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.
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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
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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
5 1
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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
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APPENDIX
299
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372
APPENDIX
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APPENDIX 373
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APPENDIX
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APPENDIX
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APPENDIX
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APPENDIX
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APPENDIX 333
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APPENDIX
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APPENDIX
387
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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
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