cent]. 



APPENDIX 393 

Whenever the abundance value for the target isotope or the assignment of 
the active isotope used by Seren et al. differs from those given in table A, the 
isotopic cross section was recalculated from the atomic cross section given by 
Seren et al. with the data from table A. This is denoted by an asterisk. 



TABLE C 

THICK-TARGET YIELDS FOR SOME NUCLEAR REACTIONS OBTAINED WITH 
14-Mfiv DEUTERONS * 

Yield in 
Rutherfords per 

Reaction microampere-hour 

Mg 24 (d, ) Na 22 0.065 

Na 23 (d, p) Na 24 410 

Mg 26 (d, a) Na 24 8.70 

Al 27 (d, pa) Na 24 1.66 

Si 30 (d, p) Si 31 37 

P 31 (d, p) P 32 8.5 

Cl 37 (d, p) Cl 38 1500 

K 41 (d, 7>)K 42 8.3 

Cr 52 (d, 2n) Mn 62 3 

Co 69 (d, p) Co 60 (5.3y) 0.040 

Cu 63 (d, p) Cu 64 92 

Cu 63 (d, 2n) Zn 63 1160 

Cu 65 (d, 2n) Zn 65 0.126 

Br 81 (d, p) Br 82 28.5 

Sr 88 (d, p) Sr 89 1.3 

Sr 88 (d, 2n) Y 88 1.4 

Te 130 (d, 2n) I 130 32 

Te 130 (d, n) I 131 3.2 

* Data from E. T. Clarke and J. W. Irvine Jr., Phys. Rev. 70, 893 (1946). 



The yields listed are those that would be obtained with thick targets of the 
pure target elements of natural isotopic composition, and in bombardments 
short compared to the product half-lives. 

TABLE D 
TABLE OF PHYSICAL CONSTANTS AND CONVERSION FACTORS * 

Velocity of light c = (2.99776 db 0.00004) X 

10 10 cm sec- 1 

Planck constant h - (6.624 =t 0.002) X 10" 27 erg 

sec 

Boltzmann constant k - (1.38047 0.00026) X 

10~ 16 ere dee~ l 



394 APPENDIX 

TABLE D (Continued) 
TABLE OF PHYSICAL CONSTANTS AND CONVERSION FACTORS 

Electronic charge e = (4.8025 dr 0.0010) X 10~ 10 

Absolute esu 

- (1.60203 =b 0.00034) X 10~ 19 

absolute coulomb 
Electron mass m (9.1066 db 0.0032) X 10~ 28 g 

- (5.4862 0.0017) X 10~ 4 

physical atomic weight unit 

Electron rest energy me 2 0.5107 Mev 

Neutron mass 1.00893 =fc 0.00004 physical atomic 

weight units 
Hydrogen atom mass 1.008125 db 0.000009 physical atomic 

weight units 
Katio of physical atomic weight to 

chemical atomic weight 1.000272 db 0.000005 

Avogadro number (chemical scale) N = (6.0228 0.0011) X 10 23 

mole" 1 
Faraday constant (chemical scale) F = 96,487 10 absolute coulombs 

g-equiv." 1 

Energy equivalent of one mass unit 931.05 0.15 Mev 
Energy in ergs of one absolute-volt- 
electron (1.60203 db 0.00034) X 10~ 12 erg 
Energy in calorics per mole for one 

absolute- volt-electron per molecule 23,052 =t 3 calis mole"" 1 

* Mostly based on values given by R. T. Birgc, Rev. Modem Phys. 13, 233 
(1941). 

SELECTED EXAMINATION QUESTIONS FROM AN 
INTRODUCTORY COURSE IN RADIOCHEMISTRY 

1. Diagram the decay of U 238 to radium. Show the names, atomic num- 
bers, mass numbers, half-lives, and modes of decay of the active substances. 

2. 2 eFe 56 is stable. 2sMn 56 is radioactive, half-life 2.59 hr, and emits nega- 
tive j8 particles of 2.9 Mev maximum kinetic energy. Mn 56 can be made by the 
action of neutrons on Fe 56 . Is the reaction exoergic? Estimate the minimum 
practical neutron energy (threshold). 

3. Suggest two different reasons for believing that attractive forces between 
nuclear particles have a very short range. 

4. Define (a) isotopes, (6) isobars, (c) isomers, (d) Oppenheimer-Phillips 
process, (e) packing fraction, (/) secular equilibrium, (g) Compton effect, (h) 
Bragg curve, (i) saturation current, (j) betatron. 

5. Why is He 6 unstable? 

6. Derive the relationship between half-life and disintegration constant. 

7. What is meant by the "three radioactive series"? Why do we say that 
one series is missing? 



APPENDIX 395 

8. Define briefly: (a) (a, 2n) reaction; (b) binding energy of a nucleus; (c) 
chemical scale of atomic weights; (d) physical scale of atomic weights; (e) 
"one over " law. 

9. In connection with nuclear reactions, why is there so much emphasis on 
slow neutrons? Why so little concern with slow protons, a. particles, etc.? 

10. Define the following: (a) internal-conversion coefficient of a 7 ray; (b) 
half-life of a radioactive substance; (c) threshold of a Geiger-Miiller counter; 
(d) probability; (e) a compound nucleus; (/) excitation function of a nuclear 
reaction; (g) exchange reaction. 

11. The stable isotopes of a hypothetical element Z and its neighbors in the 
periodic table are given in the following chart: 

Z - 2 44 45 46 48 50 

Z - 1 47 49 

Z 48 50 51 52 

Z + 1 53 

Z + 2 54 55 56 

Five radioactive isotopes of element Z have been prepared in the course of the 
bombardments listed in the following table. Their half-lives are 10 min, 
45 min, 4 hr, 20 hr, and 14 days. 

Bombard- Resulting Half-lives Chemically 

ing Bombarding Identified as Belonging to 

Target Particle Energy Element Z 

Z - 2 a 30 Mev 10 min, 45 min, 4 hr, 20 hr, 14 days 

Z - 1 p 8 Mev 10 min, 20 hr, 14 days 

Z - 1 d 15 Mev 10 min, 45 min, 20 hr, 14 days 

Z n Thermal 10 min, 4 hr, 14 days 

Z n Up to 22 Mev 10 min, 45 min, 4 hr, 20 hr, 14 days 

Z -f 1 n Up to 22 Mev 4 hr 

Z + 2 n Up to 22 Mev 4 hr 

(a) To what mass number is each of the five radioactive isotopes to be as- 
signed? State your reasoning. 

(6) What mode of decay would you expect for each of these active isotopes? 
Where several possibilities appear about equally probable from the data given, 
state all alternatives. 

12. Answer any four parts of this question. How would you distinguish 
experimentally between (a) a particles and /3 particles; (b) /3 particles and 
internal-conversion electrons; (c) X rays and 7 rays; (d) positive and negative 
electrons; (e) X-electron capture and an isomeric transition in a given element. 

13. (a) Discuss the conditions necessary for the operation of the Szilard- 
Chalmers process. 

(b) Why is the chemical separation of nuclear isomers possible in some cases 
even though the recoil energy of the 7 rays or conversion electrons is not suffi- 
cient to break the chemical bonds in question? 

14. What is the weight of 1 "curie" of H 8 ? 



396 APPENDIX 

15. 1.00 g of rubidium in the form of a thin foil is exposed to a thermal 
neutron flux of 1.00 X 10 10 neutrons cm'^ec" 1 for 48 hr. At the end of this 
irradiation what would be the total rate of ft disintegrations (from the two 
induced radioactivities, Rb 86 and Rb 88 )? The thermal neutron radiative 
capture cross sections are: for Rb 85 <r = 0.72 barn, for Rb 87 <r = 0.122 barn. 

16. In a set of data, what is meant by (a) standard deviation; (6) probable 
error; (c) dispersion? 

17. After ethyl bromide is bombarded with neutrons, a substantial fraction 
of the active bromine formed is found in the form of ethyl bromide. In what 
other chemical species would you expect to find bromine activity? How 
would you expect the "retention" of bromine activity in ethyl bromide to be 
affected by dilution of the ethyl bromide with ethyl alcohol before bombard- 
ment? Explain. 

18. What conclusions would you draw from each of the following experi- 
mental results: (a) Br2 and AsBrs are found to exchange active bromine rapidly 
in CCU solution. (6) Iz exchanges more readily with C 2 H 5 I than with C 6 H 5 I. 

19. List three neutron sources that employ natural radioactivities and two 
neutron sources made possible by accelerating devices. Write down the 
nuclear reaction responsible for the production of neutrons in each case. 

20. Describe in detail how you would best determine experimentally the 
maximum energy of the beta particles from a given radioactive substance with- 
out recourse to an electron spectrograph. 

21. The stable isp topes in the region of the hypothetical element Z are: 

Z -2 87 

Z - 1 86 88 90 

Z 89 91 

Z + 1 90 92 93 94 

Z +2 95 

Assuming you have available a cyclotron capable of producing 12-Mev 
deuterons, 6-Mev protons, or 24-Mev particles, list the reactions by which 
you would expect to be able to produce in reasonably good yield each of the 
following isotopes: Z 88 , Z 90 , and Z 92 . 

22. Discuss the Bohr theory of nuclear reactions. 

23. (a) Explain the principle of the Szilard-Chalmers reaction. (6) Sug- 
gest possible methods of applying the Szilard-Chalmers process to each of 
the following elements: iodine, lead, tellurium, arsenic. 

24. How would you explain the following experimental facts: If Ag + is 
added as a carrier to a ThB solution, and 10 per cent excess I"~ is added, 77 per 
cent of the ThB is carried by the Agl precipitate. If only 90 per cent of the 
stoichiometric amount of I ~ is added, only 4 per cent of the ThB is carried. 

25. (a) Briefly describe the main features of a cyclotron. 

(b) What is the principal distinction between an ionization chamber and a 
Geiger counter? 

26. Discuss the concept of probability and derive the radioactive decay law 
from probability considerations. 

27. Discuss briefly the difficulties in the theory of ft decay which led to the 
neutrino hypothesis. 



APPENDIX 397 

28. What is wrong with this hypothetical table of stable isotopes? 



Atomic 
Number 


Mass Numbers 


Z -2 


49 


Z - 1 


50 52 54 55 


Z 


49 51 


Z + l 


48 50 52 53 54 


Z -f 2 


51 53 



29. Name three quantized nuclear properties; give (or guess reasonably) an 
illustrative value for each. Name three nonquantized nuclear properties; 
give (or guess reasonably) illustrative values for two of them. 

30. Suggest methods for preparing samples of high specific activity of (a) 
C6H 4 C1 2 containing radiochlorine; (b) CI^BrCOOH containing radiobromine; 
(c) CeH4lNO2 containing radioiodine. 

31. What is meant by each of the following: (a) the plateau of a Geiger- 
Miiller counter; (b) a proportional counter; (c) the observed threshold of a 
nuclear reaction; (d) the standard deviation of a set of numbers; (e) the cross 
section of a nuclear reaction. 

32. Sketch and briefly describe the essential features of one of the following: 
(a) Lauritsen electroscope; (b) Geiger-Muller counter. 

33. If M is the average counting rate per minute of a given sample, what is 
the probability of obtaining m counts per minute in a particular observation: 
(a) according to the Poisson distribution law? (b) according to the Gaussian 
distribution law? (c) Under what assumptions is the Gaussian distribution 
a good approximation to the Poisson distribution? 

34. If a cyclotron is tuned to accelerate deuterons to 10 Mev, what must be 
done to the cyclotron (other than building a new one) so that it can accelerate: 
(a) a particles, (b) protons? What will be the energy of the a particles and 
protons? Explain. 

35. For what reasons do we believe that nuclei consist of protons and 
neutrons, rather than protons and electrons? 

From this point on the questions are of the "open-book" type. 

36. A counting rate is very nearly 800 counts per min. How long must it 
be counted to give an answer with a probable error of 0.3 per cent? If this is 
a total rate for sample plus background, and the background is very nearly 
100 counts per min as determined in a 10-min measurement, what is the per- 
centage probable error in the net sample rate? 



398 



APPENDIX 



37. The fission of a uranium nucleus yields about 200 Mev. This is what 
fraction of the total energy content of that nucleus? 

38. The energy of the sun is believed to result from a reaction which produces 
helium from hydrogen (that is, from protons and electrons). How many ergs 
result per gram of helium produced? 

39. On a classical picture, what is the lowest-energy a particle (in million 
electron volts) that U 238 can emit? 

40. Write several nuclear reactions that could be used to produce 24Cr 51 . 
Emphasize those that would be most practical with an average cyclotron. 
Outline how it could be made using a 0.8-Mev voltage multiplier set. 

41. Define the cross section for a nuclear reaction by means of a suitable 
differential equation (explain the meaning of each term of your notation). 
Use this equation to calculate the intensity of a neutron beam emerging after 
passing through a 0.1-mm-thick cadmium foil. The intensity of the original 
beam, normal to the surface of the foil, is 1000 neutrons per sec per cm 2 . 
The average cross section for capture of these neutrons is 2500 barns. 

42. How might you measure the half-life of thorium? Make a practical 
answer; be specific as to sample sizes, types of instruments, etc. 

43. The activity of a sample was measured as a function of time on a Laurit- 
sen electroscope: 



Time (hr) 


1 


2 


3 


4 


5 


6 


8 


10 


12 


Div. min." 1 


7.1 


5.7 


4.5 


3.7 


3.0 


2.4 


1.7 


1.2 


0.90 



Time (hr) 


15 


18 


21 


24 


27 


30 


35 


40 


Div. min.~ l 


0.63 


0.48 


0.39 


0.32 


0.27 


0.22 


0.175 


0.137 



Several absorption curves were taken at different times with aluminum ab- 
sorbers; all were similar in appearance. A typical one: 



Absorber 
(mg cm~ 2 ) 





70 


130 


200 


300 


400 


500 


600 


700 


800 


Div. min." 1 


5.8 


3.5 


2.2 


1.3 


0.6 


0.28 


0.12 


0.11 


0.11 


0.10 



Give the half-lives of the activities obviously present, and the upper energy 
limit of the spectrum. What type of decay might be suggested? 

44. (a) A disk of iodine, area = 2 cm 2 , weight = 1 mg, was exposed to a 
beam of thermal neutrons incident perpendicularly on the face of the disk. 
The beam intensity was 5 X 10 7 neutrons per cm 2 per sec. The irradiation 
was from 1 :00 P.M. to 1 :50 P.M. The activity of the I 128 produced was meas- 
ured by placing the sample directly underneath and very close to the large 
aluminum leaf window of an ' 'infinitely" large air-filled ionization chamber. 
The saturation ionization current in the chamber was 4 X 10~" 12 ampere at 
2:15 P.M. Estimate the cross section of the (n, 7) reaction on I 127 for thermal 
neutrons. (6) Discuss briefly the effect on the measured ionization current of 
varying the thickness t of the iodine sample, its area remaining 2 cm 2 . 



INDEX 



Abelson, P., 273 

Absorber position, effect of, 227 

Absorbers, 220 

Absorption coefficient, 4, 158, 170 

for soft p particles, 230 
Absorption curves, 158-163, 168-171 

extrapolation of, 227 

Feather analysis of, 161, 162 

for 7 rays, 168-171 
Absorption edges for X rays, 221-223 
Abundances, see Isotopic composition 
Accelerating tube, 82, 83 
Actinide series, 274, 275 
Actinium, discovery of, 5, 269 
Actinium series, 12, 14 
Activation analysis, 291, 292 
Adsorption in radiochemical proce- 
dures, 250 

Age of rocks, determination of, 1719 
Alcohol, use in sample mounting, 232 
Allen, M. B., 294 
Allison, F., 271 

Alpha-particle backgrounds, 185 
Alpha-particle emission, decay con- 
stant for, 129 

recoil energy in, 124 

theory of, 126-129 
Alpha-particle groups, 124-126 
Alpha-particle range, determination 
of, 150 

extrapolated, 149, 153, 154 
Alpha-particle sources, 79 
Alpha particles, absolute counting of, 
224, 225 

absorption of, 4 

back-scattering of, 225 

change of charge of, 155 

charge-to-mass ratio of, 4 

cloud-chamber tracks of, 177 

electrostatic deflection of, 4 

energy loss of, 147, 148 

identification of, as helium ions, 4, 5 



399 



Alpha particles, ionization by, 147, 
148 

long-range, 126 

magnetic deflection of, 4 

nature of, 124 

properties of, 44 

pulse analyzer for, 185 

range-energy relations for, 150 

range of, 4, 148-150 

scattering of, 22, 23, 38, 39, 40 

specific ionization of, 153-155 

stopping power for, 150-153 

straggling of, 149, 155 
Aluminum halides in exchange reac- 
tions, 285 

Americium, 274, 275 
Anderson, H. Z/., 99 
Anfinsen, C. B. } 230 
Angular correlation of ft and y rays, 

228 
Annihilation of positron-electron pair, 

131, 168 
Annihilation radiation, 131, 168, 224 

effect on 0y coincidences, 228 

in positron absorption curves, 161 
Anomalous mixed crystals, 265 
Arithmetic mean, 199 
Arsenic, exchange reactions with, 283 

Szilard-Chalmers separations for, 

256 

Arsenic trichloride, distillation of, 252 
Artificial radioactivity, discovery of, 

54 

Astatine, chemical properties of, 271, 
272 

discovery of, 271 

distribution coefficients for, 277 

isotopes of, 271 

volatilization from bismuth targets, 

268 

Aston, F. W., 45 
Atom, nuclear model of, 22, 23 



400 



INDEX 



Atomic beam technique, 42 

Atomic masses, determination of, 35 

Atomic number, 30 

measurement of, 24 
Atomic structure, quantum theory of, 

24 

Atomic weight scales, 34 
Auger electrons, following internal 

conversion, 141 
following K capture, 137 
Automatic recording equipment for 

counters, 193 
Average deviation, 199 
Average disintegration rate, 204, 205 
Average value, 199, 201 
Azimuthal quantum number, 26 

Background ionization, causes of, 193 
Background rates, 194 

of ionization chambers, 185 
Back-scattering of electrons, 165, 166, 

225, 226 

Barn, definition of, 72 
Bateman, H., 116 
Bateman solution, 116, 117 
Becquerel, E., 1 
Becquerel, H., 1, 2, 3 
Berg, 0., 269 

Berthelot-Nernst distribution, 267 
Beryllium basic acetate, chloroform 

extraction of, 252 
Beta decay, double, 139 

energetic condition for, 138 

hot-atom chemistry of, 258 

relation between maximum energy 
and half-life, 135 

selection rules for, 134 

stability against, 137-139 

theory of, 133-135 
Beta-particle counting, geometry for, 

225 

Beta-particle ranges, by Feather 
method, 161, 162 

visual, 161 
Beta-particle sources, supports for, 

225,226 
Beta particles, see Electrons 

absolute counting of, 225-228 



Beta particles, absorption coefficient 

for, 4, 158, 159 
absorption of, 4, 158-162 
back-scattering of, 225, 226 
cloud-chamber tracks of, 177 
distinction from X rays by absorp- 
tion measurements, 160, 170 
electrostatic deflection of, 4 
half -thickness for, 159 
ionization by, 156-158, 182 
magnetic deflection of, 4 
mass stopping power for, 159 
nature of, 4, 130 
range of, 159, 162 
self-absorption of, 166, 229-231 
self-scattering of, 227, 231 
sign determination of, 223, 224 

Beta-ray spectra, 132, 134 
complex, 135 
upper energy limits of, 159 

Beta-ray spectrograph, 194, 223 

Beta-ray standards, 226, 227 

Beta rays, see Beta particles 

Betatron, 91-93, 98 
electron injection in, 92 
energies attainable with, 92, 93 
target of, 92 

Betatron operation, equations for, 91 

Bethe, H. A., 150, 155, 167, 297 

Binding energy, 35-38 

effect of Coulomb repulsion on, 48 
for an additional neutron, 38, 39 
semiempirical formula for, 47 
use for decay energy calculations, 
244 

Binomial distribution, 202, 203 

Biological effects of radiation, 171, 
172 

Birge, R. T., 394 

Bismuth, carrying, by barium carbon- 
ate, 264 

by ferric hydroxide, 264 
self-diffusion of, 281 

Bleuler, E., 162 

Bohr, AT., 24, 25, 28, 61, 62, 63, 65, 66, 
71 

Bohr magneton, 41 

Bohr orbit, radius of, 25 



INDEX 



401 



Bond character and exchange reac- 
tions, 285 

Bond rupture due to nuclear recoil, 
253, 254 

following isomeric transitions, 259 
Born, M., 28 

Boron trifluoride chambers, 171 
Bose statistics, 33, 43, 132 
Bragg, W. H., 4, 6 
Bragg curve, 153, 154, 156 
Bragg's rule, 150, 151 
Branching decay, 11, 117 
Bremsstrahlung, 98, 158 

effect on 0y coincidences, 228 

effect on 0-ray absorption curves, 
159, 161 

spectrum of, 98 

Cadmium, as slow-neutron absorber, 

74, 75, 239 
volatilization from silver target, 268 

Cage effect in hot-atom chemistry, 
257 

Calorimetry for radiation measure- 
ment, 178, 225 

Calutron, 47, 241 

Calvin, M., 294 

Capture, radiative, 63, 67, 74, 236, 
237, 239 

Capture cross sections, 74 

Capture 7 rays, 253, 254 

Carbon-14, 293, 294 

Carrier-free preparations, 249-252 

Carriers, 246-249 
amounts of, 248 

nonisotopic, 246, 247, 249; 263-267 
oxidation states of, 248 
solutions of, 248 

Cascade transformer, 79, 80 

Centrifugation, isotope concentration 
by, 47 

Cerium exchange reactions, 284 

Chadwick, J., 132 

Chain reaction, 71, 101 

Chain reactors, see Nuclear reactors 

Chalmers, T. A., 253 

Charge exchange in nuclear reactions, 
67 



CharUon, E. E., 93, 95 
Chupp, W. W., 66 
Clarke, E. T. t 393 
Cloud chamber, 148, 177, 178 
Clusius, K., 46 

Cobalt, separation from copper, 247 
Szilard-Chalmers separation for, 

256 

Cockroft, J. D., 79 
Cockrof t-Walton machine, 79 
Coefficient of internal conversion, 141, 

142, 144 
Coincidence correction, 213, 214 

determination of, 220, 221 
Coincidence method, for absolute 

counting, 227, 228 
for determining counter efficiencies, 

226 

for half -life measurements, 120 
Coincidence spectrometer, 194 
Competition in nuclear reactions, 62- 

64 
Compound nucleus, 61, 63, 71, 73 

lifetime of, 61 

Compton effect, 167, 168, 170 
Compton electrons in cloud chamber, 

178 

Compton scattering, dependence on 
atomic number, 167, 168, 170 
energy dependence of, 167, 168 
Condon, E. U., 128 
Conservation laws, in /3 decay, 132, 133 

in nuclear reactions, 55 
Contaminants, radioactive, 236, 246 
Contamination, detection of, 196 
Controi rods, 101, 102 
Conversion, internal, 140, 141, 142, 

144 

Conversion coefficients, 141, 142, 144 
Conversion electrons, "absorption of, 

162, 163 
ranges of, 163 
recoil from, 259 
Converter for 7 rays, 167 
Copper, self-diffusion of, 281 

separation from zinc targets, 252 
Szilard-Chalmers separation for, 
256 



402 



INDEX 



Coprecipitation, 263-267 
Cork, J. M., 272 

Corrosion studies with tracers, 281 
Corson, D. R., 271 
CoryeU, C. D., 165, 273 
Coulomb barriers, see Potential bar- 
riers 
Counter efficiency, 212, 213 

determination of, 226 
Counters, see Geiger-Miiller counters, 
and Proportional counters 

voltage gradients in, 187 
Counting rate meter, 193 

standard deviation for, 214, 215 
Counting rates, ratios and products 
of, 209, 210 

standard deviation of, 206, 207, 
211, 212 

sums and differences of, 209 
Critical absorbers, preparation of, 223 
Critical absorption of X rays, 221-223 
Crookes, W., 6 

Cross bombardments, 237, 238 
Cross section, definition of, 71, 72 

for charged particles, 74 

for y rays, 74 

for slow neutrons, 74, 75 

for thick targets, 72, 73 

for thin targets, 72 

partial, 73 

Crystal counters, 185, 186 
Curie, definition of, 117, 118 
Curie, I., 54 

Curie, M. S., 2, 3, 4, 275 
Curie, P., 2, 3, 275 
Curium, 274r-276 
Curved-crystal spectrograph, 194 
Cyclonium, 272 
Cyclotron, 84-90 

beam contamination in, 239, 240 

Berkeley 184-inch, 89, 90 

dees of, 85 

equations of motion for, 86 

frequency-modulated, 89, 90 

relation between radius and energy 
for, 87 

shimming of, 88 

targets for, 236, 246 



Daily dose, maximum allowable, 172, 

173 

Dauben, W. G., 294 
D-c amplifier, 181-183, 218 
D,D reactions, 99 
De Broglie, L., 28 

De Broglie wave length, 28, 32, 74 
Dead time, 190, 191, 213, 214, 221 
Decay constant, 6, 7, 8, 107 

constancy of, 7 

for a decay, 129 

partial, 117 

Decay curves, analysis of, 108, 109, 114 
Decay law, statistical derivation of, 

6,7 

Decay schemes for ft and y transitions, 
136 

for isomers, 143 
Delta rays, 148 

in cloud chamber, 177, 178 
Detection coefficient, 8, 108, 112, 113, 

115, 207 
Deuterium, 31 
Deuteron, 33, 34 

binding energy of, 36, 69 

ground state of, 34, 44 

photodisintegration of, 97 

polarization in Coulomb fields, 68 

properties of, 44 
Deuterons, ranges of, 155 
Dickel, G., 46 
Dickinson, R. G., 289 
Diffusion of ions in solutions, 281 
Dipole radiation, 140, 142 
Dirac, P. A. M., 130, 131, 137 
Disintegration constant, 6, 7, 8, 107 
Disintegration rates from coincidence 

measurements, 227, 228 
Dispersion, 200 
Dodson, R. W., 212 
Doerner, H. A., 266 
Doerner-Hoskins distribution, 266, 

267 
Doughnut, betatron, 91 

Einstein, A., 35 

Electrochemistry at tracer concentra- 
tions, 268 



INDEX 



403 



Electrode potentials from tracer ex- 
periments, 268 
Electrodeposition, 233 
Electrolysis in radiochemistry, 252 
Electron, classical radius of, 32 
collection in ionization chambers, 

185 

magnetic moment of, 41 
multiplier, 178 

negative energy states of, 130 
orbits, 24 
properties of, 44 
shells, 27, 28 
size of, 32, 33 
spectrograph, 194, 223 

for measurement of 7-ray ener- 
gies, 167 
spin, 25, 26 
Electron accelerators as X-ray 

sources, 98 

Electron volt, definition of, 36 
Electrons, arrangement of, in atoms, 

27 
back-scattering of, 165, 166, 225, 

226 

energy loss of, 156-158 
injection of, in betatron, 92 

in synchrotron, 94 
interaction of, with nuclei, 68 
internal-conversion, 140, 141 
rangeenergy relations for, 163-165 
scattering of, 157, 158 
specific ionization of, 157 
straggling of, 157 
velocities of, 157 

Element 61, absence of stable iso- 
topes of, 140 
history of, 272, 273 
identification of, by ion exchange, 

251 
Elementary particles, properties of, 

44 

Elements, genesis of, 19 
Emanations, 5 

Energy-level diagrams, 125, 127 
Energy levels, terminology for, 26 
Energy loss per ion pair, 147, 148, 156, 
166 



Energy release, in fission, 103 

in nuclear reaction, 55, 56 
Equilibrium, secular, 9, 112-114 

transient, 111-112 
Evaporation of samples, 231, 232 
Even-even nuclei, 50 
Even-odd nuclei, 50 
Exchange force, 30 
Exchange reactions, heterogeneous, 
284 

importance in Szilard-Chalmers 
processes, 253, 254, 255 

mechanisms for, 282 

quantitative treatment of, 285-288 
Excitation function, 64, 65, 66, 238 

/ orbitals in lanthanides and acti- 

nides, 274, 275 
Fajans, K., 263, 264, 265 
Fajans' precipitation rule, 263, 264, 

265 

Feather, N., 161, 162, 165, 204 
Feather plot, 161, 162 
Feather relation, 165 
Fermi, E., 133, 134, 135, 273 
Fermi statistics, 33, 43, 132, 133 
Fernandes, L., 272 
Ferric chloride, ether extraction of, 

252 
Ferric hydroxide as scavenger, 249, 

263 

Fick's diffusion law, 280 
Fields, R., 99 
Film badge, 195 
Filtration of samples, 232 
Fireman, E. L., 139 
Fission, 69-71 

asymmetry of, 70, 71 

discovery of, 273 

energy release in, 71 

spontaneous, 144 

theory of, 71 
Fission chambers, 171 
Fission fragments, energy loss of, 155, 
156 

ranges of, 155, ^156 

specific ionization of, 156 
Fission neutrons, energy of, 102 



404 



INDEX 



Fission products, 69 

carrier-free, 251 
Fission yields, 70 
Focusing, in accelerating tubes, 83 

in betatron, 92 

in cyclotron, 87 
Fogg, H. C., 272 
Francium, chemical properties of, 272 

discovery of, 269, 272 

isotopes of, 272 

Friction studies with tracers, 282 
Friedlander, H. N., 390 
Fumaric acid oxidation, mechanism 
of, 293, 294 

g factor, 41 

Gallium chloride, ether extraction of, 

252 

Gamma-ray absorption, energy de- 
pendence of, 168 

experimental arrangement for, 168 
Gamma-ray emission, lifetime for, 

126, 140, 142-144 
Gamma-ray energies, by absorption 

measurements, 168-171 
from Compton electron energies, 

167 
from energies of positron-electron 

pairs, 167 

from photoelectron lines, 167, 194 
Gamma-ray intensities, units for, 118, 

119 

Gamma-ray sources, from nuclear re- 
actions, 97 
radioactive, 97 
Gamma-ray transitions, selection 

rules for, 142 

Gamma rays, absorption of, 166-171 
crystal counters for, 186 
delayed, 228 

distinction of, from X rays, 5 
efficiency of counters for, 160, 161 
energy determination of, 170, 171 
energy loss of, 166-168 
half-thickness values for, 169, 170 
ionization by, 166 
multipole order of, 140, 141, 142 
nature of, 5 



Gamma rays, scattering of, 167 

specific ionization of, 166 
Gamow, G., 128, 129, 134 
Gaussian distribution, 208, 209 
Gaussian error curve, 210, 211 
Geiger, H., 22 

Geiger-Muller counters, 188-194, 
218, 219 

backgrounds in, 193, 194 

calibration of, 220 

construction of, 190 

dead time of, 190, 191 

discharge mechanism of, 189, 190 

efficiency of, 225, 226 

filling mixtures for, 188, 190, 191 

for gas counting, 191, 192 

for soft /3 rays, 191, 192, 219 

hysteresis effect in, 191 

maximum counting rates with, 194 

plateau of, 188, 189 

portable, 196 

quench circuits for, 192 

quenching of, 190 

self-quenching, 189-191 

sensitivity of, 194 

specifications for, 191 

starting voltage of, 188, 189 

thin-window, 191, 192, 219 
Geiger-Nuttall rule, 121, 129 ^' 
Generator, electrostatic, 80-82, 90, 

98, 99, 100 
Genetic relationships as an aid in 

mass assignments, 242 
Germanium chloride, distillation of, 

252 

Ghiorso, A., 274 
Ghoshal, S. N., 64 

Glendenin, L.E., 164, 165, 169, 170, 273 
Gold, self-diffusion of, 281 

solvent extraction of, 252 

Szilard-Chalmers separations for, 

256 

Gould, R. G., 230 
Gurney, R. W. t 128 

Hahn, 0., 264, 266, 273 
Half-life, definition of, 8, 107, 117 
variation with chemical form, 137 



INDEX 



405 



Half-lives, experimental determina- 
tion of, 8, 119-121 
from decay curves, 8, 119, 120 
from delayed coincidences, 120 
from Geiger-Nuttall rule, 121 
from specific radioactivity, 120, 

121 
Half-thickness, relation to range for 

soft ft rays, 230 

values for X and y rays, 169, 170 
Half-time of exchange, 287 
Halogens, exchange reactions with, 

282-283, 284-285 
Szilard-Chalmers separations for, 

255 

Hanson, A. 0., 68 
Harris, J. A., 272 
Hassid, W. Z., 292 
Health physics instruments, 195, 196 
Heating effect of radium, 3 
Heavy water, concentration of, 47 
Heisenberg, W., 28 
Heitler, W., 167 
Helium, presence of in uranium and 

thorium ores, 5 
Helmholz, A. C. t 101 
Herb, R. G., 81 
Hevesy, G., 280, 282 
Higinbotham, W. A., 193 
Hold-back carrier, 247 
Hopkins, B. S., 272 
Hoskins, W. M., 266 
Hot-atom effects in tracer work, 279 
Hot-atom reactions, 255, 256-258 
Hydrogen atom, electron orbits in, 

25 

mass of, 35 
Hydrogen spectrum, fine structure of, 

25 
Hyperfine structure, 41, 42 

Illinium, 272 

Induction accelerator, see Betatron 
Intercalibration of instruments, 220 
Internal adsorption, 266 
Internal conversion, 140, 141, 142, 144 
Internal-conversion electrons, see 
Conversion electrons 



Iodine, separation from tellurium tar- 
gets, 251 
Ion collection, multiplicative, 180, 

186-194 
Ion exchange, for americium-curium 

separations, 275 
separations, 250, 251 
lonization, by a. particles, 147, 148 
by ft particles, 156-158, 182 
by 7 rays, 166 
specific, 153-155, 156, 157 
lonization chamber, 180-186 

for photons, 182 

lonization-chamber instruments, port- 
able, 195, 196 
lonization-chamber measurements, 

standard deviation for, 215 
Iron activity, separation of, 247 
Iron, exchange reactions of, 283, 284 
Iridium, Szilard-Chalmers separation 

for, 256 

Irvine, J. W., 393 
Isobars, 31 

stability considerations for, 51, 139 
Isomer separations, 258, 259 
Isomeric transitions, empirical for- 
mula for lifetimes of, 142 
experimental demonstration of, 221 
Isomerism, 31, 142-144 
theory of, 142, 144 
triple, 144 

Isomers, of stable nuclei, 68 
Isomorphous replacement, 264, 265 
Isotones, 31 

Isotope dilution analysis, 291 
Isotope fractionation, chemical, 46, 

278 
Isotope separations, methods for, 46, 

47 
Isotopes, definition of, 30, 31 

search for, 45 
Isotopic composition, constancy of, 

45, 278 
importance in mass assignment 

problems, 238 

in relation to time scales, 19-20 
variations in, 45, 46 
Isotopic number, 30 



406 INDEX 

Isotopic tracers, stable, 279, 280 
Isotopy, discovery of, 45 

James, C., 272 
James, R. A., 274 
Johnson, G. L., 271 
Joliot, F., 54 

K capture, see JT-electron capture 

K conversion, 141 

tf -electron capture, 16, 136, 137, 138, 

139, 221 

energetic condition for, 138 
experimental demonstration of, 221 

Kamen, M. D., 292, 293 

Kennedy, J. W., 273, 296 

Kerst, D. W., 91, 93 

Klein, O., 167 

Kohman, T. P., 31 

Kundu, D. N., 69 

Kurbatov, J. D., 272 

Labeled compounds from hot-atom 

reactions, 257 
Laughlin, J. S. r 68 
Lauritsen, C. (7., 79 
Lauritsen electroscope, 180, 181, 218, 

219 

standard deviation for, 215 
Law, II. B., 272 
Lawrence, E. O., 83, 84 
Lead, as absorber for electromagnetic 

radiation, 158-171 
carrying, by ammonium dichro- 

mate, 263 

by barium carbonate, 264 
by barium chloride, 265 
by calcium sulfate, 265 
by ferric hydroxide, 264 
by silver bromide, 265 
self-diffusion of, 280, 281 
shields for counters, 220 
Leininger, R. F., 271 
Level spacings, 62 
Level width, 62, 75 
Libby, W. F., 257, 258 
Linear accelerator, 83, 84, 96 
Linear pulse amplifier, 183-186 



Linear pulse amplifier, for fission 

counting, 185 
Liquid-drop model, 71 
Livingston, M. ., 150, 155 
Low-geometry counting arrangement, 

225 
Lutecium, radioactivity of, 16, 140 

Mackenzie, K. R., 271 

Magnetic curvature of cloud-chamber 

tracks, 177 

Magnetic moments, 41-43 
Magnetic quantum number, 26 
Manganese, exchange reactions of, 
283 

Szilard-Chalmers separation for, 

255 

Manganese dioxide as scavenger, 249 
Marinsky, J. A., 273 
Marsden, E., 22 

Mass absorption coefficient, 159, 170 
Mass assignment, 237-244 

by genetic relationship, 242 

by mass spectrograph analysis, 241 

by use of separated isotopes, 240, 
241 

by yield arguments, 238, 239 
Mass defect, 38 
Mass doublets, 35 
Mass excess, 38 
Mass number, 30, 38, 55 

assignment of, 237-244 
Mass spectrograph, 34, 35, 45, 241 
Mass spectrometer, 35 
Mass unit, definition of, 34 

energy equivalent of, 36 
Masses from Q values, 56 
Masurium, 269 
Maxwellian distribution of neutron 

velocities, 60, 61 
McKay, H. A. C., 281 
McMillan, E. M., 66, 94, 101, 273 
Mercuric nitrate, solvent extraction 

of, 252 

Meson theory of nuclear forces, 134 
Mesons, 30, 134 

production of, 67 

properties of, 44 



INDEX 



407 



Mesotrons, 30 

Meteorites, age determinations on, 

18, 19 

Meyers, 0. E., 296 
Mica window counters, 225 
Microcurie, 118 
Microwave spectroscopy, 45 
Millicurie, 118 

Minerals, uranium and thorium, 2 
Moderators for neutrons, 60 

in piles, 101, 102 

Modulation of radiation sources, 120 
Molecular beam technique, 42 
Momentum conservation, in /3 decay, 
133 

in nuclear reactions, 57 
Morgan, L. O., 274 
Moseley, H. G. y 24, 38 
Multiplication constant of a reactor, 
101 

Nahinsky, P., 293 

Neptunium, identification of, 273 

isotopes of, 273 

oxidation potentials of, 275 
Neutrino, in K capture, 137 

interaction of, with matter, 176 

mass of, 44, 134 

properties of, 44, 133 
Neutrino hypothesis, 133 
Neutron, lifetime of, 60 

magnetic moment of, 41, 43, 44 

mass of, 29, 35, 44 

properties of, 44 
Neutron capture, 63, 67, 74, 236, 237, 

239 

Neutron emission, 63, 68, 101, 145 
Neutron excess, 30 
Neutron fluxes in piles, 104 
Neutron number, 30 
Neutron-proton ratio, 48, 49 
Neutron reflector, 102, 103 
Neutron resonances, 75 
Neutron sources, radioactive, 98, 99 
Neutron yields, 99, 100 
Neutrons, delayed, 102, 145 

detection of, by induced radioac- 
tivity, 75, 171, 176 



Neutrons, detection of, by nuclear 

transmutations, 171, 176 
by recoil protons, 171, 176 
by resonance capture, 75, 171 

from nuclear reactions, 99, 100 

interaction of, with matter, 171, 176 

of high energy from stripping proc- 
ess, 101 

produced in fission, 69, 101 

scattering of, 60, 171 

slowing down of, 59 

thermal, 59-61, 74, 75, 102-104, 
237, 253 

velocity distribution of, 60, 61 
Nishina, F., 167 
Noddack, W., 269 
Nuclear charge, determination of, 23, 

24,38 
Nuclear forces, 29, 30 

saturation of, 48 
Nuclear fuels, 102, 103 
Nuclear induction, 43 
Nuclear isomerism^ee Isomerism 
Nuclear magneton, 41 
Nuclear matter, density of, 29 
Nuclear radii, determination of, 39, 40 

formula for, 40, 58 

Nuclear reactions, Bohr theory of, 61- 
64, 65, 68 

definition of, 54 

discovery of, 54 

electron-induced, 90 

7-ray-induced, 97 

mechanisms for, 60-69 
at high energies, 65-67 

notation for, 54 

secondary, 240 

yields of, 238, 239 
Nuclear reactors, 67, 101-104 

neutron fluxes in, 104 

power levels of, 104 
Nuclear shell structure, 49 
Nuclear spin, 33, 40-43 
Nuclei, dimensions of, 29 
Nucleon, definition of, 30 

emission of, 64 
Nuclide, definition of, 31 
Nuclides, stability rules for, 50, 51 



408 



INDEX 



Octopole radiation, 142 

Odd-even nuclei, 50 

Odd-odd nuclei, 50 

Oppenheimer, J. R., 68 

Oppenheimer-Phillips process, 68, 69 

Orlin, J. J., 68 

Orthohydrogen, 42 

Osmium tetroxide, distillation of, 252 

Packing fraction, 37-38 

Pair production, 131, 167, 168 

dependence of, on atomic number, 

167, 168 
on energy, 167, 168 

Paneth, F., 272 

Parahydrogen, 42 

Parallel-plate counters, 186 

Parent-daughter relations in radio- 
activity, 9, 10, 109-116 

Parity, 44, 134, 142 

Pauli, W ., 26, 133 

Pauli exclusion principle, 26, 27, 43, 
49 

Percy, M., 272 

Periodic table in relation to atomic 
structure, 27 

Perlman, /., 297 

Permanganate, hot-atom chemistry 
of, 257, 258 

Perrier, C., 269, 270 

Phase stability in synchrotron, 94 

Phillips, M., 68 

Phosphorus, exchange reactions of, 

283 

Szilard-Chalmers separations for, 
255 

Photoelectric absorption, dependence 
of, on atomic number, 166, 168 
on energy, 167, 168 

Photoelectric effect, 166, 167 

Photoelectrons in cloud chamber, 178 

Photographic film for radiation de- 
tection, 176, 177 

Photomultiplier, 178 

Photon, properties of, 44 

Photoneutrons, 99, 100 

Photosynthesis, radiocarbon studies 
of, 292, 293 



Piles, see Nuclear reactors 
Platinum, Szilard-Chalmers separa- 
tion for, 256 
Pleochroic haloes, 17 
Plutonium, 273-275 

oxidation potentials of, 275 
Pocket ionization chambers, 195 
Poisson distribution, 208, 209 
Polonium, discovery of, 3, 269 

electrodeposition of, 268 
Pool, M. L., 69, 272 
Positron emission, energetic condition 

for, 138 
Positrons, annihilation of, 168 

experimental identification of, 224 

lifetime of, 131 

prediction of existence of, 130 

properties of, 44 
Potassium, radioactivity of, 16, 140, 

291 
Potential barriers, 58, 59, 72 

effect on reaction probabilities, 64, 
74 

penetration of, 58 
by protons, 130 
in a decay, 126-129 
Preformed precipitates, 264 
Principal quantum number, 25 
Probability, addition theorem in, 201, 
202 

definition of, 201 

multiplication theorem in, 202 
Probable error, 211 
Projectiles, nuclear, 67 
Promethium, 273 
Propionate oxidation, mechanism of, 

293 

Proportional counter, 188 
Protactinium, discovery of, 269 

isotopes, 11, 12, 14, 15 
Proton, magnetic moment of, 41 

properties of, 29, 44 
Proton emission, lifetime for, 130 
Proton synchrotron, 90, 96 
Protons, ranges of, 155 
Pulse analyzer for a particles, 185 
Purification, radioactive, 247 



INDEX 



409 



Q value, 55, 56 

calculated from masses, 56, 57 
and threshold of a reaction, 57, 
58 

Quadrupole moment, 44, 45 

Quadrupole radiation, 140, 142 

Quantum numbers, 25, 26 

Quantum theory of the electron, 
130 

Quench circuits for counters, 192 

Quenching gas, function of, 190 

Quill, L. L., 272 

Raki, L /., 42 

Radiation characteristics, importance 
in mass assignments, 243 

Radiation dosage, units of, 172, 173 

Radiation dosages from various 7 
emitters, 245 

Radiation effects, biological, 171, 172 
chemical, 245, 253, 276, 279 

Radiation monitoring, 195, 196 

Radiation shielding, 245, 246 

Radiator for 7 rays, 167 

Radioactive decay and growth, 9, 
109-117 

Radioactive disintegrations, binomial 

distribution for, 203, 204 
time intervals between, 204 

Radioactive families, general equa- 
tions for, 116, 117 

Radioactive nuclides, number of, 45 

Radioactive poisons, 276 

Radioactive preparations, purity re- 
quirements for, 245, 247 

Radioactive radiations, early charac- 
terization of, 3, 4 

Radioactive samples, preparation of, 
231-234 

Radioactive series, actinium, 12, 14 
4n + 1, 11, 15, 16 
thorium, 11, 13 
uranium-radium, 10, 11, 12 

Radioactive tracers, limits of detec- 
tion for, 262 

Radioactivity, as analytical tool, 290- 

292 
discovery of, 1 



Radioactivity, rate of production of, 

by steady source, 113, 114 
recognized as subatomic process, 6 
statistical nature of, 6, 203-207 
units of, 117-119 
Radioautograph, 176, 264, 266 
Radiochemical procedures, chemical 

yields in, 245, 248 
time element in, 244, 245 
Radiocolloids, 262, 263, 266 
Radiometric analysis, 292 
Radium, carrying, by barium sulfate, 

263 

by silver chromate, 265 
discovery of, 3, 269 
first isolation of, 3 
heat produced by, 3 
Radium D, E standards, 227 
Radium E, electrodeposition of, 268 
Radium series, 10-12 
Radon, discovery of, 5, 269 

in definition of curie unit, 117 
Range-energy relations, for a parti- 
cles, 150 

for deuterons and protons, 155 
for electrons, 163-165 
Rare-earth separations by ion ex- 
change, 251 
Rate constants at equilibrium, by 

tracer study, 288, 289 
Rate of exchange, 285-288 
Reaction kinetics from tracer studies, 

288, 289 
Reaction mechanisms from tracer 

studies, 289 

Reactions, nuclear, see Nuclear reac- 
tions 

Reactors, nuclear, see Nuclear reac- 
tors 

Recoil energies for various 7-ray ener- 
gies, 254 

Recoil protons, 171, 176 
Recovery time, 213, 214 
Reid, J. C., 294 

Relativity effect in cyclotron, 88 
Remote-control methods, 246 
Reproduction factor, 101 
Resolving time, 213, 214 



410 



INDEX 



Resonance detectors, 75 

Resonance levels, 75 

Retention in hot-atom reactions, 257, 

258 

Rhenium, radioactivity of, 16, 139 
Rhodium, Szilard-Chalmers separa- 
tion for, 256 
Rice, C. N., 293 
Roentgen, W. C., 1 
Roentgen-equivalent-man, 172, 173 
Roentgen-equivalent-physical, 172 
Roentgen unit, 119 

definition of, 172 
Rotta, L., 272 
Ruben, S., 292, 293, 294 
Rubidium, radioactivity of, 16, 140 
RusseU, B., 99 
Ruthenium tetroxide, distillation of, 

252 

Rutherford, definition of, 118 
Rutherford, E., 3, 4, 5, 6, 22, 23, 24, 54, 

55, 58 
Rutherford scattering formula, 23 

deviations from, 39, 40 

Sachs, D., 99 

Samarium, radioactivity of, 16 
Sample holders, 219 
Sample-mounting techniques, 231- 

234 

Sample mounts, 231, 232 
Sargent, B. W., 135 
Sargent diagram, 135 
Sargent relation, 121 
Saturation current, 179 

collection of, 179-186 
Scaling circuits, 192, 193 
Scattering, inelastic, 67, 68 

of ot particles, 22, 23, 38-40 

of electrons, 157, 158, 165, 166, 225, 
226 

of y rays, 167 

of neutrons, 59, 60, 68, 171 
Scavenger precipitates, 249 
Schmidt, G. C., 3 
Schroedinger, E., 28 
Schweidler, E. ., 6, 204 
Scintillation counting, 178 



Screen wall counter, 191 

Seaborg, G. T., 273, 274, 297 

Secondary ionization, 148, 156, 166 

Secondary nuclear reactions, 240 

Segre, E., 137, 269, 270, 271, 272, 273 

Selection rules, Fermi, 134 
Gamow-Teller, 134 
in -y-ray transitions, 142 

Selenium, Szilard-Chalmers separa- 
tion for, 255 

Selenium tetrachloride, distillation of, 
252 

Self-absorption, 166, 229-231 
equations for, 229, 230 

Self-diffusion, 280, 281 

Self-scattering of /3 particles, 227, 231 

Senseman, R. W., 66 

Separated isotopes as targets, 240, 241 

Separation procedures, tests of, with 
tracers, 290 

Serber, R., 66, 101 

Seren, L., 390, 393 

Sewell, D. C., 101 

Shimming of cyclotron magnet, 88 

Short-lived activities, techniques for 
measurement of, 119, 120 

Silver, a-particle reactions of, 64, 65 
self-diffusion of, 281 

Skaggs, L. S., 68 

Sloan, D. It., 83 

Soddy, F., 5, 6 

Sodium, separation from magnesium 
oxide targets, 252 

Solomon, A. K., 230 

Solubility studies with tracers, 291 

Solution counting, 229, 233 

Solvent extraction at low concentra- 
tions, 267 

Solvent extraction methods in radio- 
chemistry, 252 

Sommerfeld, A., 25 

Spallation reactions, 66 ' 

Specific activity, 248, 249 

Spin, conservation of, 132 

Spin changes in ft decay, 134 

Spin quantum number, 26 

Stability rules, 50, 51 

Standard deviation, 199, 200 



INDEX 



411 



Standard deviation, for very low 
counting rates, 212 

in counting-rate meter, 214, 215 

of counting rate, 206, 207, 211, 212 

of differences, 209 

of ratios and products, 209, 210 

of sums, 209 

Standard sample mount, 218, 219 
Standard samples, 220 
Statistics, Rose, 33, 43, 132 

conservation of, 132 

Fermi, 33, 43, 132, 133 

of nuclei, 43 
Stopping power, 150-153, 156 

atomic, 150-151 
Strassman, F., 273 
Sulfur exchange reactions, 283 
Surface adsorption, 265 
Synchrocyclotron, 89 
Synchrotron, 94-96, 98 

proton, 90, 96 
Szflafd, L., 253 

Szilard-Chalmers method, discovery 
of, 253 

illustrations of, 255, 256 

principle of, 253 

Tacke, I., 269 

Target impurities, 236, 242, 246 

Target materials, purity requirements 

for, 242 

Technetium, absence of stable iso- 
topes of, 140 

chemical properties of, 270 

discovery of, 269 

isotopes of, 270 
Teller, E., 134 
Tellurium, hot-atom chemistry of, 258 

isomer separation for, 259 

Szilard-Chalmers separation for, 

255 

Thallium exchange reactions, 284 
Thermal column, 104 
Thermal diffusion, isotope concentra- 
tion by, 46, 47 
Thomson, J. J., 3, 45 
Thorium, radioactivity of, 3 
Thorium C, decay scheme of, 125, 127 



Thorium series, 11, 13 

Thorium X, 6 

Thornton, L. R., 66 

Threshold energy, 57 

Time constant of a circuit, 183, 184 

of counting-rate meter, 214 
Tracer applications, classification of, 

280 

Tracer principle, 278 
Tracers applied to tests of analytical 

procedures, 290 

Transmutation, first artificial, 55 
Transuranium elements, 273-276 
Triton, properties of, 44 
Turkel, S. H., 390 

Uncertainty principle, 28 

Universe, age of, 19 

Uranium, oxidation potentials of, 275 

separation of, from fission products, 
252 

Szilard-Chalmers separation for, 

256 

Uranium series, 10-12 
Uranium standards, 226, 227 
Uranium 234, 11, 12 
Uranium 235, 11, 14 
Uranium 238, 10-12 

decay of, 9 
Uranium X, 6 

Uranium Xi, growth and decay of, 9 
Uranyl nitrate, ether extraction of, 
252 

Van de Graaff, R. J., 80 

Van de Graaff generator, 80-82, 98, 

99, 100 

Veksler, V. /., 94 
Vibrating-reed electrometer, 183 
Volatility at tracer concentrations, 

268 
Volatilization methods in radiochem- 

istry, 251, 252 

Voltage gradients in counters, 187 
Voltage multiplier, 79 

Wahl, A. C., 273 
Walton, E. T., 79 



412 



INDEX 



Washing-out principle, 247 

Water boiler, 103 

Wallenberg, A., 99 

Wave equation, 28 

Wave function, physical interpreta- 
tion of, 28 

Wheeler, J. A., 71 

Wideroe, R., 83 

Wiedenbeck, M. L., 142 

Wiegand, C. E., 137 

Willgerodt reaction, tracer study of, 
294 

Wilson, J. N., 289 

Wilson cloud chamber, 177 

Wu, C. S., 272 

X-ray emission, following internal 

conversion, 141, 142 
following K capture, 137 
X-ray emission lines, 222, 223 
X-ray energies, measurement of, 168- 
171, 194 



X rays, absorption coefficient for, 221, 

222 

absorption of, 166-171 
critical absorption of, 221-223 
designation of, 221 
from electron accelerators, 98 
half-thickness values for, 169, 170 
versus y rays, 5 

Yankwich, P. E., 294 

Yntema, L. F., 272 

Yttrium, carrier-free, 249, 250 

carrying by lanthanum fluoride, 

264 
separation from strontium targets, 

249 
Yukawa, H., 134 

Zeeman effect, 25 

Zinc, self-diffusion of, 281 

separation, from copper targets, 252 

from iron, 247 
Zunti, W., 162