X-RAY SCATTERING IN LIQUID METALS AND ALLOYS

V. I. Danilov, Dnepropetrovsk

The study of certain physical properties of a liquid, especially near the crystallization point, makes it possible to conclude that, in its structure, a liquid must be more closely approximated (at least near the solidification temperature) to the structure of a solid, characterized by the regular arrangement of molecules, than to a gaseous state with a disordered arrangement of its structural particles.

Among such investigations should be included, for example, studies of the thermoelectric properties of bismuth near the melting point. Boydston¹, observing an irregular course of the thermoelectromotive-force curve near the melting point of bismuth, explains this by the fact that above this point, in liquid bismuth, a certain ordering in the arrangement of molecules is preserved. Sorez², repeating Boydston’s experiments, established that immediately after melting the thermocouple shows a course intermediate in comparison with solid and liquid bismuth. Getz³ comes to the same conclusions, explaining the phenomena he observed during the growth of Bi single crystals. It is interesting that a polycrystalline specimen of bismuth, when heated somewhat above the melting point and then successively cooled, had the same grain structure as before melting. And, finally, an extremely interesting phenomenon, again with bismuth, was observed by Donat and Stierstadt⁴. They slowly melted a single crystal of bismuth in a special furnace for growing single crystals, a characteristic feature of which was that increase and decrease of the specimen temperature were achieved by moving the heating winding itself, while the specimen remained motionless throughout the experiment and therefore was not subjected to mechanical shocks. The shape of the crucible in which the specimen was placed was selected in such a way that the change in volume during melting would affect as little as possible the mutual arrangement of the parts of the metal being melted. Under these conditions Donat and Stierstadt, upon subsequent cooling, obtained a single crystal of bismuth of the same orientation as the original one. This was always observed if the molten bismuth was not superheated by more than 10° above the melting point. The most natural explanation of this phenomenon may be found by assuming

existence, in the molten single crystal, of crystalline order, which is also in direct connection with the crystal lattice of solid bismuth.

A similar kind of kinship between the solid and liquid states has been observed by various authors not only in the growth of bismuth single crystals. Graf observed the same phenomenon in copper.

Further, a fact very well known in foundry practice, indicating that the crystals of a solid have some relation to the structure of the liquid, is that the size and shape of metal crystallites depend on how far the liquid metal is superheated above the melting point before solidification.

The question naturally arises whether, with the aid of X-rays, which have yielded such brilliant results in the investigation of the structure of solids, one might attempt to study also the structure of liquids.

The first attempt to investigate the structure of liquids by means of X-rays should be considered the work of Debye and Scherrer with liquid benzene. They showed that if a parallel beam of monochromatic X-rays passes through a layer of benzene, then the radiation scattered by the latter gives on a photographic film a quite definite diffraction pattern, similar to some extent to that observed in the case of finely crystalline substances. On the photographic plate there appears a rather narrow, more or less sharp ring. Subsequently it turned out that in the case of various liquids there may be not one such ring, but two, three, or more. It is characteristic of all liquids that these rings are always separated from the primary beam by a region in which the intensity of the scattered rays is extremely small. The same is known to occur in solids as well. A characteristic feature of the distribution of the intensity of X-rays scattered in gases, however, is a high intensity near the primary beam, decreasing gradually with increasing angle of scattering.

The width of the diffraction rings of a liquid is usually large in comparison with the width of the lines given by polycrystalline specimens on Debye radiograms.

Since this first application of X-rays to the investigation of the structure of liquid benzene, a large amount of material on the X-ray investigation of liquids has accumulated.

In the overwhelming majority of cases the photographic method was used, and only in the work of Hewlett[^5] and of Stewart and his collaborators did the ionization method find application, although, as we shall see below, the latter would seem to have greater advantages over the photographic method precisely in the investigation of liquids, since here the general course of the intensity plays a considerably greater role than in crystal-structure analysis, where the principal data for structural calculations are taken from the positions of the maxima of thin lines, and an accurate estimate of the intensity proves necessary only in deciphering especially complex structures.

The results of all these works led to the conclusion that the distribution of the intensity of X-rays scattered in a liquid depends not only on 1) the distribution of electric charges in the atom and 2) the arrangement of atoms in the molecule, as was assumed at first, but also on 3) the mutual arrangement of the molecules of the liquid.

The first of these three factors—the factor of atomic scattering—can be calculated theoretically and determined experimentally from observations of X-ray diffraction in crystals, or still better in vapors of monoatomic liquids, when the interference pattern is determined only by intra-atomic diffraction. This makes it possible to separate the effect of intra-atomic scattering from the general diffraction pattern given by the liquid.

Considerably more complex is the problem of separating intramolecular scattering from the effect caused by order in the mutual arrangement of the molecules of the liquid. At the same time it appears quite clear that, in studying the structure of a liquid, this problem acquires primary importance. All work on the study of the structure of a liquid by means of X-rays may be divided into three groups.

The first group comprises the works of Stewart^6 and his students, chiefly on organic liquids, which led to the view of the liquid state as a structural system in which the molecules are in a state of mobile orientation. If organic substances in the solid state are characterized by constancy of the mutual orientation of the molecules, then in the case of a liquid this is replaced by the mobility of such mutual molecular arrangements. From this point of view, at a given instant there are in a liquid many such “cybotactic” groups, the arrangement of molecules in which is not random, but represents a quite definite crystalline formation. The appearance of maxima on the X-ray diagram of liquids is due to the presence of periods of identity in such cybotactic groups. However, the possibility is not excluded of the existence of deviations in the mutual orientation of the molecules from the principal cybotactic grouping characteristic of the given liquid. This, evidently, explains the diffuseness of the diffraction maxima observed in liquids. Thus the positions of the diffraction maxima make it possible to judge the periods of identity existing in the liquid and, consequently, the structure of the liquid. In the case of organic liquids it turns out that the geometrical form of the molecules has a profound influence on their possible groupings.

The second group should include the works of the Indian school, whose starting point is the theory of Raman and Ramanathan^7, subsequently continued by Zorn^8 and Krishnamurti^9. Raman and Ramanathan proceed from the theory of density fluctuations of Smoluchowski^10 and Einstein^11. In Einstein’s theory the density fluctuations are considered purely macroscopically; they therefore relate to regions containing a large number of molecules. If now, as

this is done by Raman and Ramanathan, then, if the scattering of X-rays is to be calculated in the same way as for visible light in a continuous medium subject to local changes of density, satisfactory results can be expected only in the case when the regions in which the path difference can reach one wavelength will have dimensions large in comparison with the mean molecular distances. Since the path difference is proportional to $\sin \frac{\vartheta}{2}$, it is obvious that, in the case of X-rays, for which the wavelength is of the order of atomic distances, the conclusions of the theory of Raman and Ramanathan must be valid for small scattering angles.

And indeed, in agreement with the theory, it has been possible to establish that for very small angles the intensity of the scattered radiation is the smaller, the smaller the compressibility and the density fluctuations depending on it. Thus, as was to be expected, experiment confirms the theory of Raman and Ramanathan at small scattering angles. The limiting angle, beyond which arguments based on the theory of thermal fluctuations of Smoluchowski and Einstein lose their meaning, is the angle determined by the following relation:

\[ 2d \sin \frac{\vartheta}{2} = \lambda, \]

if $d$ is the mean intermolecular distance. As is known, this same equation approximately determines the position of the first maximum.

Thus the range of applicability of the Raman and Ramanathan theory is small angles from zero to the first maximum.

And finally, to the third group we shall assign works on the study of the diffraction of X-rays in liquid metals. The latter give X-ray diagrams whose interpretation is facilitated by the circumstance that, owing to the monoatomic nature of the metals, the molecular factor drops out.

A detailed consideration of the works devoted to the scattering of X-rays in liquid metals is the aim of the present article.

All these investigations are based to one degree or another on Debye’s theoretical works on the scattering of X-rays in amorphous bodies.

Theory of the Scattering of X-rays in Monoatomic Liquids

For monoatomic liquids, the scattering of X-rays must be determined by two factors: 1) the atomic factor, depending on the distribution of electric charges in the atom, and 2) the “structural” factor, taking into account the structure of the liquid—the mutual arrangement of the molecules.

Atomic factor. If the dimensions of the atom were small in comparison with the wavelengths of the X-rays usually employed

for structural purposes, then for such wavelengths the atom, to an admissible approximation, could be regarded as a point, and the waves scattered by the different electrons as being in one phase. The resultant amplitude of the radiation issuing from the whole atom would then be obtained by simply multiplying the amplitude given by one electron by the number of electrons (Punkttheorie). In reality, however, the wavelengths of X-rays are of the order of atomic dimensions, and therefore rays scattered in different parts of the atom possess certain phase differences, the magnitudes of which depend on the angle of scattering. In the case of heavy atoms this causes a strong decrease of the intensity with increasing angle of diffraction. The steepness of the decrease diminishes as the atomic number of the scattering atoms decreases. Therefore, in the case of organic compounds, where the principal constituent elements are oxygen, hydrogen, and carbon, the influence of the atomic factor on the general course of the intensity proves to be insignificant and can often be omitted from consideration. In the case of heavy liquids, however, the general picture of the intensity distribution of diffracted X-rays depends to a considerable degree on intra-atomic diffraction, and therefore taking this factor into account in interpreting radiograms, for example of liquid metals, is extremely essential.

As the basis for calculating the atomic factor, as follows from what has just been said, some representation of the distribution of electrons in the atom must be adopted. Debye\(^{12}\) assumes the distribution of electric charge density according to Fermi–Thomas.\(^{13}\) The latter, as is known, treat the extranuclear electrons of the atom as a degenerate gas, and the energy of the atom as zero energy in the usual sense. It then becomes possible, introducing the so-called characteristic atomic radius \(a=\dfrac{0.47}{Z^{1/3}}\,\text{Å}\), where \(Z\) is the atomic number, to express the electron density \(\nu\) through a single function \(\varphi\) common to all elements:

\[ \nu=\frac{Z}{4\pi a^3}\left(\frac{\varphi}{x}\right)^{3/2}, \tag{1} \]

where \(x=\dfrac{r}{a}\), and \(r\) is the distance from the center of the atom.

For \(\varphi\), as a function of \(x\), Fermi gives its tabulated value. The atomic factor \(\psi\) can be represented through the density \(\nu\) in the following way:

\[ \psi=\int_0^\infty \nu\,\frac{\sin ksr}{ksr}\,4\pi r^2\,dr, \tag{2} \]

where \(k=\dfrac{2\pi}{\lambda}\) and \(s=2\sin\dfrac{\vartheta}{2}\), if \(\vartheta\) is the angle between the incident and the diffracted ray.

If we now take the distribution of the density of electric charges according to Fermi–Thomas, then expression (2) assumes the following form

\[ \psi=\frac{Z}{ksa}\int_{0}^{\infty}\frac{\varphi^{3/2}}{x^{1/2}}\sin ksa\,x\,dx \]

or, putting

\[ u=ksa=4\pi\frac{a}{\lambda}\sin\frac{\vartheta}{2} \]

and, further,

\[ \Phi(u)=\frac{1}{u}\int \frac{\varphi^{3/2}}{x^{1/2}}\sin ux\,dx, \]

the atomic factor will be represented in this form

\[ \psi=Z\Phi(u). \]

The function \(\Phi(u)\) can be reduced to tables (Bewilogua)\(^{14}\), and then the calculation of the atomic factor for any atom becomes quite an easy problem. It is self-evident that the Thomas–Fermi method does not give an exact distribution of electrons in the atom, and therefore the above method of determining the atomic factor is an approximate one. For heavy atoms, however, this approximation proves to be quite sufficient. Experimental verification has shown that even in the case of comparatively light atoms, when satisfactory results cannot be expected from this statistical method, the experimental curves turn out to be very close to the theoretical ones obtained by the method just given. It turns out that Hartree’s more exact method\(^{15}\) also gives curves deviating from the experimental ones to the same degree as the Debye curves, while at the same time being incomparably more difficult and complicated.

Alongside coherent radiation, the rays scattered by atoms also contain a certain incoherent part, which, in the case of interference in liquids, characterized by broad maxima, cannot be separated experimentally from the coherent radiation. Recently Heisenberg\(^{16}\) gave a formula that makes it possible to determine the incoherent radiation for any atom by means of a single function. Verification of this formula on gases gave excellent results. We give the final form of Heisenberg’s formula:

\[ S_1=Z\left[1-\int_{0}^{r_0}\xi\,d\xi \left\{ \left(\left(\frac{\varphi(\xi)}{\xi}\right)^{1/2}-v\right)^2 \left(\left(\frac{\varphi(\xi)}{\xi}\right)^{1/2}+\frac{1}{2}v\right) \right\}\right], \]

where

\[ \xi=\frac{x}{a}N^{1/2} \quad\text{and}\quad v=\frac{ksa}{(6\pi Z)^{1/2}}. \]

In the case of heavy atoms, however, incoherent radiation is insignificant, and for the objects of interest to us the correction may remain unaccounted for.

The factor determined by the mutual arrangement of the molecules. In the theory of the scattering of X-rays in liquids, given by Debye^17, 18, 19, two cases must be distinguished: the first, when the arrangement of particles in the liquid is taken to be disordered, similar to what occurs in gases, and the second, when the regularity in the distribution of molecules is taken into account. For both cases the reasoning proceeds in exactly the same way up to the point at which the intensity of the scattered radiation is averaged over time.

Let a beam of monochromatic X-rays of wavelength \(\lambda\) illuminate a certain volume of a monatomic liquid. Denote the latter by \(V\), and the number of atoms contained in it by \(N\). The direction of the primary beam is characterized by the unit vector \(\mathfrak{S}_0\). We consider the intensity of the scattered radiation in the direction corresponding to the unit vector \(\mathfrak{S}\). If the atoms \(1, 2, \ldots, m, \ldots, n, \ldots, N\) at a given moment of time are at distances \(r_1, r_2, \ldots, r_N\) from the origin of coordinates, chosen arbitrarily in the volume \(V\), then at that moment the amplitude of the radiation scattered by \(N\) atoms in the direction \(\mathfrak{S}\) will be equal to

\[ \psi \sum_n e^{ik(s,r_n)}, \tag{3} \]

where \(\psi\) is the atomic factor, calculated, for example, by the method discussed above.

The intensity is obtained by multiplying (3) by its complex conjugate. This gives the following expression for the intensity, if the polarization of the scattered radiation is also taken into account:

\[ I = \frac{1+\cos^2 \vartheta}{2}\,\psi^2 \sum_m \sum_n e^{ik(s,r_m-r_n)} . \tag{4} \]

However, owing to the thermal motion of the atoms, (4) does not represent the experimentally observed course of the intensity. The latter is found by averaging (4) over all possible positions of the atoms that they may occupy during thermal motion. Finding the mean value \(I_m\) is the principal task of the theory of the scattering of X-rays in liquids.

Denoting the operation of averaging by \(M\), and the mean intensity by \(I_m\), we write:

\[ I_m = \frac{1+\cos^2 \vartheta}{2}\,\psi^2 \sum_m \sum_n M\left[e^{ik(s,r_m-r_n)}\right]. \tag{5} \]

If we now assume that in liquids the atoms are distributed according to the law of chance, then the probability that the center of molecule \(n\) is in the volume element \(dV_n\), and the center of molecule \(m\) in \(dV_m\), will be equal to

\[ \frac{dV_m}{V}\cdot \frac{dV_n}{V}. \tag{5a} \]

In order to obtain the mean value \(M\), it is necessary to multiply

\[ \frac{dV_m}{V}\cdot \frac{dV_n}{V} \]

by \(e^{ik(s,\mathbf r_n-\mathbf r_m)}\) and integrate over all possible positions of the atoms. For the case of a random distribution of molecules, the only restriction on the mutual arrangement of atoms will be that molecules \(m\) and \(n\) cannot approach one another at a distance smaller than the diameter of the molecule.

For \(M\) one then obtains the expression:

\[ M_{mn}=-\frac{1}{V}\frac{4}{3}\pi(2a)^3\Phi(2ksa), \tag{6} \]

where \(a\) is the radius of the molecule, and

\[ \Phi(2ksa)=\Phi(u)=\frac{3}{u^3}(\sin u-u\cos u). \]

Since \(M_{mn}\) does not depend on \(m\) and \(n\), the summation (5) in the general expression for the averaged intensity reduces to a simple multiplication of (6) by \(N(N-1)\), or, since \(N\) is always a large quantity, by \(N^2\), and then, taking into account that

\[ \frac{4}{3}\pi(2a)^3=\Omega \]

is the proper volume of the molecules for the mean intensity observed in the direction \(\mathfrak s\), we obtain the expression

\[ I_m=\frac{1+\cos^2\vartheta}{2}\,\psi^2 N\left[1-\frac{\Omega}{V}\Phi(2ksa)\right]. \tag{7} \]

Here the second term in brackets is due to the external interference of the atoms. Its influence on the general picture of scattering depends on the proper dimensions of the molecules, or more precisely on the ratio of the proper volume \(\Omega\) of the molecules to the volume \(V\) occupied by them, i.e., on the density of the scattering amorphous substance. Its very occurrence is caused by the restriction adopted in averaging, namely that the molecules cannot approach one another at a distance smaller than the sum of their radii.

In order to illustrate the conclusions obtained, we give a curve (Fig. 1) on which is represented the distribution of the intensity of radiation scattered by a diatomic substance satisfying

satisfying the conditions adopted above for the randomness of the distribution of molecules. The curve is plotted for \(\Omega/V=0\), \(\Omega/V=1/4\), \(\Omega/V=1/2\), and \(\Omega/V=3/4\). The first maximum, appearing for \(\Omega/V=3/4\) at \(12^\circ\), when the density is decreased first shifts toward small angles, and then disappears completely. The second maximum (\(\vartheta=45^\circ\)) does not change its position when the intermolecular distance is changed—it is due to the “internal” interference of two atomic molecules. In the case of monatomic liquids the second maximum disappears, while the first, caused by the dimensions of the molecules, remains. Thus, for a monatomic amorphous substance, according to this theory, at \(\Omega/V=0\) (which corresponds to a gas) we have a gradual decrease of intensity. The character of the decrease is determined by the atomic factor \(\psi\).

Fig. 1. Scattering curves of a diatomic liquid.

This follows directly from formula (7), if \(\Omega/V\) is put equal to zero. For a monatomic liquid, \(\Omega/V\) will be a quantity far from zero, and therefore the expected intensity distribution should be characterized by the presence of one maximum.

However, liquids usually give an entirely different interference pattern. This indicates that in a liquid the basic assumption made in deriving formula (7)—the disorder in the mutual arrangement of molecules—does not in fact hold. It is necessary to assume that a substance in the liquid phase possesses a certain ordered molecular structure. In the case of a monatomic liquid, assigning spherical symmetry of their physical properties to the atoms, this ordering of the arrangement of molecules

can be described by means of a certain probability function depending only on the interatomic distances.

This function can be defined as follows: let the volume of liquid illuminated by the rays be equal to \(V\). In this volume two volume elements \(dV_1\) and \(dV_2\), situated at a distance \(r\), are considered. The probability that some fixed atoms 1 and 2 will simultaneously be located, the first in \(dV_1\) and the other in \(dV_2\), is represented by the following expression

\[ W(r)\,\frac{dV_1}{V}\cdot \frac{dV_2}{V}. \tag{8} \]

Here, through the probability function \(W\), the random average regularity in the mutual arrangement of molecules is expressed.

If all possible mutual arrangements are equally probable, then \(W=1\), and the calculation of the angular distribution of intensity, as we have seen, leads to formula (7). With any regularity in the distribution of atoms, obviously, \(W=1\) for large distances and \(W=0\) for very small ones (smaller than the atomic diameter). For intermediate distances, however, \(W\) is unknown to us. One can only assert that for monatomic liquids it depends only on the distance.

Taking the above into account, we see that for all the arguments of the preceding section the conclusions of Debye remain unchanged up to the point at which the averaged intensity is determined. It is obvious that the expression for \(M\) in the present case will be obtained by replacing in it (5a) by expression (8). This leads to the following form of the averaged intensity:

\[ I_m=\frac{1+\cos^2\vartheta}{2}\,\psi^2 \sum_m \sum_n \iint e^{ik(s,r_n-r_m)}\,W\,\frac{dV_n}{V}\,\frac{dV_m}{V}, \]

and, calculating this latter expression, we obtain:

\[ I_m=\frac{1+\cos^2\vartheta}{2}\,\psi^2 N \left[ 1-\frac{4\pi}{d^3}\int (1-W)\,\frac{\sin ksr}{ksr}\,r^2\,dr \right]. \tag{9} \]

If \(W\) were a known function of \(r\), it would be possible to calculate the distribution of intensity as a function of \(\vartheta\) \(\left(s=2\sin\frac{\vartheta}{2}\right)\). The expression standing in brackets is a factor depending only on the structure of the liquid. We shall denote it by \(E(s)\). From the general course of the intensity, taking into account atomic scattering and polarization, one can always isolate \(E(s)\).

However, nothing can be said a priori about the form of the function \(W(r)\). One can pose the inverse problem: to determine \(W(r)\) from the experimentally obtained \(E(s)\). To carry this out it is more convenient to use (9)

transform \(K\) with the new variable \(\rho=\dfrac{r}{\lambda}\), then

\[ I_m=\frac{1+\cos^2\vartheta}{2}N\psi^2 \left[ 1-\frac{\lambda^3}{d^3}\frac{2}{s} \int_0^\infty \rho(1-W)\sin 2\pi\rho s\,d\rho \right], \tag{10} \]

and, taking into account what was said about \(E(s)\),

\[ s[1-E]=2\frac{\lambda^3}{d^3}\int_0^\infty \rho(1-W),\sin 2\pi\rho s\,d\rho. \tag{11} \]

This expression, according to Fourier’s theorem, can be inverted, and then one obtains

\[ \rho(1-W)=2\frac{d^3}{\lambda^3}\int_0^\infty s(1-E)\sin 2\pi\rho s\,ds. \tag{12} \]

The last equation makes it possible to determine \(W\) as a function of the mutual distance of the molecules from the experimentally obtained \(E(s)\).

Equation (12), obviously, can be strictly extended only to monatomic liquids. In cases where the molecule of the liquid consists of several atoms, in place of the atomic factor \(\psi\) in the general expression for the intensity there appears a molecular factor, depending on the orientation of the molecules relative to \(s\). The arguments proceed in the same way, and the result obtained is not complicated, although, of course, somewhat more cumbersome than for monatomic liquids.

A calculation of this kind was carried out by Menke\(^{20}\) for \(\mathrm{CCl}_4\). However, we shall not dwell on this and shall confine ourselves to the case of monatomic liquids, since our task is the consideration of metals.

DIFFRACTION OF X-RAYS IN PURE LIQUID METALS

Comparatively few works have been devoted to the experimental study of X-ray diffraction in liquid metals. In all cases the photographic method was used. In most cases the photographs were obtained from the free surface of the metals.

Debye and Menke\(^{21}\), in their work with mercury, and later Menke\(^{20}\) with mercury and liquid gallium, used a camera whose scheme is shown in Fig. 2. The angle of the diaphragm with the surface of the liquid metal \(\alpha\) could be varied at will. Usually the photographs were taken at \(\alpha=5^\circ\) or \(\alpha=15^\circ\). The flat slit of the diaphragm, depending on this, had dimensions \(0.2\times2\ \mathrm{mm}^2\) and, respectively, \(0.6\times2\ \mathrm{mm}^2\). Very important is the precise orientation of the metal surface relative to the cylindrical film and the incident beam. With this method the absorption correction can be made very accurately and simply,

In the case of liquids we do not have sharp interference lines, as occurs in radiographs of crystals. Therefore, for the interpretation of any given radiograph it is very important to know the general course of the intensity as a function of the scattering angle. This imposes great demands on the composition of the radiation used. As a result of filtering in the liquid itself, maxima caused by the continuous part of the spectrum may appear on the radiograph for one or another part of the spectral composition of the radiation. Studying the influence of monochromatization, Maier, for example, found that the filtration usually used in work with crystals, which leads to the removal of β-radiation from the total composition of the rays, is wholly insufficient for liquids.

Fig. 2. Diagram of the Debye–Menke camera.

Radiographs of water obtained by him using a filter (b) and reflection from a crystal (a) are shown in Fig. 3. It is clearly seen here that in the first case not only is the blurring of the maxima increased, but their very positions turn out to be strongly shifted. More precisely, the first maximum in Fig. 3, a, corresponding to the CuKα radiation, turns out to be a secondary maximum in Fig. 3, b. The main maximum here is due to the short-wavelength boundary of the continuous spectrum.

Fig. 3. Radiographs of water obtained by different methods of filtering the radiation.

The most reliable method of obtaining a monochromatic beam is, of course, reflection from a crystal. However, the extremely large increase in exposure associated with this is also undesirable. Debye, Menke, and Prins²³ prefer to work with filters, but at low voltage on the tube. For CuKα the best voltage is 18 kV, and for MoKα, 30 kV. The maximum voltage is determined by the condition that, while retaining a good yield of characteristic radiation, a minimum intensity of the continuous spectrum of rays be obtained. The monochromaticity of the radiation thus obtained was checked by repeated photographs in rays monochromatized on a crystal.

The photographs obtained were microphotometered. From the total intensity curve there was excluded that part of it which is caused by atomic scattering and polarization.

The experimental curve corrected in this way corresponds, evidently, to the external diffraction of the molecules of the liquid metal. In Fig. 4 such a curve is presented, obtained by Debye and Menke for mercury, and in Fig. 5—for liquid gallium, obtained by Menke. Along the abscissa axis are plotted not angles, but \(s = 2 \sin \frac{\vartheta}{2}\).

If one does not count the side maximum, lying on the outer fall of the intensity of the first maximum in Fig. 5, both curves display a similar character of the dependence of intensity on the scattering angle.

The interpretation of these curves is carried out according to the equation given above

\[ \rho(1 - W) = 2 \frac{d^3}{\lambda^3} \int_0^\infty 3(1 - E)\sin 2\pi \rho s\,ds. \]

Fig. 4. Intensity distribution curve for mercury.

The results are given in Figs. 6 and 7.

Here the ordinates indicate, for each \(r\), the probability that any two fixed atoms are simultaneously: one in the volume element \(dV_1\), the other in \(dV_2\), at a distance \(r\) from the first. For very small distances this probability is equal to zero. This corresponds to the fact that two atoms cannot approach one another to a distance smaller than the sum of their radii. For large atomic distances we have a rectilinear course of the curve parallel to the abscissa axis. This corresponds to the equal possibility of all possible positions of atoms at large distances.

Fig. 5. Intensity distribution curve for gallium.

The same course of probability, as Menke indicates, can be obtained by determining the statistics of the distances of two fixed balls situated in a mass of other steel balls. Each time after shaking the box with the balls, the distance between the fixed balls is determined. The probability curve obtained turns out to be

analogous to the curve in Fig. 6. This experiment with spheres leads Debye and Menke to the conclusion that, in order to obtain an ordering of the arrangement of molecules such as occurs in mercury, it is by no means necessary that there be any ordering forces, of the kind that occur in solid crystals. The presence of a secondary maximum in the case of gallium, in Menke’s opinion, indicates the presence of such forces here. However, this maximum rapidly disappears with increasing temperature, and the gallium curve becomes similar to the Hg curve.

Fig. 6. Probability curve for Hg.

Fig. 7. Probability curve for Ga.

The study of liquid metals was also undertaken by Sauerwald and Teske24. In their work they give curves only for mercury and thallium (Fig. 11). The intensity distribution given by liquid thallium is quite analogous to that which we have for mercury and gallium.

Model interpretation of radiographs of liquid metals

The probability curves obtained give us an idea of the statistics of atomic distances in liquid metals. From them one may conclude that in a liquid some distances prove more preferable than others. This indicates the presence of a certain ordering in the distribution of molecules. However, it is perfectly clear that these curves cannot be directly translated into the language of models of the spatial arrangement of molecules.

An attempt of this kind was made by Kratky25. He gave an interpretation of the probability curve for mercury on the basis of model concepts, to which we now turn.

The thermal motions of atoms in a crystal consist in the atoms performing oscillations relative to the lattice sites in which, we assume, they are located. Thus the average po-

positions of atoms remain unchanged, and the relative displacements of any two atoms of the crystal lattice do not depend on their mutual distance.

Kratky lays a substantially different proposition at the basis of his reasoning about a model spatial lattice possible for a liquid.

If one observes, at different moments in time, the fixed arrangement of the molecules of a liquid, then we shall always find one of those orderings which can be imagined by starting from some crystal lattice and assigning to each atom a definite scattering of its position relative to its neighbors. As a consequence of this, in contrast to what occurs in crystals, in liquids the relative displacement of atoms \(A\) and \(B\) increases with their distance. It is obtained as the result of vector summation of the separate displacements of the atoms lying between \(A\) and \(B\). The presence of such scattering of the positions of an atom characterizes its mobility in the lattice. It indicates either the frequency of changes of place or the sequence of displacements and thus has a direct relation to the phenomenon of diffusion in liquids.

All that has been said leads to the problem of an initial lattice. The initial lattice and the scattering of the individual atoms determine the complex of possible configurations of the atoms of a liquid. The X-ray pattern of a liquid may be regarded as the result of superposed instantaneous photographs of a series of momentary ordered arrangements of atoms, possible from the point of view of one or another initial spatial lattice.

The task of the model interpretation of the probability curves, consequently, reduces to this: starting from various spatial lattices, and taking into account the possible scatterings, to construct the probability curves of the atomic distances possible here and then compare them with the experimental ones.

Kratky carried out calculations for mercury, starting from five possible lattices: 1) hexagonal close-packed, 2) face-centered cubic, 3) body-centered cubic, 4) simple cubic, and 5) tetragonal.

The course of the reasoning is as follows.

Having fixed some atom, we compare it with other atoms according to their appearance in concentric spheres, at the center of which is the fixed atom. Let us define the number \(L\) as the surface density of atoms on such a sphere

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

where \(Z\) is the number of atoms in the sphere.

If along the axis of abscissas one plots the radii of the spheres, and as the ordinate one takes \(L\)—the surface density on them—then for the hexagonal close-packed lattice one obtains a dependence between \(L\) and \(r\),

presented in Fig. 8. Here lines lying very close to one another have been summed into one.

The “lattice constants,” i.e. the distances of neighboring atoms, are obtained from the specific gravity. Corresponding to coordination numbers 12.8; 6.4 we have 3.26; 3.16; 2.90; 2.57 Å. The vertical straight lines in Fig. 8 may serve as a measure of the probability of the appearance of an atom on one or another sphere. In this case, for crystalline lattices, corresponding to a strict arrangement of atoms at the nodes of the lattice, a probability different from zero can be obtained only on a discrete series of spheres. If now, however, in accordance with Kratky’s basic assumption concerning the scattering of atomic positions, one carries out a “blurring” of the lattice, then at any distance from a fixed atom some atom may appear, and therefore the density of the probability of the appearance of atoms will now already be different from zero on any sphere. In order to reconstruct the diagram of Fig. 8 in the corresponding way, let us suppose that for the first sphere the probability falls linearly on both sides from the most probable mean position.

Fig. 8. Graph of the probability of the appearance of atoms on spheres for a hexagonal close packing of a crystal lattice.

To obtain the blurring for the following spheres, it is necessary to take into account that the relative displacement of atoms \(A\) and \(B\) is obtained as the vector sum of the separate displacements of the intermediate atoms. Extending to these purely statistical summations Einstein’s diffusion equation, which states that the total displacement is proportional to the square root of the number of individual displacements, we arrive at the following proposition: in our case the mean “blurring” is proportional to the square root of the distance between the atoms under consideration. Taking here too as a basis the linear law of decrease of probability (from the mean, most probable one), we replace each segment of Fig. 8 by a triangle whose area is specified by the length \(L\), while its width is proportional to the square root of the distance from the origin. All these triangles are then superposed on one another.

In Fig. 9 the results of such a construction are given for five initial lattices.

At the top is given the probability curve of Debye and Menke. Even in the case of such a very approximate calculation, the similarity of the Debye and Menke curve to the curve obtained by the method just described for a hexagonal close-packed lattice of mercury is striking.

Kratky further refines the method for constructing probability curves and arrives at the smooth curves shown in Fig. 10.

The refinement of the calculations leads to confirmation of the results of the preceding, cruder construction. Thus the model interpretation of the probability curve leads to the conclusion that the structure of liquid mercury is based on a hexagonal, close-packed lattice.

In Kratky’s work the first attempt was made to determine the structure of a liquid metal. For the method he proposed, the most exact possible knowledge is required of the general course of the intensity as a function of the change in the scattering angle. The information presently available to us on the diffraction of X-rays in liquid metals is limited to two or three substances (mercury, gallium, thallium). Both gallium and thallium give a diffraction pattern similar to that of mercury. This allows one to suppose that these metals too, in the liquid state, possess a structure very close to that which Kratky found for liquid mercury.

It is interesting that the character of the intensity distribution on the X-ray photograph of gallium approaches that given by mercury as the temperature of gallium is raised. It is quite possible that all liquid pure metals, at temperatures far from the crystallization point, have a mercury-like structure. As the temperature is lowered, alongside such a structure there may appear regions with such a mutual arrangement of molecules as the metal has in the solid phase. One may imagine that, long before the point of solidification of the metal, in certain places, owing to temperature fluctuations, conditions may be created that are favorable for the mutual orientation of molecules in accordance with its crystalline structure in the solid state. Then maxima should appear on the X-ray pattern corresponding to periods of identity of the crystals of the solid metal. These will not be lines, of course—they will also be broadened, owing to the fact that here, alongside such crystalline formations, there will also be intermediate ones with somewhat different periods of identity.

Fig. 9. Kratky probability curves obtained for five possible initial lattices.

In reality, in gallium as well one observes the appearance of a new maximum when the temperature is lowered. Its intensity and sharpness increase as solidification is approached. The structure of solid gallium was studied by Eger, Terpstra, and Westenberg[^27]. According to their data, precisely at the place where, in liquid gallium, this maximum appears upon lowering the temperature, an intense line should be found in solid gallium.

The appearance of this new maximum does not entail a sharp change in the general course of the intensity. Thus, in liquid gallium at low temperatures there apparently coexist both the mercury-like, closely packed hexagonal structure and, alongside it, liquid crystalline formations corresponding to the structure of solid gallium crystals. At the same time, the latter are probably not fragments of solid crystallites; one may suppose that they are very short-lived formations with a lifetime of the order of the duration of existence, at a given point, of a region of molten metal of increased or decreased density as a result of thermal fluctuations.

Such an assumption seems plausible; however, the experimental material is still too scanty to permit more or less reliable general conclusions about the structure of liquid metals.

Fig. 10. Utopian curve.

It should also be mentioned that in the study of a whole series of liquid metals near the melting point, carried out by Randall and Rooksby[^26], sodium, potassium, rubidium, and cesium at a temperature slightly above the melting point give maxima on the X-ray photograph at those positions where the intense lines of these metals are located in the solid state.

However, the absence of intensity curves does not make it possible to compare these data with those obtained for other metals by Debye and Menke, Sauerwald and Teske.

Diffraction in Liquid Alloys

The investigation of the diffraction of X-rays in liquid alloys appears to be extremely interesting. For solid metallic alloys one may distinguish three cases: 1) mixed crystals—solid solutions, 2) chemical compounds, and 3) eutectic alloys. The first and second give X-ray diagrams of homogeneous crystals. In solid solutions there is usually found the lattice of that one of the constituent metals which is present here in the predominant amount. Into this lattice enter atoms of the other metal and, without in most cases changing the symmetry of the lattice, they alter the dimensions of the unit cell. The general course of the intensity in this case is similar to that observed for the principal component. In contrast to solid solutions, chemical compounds give a completely new course of intensity in comparison with their constituents, which corresponds to the fact that here we are already dealing with another crystal lattice. And, finally, eutectics are a mechanical mixture of crystallites of the components of the alloy.

Fig. 11. Curves of Zauervald.

If one takes into account that liquids also possess elements of crystalline structure, then it may be supposed that in alloys, even in the molten state, relations may occur similar to those observed in solid alloys.

At present there are still not enough works in this field to permit anything definite to be said on this question.

However, certain indications of such an analogy can already be detected from the work of Zauervald and Teske with the alloys Hg—Tl and Hg—K.

In Fig. 11 are shown the intensity curves for the alloys Hg₃Tl₂—(II and III), KHg₂—(V), and pure Hg and Tl₂—(I and IV).

The compound Tl\(_2\)Hg\(_5\) consists of metals with close ordinal numbers. Their X-ray photographs differ little from one another, apart from the fact that for thallium all three maxima are shifted toward smaller angles. This is in agreement with the fact that thallium has a larger atom than mercury.

The compound Tl\(_2\)Hg\(_5\), however, at low temperatures gives an X-ray photograph sharply different from the X-ray photographs of its constituents. Especially interesting is the circumstance that a maximum appears at small angles, corresponding, consequently, to a large period of identity. According to Zauervald’s calculations its magnitude agrees well with the dimensions of the molecule not Tl\(_2\)Hg\(_5\), but Tl\(_2\)Hg\(_4\). All this indicates that here we are dealing with a new structure of a chemical compound. With increasing temperature (curve III) the first maximum shifts toward larger angles and the whole curve becomes similar to curves I and IV, corresponding to pure mercury and thallium. It is difficult to suppose that in liquid metals there exist separate molecules consisting of metallic atoms, as occurs in the case of nonmetallic compounds. It is, however, quite natural to assume that, even long before crystallization, the alloy has a structure with crystalline formations similar to crystals of chemical compounds in solid alloys. In such formations there must be periods of identity determined by the dimensions of the molecules. The molecules themselves arise as a result of the ordered arrangement of thallium and mercury atoms in the crystalline formations of the liquid alloy.

The compound KHg\(_2\) also gives a maximum at small angles.

In the liquid state this alloy forms with a large thermal effect and a decrease in volume. This may serve as confirmation that the liquid alloy KHg\(_2\) is also a chemical compound.

In conclusion it is necessary to note that the X-ray study of liquid metals and alloys is at present in an embryonic state; there are only a few works on pure metals and, perhaps, the only work on alloys is that of Zauervald and Teske.

And yet the results obtained have already, to a considerable degree, shed light on a question of the structure of liquids that had previously been completely obscure; and it may be said with confidence that the application of X-rays to the investigation of the structure of liquids, and especially of liquid metals and alloys, will resolve this question just as it did for solids. It must be recalled, however, that the X-ray study of liquids is in somewhat less favorable conditions in comparison with X-ray analysis of solids.

The latter had as its predecessor an extremely rich crystallographic material, and the very concept of the space lattice arose long before the appearance of Laue’s work.

The X-ray study of liquids, by contrast, is beginning to grow without such predecessors.

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