ANISOTROPY OF THE OPTICAL POLARISATION FIELD IN LIQUIDS. BY IV~. RAMANADHAM, Department of Physics, Andhra University, Waltair. Received October 11, 1934. (Communicated by Nfr. S. Bhagavantam.) 7. Introduction. IT is now well known that while computing the optic moments induced in the individual molecules of a liquid one has to take into account not only the electric vector in the incident field but also the local polarisation field of the surrounding molecules. In the case of a gas, however, the latter becomes negligible as the distances separating the molecules is then considerable. If b is the optic moment induced per unit incident field in the gaseous condition the optic moment induced per the same incident field when in liquid condition is shown by Lorentz to be equal to b (l + 3 x), where, nz2--] the optical susceptibility of the liquid is equal to 4~' nz being the refractive index of the liquid medium. In deriving the above expression the assumption is made that the distribution of polarisable matter round the molecule is spherically symmetrical. The above considerations lead nl ~-1 4~r to the well-known Lorentz's equation 12+ 2 = -~ v~ b, which when com- bined with the simple relation n~2--1 ~ v~ b gives equation (1) in ~vhich n~ 4~r the refractive index of the vapour at N.T.P. is expressed in terms of that of the liquid. 3 nz~--i :M: (1) n~--l=~ nz~+-----~ d 3 v denotes the number of molecules per e.c., M, the molecular weight and d, the density of the liquid. It is now recognised that the above relation is at best only an approximation and that considerable deviations from the same are to be found in practice. The following table shows the disagreement between the observed refractive indices for a few typical vapours and those calculated on the basis of equation (1) using the known refractive index of the liquid. 281
282 M. Ramanadham TABLt~ I. Substance (no--l) 1 ~ observed (no--]) i 6 calculated Benzene.. 1"514 182 1753 Pentane.. 1"3564 1711 167 Hexan~., 1"376 232 23 Carbon disulphide.. 1"632 1485 145 It has been pointed out by Raman and Krishnan (1928) that the above discrepancies have their origin in a fundamental defect underlying I, orentz's theory, viz., the assumption of a spherically symmetrical local polarisation field. They have suggested that the local polarisation field should be regarded generally as anisotropic and it is the purpose of the present paper to see if better agreement between the observed and calculated values could be effected by introducing this conception into the calculations. Considerable amount of evidence in support of the above modified theory of Raman and Krishnan is already available and witl only be briefly referred to here. 2. Raman and Krishm~n's Theory and i/s applications. This theory supplemented Lorentz's idea of the polarisation field by the additional postulate that the local field is anisotropic, i.e., is different for different directions of the molecule, because the molecule by its anisotropic shape, evidence to which has been revealed by studies of -ray diffraction in liquids (Sogani~ 1927), cannot present the same degree of accessibility in all directions to the surrounding molecules. They supposed, therefore, that the magnitude of the polarisation field depends on the orientation of the molecule with respect to the light field. Instead of assuming, as Lorentz supposed, a spherical cavity to be scooped out they in general supposed an ellipsoid to be scooped out and knowing the dimensions of the molecule, they have shown that the magnitude of the polarisation field can be evaluated in terms of three characteristic coefficients Pl, P2 and Ps whose values are given by the integral oo s du 2rrabe Ju ~ + u r 2 + ( (b ~" % u)cd + ( where a, b, c are the dimensions of the molecule and m can assume the values of a, b, or c, according as we are finding Pl, ~b.2, or 23.
Anisotropy o~ [he @[ice/polariseho~z Field ijz Liquids 283 The integral reduces to a simple form when the ellipsoid is a spheroid of revolution. Two cases arise according as the spheroid is prolate or oblate. In the former b ---- c --- a ~/i'e 2 In the latter b= c -- (e/2)/1 l+e \ pl = -1 [ logf -l) 1 1--e 2 l+e\ = =. logf ) VJl_e ',. p~=47r ~-- 1--e'sin'~e 3 ( ~/~sin-le 1--e2~ p2=p~=2~r ~. -- e2 ] Krishnan and Ramachandra Rao (1929) calculated t51 and ~ from -ray data in some suitable cases, and have shown that the values thus calculated agree with those derived from the study of the scattering of light in liquids and gases. The latter method of deriving Jh and f12 is given here in some detail as certain of the equations contained therein will be referred to again in the following sections. We start with the two well-known equations (2) and (3) showing the relationship between the depolarisation factor and the anisotropy. 65~ rv = 5 + 7 3v (2) 6~z rl== 51~ Tfl v< + 7 ~ (3) N The letters have the usual significance and the eqlmtions refer respectively to the gaseous and liquid states. The optic moments induced in a molecule in the latter case are not identical with those in the gaseous condition but are according to the theory of Raman and Krishnan connected with them by equations of the type B~ = b~ (l+pl) (4) where the capital letters denote the effective optic moments in the liquid state and - nz2-1 4~r The equations for the refractive index in terms of the optic moments become somewhat different in as much as we have to take the optical anisotropy into account. Thus n, 2-1 b 1 + b., + b~ - (5) 4vv~ 3 and n?--i BI+B~+B 3 (6) 4~rv~ 8
284 M. Ramanadham It is possible by the aid of equations (2), (3), (5) and (6) to calculate B1, Be, and bl, be, separately in the case of molecules possessing an axis of symmetry and from them to deduce the values Pl and t52 by equations of type 4. This is what Krishnan and Ramachandl-a Rao have done. The values of 151 and p~ so calculated agree with those derived from the -ray data, thus supporting the theory of the auisotropic polarisation field. Krishnan (193) has subsequently made use of the data of the scattering of light alone in support of the theory, tie calculated the refractive index of benzene liquid from the knowledge of its refractivity in the vapour state, and the scattering data both in gaseous and liquid conditions and showed that better agreement between the calculated and the observed values could be obtained in this manner than when the calculations were made by the simple Lorentz equation. I, angevin's theory of electric and magnetic birefringence has also been modified on the basis of this idea (Raman and Krishnan, 1928) in order to explain the observed results satisfactorily in liquids. The present author (t929) has applied the same idea of the anisotropic polarisation field in deriving an expression for the magnetic birefringence in liquid mixtures. Narasimhiah (1934) has recently derived an equation for the refractivity of the liquid mixtures on this basis. A similar equation has also been derived by the present author in an earlier paper already referred to. 3. Calculation of the Refractivity of the Vapour. Krishnan's calculation of the refractivity of the liquid from that of the vapour is beset with certain difficulties, which he overcomes by the method of successive approximations. In making use of equation (3) for calculating Pl and P2 one can see that it involves the very refractive index of the liquid which we want to find out. IZrishnan first makes use of the refractive index as derived from relation (1) based on the simple Lorentz's theory and then calculates Pl and p~ by the use of equations (3) and (4). Then he calculates the refractive index by inserting these values of Pl and P2 in equation (6). He resubstitutes this in the formula (3) and gets new values for Pl and ft.2 and so on. We have in this paper avoided this method of successive approximations by calculating the refractivity of the vapour from the known refractivity of the liquid instead of the other way. This procedure is free from the above difficulties and provides a satisfactory and direct means of testing the validity or otherwise of the anisotropic polarisation field theory. The method of calculation is as follows. We have assumed for simplicity that the molecule possesses an axis of symmetry, i.e., B2=B3; b2---bs and /~-~=P3. Two cases at once present themselves, e.g., when BI> B2 and B1 <B2.
A~Hsot.ropy of the O#tical Polarisatio~z fi'idd in Lir 285 Let us suppose BI > B2. We calculate first BI and B2 from equations (3) and (6). Then we calculate Pl and 1#., by means of the equation (Narasimhiah, loc. cir.) 4~r+pl(n~'-- ~ 1 ) _ I+2~/$z 1-- r 4~r+p2(n~--l) 1-- VSz l+2vsv together with the equation p~+ 2p~= 4~. Knowing Bz and B2 and ibz and ib2 we calculate bl and b,,, which latter enable us to calculate the refractivity in the vapour using the equation nj-71 = bl + 2b~. In the case in which 47rv~ 3 B1 < B~ to calculate Pl and P2 we must make use of an equation like 4zr+io ~ (nz2--1) 1--2 ~Sz 1 + ~/(S,v -- 4~r+p2 (n[-'--l) 1 + ~/~z ] --2 ~/~v To decide the choice between the two possibilities, one may be guided to a certain extent by the doublet theory of optical anisotropy developed by Ramanathan (1925). In this connection, it is, however, of interest to mention the recent theoretical work of Mrowka (1932) according to which the optical polarisability of the hydrogen molecule along the nuclear axis is less than that in directions lying in the symmetry plane. This result is surprising and in our terminology, means B1 < B2 which is in direct conflict with the conclusions of the theory developed by Ramanathan. An experimental test between the two alternatives is only possible in those cases in which the crystal structure and birefringence of the substance in the solid state are known. In view of this both cases have been worked out in the present paper without showing any prejudice towards the one or the other, with a view to see which of the cases brings closer agreement between calculation and observation. In the table the alternative sets of values are given only in the cases of benzene and carbon disulphide as they happen to afford results distinctly in favour of the theory developed by Ramanathan. In other liquids, the two alternatives lead to more or less the same results. 4. Discussion. The first six liquids are cases where we can justifiably assume an axis of symmetry to exist for the molecule concerned. Such an assumption in the other instances is an approximation and these have been worked out only to see whether the conception of the anisotropic polarisation field improves the results or not. One can in general find from a glance at the table that the values calculated on the basis of Raman and Krishnan's theory are higher than those calculated according to Lorentz's formula. With the exception of the liquids which we have listed as third group in the table, Raman and Krishnan's theory makes the calculated value~
286 M. Rarnanadham ~x~,.._,, &x? x,m,4 d' r *--4 O ~ ~ ~ ~ ~ I r @ II $ x 4 g ~g a~ > d O~.! o @ a AV A V A V A ~ ~. ~. ~. ~.. ~ ~ ~ 9 ~ o 9. 9. 9 ' " ~ ' " " i " d < i A Q N w ww~n NN N~
.q~z/sotropy of #ze Ofifica/ Po/azisaLiosz FieM i~z Licuids 287 approach the observed values more closely than the Lorentz's theory. It must also be noted that the agreement between the calculated and observed values in the case of the first six compounds with the exception of cyclohexane (for which we do not have reliable data for the refractivity of the vapour), is excellent as may be expected. It is interesting to note that in the third group of compounds the value calculated according to Lorentz is itself generally higher than the observed value but it is not possible to draw any definite conclusions. Of the two alternatives given in benzene and carbon disulphide it will be noticed that the case in which Bl is less than B2 for the former and the case in which B1 is greater than :B2 for the latter provide decidedly better agreement with the observed values and this is in accordance with the prevailing ideas regarding these two molecules. It may further be pointed out that the mode of calculation adopted in the present paper lends a strong support to the theory of Raman and Krishnan in that it affords a kind of internal agreement without our having to fall back upon any theory to derive the values of pl and P2 from the dimensions of the molecules. The only assumption we made regarding them was that Pl + 2p2 = 4rr. 5. Summary. The theory of the anisotropic polarisation field developed by Raman and Krishnan has been applied to calculate the refractivities of a number of organic vapours from the known refractive indices in the liquid state. The observed values are found to agree better with those calculated on the above basis than those obtained with the help of Lorentz's formula. The author is highly thankful to Mr. S. Bhagavantam for his interest in the work. He is also thankful to the Syndicate of the And&ra University for having awarded him a research scholarship. REFEI%ENCES. Krishnan...... Krislmau and ~amachandra ~rowka...... Narasimhiah.. R.aman and Krishnan Do, ~amanadham 9 9 Ramanathan.. Sogani.... R ao.. Proc. Roy. Soy. (A), 193, 126, 155. Ind. Jour. Phys., 1929, 4, 396. Z. fl Phys., 1932, 76, 3. Proc. Ind. Acad. Sci., 1934, l, 34. Proc. Roy. Soc. (A), 1928, 117, 1. Ibid., 1928, 117, 589. Ind. Jour. Phys., 1929, 4, 19. 7~roc. Roy. Soe. (A), 1925, 17, 684. Ind. Jour. Phys., 1927, 1, 357.