Birefringence in Electric and Magnetic Fields*

John W. B. Virginia

A. Introduction.

B. Birefringence in an electric field:
1. The electro-optical Kerr effect. Discovery of the effect. Nature and properties of the phenomenon. Kerr’s law on the dependence of the effect on the electric-field intensity. — 2. Methods of experimental determination of the Kerr constant. Absolute measurements, relative measurements. — 3. Theory of the electro-optical Kerr effect. Havelock’s theory. Langevin’s theory, revised by Born. The new quantum-mechanical theory of Kronig and Born–Jordan. — 4. The Kerr effect in gases. Experimental data. Calculation of the Kerr constant \(B\) from the Langevin–Born theories from measurements of other quantities. Dependence of the Kerr effect on the electrical and optical properties of molecules. Dependence of the Kerr constant on the wavelength of light, temperature, and gas density. Anomalous effect near absorption lines. — 5. The Kerr effect in liquids. Experimental data. Determination of the electro-optical properties of molecules according to the Langevin–Born theory. Raman and Krishnan’s theory for ionic liquids. Change of the Kerr constant with temperature, dependence of \(B\) on wavelength. Absolute values of the coefficients \(n_p\) and \(n_s\). — 6. Relaxation time of the Kerr effect. Experimental data and theory. Applications of the effect. — 7. Further applications. Birefringence in mixtures of liquids. Birefringence in suspensions, colloids, etc. — 8. The Kerr effect in solids. Amorphous bodies and crystals.

C. Birefringence in a magnetic field:
1. The Cotton–Mouton effect. Discovery and nature of the phenomenon. Dependence on the magnetic field. — 2. Methods of experimental study. Necessity of large homogeneous fields and special apparatus for separating weak birefringence from rotation of the plane of polarization. — 3. Theory of the Cotton–Mouton effect. (Langevin–Born.) — 4. Raman and Krishnan’s theory for ionic liquids. The Cotton–Mouton constant for dissolved substances. The Cotton–Mouton effect in crystals and the electromagnetic anisotropy of these crystals. Application of the effect to the study of crystal structure. — 5. Dependence of the Cotton–Mouton constant \(C\) on wavelength and temperature. Brief comparison of experimental data with theory. — 6. Magnetic birefringence in gases. Voigt’s effect near absorption lines. Its explanation according to Voigt’s theory. Experiments with paramagnetic gases. — 7. Further applications. Magnetic birefringence in suspensions, colloids, liquid crystals, etc. — 8. Magnetic birefringence in solids. Amorphous bodies and crystals.

* Reviews of Modern Physics, 4, 133, 1932.

A. Introduction

The branches of physics known as electro-optics and magneto-optics had their origin in 1845, with Faraday’s discovery of the rotation of the plane of polarization of polarized light as it passes through a transparent medium parallel to the lines of force of a magnetic field. This epoch-making discovery by Faraday was the result of his long investigations into the connection between light, electricity, and magnetism.

In his article,^1 describing this discovery, he says: “Like many lovers of natural philosophy, I had long held the opinion, almost passing into conviction, that those various forms in which the forces of matter manifest themselves have one common origin, i.e., in other words, that they are so dependent upon one another and so connected with one another that they can be transformed one into another and that equivalents of their action exist.” This conviction of Faraday’s led him to search for further connections between light, electricity, and magnetism, but the sensitivity of his apparatus was insufficient to detect those relations which were subsequently established. It should be noted, however, that these convictions and Faraday’s earlier investigations served as a constant stimulus for applying more refined techniques in the search for new phenomena in the field of electro- and magneto-optics. Among the many examples of this influence, the most important are the discoveries of the effects bearing the names of Kerr, Zeeman, and Cotton–Mouton. Indeed, the majority of discoveries in electro- and magneto-optics (with the possible exception of the photoelectric effect) resulted from deliberate attempts by experimenters to improve their apparatus so as to detect the desired relation. It is therefore quite clear that one cannot illuminate in detail any single connection between light, electricity, and magnetism without at the same time touching on most of the others. Such an attempt would lead to unnecessary repetition and would greatly lengthen the exposition. The purpose of the present article is therefore to survey the most important works studying the effects observed when light passes through a medium perpendicular to the lines of force of electric and magnetic fields. A substantial part of the article will be devoted to the description of two effects, usually named after the researchers who discovered them: the Kerr effect and the Cotton–Mouton effect.

In all cases, references will be given to the literature, making possible further specialized study.

B. Double Refraction in an Electric Field

1. The Electro-Optic Kerr Effect

Kerr² was the first, in 1875, to observe that certain isotropic substances, when placed in an electric field, become birefringent with “optical axes” parallel to the lines of force, i.e., that a substance placed in a homogeneous electric field acquires the properties of a uniaxial crystal with its optical axis parallel to the electric lines of force. One of the typical arrangements for producing this effect is shown in Fig. 1. Monochromatic light, collected by the lens $L$ into a parallel beam, is polarized by the Nicol prism $N_1$, passes between two parallel metallic plates

Fig. 1.

$K$ (called the “Kerr cell”) and through a second Nicol prism $N_2$, crossed with the first. To obtain the maximum effect, the transmission plane of $N_1$ must make an angle of $45^\circ$ with the lines of force of the field. If both metallic plates $K$ are at the same potential, and if the medium enclosed between them is isotropic, then the light will not pass through $N_2$. However, if an electrostatic field is applied to the plates $K$, then the medium between them becomes birefringent, i.e., the components of the light vector parallel and perpendicular to the lines of force pass through it with different velocities. The light therefore emerges from $K$ elliptically polarized, and part of it can pass through $N_2$. The effect does not depend on the direction of the field and is therefore doubled if, by placing a mirror in front of $N_2$, the light is made to pass once more through $K$.

The first substance investigated by Kerr was a piece of glass with the ends of the winding of an induction coil sealed into it. In terms of the optical properties produced, the applied field proved equivalent to a mechanical stretching of the glass directed along the lines of force.

lines, i.e., the glass behaved like a positive crystal, for example quartz.

In other solid bodies, for example in colophony, the effect was the reverse, i.e., they behaved like glass compressed along the lines of force, as, for example, Iceland spar.

The effect did not appear simultaneously with the application of the field, but sometimes required several seconds (in the case of glass even about \(1/2\) minute) in order to reach its full magnitude. The same time elapsed between switching off the field and the disappearance of the double refraction.

This delay in the appearance and disappearance of the effect led to various hypotheses about its nature. It seemed plausible that it was caused by stresses produced by the field in the glass, since already from Volta’s work it was well known,\(^3\) that “the dielectric in a charged glass condenser increases in volume and is subjected to electrical stresses.”

Another hypothesis explained the effect by an increase in temperature caused by the electrical conductivity of the substance.

Kerr himself, however, was convinced that the effect he had discovered could not be explained by purely mechanical or thermal causes. He therefore began an investigation of slightly viscous liquids, in which the field could not produce any significant stresses. He immediately discovered the effect in a whole series of liquids.\(^4\) In some, such as, for example, carbon disulfide, paraffin oil, toluene, and benzene, the double refraction was positive, while in others, for example in olive oil, turpentine, and seal oil, it was negative. He was able to establish that in all these cases the effect appeared and disappeared simultaneously with the switching on and off of the field, which clearly proved the electrical nature of the phenomenon.

The discovery of this effect also in liquid bodies at once attracted to it the attention of many other investigators, and Kerr’s results were quickly confirmed and extended.\(^{5-10}\)

Already in his early works Kerr\(^11\) was able to establish the dependence between the magnitude of the refraction and the strength of the electric field. The law he established is formulated as follows: if \(n_p\) and \(n_s\) are the refractive indices of the medium in the directions perpendicular and parallel to the lines of force, then the phase difference (in radians) after passage through an electric field of intensity \(E\) (expressed in \(CGS\)-\(E\) units) will be:

\[ D=\frac{2\pi l\,(n_p-n_s)}{\lambda}=2\pi BlE^2, \tag{1} \]

where \(\lambda\) is the wavelength of the transmitted light, \(l\) is the path length of the light in the medium (in cm), and \(B\) is the so-called “Kerr constant.” The latter, however, as it turned out, depends on the substance, the wavelength, and the temperature. This Kerr law is justified with great accuracy, as has been verified by many investigators. The few exceptions\(^{12}\) had to be explained by errors in the methods of observation, since they were not confirmed in subsequent investigations.\(^{13,14}\) In carbon disulfide, Schomon\(^{15}\) found that the exponent \(E\) is equal to 2.0045, and concluded that Kerr’s law is valid for chemically pure carbon disulfide. The same was also confirmed for nonpolar, nonconducting carbon disulfide.\(^{16}\) But for some polar liquids at high field strengths, small deviations were found. Deviations from Kerr’s law in strong fields were also detected for a number of crystals.\(^{17}\)

2. Methods of experimental determination of the Kerr constant \(B\). To determine \(B\) for a given wavelength, temperature, and medium, it is necessary, as equation (1) shows, to measure \(E\), \(l\), and \(D\). Equation (1), rewritten for experimentally measured quantities in the case of a Kerr cell in the form of two plane-parallel plates at a distance \(a\), will be:

\[ D = 2\pi B l \frac{V^2}{a^2}, \tag{2} \]

where \(V\) is the voltage applied to the cell.

The different methods of determining \(B\) have thus turned out to be a consequence of improved techniques for measuring the potential on the cell, the dimensions of the cell (especially the distance between the plates and the corrections for the field strength at their ends), and the elliptic polarization of the emerging light.

In general, arrangements are divided into two types depending on whether the substance is a conductor or not. If it is a conductor, then a somewhat modified Kerr method is used; if it is a semiconductor, then the method first proposed by de Kudre\(^{18}\) is used, according to which the Kerr constant is expressed through the already known constant of some other substance.

In Kerr’s method the desired wavelength is isolated by a light filter or monochromator; in the most carefully performed works the Kerr cell is placed in a thermostat. The windows in the cell cuvette must be free from any mechanical stresses, so that when the field is switched off there is no residual birefringence. The metallic plates of the cell are usually made flat and mounted immovably on insulators parallel

each other. They must be sufficiently large to give a homogeneous field along the path of the light beam. Homogeneity of the field is also achieved by placing the plates at a distance of several millimeters from one another, so that it is necessary to determine this distance with great accuracy, especially since in equation (2) it enters squared.

Determination of the effective length of the plates (i.e., the length of the electric field) \(l\) also presents difficulties because of the end corrections that must be taken into account. The problem of distortion of the field at the edges of the capacitor has been solved by many investigators. \(^{19--22}\) In addition, it is often possible to determine the end correction experimentally by using elements of different lengths. \(^{14}\) The potential \(V\) is usually supplied to the cell by an electrostatic machine, by a transformer through a rectifier, or by batteries. The magnitude of this potential is measured with a potentiometer, or with an electrostatic or even an ordinary voltmeter. Equation (2) shows that an error of \(1\%\) in \(V\) will lead to an error of \(2\%\) in the determination of \(B\). For most substances \(B\) is comparatively small, and it is necessary to take a very large \(V\) in order to obtain \(D\) suitable for measurements. Unfortunately, the technique for obtaining and measuring high electrostatic potentials is still not sufficiently developed, so that the accuracy of determining \(B\) is limited by errors in the measurement of \(V\). In fact, the reverse measurement process is even often used, employing a Kerr cell as a voltmeter for high voltages. \(^{23}\)

The quantity \(D\), which is a measure of the double refraction, can be measured in various ways. All of them are modifications of two principal methods, one of which is based on measuring the displacement of the dark band (the Babinet compensator), \(^{24}\) and the other on comparing the intensities of different parts of the field of view. The latter method is considerably more accurate and came into general use after Brace’s work, \(^{25}\) who, with the aid of an ingenious half-shade arrangement, achieved an accuracy of \(1/20000\) of a wavelength. There are several excellent descriptions of methods for measuring double refraction and of their applications \(^{26--30}\) to the measurement of \(D\). \(^{13,14,15,45,29}\) Therefore there is no need to dwell on these methods in detail. Recently, \(^{31}\) with the aid of a photoelectric cell, great accuracy has been achieved in the determination of \(D\).

If the substance under investigation is a conductor, then the above-mentioned methods are inapplicable, since it is impossible to maintain a constant electric field in a Kerr cell without heating or decomposing the medium. Hence it is neces-

ability to switch on the field only for a very short time, in order to avoid a rise in temperature due to conductivity.

The general scheme of the arrangements proposed by de Kudre, Schmidt\({}^{33}\), and others\({}^{33}\) is shown in Fig. 2. Monochromatic light is polarized by the Nicol \(N_1\), at an angle of \(45^\circ\) to the vertical, passes through the Kerr cell \(K\) with vertical plates, then through a second horizontal cell \(K_2\), and, finally, through a second Nicol \(N_2\), crossed with the first. If a substance with a known Kerr constant \(B_1\) is placed in \(K_1\), then the unknown constant \(B_2\) can be determined for the substance placed in \(K_2\) (provided only that \(B_1\) and \(B_2\) have the same sign; if they have different signs, the plates of \(K_2\) are set parallel to the plates of \(K_1\)). The same potential is applied to both cells, and the distance between the plates of \(K_2\) is varied until the double refraction in \(K_1\) is not compensated in \(K_2\), i.e., until the light completely ceases to pass through \(N_2\). Then

Fig. 2.

\[ \frac{B_1}{B_2}=\frac{a_1^2\,l_2}{a_2^2\,l_1}. \tag{3} \]

It should be noted that the plates of both capacitors must be strictly perpendicular (or parallel) to one another, in order to avoid rotation of the plane of polarization.

3. Theory of the electro-optical Kerr effect. Although the classical theories of Drude,\({}^{34}\) Voigt,\({}^{35}\) and others\({}^{36}\) were able successfully to explain the phenomena of dispersion, absorption, and even the Faraday effect, they proved manifestly insufficient when applied directly to electric and magnetic double refraction, i.e., to the electro-optical Kerr effect and to the Cotton–Mouton effect. It was soon suspected that the discrepancy between theory and experiment in the case of these two effects was caused by the anisotropy of the optical, electrical, and magnetic properties of individual molecules in an isotropic medium.

Restricting ourselves at first only to the Kerr effect,* we see that Havelock^37 was one of the first to formulate a hypothesis that reconciled a whole series of experimental data. He proposed that each particle may be enclosed in a certain “effective cell,” within which the polarization of the medium is constant and equal in magnitude and direction to the polarization vector applied to the particle. Then the force acting on the particle may be determined as the force acting on the whole cell. In an isotropic medium, a small sphere should be taken as the cell; in an electric field it is deformed, becoming an ellipsoid. Physically, Havelock explains this by a change in the mean distance between molecules along and across the field; however, he points out that it may also be explained by the orientation of anisotropic molecules in an electric field.

Proceeding from these assumptions, he shows that:

\[ n_p = n - \frac{2}{15}\frac{(n^2-1)^2}{n}\varepsilon, \]

\[ n_s = n + \frac{1}{15}\frac{(n^2-1)^2}{n}\varepsilon, \tag{4} \]

where \(n_p\) and \(n_s\) are the refractive indices in the directions parallel and perpendicular to the field, \(n\) is the refractive index without the field, and \(\varepsilon\) is a constant independent of the wavelength.

Equations (4) and (1) give:

\[ B = \frac{c}{E^2}\frac{(n^2-1)^2}{\lambda n}, \tag{5} \]

where \(c\) is a constant, likewise independent of \(\lambda\). In addition,

\[ \left(\frac{n_p-n}{n_s-n}\right) = -2. \tag{6} \]

Equation (5) relates the dispersion under electric double refraction to the ordinary dispersion in an isotropic medium. It is known as Havelock’s law and was experimentally verified by many investigators in the visible region.^14,20,38

Equation (6) was also verified, especially by Ekerlein,^39 Potenne^40 and Larkins.^41

* Theories of one and the same type may simultaneously explain both the electro-optic Kerr effect and the Cotton–Mouton effect; in most cases the formulae derived for the former may also be applied to the latter, replacing only the electrical units by the corresponding magnetic ones.

The generally accepted theory at the present time, which seems to agree well with the principal experimental data, is based on Kerr’s idea of the orientation of the molecules of a medium in an electric field. In its definitive form this theory was first formulated by Langevin,^42 although Larmor^43 and Cotton and Mouton^44 had already experimentally substantiated the assumptions of which he made use. Langevin proposed that the molecules of a substance are anisotropic both electrostatically and optically. Under the orienting action of the field on the induced moments of the molecules, the whole medium as a whole becomes birefringent. Born^45,46 extended this theory also to the case of possible permanent electric moments. Recently these same phenomena have been investigated from the standpoint of the new quantum mechanics.^47,48

For dense liquids, Raman and Krishnan^49 modified the Langevin–Born theory, assuming that not only is each molecule itself anisotropic, but it is also surrounded by an anisotropic polarization field, or by an anisotropically distributed polarizable substance. They assumed that this distribution arises because of the orientation of anisotropic molecules in the field.

We shall first briefly set forth the Langevin–Born theory, which agrees well with the experimental data in gases. We shall start from the well-known Lorentz–Lorenz relation^50,51,46 for an isotropic medium free of an electrostatic field:

\[ \frac{n^{2}-1}{n^{2}+2} = \frac{4}{3}\pi N\alpha_{0} = \frac{4\pi N}{3}\frac{b_{1}+b_{2}+b_{3}}{3}, \tag{7} \]

where \(n\) is the refractive index, \(N\) is the number of molecules in \(\mathrm{cm}^{3}\), \(\alpha_{0}\) is the polarizability,* and \(b_{1}, b_{2}, b_{3}\) are the moments induced in the molecule along the three principal axes by a unit electric force in the light wave acting along each of the corresponding axes. If an electric field is now applied to the medium, then \(N\) will change by \(\Delta N\) owing to electrostriction, and \(\alpha_{0}\) will change by \(\Delta \alpha\), with \(\Delta \alpha\) being different in the directions parallel and perpendicular to the lines of force.

\(\Delta n\) can now be obtained with sufficient approximation by differentiating equation (7):

\[ \Delta n = \frac{(n^{2}-1)(n^{2}+2)}{6n} \left( \frac{\Delta N}{N} + \frac{\Delta \alpha}{\alpha_{0}} \right). \tag{8} \]

* The general formula for the polarizability \(\alpha\) has the form:

\[ \alpha=\alpha_{0}+\frac{\mu^{2}}{3kT}. \]

The magnitude of the electrostriction can be determined from thermodynamic considerations.^58 It is equal to

\[ \frac{\Delta N}{N}=\frac{1}{4\pi}\left(\frac{\partial \varepsilon}{\partial p}\right)_{T}\frac{E^{2}}{2}, \tag{9} \]

where \(\varepsilon\) is the dielectric constant, \(p\) is the pressure, and \(T\) is the absolute temperature. As was indicated above, the quantity \(\Delta\alpha\) depends on the direction. But since the calculation of \((\Delta\alpha)_p\) and \((\Delta\alpha)_s\) (parallel and perpendicular to the field) is rather laborious and lengthy, and since the corresponding derivations can readily be found in the literature,^45,53,54 we shall give the final expressions without proof:

\[ \begin{gathered} (\Delta\alpha)_p=2(\theta_1+\theta_2)\left(\frac{\varepsilon+2}{3}\right)^2\frac{E^2}{2},\\ (\Delta\alpha)_s=-(\theta_1+\theta_2)\left(\frac{\varepsilon+2}{3}\right)^2\frac{E^2}{2}, \end{gathered} \tag{10} \]

where

\[ \theta_1=\frac{1}{45\,kT}\bigl[(a_1-a_2)(b_1-b_2)+(a_2-a_3)(b_2-b_3)+ \]

\[ +(a_3-a_1)(b_3-b_1)\bigr], \tag{11} \]

\[ \theta_2=\frac{1}{45\,k^2T^2}\bigl[(\mu_1^2-\mu_2^2)(b_1-b_2)+(\mu_2^2-\mu_3^2)(b_2-b_3)+ \]

\[ +(\mu_3^2-\mu_2^2)(b_3-b_1)\bigr]. \tag{12} \]

Here \(T\) is the absolute temperature, \(k\) is Boltzmann’s constant; \(b_1, b_2, b_3\) have the same meanings as in equation (7), i.e., they are the moments induced in the molecule along its three principal optical axes by a unit electric force in the light wave acting along each of the corresponding axes; \(a_1, a_2, a_3\) are the moments induced in the same directions by a unit electrostatic force arising as a result of the applied electric field; \(\mu_1, \mu_2, \mu_3\) are the components, in these same three directions, of the permanent electric moment.

Since the change of the refractive index \((\Delta n)_p\) in the Kerr effect is small, we can find \((\Delta n)_p\) as the component of the light vibration parallel to the electric field by substituting expressions (9) and (10) into equation (8):

\[ (\Delta n_p)=n_p-n=\frac{(n^2-1)(n^2+2)}{12n}\left[\frac{1}{4\pi}\left(\frac{\partial\varepsilon}{\partial p}\right)_{T}+ \right. \]

\[ \left. +2\left(\frac{\varepsilon+2}{3}\right)^2\frac{\theta_1+\theta_2}{a_0}\right]E^2, \tag{13} \]

and analogously for the component perpendicular to the field,

\[ (\Delta n)_s=n_s-n=\frac{(n^2-1)(n^2+2)}{12n}\left[\frac{1}{4\pi}\left(\frac{\partial\varepsilon}{\partial p}\right)_{T}-\left(\frac{\varepsilon+2}{3}\right)^2\frac{\theta_1+\theta_2}{a_0}\right]E^2, \tag{14} \]

then

\[ n_p-n_s=\frac{(n^2-1)(n^2+2)}{4n}\frac{\theta_1+\theta_2}{a_0}\left(\frac{\varepsilon+2}{3}\right)^2E^2 \tag{15} \]

and

\[ B=B_mN=\frac{n_p-n_s}{\lambda E^2} =\frac{(n^2-1)(n^2+2)}{4n\lambda} \left(\frac{\varepsilon+2}{3}\right)^2 \frac{\theta_1+\theta_2}{a_0}, \tag{16} \]

where \(B_m\) is the Kerr constant for one molecule and \(\lambda\) is the wavelength.

Consideration of the Kerr effect from the standpoint of the new quantum mechanics\(^{47,48}\) reveals a connection between the magnitude of electric double refraction and the magnitude of the electric splitting in the Stark effect. It turns out that the parallel and perpendicular components in the Stark effect are closely connected with the refractive indices of light oscillations polarized parallel and perpendicular to the field.

Born and Jordan derived a relation connecting the Kerr constant with the frequency of light and the temperature:

\[ S_{0p}-S_{0s}=\frac{n}{2\pi}(n_p-n_s)=\frac{Bn\lambda E^2}{2\pi}=k= \]

\[ =E^2\left[C_0+\frac{1}{kT}(C_1+D_0)+\frac{D_1}{k^2T^2}\right], \tag{17} \]

where \(S_{0p}\) and \(S_{0s}\) are the optical susceptibilities* parallel and perpendicular to the field, \(k\) is Boltzmann’s constant, \(T\) is the absolute temperature, \(C_0\) and \(D_0\) are determined by the change of the quantum number “\(n\)” (depending on the arrangement of the electrons in the molecule), and \(C_1\) and \(D_1\) by changes of the quantum number “\(j\)” (depending on the total angular momentum). It should be noted that equation (17) gives, in the main, the same dependence on temperature as expression (16), provided only that \(C_0\) is small. \(C_0\) should in general be considered small, since it represents a correction for anisotropy caused by the field in the molecule itself, i.e. such an effect as can manifest itself only in completely isotropic molecules.

If, in deriving equation (16), this anisotropy is taken into account and added to the initial anisotropy of the molecule, then a new term will appear in equation (16), independent of temperature. But experiments show that such a term, if it exists at all, is negligibly small and may be discarded.

* The optical susceptibility is given by the formula:

\[ S_0=\frac{n^2-1}{4\pi}. \]

The determination of \(B\) from the quantum-mechanical equation (17), as Kronig indicates, is in principle possible for the case of certain gases, but it is practically unfeasible, since up to now, at least, the Stark effect is insufficiently known for some transitions, and especially for transitions in the continuous region of the spectrum. We shall therefore consider the experimental data from the point of view of the Langevin–Born theory.

4. The Kerr effect in gases. From the premises of the Langevin–Born theory it follows that it must be strictly justified only in those cases where the interaction of molecules may be neglected, or where it is well known. This condition is approximately fulfilled for gases and vapors, but our present knowledge of the liquid and solid states is still quite insufficient for this purpose.

Unfortunately, the experimental data on the Kerr effect in gases are very few. By contrast, liquids and solids have been studied in this respect much better. This is due to the fact that the effect in gases at atmospheric and lower pressures is small in comparison with the effect in liquids and solids and can be measured only with difficulty. Nevertheless, where it has proved possible to carry out the experiments, they have confirmed the theory. The first experimental studies in gases and vapors in a spectral region containing no absorption lines were carried out by Leiser and Gansen\(^{55,56}\) by the relative de-Coudres method. Since then, Sdivessi,\(^{57}\) Stuart,\(^{58}\) and others\(^{59}\) have also made absolute measurements of \(B\) in various gases. Table 1 gives some

TABLE 1

Optical anisotropy of molecules
according to Raman-Krishnan\(^{64}\) and Sirkar\(^{71}\)

Gas or vapor	Depolarization coefficient \(r \cdot 100\)	Approximate value \(\mu \cdot 10^{18}\), observed	Approximate value \(\mu \cdot 10^{18}\), calculated at \(\rho = 0\)	Kerr constant for a gas at 1 atm pressure and at \(20^\circ\mathrm{C}\), \(B \cdot 10^{10}\), observed	Kerr constant for a gas at 1 atm pressure and at \(20^\circ\mathrm{C}\), \(B \cdot 10^{10}\), calculated
Acetic acid . . . .	9.8	0	0	0.24 \((17.5^\circ\mathrm{C})\)	0.28
Nitrous oxide . . . .	12.2	0	0	0.48	0.48
Acetylene . . . .	4.6	0	0	0.29	0.23
Chlorine . . . .	4.1	0	0	0.35	0.30
Nitrogen . . . .	3.75	0	0	—	0.04
Oxygen . . . .	6.45	0	0	—	0.06
Carbon dioxide . . . .	3.4	0	0	—	0.05
Chloroform . . . .	1.0	1.03	1.04	0.90	—
Methyl chloride . . . .	1.52	1.69	1.68	5.45	—
“ ethyl . . . .	1.64	1.98	1.76	8.7	—

experimental data and theoretical values of \(B\), computed from equation (16) by the method described below. It is interesting to note that for polar molecules the Kerr constant is larger.

The Kerr constant is closely connected with the depolarization coefficient \(r\) of longitudinally scattered light (well studied by Rayleigh,\(^{60}\) Raman,\(^{61}\) Kabanov\(^{62}\) and others\(^{63}\)). If one investigates the polarization of light scattered at right angles to the direction of a parallel beam of light passing through a transparent medium, then, in addition to the intense component of the oscillation perpendicular to the incident ray, one can detect, for most substances, a comparatively weak component parallel to the ray. The ratio of the intensities of the second component to the first is denoted by \(r\), and it determines the optical anisotropy of the molecule.*

Many investigators\(^{64-67}\) have determined the Kerr constant \(B\) by measuring the refractive index, the dielectric constant (electric moment), the scattering coefficient \(r\), the temperature, the gas density, and the wavelength. The dependence of the Kerr effect on the structure of the molecules has also been investigated. To find an expression for the Kerr constant in terms of these directly measurable quantities, let us substitute the value of the polarizability \(a\) from equation (7) into equation (16). Then

\[ B=\frac{\pi N (n^2+2)^2(\varepsilon+2)^2}{27 n \lambda}(\theta_1+\theta_2). \tag{18} \]

In the case of a gas or vapor at atmospheric pressure or below, \((n^2+2)\) and \((\varepsilon+2)\) are approximately equal to 3, and equation (18) takes the form:

\[ B=\frac{3\pi N}{\lambda}(\theta_1+\theta_2), \tag{19} \]

and the problem reduces to the determination of \(\theta_1\) and \(\theta_2\).

For nonpolar molecules \(\mu_1=\mu_2=\mu_3=0\), and equation (12) gives \(\theta_2=0\). The value of \(\theta_1\) can be obtained from the depolarization coefficient \(r\) (which is known for a large number of substances), if only \(a_1, a_2, a_3\) in equation (11) can be expressed in terms of \(b_1, b_2\), and \(b_3\), since according to the theory of scattering:

\[ \frac{10r}{6-7r}\left[\frac{3(n-1)}{2\pi N}\right]^2 = (b_1-b_2)^2+(b_2-b_3)^2+(b_3-b_1)^2. \tag{20} \]

* Optical anisotropy is given by the expression:

\[ 2\delta= \frac{(b_1-b_2)^2+(b_2-b_3)^2+(b_3-b_1)^2}{(b_1+b_2+b_3)^2}; \qquad \delta=\frac{5r}{6-7r} \]

for spheres and gases obeying Boyle’s law.

To express \(a_1, a_2, a_3\) in terms of \(b_1, b_2, b_3\), let us adopt the quite plausible assumption of Gans,\(^{68}\) consistent with the experimental data, that

\[ \frac{a_1}{b_1}=\frac{a_2}{b_2}=\frac{a_3}{b_3}=\frac{\varepsilon-1}{n^2-1}, \tag{21} \]

which, on substitution into equations (11) and (16), gives:

\[ \theta_1=\frac{1}{45kT}\left(\frac{\varepsilon-1}{n^2-1}\right) \left[(b_1-b_2)^2+(b_2-b_3)^2+(b_3-b_1)^2\right] \tag{22} \]

and

\[ B=\frac{3(\varepsilon-1)(n-1)^2}{4kT\pi N\lambda(n^2-1)} \cdot\frac{r}{6-7r} \simeq \frac{3(\varepsilon-1)(n-1)}{4\pi N\lambda kT} \cdot\frac{r}{6-7r}. \tag{23} \]

The values of \(B\) in Table 1 were calculated from this formula.

In the case of polar molecules, \(\mu_1, \mu_2\), and \(\mu_3\) are no longer equal to zero, so that \(\theta_2\) must also be determined. In general \(\theta_2\) is considerably greater than \(\theta_1\), and the approximations made in determining \(\theta_1\) do not substantially affect the result.

In polar gases and vapors \(\theta_1\) is obtained in the same way as in nonpolar ones; only the effect caused by the influence of the permanent dipole moments of the molecules on the dielectric constant must, on substitution into equation (21), be subtracted from its observed value, i.e. in equation (21) one should replace \(\varepsilon\) by \(c\), where \(\varepsilon\) and \(c\) are related by the relation \(\varepsilon=c+\frac{a}{T}\). The value of \(c\) in the case of polar and nonpolar molecules is equal to the squares of the corresponding refractive indices extrapolated to wavelength \(\infty\).

To determine \(\theta_2\), it is necessary to know not only the relative magnitudes \(b_1, b_2\), and \(b_3\), but also the angles which they make with \(\mu\), i.e. one must know the “optical ellipsoid” of the molecule and the magnitude and direction of its permanent electric moment \(\mu\). The magnitude of the permanent moment is usually obtained from measurements of the dielectric constant,\(^{69}\) and the “optical form” of the molecule from measurements of the polarization coefficient \(r\) of transversely scattered light.

The optical form of the molecule may in general be not simple, but we shall restrict ourselves to the case where it is represented by a triaxial ellipsoid. Under this restriction the Kerr constant itself gives the orientation of the permanent electric moment with respect to the three principal axes.

Following Raman and Krishnan,\(^{64}\) we shall consider the simplest case, when the optical ellipsoid is a spheroid of revolution. Let \(b_2=b_3\), and let \(\mu\) make an angle \(\rho\) with \(b_1\). Then, projecting,

\[ \mu_1^2=\mu^2\cos^2\rho \quad \text{and} \quad \mu_2^2+\mu_3^2=\mu^2\sin^2\rho \]

and substituting these values, we obtain:

\[ \theta_2=\pm \frac{(n-1)\mu^2}{30\pi Nk^2T^2}(2\cos^2\rho-\sin^2\rho)\cdot \left(\frac{5r}{\delta-7r}\right)^{\frac{1}{2}}, \tag{24} \]

where the \(+\) sign is taken in the case \(b_1>b_2=b_3\), and the \(-\) sign if \(b_1<b_2=b_3\).

As was indicated above, \(\theta_2\) for polar molecules is much greater than \(\theta_1\), so that when \(b_1<b_2=b_3\) and \(\theta_2\) is negative, then, according to equations (18) and (19), \(B\) is also negative. Physically, a negative value of \(B\) means that the permanent electric moment is directed along the axis of minimum optical polarizability, or makes a small angle with it.

Since the permanent moment is assumed to be fixed with respect to the optical ellipsoid, the application of an electric field rotates the molecule in such a way that the axis of least polarizability is directed along the lines of force. As a result \(n_s>n_p\), and \(B\) becomes negative.

But \(B\) can turn out to be negative only for polar molecules, unless one admits that the direction of maximum electric polarizability coincides with the direction of the least optical polarizability, which would contradict the generally accepted views on electric and optical anisotropy. \(^{46,68,70}\) It should be noted that whereas Langevin’s theory, which took into account only induced moments, could in no way explain the appearance of negative \(B\), this was explained and became a logical consequence of the theory as soon as Born also took into account permanent electric moments. The presence of negative Kerr constants well confirms the usual notion that the permanent electric moments of molecules can considerably exceed the induced ones. For nonpolar substances, where \(\theta_2=0\), negative \(B\) have never been observed, which in itself is evidence of the general correctness of the assumptions made in the Langevin–Born theory. As has already been indicated, one of the advantages of formula (24) consists in the fact that it gives a convenient way of calculating the angle \(\rho\) between the permanent electric moment and the direction of the principal axes in simple molecules, provided that \(\mu\), \(r\), and \(B\) are known. For example, Raman and Krishnan showed that for \(\mathrm{HCl}\), \(\rho=0\), whereas for \(\mathrm{CH_2Cl_2}\), \(\rho=\frac{\pi}{2}\). Sirkar and others \(^{68,72}\) found that for some substances \(\rho\) may also have other values. If the angle \(\rho\) is equal to \(0\), \(\frac{\pi}{2}\), or is in general well known, then

it is obvious that the electric moment can be found from the value of Kerr’s constant and \(r\), and, moreover, with considerable accuracy, since \(\mu\) enters equation (24) to a higher power than \(B\) and \(r\); conversely, if \(B\) and \(\mu\) are known, then one can determine the depolarization coefficient \(r\) of the scattered light, the refractive index \(n\), the dielectric constant \(\varepsilon\), and the general form of the optical ellipsoid; one can also calculate the values \(b_1, b_2, b_3\). Namely, if \(\theta_1\) for polar molecules is obtained from equation (22), then equations (12), (18), and (19) determine the expression:

\[ [(\mu_1^2-\mu_2^2)(b_1-b_2)+(\mu_2^2-\mu_3^2)(b_2-b_3)+(\mu_3^2-\mu_1^2)(b_3-b_1)]. \]

If, in addition, the angles between the direction of the permanent electric moment and the three principal axes are known, then the quantities \(\mu_1, \mu_2, \mu_3\) can also be obtained.

Thus the value of a linear function of \(b_1, b_2, b_3\), expressed in terms of the constant \(B\), becomes known. For example, in the case when \(\mu\) coincides with \(\mu_1\), \(\rho=0\), and \(\mu_2=\mu_3=0\), the above-written function takes the form:

\[ (2b_1-b_2-b_3)\mu^2 \]

and, according to (18) and (19), is expressed through \(B, n, \varepsilon\), and \(r\). Further, by equation (20) it is possible to express

\[ (b_1-b_2)^2+(b_2-b_3)^2+(b_3-b_1)^2 \]

through \(r\) and \(n\), assuming that, in addition, equation (7) is given.

Thus \(b_1, b_2, b_3\) can be obtained from three equations containing, besides them, only known quantities. For the few cases in which all the necessary data were known, such a calculation was carried out by Stuart\(^{67,72}\) and applied by him to the investigation of molecular structure. If \(b_1, b_2, b_3\) are known, then Gans’s hypothesis (equation 12) gives directly \(a_1, a_2\), and \(a_3\). The change of \(B\) with temperature has been studied for several gases and vapors and has proved, in general, to agree with the theory. The experimental data, however, are insufficient for final conclusions. The change of \(B\) with wavelength has also been little studied. One can only point out that Havelock’s formula (5) is approximately justified for the visible region. It should be recalled that Havelock’s formula was derived on the basis of a rather special hypothesis and before the Langevin–Born theory; however, if the value \((n^2+2)\) from equation (7) is substituted into equation (16), one obtains:

\[ B=\frac{(n^2-1)^2}{n\lambda}\,\frac{3}{16\pi N a_0^2}\left(\frac{\varepsilon+2}{3}\right)^2(\theta_1+\theta_2), \tag{25} \]

or

\[ B=\alpha\,\frac{(n^2-1)^2}{n\lambda} \tag{26} \]

DOUBLE REFRACTION IN AN ELECTRIC FIELD

(Havelock’s law), provided that \(q_0\), \(\varepsilon\), \(\theta_1\), and \(\theta_2\) do not depend on \(\lambda\). In a few cases, especially in sodium vapor, an “anomalous” Kerr effect is observed in the immediate vicinity of lines or absorption bands. \(^{73,74}\) This, of course, was to be expected also from theory, \(^{75}\) since the refractive index too experiences in these regions the well-known “anomalous” effect. \(^{76}\) Kochermann and Ladenburg found a connection between their results in sodium vapor near the \(D\)-lines and the Stark effect for \(D_1\) and \(D_2\). They obtained fairly good agreement with experiment, if one takes into account the assumptions they had to make.

The investigation of the dependence of \(B\) on the gas density also presents considerable experimental difficulties. Increasing the pressure inside the Kerr cell produces strong stresses in the windows and makes accurate measurement of the double refraction almost impossible.

Nevertheless, Sieveking, \(^{54,77}\) measuring \(B\) in \(\mathrm{CO}_2\), went up to a pressure of \(3\ \mathit{atm}\) and found that over this small interval the Kerr effect, calculated per molecule, does not change. Lyon \(^{78}\) worked with \(\mathrm{CO}_2\) at \(78\ \mathit{atm}\) and likewise came to the conclusion that over the interval from \(10\) to \(50\ \mathit{atm}\) the effect per molecule remains unchanged.

However, quite recently it has become possible to measure \(B\) more accurately by placing the polarizer and analyzer inside the Kerr cell itself. Nonpolar molecules of \(\mathrm{CO}_2\) were investigated up to a pressure of \(0.18\ \mathrm{g/cm^3}\) at a temperature slightly above the critical one, and a small change in \(B\) with increasing density was discovered. This, however, agrees with theory. If one substitutes into equation (18) for \(\mathrm{CO}_2\) the refractive index \(^{79}\) and the dielectric constant \(^{80}\) at the corresponding densities, then the equation gives the change of the Kerr constant found experimentally.

The constant \(B\), measured with high accuracy for high pressures, can serve as a means of investigating the interaction of molecules during their relative approach.

In the author’s opinion, there is a great need for good experimental data on the Kerr effect in polar and nonpolar gases and liquids; the dependence of the effect on wavelength, temperature, and density is also of interest.

Electro-optical dispersion should, if possible, be studied in detail in the infrared and ultraviolet regions, as well as near absorption bands. Temperature and density must likewise be varied within sufficiently wide limits, so that the substance, from the gaseous state—in which the molecules are comparatively far apart and independent of one another—passes into the liquid state. The hypothesis of

electrical and optical anisotropy of the molecules can be rigorously tested only if these data are available. Then an idea of the interaction of the molecules upon transition to the liquid state will also be obtained.

5. The Kerr Effect in Liquids. The experimental investigation of the Kerr effect in liquids presents far fewer difficulties than in gases, but its quantitative explanation is much more difficult. The reason here probably lies in the insufficiency of our knowledge of the liquid state, because the data available for gases agree quantitatively well with theory. Nevertheless, in the case of liquids the theory agrees qualitatively with experiment, and in a few cases even quantitatively.

In general, the experimental data for liquids, although more numerous than for gases, can give only the general character of the phenomenon. In many cases, especially for conducting and easily contaminated liquids, the experimental data are associated with considerable errors. A good example of this is provided by the works of Illiberg,^81 Möller,^82 Tölgyessy^83 and others^84 with nitrobenzene, where very large fluctuations in the value of \(B\) are caused by contamination of the liquid. As for gases, the Kerr effect for polar liquids is, as a general rule, greater than for nonpolar liquids. As was to be expected, the presence of a permanent dipole moment \(\mu\) has a stronger effect here than for gases. This is evident, moreover, already from equation (16), where \((\varepsilon+2)^2\) enters as a factor. In general, the Kerr effect per molecule in gases and vapors is greater than in liquids, but interesting exceptions are encountered.

Raman and Krishnan^49 modified the Langevin–Born theory so that it could explain this difference in the value of \(B\) per molecule of dense liquids. But we shall first briefly set forth the theory of the Kerr effect that is contiguous with the theory already developed by us for gases, i.e., we shall assume that each molecule is anisotropic by itself, but is surrounded by an isotropically distributed polarizable medium. We must express, on the basis of equation (18), \(\theta_1\) and \(\theta_2\) in terms of quantities that can be determined experimentally. We shall first consider the simpler case of nonpolar molecules, since then \(\theta_2=0\).

As for a gas, it is necessary to express the electrical constants of the molecule \(a_1, a_2, a_3\) in terms of its optical constants \(b_1, b_2, b_3\), since the latter can be obtained from light-scattering experiments. Adopting Gans’ assumption,

\[ \frac{a_1}{b_1}=\frac{a_2}{b_2}=\frac{a_3}{b_3}=\frac{\varepsilon-1}{\varepsilon+2}\,\frac{n^2+2}{n^2-1} \tag{27} \]

and substituting into equation (11), we obtain:

\[ \theta_1=\frac{1}{45kT}\frac{(\varepsilon-1)(n^2+2)}{(\varepsilon+2)(n^2-1)} \left[(b_1-b_2)^2+(b_2-b_3)^2+(b_3-b_1)^2\right]. \tag{28} \]

For liquids, several relations have been proposed expressing \(r\) through optical constants. \(^{85,86}\) Within the limits of the approximation given by the theory, one may put:

\[ r=\frac{6\left[(b_1-b_2)^2+(b_2-b_3)^2+(b_3-b_1)^2\right]} {10(kT\beta N)(b_1+b_2+b_3)^2+7\left[(b_1-b_2)^2+(b_2-b_3)^2+(b_3-b_1)^2\right]}, \tag{29} \]

where \(\beta\) is the isothermal coefficient of compressibility of the liquid, and \(b_1+b_2+b_3\) is given by the well-known relation \(^{46}\) in equation (7):

\[ \theta_1=\frac{95}{8\pi N}\frac{(\varepsilon-1)}{(\varepsilon+2)} \frac{(n^2-1)}{(n^2+2)}\frac{r}{6-7r} \tag{30} \]

and equation (17) takes the form:

\[ B=\frac{\beta(n^2-1)(n^2+2)(\varepsilon-1)(\varepsilon+2)} {24\pi n\lambda}\frac{r}{6-7r}. \tag{31} \]

This formula was applied by Raman and Krishnan \(^{86}\) to obtain \(B\) for several nonpolar liquids, and although the theoretical values were, in general, of the same order as the experimental ones, there were also deviations exceeding the errors of measurement.

For polar liquids, the quantity \(\theta_1\) can be estimated in the same way as for nonpolar ones, except that in equation (27) one should substitute, instead of \(\varepsilon\), the square of the refractive index extrapolated to infinite wavelength.

On the other hand, \(\theta_2\) is much more difficult to determine, since one must know the shape of the optical ellipsoid and the magnitude of the electric moment. Meanwhile, there are grounds for supposing that the optical shape of a molecule, appropriate for a gaseous medium, is considerably distorted in a liquid. \(^{87-91}\) These changes of the effective electric moment are especially large in so-called “associated” liquids. It is therefore impossible to apply \(\mu\) and the constants of the optical ellipsoid determined for the gaseous state in order to obtain \(B\) in a liquid. This is confirmed, at least, by the existing large deviations between experimental values of \(B\) and theoretically calculated ones. However, it is possible to estimate \(B\) also in the liquid state, knowing the nature and magnitude of the interaction of the molecules, since \(r\) and \(\mu\) (effective) can easily be measured.

Experiments on the scattering of X-rays in liquids \(^{92-96}\) gave some indications as to the sizes and arrangement of molecules. Further, in some cases it proved possible to trace the change in the refractive index of dielectric-

ical constant upon the transition from the gaseous state to the liquid one. It would be natural for any theory explaining the change of \(B\) in the transition from the gaseous state to the liquid one also to explain the corresponding changes of \(r\) and the deviations from the Clausius–Mosotti and Lorentz–Lorenz relations. From time to time hypotheses have been proposed to explain these phenomena separately, but the most satisfactory general theory was proposed by Raman and Krishnan.^49 Therefore we shall indicate, in general outline, its application to the Kerr effect.

Raman and Krishnan assumed that not only is the molecule itself anisotropic, but that it is also surrounded by anisotropically distributed polarizable matter, i.e., in contrast to the hypothesis of Langevin–Born they allowed that the polarization field surrounding the molecule is anisotropic.

The anisotropy in the immediate vicinity of the molecule is caused, in their view, by the orientation of the surrounding anisotropic molecules. Starting from these assumptions, they obtained expressions for the constant \(B\), which we present in final form, referring the reader to the detailed derivation in the original paper:

\[ B=\frac{n^{2}-1}{4n\lambda}\frac{3}{a_{0}}(\theta'_1+\theta'_2), \tag{32} \]

where

\[ \theta'_1=\frac{1}{45kT}\bigl[(A_1-A_2)(B_1-B_2)+(A_2-A_3)(B_2-B_3)+ +(A_3-A_1)(B_3-B_1)\bigr]; \tag{33} \]

\[ \theta'_2=\frac{1}{45k^2T^2}\bigl[(B_1-B_2)(M_1^2-M_2^2)+(B_2-B_3)(M_2^2-M_3^2)+ +(B_3-B_1)(\mu_3^2-\mu_1^2)\bigr], \tag{34} \]

where

\[ \begin{aligned} A_1&=a_1(1+p_1S_e)^2, & B_1&=b_1(1+q_1S_0),\\ A_2&=a_2(1+p_2S_e)^2, & B_2&=b_2(1+q_2S_0),\\ A_3&=a_3(1+p_3S_e)^2, & B_3&=b_3(1+q_3S_0), \end{aligned} \tag{35} \]

\[ \begin{aligned} M_1&=\mu_1(1+p_1S_e),\\ M_2&=\mu_2(1+p_2S_e),\\ M_3&=\mu_3(1+p_3S_e), \end{aligned} \tag{36} \]

here \(S_0\) and \(S_e\) are the optical and electrical susceptibilities, \(p_1,p_2,p_3\) and \(q_1,q_2,q_3\) the so-called “polarization constants” of the electrostatic and light fields along the three principal axes, while all the other quantities have the same meanings as in equations (11), (12), and (16).

It is evident that when

\[ p_1=p_2=p_3=q_1=q_2=q_3=\frac{4}{3}\pi, \]

then, as for an isotropic distribution around the molecule, equation (32) reduces to equation (16).

DOUBLE REFRACTION IN AN ELECTRIC FIELD

Starting from this same theory, it can be shown,^{95,97} that the depolarization coefficient of light scattered in a liquid is

\[ r=\frac{6nN\left[(B_1-B_2)^2+(B_2-B_3)^2+(B_3-B_1)^2\right]} {90\left(\frac{n^2-1}{4\pi}\right)^2 RT\rho +7nN\left[(B_1-B_2)^2+(B_2-B_3)^2+(B_3-B_1)^2\right]} . \tag{37} \]

To obtain the Kerr constant according to the theory set forth above, it is necessary, for nonpolar molecules, to determine all \(A_{1,2,3}\) and \(B_{1,2,3}\), and, in the case of polar molecules, also \(\mu_{1,2,3}\).

In the simple nonpolar case, when \(b_2=b_3\), the quantity \(b'\) can be determined from the values of \(r\) and \(n\) in the vapor-like state. Then, according to Gans’ hypothesis, the quantity \(a\) will also be determined.

The calculation of \(p\) and \(q\) generally presents great difficulties, but with the aid of certain assumptions they can be found for special cases. For such a calculation the anisotropy of the polarization field is approximately represented in the form of a surface charge on an ellipsoidal cavity surrounding the molecule. The magnitude of the polarization field at the center of such an ellipsoid is equal to the field produced by a surface charge equal to \(-SE\cos\theta\) per unit surface of the ellipsoid, where \(\theta\) is the angle between the normal and the direction of the vector \(E\).

If, for example, it is assumed that the cavity has the form of an ellipsoid of revolution with semiaxes \(b=c=a(1-e^2)^{1/2}\), where \(e\) is the eccentricity, then the polarization constants can be obtained^{49,98} from the well-known equations:

\[ p_1=q_1=4\pi\left(\frac{1}{e^3}-1\right) \left(\frac{1}{2e}\lg\frac{1+e}{1-e}-1\right), \tag{38} \]

\[ p_2=p_3=q_2=q_3 =2\pi\left(\frac{1}{e^3}-\frac{1-e^2}{2e^3}\lg\frac{1+e}{1-e}\right), \]

provided only that \(a\), \(b\), and \(c\) are known. These semiaxis magnitudes are obtained in experiments on the scattering of light in liquids,^{95} and therefore the formulas given above make it possible to calculate \(B\).

The influence of the anisotropy of the polarization field is clearly seen in the example of pentane, for which \(B\), calculated according to the Raman–Krishnan theory, proved to be \(5.6\cdot10^{-9}\), according to the Langevin–Born theory \(17.9\cdot10^{-9}\), while the experimental value proved to be \(5.0\cdot10^{-9}\). Let us note that, in the case of pentane, the anisotropy of the polarization field decreases the double refraction. This is a completely general rule, provided that no associations are formed in the liquid. The Raman and Krishnan theory, in those cases in which it was applied, proved to be in good qualitative accord-

agrees with experiment and for dense liquids proves to be considerably better than the theory of Langevin–Born.

Dependence of Kerr’s constant in a liquid on temperature. Electric double refraction is caused, as we already know, by the orienting influence of the applied external electric field on the permanent and induced moments of anisotropic molecules. This orientation of molecules is continually disturbed by thermal motion, so that with an increase in temperature, when collisions between molecules become more frequent and stronger, the number of oriented molecules decreases. Therefore the Kerr effect too must decrease with increasing temperature. The change of \(B\) with temperature has been studied in many liquids by various investigators, \(^{99-107}\) and the experimental data have well confirmed the theory.

The dependence of \(B\) on temperature, as well as changes in the scattering of X-rays caused by an electric field, are the best proof that orientation of molecules actually takes place. \(^{108}\) In nonpolar liquids \(B\) changes with temperature less than in polar ones. This agrees well with the fact that \(T\) enters equation (11) to the first power, whereas in equations (12) and (34) for \(\theta_2\) (which for nonpolar molecules is equal to zero) it enters to the second power. Recent investigations of the dependence of the depolarization coefficient \(r\) on temperature show that \(r\), reduced to one molecule, usually increases with temperature. From this it may be concluded that the anisotropy of the molecules themselves remains almost constant, while the general increase of the anisotropy of the substance with temperature is caused by a decrease of the anisotropy of the polarization field produced by the surrounding molecules. According to the preceding theory, a decrease of the anisotropy of the polarization field increases the Kerr effect, so that a careful study of the dependence of \(B\) on temperature may give indications of a change in the polarization field. Indeed, the experimental data show that \(B\) changes with temperature more strongly than follows from equation (16), where the polarization field is assumed to be homogeneous and, consequently, unchanged.

Let us note that one of the few exceptions is nitrobenzene. Its peculiarity is manifested not only with respect to \(B\), but also with respect to \(r\), which probably indicates polymerization of the molecules. Of all liquids it has the largest Kerr constant for the visible region of the spectrum.

Dependence of \(B\) on wavelength. As early as 1892 Kerr discovered a change in the magnitude of double refraction with change in wavelength, but precise quantitative

the data were obtained only in the works of Bléculua \(^{109}\) and Mac-Comb. \(^{14}\) These investigators found that Havelock’s law (26) agrees excellently with the experimental data. This was also confirmed in later investigations. \(^{110}\)

However, in one of their most recent works, Szivessy and Dierkesmann, \(^{111}\) investigating the Kerr phenomenon in the visible and ultraviolet regions, came to the conclusion that the curve of electro-optical dispersion is not represented with complete accuracy by Havelock’s formula. Ingersoll, \(^{112}\) in an important, recently begun investigation of the Kerr effect in the infrared region, found that for \(\mathrm{CS}_2\), from \(0.5\,\mu\) to \(2\,\mu\), Havelock’s formula is justified. The study of electro-optical dispersion is of considerable interest, since it indicates which of the quantities entering equations (16) and (32) do not depend on the wavelength. For example, if for some region of the spectrum Havelock’s law is valid, then, according to equation (29), \(a_1\), \(\varepsilon_1\), \(\theta_1\), and \(\theta_2\) either do not depend on \(\lambda\) in this region or vary in such a way as to compensate one another completely.

Absolute values of \(n_p\) and \(n_s\). The question of the absolute values of the refractive indices in the directions perpendicular and parallel to the field was raised long ago. By the methods described above, which give the magnitude of the birefringence, it was possible to measure only the difference \(n_p-n_s\), but not the values of \(n_p\) and \(n_s\) themselves. The general method for determining these refractive indices consists in the following: the Kerr cell is placed in one arm of an interferometer (Michelson, Jamin, or others) in such a way that the light passing through can be plane-polarized parallel or perpendicular to the lines of force. In the other arm a glass cell with a liquid having the same optical length is placed, and in such a way that the light passing through it is not subjected to the action of the field. This is necessary if it is desired to use the bands of undecomposed light for the readings. When a field is applied to the Kerr cell, the displacement of the bands for vibrations perpendicular and parallel to the lines of force gives \(n_p-n\) and \(n-n_s\).

This also gives \(n_p\) and \(n_s\) separately, since \(n\) can be measured independently. Equation (6) gives the expression derived by Havelock

\[ \frac{n_p-n_s}{n_s-n}=-2. \]

Focht \(^{113}\), considering that birefringence is caused not by the orientation of molecules by an external field but by a change in intra-atomic and intramolecular forces, obtained that

\[ \frac{n_p-n}{n_s-n}=3. \]

Generalizing this theory, Enderle^114 found that this ratio depends on \(\lambda\) and approaches 3 for frequencies small in comparison with electron frequencies.

Dividing equation (13) by (14), one may obtain Havelock’s formula (6) from the Langevin–Born theory, provided one neglects the terms due to electrostriction. However, these terms are usually not so small that they can be discarded, so that equation (6) may also fail for constant applied fields. But usually, when the field is switched on, the Kerr effect appears before electrostriction is established, so that \(n_p - n\) and \(n_s - n\) can be measured before its appearance. In this way one can avoid heating of the liquid, which is always an interfering factor in these measurements.

Experiments carried out by many different investigators,^39–41 have shown that equation (6) is approximately justified.

6. Relaxation time of the Kerr effect. In his first experiments, performed with glass, Kerr observed that the double refraction appears and disappears not immediately, but several seconds after the field is switched on or off. But in experiments with liquids he was unable to observe any lag of the effect behind the field. Later, Blondlot^115 showed that for the liquids studied the relaxation time is less than \(2.5 \cdot 10^{-5}\) sec., and Leumann and Abraham^116 came to the conclusion that the Kerr effect with \(\mathrm{CS}_2\) lasts no longer than \(10^{-8}\) sec. after the field is switched off. Other experimenters^117,118 found for various liquids, including also \(\mathrm{CS}_2\) and nitrobenzene, relaxation times in the range \(10^{-8}\)—\(10^{-10}\) sec.

However, Lawrence, together with the author,^119,120 after a series of experiments, found no experimental confirmation of the existence of such a relaxation time of the Kerr effect for liquids with relatively small molecules and low viscosity. Recently Raman^121 arrived essentially at the same conclusion. On the other hand, for polar viscous liquids comparatively large relaxation times were found.^122–124 Applying a high-frequency electric field to a Kerr cell filled with octyl alcohol, having \(B < 0\), Raman and Sirkar found that with increasing field frequency the double refraction first decreases, then disappears altogether, then appears and again begins to increase. Sirkar^124 found that in alcohol (undecyl alcohol) the relaxation time of the Kerr effect lies between \(10^{-8}\) and \(10^{-9}\) sec.

Experiment shows that, as a rule, the Kerr constant in polar liquids is greater than in nonpolar ones, and therefore in the former \(\theta_1\) is small in comparison with \(\theta_2\). If this were not

Thus it would be difficult to explain satisfactorily, on the basis of the Langevin–Born or Raman–Krishnan theory, the existence of a negative Kerr constant. This fact shows that the increase of the Kerr effect caused by the influence of the field on permanent moments is considerably greater than that caused by its influence on the induced dipole moments of the molecules.

Debye^125 derived a formula by which one can determine the time \(t\) during which, after the field is switched off, the oriented moments receive an arbitrary distribution. For molecules having a spherical form this will be:

\[ t=\frac{4\pi \eta a^3}{kT}, \tag{39} \]

where \(\eta\) is the coefficient of viscosity, \(a\) is the radius of the molecule, \(k\) is Boltzmann’s constant, and \(T\) is the absolute temperature. Although equation (39) is correct only approximately, owing to the simplifying assumptions that the molecules have a spherical form and that Stokes’ law is applicable to particles as small as molecules, it gives the correct order of magnitude for \(t\) and the correct course of its variation with increasing \(\eta\) and \(a\).

It also becomes obvious that, with sufficiently rapid switching on and off of the electric field, the Kerr effect must decrease and disappear. Indeed, if a liquid having a negative Kerr constant is placed in a rapidly oscillating electric field, then the negative double refraction caused by the orientation of the permanent moments in the molecules will decrease to zero as the frequency increases, whereas the other, ordinarily positive, part of the Kerr effect, which is due to the influence of the field on the induced moments, will not change so strongly.

According to Raman and Sirkars, the Kerr effect in negative liquids at first decreases with increasing frequency of the applied field, passes through zero, and then begins to increase again. Thus, with increasing frequency of the field, the electric double refraction changes from negative to positive. Bramley^126 described several curious phenomena occurring in water at certain definite frequencies. At first he thought that these frequencies produced an enhanced Kerr effect, but later came to other conclusions. He found that light, on passing through a Kerr cell with an applied oscillating field, is scattered very intensely and that certain spectral lines are displaced toward the red part. It turned out that this phenomenon, in general, is not related to the Kerr effect, but is caused by a change in the concentration of salts dissolved in the water. Sirkar^124 also discovered peculiar phenomena in alcohols

for those frequencies at which strong electric absorption occurred. At these frequencies the liquids transmitted light between two crossed Nicols. The nature of this phenomenon has not yet been clarified, but apparently it should be attributed not to the Kerr effect, but rather to some special kind of scattering.

The short relaxation time of the Kerr effect in non-viscous liquids has found application in various branches of physics. A Kerr cell between two crossed Nicols, as, for example, is shown in Fig. 1, can serve as an excellent light relay, almost instantaneously obeying electrical control. With the aid of Kerr cells, flashes of light of duration \(10^{-9}\) sec were produced, and phenomena of duration \(10^{-8}\) sec were investigated.

Applications of Kerr cells to the measurement of small time intervals and to the production of instantaneous light flashes have been described repeatedly in the literature,\(^{127,128}\) so that in the present article we shall not dwell on this.

7. Further applications. The Kerr effect in mixtures of liquids is still insufficiently studied, but the available data\(^{129,130}\) show that it cannot be calculated from the effects of the constituent parts according to their percentage content. Such investigations in mixtures are of great interest, because, by extrapolating to very weak concentrations the graph of the dependence of the Kerr constant on concentration, one can estimate this effective Kerr constant when the given molecule is surrounded not only by identical molecules, but also by different molecules. Of particular importance are the results obtained in solutions of the liquid under investigation in another normal, non-polar liquid, i.e. when the anisotropic molecules under study are practically surrounded by molecules with small anisotropy.*

In the case of an almost isotropic solvent, the Kerr constant according to Raman–Krishnan should increase and approach the value determined by the Langevin–Born theory.

Electric double refraction in mixtures of optically active substances and its theory were investigated in detail by P. de Mallemann.\(^{131}\) Solutions of solids in liquids with a small Kerr constant are also of great interest, since they make it possible to clarify certain questions of intermolecular interaction. No less important are experiments on the investigation of the orienting action of the field on small particles in suspensions. The first major work in this area belongs to Cotton and Mouton,\(^{132}\) who observed the Kerr phenomenon in a solution of hydrate,

* Experiments on the scattering of light show that optically completely isotropic substances do not exist.

iron oxides. Subsequently, dispersed crystalline substances were studied, such as, for example, quartz, spar, etc., suspended in organic liquids.^133 Recently the Kerr effect has been applied to colloids^134 and to the study of the crystallographic properties of small, finely comminuted particles.^135 It is supposed that the birefringence in them is also caused by the orientation of optically inhomogeneous particles under the action of an external field. In general, the orientation in these cases is probably caused by the shape of the particles (crystals) and by their different dielectric properties in different directions.

The influence of shape on orientation in an electric field is also demonstrated by an electrical and mechanical effect. Thus, for example, if the dielectric constant of a particle is greater than that of the liquid in which it is suspended, then the particle will tend to turn its long axis along the lines of force, while the so-called “cataphoretic” motion exerts a mechanical effect on its position. Although the orientation of the particles is, of course, disturbed by Brownian motion, nevertheless in some cases an almost complete regrouping can be achieved. It is interesting to note that, from the lag of the birefringence behind the electric field, one can determine the sizes of these ultramicroscopic particles. In many cases, in addition to electric birefringence, such mixtures and suspensions also exhibit dichroism, i.e., they have different absorption coefficients for component light vibrations parallel and perpendicular to the lines of force. This phenomenon is usually encountered in limited spectral regions, where selective absorption occurs.

8. The Kerr effect in solids. The study of the Kerr effect in solids is complicated by birefringence caused by electrostriction and by stresses due to temperature gradients. In nonconducting bodies the latter may be neglected, but the birefringence caused by electrostriction is usually superposed on the Kerr effect. In contrast to liquids, the relaxation time of the Kerr effect in solids is large, and it is difficult to separate the Kerr effect from the effect caused by electrostriction, as Potenier successfully did for liquids. However, the birefringence caused by electrostriction can be measured experimentally and even calculated theoretically. Taughard,^136 calculating \(B\) for various kinds of glass, arrived at the following expression:

\[ \frac{n_p - n_s}{\lambda} = E^2\left(B - \frac{c\varepsilon}{8\pi}\right), \tag{40} \]

where \(c\) is a substance-dependent constant that can be determined by special experiments. For different kinds of glass he found that Kerr’s constant increases with increasing percentage content of lead, whereas for pure silicates \(B\) is very small.

Amorphous solids have been studied, but comparatively little. Electrical double refraction in quartz crystals was observed by Kerr[^137] soon after his discovery of the effect in amorphous bodies. He studied it for the case of an electric field parallel and perpendicular to the optical axes. These investigations have since been considerably extended and supplemented by Röntgen,[^138] Kundt,[^139] Pockels[^140] and others.[^141],[^142] It turned out that the phenomenon is closely connected with the type and structure of the crystals. A theory of this effect was developed, relating its magnitude to various constants of the crystal.[^142],[^140],[^143]

C. Double refraction in a magnetic field

1. The Cotton–Mouton effect

In 1901 Kerr[^144] noticed that finely divided \(Fe_3O_4\), suspended in water, becomes doubly refracting when light passes perpendicular to the lines of force of a magnetic field. At about the same time Majorana[^145] independently discovered this phenomenon in various colloidal solutions of iron. Several years later Cotton and Mouton[^146] began a detailed study of these phenomena, which led to the discovery of a very important effect bearing their name. They found that very many pure liquids become doubly refracting when light passes perpendicular to the lines of force of a magnetic field,[^146]–[^148] i.e., isotropic liquids placed in a magnetic field acquire the optical properties of uniaxial crystals, with optical axes directed along the lines of force. The effect is, in general, not large, is easily masked by the Faraday effect (rotation of the plane of polarization when light passes through a medium parallel to the lines of force of the magnetic field), and requires special analysis of the emerging light.

Soon Cotton and Mouton were able to show that the new effect is analogous to Kerr’s electro-optical effect and obeys relations of the same type. If \(n_p\) and \(n_n\) are the refractive indices of the components of the light vibration parallel and perpendicular to the lines of force of the magnetic field, then the phase difference (which gives the magnitude of the double refraction) after traversing a path \(l\) in a homogeneous magnetic field \(H\) will be:

\[ D=\frac{2\pi l\left(n_p-n_n\right)}{\lambda}=2\pi ClH^2, \tag{41} \]

where \(\lambda\) is the wavelength of the transmitted light and \(C\) is the Cotton–Mouton constant. \(C\) may be positive or negative and varies depending on the substance, the wavelength, and the temperature. The validity of this Cotton–Mouton law has been confirmed by many investigations.\(^{149,150}\)

2. Methods of experimental study. Experimental methods for studying the Cotton–Mouton effect are quite analogous to those used for the Kerr effect (Fig. 1), except, of course, that a magnetic field is applied instead of an electric one. In both cases plane-polarized monochromatic light passes through the medium perpendicular to the lines of force. (To obtain the maximum effect, the plane of polarization of the light must make an angle of \(45^\circ\) with the direction of the field.) The resulting double refraction is measured by the various standard methods mentioned earlier. Here, however, special precautions are necessary in order to separate the almost unavoidable superposed Faraday effect. Experimentally it is rather difficult to direct a beam of polarized light exactly at right angles to the lines of force. Therefore there is always a weak component of the field parallel to the path of the light beam. As a result, whatever method is used to analyze the emerging polarized light, it is always necessary to separate the rotation of the plane of polarization due to the rather strong Faraday effect from the comparatively weak Cotton–Mouton effect. (It should be noted that in the case of the Kerr effect this difficulty plays no role, since any rotation of the plane of polarization on passing through a medium parallel to the electric field—that is, the electrical analogue of the Faraday effect—if it exists at all, is so insignificant that it is not observable.) Methods for analyzing the emerging light that make it possible to separate the Cotton–Mouton effect from the Faraday effect have been described in detail in the literature.\(^{151—156}\) Equation (41) shows that the magnetic field must be not only sufficiently intense, but also, as far as possible, long and homogeneous. As already indicated, the Cotton–Mouton effect is small in many substances, so that specially designed magnets with large pole pieces are required to obtain good results.\(^{157,158}\) All this leads to the fact that investigations in this field are carried out by a very limited number of laboratories.

3. Theory of the Cotton–Mouton effect (Langevin–Born). The theory of the Cotton–Mouton effect is basically analogous to the theory of the Kerr effect. Indeed, in almost all theoretical works these two phenomena were considered together; only in the formula giving the Kerr constant were the corresponding magnetic quantities substituted for the electric quantities.

magnetic. The Langevin–Born theory can be applied also to the Cotton–Mouton effect if it is assumed that, in addition to electric and optical anisotropy, the molecules also possess magnetic anisotropy. If the molecules are magnetically anisotropic, then the influence exerted by the magnetic field on their permanent and induced moments will cause their orientation. This orientation, of course, is disturbed by thermal motion, so that a certain equilibrium position is reached in which the direction of the greatest component of the magnetic moment of the molecule coincides predominantly with the direction of the magnetic field, and therefore, since each molecule is anisotropic, the whole medium as a whole also becomes birefringent. It can be shown \(^{42,45,150,159}\) that, similarly to the Kerr constant (equation 16), the Cotton–Mouton constant is given by the equation:

\[ C=\frac{(n^{2}-1)(n^{2}+2)}{4n\lambda}\,\frac{\theta_{1}+\theta_{2}}{a_{0}} \left(1+\frac{4}{3}\pi S_{m}\right)^{2}, \tag{42} \]

where, as before, \(n\) is the refractive index of the medium outside the magnetic field, \(\lambda\) is the wavelength, \(a_{0}\) is the polarizability, and \(S_{m}\) is the magnetic susceptibility.

\[ \theta_{1}=\frac{1}{45kT}\left[(w_{1}-w_{2})(b_{1}-b_{2})+(w_{2}-w_{3})(b_{2}-b_{3})+ (w_{3}-w_{1})(b_{3}-b_{1})\right], \tag{43} \]

\[ \theta_{2}=\frac{1}{45k^{2}T^{2}}\left[(m_{1}^{2}-m_{2}^{2})(b_{1}-b_{2})+(m_{2}^{2}-m_{3}^{2})(b_{2}-b_{3})+ (m_{3}^{2}-m_{1}^{2})(b_{3}-b_{1})\right], \tag{44} \]

where \(k\) is the Boltzmann constant, \(T\) the absolute temperature, \(b_{1}, b_{2}, b_{3}\), as in equations (11) and (12), are the moments induced in the molecule along the three principal axes of optical anisotropy by a unit electric force acting along these three directions; \(w_{1}, w_{2}, w_{3}\) are the magnetic moments induced in the molecule by a unit magnetic force in these same directions; \(m_{1}, m_{2}, m_{3}\) are the components of the permanent magnetic moment along these same directions. Since in many substances \(S_{m}\) is very small, equation (42) can be rewritten in the form:

\[ C=\frac{(n^{2}-1)(n^{2}+2)}{4n\lambda}\cdot\frac{\theta_{1}+\theta_{2}}{a_{0}}. \tag{45} \]

Let us now consider the case when \(m=0\) and, consequently, by equation (44), \(\theta_{2}=0\). If the value of \(a_{0}\) from the Lorentz–Lorenz relation (7) is substituted into equation (45), then

\[ C=\frac{3(n^{2}-1)^{2}}{80\pi N\lambda kT}\cdot \frac{(w_{1}-w_{2})(b_{1}-b_{2})+(w_{2}-w_{3})(b_{2}-b_{3})+(w_{3}-w_{1})(b_{3}-b_{1})}{(b_{1}+b_{2}+b_{3})^{2}}. \tag{46} \]

To determine \(C\) from equation (46), which contains \(w_1, w_2, w_3\) and \(b_1, b_2, b_3\), one must know the optical and magnetic constants of the molecule along its axes of anisotropy. Data concerning optical anisotropy can be obtained from the Kerr effect and scattering experiments, but the magnetic anisotropy is best elucidated by the Cotton–Mouton effect itself. Since this magnetic anisotropy is one of the most important factors determining the structure of the molecule, it is necessary to indicate here how it can be obtained from the Cotton–Mouton constant \(C\). Here, just as in the Kerr effect, the Langevin–Born theory is strictly valid only for gases; but here, for them, experimental data are almost entirely lacking (except in the immediate vicinity of an absorption line or band). Therefore, for calculations one will have to restrict oneself to the case of liquids, which at present can be done only approximately. Raman and Krishnan proposed modifications of the theory set forth above which make it more applicable to liquids, but at the same time they considerably complicate it and therefore will be considered below.

We shall begin with simple cases, when the electric and magnetic anisotropy of the molecule can be represented by ellipsoids of revolution, so that \(b_1=b_2\) and \(w_1=w_2\). Then from (45):

\[ C=-\frac{(n^2-1)(n^2+2)}{60n\lambda kT}\cdot \frac{b_1-b_2}{2b_1+b_2}\,[3w_3-(w_1+w_2+w_3)], \tag{47} \]

on the other hand, the optical anisotropy gives:

\[ \delta=\frac{(b_1-b_2)^2+(b_2-b_3)^2+(b_3-b_1)^2}{2(b_1+b_2+b_3)^2} =\frac{(b_1-b_3)^2}{(2b_1+b_3)^2}, \tag{48} \]

therefore,

\[ w_1+w_2+w_3=3\frac{S_m}{N}, \tag{49} \]

where \(S_m\) is the magnetic susceptibility per unit volume. Thus

\[ C=-\frac{(n^2-1)(n^2+2)}{60n\lambda kT}\, 3\left(w_3-\frac{S_m}{N}\right)\delta^{\frac12}, \tag{50} \]

or

\[ C=-\frac{(n^2-1)(n^2+2)}{60n\lambda kT}\, 2(w_3-w_1)\delta^{\frac12}, \tag{51} \]

or

\[ \frac{w_3}{w_1}= \frac{S_m\delta^{\frac12}(n^2-1)(n^2+2)-20NCn\lambda kT} {S_m\delta^{\frac12}(n^2-1)(n^2+2)+10NCn\lambda kT}; \tag{52} \]

this last equation makes it possible to determine the “magnetic anisotropy”\(^{162}\) \(\frac{w_3}{w_1}\) of the molecule, since all the quantities …

the quantities on the right-hand side of the equation are known, or can be measured.¹⁶⁰, ¹⁶¹ Besides the ratio $\dfrac{w_3}{w_1}$ from equation (52), from equations (50) and (51) one can also obtain the absolute values of $w_1$ and $w_3$. Further, it should be noted that if the quantities $b_1, b_2, b_3$ are determined (for example through the Kerr constant, as was indicated above), then equations (46) and (49) also give the quantities $w_1$ and $w_3$, if $C$ and $S_m$ are known and $w_1 = w_2$. However, even if these assumptions cannot be made, i.e. if $w_1, w_2, w_3$ are different, equations (46) and (49) nevertheless give the magnitude of the ratios between any components of $N$, taken pairwise.

Table 2 gives the magnitudes of the magnetic anisotropy calculated by Ramanathan by the methods set forth above. These values are only approximate, since the assumption made in deriving equation (52) that the molecule has an axis of symmetry may not prove valid in all cases.

Nevertheless, they do give an idea of the order of magnitude of the magnetic anisotropy. From the table one may observe that the Cotton–Mouton constant $C$ for aromatic compounds is greater than for aliphatic ones. This entirely general rule explains the fact that magnetic double refraction in many aliphatic compounds was measured only in the very most recent years. Likewise, the magnetic anisotropy in aromatic compounds is much greater than in aliphatic ones. Table 2 and equation (52) indicate that the insignificant Cotton–Mouton effect in aliphatics is due not to their small optical anisotropy, but rather to their very small magnetic anisotropy. The table permits one to make one more interesting observation. In substances with positive double refraction $w_3 > w_1$, while in substances with negative double refraction $w_3 < w_1$, i.e., as is seen from equation (47), for positively refracting diamagnetic substances the direction of maximum optical susceptibility coincides with the direction of minimum magnetic susceptibility (or makes a small angle with it). For negatively refracting substances the reverse is observed, i.e. the direction of the axis of maximum optical susceptibility coincides with the direction of likewise maximum magnetic susceptibility.

Special attention should be paid to the fact that saturated compounds have negative Cotton–Mouton constants, while unsaturated compounds containing a carbonyl group, as well as cyclic compounds such as, for example, benzene, have positive $C$. Indeed, Ramanathan ascribes negative double refraction to a saturated chemical bond and positive double refraction to an unsaturated one. However, as he himself admits, the data are still far from sufficient to make it possible

TABLE 2

Values of the magnetic anisotropy \(\dfrac{w_3}{w_1}\) according to Raman. Values of the optical anisotropy \(\delta\) are taken from Rao’s work

Substance	\(\delta \cdot 10^3\) in liquid	\(C \cdot 10^{14}\) of liquid	\(\dfrac{w_3}{w_1}\)
Saturated hydrocarbons:
1) Pentane	3.1	−1.8	0.96
2) Heptane	2.3	−2.5	0.91
3) Octane	2.2	−3.0	0.89
Water and saturated alcohols:
1) Water	5.53	−1.1	0.81
2) Methyl alcohol	3.90	−1.8	0.75
3) Ethyl alcohol	2.2	−1.1	0.88
4) Propyl alcohol	2.1	−1.1	0.91
5) Butyl alcohol	2.1	−2.2	0.84
Ethers:
1) Ethyl ether	3.2	−2.2	0.88
Ketones:
1) Acetone	8.6	4.1	1.28
2) Diethyl ketone	13.4	2.7	1.06
Fatty acids:
1) Formic	47.0	6.5	1.29
2) Acetic	36.2	2.7	1.04
3) Propionic	20.1	2.7	1.06
4) Butyric	16.5	1.8	1.04
Esters of fatty acids:
1) Ethyl ester of formic acid	11.3	0.7	1.01
2) Propyl ester of formic acid	5.7	2.7	1.10
3) Ethyl ester of acetic acid	7.4	1.0	1.04
4) Propyl ester of acetic acid	6.4	1.4	1.05
Aromatic compounds:
1) Benzene	22.5	75.0	2.10
2) Toluene	21.5	67.1	1.94
3) Metaxylene	24.0	68.3	1.80
4) Paraxylene	26.0	65.3	1.80
5) Chlorobenzene	26.5	81.4	1.90
6) Bromobenzene	31.5	72.7	1.50
7) Nitrobenzene	—	235.0	—

to establish this as a rule. Sherer[^163] recently investigated the magnetic double refraction in hydrocarbons of the series \(C_nH_{2n+2}\) and \(C_nH_n\) and also came to the conclusion that the simple \(C—H\) bond gives an effect opposite to that of the double bond.

The table indicates, moreover, certain interesting facts, such as, for example, that the negative magnetic anisotropy increases with increasing length of the hydrocarbon chain; that the radicals \(CH_3\) and \(OH\) predominantly give negative magnetic refraction; that the positive magnetic anisotropy of cyclic hydrocarbons is large in comparison with the negative anisotropy of chain hydrocarbons. Thus, for example, for benzene the magnetic susceptibility of the molecule along the axis perpendicular to the ring is 2.1 times greater than along the axis lying in the plane of the ring. Raman and Krishnan[^164] calculated the value \(C\) for benzene derivatives, assuming that the expression \(\left[w_3-\dfrac{S_m}{N}\right]\) in equation (50) is the same both for benzene and for its derivatives. This is true only approximately, but the calculated and observed values agree in order of magnitude. The calculation shows that the anisotropy of benzene derivatives is in general large in comparison with the anisotropy of the individual substituent groups.

By raising the temperature of substances that are solid at room temperature above the melting point and observing their transition into the isotropic liquid state, Salsenio[^165] found that the magnetic double refraction increases with increasing number of benzene rings in the molecule and, thus, for example, the Cotton–Mouton effect in phenanthrene is greater than in naphthalene, and that in turn is greater than in phenol.[^166]

4. Raman–Krishnan theory for dense liquids. This more exact theory of the Cotton–Mouton effect differs from that set forth above only in that it introduces the anisotropy of the field of the molecule, caused by the influence of neighboring molecules. We recall that the Langevin–Born theory assumed this field to be isotropic. In order for this to be correct, it is necessary that the anisotropic molecules surrounding each individual molecule not only be symmetrically distributed, but also have, in addition, arbitrary orientation.

Scattering experiments have established that in the majority of substances the molecules are anisotropic, and scattering of X-rays in liquids has shown that the molecules around each given molecule are not oriented arbitrarily, although in the substance as a whole the orientation is arbitrary. Moreover, the inconstancy of the expressions \((n^2-1)(n^2+2)\) and \(\dfrac{\varepsilon-1}{\varepsilon+2}\) upon transition from the gaseous state to the liquid likewise indi—

DOUBLE REFRACTION IN AN ELECTRIC FIELD

They point to the inaccuracy of the assumptions of the Langevin–Born theory. Raman and Krishnan give an expression analogous to equation (32). In general,

\[ C=\frac{n^2-1}{4\pi n\lambda}\cdot \frac{3(\theta'_1+\theta'_2)}{a_0}; \tag{53} \]

\[ \theta'_1=\frac{1}{45kT}\left[(W_1-W_2)(B_1-B_2)+(W_2-W_3)(B_2-B_3)+\right. \]

\[ \left.+(W_3-W_1)(B_3-B_1)\right]; \tag{54} \]

\[ \theta'_2=\frac{1}{45k^2T^2}\left[(M_1^2-M_2^2)(B_1-B_2)+(M_2^2-M_3^2)(B_2-B_3)+\right. \]

\[ \left.+(M_3^2-M_1^2)(B_3-B_1)\right]; \tag{55} \]

\[ \begin{aligned} W_1&=w_1(1-p_1S_m)^2, & B_1&=b_1(1+q_1S_0),\\ W_2&=w_2(1-p_2S_m)^2, & B_2&=b_2(1+q_2S_0),\\ W_3&=w_3(1-p_3S_m)^2; & B_3&=b_3(1+q_3S_0); \end{aligned} \tag{56} \]

\[ M_1=m_1(1+p_1S_m);\quad M_2=m_2(1+p_2S_m);\quad M_3=m_3(1+p_3S_m), \]

where, as before, \(S_m\) and \(S_0\) are the mean values of the magnetic and optical susceptibilities, \(q_1,q_2,q_3\) are the electro-optical, and \(p_1,p_2,p_3\) the magnetic coefficients of polarization along the three principal axes of the molecule. In order to determine \(C\) from equation (53), one must know \(\theta'_1\) and \(\theta'_2\). Unfortunately, experimental data are available only in a few cases, and even then are not free from objections. In those simple cases in which \(\theta'_2=0\), and the optical and magnetic anisotropy are represented by ellipsoids of revolution, \(B_1,B_2,B_3\) can be obtained from scattering data, refractive indices, and Kerr constants. \(q_1,q_2,q_3\) can be estimated from the scattering of X-rays in liquids or by calculations\({}^{166}\) analogous to equation (38). It is therefore possible to obtain a direct idea of \(W\) and of the mean magnetic susceptibility \(S_m\) from the Cotton–Mouton constant \(C\). In general, the changes introduced by Raman and Krishnan into the Langevin–Born theory lead to a diminution of magnetic double refraction. The expressions given by this new theory agree better with experiment, but they contain quantities that do not admit of exact determination without some additional and not always reliable assumptions.

The Cotton–Mouton constant in solutions. The magnetic properties of certain crystals\({}^{167}\) show that their magnetic susceptibility is different in different directions. This magnetic anisotropy of crystals is sometimes very clearly expressed and appears both in organic and in inorganic substances. Almost always these magnetically anisotropic crystals are also optically anisotropic, as is shown by double refraction. Ramanadan\({}^{168}\) attempted to connect these anisotropic properties with the Cotton–Mouton constants of these crystals by dissolving the latter in liquids with ma-

lim double refraction. In this way he found that, for certain aromatic substances, the observed Cotton–Mouton effect coincided with the calculated one. This confirmed his assumption that the magnetic and optical properties of a crystal are explained by the properties of the molecules composing it. However, in order to detect this connection directly, one must know the mutual orientation of the molecules or ions in the crystal lattice, their orientation relative to the crystallographic axes, their number in each unit cell, and the distances between them. Conversely, if the Cotton–Mouton constant is known, then it can evidently serve as a valuable supplement to X-ray analysis of crystals. Indeed, Raman had already emphasized that the Cotton–Mouton effect in solutions, together with a general investigation of the optical and magnetic properties of these same crystals, can also reveal such details of crystalline structure as are still inaccessible to X-ray analysis. It should not be forgotten, however, that for quantitative agreement with experiment the optical and magnetic properties of crystals, upon passing into the dissolved state, must either not change at all or must change only in a previously known manner. Nevertheless, in many cases even purely qualitative results are of great interest, as has recently been shown by members of Raman’s laboratory.^167

Magnetic double refraction in solutions not only facilitates structural analysis of a crystal, but also reveals a number of interesting details of the structure of molecules and of the properties of the liquid state. Thus, for example, from the Cotton–Mouton constant \(C\) and the scattering coefficient \(r\) in nitric acid, the magnetic anisotropy of the \(NO_3\) ion was determined.^169 On this basis one can, for example, directly obtain the magnetic effect of the alkali-metal ion, since the Cotton–Mouton constant for solutions of alkali nitrates is known. It was also found that nitrates have smaller double refraction than nitrites; this confirms the previously expressed assumption that an unsaturated chemical bond causes enhanced double refraction. The influence of the aggregated state of a substance* on the Cotton–Mouton effect is well illustrated by the decrease in the specific magnetic double refraction for nitro compounds in dilute solutions.^170,171,172,168 In connection with this, let us recall that the specific optical anisotropy of nitrobenzene also decreases when it passes from the liquid state into the gaseous state.

* The mean magnetic susceptibility \(S_m\) changes when a substance passes from the crystalline state into the liquid one. Oxley, “Phil. Trans.,” 214, 109, 1914; 215, 79, 1915; 220, 247, 1928.

In most “normal” liquids, however, the opposite effect is observed.

5. Dependence of the Cotton–Mouton constant \(C\) on wavelength and temperature. Cotton and Mouton\(^{173}\), in their first paper, found that the change in magnetic birefringence with change in the wavelength of the transmitted light for nitrobenzene is the same as the change in electric birefringence. Schipper,\(^{149}\) continuing this work, found that the ratio of the Cotton–Mouton constant \(C\) found by him to the Kerr constant \(B\), measured by MacComb, is, to a first approximation, independent of wavelength. He also found that the dispersion in magnetic birefringence follows Havelock’s law:

\[ C = h \frac{(n^{2}-1)^{2}}{n\lambda}, \tag{57} \]

which is analogous to equations (5) and (26) for dispersion in electric birefringence. Let us note that equation (47) passes directly into equation (57) if only the optical and magnetic anisotropy of the molecule do not depend on wavelength. Havelock’s law was also checked by later investigators; only slight deviations were found. In connection with the recent work of Szivessy\(^{174}\) and his collaborators on determining the dispersion in electric birefringence in the ultraviolet, and the same work by Ingersoll in the infrared region, it would be interesting to have data also on magnetic birefringence in these regions.

The dependence of the Cotton–Mouton constant on temperature\(^{175}\) at constant wavelength was studied by Szivessy in several liquids.\(^{176–177}\) His results are, in general, consistent with Langevin’s theory, although in individual cases there are noticeable discrepancies. Langevin’s formula can easily be obtained from equation (46), if one substitutes \(a_{0}\) from (46) and sets \(\theta_{2}=0\). Then

\[ C = k \frac{\rho (n^{2}+2)^{2}}{n\lambda T}, \tag{58} \]

where \(\rho\) is the density and \(k\) is a certain constant independent of temperature. The latter assumption may also prove incorrect, since it would mean that the effective electric and magnetic anisotropy of the molecule does not depend on temperature, or—which is likewise improbable—that their changes mutually compensate in equation (58). It was found that the effective optical anisotropy of many liquids changes with temperature. In most “normal” liquids the optical anisotropy increases with increasing temperature, but in others, which are assumed to be “associ-”

oriented” over certain temperature intervals, the opposite effect is observed. The increase of anisotropy with increasing temperature agrees with the theory of Raman and Krishnan. It would be important to have more accurate measurements of \(C\) as a function of temperature, since, knowing also the course of the optical anisotropy with temperature, one could trace the changes of the magnetic anisotropy as well.

6. Magnetic double refraction in gases. Magnetic double refraction in gases and vapors was predicted by Focht \(^{178}\) from his theory of magneto-optical phenomena. Subsequently he succeeded in observing this phenomenon in sodium vapor near the \(D\)-lines. Following this work, Zeeman \(^{179}\) and others showed that Focht’s theory is quantitatively justified in the immediate vicinity of absorption lines, where anomalous dispersion is observed. It is now generally accepted that this magnetic double refraction near an absorption line is directly connected with the Zeeman effect. \(^{179,181}\) Magnetic double refraction in spectral regions far from absorption lines, if it exists at all, is negligibly weak. The experiment of Stern and Gerlach \(^{182}\) and the theory \(^{183,184}\) of “quantization of directions in a magnetic field” indicate that, under suitable conditions of temperature and pressure, the atoms of paramagnetic gases should be oriented by the magnetic field along straight lines. Then, if the atom is optically anisotropic, magnetic double refraction should also be observed for wavelengths far from absorption lines, where the refractive index does not assume anomalous values. The magnitude of the double refraction would be almost independent of the strength of the magnetic field, provided only that the latter exceeds some comparatively small value. Experiments \(^{185,186,187}\) carried out to test this hypothesis showed that in sodium and potassium vapors, oxygen, and nitric oxide the magnetic double refraction for the investigated regions of the spectrum is so small that it cannot be observed, although according to the theory it should have had an easily measurable value. Krishnan \(^{187}\), on the basis of his experiments, came to the conviction that paramagnetic molecules and atoms, as a rule, are not oriented by a magnetic field in the way indicated above. He points to the possibility of avoiding the contradiction if one assumes that in each individual case the axes of the optical ellipsoid of the molecule or atom make an angle of suitable magnitude with the direction of the permanent electric moment. Then the orientation of molecules could occur without causing double refraction. However, he considers this very improbable and points to the negative results of the experiments of Debye and Huber. These investigators \(^{188,190}\) were unable to detect any electric potential on

DOUBLE REFRACTION IN AN ELECTRIC FIELD

two metal plates placed perpendicular to the magnetic field, between which a polar paramagnetic gas was placed. If the orientation of the molecules actually occurred, then a measurable potential difference should have appeared between the plates, unless the electric moment of the molecules is strictly perpendicular to the magnetic one, which would be highly improbable. It is a great pity that the existing apparatus is apparently not sensitive enough to measure magnetic double refraction in gases far from absorption lines or bands, since it is precisely in gases that the unknown influence of the molecules surrounding a given molecule is small or isotropic, and the Langevin–Born theory ought to be justified.

7. Further proposals. Let us return to the effect mentioned at the very beginning. Finely ground suspended particles in liquids, being placed in a magnetic field, produce strong double refraction when light passes perpendicular to the lines of force. “Fer Bravais”[^145] and many other iron colloids give very strong double refraction, and in some so-called “liquid crystals” the effect exceeds the Cotton–Mouton effect in “normal” liquids by more than a million times. Experiments show that the particles of these substances are anisotropic both optically and magnetically and therefore are oriented by the magnetic field, producing double refraction. In the case of liquid crystals and some colloids, even comparatively weak magnetic fields already cause almost complete orientation. These phenomena are in some cases rather complex, as is shown by certain changes with time and especially those observed as a function of wavelength. Dichroism is often even present. Sometimes a relaxation time of the order of several seconds is required, which indicates an actual orientation of the particles by the field. Magnetic double refraction in optically active liquids was studied in detail, theoretically and experimentally, by de Mallemann.[^192]

8. Magnetic double refraction in solids. The magnetic double refraction of amorphous solids has not been studied in detail. These experiments are always complicated by the superposition of double refraction due to magnetostriction, although, in contrast to the Faraday effect, it is rather weak. On the other hand, the phenomena observed when light passes through crystals perpendicular to the lines of force have been studied in detail. Becquerel[^193–^195] and his collaborators carried out a number of interesting experiments on magneto-optical effects in crystals of the rare earths. At low temperatures these crystals have fine absorption lines, near which Becquerel[^196] and

carried out investigations, varying the direction of the magnetic field with respect to the axes of the crystals. The observed phenomena are rather complex, and we are compelled to refrain from presenting them. Referring the reader to the original papers, we shall note only that the observed effects are probably related to the Zeeman effect.

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