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Full text of "On Haidinger's Rings in Mica"

176 Mr. T. K. Chinmayanandam. 

5. Saturated vapours, even when very dense, show no increase of scattering 
power beyond what the density would lead one to expect. If molecular 
aggregates are formed, they are not numerous enough to show by this 
method. 

6. Liquid ether apparently scatters about seven times less light than a 
corresponding mass of ether vapour. 

DESCRIPTION OF PLATE 2. 

1. Photographs of the two polarisations in ether vapour. Upper image lias vibration 

vertical, lower horizontal. Screens over the upper image of opacity increasing from 
left to right. Equality at No. 5. 

2. Similar photographs for nitrous oxide, using same opaque screens. Equality falls 

between Nos. 1 and 2. 

3. Ether vapour. Nos. 1 and 5 at low pressure ; 2, 3, and 4 at saturation pressure, but 

with successively reduced lens aperture (see p. 172). Note that No. 3 approximately 
equals Nos. 1 and 5. 



On Haidinger's Rings in Mica. 
By T. K. Chinmayanandam, M.A. (Hons.). 

(Communicated by Dr. Gilbert T. Walker, C.S.I., D.Sc, F.R.S. Received 

July 10, 1918.) 

1. Introduction. 

The interference rings observed between plane parallel surfaces under diffuse 
monochromatic illumination have recently acquired much importance, in view 
of their practical application in the construction of spectroscopes of high 
resolving power. These rings were first observed by Haidinger in mica. This 
circumstance and the fact that mica is readily available, and by its natural 
cleavage furnishes plates with absolutely parallel faces, make the study of 
the rings in that substance one of considerable interest. In a paper published 
in the ' Philosophical Magazine ? for November, 1906, Lord Rayleigh described 
some interesting features of the rings observed in natural light. Mica being 
a doubly refracting substance, there are in fact two systems of rings super- 
posed upon each other, being each due to one of the beams polarised in planes 
at right angles to each other. It may be expected, therefore, that there must 
be regions of maximum and minimum visibility in the field of view. This is 
exactly what happens, and Lord Rayleigh has noted that in a plate of mica 



On Haidingers Rings in Mica. 177 

0*213 mm. thick, observed in the light of a salted Bunsen flame, the rings 
were indistinct along four directions apparently at right angles, radiating 
from the centre of the field. He has remarked in conclusion that it would 
be of interest to undertake a more detailed discussion of the subject, especially 
in comparison with observations on a plate whose optical constants are 
known. It is proposed in this paper to give an account of some further 
observations made by the writer on the rings in mica, and also to attempt a 
theoretical discussion of the phenomenon. 

Mica, as is generally known, occurs in many different varieties with 
different optical constants. In muscovite, which is the most common variety, 
the apparent angle between the optic axes is about 70°. Most of the observa- 
tions of the present writer were made with this variety of mica, of which 
sheets of different thicknesses were available. In some other varieties, such 
as biotite and phlogopite, the axial angle is much smaller, less than 10°, so that 
these varieties are nearly uniaxial. Of these, biotite, being strongly absorbent 
to light, is not very suitable for observation of Haidinger's rings ; but by the 
courtesy of the Director of the Geological Survey of India, the writer was 
furnished with a few good specimens of phlogopite from Burma, with which 
some observations, to be mentioned later in the paper, were also made. The 
phenomena observed differ, as may be expected, with different varieties of 
mica. As regards the source of light, Lord Eayleigh used an ordinary sodium 
flame. It was found by the writer that a much more convenient source is a 
Cooper-Hewitt lamp with some suitable ray filter ; the light being powerful, 
the observations are very easily made. 

2. Experimented Methods and Results. 

The methods of observing and photographing the phenomena in transmitted 
light as well as in reflected light may now be described. All the observations 
were made in natural light. 

Transmitted System. — The rings formed by the transmitted light are very 
easily observed. The mica plate is simply held close to the eye, which is 
then directed towards the Cooper-Hewitt lamp, a ground glass plate and a 
green ray filter being interposed between the mica plate and the lamp, to 
diffuse the light and to render it monochromatic. If a photograph is to 
be taken, the camera lens is brought close to the mica, the best lens to use 
being a short-focus one, with as large an aperture as possible, consistent 
with good definition. This arrangement, though suitable for visual observa- 
tion, does not give a good photograph, on account of the feebleness of 
contrast between the bright and the dark rings. The contrast is very much 
greater in the rings formed by reflection from the surface of the mica, and it 

Q A 



178 Mr. T. K. Chinmayanandam. 

was therefore desirable to devise some simple means of observing and 
photographing the reflected systemof rings. 

Reflected System of Rings. — The chief difficulty in the observation of this 
system is that the head of the observer screens the light off from the mica 
plate, so that the centre of the system never comes into the field of view. 
The usual way of getting over this difficulty is to use a plate of glass at an 
inclination of 45° to the surface of the mica, and to have the source of light 
on one side. This arrangement is, however, not quite suitable for the 
present purpose, for the eye, or the camera lens in case a photograph is to 
be taken, cannot be brought quite close to the mica plate, and has to be kept 
at a distance. The result is not only that the field of view becomes corre- 
spondingly limited, but also that, a larger area of the crystal being used, the 
smallest defect or non-uniformity, or want of flatness in the mica, results in 
the rings becoming distorted or irregular. There is also a loss of light in 
reflection at the glass plate. To get over these difficulties, the following 
simple arrangement has been used in the present investigation, to observe 
and photograph the reflected rings. A white sheet of cardboard is taken, 
and a small hole bored in it in the centre. This is mounted vertically on a 
stand, and the mica plate is fixed immediately in front of it, with its face 
parallel to the cardboard. The side of the cardboard facing the mica is 
lighted by one or more Cooper-Hewitt lamps, and the rings are viewed 
from the other side through the hole in the cardboard against a dark 
background. The reflected rings can be seen in this way very easily and 
clearly. To prevent extraneous scattered light from entering the eye or the 
camera lens, the edges of the hole and the back of the cardboard may be 
blackened. For visual observation, a green ray filter may be fixed behind 
the cardboard, though the effects described in this paper are, indeed, seen 
very well without it. For photographing the rings the green light is not 
very suitable, but a cell containing a solution of sulphate of quinine may be 
used with great advantage. The function of this cell is to cut of! the 
ultra-violet rays and the two lines at 4047 and 4078 A.TJ. without in any 

o 

way affecting the bright violet line at 4358 A. IT., and thus to secure the 
homogeneity of the violet light. The green and the yellow rays are, indeed, 
transmitted freely by the cell, but they do not affect the photographic plate 
appreciably. The cell may be fixed behind the camera lens. Process plates, 
with an exposure of 10-15 minutes, give very satisfactory pictures, one of 
which is reproduced in fig. 1. 

Some of the experimental results obtained with mica may now be 
described. A number of plates of muscovite mica of different thicknesses 
were examined. The dimensions of the rings vary, of course, with the 



On Haidinger's Rings in Mica. 



179 



thickness of the plate and the wave-length of the light. The general 
form of the curves of minimum visibility is, as seen in the figure, roughly 
a system of hyperbolas. As the entire field that could be visually \ observed 




Fig. l. 

was not covered by a single negative, a diagram (fig. 2) has been drawn 
(copied from two or three negatives taken in different positions) to show the 
curves observed. It will be seen that there are two series of curves with 
their axes nearly at right angles. In each set, at least two curves, sometimes 



180 



Mr. T. K. Chinmayanandam. 



more, could be observed on either side of the centre. The plane containing 
the optic axes of the crystal was parallel to the direction AB. The curves 
are fixed relatively to the crystal, and rotate with the latter as it is turned 
round. The general form of the curves does not vary much with the wave- 
length of the light or with the thickness of the crystal, but their actual 
dimensions and position relatively to the centre of the system do depend 
upon both. As the thickness of the mica plate is varied, one of the pairs of 
hyperbolas moves towards the centre, and for certain thicknesses meets at 
that point, forming a cross in the field, as shown by the dotted lines in 






Fig. 2. 



fig. 2. The appearance is then somewhat similar to that described by 
Lord Eayleigh, except that the other hyperbolas remain in the field of view, 
and a single cross is not observed in any case. A slight asymmetry may be 
noticed with regard to the curves on the same axes situated on either side of 
the centre; this is apparently due to the fact that the acute bisectrix of the 
angle between the optic axes is not exactly perpendicular to the plane of 
cleavage. 

The rings do not actually vanish at any point of the field, though 
the lines of minimum visibility are seen well defined. A scrutiny of 



On Haidingers Rings in Mica. 181 

the photograph reproduced in the figure will show that the rings 
appear doubled where a line of minimum visibility meets them at a small 
angle, as is the case near the vertices of the hyperbolas, but where it 
cuts across them nearly at right angles, we find the bright bands on one 
side running into and meeting the dark bands on the other side, producing a 
pretty ( system of dislocated rings. It can easily be verified that these 
phenomena are due to the superposition of two independent sets of rings, by 
looking at the ring system through a nicol. As the latter is rotated, the 
lines of minimum visibility disappear absolutely in four positions of the nicol 
at right angles to one another. 

3. Theory. 

The fact that in mica the acute bisectrix of the angle between the optic 
axes is very nearly normal to the plane of cleavage simplifies the theory 
considerably. The rings observed by the light reflected from the crystal 
are, of course, due to the interference of the rays reflected from the first 
surface of the crystal with those that enter the plate and, after undergoing 
one or more internal reflections at the second surface, emerge from the first 
surface in a direction parallel to the former. Now since any ray incident on 
the crystal is generally split up into two rays polarised in and perpendicular 
respectively to the principal plane, and travelling in slightly different 
directions with different velocities, there will be two rays, polarised in planes 
perpendicular to each other, emerging in any given direction from the first 
surface after one internal reflection. Suppose Si, 82 are the respective path 
differences in the two cases between the directly reflected and the internally 
reflected rays. Then the dark rings of the two sets are given respectively by 

Si = n\, 

$ 2 = m\, 

where n, m are integers. A point of minimum visibility will correspond to 
the condition that 81 = NX, and 8 2 = (M + |) X, where (N—M) is an integer. 
A line of minimum visibility evidently satisfies the condition that, at every 
point on it, the respective orders of the rings of the two sets have a constant 
difference equal to N— (M + |), i.e., the condition that the relative path 
difference of the two rays polarised in perpendicular planes and emerging in the 
same direction is constant over all points on the line. The problem of 
determining the form of such a line, therefore, reduces to finding (Si— S 2 ) 
corresponding to different directions. For an isotropic medium, the path 
difference S between the interfering rays is simply 2 pe cos r, where jn is the 
refractive index and e the thickness of the plate, and r is the angle of refrac- 
tion. Since, in mica, the normal to the plate is practically the bisectrix of 



182 Mr. T. K. Chinmayanandam. 

the angle between the optic axes, the velocities of each of the two rays into 
which the incident ray is split up is, by symmetry, the same before and 
after internal reflection at the rear surface of the plate, and we may to a 
sufficient degree of approximation write 

81 = 2e ~ cos ri, 

v 

S 2 = 2e — cos r 2 , 

v 

where % is the velocity in air, and v f and v" are the velocities of the 
refracted waves in the crystal. Thus we have 

s ^ o fcosri cosr 2 

L v v 

Even without evaluating this expression in terms of the angle of emergence 
of the rays, it is seen that it is identical with the well-known formula giving 
the form of the isochromatic lines in convergent polarised light for a plate of 
thickness 2e* Since mica is biaxial, the optic figure should resemble a series 
of Cassini's ovals and leminiscates. The angle between the optic axes in 
muscovite is, however, fairly large, and in the centre of the field the 
isochromatic lines look very much like hyperbolas. The analogy between 
the form of the lines of minimum visibility and of the isochromatic lines in 
convergent polarised light pointed out here is strikingly verified by an 
examination of the rings in phlogopite.f In this variety of mica the optic 
axes are not widely, separated, and the lines of minimum visibility in 
Haidinger ? s rings are found to be closed curves of the form of Cassini's ovals, 
exactly as indicated by theory. 

It is interesting to consider the actual form of the rings, and see exactly 
how the two systems of rings give rise to the observed lines of minimum 
visibility. For simplicity, the difference between the wave velocities and the 
corresponding ray velocities may be assumed to be negligible. Let <f> be the 
azimuth and £0 the angle of incidence of any ray falling on the plate, and 
<£', K > V* ?"> similar co-ordinates for the corresponding refracted wave- 
normals. Also let v > v'> v"> be the velocities of the two waves in air and in 
the crystal respectively. Then 

4> = <f>' = 4>" (1) 

, sin f __ sin £" __ sin £ /r>\ 

and. — -. .. — . ("") 

* See Drude's i Optics,' English translation by Millikan, p. 350. 

t This analogy between the lines of minimum visibility and the isochromatic lines 
in polarised light is not applicable to all crystals cut in any manner, but only to those 
which cleave or which are cut perpendicular to one of the axes of optic symmetry in 
the crystal. 



On Haidingers Rings in Mica. 183 

The wave-velocities in the direction <j>, f, are given by the equation 

cos 2 <j> sin 2 cj> cot 2 £ _ r. ,ox 

v 2 — a 2 v 2 — b 2 v* — e 2 

the solution of which is 

2^ = a 2 + b 2 - sin 2 £ ( Ai cos 2 <£ + A 2 sin 2 <f>)±* (4) 

where Ai = a 2 — c 2 , A 2 = b 2 -~c 2 , and A 3 = & 2 — & 2 ; 

and 

a: 2 = A 3 2 - 2A 8 sin 2 f (A x cos 2 0- A 2 sin 2 <-&) + sin 4 £ (A x cos 2 <£ + A 2 sin 2 <j>) 2 . (5) 

Cte ^Aere ^6 Angle between the Optic Axes is Large. — In this case, if £ be 
small, we can neglect the fourth and higher powers of sin £ and (5) reduces to 

k = A 3 -- sin 2 f(AiCos 2 ^> — A 2 sin 2 $). (6) 

Prom (4) and (6) we get for v' and v" 

2v' 2 = ^ 2 + & 2 + A3-2AiCos 2 <^sin 2 r J 

or v' 2 = a 2 — Ai cos 2 <£ sin 2 f ; "1 

•2 w r* 



and similarly v" 2 = b 2 —A 2 sin 2 $ sin 2 f 

But from (2), 

v' 2 = -7-0-t- ^ and ^*" 2 — ~^V ^> 2 , 
sm J £0 sm J 5) 

so that on eliminating v', v" ', we get from (7) 

sin 2 P = ^ 2 sin2 6) 

^o 2 -f- Ai cos 2 <£ sin 2 £0 

and sin* f" = ^^"'f . ,,. 

^o •+■ A 2 sm J <p sm J £ 



(7) 



}>■ (8) 



If now Si, S 2 , be the path differences between the rays reflected at the first 
surface and the rays that emerge from the same surface after refraction and 
one internal reflection, 



^0 



Si = 2e cos f ' ~ = 2e sin f cot f '. 



v 



Similarly S 2 = 2e sin £ cot f ". 

From (8) cot* £' = V + AiCOS 2 <ftsin 2 g- -aW£ 

S 2 1 
Hence ~ = — {-y 2 — sin 2 g, (a 2 sin 2 <£ -f c 2 cos 2 <£)}, (9) 

tx.e Lb • 

and similarly 

S 2 1 

-^ = 75 {% 2 — sin 2 go (& 2 cos 2 <£ + c 2 sin 2 <£)}. (10) 



184 Mr. T. K. Chinmayanandam. 

The forms of the two sets of Haidinger's rings observed in the crystal are 
thus determined respectively by 

sin 2 £n 

— — -(& 2 sin 2 <£ + c 2 cos 2 <£) = const. (11) 

ci 

and — -~^ (b 2 cos 2 $ + e 2 sin 2 (j>) = const., (12) 

or, in Cartesian co-ordinates, by 

a 2 y 2 +b 2 x 2 = const. and b 2 x 2 -\-c 2 y 2 — const. 

Both sets of rings are hence, to the first order of approximation, ellipses. 
The lines of minimum visibility are given by 81 — S2 = const., or 

- {^o 2 — sin 2 £0 (a 2 sin 2 <£ -h c 2 cos 2 <£)p 



a 



■—• - {^o 2 — sin 2 £ (b 2 cos 2 $ + c 2 sin 2 <£)p = const., 



or, approximately, 



sin 2 f ( 







- (a 2 sin 2 ^ + c 2 cos 2 <£) — - (6 2 cos 2 <£ + c 2 sin 2 <£) 



= const., 



which in Cartesian co-ordinates becomes 



- (a 2 y 2 4- c 2 x 2 ) — - (& 2 # 2 -f (?y 2 ) = const., (13) 

representing a series of hyperbolas, whose asymptotes make an angle 2 a with 

each other, where 

tan 2 a= b/a. 

Since a and b are nearly equal in mica, the asymptotes are practically at 
right angles. This is not, however, exactly what is observed in the photo- 
graph, nor are the curves strictly hyperbolas. The deviation must be 
ascribed to the fact that the fourth power of sin f in equation (5) is not 
entirely negligible. 

Case where the Angle betiveen the Optic Axes is Very Small.- — This is the case 
with the phlogopite variety of mica, which is practically uniaxial. In this 
case, A3 becomes so small that in equation (5) the term A3 2 becomes less 
important than that involving sin 4 f, so that the approximation made in the 
previous case would no longer be valid. We can, however, write equation (5) 
in the approximate form 

K 2 = A3 2 - A 3 {a 2 + b 2 ) sin 2 & ( Ai cos 2 <£ - A 2 sin 2 <£) 

+ ^!±ZZ sin 4 f (Ai cos 2 4> + A 2 sin 2 cf>) 2 , (14) 

4: 

which has been obtained by writing sin 2 £ = (a? + b 2 ) sin 2 £o/2 in equation (5). 



On Haidingers Rings in Mica. 185 

Proceeding exactly as in the previous section, the form of the rings is found 
to be given by the equations 

B 2 _ 2 v 2 + (Ai cos 2 6 + A 2 sin 2 </>) sin 2 £ • 2 y 
&2- a 2 + b 2 ±* -sin ft, 

that is, sin 2 f ^ 1 — — — — -,o 2 ^ r = const., 

L a 2 + 6 2 4 : /c J 

or, since k is small compared with <x 2 +6 2 , 

If we neglect the squares of A 3 in (14), we have 

& 2 +& 2 . 2!//A a -l • a • 2 i\ a AiCos 2 <f>— A 2 sm 2 <f> 

* = — ~— sm 2 fo (Ai cos 2 + A 2 sin 2 <£)-A 3 . -~ 2 T , A • 2 T 

2 ^ ^ Ai cos J <£ +- A 2 sm J cf> 

so, that equation (15) becomes on being transformed to Cartesian co-ordinates, 

x 2 + y 2 A,x 2 + A 2 y 2 f ( A 1 x 2 + A 2 y 2 A 3 Aj^-A^n _ p . 
/a /a ( a a + b 2 ) \ x + ^ 2/ 2 a 2 + Z> 2 * A^ + A 2 */ 2 / J ~" ' 

which reduces to 

where / denotes the focal length of the lens when the rings are photo- 
graphed. When A 3 vanishes absolutely, i.e., in the case of a uniaxial crystal 
cut perpendicular to its optic axis, a = b, Ai = A 2 , and hence equation 
represents two sets of circles of slightly different radii. In the general case, 
the equation represents two series of curves of the fourth degree. 
The lines of minimum visibility are given approximately by 

, [*+*p> {l + AlCOs2 t + £ 2Sin2< H l = const., (17) 

_a 2 + b 2 2v I a 2 + b 2 J J K ' 

which becomes, on squaring and reducing, 
B 2^k (a 2 + W) 2 ( Ai cos 2 <£ + A 2 sin 2 0) 2 

-sin 2 ^ • A 3 (a 2 + & 2 )(AiCos 2 ^-A 2 sin 2 <^>) = const., (18) 

where squares of A 3 have been neglected. In Cartesian co-ordinates 
equation (18) may be written, if x 2 + y 2 =/ 2 sin 2 g), 

(»■ + Vf <^+^g -(»■ + y) A, A -y '£ = const, 



186 Mr. T. K. Chinmayanandam. 

This is of the form 

{(oc-ct) 2 + (l + /3)y 2 } {(x + *Y + (l + lS)y*} = const 
which is the equation of an oval, similar to that of Cassini. 

4. Numerical Besults. 

If we confine ourselves to the plane containing the optic axes and the 
plane perpendicular to it, the positions of successive rings can be easily worked 
out directly without any approximations. Also, the observation of the 
number of rings between the intersections of the two successive lines of 
minimum visibility with those planes enables us to calculate with very fair 
accuracy the ratios of the principal refractive indices, if we only know 
roughly the mean refractive index of the specimen and the angle between the 
optic axes. 

Consider first the plane containing the optic axes. The section of the wave 

surface by that plane is a circle of radius b and an ellipse whose axes are a 

and c. Bays incident in this plane are refracted and reflected, and finally 

emerge in the same plane ; and if 8 be the path difference for a ray emerging 

at an angle ft, we have, for the refracted ray corresponding to the circular 

section, 

4yu,Veos 2 r = 46 2 (/* 2 2 -sin 2 ft) = S 2 =' n 2 \ 2 , (20) 

or sin 2 ft = A*a 2 --*-?■• ( 21 ) 

4er 

Corresponding to the elliptic section, we have 

ix 2 = fjL ± 2 cos 2 r -f- fis 2 sin 2 r. (22) 

It can easily be shown also that 



sin ft = fjij 
sin r fi' 

Eliminating r from (22) and (23), we get 



(23) 



a 2 = . f^J^ (24) 

^ ^ 4 + (^i 2 -^3 2 )sin 2 ft- V ; 

Also 8 = 2e seer {//,-- sin ft sin r}, (25) 

which after reduction becomes 

sin* £•„ = ^-^ . ^. (26) 

It may be noted that equations (21) and (26) are identical with the general 

equations (9) and (10) when cf> = 0. For the plane at right angles to that 
containing the optic axes, it can be similarly shown that 



n 2 X 2 t . o o o n 2 X 2 



2 



sir ft = /if-77 and sir ft = ^ — — . ^— respectively. 

fl2 



4e» iU ~ ^ 4e* * •■-» 



On Haidingers Rings in Mica. 187 

We proceed to show how these simple formulae can be made use of to 
determine the ratios of the principal refractive indices of mica. Suppose 
the first line of minimum visibility intersects the plane <fi = at a point 
which corresponds to order n of one set and order (m + |) of the other (m^>n). 
Let the next line of minimum visibility lie on orders (n — r) and [m -f- \ — (r + 1)] 
respectively of the two sets of rings. Then, 



. 2 . o n 2 X 2 o (m-f4) 2 \ 2 u$ 

sm 2 Jb = /* 2 2 — ^ = /*3 2 - v A % } . ^ 



2 



4^2 A* 4^2 * f ._2> 



sin 2 ft/ - 2_ 0-Q 2 ^ 2 _ 2 _[m + l-(^+l) ] 2 X 2 M3 2 



so that 2m - r 2 = { 2 (m + i) (r + 1) - (r + 1) 2 } ^, 

fix 



or since, in a fairly thick plate, r is small compared with n or m, 



/i3 2 __ W, r 



2- • ^t (2"7) 

Similarly for the plane perpendicular to that containing the optic axes we get 

^i_rn/_ r' 

/*2 2 " n f * 7+T {M) 

In equations (27) and (28), the quantities r, r' are known, being simply 
counted off on the ground glass plate of the camera or the photographic 
negative ; m, n, m', and n\ are all of the order 2fie/\, where jjl is the mean 
refractive index of the mica, e its thickness, and X the wave-length of the 
incident light. In the actual specimen used, //, was about 1*62, e = 0*0144 cm., 
and X = 4*36xl0~ 5 cm., so that m, n, etc., were of the order 1050. The 
differences m—n, w! —n f , being very small compared with m, n, it is sufficient 
for our purpose to know the absolute values of the latter roughly, provided 
the differences m—n and m f — n\ which are small and integral, are known 
accurately. Now, in the direction f = in the plane <£ = which corre- 
sponds to that of the single ray velocity in the crystal, the respective orders 
of the two sets of rings are the same. Further, the difference between the 
respective orders of the rings will increase by unity each time as we pass 
successive lines of minimum visibility. Thus if z be the number of lines of 
minimum visibility between the directions § i> = and f = #, we have 

m — n = z — 1 and m'— n' = z — 1 + 1 = z. 

If the field of view covers the direction f = 0, z can be directly counted off. 
Otherwise, we can calculate the same by a simple method. It can easily be 
shown that the number of rings between successive lines of minimum 



188 Mr. T. K. Chinmayanandam. 

visibility is nearly the same, so that if we know the number N" of rings lying 
between the first and the last of those lines in the region 0<£o<# 5 we have 

r{z—l) = N". 

N is determined by the equation 

XT 2e f 1 /- sin 2 0\*"l r 

N = T /i i 1 -( 1 — W-lS~~2~ no ' 

where n is the number of rings from the centre up to the first line of 
minimum visibility. It is sufficient if we use the mean value for /x and an 
approximate value for 0. In the specimen used by the writer, the angle 
between the optic axes was about 73°, so that 6 = 37° nearly. The values of 
the other quantities are as follows :— 

r = 21*5, r f = 20-5, 
n = 4, N = 61, 3—1 == m—n = 3, m' —n' = 4, 

fisffjLi, fi^j fX2y can now be calculated from equations (27) and (28). The 
refractive indices were also measured with a ref ractometer, and the calculated 
and observed values of their ratios are given below. 






Observed. 


Calculated. 


0-9761 


0-9760 


0-9790 


0-9785 



5. Summary and Conclusion, 

This paper is an attempt at a fuller study of Haidinger's rings in mica, 
some special features of which due to double refraction were described by 
Lord Eayleigh in a paper in the 'Philosophical Magazine' for November, 1906. 
The principal results obtained here are as follows : — 

(1) The regions of minimum visibility in the field, due to the superposition 
of two independent sets of rings, lie along a series of curves which, in 
muscovite, are approximately hyperbolas. Lord Kayleigh's observation that 
the lines of minimum visibility are crosses traversing the centre of the field 
is true only for particular thicknesses, and, even then, is not a complete 
description of the phenomena, as two series of hyperbolas are seen in addition 
to the cross. In phlogopite, the rings are indistinct along closed curves of 
oval shape. 

(2) The usual method of observing reflected systems of rings, using a plate 
of glass inclined at 45° to the plate under observation, is not satisfactory for the 
observation of Haidinger's rings in mica. The effects of a want of flatness, or 
other defect, in the mica plate, were considerably minimised in a new method 
devised in the present investigation for observing and photographing the 



On Haidinger's Rings in Mica. 189 

reflected system of rings. A diffusing screen is placed close to the mica 
with its surface parallel to it, and the rings are observed through a hole in the 
centre of the former. 

(3) In mica, as in all other crystals which cleave or which are cut perpen- 
dicular to one of the axes of optic symmetry, theoretical considerations 
indicate that the lines of minimum visibility in Haidinger's rings should be 
practically the same as the isochromatic lines observed in a plate of twice the 
thickness in convergent polarised light. This conclusion is in agreement with 
the observations described above. 

(4) In such crystals, if the angle between the optic axes is also large, the 

rings are shown to be approximately two sets of ellipses, given respectively 

by the equations 

a 2 y 2j tc 2 x 2 = const., 

c 2 y 2 + b 2 x 2 = const., 

where a, h, c are the principal velocities in the crystal. The major axes of 
one set are in the same direction as the minor axes of the other. These 
results are not far from the truth in the case of muscovite. 

(5) The ratios of the principal refractive indices in mica can be found with 
fair accuracy by observing the number of rings lying between successive lines 
of minimum visibility in the plane containing the optic axes and in a perpen- 
dicular plane. 

The investigation was carried out in the Physical Laboratory of the Indian 
Association for the Cultivation of Science, Calcutta, and the writer's best 
thanks are due to Prof. C. V. Raman, who suggested the investigation and took 
much interest in its progress. He has also much pleasure in acknowledging 
the kindness of Dr. H. H. Hayden, F.R.S., who put some specimens of 
phlogopite mica at the author's disposal, and of Dr. G. T. Walker, F.R.S., 
whose interest in the work and encouragement have been of the greatest 
value to the author of this paper.