STRUCTURE OF LIQUID METALS*)

I. V. Radchenko

INTRODUCTION

1. Short- and Long-Range Order

An essential feature of the arrangement of atoms in a crystal is long-range order. It is characterized by the fact that in the lattice only discrete interatomic distances are possible. For example, for a close-packed lattice the possible distances are \(a\sqrt{n}\), where \(n\) is an integer and \(a\) is the shortest interatomic distance.

Since there is no long-range order in a liquid, the arrangement of atoms in it cannot be described by means of a space lattice. For this purpose one uses the distribution-density function \(\rho(r)\), or the probability function \(W(r)\), which represents the relative probability of simultaneously finding one atom in the volume element \(dV_1\) and another in the volume element \(dV_2\), at a distance \(r\) from \(dV_1\).

Let us suppose that an arbitrary atom, participating in thermal motion, moves together with a spherical layer formed by spheres described about it, one of which has radius \(r\), and the other a radius greater by \(dr\). As a result of thermal motion the number of atoms in the layer will vary as a function of time and of the position of the central atom. If the average, over time and space, number of atoms in the spherical layer is equal to \(n_r\), then, dividing \(n_r\) by the volume \(4\pi r^2 dr\), we obtain the average number of atoms per unit volume at a distance \(r\) from an arbitrarily chosen atom:

\[ \rho=\frac{n_r}{4\pi r^2dr}. \]

At distances of several atomic diameters, as shown by X-ray and model investigations\(^{8,9}\), \(W(r)=1\) and \(\rho=\rho_0\), where \(\rho_0\) is the average number of atoms per unit of a sufficiently large volume of liquid.

Graphically, the distribution of particles in a liquid is represented by a curve giving the function \(W(r)\) or \(\rho(r)\). These curves oscillate about the straight line \(W=1\), or, respectively, \(\rho=\rho_0\), which indicates deviations of the arrangement from a uniform one toward greater or lesser density compared with the average. In another method of graphical representation of the atomic distribution, \(r\) is plotted on the abscissa, and \(4\pi r^2\rho\) and \(4\pi r^2\rho_0\) on the ordinate. Then the curve \(4\pi r^2\rho\) oscillates about \(4\pi r^2\rho_0\).

The characteristics of the arrangement of atoms in liquids are the mean coordination number and the most probable radius of the coordination sphere. The mean coordination number is the average number of atoms located on the surface of a sphere described about an arbitrary atom with radius,

*) See also\(^{1-7}\).

equal distance from some other atom. The radius of the first coordination sphere is equal to the shortest distance between atoms. According to the theory, the mean coordination number on the first coordination sphere is determined by the area under the first maximum on the radial distribution curve\(^{10,11}\), and the mean radius of the coordination sphere by the distance from the origin of coordinates to the foot of the perpendicular dropped from the apex of the maximum to the abscissa axis.

2. On the method of the X-ray determination of the functions \(\rho(r)\) and \(W(r)\)

The functions \(\rho(r)\) and \(W(r)\) are determined from the formulas\(^{11-13}\)

\[ 4\pi r^{2}\rho(r)=4\pi r^{2}\rho_{0}(r)+\frac{2r}{\pi}\int_{0}^{\infty} si(s)\sin rs\,ds, \tag{1} \]

\[ W=\frac{\rho(r)}{\rho_{0}(r)}, \tag{2} \]

where

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

\(\lambda\) is the wavelength, and

\[ i(s)=\frac{\dfrac{I(s)}{N}-F^{2}}{F^{2}}, \tag{3} \]

where \(\dfrac{I(s)}{N}\) is the intensity of the unmodified radiation in electron units per atom, and \(F\) is the atomic factor.

\(I(s)\) is found as follows. A beam of X-rays is directed either onto a free surface of the liquid, or onto a liquid enclosed in a thin-walled glass capillary or between two parallel mica sheets. By means of photographic film or a Geiger counter, the angular distribution of the intensity of the scattered rays is measured. In the photographic method, the resulting photograph is microphotometered. From the microphotogram the intensity curve is determined as a function of

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

where \(2\theta\) is the scattering angle. After introducing into this curve corrections for absorption\(^{10,14}\) and for polarization by the sample and monochromator\(^{15}\), it is reduced to electron units, using tabulated data for the atomic factor\(^{16,17}\) and taking into account that at large \(s\) there are no interference effects, and the observed scattering corresponds to scattering by a disordered arrangement of particles. Then the integral (1) is found by means of a harmonic analyzer\(^{10}\), graphical integration, trigonometric interpolation\(^{18-20}\), or with the aid of a photomechanical device\(^{21}\). As a result, the desired curves \(4\pi r^{2}\rho(r)\) and \(W(r)\) are obtained, and from them the mean coordination number and the most probable radius of the coordination sphere are calculated.

Recently, the theoretical foundations of the method of integral analysis of intensity curves in the interpretation of X-ray patterns of liquids have been subjected to criticism\(^{22,23}\). One may disagree with this criticism\(^{24}\), but there is no doubt that the possibilities of this method are limited, and therefore, for revealing

to other methods for determining the structure. Here it is appropriate to recall that even before the method of integral analysis of intensity curves was developed, certain conclusions of a qualitative nature were already being drawn from the radiograph of a liquid. The methods used at that time have not lost their significance even today as an auxiliary means of determining structure. One such method is the calculation of the period by the Wulff—Bragg or Ehrenfest—Keesom formula^25,26. Another method consists in comparing the radiograph of the liquid with the radiograph of the same substance in the crystalline state. Mention should also be made of the method of calculating intensities on the basis of model representations^27.

3. Experimental conditions for radiography of liquid metals

The obtaining of X-ray photographs from liquid metals suitable for integral analysis of intensity curves is associated with great experimental difficulties.

To obtain a flat free surface of a metal whose surface-tension coefficient is large, it is necessary to create a surface of large dimensions^28; for example, in^29, for mercury, \(3 \times 5\ \text{cm}^2\). In^30, for the same purpose, the molten metal was pressed against a sheet of mica \(0.005\ \text{mm}\) thick, stretched on a frame previously oriented along the axis of the camera. All this complicates the design of the X-ray camera.

Owing to the great width of the maxima of the liquid and the possibility of their overlapping one another^31, the radius of the camera must be sufficiently large—\(70\ \text{mm}\) or more. This leads to long exposures—several tens of hours.

To avoid oxidation, photographs from a free surface must be taken in vacuum or in a protective atmosphere, which creates additional difficulties: elimination of fogging of the photographic film by the action of hydrogen filling the camera on the sulfur of the photographic layer^32, maintenance of a constant metal level^33–35, removal of the oxides formed^29,35, etc.

Obtaining photographs from a free surface has a number of advantages; however, this method is applicable only to metals with low vapor pressure.

At high temperatures, cooling of the photographic film and removal of the vapors formed are required^29. At a low specimen temperature it is necessary to prevent the appearance of dew or frost on its surface and on the walls of the camera^29,36.

If filtered radiation is used, then with an insufficiently thick filter false maxima may appear on the radiograph^37,33; with a thick filter the exposure must be longer than with a crystalline monochromator^39. Danilov developed and successfully applied, in the X-ray study of liquid lead, tin, and bismuth, the method of differential filters^40. However, this method is more complicated than monochromatization by reflection from a crystal.

For exact determination of the positions of the maxima on the radiograph, it is necessary to resort to calibration of the camera by means of aluminum powder, diamond, rock salt, or magnesium oxide sprinkled on the surface of the molten metal^32,41. Sometimes, for the same purpose, oxides of the metal whose radiograph is being taken are used^33. However, there is reason to suppose that this method of calibrating the camera may distort the diffraction pattern^42.

For the transition from the microphotometric curve to the intensity curve it is necessary to have a blackening scale expressing the dependence of the blackening of the photographic film on the intensity of the rays incident upon it for the given exposure time^43–45. It is also important to choose the exposure time correctly, since with underexposure or overexposure the ratio of the maxima may turn out to be incorrect^46,47.

In addition to errors connected with the experimental technique, errors may arise in calculating the radial distribution curve: a) from incorrect allowance for absorption, especially when different wavelengths are used successively to detect distant maxima \(^{46,48}\); b) from insufficient resolving power of the X-ray method \(^{49,50}\); c) from inaccurate knowledge of the intensity curve \(^{50-54}\) in the region of large and small values of \(s\); d) from the fact that the intensity curve is known not for values of \(s\) from 0 to \(\infty\), as theory requires, but for a finite interval, never greater than \(10^{-1}\) Å.

Prins \(^{56}\) considers these errors so significant that recently he has called into question the expediency of the method of integral analysis of intensity curves with construction of distribution functions, proposed in 1927 by the same authors (together with Zernike). Since experimental errors are unavoidable, the development of methods for detecting and allowing for them acquires great importance \(^{50,51}\).

The X-ray method using counters for determining intensity curves \(^{47}\), and the method of neutron-diffraction investigation \(^{57}\), are free from many systematic errors.

Finally, it should be noted that attention must be paid not only to the elimination of errors in obtaining the intensity curve and in processing the results, but also to the identity of the experimental conditions in repeated investigations. The significance of this was clarified by Sauerwald and Osswald \(^{42}\). They compared the periods for mercury calculated by various authors who had studied mercury by X-ray methods, and suggested that the differences are explained not only by experimental errors, but also by real differences in structure connected with the conditions under which the radiogram was obtained. This conclusion was confirmed by the fact that in vacuum and gas-filled cameras Sauerwald and Osswald obtained results differing somewhat from one another. In addition, it turned out that the surface tension of mercury is noticeably affected by the presence in the camera of gas, as well as of rock-salt powder or another substance used for calibrating the camera. These changes in surface tension are in some way connected with changes in the structure of the liquid, which receive a known reflection in the radiogram.

Fig. 1. Probability distribution curve for gelatin spheres in a gelatin solution, in comparison with the probability curve for mercury: —— for mercury, ·—·—·— for gelatin spheres.

MAIN RESULTS OF INVESTIGATIONS OF THE STRUCTURE OF LIQUID METALS

Mercury. Earlier and more fully than other liquid metals, mercury has been studied by X-ray methods. It is of interest because, in its thermodynamic properties, it is close to an ideal liquid, and one may expect that such a liquid has the densest packing of atoms.

The initial X-ray studies of mercury were reduced to obtaining satisfactory photographs \(^{58-63}\) and to calculating periods

identity by the Wulff–Bragg or Keesom–Ehrenfest formulae \(^{25,26}\). The basis for such a calculation was the assumption, already expressed in one of the very earliest papers \(^{59}\), that the diffraction pattern obtained from mercury is explained by the presence of a certain ordering in the arrangement of atoms.

Prins \(^{64}\) found that the theoretical intensity curve of scattered X-rays which he obtained for a close packing of atoms is very close to the experimental curve for mercury.

Debye and Menke \(^{65}\), for the first time, calculated the radial distribution function of mercury atoms from the experimental intensity curve. This curve was compared by Debye with the distribution curve of small spheres obtained by him in a model experiment, there being no interaction forces between the spheres \(^{10,28,65}\). A good agreement of these curves was found. It was also confirmed by subsequent experiments \(^{8,66}\). For clarity, a figure from the work of Morell and Hildebrand \(^{66}\) is reproduced (Fig. 1). The solid line shows the probability curve of the distribution for mercury atoms, the dotted line that for hard gelatin spheres suspended in a gelatin solution. Details may be found in \(^{6}\).

Krátký \(^{67}\) proposed that the arrangement of atoms in a liquid may be represented as arising from an arrangement in some ideal crystal lattice by its blurring as a result of thermal motion. Comparing Debye’s experimental curve for mercury with curves obtained by blurring various ideal lattices, Krátký found an obvious similarity with the curve for a blurred hexagonal close packing (Fig. 2).

Prins \(^{68}\) developed a theory that makes it possible to calculate intensity curves of scattered X-rays for blurred structures and thus, directly, i.e. without calculating \(W(r)\), to determine the structure of a liquid by comparing the experimental intensity curve with theoretical ones. By this method also, a similarity was found between the structure of liquid mercury and the structures of blurred close packings.

Danilov and Neimark \(^{36}\) investigated mercury by X-ray methods at lowered temperatures and, on the basis of a comparison of the intensity curves for liquid mercury with the X-ray diffraction pattern for solid mercury, concluded that near the crystallization point the structure of liquid mercury differs from a close-packed one and approaches its structure in the crystalline state, i.e. the rhombohedral one \(^{69}\). In order to verify the correctness of this conclusion, it was necessary to obtain atomic-distribution curves at different temperatures and to trace their change as the crystallization point is approached. The first investigations of this kind were carried out by Boyd and Uecker \(^{29}\).

Boyd and Uecker obtained photographs of liquid mercury in filtered radiation at various temperatures. In \(^{29}\) the intensity curves are presented for the temperatures \(-34^\circ\) and \(+175^\circ\) (Fig. 3) and distribution curves for the temperatures \(-36^\circ\), \(-34^\circ\), \(0^\circ\), \(30^\circ\), \(75^\circ\), \(125^\circ\), \(175^\circ\), \(250^\circ\) (Fig. 4). The authors took credit

Fig. 2. Theoretical probability curves for five packings of spheres and the probability curve for mercury according to Debye and Menke. \(Z\) denotes the coordination number for the first sphere: a) curve for mercury according to Debye and Menke, b) for hexagonal close packing, c) for cubic close packing, d) for a body-centered lattice, e) for a simple cubic lattice, f) for a tetragonal lattice.

discovery of a maximum on the intensity curve at

\[ \frac{\sin \theta}{\lambda} \approx 0.1, \]

which had not been found by anyone before them, and attributed this success to the correct choice of the angle between the mercury surface and the beam of X-rays incident upon it \((\alpha = 4^\circ 11')\). Since, however, other authors \(^{10,60}\), obtaining photographs of mercury under analogous experimental conditions, did not detect this maximum, it may be assumed that it is spurious and that its appearance in the experiments \(^{29}\) is explained by the use of filtered radiation \(^{41}\), as in the experiments with liquid sodium \(^{39}\). It is strange, however, that Menke \(^{10}\) did not detect splitting of the principal maximum in the study of mercury, although he, like Boyd and Uokeryem, used

Fig. 3. Intensity curves for mercury according to Boyd and Uokeryem.

Fig. 4. Curves of the radial distribution of mercury according to Boyd and Uokeryem.

radiation from a molybdenum anticathode filtered by zirconium at a voltage of 37 kV. To resolve the question it would have been necessary to use strictly monochromatic radiation.

Campbell and Hildebrand \(^{33}\) used Mo-\(K_\alpha\) radiation, monochromatized by reflection from a crystal, and obtained photographs of mercury at temperatures: \(-38^\circ\), \(0^\circ\), \(50^\circ\), \(100^\circ\), \(150^\circ\), \(200^\circ\). As in work \(^{29}\), splitting of the principal maximum on the intensity curve was found (Fig. 5). However, Hendus \(^{41}\), taking into account that the results \(^{29}\) contradicted the investigations of other authors (he does not mention work \(^{33}\)), again carried out an X-ray study of mercury in Cu-\(K_\alpha\) and Ag-\(K_\alpha\) radiations, monochromatized by reflection from a crystal. In this case an intensity curve was obtained similar to Menke’s curve \(^{10}\) and differing from the curve of Boyd and Uokeryem \(^{29}\)—without a maximum at

\[ \frac{\sin \theta}{\lambda} = 0.1 \]

(Fig. 6). It should be noted that Boyd and Uokeryem themselves \(^{29}\), using monochro-

...radiation, did not find*) the indicated maximum. Since, however, this maximum has been found in some studies of liquid mercury and not in others, and moreover not only in filtered but also in monochromatic radiation, it is natural to suppose that it is false and is due to the peculiarities of the photographic method of measuring intensity. Indeed, Jennings \(^{70,109}\), using a Geiger counter to measure the intensity, obtained a curve for mercury at \(22.7^\circ\) without an internal peak on the principal maximum. In Fig. 7 this curve is given as a function of \(kR\), where

Fig. 5. Intensity curves for mercury according to Campbell and Hildebrand.

Fig. 6. Intensity curve for mercury according to Hendus in comparison with the intensity curves of other authors: — according to Hendus, — — — according to Boyd and Wakeham, ·—·—·— according to Menke.

\[ k=\frac{4\pi}{\lambda}\sin\theta \]

and \(R\) is the radius of the mercury atom.

The unreality of the peak under consideration was also confirmed by neutron-diffraction studies of Vineyard \(^{71}\) (Fig. 8) and by X-ray studies carried out by several authors \(^{72}\) in 1955.

Considering the radial-distribution curve for liquid mercury calculated by Hendus (Fig. 9), we see that the position of the principal maximum on this curve is the same as in the crystal, namely at \(3\,\text{\AA}\), while the small maximum at \(r=3.47\,\text{\AA}\) coincides exactly with the position of the second line in the X-ray pattern of crystalline mercury,

Fig. 7. Intensity curve for mercury according to Jennings.

*) See the note in \(^{41}\) on p. 419.

By measuring the area under the first maximum, Hendus found 6 atoms in the first coordination sphere, i.e., the same number as in the crystal. In the second coordination sphere, with radius 3.47 Å, in the liquid

Fig. 8. Neutron-scattering curve for mercury according to Vineyard.

4 atoms were found, whereas in the crystal there are 6. We note that Boyd and Waker^29 also found the coordination number for mercury at \(-38^\circ\) to be equal to 6, despite the splitting of the main maximum. Thus, in the present case the shape of the intensity curve in the small-angle region did not affect the coordination number, which apparently is accidental.^53

Fig. 9. Radial distribution curve of mercury atoms according to Hendus.

The second maximum on the radial distribution curve of mercury coincides with the position of two coordination spheres of the crystal, with 12 atoms in each, and lies at approximately a distance of 6 Å. At still greater distances the distribution approaches a uniform one.

In a neutronographic investigation, Vineyard^71, from the radial distribution curve (Fig. 10), found 8.3 atoms in the first coordination sphere, whereas, carrying out a recalculation from the curve of Menke^10, he obtained 10 atoms. Vineyard considers the small subsidiary maxima on both sides of the main one (Fig. 10) to be false, on the grounds that their occurrence was predicted by theory^51 precisely at the places where they were found, namely at 2.26 Å and 3.94 Å. It should be noted that a subsidiary maximum on the radial distribution curve of mercury at \(r = 4\) Å was also found in the work of Hendus^41 (Fig. 9) and in the work of Campbell and Hildebrand^33 (Fig. 11),

Fingbar[^50], using the intensity curve of Campbell and Hildebrand (Fig. 5), confirmed by calculation the suggestion they had made that the secondary maximum at \(r = 4 \text{ Å}\) is due to the insufficient resolving power of the X-ray method.

The question of the reality of the secondary maxima on the radial-distribution curves of mercury and other liquid metals deserves special attention, because conclusions about the nature of the translational motion of atoms in a liquid are drawn from their position and temperature dependence. For example, Campbell and Hildebrand explain the secondary maximum at \(r = 4 \text{ Å}\) on the curve for mercury by saying that when atoms from the first and second coordination

Fig. 10. Radial-distribution curve of mercury atoms according to Wainard.

Fig. 11. Radial-distribution curves of mercury atoms according to Campbell and Hildebrand.

spheres exchange positions, they must overcome a certain potential barrier and for some time are delayed at the top of this barrier[^33]. Samoilov[^73],[^102] proposes a more plausible and graphic explanation, connecting the appearance of distances that could or could not exist in the equilibrium arrangement of atoms with the temporary delay of a moving atom in a void formed by the equilibrium distribution of the atoms surrounding it.

Considerations as to how the translational motion of atoms is reflected in the X-ray pattern can evidently be sufficiently justified only under the condition that reliable experimental data are obtained.

On the basis of the works considered on the study of the structure of mercury, the following conclusion may be drawn. At a temperature close to the crystallization point, the structure of liquid mercury corresponds to its structure in the crystalline state, and the coordination number is the same as in the crystal, namely 6; moreover, the most probable radius of the coordination sphere is \(3 \text{ Å}\). With increasing temperature the coordination number increases to 8–10, at high temperatures reaches 12, and the structure becomes close-packed.

I. V. RADCHENKO

METALS WITH DENSE PACKING OF ATOMS IN THE SOLID STATE

Gold. Gold is an excellent example of the similarity of structure in the solid and liquid states. This is manifested in the following.

A. The principal maximum on the intensity curve[^32] for 1100° (Fig. 12) coincides with the first line on the X-ray photograph of gold powder. With all the other maxima of liquid gold there coincide the positions of the centers of gravity of groups of lines located under these maxima, determined by the formula

\[ X=\frac{\sum J_i x_i}{\sum J_i}, \]

where \(J_i\) is the intensity of a line, and \(x_i\) is the position of the line.

B. The mean radius of the first coordination sphere of liquid gold is equal to the shortest interatomic distance in the gold crystal, namely \(2.86\) Å.

C. From determining the area under the first maximum on the atomic distribution curve[^32] (Fig. 13), it follows that in the first coordination sphere of liquid gold there are, on average, 11 atoms, i.e., almost as many as in the crystal, 12.

Fig. 12. Intensity curve for gold according to Hendus.

Fig. 13. Radial distribution curve of gold atoms according to Hendus.

Lead. In the initial X-ray studies of liquid lead[^34,^35,^74] it was concluded that it has a close-packed structure. This conclusion corresponded to the assumption that metals having dense packing of atoms in the solid state retain it also after melting[^34,^35]. Subsequently the conclusion was confirmed by electron-diffraction studies[^75]. However, Gloker and Hendus[^32,^76], on the basis of integral analysis of the intensity curve (Fig. 14), came to the conclusion that the structure of lead changes sharply on melting: each atom of liquid lead has 8 neighbors at a distance \(r=3.40\) Å and 4 neighbors at \(r=4.37\) Å, whereas in the crystal there are 12 atoms at \(r=3.49\) Å and 6 at \(r=4.95\) Å. Thus,

thus, according to Glocker and Hendus, upon melting of lead the coordination numbers and the shortest interatomic distance decrease.

Chamberlen^57 investigated liquid lead neutronographically at a temperature of 390° C. By measuring the area under the first maximum on the distribution curve (Fig. 15), it was found that each atom has, on average, \(12 \pm 1\) neighbors at a distance of 3.40 Å, i.e. 0.8 Å less than the radius of the first coordination sphere in the crystal. This decrease in the shortest interatomic distance with the coordination number unchanged can be reconciled with the fact that the volume of lead increases upon melting^77,78 by 3.5% only on the condition that the local rarefactions^73 in liquid lead are sufficiently large.

Fig. 14. Intensity curve for lead according to Hendus.

Danilov and his coworkers studied liquid lead radiographically at a temperature 10–20° above the melting point^30. On the basis of an analysis of the radial distribution curve and a comparison of its maxima with the distribution of lines characterizing the close packing of atoms in solid lead, the authors concluded that there are no sharp changes in lead upon its melting and that the work^32 is erroneous. The coordination—

Fig. 15. Radial distribution curve of atoms in liquid lead according to Chamberlen.

—number was found to be equal to 11, and the mean interatomic distance 3.43 Å, i.e. closer to the interatomic distance in the crystal than in^32.

Fig. 16. Curves of the radial distribution of atoms in liquid lead according to Sharpe and Smith.

Fig. 17. Intensity curves for liquid lead according to various authors.

Fig. 18. Curves of \(W\) for liquid lead according to various authors.

Sharp and Smith \(^{79}\) carried out a neutron-diffraction study of liquid lead at two temperatures: \(350^\circ\) C and \(550^\circ\) C. The radial distribution curves they obtained are given in Fig. 16. The radius of the first coordination sphere proved to be \(3.40\ \text{Å}\) both at \(350^\circ\) and at \(550^\circ\), i.e., the same as in \(^{32}\) and \(^{57}\), while the coordination number for the first sphere was 9.4 at \(350^\circ\) and 9.5 at \(550^\circ\). These values of the coordination number for liquid lead are intermediate between 8, found in \(^{32}\), and 11—12 in \(^{30,57}\).

Thus, the question of the coordination number for liquid lead remains unresolved. Attention is drawn, however, to the fact that the intensity curves obtained by different authors and by different methods differ little from one another, at least in the positions of the maxima and in their general appearance*) (Fig. 17), whereas there are substantial differences in the probability curves (Fig. 18). It is therefore natural to suppose that the discrepancies in the estimate of the coordination number in this case arise in the process of the integral analysis of the intensity curves.

Thallium. Without dwelling on the initial study of liquid thallium \(^{34}\), let us consider the results of Hendus’s work \(^{32}\). As with gold and lead, the principal maximum on the intensity curve of thallium (Fig. 19) coincides with the line corresponding to the distance between the (002) planes for a hexagonal lattice.

Fig. 19. Intensity curve for thallium according to Hendus.

Fig. 20. Intensity curve for indium according to Gamertsfelder.

The coordination number proved to be 8 at a radius of the coordination sphere \(3.30\ \text{Å}\), while the next coordination number is equal to 4 at \(r = 4.42\ \text{Å}\), whereas in the crystal it is equal to 12 at \(r = 3.45\ \text{Å}\). In subsequent investigations it should be ascertained whether this result is not due to the fact that the intensity curve was determined only up to comparatively small \(s\).

Indium. Gamertsfelder \(^{80,81}\) obtained intensity curves for liquid indium at two temperatures: \(160^\circ\) and \(390^\circ\) C (Fig. 20). From the radial distribution curve (Fig. 21) for \(160^\circ\), 8.5 atoms were found at \(r = 3.17\ \text{Å}\) and 4 atoms at \(r = 3.37\ \text{Å}\), or 12 atoms at the mean distance \(3.32\ \text{Å}\). Thus, according to this study, the coordination number of indium upon melting decreases from 12 to 8.

*) In the curve of work \(^{79}\) the atomic factor is not excluded.

Fig. 21. Curves of the radial distribution of indium atoms according to Gamertsfelder.

Fig. 22. Intensity curve for indium according to Hendus.

Fig. 23. Intensity curve for cadmium according to Gamertsfelder.

Fig. 24. Curve of the radial distribution of cadmium atoms according to Gamertsfelder.

When comparing the intensity curves for \(160^\circ\) and \(390^\circ\), we see that they differ little from one another: only the first and third maxima are somewhat broader and higher at the higher temperature. It follows from this that the short-range order in the liquid is preserved even when the temperature is raised by more than \(200^\circ\).

It is interesting to note that on the atomic distribution curve for \(390^\circ\) (Fig. 21), at \(r = 4.3 \text{ Å}\), there appears a maximum which was not present on the curve for \(160^\circ\). As Samoilov pointed out \(^{102}\), this maximum is apparently connected with the translational motion of the atoms and, consequently, with local rarefactions.

Glocker and Hendus \(^{32,76}\) investigated liquid indium at \(165^\circ\). The intensity curves obtained by them (Fig. 22) differ hardly at all from those found in \(^{80}\) and from the microphotometric curve of the electron diffraction pattern \(^{75}\) of liquid indium. In \(^{32}\), as also in \(^{80}\), it was found that upon melting indium the coordination number decreases from 12 to 8.

Cadmium. The intensity curve \(^{80,81}\) for liquid cadmium (Fig. 23) is similar to the curve for indium at \(350^\circ\text{C}\). On the radial distribution curve (Fig. 24) a maximum was found at \(r = 4 \text{ Å}\), to which no coordination sphere of the crystal corresponds. In liquid cadmium, 8.3 neighbors of each atom were found at \(r = 3.06 \text{ Å}\), whereas in the crystal there are 6 at \(r = 3.16 \text{ Å}\) and 6 at \(r = 3.34 \text{ Å}\). All this may indicate a change in the structure of cadmium upon melting. But for this it is necessary to prove the reality of the maximum at \(r = 4 \text{ Å}\).

Aluminum and zinc. In the work of Randall and Rooksby \(^{74}\) it was found that the position of the principal maximum on the X-ray diffraction pattern of liquid aluminum coincides with the first line on the X-ray diffraction pattern of solid aluminum.

Fig. 25. Intensity curve for aluminum according to Gamertsfelder.

Fig. 26. Intensity curve for liquid zinc according to Gamertsfelder.

Gamertsfelder \(^{80,81}\) obtained photographs of liquid aluminum and zinc in monochromatic radiation. A special feature of the intensity curves of these two metals is the presence of an additional maximum on the inner side of the principal one (Figs. 25 and 26). From the radial distribution curves (Figs. 27 and 28), for both aluminum and zinc, 10.8 atoms were found...

in the first coordination sphere. The results of X-ray studies of aluminum and zinc make it possible to say that, in the liquid state, the atomic distribution in these two metals is almost close-packed.

Fig. 27. Radial distribution curve of aluminum atoms according to Gamertsfelder.

Fig. 28. Radial distribution curve of zinc atoms according to Gamertsfelder.

ALKALI METALS

Sodium. Qualitative X-ray studies of liquid sodium were carried out by Keesom^82 and by Randall and Rooksby^83. The first quantitative investigations belong to Tarasov and Warren^84. Let us consider the results of the later investigation by Trimble and Gingrich^85. These authors obtained X-ray photographs of liquid sodium at eight temperatures, using Mo-\(K_{\alpha}\) radiation and placing the liquid in glass capillaries. In work^85, intensity curves (Fig. 29) and radial distribution curves for two temperatures, \(100^\circ\) and \(400^\circ\) (Figs. 30 and 31), are given. It was found that, for liquid sodium at \(100^\circ\)C, there are eight atoms at \(r = 3.83\) Å, whereas in the lattice there are 8 atoms at \(r = 3.72\) Å. Thus, upon melting of sodium its coordination number does not change. The radial distribution curves for sodium were used by Wall^86 to calculate the heat of fusion, and satisfactory results were obtained.

Fig. 29. Intensity curves for liquid sodium according to Trimble and Gingrich.

Potassium. Without dwelling on the initial studies of liquid potassium near the melting point^81, ^82, let us consider the results of quantitative,

Fig. 30. Radial distribution curve for liquid sodium at 100° according to Trimble and Gingrich.

Fig. 31. Radial distribution curve of the atoms of liquid sodium at 400°C according to Trimble and Gingrich.

Fig. 32. Intensity curve for liquid potassium at 70°C according to Thomas and Gingrich.

Fig. 33. Intensity curve for liquid potassium at 395°C according to Thomas and Gingrich.

studies of Thomas and Gingrich[^87]. From the intensity curves (Figs. 32 and 33) these authors constructed distribution curves (Figs. 34 and 35). From these curves eight neighbors were found for each atom both at \(70^\circ\mathrm{C}\) and at \(390^\circ\), i.e., the same number as in the crystal. Thus even at a temperature more than \(300^\circ\) above the melting point, the structure of liquid potassium differs from close-packed. This conclusion is supported by the following considerations.

The first maximum on the atomic-density curve shifts from \(r = 4.67\ \text{\AA}\) (at \(70^\circ\)) to \(4.76\ \text{\AA}\) (at \(390^\circ\)). If the structure of potassium, after melting, changed from a blurred body-centered arrangement to a close-packed one, then the shift in the position of the first maximum would be larger, namely, according to Danilov’s calculations[^88], from \(4.67\) to \(4.95\ \text{\AA}\). If, however, it is assumed that the packing of the body-centered type is preserved also at the higher temperature, then the calculated change in the position of the maximum agrees with that found experimentally.

Fig. 34. Radial distribution curve of potassium atoms at \(70^\circ\mathrm{C}\), according to Thomas and Gingrich.

Fig. 35. Radial distribution curve of potassium atoms at \(395^\circ\mathrm{C}\), according to Thomas and Gingrich.

X-ray investigation of liquid potassium represents one of the few examples of the use of experimental atomic-distribution functions for calculating physical constants. Thus, Hildebrand[^89] calculated from the data of work[^87] the ratio of the heats of evaporation at \(70^\circ\) and at \(390^\circ\), and obtained good agreement with experiment, although this, in the opinion of Porai-Koshits[^90], may be regarded as accidental, since the calculation was based on an assumption, clearly at variance with reality, that the interaction is due to dispersion forces. Further, Gingrich and Wood[^91], from the temperature dependence of the atomic-distribution curves, calculated the entropies at the melting point and at the boiling point, as well as the heats of melting and evaporation, obtaining satisfactory agreement with experiment.

Lithium, rubidium, and cesium. Gamertsfelder[^81] obtained atomic-distribution curves for liquid lithium at \(200^\circ\mathrm{C}\) and found \(9.8\) neighbors for each atom at \(r = 3.24\ \text{\AA}\), whereas the coordination number in the crystal is 8. However, because of the imperfection of the experimental technique, the results of this investigation cannot be regarded as sufficiently accurate[^81],[^50].

Rubidium and cesium were studied only qualitatively[^83].

METALS WITH LOOSE PACKING OF ATOMS IN THE SOLID STATE

Gallium. Menke[^10] obtained intensity curves (Fig. 36) and probability curves (Fig. 37) for two temperatures: \(18^\circ\) and \(45^\circ\mathrm{C}\). At \(18^\circ\) gallium was in the supercooled state. Menke associated the subsidiary maximum on the outer slope of the main one (Fig. 36) with the presence of double atomic layers in the crystal[^92]. A subsidiary maximum is also present on the intensity curve of liquid gallium obtained by Hendus[^32] (Fig. 38). The reality of the subsidiary maximum is confirmed, moreover, by electronographic investigations of Richter[^75].

Fig. 36. Intensity curve for liquid gallium according to Gamertsfelder.

On the right-hand side of the main maximum in Fig. 38 a dotted line has been drawn symmetrically to its left branch, and the difference of ordinates has been constructed for the intensity curve and this dotted line. The position of the subsidiary maximum thus isolated coincides with the \((200)\) and \((113)\) lines of the crystal. From this one may conclude that in liquid gallium near the crystallization point there are preserved “traces” of the arrangement of atoms in the crystal. The disappearance of the subsidiary maximum at a temperature \(15\text{–}20^\circ\) above the melting point indicates that the forces responsible for the loose lattice of the crystal are comparatively weak, and therefore a rather small increase in temperature after melting is sufficient for the related features of the structures in the solid and liquid states to be smoothed out. The absence of similarity between the course of the probability curve and the distribution of atoms over the coordination spheres of the crystal (Fig. 39) also speaks in favor of such a representation; this is also indicated by analysis of the distribution curves.

Fig. 37. Curve \(W\) for gallium according to Menke.

From the distribution curve Hendus[^32] found in the first coordination sphere of liquid gallium 11 atoms at \(r = 2.77\,\text{\AA}\), whereas in the crystal there is 1 atom at \(r = 2.48\,\text{\AA}\) and 6 atoms at \(r = 2.71\text{–}2.77\,\text{\AA}\), i.e. 7 atoms at approximately the same distance as in the liquid. Thus, upon melting, the structure of gallium changes substantially and approaches a close-packed one with coordination number 12.

Since the volume of gallium decreases by \(3.2\%\) on melting, the increase in coordination number is apparently connected with the fact that, during melting, the homopolar bonds that existed in the crystal are almost entirely replaced by metallic bonding[^93].

Bismuth. In 1936 Danilov and Radchenko[^35], on the basis of a comparison of the intensity curve for liquid bismuth with the intensity curves for

Fig. 38. Intensity curve for liquid gallium according to Hendus.

Fig. 39. \(W\) curve for gallium according to Hendus.

blurred ideal structures^68 came to the conclusion that the structure of liquid bismuth near the crystallization point differs from close-packed structure and preserves certain features of the mutual coordination that existed in the crystal.

Randall and Rooksby^74, comparing the radiographs of lead and bismuth, made, apparently, the erroneous conclusion that the structures of these metals in the liquid state near the crystallization point are completely identical.

Richter^75 obtained an electron diffraction pattern of a molten solidified bismuth film. In this case, between two sharp interference rings, one more, weaker ring was found, in full agreement with the results of work^35.

The first X-ray study of liquid bismuth in strictly monochromatic radiation was carried out by Hendus^32. The intensity curve obtained by him is presented in Fig. 40. In the first coordination sphere at 390° there were found 7–8 atoms at \(r = 3.32\) Å, whereas in the crystal there are two very close coordinations of 3 atoms each at distances of 3.09 Å and 3.46 Å. On the basis of this and of a comparison of the probability curve for liquid bismuth with the distribution of atoms in the lattice (Fig. 41), the conclusion was drawn that there is no correspondence in the arrangement of atoms in the solid and liquid states.

Fig. 40. Intensity curve for liquid bismuth according to Hendus.

Fig. 41. \(W\) curve for bismuth according to Hendus.

Chamberlain^57 investigated liquid bismuth by neutron diffraction at 310° C and obtained results almost coinciding with those found in work^32. The scattering curve found by Chamberlain is presented in Fig. 42, and the radial distribution curve in Fig. 43.

In 1952, Danilov and co-workers^30 again investigated liquid bismuth by X-rays under more careful experimental conditions than before^35. Much attention was paid to proving the reality of the secondary maximum, appearing as a small bend on the descending branch of the main one. In work^30 the same intensity curve was obtained as in work^32. However, when it was compared with the radiograph of the crystal, a conclusion was drawn opposite to the conclusion in work^32: the atomic packing in liquid bismuth is a blurred variant of the packing characteristic of the bismuth crystal, but upon melting there occur

Fig. 42. Neutron scattering curve in liquid bismuth according to Chamberlain.

Fig. 43. Radial distribution curve of bismuth atoms according to Chamberlain.

Fig. 44. Radial distribution curves of atoms of liquid bismuth at two temperatures according to Sharrah and Smith.

disruptions of the loose packing of the crystal, leading to an increase in the coordination number from 6 in the crystal to 7–7.5 in the liquid.

Sharrah and Smith\(^ {79}\) carried out a neutronographic investigation of liquid bismuth, using a more refined method of monochromatization of the neutron beam than Chamberlain\(^ {57}\). In work\(^ {79}\) radial-distribution curves are given for two temperatures: \(350^\circ\) and \(550^\circ\) (Fig. 44). In the authors’ opinion\(^ {79}\), the results of their work agree with those found by Danilov et al.\(^ {30}\).

Takagi\(^ {103}\) investigated liquid bismuth by the method of electron diffraction above the melting point \((271^\circ\text{C})\) and in the supercooled state at five temperatures: \(400^\circ\), \(271^\circ\), \(175^\circ\), \(130^\circ\), \(110^\circ\), and constructed atomic-distribution curves. It was found that in the supercooled state at \(110^\circ\) the arrangement is closely connected with the arrangement in the solid state, while above the melting point a tendency toward close packing is observed.

A considerable change in the structure of bismuth on melting is also indicated by thermodynamic data. The volume of bismuth on melting decreases\(^ {77,94}\) by 3.5%; the entropy changes\(^ {95}\) by \(4.78\ \text{cal}/\text{g-atom}\), i.e., by a value considerably larger than for gold\(^ {95}\) \((\Delta S = 2.22)\), in which the arrangement of atoms does not change substantially on melting.

Germanium. Comparing the intensity curve for liquid germanium obtained by Hendus\(^ {32}\) (Fig. 45) with the distribution of interference lines of the crystal, we find no correspondence whatever. The same conclusion is reached by comparing the probability curve for liquid germanium with the distribution of atoms in the crystal by coordination spheres (Fig. 46). The coordination number of liquid germanium is equal to 8 at an average radius of the coordination sphere of \(2.7\ \text{Å}\). In the crystal each atom has 4 neighbors at a distance of \(2.43\ \text{Å}\). Thus, on melting germanium its structure changes sharply.

Fig. 45. Intensity curve for liquid germanium according to Hendus.

Fig. 46. Curve \(W\) for liquid germanium according to Hendus.

According to Richter’s investigations\(^ {96}\), in amorphous germanium there are 4 atoms at \(r = 2.40\ \text{Å}\) and 12 at \(r = 3.95\ \text{Å}\); therefore the amorphous phase of germanium is more closely related to the crystalline phase than to the liquid phase. König\(^ {97}\) suggested that amorphous germanium represents a disordered arrangement of tetrahedral atomic groups. Richter\(^ {96}\) proposed a scheme of arrangement of tetrahedra satisfying the concept of the preservation in the amorphous phase of the near order present in the crystalline phase, and of its complete disappearance on melting.

Antimony. Only one X-ray investigation of liquid antimony is known, carried out by Hendus and Müller\(^ {104}\) in monochromatic

in W-\(K_\alpha\) rays. As a result of comparing the periods found from a large number of photographs of liquid antimony with the periods for amorphous\(^{105,106}\) and crystalline antimony, a conclusion was drawn concerning the different arrangement of atoms in these phases.

Selenium. Liquid selenium was investigated only by Prins\(^{107}\). Subsequently Hendus\(^{105}\) showed that the periods found by Prins for liquid selenium are close to the periods for the glassy form. But the most interesting result of this investigation was that, despite the smallness of the coordination number (2), neither this number nor the radius of the coordination sphere changes upon crystallization or vitrification.

Tellurium. Tellurium\(^{108}\) was investigated in monochromatic rays in a helium atmosphere. The intensity curves obtained with argon and krypton counters were subjected to analysis by the Fourier method. From the radial-distribution curves it was found that in the first coordination sphere of liquid tellurium there are 2 atoms at \(r=2.9\) Å. Since in the solid state the distance between neighboring atoms of the chain is equal to 2.86 Å, it was concluded that chains of atoms continue to exist after melting.

Tin. The structure of liquid tin has been investigated by many authors. Since mutually consistent results were obtained, we shall dwell in detail on only one work\(^{32}\). Hendus obtained photographs of liquid tin at temperatures of \(280^\circ\) and \(480^\circ\). The intensity curves for these two temperatures differ little from one another (Fig. 47). On the principal maximum a subsidiary maximum is detected, as in bismuth and thallium. The reality of this maximum is confirmed by X-ray\(^{30,34,35,74,83,98,99}\), as well as electron-diffraction\(^{75}\), investigations of other authors. It turns out that both this subsidiary maximum and the principal one coincide in position with the lines on the X-ray photograph of powdered tin (Fig. 47). This gives grounds for asserting that liquid tin is characterized by an atomic packing which differs to a known extent from a dense one and is somewhat related to the packing of the crystal. But the higher the temperature, the more the arrangement of atoms in liquid tin approaches dense packing. This assertion, made already in the original work\(^{35}\) on the basis of a simple study of the intensity curves, is confirmed by subsequent investigations and, in particular, by analysis of the atomic-distribution curves.

Hendus found for liquid tin 10 atoms at \(r=3.20\) Å, whereas in the crystal of white tin there are 4 atoms at a distance of 3.76 Å, 2 atoms at \(r=3.15\) Å, and 4 at \(r=3.76\) Å, or 10 atoms at an average distance of 3.34 Å. A similar result was obtained earlier by Gamertsfelder\(^{98}\), and subsequently by Danilov\(^{32}\), as well as by Tsvetkov\(^{99}\).

Glauberman\(^{100}\), on the basis of Prins’s\(^{68}\) ideas, which he further developed, on the smearing of theoretical lattices, found a similarity between the experimental atomic-distribution curve in liquid tin and the theoretical curve for smeared hexagonal close packing\(^{101}\).

In Fig. 48 the probability curves obtained by two different authors are compared. It is interesting to note that there is no small maximum at \(r=4.5\) Å on Danilov’s curve. In addition, we see that Hendus’s curve has 6 small maxima superposed on 2 broad ones at 6.2 Å and at 9.2 Å. Finbak\(^{50}\) drew attention to the fact that these 6 maxima are at equal distances from one another, suggested that they are spurious, and explained their appearance by an error in some narrow interval of the intensity curve, since this error introduces into the Fourier transform a periodic term. Knowing the distance between the periodic maxima, one can calculate the position of that interval on the intensity curve which caused their occurrence. In the present case the period was equal to 0.95 Å, and the desired interval is approximately equal to

\[ s=\frac{2\pi}{0.95}\simeq 2\pi \text{ Å}^{-1}\quad \text{or}\quad \frac{\sin \Theta}{\lambda}=0.5 \text{ Å}^{-1}. \]

On the intensity curve (Fig. 47), at \(\dfrac{\sin \Theta}{\lambda} = 0.5\ \text{\AA}^{-1}\) there is a maximum. Since the height of this maximum is rather large, then, as calculation shows, it introduces into the distribution a periodic term with maxima whose positions

Fig. 47. Intensity curve for liquid tin according to Hendus.

Fig. 48. \(W\) curves for liquid tin according to Danilov and Hendus.

almost exactly coincide with the positions of 6 maxima on the experimental curve. Hence the conclusion was drawn that the reason for the appearance of 6 periodic maxima on the atomic-distribution curve may be an error in determining the height of the maximum on the intensity curve at

\[ \frac{\sin \Theta}{\lambda} = 0.5\ \text{\AA}^{-1}. \]

CONCLUSION

On the basis of X-ray and neutron-diffraction data it may be asserted that the packing of atoms in a liquid metal is related in a definite way to its packing in the solid state, and that the nature of this relation depends on the kind of metal.

The coordination number of all metals after melting either increases or remains unchanged. The conclusion that the coordination number decreases in lead, thallium, and indium is insufficiently reliable, since this conclusion is connected with a subsidiary maximum on the atomic-distribution curve, and there is reason to suppose that this maximum is the result of an error in determining the intensity curve.

A study of the temperature dependence of the intensity curves of potassium, sodium, tin, and certain other metals shows that in liquid metals the type of packing is preserved up to high temperatures, considerably exceeding the melting temperature.

Further investigations are needed to study the causes of the errors that arise in finding atomic-distribution curves, and to develop methods for detecting and eliminating these errors; otherwise conclusions about the features of one or another structure of a liquid may give rise to objections.

References

1. F. Sauerwald, Zeits. Metallkunde 41, 97, 214 (1950).
2. V. I. Danilov, UFN 14, 449 (1934).
3. A. Latin, J. Inst. of Metals 66, 177 (1940).
4. B. R. T. Frost, Progress in Metal Physics, ed. 1954, vol. 5, pp. 96–125.
5. N. S. Gingrich, Rev. Mod. Phys. 15, 90 (1943), Russian translation in Uspekhi khimii 15, 297 (1946).
6. I. V. Radchenko, UFN 17, 319 (1937).
7. V. I. Danilov, Problems of Metallurgy and the Physics of Metals, Collection 4, Metallurgizdat, Moscow, 1955, pp. 15–33.
8. W. H. Morell and J. Hildebrand, J. Chem. Phys. 4, 224 (1936).
9. I. V. Radchenko and F. K. Shestakovskii, Collection “Structure and Physical Properties of Matter in the Liquid State,” KGU Press, Kiev, 1954, pp. 45–61.
10. H. Menke, Phys. Zeits. 33, 593 (1932).
11. B. E. Warren, H. Krutter, O. Morningstar, J. Amer. Ceram. Soc. 19, 202 (1936).
12. B. E. Warren, J. Appl. Phys. 8, 645 (1937).
13. F. Zernicke and J. A. Prins, Zeits. f. Phys. 41, 184 (1927).
14. P. C. Blak, Rev. Mod. Phys. 5, 180 (1933).
15. W. Van der Grinten, Phys. Zeits. 34, 609 (1933).
16. A. H. Compton and S. K. Allison, X-Rays in Theorie and Experiment, 2nd ed., Russian translation, OGIZ, 1941.
17. R. W. James and G. W. Brindley, Phil. Mag. 12, 81 (1931).
18. C. Lanczos and G. C. Danielson, Phys. Rev. 55, 242 (1939).
19. G. C. Danielson and C. Lanczos, J. Frankl. Inst. 233, 365, 435 (1945).
20. A. S. Lashko, DAN USSR, No. 3, 150 (1953).
21. H. Krebs and R. Thees, Zeits. Elektrochemie 57, 741 (1953).
22. M. I. Petrashen and E. A. Porai-Koshits, ZhETF 21, 881 (1951).
23. G. Fournet, Acta Crist. 4, 293 (1951).
24. A. S. Lashko, Collection “Questions of the Physics of Metals and Metallurgy,” Kiev, No. 5, 1954, pp. 15–20.
25. W. H. Keesom and J. de Smedt, Proc. Amsterd. Acad. 25, 118 (1922); 26, 112 (1923).
26. P. Ehrenfest, Proc. Amsterd. Acad. 17, 1132 (1915).
27. I. D. Bernal and R. H. Fowler, J. Chem. Phys. 1, 515, 1933.
28. P. Debye and H. Menke, Fortschritte der Technischen Röntgenkunde, Bd. 2, Akad. Verlagsges., Leipzig, 1931.
29. R. N. Boyd and H. R. R. Wakeham, J. Chem. Phys. 7, 958 (1939).
30. A. I. Danilova, V. I. Danilov, and E. Z. Spektor, “Problems of Metallurgy and the Physics of Metals,” Collection 3, Metallurgizdat, Moscow, 1952, pp. 336–354.
31. H. Mark, Zeits. f. Physik 54, 505 (1929).
32. H. Hendus, Zeits. Naturforschung 2a, 9, 505 (1947).
33. J. A. Campbell and J. H. Hildebrand, J. Chem. Phys. 11, 330 (1943).
34. F. Sauerwald and W. Teske, Zeits. anorg. allg. Chemie 210, 249 (1933).
35. V. I. Danilov and I. V. Radchenko, ZhETF 7, 1153 (1937); Phys. Zeits. Sowjetunion 12, 745 (1937); 10, Heft 2 (1936).
36. V. I. Danilov and V. E. Neimark, ZhETF 5, 724 (1935).
37. J. Thibaud and J. J. Trillat, Zeits. f. Phys. 61, 816 (1930).
38. J. Fankuchen, Nature 139, 193 (1937).
39. R. Q. Gregg and N. S. Gingrich, Rev. Sci. Instr. 11, 305 (1940), Phys. Rev. 57, 1055 (1940).
40. A. I. Danilova and V. I. Danilov, Problems of Metallurgy and the Physics of Metals, Collection 2, Metallurgizdat, Moscow, 1951, pp. 31–47.

1. H. Hendus, Zeits. Naturforschung 3a, 416 (1948).
2. F. Sauerwald and E. Osswald, Zeits. anorg. allg. Chem. 257, 195 (1948).
3. A. Prietzschk, Zeits. f. Phys. 117, 482 (1941).
4. F. Cennamo, Nuovo Cimento 40, 395 (1953).
5. I. I. Yamzin, Transactions of the Institute of Crystallography, Academy of Sciences of the USSR, issue 5, 69 (1949).
6. Ch. Finbak, Acta Chem. Scand. 3, 1285 (1949).
7. A. Eisenstein and N. S. Gingrich, Rev. Sci. Instr. 12, 582 (1941).
8. O. Bastiansen and Ch. Finbak, Arch. for Math. Naturvidenskab B 47, No. 12 (1944).
9. V. N. Filippovich and E. A. Porai-Koshits, Doklady of the Academy of Sciences of the USSR 105, 968 (1955).
10. Ch. Finbak, Acta Chem. Scand. 3, 1293 (1949).
11. T. Sugawara, Sci. Rep. Tohôku Univ. A3, 39 (1951).
12. T. Sugawara, Sci. Rep. Tohôku Univ. A4, 53 (1952).
13. A. S. Lashko, in the collection “Structure and Physical Properties of Matter in the Liquid State,” Kiev, KGU Publ., 1954, pp. 39–44.
14. E. Kànda, T. Sugawara, Sci. Rep. Tohôku Univ. A3, 34 (1951).
15. W. J. Bragg and J. West, Phil. Mag. [7] 10, 823 (1930).
16. J. A. Prins, in: Bouman, “Selected Topics in X-Ray Cristallographie,” Amsterdam, 1951, p. 192.
17. O. Chamberlain, Phys. Rev. 77, 305 (1950).
18. D. Coster and J. A. Prins, Nature 122, 314 (1928); Journ. de Physique et le Radium 9, 153 (1928).
19. P. Debierne, Comptes Rendus 173, 140 (1921).
20. R. W. Wycoff, Amer. J. Sci. 5, 455 (1923).
21. J. A. Prins, Physica 6, 315 (1926).
22. C. V. Raman and C. M. Sogani, Nature 120, 514 (1927); C. M. Sogani, Ind. Journ. Phys. 2, 97 (1928).
23. J. A. Prins, Zeits. f. Phys. 56, 617 (1929).
24. J. A. Prins, Naturwissenschaften 19, 435 (1931).
25. P. Debye and H. Menke, Phys. Zeits. 31, 797 (1930).
26. K. Mendelssohn, Nature 154, 227 (1944).
27. O. Kratky, Phys. Zeits. 34, 482 (1933).
28. J. A. Prins and H. Petersen, Physica 3, 147 (1936).
29. L. W. McKeehan and P. P. Gioffi, Phys. Rev. 19, 444 (1922); H. Terrey and C. M. Wright, Phil. Mag. 6, 1055 (1928); M. Wolf, Nature 122, 314 (1928); M. Wolf, Zeits. f. Phys. 53, 72 (1929).
30. G. Oster and P. P. Riley, Acta Cryst. 5, 1 (1952).
31. G. H. Vineyard, J. Chem. Phys. 22, 1665 (1954).
32. J. S. Lukesh, W. H. Howeland, L. F. Epstein, M. D. Powers, J. Chem. Phys. 23, 1923 (1955).
33. O. Ya. Samoilov, Doklady of the Academy of Sciences of the USSR 61, 665 (1948).
34. J. T. Randall and H. P. Rooksby, Trans. Faraday Soc. 33, 109 (1937).
35. H. Richter, Phys. Zeits. 44, 406 (1943).
36. R. Glocker and H. Hendus, Ann. d. Phys. 43, 513 (1943).
37. H. Endo, Sci. Rep. Tohôku Univ. 13, 193 (1924).
38. I. Matuyama, Sci. Rep. Tohôku Univ. 17, 1 (1928).
39. C. Sharrah and C. P. Smith, J. Chem. Phys. 21, 228 (1953).
40. C. Gamertsfelder, J. Chem. Phys. 9, 450 (1941).
41. C. Gamertsfelder, Phys. Rev. 57, 1055 (1940).
42. W. H. Keesom, Proc. Amsterd. Akad. Sci. 30, 341 (1927).
43. J. T. Randall and H. P. Rooksby, Nature 130, 473 (1932).
44. L. P. Tarasow and B. E. Warren, J. Chem. Phys. 4, 236 (1936).
45. F. H. Trimble and N. S. Gingrich, Phys. Rev. 53, 278 (1938).
46. C. N. Wall, Phys. Rev. 54, 1062 (1938).
47. C. D. Thomas and N. S. Gingrich, J. Chem. Phys. 6, 411 (1938).

1. V. I. Danilov, Izv. AN SSSR, ser. fizich. 5, 30 (1941).

2. S. Hildebrand, J. Chem. Phys. 7, 1 (1939).

3. H. C. Gingrich, Uspekhi Khimii 15, 297 (1946).

4. N. S. Gingrich and C. N. Wail, Phys. Rev. 55, 1116 (1939); 56, 336 (1939).

5. F. M. Jäger, P. Terpsta, H. G. Westenbrink, Zeits. Kristalogr. 66, 195 (1928).

6. O. Kubaschewski, Trans. Farad. Soc. 45, 931 (1949).

7. Blok, Zeits. Phys. Chem. 78, 385 (1912).

8. O. Kubaschewski, P. Brizgis, O. Huchler, R. Jauch, K. Reinartz, Zeits. Elektrochem. 54, 275 (1950).

9. O. Fürst, R. Glocker, H. Richter, Zeits. Naturforschung 4, 540 (1949).

10. König, Optik 3, 201 (1948).

11. C. Gamertsfelder, Phys. Rev. 55, 1116 (1939).

12. V. P. Tsvetkov, Reports and Communications of Lviv University 2, 82 (1955).

13. A. E. Glauberman, ZhETF 22, 249 (1952).

14. A. E. Glauberman and V. P. Tsvetkov, DAN SSSR 106, 623 (1956).

15. O. Ya. Samoilov, DAN SSSR 53, 447 (1952).

16. M. Takagi, J. Phys. Soc. Japan 11, 396 (1956).

17. H. Hendus, H. Miller, Zeits. Naturforsch. 10a, 254 (1955).

18. H. Hendus, Zeits. f. Physik 119, 265 (1942).

19. L. I. Tatarinova and Z. G. Pinsker, DAN SSSR 95, 265 (1954).

20. K. Lark-Horovitz and E. P. Miller, Phys. Rev. 51, 380 (1937).

21. R. Buschert, J. G. Geib, K. Lark-Horovitz, Phys. Rev. 98, 1157 (1955).

22. R. E. Jennings, Acta crystallographica 5, 1 (1952).

23. V. I. Danilov, A. I. Danilova, E. Z. Spektor, in the book: V. I. Danilov, Structure and Crystallization of Liquids, Kiev, Publishing House of the Academy of Sciences of the Ukrainian SSR, 1956, pp. 225–231.

24. R. E. Smallman and B. R. Frost, Acta metallurgica 4, 611 (1956).