ON METHODS FOR DETERMINING THE STRUCTURE OF A LIQUID FROM X-RAY DIFFRACTION PATTERNS*

I. V. Radchenko, Dnepropetrovsk

Introduction

For a long time it was universally acknowledged that in a liquid the molecules are arranged in a completely disordered manner, and that, in its properties, a liquid is much closer to the gaseous state than to the solid. Recently, however, a number of facts have become known that compel one to assume the existence of order in the arrangement of the molecules of a liquid, and at times even the continuity of the transition from the solid aggregate state to the liquid one (Ya. I. Frenkel).

Some indications of the presence of an ordered structure in a liquid are given to us by the nature of the changes in the physical properties of the liquid during the transition from the solid state to the liquid state (specific heat, electrical conductivity, thermal conductivity, thermo-e.m.f., Raman spectra, etc.). The most complete information about the structure of a liquid, however, can be provided by X-ray investigations. The methods for determining the structure of a liquid from X-ray data constitute the subject of the present article.

Before beginning consideration of the question that interests us, we shall clarify what is usually understood by the “structure of a liquid.” It should be noted that there is still no unanimity on this question.

According to the ideas of the American physicist Stewart, a liquid at any given moment consists of a large number of regions possessing a certain orderliness in the arrangement of molecules, close to that which exists in the crystalline state. At the same time, sharp boundaries between the regions are absent; regions with a higher degree of order gradually pass into regions with a lesser degree of order, and the molecule under consideration proves to be now in a region of greater order, now in a region of lesser order. The ordered regions na-

* On the scattering of X-rays by liquids, see Korsunsky, Uspekhi fizich. nauk, X, No. 5—6, 1930; Daniels and Davisson, Uspekhi fizich. nauk, XIV, No. 4, 1934; Debye, Molecular Structure of Liquids, Uspekhi fizich. nauk, XIV, No. 7, 1934.

called by Stewart¹ “cybotactic groups.” Studying the scattering of X-rays, chiefly by organic liquids, gases near the critical temperature², and salt solutions³, Stewart and his collaborators accumulated a vast amount of experimental material in support of their theory.

The experimental basis of Stewart’s theory consisted of the following facts:

1. The blackening on radiographs of a liquid in the region of small angles is just as small as in the case of crystals (in monochromatic rays), whereas in the case of gases the blackening in the region of small angles is especially intense and decreases with increasing diffraction angle.

2. The diffraction rings on radiographs of a liquid appear near those diffraction angles at which, in the case of crystalline powders, especially intense lines appear.

3. A certain experimental confirmation of Stewart’s theory is also provided by the well-known fact that the lines on Debye–Scherrer radiographs begin to broaden⁴ as the crystallites decrease in size, and thus the radiograph of a finely dispersed solid approaches the radiograph of a liquid.

On the basis of experimental data Stewart came to the conclusion that a cybotactic group contains from 100 to 1 thousand molecules⁵. At the same time he expressed the idea that such groups may already exist in the gaseous state at temperatures considerably above the critical one, and that the decisive significance for their formation is possessed not so much by pressure and temperature as by specific volumes⁵, ⁶.

Somewhat different features were introduced into the conception of the structure of a liquid by Debye⁷ and, somewhat earlier and independently, by Prins and Zernike⁸. The starting point of this theory is the assertion that, although the distribution of molecules in a liquid or in a sufficiently dense gas cannot be specified exactly, nevertheless it can be represented with the aid of a certain probability function, the so-called “distribution function” \(W(r)\). This function determines the probability of the appearance of atoms at a given distance from an arbitrarily chosen atom. According to the theory, between \(W(r)\) and the angular distribution of the intensity of scattered X-rays there exists the following dependence:

\[ \rho(1 - W) = 2 \frac{d^{3}}{\lambda^{3}} \int_{0}^{\infty} s(1 - E)\sin 2\pi \rho s\, ds, \tag{1} \]

where \(\rho = \frac{r}{\lambda}\), \(r = r_n - r_m\) is equal to the distance between the \(n\)-th and \(m\)-th atom, \(s = 2\sin\frac{\Theta}{2}\), \(\lambda\) is the wavelength, and \(E(s)\) is the observed intensity, corrected for absorption and polarization of the X-rays. With the aid of this formula, from a given function

distributions one can calculate the intensity curve, and, conversely, from the known curve of the angular distribution of intensity one can calculate the distribution function of the molecules. By this method Debye succeeded, from the experimental intensity curve, in calculating the distribution of atoms in liquid mercury⁹ and in showing the existence of preferred distances at which the appearance of atoms is more probable than at distances slightly greater or slightly smaller. Subsequent experiments established that this arrangement of molecules in mercury corresponds to a close packing of spheres between which there are no interaction forces.

In Debye’s theory the appearance of an interference pattern (as a consequence of scattering by different atoms) is by no means connected with the existence of any quasi-crystalline regions or submicroscopic groups, but only with a definite density distribution (in atoms per cm³). Regardless of whether such regions exist or not, the diffraction pattern in both cases will be the same if the distribution of atoms is expressed by one and the same distribution function. Consequently, the indicated theory can also be used to determine the distribution of molecules in solids. By this method Gingrich and Warren found the distribution of atoms in rhombic sulfur¹⁰. Warren found the distribution function for soot¹¹.

At the same time Debye¹² showed that even with complete disorder in the arrangement of atoms*, when the probability of the appearance of some atom at any point near a given atom, with the exception of a sphere of volume \(\frac{4\pi}{3}a^3\) (\(a\) is the molecular diameter), is equal to unity, a diffraction pattern appears which has all the features typical of scattering by a liquid. In this case the scattering intensity can be calculated by the formula

\[ I = N \left\{ I_i + I_m \left[ 1 - \frac{\Omega}{V}\frac{3}{u^3}(\sin u - u \cos u) \right] \right\}, \tag{2} \]

where \(u = 4\pi \frac{a}{\lambda}\sin \frac{\vartheta}{2}\), \(\Omega = N \frac{4\pi}{3}a^3\) is equal to the sum of the effective volumes of all molecules, and \(I_i\) is the intensity of the interference of X-rays scattered by the electrons composing one molecule. From this formula it follows that, even in the absence of any groupings of molecules, at a sufficient density of the substance there must be observed an intensity maximum, becoming the sharper the larger \(\frac{\Omega}{V}\), i.e. the density of the substance. Here one has in mind the appearance of a maximum caused by the interference of rays scattered by different molecules. Harvey’s experiments¹³ with the scattering of X-rays by gases under pressure gave quite satisfactory qualitative agreement

* In the case of a dense gas.

with the results of equation (2). Garvey, on the basis of his experiments with gases at temperatures above the critical temperature, came to the conclusion that the intensity maxima obtained on radiographs are due exclusively to the density of the gas, and not to the presence of any ordered regions, the existence of which at temperatures above the critical temperature is difficult to assume.

Some authors, on the basis of their experiments, have come to the conclusion that “quasi-crystalline regions” are entirely absent in a liquid. For example, Warren¹⁴, who investigated fatty acids, normal alcohols, and paraffins by X-ray methods, came to the conclusion that, in order to obtain a diffraction pattern consistent with experiment, it is sufficient that only the nearest neighbors be arranged approximately parallel to one another. Katz investigated water, pentane, decane, benzene¹⁵, and cyclohexane and showed by calculation that the observed diffraction pattern can be explained if only the nearest four molecules are oriented in a known manner relative to one another. Zachariasen¹⁶, studying the scattering by methyl (and nonyl) alcohol, believes that if one takes into account the “shape of the molecule” and the orientation of the nearest neighbors, then the observed diffraction pattern should result, and there is no need to explain it by any “quasi-crystalline” structure.

However, one may think that the results of the works just cited do not deny altogether the significance of ordering and orientation for the formation of the diffraction pattern, but prove only that the principal role in producing the interference maxima is played by the nearest neighbors. In most cases this is indeed so. It is precisely for this reason that Keesom’s original primitive theory¹⁷ of scattering by a liquid, in which only the nearest neighbors are taken into account, often gave satisfactory agreement with experiment. However, as Prins and Petersen¹⁸ showed, cases are possible in which the appearance of the principal maximum is due not to the nearest neighbors, but to more distant ones (for example, in the case of a diamond-like structure).

In addition, it should be noted that, although—even with complete disorder in the arrangement of the molecules—the appearance of diffraction rings is possible, these maxima are nevertheless considerably less intense than in the case of the presence of some ordering. If, for example, one calculates from equation (2) the intensity curve for mercury on the assumption of a disordered arrangement of atoms and compares it with the intensity curve for close packing, one finds a certain similarity. Nevertheless this similarity is only qualitative and is confined to the region of the first maximum (Fig. 1). When ordering appears in the liquid, the first maximum becomes sharper, and further maxima arise.

Some clarity in the question of the structure of liquids may apparently be introduced by the study of changes in the X-ray diffraction pattern caused in it by various influences: by an electric field¹⁹, by a magnetic field²⁰,²¹, and by an increase in temperature³⁵.

and so forth. The study of mixtures of liquids\(^2\), as well as of metallic alloys, may also prove useful for this purpose.

The existence of two points of view on the structure of a liquid makes it interesting to consider the various methods of X-ray study of the structure of liquids and the results obtained by these methods.

Fig. 1. Intensity curve for mercury.
The solid curve is experimental.
The dashed curve was calculated by formula (2) under the assumption of a disordered arrangement of molecules.

METHODS FOR DETERMINING THE ARRANGEMENT OF MOLECULES IN A LIQUID FROM X-RAY DIFFRACTION PATTERNS

First of all it should be noted that all methods for determining the arrangement of molecules in a liquid from X-ray diffraction patterns require a well-founded and verified theory of the scattering of X-rays in liquids. However, in order to create such a theory, one must already have as accurate an idea as possible of the arrangement of molecules in a liquid. It is known that the rapid development of the X-ray study of crystals is explained by the fact that, by the time of the first attempts to apply the X-ray method to the investigation of the structure of crystals, an approximate and, incidentally, quite accurate picture of their structure, found by other methods, was already available. However, at the time when X-ray study of liquids arose, the arrangement of molecules in a liquid was completely unknown. Moreover, the question of this arrangement was not even raised, since, in accordance with van der Waals’ theory, it was believed that the arrangement of molecules in a liquid was disorderly, analogous to their arrangement in a gas. The first sufficiently accurate theory making it possible to determine the arrangement of molecules in a liquid was the theory of Prins and Zernike\(^8\) and of Debye\(^7\).

The methods for determining the arrangement of molecules in a liquid from X-ray diffraction patterns may be divided into two groups:

To the first group we shall assign those methods in which, on the basis of the experimentally found intensity distribution, the distribution function is calculated.

To the second group we shall assign all those methods in which, on the basis of certain assumptions, the distribution function is calculated or selected so that the intensity distribution obtained from it corresponds to that found experimentally.

a) Calculation of the arrangement of molecules in a liquid from the experimental intensity curve of X-ray scattering. Such a calculation was first carried out by Debye and Menke.[^9] According to Debye’s theory,[^7] the scattering intensity can be expressed by the following equation:

\[ s[1-E(s)] = 2\frac{\lambda^{3}}{d^{3}}\int_{0}^{\infty}\rho[1-W(\rho)]\sin 2\pi s\rho\,d\rho, \tag{3} \]

where \(s=2\sin\frac{\Theta}{2}\), \(d^{3}\) is the volume assigned to one atom,

Fig. 2a. Intensity curve for mercury

Fig. 2b. Radial distribution of atoms in mercury. The solid curve was obtained by calculating from the experimental intensity curve (Fig. 2a). The dotted curve corresponds to the close packing of spheres

\(W(\rho)\) is the distribution function and \(\rho=\frac{r}{\lambda}\). \(E(s)\) is related to the experimentally determined scattering-intensity function \(I(s)\) by the equation

\[ I=\frac{1+\cos^{2}\theta}{2}\,N\psi^{2}E(s), \]

where \(N\) is the number of atoms in the scattering volume and \(\psi\) is the atomic factor. By inverting this Fourier integral we obtain the expression

\[ \rho[1-W(\rho)] = 2\frac{d^{3}}{\lambda^{3}}\int_{0}^{\infty}s[1-E(s)]\sin 2\pi\rho s\,ds, \tag{4} \]

Using this equation and the experimentally obtained intensity curve for mercury, Debye and Menke obtained the distribution function \(W(r)\) of molecules in mercury (Figs. 2a and 2b). By the same formula Menke also determined the arrangement of molecules in liquid gallium (Figs. 3a and 3b). Tarasov and Warren \(^{23}\), by the same method, computed the arrangements of molecules in liquid sodium.

Fig. 3a. Experimental intensity curve for liquid gallium

Fig. 3b. Radial distribution of molecules in liquid gallium, found by calculation from the experimental intensity curve (Fig. 3a)

However, the attempt to use the indicated method for determining the arrangement of complex molecules, such as, for example, \(\mathrm{CCl}_4\), could not be successful. In this case the expression for the scattering intensity has the form \(^{24}\)

\[ I=\frac{1+\cos^2\theta}{2}N\left\{F_i-F_a\frac{\lambda^3}{d^3}\frac{2}{s}\int_0^\infty [1-W(\rho)]\sin 2\pi s\rho\,d\rho\right\}, \tag{5} \]

where \(F_i\) is the intensity term caused by interference within a single molecule. In view of the fact that \(F_i\ne F_a\), the integral cannot be inverted, and consequently \(W(\rho)\) cannot be directly determined. Note, however, that Masakatsu Ogura \(^{25}\) indicated the possibility of inverting the indicated integral in the case of \(\mathrm{CCl}_4\), taking into account,

besides scattering by atoms with unchanged wavelength (unmodified radiation), also incoherent radiation with changed wavelength (Compton radiation).* The expression obtained by him has the form

\[ W(\rho)=1+\frac{v}{2\rho\lambda}\int_{0}^{\infty}\frac{k(I-I_i)}{I_a}\sin \pi k\lambda \rho\,dk. \tag{6} \]

Fig. 4. Distribution functions for possible types of arrangement of water molecules
(Bernal and Fowler)

where \(I\) is the observed intensity, \(I_i\) is the intensity of scattering by the atoms comprising one molecule, with allowance for both modified and unmodified radiation, \(I_a\) is the intensity due to external interference, and

\[ k=\frac{4}{\lambda}\sin\frac{\Theta}{2}. \]

It is difficult to say what results the application of this equation may give for determining the arrangement of molecules in a liquid from radiographs: no such attempts have yet been made. One of the advantages of the method we have considered is that it is not connected with any model representations or arbitrary assumptions.

At the basis of the second group of methods for determining the arrangement of molecules in a liquid from radiographs, as we have already mentioned, lies the preliminary calculation or selection of the distribution function.

* In this case it is assumed that the intensities are measured in absolute units.

b) Selection of the distribution function. An example of determining the arrangement of molecules in a liquid by selecting a distribution function may be the determination of the arrangement of molecules in water carried out by J. Bernal and Fowler^26. Comparison of the experimental curve of X-ray diffraction in water* with the theoretical curve for the case of close packing (mercury-like) leads to the conclusion that the arrangement of molecules in water is substantially different from a mercury-like arrangement.

Fig. 5. X-ray scattering curves in water (Bernal and Fowler): 1—theoretical curve for close packing, 2—experimental curve, 3—theoretical curve for a quartz-type distribution, 4—theoretical curve for an ice–tridymite-type distribution

Therefore Bernal and Fowler made a number of assumptions about a possible structure of water by which its physical properties could be explained, for example, volume changes on approaching the crystallization point and in the transition from the liquid state to the solid state, the structure in the crystalline state, the high dielectric constant, etc. They then constructed distribution functions for various possible types of arrangement of molecules, namely: for an arrangement of the ice–tridymite type (curve b in Fig. 4) and for an arrangement of the quartz type (curve c).

* The most complete and precise results of the investigations belong to Meyer, Stewart, and Amaldi.

Further, from these distribution functions \(g(r)\), with the aid of Prins’ formula,

\[ I_\theta=\int_0^\infty 4\pi r^2 [g(r)-\rho]\frac{\sin sr}{sr}\,dr, \tag{7} \]

where

\[ s=\frac{4\pi \sin \frac{\Theta}{2}}{\lambda} \]

and \(\rho\) is the mean value of the density, intensity curves were calculated. Comparison of the experimental intensity curve with the theoretical curves constructed in this way (Fig. 5) showed that the experimental curve is closest to curve \(c\).

Fig. 6. Modified theoretical curve of the intensity of scattering of X-rays in water and the experimental curve
(Bernal and Fowler)

Further, Bernal and Fowler succeeded in obtaining still better agreement with the experimental curve when they somewhat modified the distribution function \(c\), assuming the presence of a certain deviation from the regular arrangement of the water molecules, especially at large (\(>4\ \text{Å}\)) distances (from an arbitrarily chosen molecule) and at elevated temperatures. The distribution \(g(r)\) for the modified quartz-like structure gives curve \(d\) (Fig. 4). The intensity curve calculated from this modified function \(g(r)\) is very close to the experimental one (Fig. 6). A certain difference in the relative intensity of the second maximum, according to Bernal and Fowler, is explained by thermal motion, which destroys the ideal structure and brings the arrangement of the molecules closer to a disordered one.

c) Calculation of the distribution function. The calculation of the distribution function \(W(r)\), or, in Prins’ notation, \(g(r)\), is associated with great mathematical difficulties. Such a calculation was carried out, for example, by Kirkwood\(^{31}\), although not for the purposes of X-ray analysis of a liquid. Applying the method of statistical mechanics to fluid mixtures, he found an approximate

the distribution function of pairs of molecules, typical for dense fluid states of matter. For the particular case of spherical molecules, between which attractive forces of the type \(-\dfrac{\gamma}{r^6}\) act, Kirkwood obtained a distribution curve \(g(r)\) which is, to a certain extent, similar to the curve of Debye and Menke for mercury. It should be noted that in this theory van der Waals forces play the chief role in producing the second maximum. In the absence of attractive forces there would be only one maximum at \(r\) equal to the diameter of the molecule. The absence of a second maximum and minimum for hard spheres is, of course, a weak point of the theory. However, this shortcoming, according to Kirkwood, is removed if a more exact calculation is carried out, which is possible, although associated with great mathematical difficulties. In the case of more exact calculations, not only the first and second maxima and minima, but also the subsequent ones, may be found theoretically.

d) Determination of the distribution function on a model. Another path to determining the distribution function is based on magnification of scale.

This method differs from the others in its simplicity and great visual clarity. Prins was one of the first to use it to clarify how the diffraction pattern arises in scattering by a liquid.

The role of molecules in Prins’ two-dimensional model of a liquid^27 was played by seeds poured in a single layer onto a glass plate. By gradually changing the density of coverage of the plate with seeds, one could directly observe their distribution. Moreover, with the aid of this model one can obtain a diffraction pattern similar to the pattern of X-ray scattering. Indeed, if the glass plate with seeds is placed on the objective lens of a vertical telescope and aimed at a distant light source, then blurred diffraction rings will appear around it (Fig. 7). In Fig. 7 it is seen how the diffraction pattern typical of a gas, with increasing density, passes into a diffraction pattern typical of a liquid. The quantitative relation between the distribution and the diffraction pattern is especially easy to establish in the case of a “gaseous state.” Simple considerations show that in this case the diffraction pattern in scattering by \(N\) particles differs from the diffraction pattern of scattering by only one particle only in that it is \(N\) times stronger. Of greatest interest is the relation between the diffraction pattern and the distribution of particles in close packing. To establish the law of distribution of the particles in this case, one may proceed as follows. Having pricked with a needle the centers of gravity of the grains, we lay upon the resulting pattern of the placement of these centers a transparent “circular scale” with concentric circles and count how many grains fall into each annular strip, and then divide the numbers obtained by

To the article by I. V. Radchenko

Fig. 7,a. Distribution of seeds on a glass plate (Prince)
Fig. 7,b. Experimental diffraction patterns of visible light corresponding to the distributions in Fig. 7,a (Prince)

Fig. 8. On determining the distribution function (Prince)

the area of the corresponding ring. This operation was repeated by Prins at 10 different places of the particle-distribution pattern available to him, and then the mean of the 10 measurements was found (Fig. 8). In this way the average density \(g(r)\) of points is determined as a function of the distance from an arbitrarily chosen point. The distribution functions \(g(r)\) for the four cases \(I, II, III, IV\) of Fig. 7 are presented in Fig. 9,a. From the known function \(g(r)\)

Fig. 9,a. Distribution functions for the four cases \(I, II, III, IV\), Fig. 7.
Fig. 9,b. On the calculation of intensity curves for the distributions of Fig. 9,a.

one can calculate the intensity of light scattering as a function of the diffraction angle, or as a function of another independent variable

\[ s=\frac{2\pi\sin\frac{\theta}{2}}{\lambda}: \]

\[ I=I_0(s)\left[1+\int_0^\infty dr\,2\pi r g(r)J(rs)\right] =I_0(s)[1-G(s)], \tag{8} \]

where \(I_0\) is the intensity that would be obtained in the case of a disordered arrangement of particles, and \(J(rs)\) is the Bessel function of zero order. The factor \([1+G(s)]\) takes into account the effect of orderliness in the arrangement of molecules. The value \([1+G(s)]\) was calculated for all four cases, and the results are presented graphically in Fig. 9,b. If the resulting expressions \([1+G(s)]\) are multiplied by

\(I_0(s)\) and compare these theoretical curves with the experimental intensity-distribution curves that can be obtained by microphotometry of the photographs in Fig. 7,a, a satisfactory agreement with experiment is obtained.

Analogous experiments were also carried out by Debye\(^{28}\) in order to study how preferred distances arise in the arrangement of mercury atoms. Steel balls were placed in a single layer in a small box with a glass lid in such a way that free spaces still remained between them. Two balls were painted black. The box was shaken repeatedly, and after each shaking the distance between the black balls was measured. Such an experiment was repeated 6 thousand times. Then, on the basis of the numerical material obtained, statistics were found for the recurrence of various distances. The result was a curve analogous to the distribution curve of atoms in mercury, found by calculation from the scattering of X-rays. These experiments confirmed the presence of close packing in mercury.

A particularly interesting model experiment is Morell’s method.\(^{29}\) In this experiment the model for the molecules consisted of solid gelatin balls about 4 mm in diameter. About a thousand such balls were placed in a gelatin solution of the same density, poured into a glass plane-parallel cuvette. The solid balls in the solution neither sank nor floated up and practically remained invisible. Among the multitude of such invisible balls, several balls (6–10) were dyed a dark-red color. A photographic plate was pressed tightly against two adjacent sides of the cuvette. An instantaneous photograph was taken simultaneously in two projections. After the photograph was obtained, the cuvette was shaken and a photograph was taken again. The total number of photographs in one series of experiments was about a thousand. From the photographs the distance between each pair of balls was determined, and the relative frequency of recurrence of one or another distance in one series of experiments was calculated. The average number of molecules \(W\) in the volume \(\frac{V}{N}\) at a distance \(r\) from the center was determined by the formula:

\[ W = \frac{ 3V(N-1)\displaystyle\sum_{r_1}^{r_2} x }{ 4\pi N \displaystyle\sum_{0}^{\infty} (x)(r_2^3-r_1^3) }, \tag{9} \]

where \(N\) is the number of molecules, \(V\) is the volume, \(r_1\) and \(r_2\) are radii, \(\displaystyle\sum_{0}^{\infty} x\) is the total number of measurements in the series, and \(x\) is the number of recurrences of each of the distances between the balls. The results of four series of experiments are given in Fig. 10. At the top, for comparison, is placed Menke’s curve for mercury, which was calculated from the experimental curve

of the angular intensity distribution. The experimental conditions are given in Table 1.

TABLE 1

By the “degree of dilution” one should understand the ratio of the total volume of the mixture to the volume that the spheres would occupy in the closest packing. Since the dilutions in series $B$ and $B'$ are approximately the same, the results of the measurements for these two series are plotted on one and the same graph $B$ in Fig. 10.

Comparison of the curves obtained with Menke’s curve for mercury confirms the fact that mercury has a hexagonal close packing, or, still more probably, in Morell’s opinion, a mixture of hexagonal and cubic close packings.

Morell’s model of a liquid offers broad possibilities for determining the distribution in the case of particles of unequal size, rod-like shape, etc.

e) Determination of $W(r)$ by calculation on the basis of model representations. We shall now consider a method for determining the distribution function by calculation on the basis of model representations. The first steps in the application of this method were already made by Prins^30 in his work of 1929. He assumed that: 1) the molecules have the form of hard spheres, 2) the forces of interaction between them are absent, and 3) the freedom of motion of the molecules is so restricted that, in a first approximation, the mean distance between molecules may be taken as equal to their diameter. Then, having assumed in advance the existence of a close packing and having used, as an auxiliary device, the distribution curves in the one-dimensional case, he found an approximate form of $W(r)$ in the three-dimensional case.

In the cases of determining the arrangement of molecules that we have just considered, it was assumed that the forces of interaction either are absent or do not play an essential role in the arrangement of molecules that arises. We shall now turn to a consideration of methods for determining the distribution function in the case when the forces of interaction cannot be neglected.

The arrangement of molecules present in a liquid can naturally be regarded as a distribution arising from the arrangement in

Fig. 10. A — experimental distribution curve of atoms in mercury (Menke); B — superposed arrangement functions of gelatin spheres at “degrees of rarefaction” 1.84 and 1.92; C — arrangement of spheres at a “degree of rarefaction” 2.84; D — arrangement of spheres at a “degree of rarefaction” 1.45; E — distributions in ideal hexagonal close-packed and cubic close-packed crystal lattices. I — hexagonal close packing. II — cubic close packing.

“gaseous state,” as, with increasing gas density, ordering forces of interaction began to act appreciably. This is the route followed, for example, by Peterlin[^31]. On the other hand, one may proceed from the fact that the given arrangement in the liquid arose from the arrangement in the crystalline state by the gradual weakening of the bonds between the atoms of the crystal as the temperature rose above the melting point. One may think that in the liquid there is then preserved, at least near the melting point, some resemblance to the structure that existed in the solid state (the so-called “blurred” structure). This idea underlies Kratky’s method, to the consideration of which we now turn.

The independence of Debye’s method from model representations or from any assumptions is, without doubt, an advantage. However, in Kratky’s opinion[^32], this is precisely its weak point. Being free of model representations, Debye’s method is incomplete, since it gives no answer to the question of how the given arrangement of molecules arose, and what part the individual atoms played in its creation. Kratky attempted to fill this gap in the theory in the following way. He assumed that the arrangements present in the liquid arose from arrangements in ideal crystal lattices by “blurring” under the action of thermal motion. Having at hand the distribution curve calculated from the experimentally found intensity distribution, and comparing it with theoretical “patterns” of distribution curves, one can decide the question of the “initial lattice,” i.e., indicate from which ideal arrangement the given arrangement could have been obtained, if it is taken into account that, owing to thermal motion, the molecules may be displaced somewhat from the positions assigned to them in the ideal structure, forming a “blurred” structure bearing traces of the original one. As the initial structure Kratky takes such ideal crystalline structures (hexagonal, cubic, cubic close-packed, body-centered cubic, tetradrical lattices), in the case of which the liquid may be regarded as monoatomic and isotropic, and then carries out their blurring first by an approximate method and then by a more exact one. Let us examine the approximate method in greater detail, since in it the fundamental ideas underlying the method appear most clearly.

Taking one atom as the central atom in the given ideal structure, we describe around it a series of concentric spheres and determine the surface density of atoms on each sphere by the formula:

\[ L=\frac{Z}{4\pi R^{2}}, \]

where \(R\) is the distance from the atom selected by us to the surface of the sphere, and \(Z\) is the number of atoms in the spherical layer under consideration. The calculated values of \(L\) are plotted as ordinates for the corresponding values of \(R\). The graph thus obtained (Fig. 11)

must be “smeared out.” Then atoms will be located not only at discrete distances from the central atom, but also at any other distances. For smearing, one must know the law of smearing. Kratky chooses, as a first approximation, a linear decrease in the probability density of the appearance of atoms as they deviate to one side or the other from the most probable position. If one then uses Einstein’s diffusion equation and takes into account that the displacement of some atom with respect to a given one is the geometric sum of all displacements of the atoms located between them, it may be assumed that the deviation of atoms from the most probable position is proportional to the square root of the distances from the initial atom. Under such conditions the smearing is carried out in the following way. Each of the vertical straight lines in Fig. 11 is replaced by a triangle whose height is equal to the corresponding value of \(L\), and whose base is proportional to the square root of the distances from the origin. The triangles thus obtained are superposed on one another, and as a result the curves shown in Fig. 12 are obtained. Comparing Debye’s experimental curve for Hg with the smeared curves of ideal structures, we find an evident similarity between the curve for Hg and the curve of the smeared hexagonal close packing, which agrees with other investigations.

Fig. 11.

We shall not dwell here on a more exact method of smearing. We note only that even in this method, in order to determine the initial structure of a liquid, it is necessary to know, by some means, the curve obtained for the distribution of atoms. Thus Kratky’s method has no independent significance and is essentially only a supplement to the existing methods of determining the arrangement of molecules in a liquid from X-ray diagrams. Kratky’s merit lies in the fact that he further developed the ideas of Prins, Mark, and others on the initial lattice[^8][^30][^36]. Prins[^18] used the idea of “smearing” the structure as an independent means of recognizing structure. Let us consider this method.

The general expression for the angular distribution of intensity in a beam of scattered X-rays according to Prins is given by

\[ i(s)=1+\int_0^\infty dr\,4\pi r^2 g(r)\,\frac{\sin(sr)}{sr}, \tag{10} \]

where

\[ s=\frac{4\pi\sin\frac{\theta}{2}}{\lambda}, \]

\(i\) is the intensity divided by the intensity that would be obtained if the same number of molecules scattered without phase relations; \(dr\,4\pi r^2 g(r)\) is the probability of finding a molecule within a thin spherical layer of thickness \(dr\) and radius \(r\), whose center is an arbitrarily chosen molecule.

In the case of an ideal structure, in the absence of thermal motion, the function \(g(r)\) for crystals is determined by purely geometrical considerations. For example, if the nearest distance between molecules is equal to \(a\), then for a simple cubic lattice we find six neighbors at distance \(a\), twelve neighbors at distance \(a\sqrt{2}\), eight neighbors at distance \(a\sqrt{3}\), etc.

In the case of the crystalline state, the expression for \(g(r)\) will not be a proper function; therefore formula (10) is better represented in the form

\[ i(s)=1+\sum_k n_k\,\frac{\sin sr_k}{sr_k}, \tag{11} \]

where the index \(k\) refers to the successive layers of molecules surrounding the given molecule, and \(r_k\) and \(n_k\) are respectively the radius and the number of molecules in the \(k\)-th spherical layer. The values of \(r_k\) and \(n_k\) for various spherical layers and various types of lattices are given in Table 2. In this table \(V\) denotes the molar volume, and \(a\) is the shortest distance between molecules. In addition, the table indicates the relation between mo-

Fig. 12. “Surface density” of atoms (according to Kratky’s theory) in the case of hexagonal close packing.
I — experimental distribution curve for atoms in mercury (Menke).
II — distribution curve with an initial hexagonally close-packed lattice.
III — distribution curve with an initial body-centered lattice.
IV — distribution curve with an initial simple cubic lattice.
V — distribution curve with an initial tetragonal lattice.

molar volume and the shortest distance between atoms for different types of lattices.

In order to derive a formula for liquids, let us suppose that in a liquid the same forces act as in the solid state. In this case \(g(r)\) will be a proper function. Let us expand it in a series of partial functions

\[ g = g_1 + g_2 + g_3 + \cdots + g_n \ \text{etc.}, \]

which pertain to the successive layers of the initial ideal structure.

The value of \(g_k\) is determined from the assumption that

\[ (r-r_k)^2=\frac{\beta r_k kT}{a^2}, \]

where \(\beta\) is the macroscopic compressibility, \(T\) the absolute temperature, \(k\) the Boltzmann constant, and \(r-r_k\) the deviation from the position corresponding to the position in the ideal structure.

Fig. 13a. Theoretical intensity curve for different hexagonal and cubic close packings (Prince)

Fig. 13b. Theoretical intensity curve for different intensities of a simple cubic lattice (Prince)

Supposing further that the “smearing” is the same along all radial directions from the molecule chosen by us, one may obtain an expression for \(g_k\)

\[ g_k=\frac{n_k}{4\pi r_k^2}\sqrt{\frac{a^2}{2\pi \beta r_k kT}}\,e^{-\frac{a^2(r-r_k)^2}{2\beta r_k kT}} . \tag{12} \]

TABLE 2

If \(g_k\) is substituted into (10) and, in the integration, \(r = r_k\) is set for the slowly varying factor \(4\pi r^2\) and for \((sr)\), but not for \(\sin(sr)\), then the calculation is completed in a rigorous manner, and in

as a result one obtains

\[ i(s)=1+\sum_k n_k \frac{\sin (sr)_k}{sr_k}\, e^{-\frac{RT\beta}{2V}(as)^2\frac{r_k}{a}}, \tag{13} \]

where \(\dfrac{RT}{2V}\) is written instead of \(\dfrac{kT}{a^3}\). The expressions \(\dfrac{r_k}{a}\), \(\dfrac{RT\beta}{2V}\), and \((sr)_k\) have zero dimension. The parameter \(\dfrac{RT\beta}{2V}\) varies for different substances within the limits from 0.01 (benzene) to 0.03 (mercury). In order to obtain an approximate idea of what the expression given above for \(i(s)\) can give (formula 13), Prince used not expression (13), but its simplified expression

Fig. 13c. Theoretical curve for a blurred body-centered cubic lattice (Prince).

Fig. 13d. Theoretical curve for the blurred structure of diamond (Prince).

\[ i(s)=1+\sum_k n_k \frac{\sin sr_k}{sr_k} e^{-0.0025(r_k s)^2}, \tag{14} \]

which has the advantage over expression (13) that in it each term is a function only of \(sr_k\). The substitution of \(r_k^2\) for \(r_k\) is not justified, but it does not have a great influence on the result.

The theoretical curves of the distribution of the intensity of X-ray scattering in liquids, obtained in this way for the various structures indicated above, are given in Fig. 13a, b, c, and d.

Fig. 14. Experimental intensity curves for mercury and gallium (Menke).

Fig. 15. Experimental intensity curve for liquid lead (Danilov and Radchenko).

Let us now note some curious features which may be found in the calculation and construction of intensity curves by this method.

In the case of structures of the type of Fig. 13a, three successive maxima owe their appearance to the nearest twelve neighbors. The effect of the more distant neighbors consists in making the first maximum sharper and in reducing the intensity for small values of \(s\).

In a structure of the type of Fig. 13b, the appearance of the first maximum is due to the 5th, 6th, and 7th terms of \(\Sigma\).

In the diamond structure the second maximum is due to the first four neighbors, and the first—to the following, more numerous neighbors.

We shall now give several examples of the application of Prins’s method for determining the structure of liquids:

1. Fig. 14 presents the experimental intensity curves

for mercury and gallium (from Menke’s work). Comparing them with Prins’s theoretical curves, we find that the experimental intensity curves for mercury and gallium are very close to the theoretical curve for a dense packing of molecules. At the same time, for gallium there is also some difference, apparently due to the character of the forces of interaction.

1. In Figs. 15 and 16 we present the experimental intensity curves for liquid lead and liquid bismuth at a temperature close to the crystallization point. These curves are taken from the as yet unpublished work of V. I. Danilov and the author of the present article.

Fig. 16. Experimental intensity curve for liquid bismuth (Danilov and Radchenko).

Fig. 17. Experimental intensity curve for liquid sodium (Tarasov and Warren).

article. Comparing these curves with Prins’s theoretical curves, we may conclude that lead in the liquid state has a packing close to dense packing, whereas in bismuth, at least near the crystallization point, the arrangement of the molecules corresponds to their arrangement in a simple cubic lattice.

1. Comparing the intensity curve for liquid sodium (Tarasov and Warren23, Fig. 17) with the theoretical curves

Prins, we find a great similarity with the blurred curve for the body-centered cubic lattice.

Thus, on the basis of the data just considered, it may be asserted that, at least near the crystallization point, traces of the “initial lattice” are preserved in the distribution of the molecules of a liquid; moreover, this distribution is determined by interaction forces apparently of the same character as in the solid state.

Comparing all that has been said about methods for interpreting X-ray diffraction patterns of liquids, the following conclusions may be drawn:

1. All methods for interpreting X-ray diffraction patterns of liquids make it possible to determine only the average distribution of the molecules of a liquid.

2. Apparently, the available methods for interpreting X-ray diffraction patterns cannot decide the question of whether we are dealing with quasi-crystalline groups (Stewart) or with some time-averaged ordering of nearest neighbors.

3. The distribution function at present serves as a convenient way of describing the “structure of a liquid,” equally suitable both in the case where the presence of sub-tactic groups is assumed and when their existence is denied.

4. Comparison of experimental data (bismuth, gallium, water, lead, sodium) with theoretical data leads us to the conclusion that interaction forces play a substantial role in the formation of order in a liquid.

LITERATURE

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