Second-Order Raman Scattering of Light

E. F. Gross, P. P. Pavinskii, A. I. Stepanov

Introduction

It is well known what an important role the natural vibrations of matter play in a number of physical processes. The thermal behavior of a body, for example, or its optical properties, are determined basically by the spectrum of frequencies of the natural vibrations of the substance.

Knowledge of the natural frequencies therefore has great significance. Determination of the natural frequencies is important not only for theoretical questions concerning the structure and properties of matter, but also for practical purposes—the analysis of the chemical composition of substances from their characteristic spectra.

There exist a number of methods for approximate estimation of the natural frequencies of vibrations; however, the most direct and differentiated methods here are those associated with optical phenomena and, first of all, spectroscopic ones. The study of the absorption of light by a substance in different regions of the spectrum directly gives the frequencies of the natural vibrations of the substance. Absorption in the ultraviolet and visible parts of the spectrum makes it possible to determine the frequencies of electronic transitions in the substance.

The discovery by Landsberg, Mandelstam¹ and Raman of Raman scattering of light (first-order scattering) provided a new method for studying the natural vibrations of matter. Absorption in the infrared region of the spectrum and first-order Raman scattering of light make it possible to determine the frequencies of the natural vibrations of nuclei and mutually complement one another.

A special type of Raman scattering is second-order light scattering. This phenomenon, still little studied, makes it possible to obtain more complete information about the spectrum of the natural vibrations of matter.

In what follows, the essence of the phenomenon of combination scattering of the second order is set forth, and the results of its experimental and theoretical investigations carried out up to the present time are reported. The exposition will relate chiefly to crystals, since in them the phenomenon has been studied most fully.

1. THE ROLE OF POLARIZABILITY IN COMBINATION SCATTERING OF LIGHT

The basic cause of combination scattering of light is the interaction between light, the electron shells of atoms, and the motion of nuclei. Owing to the fact that the frequency of the irradiating light in light-scattering experiments is always much greater than the natural frequencies of nuclear motion and is comparable with the frequencies of electronic transitions, we may, in describing the scattering of light, apply the theory of dispersion, according to which atoms or molecules situated in the alternating field of an electromagnetic light wave acquire an electric moment. The magnitude of this moment is essentially determined by the polarizability of the electron shells for a given arrangement of the nuclei. The polarizability, in principle, can be calculated from the formula of dispersion theory: it explicitly depends on the frequency of the irradiating light and contains, as parameters, the natural frequencies of electronic transitions of atoms or molecules, as well as the so-called oscillator strengths, i.e., quantities proportional to the probabilities of transitions between the ground and various excited electronic states. The dispersion formula is characterized by a sharp dependence on frequency near the natural frequencies of the atoms, i.e., in the region of strong absorption. Far from the natural frequencies of the scattering substance, under monochromatic illumination and when the motion of the nuclei is neglected, the polarizability is a constant quantity (a tensor), and therefore the electric moment of the medium varies synchronously with the electric vector of the wave, and the light emitted by it has the very same frequency as that of the incident light. As is known from molecular optics, from this one may derive both the existence of a refracted wave in a homogeneous medium and classical (Rayleigh) molecular scattering of light with unchanged frequency. For the existence of the latter, the presence of inhomogeneities (density fluctuations) is necessary, which, as is known, are connected with the existence of molecular motion. However, for the description of Rayleigh scattering there is no need explicitly to take into account the motion of the nuclei: it would be sufficient to imagine the state of molecular motion as “frozen” at some random instant of time for the phenomenon of Rayleigh scattering to receive an explanation.

A different situation obtains if we take into account the motion of the nuclei. Then, as is well known, the frequency of the scattered light will differ from the frequency of the incident light by amounts connected with the frequencies of motion of the nuclei (combination scattering). Since, in general, the nuclei move slowly, i.e., the change in their positions over one period of the light wave is small, we may apply here the parametric method of description, which is essential for understanding combination scattering. Namely, we may use the already introduced concept of polarizability, regarding it as a function of time, i.e., in the first approximation we shall describe the motion of the nuclei classically, assuming that the parameters of the theory of dispersion (the proper frequencies of the electrons and the strengths of the oscillators) are functions of the positions of the nuclei, i.e., in the presence of nuclear motion—functions of time. Such a description is essentially based on the assumption that the proper frequencies of nuclear motion are small in comparison with the frequency of light, and it is not applicable, for example, to the description of absorption and dispersion in the infrared-frequency region. The parametric method of description*), however, is also applicable in quantum mechanics, which creates an almost complete parallelism in the description of the phenomenon from the classical point of view and from the quantum-mechanical point of view.

Let us now suppose that the dependence of the polarizability \(\kappa\) on the position of the nuclei is known to us, and that the law of motion of the nuclei is also known. Then \(\kappa(t)\) will be a known function of time. Suppose that the electric field in some region is represented by a monochromatic wave of frequency \(\omega_0\):

\[ \mathcal{E}(t)=\mathcal{E}_0 e^{-i\omega_0 t}. \tag{1} \]

The question of the frequencies of the scattered light, i.e., of the frequencies of oscillation of the electric vector

\[ p(t)=\kappa(t)\mathcal{E}(t) \tag{2} \]

is resolved by considering the Fourier spectrum of the polarizability \(\kappa(t)\). Let the frequencies of this spectrum be \(\Omega_j\):

\[ \kappa(t)=\sum_j c(\Omega_j)e^{i\Omega_j t}. \tag{3} \]

*) In the parametric (approximate) method of describing a molecular system it is assumed that all quantities characterizing the states or changes of state of the electrons may be referred to a definite arrangement of the nuclei and vary continuously as the nuclei move. The coordinates of the nuclei are, consequently, parameters in describing the state of the electrons.

Then we have:

\[ p(t)=\mathscr{E}_0 \sum_j c(\Omega_j)e^{-i(\omega_0-\Omega_j)t}, \tag{4} \]

i.e., the frequencies of the scattered light \(\omega'_j\) will be shifted by the amounts \(\Omega_j\)

\[ \omega'_j=\omega_0-\Omega_j=\omega_0\pm|\Omega_j| \tag{5} \]

(it is assumed that the frequencies \(\Omega_j\) may be either positive or negative).

We have thus obtained a rule for determining the shifted frequencies: the shifted frequencies \(\omega'_j\) in the spectrum of combination scattering are formed additively from the frequency of the incident line \(\omega_0\) and the frequencies \(\Omega_j\) of the Fourier spectrum of the polarizability, considered as a function of time through the motion of the nuclei. This definition of the classical theory is applicable to the spectrum of combination scattering of any order, and it includes the selection rules. However, it makes no mention of the frequencies of the mechanical motions of the nuclei. To characterize the relation between the observed frequencies and the frequencies of the mechanical motions of the nuclei, a more concrete consideration of the polarizability as a function of the displacements of the nuclei is necessary. Before turning to this question, let us consider the refinement introduced by quantum mechanics into formula (5) and into the general treatment of the problem of combination scattering.

As is well known, relation (5) in quantum mechanics (after multiplication by \(\hbar\)) expresses the law of conservation of energy in the elementary act of scattering of a light quantum. The quantity \(\hbar\Omega_j\), for positive \(\Omega_j\), is the amount of energy transferred to the molecular system in the scattering of a quantum (Stokes shifted frequency); for negative \(\Omega_j\), the quantity \(\hbar\Omega_j\) denotes the amount of energy borrowed from the system in the act of scattering (anti-Stokes frequency). The frequencies \(\Omega_j\) are related to the energy levels of the system:

\[ \Omega_j=\frac{E_{mj}-E_{nj}}{\hbar}, \tag{6} \]

where \(E_{nj}\) is the energy (of the nuclei) in the initial state (before scattering), and \(E_{mj}\) is the energy in the final state (after scattering). Thus the frequencies \(\Omega_j\) are not the frequencies of any “actual” motion of the nuclei (in the sense of classical mechanics), but rather the frequencies of possible transitions of the nuclear system from one state to another.

Quantum mechanics also gives rules for calculating the amplitudes \(c(\Omega_j)\), which determine the intensity of scattering. These quantities are precisely the matrix elements of the polarizability, calculated with the aid of Schrödinger wave functions (not depending on time) that describe the initial and final states of the system.

nuclei. Instead of \(c(\Omega_j)\), we must therefore write:

\[ c(\Omega_j)=\int \Psi_{mj}^{0*}\chi\Psi_{nj}^0\,d\tau . \tag{7} \]

Quantum mechanics thus provides an interpretation of the individual terms of formula (3).

We have direct, visual confirmation of the correctness of such a quantum-mechanical interpretation, for example, in the case when transitions are possible only from the ground state (low temperatures). In this case all frequencies \(\Omega_j\) are positive, since \(E_{nj}<E_{mj}\). Consequently, only Stokes shifted frequencies occur, whose intensity is determined by the matrix elements (7) for the transition from the ground state. Classical theory, on the contrary, would in this case give an intensity equal to zero both for the Stokes and for the anti-Stokes frequencies, since in the absence of nuclear motion the amplitudes \(c(\Omega_j)\) must be equal to zero. In general, it would not be difficult to show (using the reality of \(\chi\)) that classical theory always leads to equality of the intensities of the Stokes and anti-Stokes shifted frequencies:

\[ |c(-\Omega_j)|_{\mathrm{class}}=|c(\Omega_j)|_{\mathrm{class}} . \tag{8} \]

Experiment, as is known, confirms quantum mechanics, although in the limit (at very high temperatures) the classical equality (8) also holds. We must here make an important qualification for what follows: the intensities observed experimentally must in reality be compared not with the squares of the moduli of the quantum quantities (7), but with their so-called thermal averages. In other words, we must imagine that we are dealing not with a single quantum-mechanical system (a molecule or a crystal) situated in a definite quantum state \(E_n\), but with a representative of a very large number of identical systems (a statistical ensemble), in which the states \(E_n\) are distributed with probabilities calculated with the aid of statistical physics. This gives us the possibility of taking into account the dependence of the intensity on temperature. In classical theory this dependence is determined by the square of the amplitude of vibration (in other words, by the energy). After averaging it will be determined by the mean thermal energy of vibration.

2. COMBINATION SCATTERING OF THE FIRST ORDER

A detailed exposition of the well-developed theory of first-order combination scattering is not part of our task. However, for understanding what follows it is necessary to recall the basic propositions of this theory. As is known, the basic assumption of the theory of first-order combination scattering consists

in that the polarizability may be regarded as a linear function of the displacements of the nuclei \(Q_i\):

\[ \varkappa=\varkappa_0+\sum_i a_i Q_i \equiv \varkappa_0+\varkappa_1 . \tag{9} \]

The term \(\varkappa_0\), which does not depend on the displacements of the nuclei, will not interest us here, since it does not lead to a shift of the frequency of the light, i.e. it does not give combination scattering, but is connected with Rayleigh scattering. It is obvious that assumption (9) already leads to an explicit dependence of the frequencies of the Fourier spectrum \(\Omega_j\) of the polarizability \(\varkappa\) on the frequencies of the mechanical vibrations of the system (in the classical description).

Indeed, as follows from (9), the Fourier spectrum of the polarizability \(\varkappa\) can now consist only of the same frequencies of which the Fourier spectrum of the vibrations of all the coordinates (displacements) \(Q_i\) of the nuclei consists. Other frequencies cannot enter into \(\varkappa\), owing to the linear dependence of \(\varkappa\) on \(Q_i\). Thus, at least part of the frequency spectrum \(\omega_j\) of the mechanical vibrations of the system is directly observed as shifts of the frequencies \((\Omega_i)\) of first-order combination scattering.

In the quantum description, assumption (9) leads to an explicit dependence of the observed frequencies \(\Omega_j\) on differences between the energy levels of the system (molecule or crystal), considered as an aggregate of nuclei. At first sight, in the quantum description, we have no simplification here, since the general theory (see equation (6)) leads to exactly the same relation between the observed frequencies and differences of the energy levels of the system. In reality, however, there is a simplification, owing to the selection rules.

Indeed, starting from the quantum interpretation of the Fourier amplitudes \(c(\Omega_j)\), we can now establish that the only differences between energy levels of the system that can be observed in first-order combination scattering are those to which there corresponds a matrix element, different from zero, of at least one of the coordinates \(Q_i\) of the system. Especially simple relations are obtained in those cases in which, in describing the motion of the nuclei, we may confine ourselves to considering harmonic vibrations, i.e. take into account in the potential energy only terms quadratic with respect to \(Q_i\). In this case, for describing the motion (or the quantum-mechanical state), one can introduce normal coordinates, linearly expressible through \(Q_i\). Without changing the form of formula (9), we may then regard the \(Q_i\) in (9) as already being the normal coordinates of the system. The Fourier spectrum of each of the normal coordinates \(Q_i\) consists of only one frequency \(\omega_i\), which

and will be observable only if the coefficient \(a_i\), as a result of the transformation to normal coordinates, does not become zero*).

In the quantum-mechanical description the energy of the system is represented as a sum of terms referring to the individual normal coordinates (vibrators) \(Q_i\), the energy levels of each vibrator being spaced from one another by the amount \(\hbar \omega_i\). The selection rules permit transitions between neighboring levels of only one vibrator. Thus, the quantum-mechanical spectrum of eigenfrequencies coincides with the classical one in this case.

It is important to note that the calculation of the frequencies of nuclear vibrations can sometimes be greatly simplified in those cases when the initial equilibrium arrangement of the nuclei possesses some spatial symmetry (a plane of symmetry, an axis of rotation, etc.). The prediction of the disappearance of lines (inactivity) in these cases can be made on the basis of symmetry. This circumstance is of especially great importance for crystals, which are necessarily characterized by translational symmetry, i.e. coincide with themselves under a displacement by any lattice vector. Consideration leads to the conclusion that in crystals, in first-order combination scattering, the frequencies of all normal coordinates disappear (are inactive), except those which correspond to parallel displacements of all homologous atoms in all elementary cells of the crystal (the so-called limiting frequencies).

A more exact consideration \(^{27,28}\) shows that in the first-order spectrum the observable frequencies can be those of elastic vibrations whose wavelength satisfies the condition: \(\lambda_{\text{light}} = 2\lambda_{\text{el}} \sin \frac{\vartheta}{2}\), where \(\vartheta\) is the angle of scattering of the light. Thus, the wavelengths of the elastic waves must be comparable with the wavelength of light \(\lambda_{\text{light}}\), i.e. they must be very large in comparison with the lattice constant \(a\). We see that in this case the observed spectrum of polarizability frequencies consists of a negligible part of the spectrum of mechanical vibrations of the system (crystal). It may happen (this, for example, occurs in crystals of alkali-halide compounds) that, from the spectrum of first-order combination scattering, all frequencies of mechanical vibrations disappear altogether. This occurs in those cases when the crystal is characterized (in addition to the obligatory translational symmetry) by a sufficiently high degree of rotational and mirror symmetry. In this case we may say that the polarizability in first order does not depend on the displacements of the nuclei (\(k_1 = 0\)).

*) Such frequencies of the spectrum of mechanical vibrations, to which the coefficient \(a_i = 0\) corresponds, we call inactive in first-order combination scattering.

Let us note that for an anharmonic vibrator the selection rules allow transitions not only between neighboring energy levels of the vibrator, but also between other levels. This leads to the appearance of new frequencies (overtones, etc.) in the scattering spectrum. The intensity of these frequencies, however, will be the smaller the smaller the anharmonicity, and, generally speaking (in combination scattering), is very small.

For what follows we must recall the well-known results concerning the temperature dependence of the intensity of first-order combination-scattering spectra. This dependence is characterized by a factor \(R\), different for a shifted frequency in the Stokes or anti-Stokes regions. For the Stokes frequency \(\Omega\)* we have:

\[ R_c^{(1)}=\frac{1}{1-e^{-\frac{\hbar |\Omega|}{kT}}}, \tag{10} \]

and for the anti-Stokes one:

\[ R_a^{(1)}=\frac{1}{e^{\frac{\hbar |\Omega|}{kT}}-1}. \tag{11} \]

These factors behave differently at low temperatures \((kT \ll \hbar |\Omega|)\); as \(T \to 0\)

\[ R_c^{(1)} \to 1,\quad R_a^{(1)} \to 0\ \left(\text{as } e^{-\frac{\hbar |\Omega|}{kT}}\right). \tag{12} \]

At high temperatures \((kT \gg \hbar |\Omega|)\) both factors become almost identical and increase approximately linearly with temperature,

\[ R_c^{(1)} \sim R_a^{(1)} \sim \frac{kT}{\hbar |\Omega|}. \tag{13} \]

This last case corresponds to the classical approximation (cf. equation (8)). Indeed, according to the classical theory, the intensity increases in proportion to the vibrational energy, i.e. (for the mean thermal energy) in proportion to \(kT\).

3. SECOND-ORDER COMBINATION SCATTERING.

More complicated is the dependence of the spectrum of polarizability frequencies on the spectrum of mechanical vibrations in the case of combination scattering of the second and higher orders. Continuing the expansion of the polarizability in formula (9), we may proceed

* Let us recall that we regard Stokes frequency shifts \(\Omega\) as positive, and anti-Stokes shifts as negative.

E. F. Gross, P. P. Pavinsky, A. I. Stekhanov

further and consider the quadratic terms with respect to the displacements of the nuclei \(Q\):

\[ \left. \begin{aligned} x &= x_0 + x_1 + x_2 + \cdots \\ x_2 &= \sum_{i,k} b_{ik} Q_i Q_k . \end{aligned} \right\} \tag{14} \]

We shall confine ourselves to the consideration of harmonic vibrations. Then \(Q_i\) may be regarded as normal coordinates and, consequently, their frequency spectrum as consisting of a single frequency \(\omega_i\). Formula (14) then leads, both in the classical and in the quantum-mechanical interpretation, to the result that the frequency spectrum of the polarizability will consist of sums and differences of the natural frequencies of the substance \(\omega_i\). We therefore obtain for the displacement frequencies \(\Omega\) the general expression

\[ \Omega = \pm \omega_i \pm \omega_j , \tag{15} \]

where the signs may be chosen arbitrarily, as may the indices \(i\) and \(j\) of the natural vibrations. We thus see that the spectrum of displacement frequencies in second-order combination scattering can, generally speaking, be much richer in frequencies than the spectrum of frequencies of the mechanical vibrations. Indeed, each frequency may occur both in the octave (in the form of a doubled frequency)—when the indices \(i\) and \(j\) and the signs in formula (15) are the same—and in combination (in the form of a sum or difference) with other frequencies.

However, although the selection rules thus permit the combination of any two frequencies of the spectrum of mechanical vibrations to form possible (sum or difference) component frequencies \(\Omega\) of the second-order spectrum, by no means can all these frequencies be observed. As in the case of the spectrum of first-order combination scattering, a very important question is that of the activity of mechanical vibrations in the spectrum of combination scattering—the possibility of deriving rules for determining activity from the symmetry properties of the unperturbed system.

Here we shall consider the question of the activity of frequencies of the second-order spectrum in crystals, i.e., the question of the influence of translational symmetry\(^4\). For this it is necessary to recall the basic propositions of the theory of vibrations of a crystal lattice.

As is known, owing to the existence of spatial translational periodicity, the harmonic vibrations of an ideal crystal can be represented as a superposition of elastic traveling waves characterized by the wave vector \(\mathbf{k}\) (with wavelength \(\lambda = \dfrac{2\pi}{k}\)). This wave vector assumes not arbitrary

values: it is expressed in the form of a rational linear combination of the vectors \(\mathbf b_1,\ \mathbf b_2,\ \mathbf b_3\) of the reciprocal lattice\(^*\):

\[ \mathbf k \equiv \mathbf k_\nu = 2\pi\left[\frac{\nu_1}{N}\mathbf b_1+\frac{\nu_2}{N}\mathbf b_2+\frac{\nu_3}{N}\mathbf b_3\right], \tag{16} \]

where \(\nu_1,\nu_2,\nu_3\) are integers taking the values \(0,1,2,\ldots,N-1\); \(N\) is a very large integer, \(N^3\) is the number of cells in the volume of the crystal under consideration. Thus the number of different values of \(\mathbf k\) is \(N^3\), i.e. is equal to the number of cells in the volume of the crystal. To each value of \(\mathbf k\) there correspond as many vibrational frequencies as there are degrees of freedom of all the atoms forming the elementary cell of the crystal, i.e. \(3s\), if \(s\) is the number of atoms in the elementary cell. Thus, in all, we shall have \(3sN^3\) frequencies. These frequencies are usually grouped into branches: since the vector \(\mathbf k\), according to (16), changes almost continuously for very large \(N\), the frequency within each branch is an almost continuous function of \(\mathbf k\). Within each branch \(\mathbf k\) runs through all \(N^3\) values. Among the \(3s\) branches of the elastic spectrum there are three acoustic branches, whose frequencies extend down to zero frequencies. The remaining \(3s-3\) branches are called optical. In the first-order spectrum, for purely harmonic vibrations, one observes (if the immediate neighborhood of the Rayleigh line is excluded) mainly the frequencies of the optical branches, and moreover only long-wave vibrations, with wavelength comparable with that of light (limiting frequencies). In the second-order scattering spectrum, as we shall see, the entire elastic spectrum may be observed, although here too a certain selection rule obtains.

In what follows it will be convenient for us to distinguish the small displacements of atoms \(Q\) from their equilibrium positions and the normal coordinates \(q\), which are linear functions of \(Q\). Further, it is convenient to denote the normal coordinates and frequencies by indicating the wave vector \(\mathbf k\) and the branch number \(j\) to which \(q\) or \(\omega\) belongs. Thus, the notation for a normal coordinate will be

\[ q \equiv q(\mathbf k,j). \tag{17} \]

To each normal coordinate \(q(\mathbf k,j)\) there corresponds one frequency \(\omega(\mathbf k,j)\), which is a function of the wave vector \(\mathbf k\) and

\(^*\) Let us recall the definition of the vectors of the reciprocal lattice. If \(\mathbf a_i\) \((i=1,2,3)\) are the vectors characterizing the elementary cell of the crystal, then \(\mathbf b_i\) are defined as the vector products of the vectors \(\mathbf a_i\):

\[ \mathbf b_1=\frac{1}{v}[\mathbf a_2\mathbf a_3] \]

and so on, where \(v\) is the volume of the cell.

the number of the branch \(j\). We shall denote the position of an atom in the lattice by the vector \(\mathbf n\), indicating the position of the cell in the crystal:

\[ \mathbf n=n_1\mathbf a_1+n_2\mathbf a_2+n_3\mathbf a_3 \tag{18} \]

\((n_1,n_2,n_3\) are integers: \(0\le n_i\le N-1\); \(\mathbf a_i\) are the cell vectors), and also by the number \(\sigma\) of the atom in the cell \((\sigma=1,2,\ldots,s)\). Then the transformation from normal coordinates to atomic displacements can be written in the form

\[ Q_{n\sigma x}=\sum_{\mathbf k,j} a_{\sigma x}(\mathbf k,j)e^{i(\mathbf n\mathbf k)}q(\mathbf k,j), \tag{19} \]

where the coefficients \(a_{\sigma x}(\mathbf k,j)\) are, generally speaking, complex,* and the subscript \(x\) characterizes the projection of the displacement of the atom onto the Cartesian axis \(x\).

Let now the polarizability \(\chi\) be known as a function of the atomic displacements \(Q\), represented in the form of a power series in \(Q\):

\[ \chi=\chi_0+\chi_1+\chi_2+\cdots \tag{20} \]

We shall consider only the quadratic terms, in accordance with the theory of second-order combination scattering,

\[ \chi_2= \sum_{\mathbf n\sigma x,\mathbf n'\sigma' x'} c_{\mathbf n-\mathbf n',\,\sigma x\sigma' x'}\, Q_{\mathbf n\sigma x}Q_{\mathbf n'\sigma' x'} . \tag{21} \]

For brevity we omit the additional subscripts indicating that \(\chi\) is a tensor. Here we have used the property of periodicity of the crystal, thanks to which the coefficients \(c\) depend only on the relative distance \(\mathbf n-\mathbf n'\) between two cells \(\mathbf n\) and \(\mathbf n'\). Substituting (19) into (21), we can perform one summation over \(\mathbf n\). As a result, one summation over \(\mathbf k\) also drops out, since the relation holds

\[ \sum_{\mathbf n} e^{i(\mathbf n,\mathbf k-\mathbf k')}=N^3\delta(\mathbf k-\mathbf k'), \tag{22} \]

where \(\delta(\mathbf k-\mathbf k')\) denotes the well-known \(\delta\)-function, i.e. \(\delta(\mathbf k-\mathbf k')=0\) for \(\mathbf k\ne\mathbf k'\) and \(\delta(\mathbf k-\mathbf k')=1\) for \(\mathbf k=\mathbf k'\). Hence it follows directly that, for the formation of the displacement frequency \(\Omega\), the following selection rule must be satisfied: only such frequencies of elastic vibrations combine with one another as correspond to the same value of the wave vector \(\mathbf k\).

\[ \text{*} \]

* The coefficients \(a_{\sigma x}(\mathbf k,j)\) are the complex displacements of the atoms of the “zero” cell \((\mathbf n=0)\) in the case when only one vibration with frequency \(\omega(\mathbf k,j)\) is excited in the lattice. The phases of the coefficients \(a_{\sigma x}(\mathbf k,j)\) in this case give the relative phases of the motions of atoms, which execute coherent spatially periodic motions about the equilibrium positions.

Thus, as a result, only products of normal coordinates with one and the same wave vector \(\mathbf{k}\) will enter into \(x_2\). We finally obtain:

\[ x_2=\sum_{\mathbf{k}jj'} d_{jj'}(\mathbf{k})q(\mathbf{k},j)q(\mathbf{k},j'), \tag{23} \]

where it is denoted

\[ d_{jj'}(\mathbf{k})=N^3 \sum_{n\alpha x\alpha' x'} c_{n\alpha x\alpha' x'} a_{\alpha x}(\mathbf{k},j)\overline{a}_{\alpha' x'}(\mathbf{k},j')e^{i(\mathbf{n}\mathbf{k})}. \tag{24} \]

The normal coordinates \(q(\mathbf{k},j)\) and \(q(\mathbf{k},j')\) correspond to the frequencies \(\omega(\mathbf{k},j)\) and \(\omega(\mathbf{k},j')\). We see, therefore, that in the second-order spectrum there appear frequencies of mechanical vibrations entering pairwise from two different branches \(j\) and \(j'\) (\(j\) may be equal to \(j'\), and then we have either an octave vibration with doubled frequency \(2\omega(\mathbf{k},j)\), or a vibration with zero frequency, entering into the Rayleigh line). In this case a selection rule is obeyed, according to which only vibrations with the same wave vector \(\mathbf{k}\) can combine.

Let us now see how a vibration of the displacement \(Q\) with a given frequency is obtained. In contrast to the first-order spectrum, where the frequency of the displacement must necessarily coincide with one of the limiting frequencies of the optical branches (discrete spectrum), in second order we have a quasi-continuous spectrum. Indeed, each \(\Omega\) must consist of frequencies \(\omega(\mathbf{k},j)\) and \(\omega(\mathbf{k},j')\) satisfying the equation

\[ \Omega=\pm\omega(\mathbf{k},j)\pm\omega(\mathbf{k},j') \tag{25} \]

with some \(\mathbf{k}\), \(j\), and \(j'\), and with some choice of signs. In view of the fact that the frequencies even of the optical branches fill some interval of frequencies quasi-continuously (and the branches often overlap one another), equation (25) has, generally speaking, several solutions for \(\Omega\) lying in the interval

\[ 0<|\Omega|<2\omega_{\max}, \tag{26} \]

where \(\omega_{\max}\) denotes the greatest frequency in the elastic spectrum of the crystal. In general, of course, it may happen that for some frequencies in the interval (26) there will be no solutions of equation (25) at all. The spectrum will then consist of separate more or less broad bands, separated by intervals in which the intensity is zero. However, if in some region of frequencies of some branch of the spectrum the spectrum of mechanical vibrations is very intense, i.e., if a very large number of normal coordinates fall within the given narrow interval of frequencies, then this

will almost inevitably be reflected in the form of the spectrum of combinational scattering of the second order, where in the region of octave frequencies a corresponding maximum will appear. In exactly the same way, to two such sharp maxima in the frequency region \(\omega(\mathbf{k},j)\) and \(\omega(\mathbf{k},j')\) in different branches of the mechanical spectrum there will for the most part correspond more or less sharp maxima of the combinational-scattering spectrum at places corresponding to the combination frequencies:

\[ \begin{split} \pm\bigl(\omega(\mathbf{k},j)+\omega(\mathbf{k},j')\bigr),\\ \pm\bigl(\omega(\mathbf{k},j)-\omega(\mathbf{k},j')\bigr). \end{split} \tag{27} \]

Generally speaking, apart from such sharp maxima, or for a comparatively diffuse spectrum of mechanical frequencies, the question of the intensity of the spectrum of the 2nd order is difficult to resolve. It leads to complicated relations not only because of the possibility of different pairs of branches being superposed on one another, but also because of the unknown coefficients \(d_{jj'}(\mathbf{k})\) in formula (23), which determine the scattering intensity. Therefore a detailed calculation of the intensity of the spectrum of the 2nd order, even for simple crystals, at present appears extremely difficult*). It is possible, however, by renouncing a theoretical prediction of the exact form of the spectrum, i.e. of the distribution of intensity over frequencies, to obtain information on the dependence of the intensity of the scattering spectrum of the 2nd order on temperature in a given frequency region. As we shall see below, definite conclusions about the temperature dependence can be drawn in two cases. First, when in some frequency region of the spectrum the contribution from one pair of branches of the elastic spectrum (or an octave from one branch) so considerably exceeds the contribution from the remaining pairs of branches that the latter may be neglected and only the former considered. In this case we have a characteristic intensity peak, whose temperature behavior we can investigate over the whole temperature range. The second case is encountered in the region of sufficiently high temperatures, when the relations are so simplified that we can derive general formulas concerning the temperature behavior over the whole region of the spectrum. In both cases, the temperature behavior of the intensity of the second-order spectrum reveals sufficiently characteristic features distinguishing it from the behavior of the first-order spectrum. This circumstance justifies the special consideration of the temperature dependence of the second-order spectrum, which we carry out in Section 6.

*) In the work of Born and Bradburn\(^{10}\), which will be discussed in Section 5, owing to the indicated difficulties, empirical values of the coefficients were used in order to obtain agreement with experiment.

4. EXPERIMENTAL TECHNIQUE FOR STUDYING SECOND-ORDER COMBINATION SCATTERING

The experimental technique for obtaining spectra of first-order combination scattering of light has been well developed and mastered over the 20 years of its existence. The same cannot be said of the experimental technique for obtaining second-order scattering spectra, which is due mainly to the very low intensity of second-order spectra. The intensity of second-order scattering spectra is hundreds and thousands of times smaller than the intensity of first-order scattering spectra. Hence it is quite obvious what experimental difficulties are involved in obtaining second-order scattering spectra. Therefore, a detailed study of second-order scattering spectra can be carried out only with the careful development of a number of questions of experimental technique, connected mainly with increasing the luminosity of the setup and with weakening the parasitic scattered light.

In this case the question of light sources acquires primary importance. As in the experimental technique for obtaining a first-order spectrum, so also for exciting second-order spectra, a mercury lamp is used as the light source. In the spectrum of a mercury lamp there are a number of intense lines, for example, $\lambda 5461\,\text{\AA}$, $\lambda 4358\,\text{\AA}$, $\lambda 4047\,\text{\AA}$, $\lambda 3660\,\text{\AA}$, and $\lambda 2537\,\text{\AA}$, which are used to excite light-scattering spectra. The choice of one or another exciting radiation depends on the transparency of the substance under investigation.

In those cases when the substance is transparent to ultraviolet light, it is more advantageous to use the ultraviolet line $\lambda 2537\,\text{\AA}$ for exciting second-order spectra. As is well known, the intensity of molecular scattering increases inversely proportionally to the fourth power of the wavelength of the exciting light. Therefore, for example, when the scattering spectrum is excited by light of wavelength $\lambda 2500\,\text{\AA}$, the intensity of the scattering spectrum will be 16 times greater than when it is excited by light $\lambda 5000\,\text{\AA}$. The strong increase in the intensity of light scattering when objects are illuminated with ultraviolet light makes it possible to observe very weak second-order scattering in crystals.

Ordinary mercury lamps used to excite first-order scattering emit the line $\lambda 2537\,\text{\AA}$ with very low intensity because of its strong self-reversal and therefore cannot be used as light sources for exciting second-order scattering.

In the technique for studying second-order scattering, mercury lamps of special design are used. To avoid self-reversal of the resonance line \(\lambda 2537 \text{ Å}\), they are cooled with running water and placed in a magnetic field. Such a powerful mercury lamp with water cooling and a magnetic field was also used by the authors of the present paper\(^6\) in investigating second-order spectra. It gave very intense resonance radiation of mercury of wavelength \(\lambda 2537 \text{ Å}\), which in intensity amounted to approximately \(80\%\) of the entire spectrum of the lamp.

Various methods are used to illuminate the object with the light source; they reduce essentially to two types:

1) the method of concentrating the light of the source on the object under study by means of condenser lenses, and 2) the so-called high-aperture method of Wood. In the latter, the light source directly illuminates the object when the source and the object are brought very close together. The first method has a very small luminosity, but on the other hand gives little stray light. The second method, although it suffers from an abundance of stray light, has great advantages in the study of weak scattering, since it makes it possible to illuminate the object strongly. Therefore all investigations of second-order scattering spectra were carried out by Wood’s method. The arrangement of the mercury lamp and the object is shown in Fig. 1. The vessel with the object \(O\) is placed above the mercury lamp \(L\) so that its axis coincides with the optical axis of the spectrograph collimator. The light scattered by the object is, as usual, collected on the slit of the spectrograph \(Cn\) by means of the lens \(K\).

Fig. 1. Diagram of the apparatus used in investigations of second-order light-scattering spectra.

A considerable increase in the illumination of the object is obtained by the use of reflectors. For this purpose an aluminum cylindrical mirror is placed over the object under study, which has the form of a cylinder. In this case the light from the mercury lamp passing through the object falls on the mirror and is concentrated by it again inside the object. In exactly the same way the mercury lamp is surrounded by an aluminum cylindrical mirror, which makes it possible, by reflection of light from the mirror, to make fuller use of the light power of the mercury lamp.

One of the chief obstacles hindering the study of second-order scattering spectra is stray scattering, which increases with increasing luminosity of the apparatus.

The reasons for the appearance of parasitic light are due to the presence of macroscopic inhomogeneities in the objects and to reflection of light from the walls of the vessel in which the object is placed, as well as reflection from the surface of the objects themselves. Even with very good objects and their careful installation with all possible precautions and devices (diaphragms), it is not possible to weaken sufficiently the parasitic light from the very intense exciting radiation \(\lambda 2537\) Å; this parasitic light causes fogging of the photographic plate and strong overexposure of the exciting line, which makes it completely impossible to study the low frequencies of the scattering spectrum.

A very effective means of weakening the exciting frequency in the scattered light proved to be a mercury filter, first used by Landsberg and Mandelstam \(^{1}\) in experiments with quartz, and also by Rasetti \(^{2}\) in his experiments on the spectra of light scattering in gases and on second-order scattering in rock salt. If the line \(\lambda 2537\) Å is used as the exciting frequency, then in the scattered light it can be absorbed by mercury vapor. For this purpose, a quartz vessel with mercury vapor must be placed in the path of the scattered light between the object and the photographic plate. In some cases, to weaken the line \(\lambda 2537\) Å it is sufficient to place a cuvette with an open surface of mercury in the spectrograph chamber.

5. EXPERIMENTAL INVESTIGATIONS OF SECOND-ORDER COMBINATION SCATTERING

Investigations of second-order light scattering carried out up to the present time are very few in number.

Rasetti \(^{3}\), in investigating the scattering spectrum of rock salt, obtained an extremely weak scattering spectrum, very characteristic in its appearance. According to his data, the scattering spectrum of rock salt consists of a continuous band with sharp boundaries, extending from \(165\ \mathrm{cm}^{-1}\) to \(365\ \mathrm{cm}^{-1}\). Within this band one can note a fairly sharp line with a frequency of \(235\ \mathrm{cm}^{-1}\).

Subsequently, a joint paper by Fermi and Rasetti \(^{4}\) was published, in which some details are reported concerning the appearance of the scattering spectrum of rock salt. According to their observations, the scattering spectrum of NaCl is a continuous spectrum with the most intense region extending from \(165\ \mathrm{cm}^{-1}\) to \(365\ \mathrm{cm}^{-1}\), against which separate broad maxima of intensity are noticeable. With less certainty the authors indicate the existence of a band of the continuous spectrum immediately adjoining the Rayleigh scattering line and extending to \(60\ \mathrm{cm}^{-1}\). In the interval from \(60\ \mathrm{cm}^{-1}\) to \(165\ \mathrm{cm}^{-1}\) a continuous spectrum is also observed, but of considerably lower intensity. All these

the characteristic features of the NaCl spectrum can be seen in the microphotogram of Fig. 2, borrowed from the work of Fermi and Rasetti and giving qualitatively the intensity distribution in the scattering spectrum.

Fermi and Rasetti regard the spectrum obtained as a second-order scattering spectrum and interpret it on the basis of Born’s theory of the crystal lattice.

Ten years after the work of Fermi and Rasetti, new studies of the scattering spectrum of rock salt were carried out in Raman’s laboratory by the Indian physicist Krishnan. In his papers Krishnan^5 reports that the second-order spectrum of NaCl that he found is a discrete spectrum consisting of nine sharp lines, whose frequencies lie in the region from 134 cm\(^{-1}\) to 350 cm\(^{-1}\). According to the author, he did not find the continuous spectrum indicated by Fermi and Rasetti, either within the interval from 134 cm\(^{-1}\) to 350 cm\(^{-1}\), or outside it. The author emphasizes in his work that the spectrum he found has a discrete character.

Fig. 2. Microphotogram of the second-order scattering spectrum of rock salt according to Fermi and Rasetti.

Thus, the very first experiments already showed that the scattering spectrum of an NaCl crystal has a very peculiar character, unlike ordinary combination spectra, and that it must be regarded as a spectrum of second-order combination scattering.

Studies of second-order scattering, as already noted above, are of very great interest for the most complete investigation of the dynamics of the crystal lattice. In this connection we undertook systematic investigations of the spectra of second-order combination scattering of a number of crystals, and first of all of crystals of alkali-halide salts,^6 as having a simple crystal-lattice structure. Of special interest was a detailed investigation of the crystal of rock salt, since for this substance the infrared spectrum is fairly well known and, in addition, for the structure of NaCl the most complete theoretical calculations of the spectrum of the lattice’s natural elastic vibrations have been carried out. Comparatively recently, on the basis of these calculations, Born and Bradburn^10 calculated the second-order scattering spectrum for rock salt. Therefore experimental-

...studies of the scattering spectrum of rock salt can be directly compared with the theoretical calculation and make it possible to judge how completely and correctly Born’s theory of the crystal lattice interprets the second-order spectrum of the NaCl crystal.

The experiments we carried out with several specimens of single crystals of natural rock salt made it possible to study its scattering spectrum more fully. From the peculiar form of the spectrum obtained (see Fig. 13, a) one may with certainty conclude that the second-order scattering spectrum of a rock-salt crystal is continuous in character; against this background it was possible to distinguish and measure 10 intensity maxima. The total number of intensity maxima in the scattering spectrum of rock salt is apparently more than ten. The frequencies of the observed intensity maxima and their approximate widths are given in Table I. The sharpest and most intense of the maxima listed has a frequency of 233 cm\(^{-1}\).

Table I

Ω in cm\(^{-1}\)	Width of maxima in cm\(^{-1}\)
54	10
182	37
204	10
233	8
259	10
285	12
301	10
316	10
345	10
353	15

Beyond the maximum at 353 cm\(^{-1}\) the intensity of the spectrum falls rather sharply; then the decrease in intensity proceeds more slowly. At a frequency of about 560 cm\(^{-1}\) the intensity of the spectrum becomes close to zero, i.e., the total extent of the second-order scattering spectrum of rock salt may be estimated as 560 cm\(^{-1}\).

Near the Rayleigh line there is also observed a region of continuous spectrum extending to 60 cm\(^{-1}\). At the end of this band of continuous spectrum an intensity maximum with frequency 54 cm\(^{-1}\) is observed. In the frequency region from 60 to 200 cm\(^{-1}\) in the scattering spectrum of rock salt there is a sharp dip in the intensity of the continuous spectrum.

Recently another paper\(^{7}\) has appeared in the literature on the investigation of second-order scattering in rock salt. The results obtained in that work agree with ours, except for two additional maxima with frequencies 31 and 415 cm\(^{-1}\), which the authors observed with a spectrograph of high dispersion in the continuous scattering spectrum of rock salt.

The scattering spectrum of rock salt, characteristic in its appearance, can be interpreted on the basis of the spectrum of elastic vibrations of the lattice.

According to Born’s theory of the crystal lattice, in a crystal there should occur \(3N\) natural vibrations of the crystalline...

lattice ($N$ is the number of particles in the crystal). The spectrum of elastic vibrations of the crystal is, therefore, quasi-continuous. This applies both to the acoustic and to the optical branches of the elastic spectrum of the crystal.

In Fig. 3, a is shown the distribution, calculated by Kellermann^9 for the NaCl crystal, of the natural frequencies of the lattice, which form the 6 branches of the elastic spectrum of the crystal: 3 optical and 3 acoustic. The transverse vibrations in two mutually perpendicular directions differ little; therefore in Fig. 3, a the frequency-distribution curves are given, by branches, only for four branches: 1) transverse acoustic (t. a.), 2) transverse optical (t. o.), 3) longitudinal acoustic (l. a.), 4) longitudinal optical (l. o.). As a result of the superposition of all the acoustic and optical branches, one obtains the curve of the distribution of the natural frequencies of lattice vibrations of NaCl, representing the entire elastic spectrum of the crystal (Fig. 3, b), with an extent of $318\ \mathrm{cm}^{-1}$.

Fig. 3. Elastic spectrum of the NaCl crystal.

From Fig. 3, a it follows that in the spectrum of elastic vibrations of the rock-salt crystal there are four maxima, lying near the following frequencies: $\omega_1 = 101\ \mathrm{cm}^{-1}$ (transverse acoustic branch), $\omega_2 = 159\ \mathrm{cm}^{-1}$ (longitudinal acoustic branch), $\omega_3 = 151\ \mathrm{cm}^{-1}$ (transverse optical branch), $\omega_4 = 217\ \mathrm{cm}^{-1}$ (longitudinal optical branch).

The features of the scattering spectrum of rock salt, its continuous character, and the existence of intensity maxima can be interpreted^8 from the elastic spectrum of the NaCl crystal.

It is natural to suppose that the places of greatest density of vibrations in the elastic spectrum will correspond to intensity maxima in the scattering spectrum of the crystal. These maxima, as may be assumed, will be located approximately at those places of the spectrum which correspond to octaves and combinations of the maxima of the elastic spectrum. This supposition, expressed in such a general form, does not take into account the circumstance that not

all combinations of elastic vibrations are possible, and that the result of the interaction of elastic vibrations with the light wave in scattering will depend on the wave vectors and directions of the vibrations.

Therefore the assumption made must, of course, be regarded only as a certain approximation to the exact solution of the problem of the distribution of intensity in the spectrum of second-order combination scattering in rock salt. In Table II are pre-

Table II

Experiment: width in cm\(^{-1}\)	Experiment: mean frequency in cm\(^{-1}\)	Calculated frequency in cm\(^{-1}\)	Combination
	60	50	\((\omega_3-\omega_1)\)
	60	58	\((\omega_2-\omega_1),\;(\omega_4-\omega_2)\)
	60	66	\((\omega_4-\omega_3)\)
diffuse	134	116	\((\omega_4-\omega_1)\)
25	186	202	\(2\omega_1\)
6	237	252	\((\omega_1+\omega_3)\)
13	260	260	\((\omega_1+\omega_2)\)
15	279	302	\(2\omega_3\)
15	291	310	\((\omega_2+\omega_3)\)
15	314	318	\(2\omega_2\)
15	314	318	\((\omega_1+\omega_4)\)
10	340	368	\((\omega_3+\omega_4)\)
10	350	376	\((\omega_2+\omega_4)\)
		434	\(2\omega_4\)

sented are the combination frequencies formed from the four frequencies corresponding to the maxima of the elastic spectrum. Here are also given the experimentally observed frequencies of the intensity maxima of the second-order scattering spectrum and their widths in cm\(^{-1}\).

As is seen from the table, each combination of the frequencies \(\omega_1\), \(\omega_2\), \(\omega_3\), and \(\omega_4\) does indeed correspond to intensity maxima in the second-order spectrum of rock salt. The most complete theoretical calculation of the second-order scattering spectrum of rock salt was carried out by Born and Bradburn\(^{10}\). In doing so they used Kellermann’s\(^{9}\) data on the elastic spectrum of the NaCl lattice. In Fig. 4, borrowed from the work of Born and Bradburn, the bold line shows

shown is the computed total curve of the intensity distribution in the second-order scattering spectrum of an NaCl crystal. The dotted lines show the contribution to the scattering from various pairs of branches of the elastic spectrum. For comparison, the authors give the experimental curve of the intensity distribution in the second-order scattering spectrum of rock salt, taken from Krishnan’s work (the solid thin line). As can be seen from Fig. 4, the calculated second-order spectrum of rock salt is a continuous spectrum extending from the Rayleigh line to \(500\ \mathrm{cm}^{-1}\), the intensity maxima of which are represented in the form of small and broad peaks on the curve and lie at frequencies: 12, 50, 206, 252, 300, \(350\ \mathrm{cm}^{-1}\).

Fig. 4. Theoretical and experimental curves of the intensity distribution in the second-order scattering spectrum of an NaCl crystal according to Born and Bradburn.

Comparison of the data on the form of the scattering spectrum of rock salt, obtained in our experiments, with the above theoretical calculations shows that the theoretical curve in general correctly describes the second-order spectrum of rock salt. It undoubtedly indicates the second-order spectrum of rock salt as a continuous spectrum with intensity maxima whose frequencies agree rather well with some of those observed experimentally. However, the theoretical curve of the intensity distribution in the second-order spectrum of NaCl does not convey the details of the scattering spectrum sufficiently fully. The number of intensity maxima observed by us is more than 10, whereas the number of peaks on the theoretical curve is only 6. The most intense maximum with frequency \(233\ \mathrm{cm}^{-1}\) is entirely absent. The calculated extent of the whole spectrum is \(500\ \mathrm{cm}^{-1}\), while the extent of the spectrum observed experimentally is more than \(560\ \mathrm{cm}^{-1}\). This

is apparently explained by a certain inadequacy of the approximations adopted by Born and Bradburn in calculating the second-order spectrum. Instead of considering the 36 possible combinations of branches of the elastic spectrum, they included only 18 in the calculation.

Along with the careful investigations of the second-order scattering spectrum of rock salt, the scattering spectrum of potassium chloride—sylvine—was also studied in detail.^{6,11,12} Figure 5 gives a microphotogram that provides a clear picture of the intensity distribution in the scattering spectrum of KCl.

Fig. 5. Microphotogram of the spectrum of second-order combination scattering of a KCl crystal.

In the scattering spectrum of sylvine, the continuous spectrum stands out especially strongly, with three broad intensity maxima. One of them, comparatively narrow, lies near the frequency \(280\ \mathrm{cm}^{-1}\); the other two, broader ones, are located in the frequency regions \(100\text{—}130\ \mathrm{cm}^{-1}\) and \(179\text{—}216\ \mathrm{cm}^{-1}\). Near the Rayleigh line in the KCl spectrum a continuous band is observed, extending over \(58\ \mathrm{cm}^{-1}\). The extent of the entire scattering spectrum is about \(450\ \mathrm{cm}^{-1}\).

Because of the absence at present of theoretical calculations of the second-order scattering spectrum of KCl, it is not possible to interpret the observed picture in all its details. One can only try to explain it on the basis of the elastic spectrum of the crystal, as was done in analyzing the second-order scattering spectrum of rock salt.

According to the calculations of Iona[^13], which were made on the basis of the Born theory of the crystal lattice, the elastic spectrum of a KCl crystal is, as is evident from Fig. 6, a quasi-continuous frequency spectrum with four maxima. Two of them, the sharper ones, have frequencies \(\omega_2 = 87\ \mathrm{cm}^{-1}\), \(\omega_4 = 140\ \mathrm{cm}^{-1}\), and two, much less distinctly expressed, lie at the frequencies \(\omega_1 = 43\ \mathrm{cm}^{-1}\) and \(\omega_3 = 67\ \mathrm{cm}^{-1}\). The total extent of the elastic spectrum of KCl is about \(250\ \mathrm{cm}^{-1}\).

Fig. 6. Elastic spectrum of a KCl crystal.

Figure 6 also gives the distribution curve of the natural frequencies for the Debye continuum, for which the number of vibrations in a given frequency region is proportional to the square of the frequency.

As was already indicated above, the second-order spectrum reproduces, in the form of a spectrum of octaves and combination frequencies, the entire elastic spectrum of the crystal. Therefore, in order to interpret the spectrum, it is necessary to compile combinations of the frequencies of the maxima corresponding to the greatest densities of vibrations in the elastic spectrum of the KCl crystal. The obtained combination frequencies must correspond to the intensity maxima in the scattering spectrum of KCl. Such calculations of the maxima of the KCl scattering spectrum are presented in Table III.

In the first column of the table are indicated the possible combinations of the frequencies of the maxima of the elastic spectrum of KCl; in the second column, the frequencies of these combinations; in the third, the experimentally observed boundaries of the intensity maxima of the second-order spectrum. In the fourth column the measured width of the maxima is given.

Comparing the observed and calculated frequency values, one can interpret the band of extent \(58\ \mathrm{cm}^{-1}\) as the result of the superposition of difference combination frequencies of the elastic spectrum. The intensity maximum lying at the frequency \(280\ \mathrm{cm}^{-1}\) is the octave of the greatest density of vibrations in the elastic spectrum of the crystal. The two other intensity maxima in the KCl scattering spectrum, with boundaries \(100\text{--}130\) and \(179\text{--}216\ \mathrm{cm}^{-1}\), apparently appeared in the scattering spectrum as a result of the superposition of several pairs of combination frequencies of the elastic spectrum (see Table III).

The extent of the entire scattering spectrum determined from the elastic spectrum of the crystal (as the octave of the maximum frequency in the elas-

Table III

Combinations	Calculated frequencies in cm\(^{-1}\)	Observed limits of maxima in cm\(^{-1}\)	Width of maximum in cm\(^{-1}\)
\(\omega_3-\omega_2\)	20	From 0 to 58	58
\(\omega_2-\omega_1\)	24	From 0 to 58	58
\(\omega_3-\omega_1\)	44	From 0 to 58	58
\(\omega_4-\omega_3\)	53	From 0 to 58	58
\(\omega_4-\omega_2\)	73	From 0 to 58	58
\(2\omega_1\)	86	88
\(\omega_4-\omega_1\)	97	100–130	30
\(\omega_2+\omega_1\)	110	100–130	30
\(\omega_3+\omega_1\)	130	100–130	30
\(2\omega_2\)	134	100–130	30
\(\omega_3+\omega_2\)	154
\(2\omega_3\)	174	179–216	37
\(\omega_4+\omega_1\)	183	179–216	37
\(\omega_4+\omega_2\)	207	179–216	37
\(\omega_4+\omega_3\)	227
\(2\omega_4\)	280	280	21

in the KCl spectrum proved to be equal to 500 cm\(^{-1}\). This value is in agreement with the measured value of 450 cm\(^{-1}\).

Thus, proceeding from the elastic spectrum of the crystal, calculated on the basis of Born’s theory of the crystal lattice, it is possible to interpret the observed second-order scattering spectra of crystals of rock salt and sylvine.

In addition to the second-order scattering spectra of NaCl and KCl crystals described here, the scattering spectra of NaBr, KBr, and KI crystals were also investigated\(^{6,14}\). Microphotograms of these spectra are shown in Figs. 7, 8, 9. It is not difficult to see that the continuous character of the second-order spectrum is also manifested in these crystals; however, the observed intensity maxima are sharper than in the spectra of NaCl and KCl.

Table IV

Crystals	Frequencies in cm\(^{-1}\)	Extent of the scattering spectrum in cm\(^{-1}\)
NaBr	31, 64, 152, 181, 254	400
KBr	79, 119, 164, 217, 272	380
KI	63, 91, 105, 172, 255	300

Table IV gives the frequencies of the intensity maxima in the scattering spectra of NaBr, KBr, and KI crystals and the extents of the entire scattering spectrum.

Fig. 7. Microphotogram of the second-order combination-scattering spectrum of an NaBr crystal.

Fig. 8. Microphotogram of the second-order combination-scattering spectrum of a KBr crystal.

For NaBr, KBr, and KI crystals, there are at present no theoretical calculations that could be compared with the second-order spectra obtained. But, taking into account the identity of the lattice type of these crystals with the lattices of NaCl and KCl, one may naturally expect that their elastic spectra will also be similar.

Indeed, this is confirmed by all the scattering spectra obtained for crystals of alkali-halide salts, which turned out to be spectra of one type: continuous, with broad intensity maxima, which undoubtedly indicates the quasi-continuous character of their elastic spectra. However, comparison of the scattering spectra of the crystals investigated shows that, while retaining their general character, the spectra differ both in their extent and in the distribution of intensity. This shows that the elastic spectra of alkali-halide salt crystals must differ in their form.

Fig. 9. Microphotogram of the spectrum of second-order combination scattering of a KI crystal.

Thus, the study of second-order scattering spectra can provide certain information on the distribution of natural frequencies in the elastic spectrum of crystals.

It is also of interest to study the second-order scattering spectra of fluoride compounds, in particular LiF and NaF.

However, despite repeated attempts and comparatively long exposures (54 hours at high temperature), we were unable to obtain scattering spectra of LiF and NaF crystals. This is apparently connected with the extremely low intensity of the second-order spectra of fluoride compounds.

The extremely weak scattering of fluoride compounds becomes understandable if one considers the polarizability of the ions, whose magnitude is given in Table V according to the data of Pauling^15 and Mayer^16.

Table V

Halide	$\chi \cdot 10^{24}$	Metal	$\chi \cdot 10^{24}$
F	0.99	Li	0.025
Cl	3.05	Na	0.17
Br	4.17	K	0.8
I	6.28

From this table it is seen that the polarizability of the fluorine ion is 3.1 times smaller than that of the chlorine ion, 4.2 times smaller than that of the bromine ion, and 6.3 times smaller than that of the iodine ion. Further, the polarizability of the halide ions is greater than that of the metal ions, sometimes by many times. Thus, the polarizability of the fluorine ion is 40 times greater than the polarizability of the lithium ion.

The intensity of second-order scattering is determined by the square of the second derivative of the crystal polarizability with respect to the normal coordinates.

It may be assumed that large values of the polarizability will correspond to large values of its second derivative. Therefore, on the basis of Table V, it may apparently be considered that the greatest contribution to the intensity of the second-order scattering spectrum of the alkali-halide salt crystals considered is made by the negative ions of the crystal.

From this point of view the failure of the experiments with LiF and NaF crystals becomes understandable. These ideas are confirmed by studies of the spectra of other crystals: NaCl, NaBr, KCl, KBr, and KI. In these crystals the intensities of the second-order spectra, if estimated by exposure time, are arranged in the following series: $I_{\mathrm{KI}} > I_{\mathrm{KBr}} > I_{\mathrm{KCl}}$ and $I_{\mathrm{NaBr}} > I_{\mathrm{NaCl}}$, i.e., they decrease as the polarizability of the halide decreases.

The very low intensity of the scattering spectra of LiF and NaF, following from our experiments, is in agreement with measurements of the intensity of molecular scattering of an NaF crystal carried out by Matulevich^17.

In connection with investigations of the scattering spectra of crystals of alkali-halide salts, it is of interest to compare the results obtained from second-order spectra with data from investigations of the infrared absorption spectra of these crystals.

As is known18, 19, 20, crystals of alkali-halide salts have, in the far infrared region of the spectrum (from 20 to 150 μ), absorption bands caused by vibrations of the ions in the crystal lattice relative to one another. The frequencies of these vibrations are the proper frequencies of the lattice, characteristic for each ionic crystal. Along with the existence in these crystals of the main absorption maximum \(\omega_{\mathrm{ost}}\) (residual rays), a number of secondary maxima are also observed in them.

Despite the importance, for the theory of the crystal lattice, of the question of the existence of secondary maxima in infrared absorption spectra, at present there is still no satisfactory explanation of their origin. Born21 regards them as combination frequencies.

In comparing the second-order combination-scattering spectra and the infrared absorption spectra of crystals of alkali-halide salts, one should first of all try to establish the presence in the scattering spectrum of the octave of the principal absorption maximum of the infrared spectrum. For this purpose, in Table VI, for all the crystals studied, there are given the octaves of the principal absorption maximum (\(2\omega_{\mathrm{ost}}\)) and the frequencies of the intensity maxima \(\Omega\) observed in the second-order scattering spectra of crystals, which are close to them in magnitude.

Table VI

Crystals	\(2\omega_{\mathrm{ost}}\ \mathrm{cm}^{-1}\)	\(\Omega\) in \(\mathrm{cm}^{-1}\)
NaCl	328	316
NaBr	268	254
KCl	282	280
KBr	226	217
KI	196	188

Taking into account the comparatively low accuracy of measurements in the far infrared region of the spectrum, the agreement of the frequencies given in Table VI is quite good. This apparently makes it possible to identify some maxima of the second-order spectrum with the octaves of the principal maxima of the infrared absorption spectrum.

The secondary absorption maxima in the infrared spectrum may also find their reflection in second-order scattering. Starting from Born’s assumption21 that they belong to the combination type, one could look for them among the maxima in the second-order spectrum.

Table VII gives a comparison of the frequencies of the subsidiary maxima \(\omega_{\mathrm{pb}}\) with the frequencies of the intensity maxima \(\Omega\) of the second-order scattering spectrum for the investigated crystals of alkali-halide salts. The frequencies \(\omega_{\mathrm{pb}}\) and \(\Omega\) are expressed in \(\mathrm{cm}^{-1}\).

Table VII

NaCl	NaCl	NaBr	NaBr	KCl	KCl	KBr	KBr	KI	KI
\(\omega_{\mathrm{pb}}\)	\(\Omega\)	\(\omega_{\mathrm{pb}}\)	\(\Omega\)	\(\omega_{\mathrm{pb}}\)	\(\Omega\)	\(\omega_{\mathrm{pb}}\)	\(\Omega\)	\(\omega_{\mathrm{pb}}\)	\(\Omega\)
25	band	149	152	167	—	286	272	96	93
196	204			213	204	119	122	100	—
247	259			235	—			110	—
294	301			244	—
				303	—

As Table VII shows, it is indeed possible to associate certain frequencies of the intensity maxima in the second-order scattering spectrum with the frequencies of the subsidiary maxima. For KCl, of the five subsidiary maxima, only one can be identified in the scattering spectrum. This is possibly connected with the diffuse character of the second-order spectrum of KCl, in which the continuous spectrum appears especially strongly.

When comparing the intensity maxima of the scattering spectrum with the subsidiary maxima of the infrared spectrum, in view of the unclear nature of the latter, one could also consider another possibility for their appearance in the second-order spectrum, assuming that they are proper frequencies of vibration of the lattice. This assumption would cast doubt on the accuracy of the calculation of the elastic spectra of NaCl and KCl and, apparently, is rather unlikely. In any case, further investigations of second-order scattering spectra and their comparison with infrared spectra may contribute significantly to clarifying the nature of the subsidiary maxima and, thus, provide valuable information on the vibrations of the crystal lattice.

Briefly summarizing the above results of the study of the spectra of combinational light scattering of the second order in crystals of alkali-halide salts, one may undoubtedly conclude that, by their character, their spectra are continuous, with separate broad and diffuse intensity maxima. The extent of the spectra and the distribution of intensity differ for all the investigated crystals. The scattering spectra of the crystals can be explained by Born’s theory of the crystal lattice, but because of the extreme difficulty of theoretical

of calculations, there is at present still no sufficiently complete theory of second-order spectra even for such simplest lattices as the lattices of alkali-halide salts.

Along with investigations of the spectra of combination scattering of light in crystals of alkali-halide salts, the literature also contains investigations of the scattering spectra of other crystals. Thus, recently Krishnan \(^{22}\) studied the scattering spectra of crystals of quartz, barite, gypsum, calcite, and diamond. Using in his experiments a high-aperture light-gathering apparatus with long exposure times (of the order of several days), Krishnan succeeded in obtaining the most complete scattering spectra of these crystals. The scattering spectra discovered by him consist of a large number of separate, comparatively sharp lines. Some weak lines in the spectrum of these crystals were interpreted by Krishnan as octaves and sum combination frequencies of certain other lines observed in the scattering spectrum, and were assigned by him to the second-order spectrum.

Fig. 10. Microphotogram of the second-order scattering spectrum of a diamond crystal according to Krishnan.

Krishnan \(^{23}\) investigated the scattering spectrum of diamond in great detail. In the scattering spectrum of diamond one sharp line with frequency \(1332\ \mathrm{cm}^{-1}\) and a broad band lying in the spectral region from \(2176\) to \(2666\ \mathrm{cm}^{-1}\) are observed. The line with frequency \(1332\ \mathrm{cm}^{-1}\) belongs to first-order combination scattering and was observed earlier \(^{24}\). The band \(2176\)—\(2666\ \mathrm{cm}^{-1}\), located in the high-frequency part of the spectrum, according to Krishnan consists of 13 sharp lines, which he regards as second-order combination scattering. In Fig. 10 a microphotogram of this band is given, borrowed from Krishnan’s work. Krishnan asserts that in his experiments with diamond, as with other crystals, he did not observe the second-order scattering spectrum in the form of a continuous spectrum.

As indicated above, according to our experiments, the continuity of the spectrum, with separate intensity maxima upon it, is a characteristic feature for the second-order spectrum of all the alkali-halide crystals investigated by us.

Turning to the results for diamond, we believe that the continuous character of the second-order scattering spectrum is clearly expressed on Krishnan’s spectrogram. In general outline this spectrum is very similar to the second-order spectra of alkali-halide crystals. To be convinced of this, it is sufficient to look at the microphotogram of the spectrum of diamond (Fig. 10) and compare it with the spectrum, for example, of NaCl at room temperature (Fig. 13). At the same time it must also be taken into account that, because of the high elastic constants of diamond, its scattering spectrum, caused by the optical branches of elastic vibrations, is located in the region of high frequencies and is therefore much farther from the Rayleigh line than the second-order spectra of alkali-halide crystals.

\[ \left(\nu=\frac{\omega}{2\pi},\quad N\text{—Avogadro’s number}\right) \]

Fig. 11. Elastic spectrum of a diamond crystal.

The remarks made concerning the form of the second-order spectrum of diamond are confirmed by theoretical calculations of the elastic spectrum of diamond and by calculations, based on them, of the form of the second-order scattering spectrum of diamond, carried out by N. Smith46. In Fig. 11 we give the theoretically calculated curve of the distribution density of the natural frequencies of the mechanical vibrations of the diamond lattice. The solid line shows the resulting frequency density of all six branches of the spectrum of the mechanical vibrations of diamond; the dashed lines correspond to the densities

of natural frequencies belonging to individual branches of the spectrum. A characteristic feature of this spectrum is the small extent and the almost identical frequency of the maxima of the optical branches, which all together form a very high, comparatively narrow peak of the intensity of the elastic spectrum in the region of the frequency \(1332\ \mathrm{cm}^{-1}\), giving the frequency characteristic of diamond of the intense

Fig. 12. Theoretical and experimental curves of the intensity distribution in the second-order scattering spectrum of a diamond crystal according to \([25]\).

line in the first-order scattering spectrum. On the basis of the general theory of the second-order spectrum one may expect the reproduction of this peak of the elastic spectrum in the region of approximately doubled frequencies, near the frequency \(2660\ \mathrm{cm}^{-1}\), where the overtones and sum frequencies of the optical branches of the elastic spectrum should be located.

This expected theoretical intensity peak in the second-order spectrum of diamond is clearly visible on Krishnan’s experimental curve (Fig. 10) at the frequency \(2666\ \mathrm{cm}^{-1}\).

Further, the calculations of the intensity distribution in the second-order spectrum of diamond, carried out by Smith for the frequency region from 1800 to \(2700\ \mathrm{cm}^{-1}\), agree well with experiment. Indeed, in Fig. 12, borrowed from Smith’s work and showing the calculation of the intensity distribution in the second-order spectrum of diamond, it is seen that in the region from 2200 to \(2700\ \mathrm{cm}^{-1}\)

the theoretical intensity curve (bold line) agrees rather well with Krishnan’s experimental curve (thin line). The dashed lines in Fig. 12 schematically indicate the contribution of individual pairs of branches of the elastic spectrum of diamond to the intensity of its scattering spectrum.

6. TEMPERATURE DEPENDENCE OF SECOND-ORDER COMBINATION SCATTERING

As already indicated above, it is well known^26, 27, 28 that first-order combination scattering has a clearly expressed dependence on the temperature of the scattering body.

The question arises as to the character of the temperature dependence of the intensity of second-order scattering. What will be the behavior of the sum and difference combination frequencies? Will the ratios between the intensities of the Stokes and anti-Stokes frequencies of the second order be the same as for the first-order spectrum?

Our very first investigations undoubtedly showed that the regularities for the second and first orders would be different.

Second-order light scattering increases with temperature considerably faster than according to the linear law characteristic of the first order. An approximate estimate of the increase in intensity showed that the dependence for the second order is close to quadratic.

In addition to a strong overall increase in intensity, changes in the intensity distribution^29 were established in the second-order spectrum of heated crystals.

Figure 13 presents microphotograms of the second-order scattering spectra of a rock-salt crystal heated to temperatures from 30 to 800°C. The frequency region from 60 to 200 cm\(^{-1}\) increases sharply in intensity as the temperature of the crystals rises.

Let us turn to an interpretation of these experiments. Consideration of simple schemes for the origin of the second-order scattering spectrum immediately shows that the behavior of the sum and difference frequencies in the spectrum under temperature variation must be different. The energy transitions occurring in the crystal lattice and leading to the formation of sum and difference frequencies in the second-order spectrum are illustrated by the schemes shown in Figs. 14 and 15. When the second-order scattering spectrum arises, incident radiation of frequency \(\omega_0\) (the exciting quantum \(\hbar\omega_0\)) interacts simultaneously with two eigenvibrations of the lattice with frequencies \(\omega_i\) and \(\omega_k\). The exchange of energy between the light quantum and both lattice vibrations may lead to a Stokes or anti-Stokes shift in the scattered light

Fig. 13. Microphotograms of second-order scattering spectra of a heated NaCl crystal.

As is known, the anti-Stokes lines in the first-order spectrum are considerably more sensitive to changes in temperature than the Stokes lines. The appearance of anti-Stokes lines is due to the existence of excited vibrational states in the crystal lattice (or in the molecule), and therefore their intensity is proportional to the expression \(e^{-\frac{\hbar \omega_i}{kT}}\).

For sum combination lines of the second-order scattering spectrum, intensity ratios should be observed that are close to the intensity ratios in the first-order scattering spectrum. The intensity of the Stokes sum lines should change little with temperature (although more strongly than for Stokes lines of the same frequency in the first-order scattering spectrum). The anti-Stokes lines of the sum frequencies will depend strongly on temperature (though somewhat differently than in the first-order scattering spectrum).

Fig. 14. Scheme of formation of sum combination frequencies in the second-order scattering spectrum.

Stokes difference frequencies differ sharply in their temperature dependence from the Stokes sum frequencies of the second-order spectrum. As is seen from Fig. 15, for difference frequencies the appearance even of Stokes lines is connected with the existence of excited states of the lattice. Therefore, for difference frequencies the temperature dependence of Stokes lines in the scattering spectrum will be determined by the factor \(e^{-\frac{\hbar \omega_i}{kT}}\), which usually characterizes the temperature dependence only of anti-Stokes lines. As a result, the intensity of the Stokes difference lines increases very strongly with temperature.

Fig. 15. Scheme of formation of difference combination frequencies in the second-order scattering spectrum.

The intensity of the anti-Stokes difference frequencies of the second-order spectrum will depend somewhat more strongly on temperature than for the lines of the ordinary first-order scattering spectrum having the same frequency.

These simple visual representations show that the distribution of intensity in the spectrum must change with increasing temperature, owing to the different temperature dependence of the sum and difference frequencies.

The qualitative arguments set forth are, of course, valid under the condition that \(kT < \hbar \tilde{\omega}\), where \(\tilde{\omega}\) denotes the smallest of the frequencies of the elastic spectrum from which the difference (or sum) frequencies under consideration are formed. To a large extent these arguments are applicable to crystals with a high-frequency elastic spectrum, such as, for example, diamond, whose optical branches of the elastic spectrum lie in the frequency region above \(1000\ \mathrm{cm}^{-1}\). For such crystals, even at temperatures of about \(1000^\circ\mathrm{K}\), \(kT < \hbar \tilde{\omega}\), and the temperature dependence of the second-order spectrum will be determined by the considerations given above.

For crystals of alkali-halide salts these qualitative arguments are applicable at low temperatures, since the elastic spectrum of these crystals does not extend beyond \(300\ \mathrm{cm}^{-1}\), and the fulfillment of the inequality \(kT < \hbar \tilde{\omega}\) is achieved at temperatures not exceeding \(400^\circ\mathrm{K}\) (i.e. \(100^\circ\mathrm{C}\)).

Our experiments on heating rock-salt crystals, however, were carried out at higher temperatures (\(T > 400^\circ\mathrm{K}\)), when the above-mentioned inequality was not satisfied; on the contrary, the opposite inequality \(kT > \hbar \tilde{\omega}\) was satisfied. Therefore both the observed strong increase in the intensity of the entire spectrum and the changes in the distribution of intensity in the scattering spectrum of rock salt at high temperatures require, for their explanation, a deeper theoretical investigation.

Theory makes it possible to determine unambiguously the temperature dependence of the spectrum of second-order combination scattering\({}^{30}\) in the region of sufficiently high temperatures at which the experiments indicated above were performed. To do this we must first compose an expression for the intensity of the scattered light. With the aid of formula (23) we can find the magnitude of the projection of the electric moment \(\mathbf{P}\) of the medium on the direction \(\mathbf{e}\) of polarization of the quantum of scattered light. The vector \(\mathbf{P}\) is produced by the action of the electric field of the incident light wave \(\mathbf{E}^0=\mathbf{e}^0E^0\). The unit vector \(\mathbf{e}^0\) denotes the polarization of the quantum of the incident light. We obtain for the projection \((\mathbf{P},\mathbf{e})\) of the electric moment the expression

\[ (\mathbf{P},\mathbf{e})=\left(\sum_{x,y}\varkappa_{2}^{xy} e_x e_y^0\right)E^0, \tag{28} \]

where \(\varkappa_{2}^{xy}\) is determined by formula (23), and we now write out

explicitly tensorial quantities \(x_\nu\). To find the intensity of the spectrum at a given temperature, we must compute the matrix element from (28) and take its average over the thermal motion. As a result, accurate up to constant factors, we obtain the following expression for the intensity of the spectrum in the region of the shifted light frequency \(\omega'=\omega_0+\Omega\):

\[ I(\omega')\sim \omega'^4 \sum_{jj'} \left| \sum_{x,y} d_{jj'}^{xy} e_x e_y^{0} \right|^2 \overline{|q(\mathbf{k},j)|^2\,|q(\mathbf{k},j')|^2}. \tag{29} \]

Here \(q(\mathbf{k},j)\) denotes, in abbreviated form, the matrix element of the normal coordinate, and the bar denotes statistical averaging. The shift frequency \(\Omega\) is determined by formula (25). In doing so, we have in mind that the plus sign before \(\omega(\mathbf{k},j)\) in (25) corresponds to emission by the lattice of a quantum of vibration of frequency \(\omega(\mathbf{k},j)\), while the minus sign corresponds to absorption of a quantum of the same frequency. Thus positive frequencies \(\Omega\) correspond to observation of the shifted frequency in the Stokes region, and negative frequencies \(\Omega\) to observation in the anti-Stokes regions of the spectrum (the frequencies \(\Omega\) in both cases may be either sums or differences). In order to determine in each case the magnitude of the matrix elements of the normal coordinates \(q(\mathbf{k},j)\) in (29), we must use the following two possible values of the squared moduli of the transition matrix elements:

\[ |q_{n,n-1}(\mathbf{k},j)|^2=\frac{\hbar}{m\omega(\mathbf{k},j)}\,n \qquad \text{(transition with absorption of a quantum} \]

\[ \text{of vibration),} \tag{30} \]

\[ |q_{n,n+1}(\mathbf{k},j)|^2=\frac{\hbar}{m\omega(\mathbf{k},j)}\,(n+1) \qquad \text{(transition with emission of a quantum} \]

\[ \text{of vibration),} \tag{30′} \]

where \(n\equiv n(\mathbf{k},j)\) is the number of vibrational quanta associated with the normal coordinate \(q(\mathbf{k},j)\) in the initial state of the lattice, \(m\) is the mean mass of the atoms of the lattice, and \(\omega(\mathbf{k},j)\) is the frequency entering into (25). We can combine both cases (30) and (30′) and relate them to the choice of signs in formula (25) in the following way. Write formula (25) in the form

\[ \Omega=(-1)^{\delta_j}\omega(\mathbf{k},j)+(-1)^{\delta_{j'}}\omega(\mathbf{k},j'), \tag{25′} \]

where \(\delta_j\) and \(\delta_{j'}\) will be taken equal to zero or one. Thus, if \(\delta_j=0\), then before \(\omega(\mathbf{k},j)\) we have a plus sign in (25) and at the same time must use formula (30′), since the quantum \(\hbar\omega(\mathbf{k},j)\) is emitted by the lattice. The opposite case \(\delta_j=1\) requires the use of formula (30) for absorption of the quantum by the lattice. Therefore, if the composition of the frequency \(\Omega\) is given by the form-

… (25), then for the matrix element \(|q(\mathbf{k},j)|^2\) in (29) we must take the expression

\[ |q(\mathbf{k},j)|^2=\frac{\hbar}{m\omega(\mathbf{k},j)}\bigl(n(\mathbf{k},j)+1-\delta_j\bigr) \tag{31} \]

and for \(|q(\mathbf{k},j')|^2\) the corresponding expression with \(j\) replaced by \(j'\). Upon averaging over thermal motion we obtain, as is known,

\[ \overline{n(\mathbf{k},j)}=\frac{1}{e^{\frac{\hbar\omega(\mathbf{k},j)}{kT}}-1}. \tag{32} \]

The total intensity of the spectrum in the region of the shifted frequency \(\omega'\), according to (29), (31), and (32), is written in the form:

\[ I(\omega')\sim \omega'^4 \sum_{jj'} B_{jj'}\bigl(\overline{n}_j+1-\delta_j\bigr)\bigl(\overline{n}_{j'}+1-\delta_{j'}\bigr), \tag{33} \]

where \(\overline{n}_j=\overline{n(\mathbf{k},j)}\) is determined by formula (32). The coefficients \(B_{jj'}\) are related to \(d_{jj'}\) by the relation:

\[ B_{jj'}=\left(\frac{\hbar}{m}\right)^2\frac{1}{\omega_j\omega_{j'}}\left|\sum_{x,y} d_{jj'}^{xy} e_x e_y^0\right|^2 . \tag{34} \]

These coefficients do not depend on the temperature, which enters into (33) only through \(\overline{n}_j\) and \(\overline{n}_{j'}\).

Formula (33) gives a complete description of the temperature behavior of the intensity of the spectrum of second-order combination scattering. However, in its general form this formula cannot be used, since the coefficients \(B_{jj'}\) are unknown and calculating them is practically almost impossible. We shall show that, nevertheless, formula (33) gives an asymptotic law for the temperature dependence that is valid over a sufficiently wide range of temperatures and does not depend on the values of the coefficients \(B_{jj'}\). For this we first consider the ratio \(I_a/I_c\) of the intensities of the anti-Stokes and Stokes components corresponding to one and the same composite frequency \(|\Omega|\). As is easy to show from an analysis of formula (33) (see also (36)), this ratio is the same for the temperature factors

\[ R_{jj'}=(\overline{n}_j+1-\delta_j)(\overline{n}_{j'}+1-\delta_{j'}) \tag{35} \]

for all terms of formula (33). It characterizes the entire spectrum, independently of the numerical values of the coefficients \(B_{jj'}\). This ratio for the entire spectrum proves to be equal to

\[ \frac{I_a}{I_c}=\left(\frac{\omega_0+|\Omega|}{\omega_0-|\Omega|}\right)^4 e^{-\frac{\hbar|\Omega|}{kT}}, \tag{36} \]

where \(|\Omega|\) is the displacement frequency for the scattering line. Relation (36) coincides with the well-known formula for a first-order spectrum.

We see from this that the ratio of the intensities of the anti-Stokes and Stokes components with one and the same displacement frequency \(|\Omega|\) for a second-order spectrum is equal to the corresponding ratio for a first-order spectrum. (As was shown in \(^{30}\), this proposition can be extended to the case of a spectrum of arbitrary order.) It is therefore expedient, instead of considering the particular temperature factors \(R_{jj'}\) of the second-order spectrum, characteristic for each pair of branches of the elastic-vibration spectrum, to single out a temperature factor of the first-order spectrum common to all of them, respectively equal to:*

\[ R_c^{(1)}=\frac{1}{1-e^{-\frac{\hbar \Omega}{kT}}} \quad \text{for the Stokes component} \tag{37} \]

or

\[ R_a^{(1)}=\frac{1}{e^{\frac{\hbar |\Omega|}{kT}}-1} \quad \text{for the anti-Stokes component} \tag{38} \]

of the frequency \(\Omega\). We can then form the exact temperature factor (35) by multiplying \(R_c^{(1)}\) or \(R_a^{(1)}\) by one and the same additional factor \(R_{jj'}^*\):

\[ R_{jj'}^c=R_c^{(1)}R_{jj'}^* \quad \text{for the Stokes component }(\Omega>0), \]

\[ R_{jj'}^a=R_a^{(1)}R_{jj'}^* \quad \text{for the anti-Stokes component }(\Omega<0). \]

The advantage of this method of notation is seen from the following. It is easy to show that

\[ R_{jj'}^*=\frac{1}{2}\, \frac{\operatorname{sh}\dfrac{\hbar|\Omega|}{2kT}} {\operatorname{sh}\dfrac{\hbar\omega_j}{2kT}\cdot \operatorname{sh}\dfrac{\hbar\omega_{j'}}{2kT}}. \tag{39} \]

At high temperatures we can expand the quantity \(R_{jj'}^*\) in powers of \(\dfrac{1}{T}\). Writing explicitly only the first term of the expansion, we obtain:

\[ R_{jj'}^*= \frac{|\Omega|}{\omega_j\omega_{j'}}\,\frac{kT}{\hbar} +O\!\left(\frac{1}{T}\right). \tag{40} \]

Thus, at high temperatures the additional temperature factor becomes proportional to the absolute

temperature. This also holds for \(R_c^{(1)}\) and \(R_a^{(1)}\):

\[ R_c^{(1)}=\frac{kT}{\hbar\Omega}+\frac{1}{2}+O\left(\frac{1}{T}\right),\qquad R_a^{(1)}=\frac{kT}{\hbar|\Omega|}-\frac{1}{2}+O\left(\frac{1}{T}\right). \tag{41} \]

However, at finite \(T\) the residual term of formula (40) is of order \(T^{-1}\) and tends to zero as \(T\to\infty\), whereas the residual term of formula (41) has a finite value. Therefore, with considerably greater accuracy and over a larger temperature interval, we may approximate \(R^{*}_{jj'}\) by its limiting formula (40):

\[ R^{*}_{jj'}\simeq \frac{|\Omega|}{\omega_j\omega_{j'}}\,\frac{kT}{\hbar}, \tag{40a} \]

than we could do for the expressions \(R_c^{(1)}\) and \(R_a^{(1)}\), and consequently also for the entire expression (35) of interest to us. We see, therefore, that for crystals with a not very extended elastic spectrum, such as, for example, NaCl, there exists a rather broad temperature interval in which the temperature dependence of the entire second-order spectrum is determined by temperature factors:

\[ \left. \begin{aligned} R^{c}_{jj'} &\sim T R_c^{(1)} && \text{for the Stokes component,}\\ R^{a}_{jj'} &\sim T R_a^{(1)} && \text{for the anti-Stokes component,} \end{aligned} \right\} \tag{42} \]

but does not depend on the detailed structure of the crystal.

At very high temperatures \(\bigl(kT\gg \hbar|\Omega|\bigr)\) both expressions (42) become proportional to \(T^2\). For rock salt, whose elastic spectrum does not extend beyond \(300\ \mathrm{cm}^{-1}\), one may approximately use formula (42) already at temperatures of the order of and above \(400^\circ\mathrm{K}\).

In summary, we may say that if the condition is satisfied under which formula (42) is valid, then the shape of the second-order spectrum should change only slightly as the temperature is raised. Its temperature dependence is determined by the factors \(R_c^{(1)}\) and \(R_a^{(1)}\) entering formula (42), i.e., by the temperature factors for the first-order spectrum, which depend little on frequency at sufficiently high temperatures. At very high

*) This is also facilitated by the circumstance that in the expansion

\[ \operatorname{sh} x=x-\frac{x^3}{6}+\cdots \]

the coefficient \(1/6\) enters, and also by the fact that in (39) the denominator contains \(2kT\) instead of \(kT\). We may therefore approximately apply formula (42) if the inequalities

\[ kT>\hbar|\omega_j|_{\max} \]

and

\[ kT>\hbar|\Omega| \]

are satisfied. We could also, abandoning the expansion of \(\operatorname{sh}\dfrac{\hbar|\Omega|}{2kT}\) and the last inequality, derive a slightly modified formula for a more accurate estimate at the ends of the spectrum, where \(|\Omega|\sim 2\omega_{j,\max}\).

The intensity of the spectrum here, as shown by the measurements given in Tables VIII and IX, increases proportionally to \(T^2\), and the shape of this part of the intensity curve remains similar to itself as the temperature is raised, as is seen for NaCl from Fig. 13. With less certainty this is also observed for the low frequencies of the spectrum in the region from the Rayleigh line to \(60\ \mathrm{cm}^{-1}\).

As for the results for KBr and NaBr (Tables X and XI), for them there is observed a systematic decrease in the ratio

\[ \frac{I_{T_2}}{I_{T_1}} \]

with increasing frequency. This is explained by the fact that the temperatures at which the measurements were made here were lower than in the case of NaCl and KCl. At such temperatures the \(T^2\) law is not yet fulfilled, since the first-order temperature factor \(R_c^{(1)}\) has not yet reached the asymptotic value

\[ \frac{kT}{\hbar \Omega}. \]

For the calculation here one should use formula (42), where the exact value for \(R_c^{(1)}\) according to formula (10) must be taken. These calculations are given in the third column of Tables X and XI.

An anomalous behavior of the intensity curve of the second-order scattering spectrum is observed for NaCl in the region \(60\text{--}200\ \mathrm{cm}^{-1}\), where the intensity of the spectrum increases faster than according to the law \(T^2\), and the shape of the curve changes strongly when the crystal is heated to \(700^\circ\mathrm{C}\) (see Table VIII and Fig. 13). The increase of the intensity here does not fit within the framework of the theoretical dependence of the second-order spectrum on temperature.

The general form of the spectrum at frequencies exceeding \(200\ \mathrm{cm}^{-1}\) changes so insignificantly that we may regard the anomaly as localized in the region up to \(200\ \mathrm{cm}^{-1}\). Hence, apparently, one may conclude that, when the crystal is heated, another phenomenon is superimposed on the second-order spectrum, one which depends on temperature more strongly than the phenomenon of second-order scattering, and which leads to a deviation from the \(T^2\) law in the frequency region from 60 to \(200\ \mathrm{cm}^{-1}\) for NaCl.

At present it is still difficult to draw any definite conclusions about the nature of this phenomenon.

We consider the following mechanism of the anomaly observed by us to be possible. As is known, in the scattering spectrum of alkali-halide crystals the first-order scattering spectrum should be absent. However, the consideration that the first-order scattering spectrum is inactive as a consequence of the structure is valid only for an ideal crystal lattice. In reality, however, we always deal with a real lattice, which has various randomly distributed structural defects that disturb the regular construction of the ideal lattice. In the present case, of importance to us are those defects of the crystal lattice which strongly depend on temperature. These defects introduce

share in the polarizability and make possible the appearance of a first-order scattering spectrum. Indeed, as a consequence of the random arrangement of these defects, the selection rules are removed, since at the sites of such defects the lattice must differ greatly from an ideal one. The symmetry of the lattice cells will be disturbed, which will be reflected in the local polarizability. Upon averaging over the whole crystal, these local changes in polarizability may not lead, in contrast to an ideal lattice, to the disappearance of the first-order scattering spectrum.

Thus, we are inclined to interpret the observed strong increase of intensity in the low-frequency region in the second-order scattering spectrum as first-order scattering caused by defective sites in the rock-salt lattice. Indeed, the first-order scattering spectrum should be situated (if it were observed) precisely in that region of the spectrum in which the anomalous increase of intensity is observed when the crystal is heated. The intensity of the first-order scattering spectrum should then be determined by the number of defects of the crystal lattice and, consequently, should increase strongly with temperature in parallel with the increase in the number of defects*).

Obviously, the form of the first-order scattering spectrum in this case should differ greatly from the usual first-order scattering spectra of crystals in which only the limiting frequencies of the elastic spectrum of the crystal are observed. The scattering spectrum from defects of the crystal lattice should have the character of a continuous spectrum, since it must reproduce, to a significant degree, the entire elastic spectrum of the crystal. The form of this spectrum, however, may differ somewhat from the form of the elastic spectrum of a defect-free lattice.

We note, in particular, that the extent of the spectrum with anomalous temperature dependence is somewhat smaller than the extent of the elastic spectrum of the crystal. Thus, for example, for NaCl the region of the anomalous spectrum extends approximately from 60 to 200 cm\(^{-1}\), whereas the elastic spectrum of NaCl extends from 0 to 300 cm\(^{-1}\). An analogous observation can be made for other crystals as well. Comparison with Kellermann’s data for NaCl shows that in the regions of the spectrum of elastic vibrations not represented in the anomaly region in the scattering spectrum, there are frequencies of long-wavelength vibrations of the acoustic branches and of the longitudinal optical branch. This circumstance, it seems to us, requires theoretical interpretation. It is possible that the same properties of the reflecting

*) The appearance of defects in the lattice that give rise to a first-order scattering spectrum may be due not only to temperature, but also to the illumination itself or to the combined action of both these causes.

symmetry of alkali-halide compound crystals, owing to which the long-wavelength spectrum of limiting frequencies, usually observed in first-order combination scattering in crystals, in the present case proves to be forbidden. Very interesting additional information about this phenomenon could be provided by observations of polarization.

7. POLARIZATION OF THE SECOND-ORDER SCATTERING SPECTRUM OF A ROCK-SALT CRYSTAL

The study of the state of polarization for second-order spectra is of considerable interest, especially in connection with the anomalous increase in intensity discovered at high temperatures in the scattering spectrum of rock salt in the frequency region from 60 to 200 cm\(^{-1}\).

If first-order scattering occurs here, then the polarization in this frequency region should differ from the polarization in the rest of the scattering spectrum, owing to differences in the polarization of first- and second-order scattering.

In studying the state of polarization of the second-order scattering spectrum of a rock-salt crystal we used the setup described above, with the modification that the scattered light was collected by means of a fluorite lens and passed through a calcite crystal placed in front of the slit of the spectrograph. Such an arrangement of the polarizer made it possible simultaneously to illuminate the slit of the spectrograph with two sharply separated beams of scattered light, polarized in two mutually perpendicular directions.

Because of the very weak intensity of second-order scattering, it did not seem possible to use illumination by a narrow-angle beam directed at the scattering object at a definite angle, as is always done in polarization studies. Such a setup, necessarily of low luminosity, would have increased the difficulty of the experiment and would have greatly lengthened the exposures when photographing the spectrum. Therefore, at the present stage of the experiments, wide-angle beams of light incident at different angles were used to illuminate the object, as in the setup ordinarily employed for studying the second-order spectrum in unpolarized light.

Under these experimental conditions, the results obtained\(^{31}\) can be regarded only as preliminary, giving an approximate idea of the state of polarization of the second-order scattering spectrum of a heated rock-salt crystal.

In Fig. 16 are presented microphotograms of the second-order scattering spectra of a rock-salt crystal heated to 500°, corresponding to two different components of polariza-

…of scattered light. From examination of these microphotograms it follows that the second-order scattering spectrum of a rock-salt crystal is polarized. The spectrum of the component with the electric vector directed perpendicular to the axis of the scattered beam of light is more intense than the spectrum of the component with the electric vector directed parallel to the axis of the scattered beam. The intensity distribution in the spectra of the two components is different, which undoubtedly indicates a different state of polarization of different parts of the second-order scattering spectrum of the rock-salt crystal. In its character the spectrum of the perpendicular component differs little from the spectrum obtained with unpolarized light

Fig. 16. Microphotograms of the spectra of the mutually perpendicular-plane polarized components of light scattered by a NaCl crystal.
a—the electric vector is perpendicular to the axis of the scattered light beam; b—the electric vector is parallel to the axis of the beam.

at high temperature. The spectrum of the parallel component, on the contrary, is similar to the spectrum obtained in experiments with unpolarized light at low temperatures.

The results obtained in polarization studies of the scattering spectrum of rock salt at high temperatures show that the most polarized part of the spectrum is the low-frequency region from 60 to 200 cm\(^{-1}\), whose anomalous temperature behavior was pointed out above.

Thus, it turns out that the region of the scattering spectrum of a rock-salt crystal that increases anomalously with temperature (more strongly than according to the \(T^2\) law) also differs in its state of polarization from the rest of the spectrum; namely, the degree of polarization in this region is higher than in the rest of the spectrum. This observation also speaks in favor of the assumption that in the region of the scattering spectrum of a rock-salt crystal from 60 to 200 cm\(^{-1}\) there is superposed light scattering that differs from second-order scattering.

Recently, one more study was published\({}^{12}\) on the polarization state of second-order scattering spectra of alkali-halide crystals; it introduced nothing essentially new into the state of the question, having extended the polarization investigations only to KCl and KBr crystals.

8. OVERTONES IN THE COMBINATION SCATTERING OF LIQUIDS

Above we considered second-order combination scattering in crystals. The investigation of the temperature dependence in alkali-halide crystals and the discovered law of the temperature dependence of the spectrum intensity—proportionality to the square of the absolute temperature (the \(T^2\) law)—proved beyond doubt that the spectrum observed in these crystals is caused by second-order combination scattering.

Is second-order scattering observed in liquids? In some cases, in the combination-scattering spectra of liquids at long exposures, weak lines were observed whose frequencies could be interpreted as the sum and difference frequencies of other, more intense lines observed in the spectrum. Thus, Ornstein and Went\({}^{32}\) observed in liquid CCl\(_4\) an intense overtone with frequency \(1550\ \mathrm{cm}^{-1}\), which is interpreted as an overtone of the doublet with frequencies \(760\)—\(775\ \mathrm{cm}^{-1}\).

Anantakrishnan\({}^{33}\) extended these experiments and, besides CCl\(_4\), also investigated SiCl\(_4\). He found two new lines in liquid CCl\(_4\), with frequencies \(434\ \mathrm{cm}^{-1}\) and \(145\ \mathrm{cm}^{-1}\). The first line is interpreted as an overtone of the fundamental frequency \(217\ \mathrm{cm}^{-1}\). The second line is ascribed to the difference combination of the fundamental lines \(459\ \mathrm{cm}^{-1}\) and \(314\ \mathrm{cm}^{-1}\). In liquid SiCl\(_4\), Anantakrishnan observed the frequency \(440\ \mathrm{cm}^{-1}\), which he interpreted as an overtone of the line \(221\ \mathrm{cm}^{-1}\).

More detailed studies of the scattering spectra in liquid CCl\(_4\) and SnBr\(_4\) were carried out by Landsberg and Malyshev\({}^{34}\). They observed in these substances a number of new overtones and combination frequencies.

The most complete investigation to date with liquids in this direction was carried out by Welsh, Crawford, and Scott\({}^{35}\). In the tables given below (see Tables XII, XIII, and XIV), a summary is given of the frequencies of the spectra of liquid CCl\(_4\), SnCl\(_4\), and SnBr\(_4\) observed by these authors, and the interpretation of these frequencies.

As all these few investigations show, the intensity of the observed overtones and combination frequencies is very small. Landsberg and Malyshev estimate the intensity of such lines as being of the order of several thousandths \((3 \cdot 10^{-3})\) of the intensity of first-order lines.

Table XII

SnBr₄

	Frequency (cm⁻¹), measured	Frequency (cm⁻¹), calculated	Intensity peak
$\nu_3-\nu_2$	—	20	—
$\nu_4-\nu_1$	—	59	—
$\nu_2$	68	—	370
$\nu_3$	88	—	560
$\nu_1-\nu_3$		133	—
$2\nu_2$	134	—	3.5
$\nu_1-\nu_2$		136	—
$\nu_2+\nu_3$	151	153	1.5
$2\nu_3$		156	—
$\nu_4-\nu_3$	176	176	0.7
$\nu_4-\nu_2$	193	192	6.5
$\nu_1$	—	213	—
$\nu_4$	221	—	1000
$\nu_1+\nu_2$	280	—	180
$\nu_1+\nu_3$	—	289	—
$\nu_2+\nu_4$	303	309	—
$\nu_3+\nu_4$	—	348	—
$2\nu_1$	367	368	1.5
$\nu_1+\nu_4$	442	442	2.5
$2\nu_4$	502	501	0.9
	565	560	0.1

Table XIII

SnCl₄

	Frequency (cm⁻¹), measured	Frequency (cm⁻¹), calculated	Intensity peak
$\nu_3-\nu_2$	—	25	—
$\nu_4-\nu_1$	—	35	—
$\nu_2$	106	—	475
$\nu_3$	131	—	520
$2\nu_2$	213	212	—
$\nu_1-\nu_3$		237	—
$\nu_2+\nu_3$	241	—	—
$2\nu_3$	—	237	—
$\nu_1-\nu_2$	—	262	—
$\nu_4-\nu_3$	—	262	—
$\nu_4-\nu_2$	—	272	—
$\nu_1$	368	297	—
$\nu_4$	403	—	1000
$\nu_1+\nu_2$	475	473	265
$\nu_1+\nu_3$	—	499	0.9
	502	—	—
$\nu_2+\nu_4$	—	509	0.4
$\nu_3+\nu_4$	534	534	—
$2\nu_1$	737	736	0.8
$\nu_1+\nu_4$	773	771	0.6
$2\nu_4$	814?	806	0.05?

Table XIV

\(\mathrm{CCl}_4\)

	Frequency (\(\mathrm{cm}^{-1}\)) measured	Frequency (\(\mathrm{cm}^{-1}\)) calculated	Intensity peak
\(\nu_3-\nu_2\)	94	96	—
\(\nu_1-\nu_3\)	143	145	—
\(\nu_2\)	218	—	890
\(\nu_1-\nu_2\)	—	241	—
\(\nu_3\)	314	—	915
\(2\nu_2\)	—	436	—
\(\nu_1\)	459	—	1000
\(\nu_2+\nu_3\)	535	532	0.7
\(\nu_4'-\nu_2\)	—	544	—
\(\nu_4''-\nu_2\)	580	572	—
\(2\nu_3\)	630	628	0.7
\(\nu_1+\nu_2\)	673	677	0.4
\(\nu_4'\)	762	—	210
\(\nu_4''\)	790	—	210
\(2\nu_1\)	915	918	0.1
\(\nu_2+\nu_4'\)	989	980	0.1
\(\nu_2+\nu_4''\)	—	1008	—
\(\nu_3+\nu_4'\)	1072	1076	0.5
\(\nu_3+\nu_4''\)	1110	1104	0.6
\(\nu_1+\nu_4'\)	1221	1221	0.3
\(\nu_1+\nu_4''\)	1250	1249	0.3
\(2\nu_4\)	1540	1524	—
\(2\nu_4\)	1540	1552	17
\(2\nu_4\)	1540	1580	—
\(\nu_2+2\nu_4\)	1749	1758	0.03
\(\nu_3+2\nu_4\)	1860	1854	0.03
\(\nu_1+2\nu_4\)	1995	1999	0.04

In the work of Welsh, Crawford, and Scott this estimate is generally confirmed. This work also gives a very interesting theoretical estimate of the influence of mechanical anharmonicity on the intensity of overtones. According to their calculations, the intensity of the first overtone is about \(1:2000\) of the intensity of the fundamental tone. In their calculations the above-mentioned authors rely on the work of M. V. Wolkenstein\({}^{36}\) and make extensive use of his theory of the intensities of combination-scattering lines, based on the assumption of the additivity of the polarizability of chemical bonds.

CITED LITERATURE

1. G. Landsberg and L. Mandelstam, Naturwiss., 13, 557 (1928).
2. F. Rasetti, Phys. Rev., 34, 367 (1929).
3. F. Rasetti, Nature, 127, 626 (1931).
4. E. Fermi and F. Rasetti, Zeits. f. Physik, 71, 689 (1931).
5. R. S. Krishnan, Proc. Ind. Acad. Sci., A. 18, 298 (1943); Proc. Roy. Soc., A. 187, 188 (1946); Nature, 159, 266 (1947); Proc. Ind. Acad. Sci., A. 26, 419 (1947); Nature, 157, 623 (1946).
6. E. Gross and A. Stekhanov, Izv. AN SSSR, ser. fiz. 11, 364 (1947).
7. H. L. Welsh, M. F. Crawford and W. J. Stapll, Nature, 164, 737 (1949).
8. E. Gross, DAN, 54, 787 (1946).
9. E. W. Kellermann, Phys. Trans. Roy. Soc., 238, 513 (1940); Proc. Roy. Soc., A. 178, 17 (1941).
10. M. Born and M. Bradburn, Nature, 156, 567 (1945); M. Born and M. Bradburn, Proc. Roy. Soc., A. 188, 161 (1947).
11. A. Stekhanov, ZhETF, 20, 330 (1950).
12. C. Menzis and J. Skiner, J. phys. et radium, 9, 93 (1948); Proc. Phys. Soc., 60, 498 (1948).
13. M. Jona, Phys. Rev., 60, 824 (1941).
14. A. Stekhanov and M. Petrova, ZhETF, 19, 1108 (1949).
15. L. Pauling, Proc. Roy. Soc., A. 114, 181 (1927).
16. J. E. Mayer und M. Göppert-Mayer, Phys. Rev., 43, 605 (1933).
17. G. Motulevich, Izv. AN SSSR, ser. fiz. 11, 390 (1947).
18. M. Czerny, Zeits. f. Physik, 65, 600 (1930).
19. R. B. Barnes, Zeits. f. Physik, 72, 447 (1931); Zeits. f. Physik, 75, 723 (1932).
20. A. Mentzel, Zeits. f. Physik, 88, 178 (1934).
21. M. Born und M. Blackman, Zeits. f. Physik, 82, 551 (1933).
22. R. S. Krishnan, Proc. Ind. Acad. Sci., A. 22, 274 (1945); 22, 329 (1945); Nature, 155, 452 (1945); Proc. Ind. Acad. Sci., A. 24, 25, 40 (1946); 23, 288 (1946).
23. R. S. Krishnan, Nature, 155, 171 (1945); 160, 230 (1947); Proc. Ind. Acad. Sci., A. 24, 25, 45 (1946); Proc. Ind. Acad. Sci., A. 26, 399 (1947).
24. Robertson and Fox, Nature, 125, 704 (1930); Robertson, Fox and Martin, Phyl. Trans. Roy. Soc., A. 232, 463 (1934).
25. H. M. J. Smith, Phyl. Trans. Roy. Soc., A. 241, 105 (1948).
26. G. Landsberg and L. Mandelstam, Zeits. f. Physik, 50, 769 (1928).
27. L. Mandelstam, G. Landsberg and M. Leontovich, Zeits. f. Physik, 60, 334 (1930).

1. I. Tamm, Zeits. f. Physik, 60, 345 (1930).
2. E. Gross and A. Stekhanov, ZhETF, 17, 947 (1947).
3. E. Gross, P. Pavinskii, and A. Stekhanov, Izv. AN SSSR, ser. fiz., 12, 711 (1948).
4. E. Gross, P. Pavinskii, and A. Stekhanov, DAN, 68, 27 (1949).
5. L. S. Ornstein and J. J. Went, Proc. Amst. Acad. Sci., 35, 1204 (1932); A. Carrelli and J. J. Went, Zeits. f. Physik, 76, 236 (1932).
6. R. Anantakrishnan, Proc. Ind. Acad. Sci., 2, 425 (1935).
7. G. Landsberg and V. Malyshev, DAN, 3, 365 (1936).
8. H. L. Welsh, M. F. Crawford, and G. D. Scott, J. Chem. Phys., 16, 97 (1948).
9. M. Wolkenstein, J. Phys. USSR, 5, 185 (1941); M. Wolkenstein, M. Elyashevich, and B. Stepanov, Molecular Vibrations, vols. I and II, Gostekhizdat, 1949.