Application of X-rays to the Study of Molecular Structure

M. A. Levashevich, Dnepropetrovsk

The concept of “molecular structure” includes, on the one hand, the establishment of the mutual connection between its constituent parts—atoms, ions—and, on the other hand, the determination of the spatial arrangement of the atoms and the distances between them. Whereas the first line of inquiry is closely connected with views on the nature of atoms and ions, the second can to a considerable extent be developed independently of these views. We shall be interested chiefly in the second line of inquiry, i.e., the determination of the distances between atoms and, consequently, the spatial arrangement of the latter in the molecule.

There exist several methods for determining molecular structure: for example, stereochemical, spectroscopic, X-ray, and others. A comparison of the results obtained by different methods shows that the X-ray method, in its accuracy, is not inferior to other methods, including the electron-diffraction method.

The X-ray method in the field of molecular structure first found application with crystals. However, X-ray study of molecules in crystals is convenient only in those cases when, upon transition of a substance from the liquid state to the solid state, the molecules retain their individuality. Otherwise one cannot be certain whether the distances between atoms remain constant (atomic or ionic lattices) in comparison with their values in free molecules.

Recently, with the development of X-ray analysis of liquids and gases, thanks to the work of Debye and other authors, it has become clear that the measurement of molecules can also be carried out in liquids and gases; moreover, in the case of a gas it is often possible to obtain results no less valuable than in crystals.

The purpose of the present article is to give a survey of the literature on the X-ray study of molecules in gases.

As for the electron-diffraction method of measuring molecules, it is in principle similar to the X-ray method. The advantage of each of these methods, as Bevilacqua has shown¹˒², depends in each individual case on particular circumstances. It is beyond dispute only that the exposure in the electron-diffraction method is much shorter.

Many authors in their works express themselves in favor of the electron-diffraction method in the case when the substance under investigation

consists of “light” atoms. However, Bevilogua¹ showed, using benzene as an example, that this is not quite correct.

In the case of polyatomic molecules containing “heavy” atoms, preference should be given to the radiographic method. In short, these two methods by no means exclude one another but, on the contrary, complement one another. They are a very valuable and powerful tool in the investigation of the structure of matter.

Theory of the Scattering of X-Rays in Amorphous Bodies

Let us imagine that a beam of monochromatic X-rays of wavelength \(\lambda\) illuminates some volume \(V\) of a monatomic amorphous substance, whose atoms are spheres of radius \(a\) and scatter X-rays independently of one another. If we now assume that the molecules (atoms) may occupy all possible positions relative to one another, with the exception of mutual penetration (i.e., they cannot approach one another closer than a distance equal to two radii), then, on the basis of Debye’s theory³, we shall have the following expression for the mean intensity of the scattered X-rays:

\[ I_m \sim N \frac{\psi^2}{R^2}\left[1-\frac{\Omega}{V}\Phi(2ksa)\right]. \tag{1} \]

Here \(N\) is the number of atoms in the volume \(V\) of the scattering substance; \(\psi\) is the atomic factor, which in Debye’s original theory was taken equal to the number of electrons \(Z\), it being assumed that all electrons are concentrated at the center of the atom; \(R\) is the distance from the center of radiation to the observer;

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

is the volume of the “spheres of action” of all the atoms; \(\Phi\) is the scattering function, equal to

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

where

\[ k=\frac{2\pi}{\lambda}, \qquad s=2\sin\frac{\vartheta}{2} \]

and \(\vartheta\) is the angle between the incident and the diffracted ray. The appearance in formula (1) of the ratio \(\frac{\Omega}{V}\) is connected with the condition adopted at the outset that the molecules cannot approach one another to distances smaller than the sum of their radii. The influence of this ratio on the general scattering pattern will obviously depend on the density of the scattering substance.

If we now turn to diatomic and polyatomic gases, then the expression for the intensity takes on a somewhat more complicated form.

Debye showed in 1915 that if a certain volume is illuminated which contains groups of atoms, the distances between which within the group remain constant, and the groups are oriented randomly relative to one another, then the general expression for the intensity of radiation scattered in various directions is represented in the following form:

\[ I \sim \sum_i \sum_j \psi_i \psi_j \frac{\sin ksr_{ij}}{ksr_{ij}}, \tag{2} \]

where \(\psi_i\) and \(\psi_j\) are the scattering powers of the \(i\)-th and \(j\)-th atoms belonging to the group, and \(r_{ij}\) is the distance between them.

However, here intergroup (intermolecular) diffraction has not been taken into account. If, analogously to what was done for a monatomic gas, intermolecular (“external”) diffraction is also taken into account, then for the case of a diatomic gas, as Debye\(^3\) showed, the following expression for the intensity is obtained:

\[ I_m = 4N \frac{\psi^2}{R^2} \left\{ \frac{1}{2} \left[ 1+\frac{\sin ksl}{ksl} \right] - \frac{\Omega}{V} \left( \frac{\sin \frac{ksl}{2}}{\frac{ksl}{2}} \right) \Phi(2ksl) \right\}. \tag{3} \]

Fig. 1.

The quantity \(\frac{\Omega}{V}\) plays here the same role as before. To give a clear representation of its influence on the scattering of X-rays, we present the curve, drawn on the basis of formula (3), for the intensity \(I_m\) as a function of the angle \(\vartheta\) for

\[ \frac{\Omega}{V}=0,\quad \frac{1}{4},\quad \frac{1}{2},\quad \frac{3}{4} \]

(Fig. 1). From this curve it is seen that the first maximum, appearing at \(12^\circ\), when \(\frac{\Omega}{V}\) decreases, first shifts toward smaller angles and then disappears completely.

As for the second maximum, corresponding to the angle \(\vartheta = 45^\circ\), it is caused by intramolecular scattering and, consequently, its position does not depend on the density of the substance.

Debye’s conclusions regarding the dependence of the X-ray scattering pattern at small angles on the density of the gas were verified experimentally by Harvey.\(^{4,5}\) He carried out experiments with nitrogen

at various pressures. In Fig. 2 the results of these experiments are presented. The continuous curves, calculated according to Debye, give the course of the intensity for nitrogen at small angles and at various pressures. Curve I corresponds to normal pressure (in the calculation \(\frac{\Omega}{V}\) was taken equal to zero). Curve II corresponds to a pressure of 60, III to 80, and IV to 100 atm. The experimental course of the intensity is shown by triangles, crosses, and circles, corresponding respectively to pressures of 60, 80, and 100 atm.

Fig. 2.

Fig. 3.

A discrepancy is obtained between the experimental curves and the theoretical ones. Geinrich and Varren\(^6\) showed that in this case better agreement can be achieved if, when calculating the intensity curve for a gas at high pressures, one takes into account the possible arrangements of molecules in the gas as they approach one another at small distances, instead of assuming complete randomness in the distribution of molecules, as was done by Debye. The curves of Fig. 9 are calculated on the basis of Debye’s assumption.

In the case of gases at atmospheric pressure the ratio \(\frac{\Omega}{V}\) corresponds to a quantity of order \(10^{-3}\); consequently the second term in formula (3) may be neglected. Then the remaining part assumes the form:

\[ I_m = 2N \frac{\psi^2}{R^2}\left[1+\frac{\sin ks}{ks}\right]. \tag{4} \]

It is caused by “internal” interference and makes it possible to determine the structure of the molecule, provided only that the substance under investigation has been chosen successfully.

It must be noted that, in constructing curve (3), two factors independent of the structure of the molecule have not been taken into account: first, the polarization factor, due to the polarization of the scattered X-rays and equal to \(\dfrac{1+\cos^2\vartheta}{2}\), and, second, the atomic factor, which takes account of the circumstance that atoms are not radiating points (the point theory), but have a finite size, which affects the scattering effect. Debye\(^7\) showed how this influence manifests itself. In Fig. 3 a series of curves is presented for the intensity of the scattered rays as a function of the quantity

\[ x = ksl = 4\pi \frac{l}{\lambda}\sin\frac{\vartheta}{2}, \]

where \(l\) is the distance between the atoms. Beside each curve is shown a model of a diatomic molecule, in which the atoms are represented by circles of radius \(r\). It turns out that the larger the ratio \(\dfrac{r}{l}\) (\(l\) remaining constant), the more steeply the curve falls with increasing scattering angle \(\vartheta\). In addition, for small values of this ratio the molecule gives noticeable maxima and minima, whereas for large values the maxima and minima completely disappear; in practice a smooth curve is obtained.

To calculate the atomic factor it is necessary to know the distribution of electrons in the atom. In the work of Debye and his school one usually uses a method based on the Fermi–Thomas atomic model. Taking the distribution of electrons in the atom according to Fermi–Thomas, the expression for the atomic factor can be represented in the following form:

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

where

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

\(a\) is the characteristic radius, equal to \(\dfrac{0.47}{Z^{1/3}}\), and \(\Phi(u)\) is the scattering function characteristic of the atom; it is shown in Fig. 4. The quantities \(k\), \(s\), and \(\vartheta\) have the same meanings as before.

Fig. 4.

From the expression for the atomic factor it is clear that it is not simply equal to the number of electrons \(Z\) in the atom, as was assumed by the point theory, but is the product of the number of electrons by a certain function \(\Phi(u)\). This function was tabulated by Bewilogua.

The data borrowed from his work^8 are given in Table 1. Using this table, one can easily find the atomic factor for any elements. It should be noted that this method gives a sufficient approximation for the charge distribution in atoms that are not very “light.”

In the case of interest to us, i.e., in determining the structure of molecules, taking the atomic factor into account in the interpretation of X-ray diffraction patterns is very substantial, especially for heavy atoms.

TABLE 1

$u = ksa$	$\Phi$	$u = ksa$	$\Phi$
0,00	1,000	1,71	0,284
0,16	0,922	1,86	0,264
0,31	0,796	2,02	0,240
0,47	0,684	2,17	0,224
0,62	0,589	2,33	0,205
0,78	0,522	2,48	0,189
0,93	0,469	2,64	0,175
1,09	0,422	2,80	0,167
1,24	0,378	2,95	0,156
0,40	0,342	3,11	0,147
1,55	0,309

Thus, if one takes into account the atomic and polarization factors and bears in mind that the determination of molecular structure in gases is usually carried out at relatively low densities—when $\Omega/V$ turns out to be a small quantity—then in the general case we arrive at the following expression for the intensity:

\[ I \sim \frac{1+\cos^2 \vartheta}{2} \sum_i \sum_j \psi_i \psi_j \frac{\sin x_{ij}}{x_{ij}}, \tag{5} \]

where

\[ x_{ij}=\frac{4\pi}{\lambda}\,l_{ij}\sin\frac{\vartheta}{2}, \]

and $l_{ij}$ is the distance between atoms $i$ and $j$.

In this form the formula is valid not only for two, but also for any number of atoms in a single molecule. On its basis, for each molecule one can calculate the scattering intensity. On the other hand, using data obtained from experiment, one can determine the structure of the molecule. Since to each observed value of the angle $\vartheta$ (maxima and minima on the curve) corres-

corresponds to a quite definite value of the quantity $\dfrac{x}{\pi}$, then from the relation

\[ \frac{x}{\pi}=\frac{4l_{ij}}{\lambda}\sin\frac{\vartheta}{2} \]

it is easy to determine the distance between the atoms $l_{ij}$.

All the experimental data obtained so far agree very well with these conclusions; the whole question is only that the choice of the substance for the experiment should be made successfully. The substance must consist of sufficiently “heavy” atoms, so that there is no sharp drop in the curve of the function $\Phi$ and, moreover, so as to avoid Compton scattering as far as possible. Further, it (the substance) must not selectively absorb X-rays, in order to avoid its own scattered radiation of greater or lesser intensity. Finally, the molecules must contain as large a number as possible of homogeneous atoms.

$CCl_4$ satisfies these requirements very well, which is apparently why it was chosen for the first investigations.

Experimental Conditions for the X-ray Photography of Molecules in Gases

Debye’s theory of X-ray scattering in amorphous bodies indicated ways of applying X-rays to the determination of molecular structure.

At the present time a rather extensive body of experimental material has already accumulated in this field, fitting well within the framework of Debye’s theory. At the same time, all the experimental and theoretical data undoubtedly confirm the fruitfulness of the X-ray method for determining the structure of molecules in gases and, in some cases, in liquids.

As regards crystals, as has already been mentioned, they often do not give the desired results for this purpose, since in many cases in crystals the concept of a molecule loses its meaning.

Gas at not very high pressures is best suited for this purpose, since in it the scattering by individual molecules may be regarded as independent of one another, i.e., the interaction between molecules may be neglected. This leads to a reduction (or complete elimination) of the influence of “external” interference on the overall scattering pattern and thus greatly facilitates the problem of determining molecular structure.

For this reason many of the works now existing in this field, in particular the works of Debye’s school, had gas as the object of their investigations.

Let us dwell briefly on the method of obtaining photographs from gases. The most widespread method should be considered to be that developed by Bevilacqua\(^9\), which he used in experiments with \(CCl_4\), \(CHCl_3\), \(CH_2Cl_2\), and \(CH_3Cl\). In Fig. 5 the chamber used in these experiments is shown schematically. The rays from the X-ray tube

Fig. 5.

enter a diaphragm \(60\ mm\) long, with a rectangular aperture \(4 \times 1\ mm^2\). They then travel a path of \(1.5\ cm\) in the gas and are absorbed by a shutter after the primary beam has given a zero mark on the film. The rays scattered in the gas (secondary rays) pass through a slit \(6 \times 3\ mm^2\), located in the bottom of the chamber and as close as possible to the primary beam. On emerging from the slit the rays fall on a cylindrical film, which is located in a cassette cooled by water. In the cassette, holes are made every \(10^\circ\), giving images on the film and thereby indicating the angular scale. The gas under investigation enters the chamber from a boiler (flask) provided on one side with a condenser, and on the other with a metal tube passing into the chamber. Both the tube and the chamber must be heated \(15\text{--}20^\circ\) above the boiling point in order to avoid condensation. The entire space in which the gas under investigation is placed is separated from the diaphragm and the window by nickel foil of \(5\ \mu\), or aluminum foil of \(10\ \mu\). Monochromaticity of the radiation was achieved with the aid of a filter: nickel for copper rays, zirconium for molybdenum rays, etc.

The chamber described above must satisfy the following requirements:

1) All rays must fall on the film at right angles, in order to avoid corrections.

2) The exposure must be as short as possible, and consequently so must the path of the rays; therefore the diaphragm must not be very long.

3) At the same time, the dimensions of the diaphragm must be such as to eliminate undesirable secondary radiation, and also to make tertiary rays imperceptible.

4) The irradiating volume must be sufficiently small in order to avoid a large divergence of the rays, leading to considerable blurring of the maxima on the radiograph.

5) The angle of coverage must be as large as possible.

6) Equal volumes should radiate at different angles.

7) The path of the rays in the gas under investigation should be short.

8) Undesirable secondary and tertiary rays should be eliminated.

Deviations from these requirements were of practically no significance owing to their smallness and therefore were not taken into account.

Special attention should be paid to the monochromaticity of the radiation, since it plays a large role in discussing the course of the intensity as a function of the scattering angle when interpreting radiographs of gases and liquids. In most cases, in order to obtain monochromatic rays, ordinary filters are used to shorten the exposure—for example, nickel filters for copper rays, zirconium filters for molybdenum rays, etc., working at low voltages (for Cu—18, for Mo—30 kV). However, such filtration, according to the investigations of Maher and other authors, can often also lead to distortion of the radiographs, namely in those cases where the voltage on the tube is high. In Fig. 6 are shown radiographs of water obtained by Maher¹⁰ in copper rays without a filter, with a nickel filter, and in radiation reflected from a crystal.

Fig. 6.

It turns out that the nickel filter enhances the false maximum at \(15^\circ\), which appears from the continuous part of the spectrum.

Van der Grinten¹¹, in his work on the question of the use of monochromatic radiation in the scattering of X-rays in gases, showed how important monochromaticity of the radiation is in radiographic investigations. He obtained Cu—\(K_\alpha\) rays by reflection from a crystal and performed experiments with \(\mathrm{CCl}_4\), using a method generally similar to that used by Bewilogua. Only in his case the gas under investigation was in a glass reservoir instead of a metal one, and the points of entry and exit of the rays were closed with thin disks of mica attached to the glass with a special cement. Filtration of the rays by reflection from the crystal increased the exposure to 20 hours, despite an increase in the slit in the direction of the primary rays. In calculating the radiographs Van der Grinten used the polarization factor in the form

\[ \frac{1+\cos^2 2\varphi \cos \vartheta}{1+\cos^2 2\varphi} \]

instead of the usual

\[ \frac{1+\cos^2 2\varphi}{2}, \]

since the primary rays are partly polarized upon reflection from the crystal. Here \(\varphi\) is the glancing angle, and \(\vartheta\) is the scattering angle in the gas.

The results obtained by Van der Grinten are presented

in Figs. 7 and 8. In the first case (Fig. 7) monochromatization of the radiation was achieved by reflection from a crystal, and in the second (Fig. 8)—by the usual method (a nickel filter). Alongside the experimental curves the theoretical ones are also shown. From comparison of the experimental curves with the theoretical ones it is clearly seen that, in the case of filtration of the rays by reflection from a crystal, the agreement of theory with experiment is much better.

In determining the structure of molecules by means of X-rays one has to take into account incoherent radiation, i.e. that part of the rays scattered by the atoms which does not take part in the formation of the interference pattern. Experimentally this radiation cannot be separated from the coherent part. However it can be allowed for by using the Heisenberg formula\(^{12}\):

Fig. 7.                Fig. 8.

\[ S_i=N\left\{1-\frac{32\pi^2}{3}\int_{0}^{x_0}\frac{dx\,x^2}{h^3} \left[\sqrt{2meF(x)}-\frac{s_0h}{4\pi}\right]^2 \left[2meF(x)+\frac{s_0h}{8\pi}\right]\right\}. \]

This formula, as Bewilogua\(^{8}\) has shown, gives very good results for gases. It may be expressed in another form, more convenient for calculation:

\[ S_i=Z\left\{1-\int_{0}^{\xi_0}\xi^2\,d\xi \left[\left(\frac{\varphi(\xi)}{\xi}\right)^{\frac12}-v\right]^2 \left[\left(\frac{\varphi(\xi)}{\xi}\right)^{\frac12}+\frac12 v\right]\right\} \tag{6} \]

where

\[ \xi=\frac{x}{a}N^{\frac13}; \]

\[ \frac{\varphi(\xi)}{\xi}=\frac{F(x)}{F_0} \quad\text{and}\quad v=\frac{ksa}{(6Z)^{\frac23}}. \]

Substituting into the last expression instead of \(a\) its value
\(\dfrac{0.47}{Z^{1/3}}\,\text{\AA}\), we obtain for \(v\):

\[ v = ks\,\frac{0.176}{Z^{3/2}} = ksb, \]

where \(b\) is the characteristic length for incoherent radiation.

\(\xi_0\) is determined from the equation

\[ \left[\frac{\varphi(\xi_0)}{\xi_0}\right]^{1/2} - v = 0. \]

The expression in braces (6), calculated for various values of \(v\), is given in Table 2, borrowed from Bewilogua’s work.

TABLE 2

\(v = ksb\)	\(s\)
0.05	0.319
0.1	0.486
0.2	0.674
0.3	0.776
0.4	0.839
0.5	0.880
0.6	0.909
0.7	0.929
0.8	0.944
0.9	0.954
1.0	0.963

Fig. 9.

Multiplying the values of \(s\) from the second column of this table by \(Z\) and substituting the corresponding values of the quantity \(b\), one can express the intensity of incoherent radiation as a function of

\[ \frac{\sin \dfrac{\vartheta}{2}}{\lambda}. \]

In the case of heavy atoms, the influence of incoherent radiation on the interference pattern is insignificant and therefore may remain unaccounted for. For comparison we give the intensity curves, from Bewilogua’s work\(^8\), for the molecule \(\mathrm{CCl_4}\) (Fig. 9), i.e., with heavy atoms (Cl), and for a molecule with light atoms \((\mathrm{C_6H_6})\) (Fig. 10).

From consideration of these curves it is clearly seen that for benzene the influence of incoherent radiation is more significant than for CCl$_4$, and this corresponds to the observations.

RESULTS OF THE X-RAY INVESTIGATION OF THE STRUCTURE OF MOLECULES IN GASES

Let us now turn to consideration of the results obtained by various authors in the field of determining interatomic distances in molecules.

According to Debye’s theory, the exact distances in a molecule must also determine the exact interference pattern of the scattered X-rays, from which one may judge the magnitude of these distances.

Fig. 10.

Fig. 11.

X-rays.

The molecule CCl$_4$ was studied especially carefully. Debye, Bewilogua, and Ehrhardt$^{13}$ in 1929 were the first to demonstrate the possibility of interference of X-rays scattered by the individual atoms of molecules, carrying out experiments on the CCl$_4$ molecule. Subsequently the investigation of this molecule was continued by Debye$^{7}$ and Bewilogua$^{9}$. Their works contain a detailed discussion of the scattering of X-rays by CCl$_4$ molecules. In view of the importance of the conclusions of Debye and Bewilogua, we shall dwell on them in somewhat greater detail.

From the formula

\[ I \sim \frac{1+\cos^{2}\vartheta}{2} \sum_i \sum_j \psi_i \psi_j \frac{\sin x_{ij}}{x_{ij}} \]

for the CCl$_4$ molecule, assuming a tetrahedral model, after summation the following expression is obtained:

\[ I=\frac{1+\cos^{2}\vartheta}{2} \left\{ 4\psi_1^{2}\left[1+3\frac{\sin x}{x}\right] +8\psi_1\psi_2\frac{\sin x'}{x'} +\psi_2^{2} \right\}. \]

Here \(x\) and \(x'\) have, respectively, the following values:

\[ x = ksl = 4\pi \frac{l}{\lambda}\sin \frac{\vartheta}{2}, \]

\[ x' = ksl' = 4\pi \frac{l'}{\lambda}\sin \frac{\vartheta}{2}, \]

where \(l\) is the distance between the chlorine atoms, and \(l'\) is the distance between the carbon and chlorine atoms. \(\psi_1\) and \(\psi_2\) are the atomic factors, respectively equal (according to Debye—Fermi): for Cl with 17 electrons

\[ \psi_1 = 17\Phi\left(ks\frac{0.47}{17^{1/3}}\right) \]

and for C with 6 electrons

\[ \psi_2 = 6\Phi\left(ks\frac{0.47}{6^{1/3}}\right). \]

Fig. 12.

The quantity

\[ k=\frac{2\pi}{\lambda}, \quad \text{and} \quad s=2\sin\frac{\vartheta}{2}, \]

where \(\lambda\) is the wavelength in Å.

The curve calculated in this way is given in Fig. 11. Alongside it the experimental curve is also placed. On the abscissa axis, in the case of the theoretical curve, the values of the quantity \(\frac{x}{\pi}\) are plotted, while in the case of the experimental curve, the values

\[ s = 2\pi \sin\frac{\vartheta}{2}. \]

In addition, Fig. 12 shows the photometric curve. Microphotometry was carried out with the aid of a Zeiss recording photometer.

Both curves, experimental and theoretical, have three sharply expressed maxima and in general agree very well with each other. In detail, the difference between them consists in the fact that on the theoretical curve the maxima are sharper than on the experimental one. This apparently depends on the fact that the filtration of the rays was insufficient.

Six characteristic points (three maxima and three minima) on the experimental curve and the same number on the theoretical curve make it possible, for each observed value of \(s\), to find the corresponding value of \(\frac{x}{\pi}\) of the theoretical curve, and then from the relation \(\frac{x}{\pi}=2s\frac{l}{\lambda}\) to calculate the distance \(l\) between the atoms.

Thus, for example, the mean distance \(l_{\mathrm{Cl}-\mathrm{Cl}}\) between chlorine atoms turns out to be \(2.99\ \text{Å} \pm 1\%\), if for the \(CCl_4\) molecule a tetrahedral model is adopted. Taking into account that the accuracy of the results for interatomic distances depends on the number of maxima and minima on the curve, and consequently also on the number of atoms of the molecule participating in the scattering of X-rays, then for \(CCl_4\), with three sharply expressed maxima, one may rely on two decimal places.

It is interesting that this same distance, calculated with the aid of Bragg radii \((0.77\ \text{Å}\) for C and \(1.05\ \text{Å}\) for Cl), on the basis of the same tetrahedral model, is equal to \(2.98\ \text{Å}\), i.e. a very good agreement is obtained.

For illustration we give a table, borrowed from the above-cited work of Debye\(^7\), in which are placed the results he obtained

TABLE 3

	Theoretical \(x/\pi\)	Theoretical \(x/\pi\)	Theoretical \(x/\pi\)	\(s\) observed	\(l_0\) in Å
	\(Cl_4\) (●)	\(CCl_4\) (●)	\(CCl_4\) (0)
1 min.	1.43	1.50	1.60	0.411	3.00
1 max.	2.47	2.48	2.38	0.618	2.97
2 min.	3.47	3.42	3.60	0.93	2.98
2 max.	4.48	4.44	4.30	1.10	3.01
3 min.	5.48	5.50	5.70	1.46	3.00
3 max.	6.48	6.56	6.30	1.64	2.96

for \(CCl_4\) (Table 3). Here six independent determinations of the distance \(l_{\mathrm{Cl}-\mathrm{Cl}}\) are given. In the first column of the table the order of the maxima and minima on the scattering curve is given. In the second and third columns are placed the values of \(x/\pi\) obtained on the basis of point theory, which is indicated by a dot in parentheses; in the second column the values of \(x/\pi\) are given without taking account of the C atom. In the fourth column are given the values of \(x/\pi\) taking into account the size of the atoms, which is indicated by a zero in parentheses. In the fifth column are placed the values \(s\), obtained from experiment, and, finally, in the last column—the values of the distances between chlorine atoms in Å. From all these values for the mean

the distance \(l_{\mathrm{Cl-Cl}}\) between the chlorine atoms is found to be 2.99. The distance between the carbon atom and the chlorine atom is easily calculated from the relation:

\[ l_{\mathrm{C-Cl}}=\frac{1}{2}\sqrt{\frac{3}{2}}\,l_{\mathrm{Cl-Cl}}. \]

It is equal to 1.83 Å. For the angle between the \( \mathrm{C-Cl} \) bonds the value \(109^\circ.5\) was obtained.

These data are a good confirmation of Debye’s theory on the scattering of X-rays in amorphous bodies and served as the occasion for a whole series of works on the measurement of molecules by the indicated method.

Bewilogua\(^9\), in addition to \(\mathrm{CCl}_4\), investigated \(\mathrm{CHCl}_3\), \(\mathrm{CH}_2\mathrm{Cl}_2\), and \(\mathrm{CH}_3\mathrm{Cl}\) and arrived at the following results concerning interatomic distances and the arrangement of atoms in the molecule.

Using the distances between atoms obtained for \(\mathrm{CCl}_4\), he calculated the scattering intensities for these molecules (\(\mathrm{CHCl}_3\), \(\mathrm{CH}_2\mathrm{Cl}_2\), and \(\mathrm{CH}_3\mathrm{Cl}\)). In doing so, the scattering of the hydrogen atom, as well as thermal motion, were not taken into account. Under these conditions he, on the basis of formula (5), obtained the following expression for the intensity of the scattered X-rays:

For \(\mathrm{CHCl}_3\)

\[ I=3\left\{\Phi_{\mathrm{Cl}}^{2}\left[1+2\frac{\sin x}{x}\right]+0.706\,\Phi_{\mathrm{Cl}}\Phi_{\mathrm{C}}\frac{\sin x'}{x'}+0.042\,\Phi_{\mathrm{C}}^{2}\right\}. \]

For \(\mathrm{CH}_2\mathrm{Cl}_2\)

\[ I=2\left\{\Phi_{\mathrm{Cl}}^{2}\left[1+\frac{\sin x}{x}\right]+0.706\,\Phi_{\mathrm{Cl}}\Phi_{\mathrm{C}}\frac{\sin x'}{x'}+0.062\Phi_{\mathrm{C}}^{2}\right\}. \]

For \(\mathrm{CH}_3\mathrm{Cl}\)

\[ I=\left\{\Phi_{\mathrm{Cl}}^{2}+0.706\,\Phi_{\mathrm{Cl}}\Phi_{\mathrm{C}}\frac{\sin x'}{x'}+0.124\,\Phi_{\mathrm{C}}^{2}\right\}, \]

where \(x=ksl_{\mathrm{Cl-Cl}}\), and \(x'=ksl_{\mathrm{C-Cl}}\).

The curves corresponding to these formulas are presented in Figs. 13, 14, and 15; the experimental curves are also placed there. Comparing the theoretical curves with the experimental ones, it is possible, as in the case of \(\mathrm{CCl}_4\), for each observed value of \(s\) to find the corresponding value \(\frac{x}{\pi}\) of the theoretical curve and then, from the relation \(\frac{x}{\pi}=2s\frac{l}{\lambda}\), to calculate the distances between atoms. The results for copper (\(\mathrm{Cu}-K_{\alpha}\)) radiation are presented in Tables 4, 5, and 6, borrowed from Bewilogua’s work.

Fig. 13.

Fig. 14.

Fig. 15.

TABLE 4

CHCl₃

	\(s\)	\(\dfrac{x}{\pi}\)	\(l\) in Å	mean \(l\) in Å
1 min.	0.398	1.62	3.13	\(3.11 \pm 0.5\)
1 max.	0.576	2.36	3.15	\(3.11 \pm 0.5\)
2 min.	—	3.72	—	\(3.11 \pm 0.5\)
2 max.	1.06	4.30	3.14	\(3.11 \pm 0.5\)
3 min.	1.44	5.84	3.10	\(3.11 \pm 0.5\)
3 max.	1.62	6.28	3.02	\(3.11 \pm 0.5\)

TABLE 5

CH₂Cl₂

	\(s\)	\(\dfrac{x}{\pi}\)	\(l\) in Å	mean \(l_{\mathrm{Cl-Cl}}\) in Å
1 min.	—	—	—	\(3.23 \pm 0.1\)
1 max.	0.51	2.20	3.29	\(3.23 \pm 0.1\)
2 min.	—	—	—	\(3.23 \pm 0.1\)
2 max.	0.98	4.18	3.26	\(3.23 \pm 0.1\)
3 min.	—	—	—	\(3.23 \pm 0.1\)
3 max.	1.53	6.20	3.13	\(3.23 \pm 0.1\)

TABLE 6

CH₃Cl

	\(s\)	\(\dfrac{x}{\pi}\)	\(l\) in Å	mean \(l_{\mathrm{Cl-Cl}}\) in Å
1 min.	—	—	—	\(1.8 \pm 0.1\)
1 max.	0.6	1.4	1.8	\(1.8 \pm 0.1\)
2 min.	—	—	—	\(1.8 \pm 0.1\)
2 max.	0.9	2.2	1.9	\(1.8 \pm 0.1\)
3 min.	—	—	—	\(1.8 \pm 0.1\)
3 max.	1.2	2.8	1.8	\(1.8 \pm 0.1\)

The last columns of these tables contain the mean values for the distances between chlorine atoms. As for the accuracy of these results, whereas for \(\mathrm{CCl_4}\), with three sharply expressed maxima, one could rely on two decimal places, for \(\mathrm{CH_3Cl}\), for example, with one weakly expressed maximum (see Fig. 15), even the first digit will be inaccurate.

For the angles between the \(\mathrm{C-Cl}\) bonds the following values were obtained: for \(\mathrm{CHCl_3}\) — \(116^\circ \pm 3^\circ\), and for \(\mathrm{CH_2Cl_2}\) — \(124^\circ \pm 0^\circ\), i.e. greater than

for CCl$_4$. The increase in the distance between chlorine atoms for the molecules CHCl$_3$ and CH$_2$Cl$_2$ in comparison with CCl$_4$ is explained by a change (expansion) of the tetrahedral angle, which is confirmed by measurements of the electric moments of the chloromethane series. Table 7 gives the relevant data.

TABLE 7

Substance	Observed	Calculated from tetrahedron	Calculated from X-ray data
CCl$_4$	0	0	0
CHCl$_3$	1.0	1.9	1.2
CH$_2$Cl$_2$	1.6	2.2	1.9
CH$_3$Cl	1.9	1.9	1.9

$\mu \cdot 10^{18}\ CGS$

The first column of the table contains values of the electric moments borrowed from Debye’s tables$^{14}$; the second column gives values for the tetrahedral model, calculated from the moment of CH$_3$Cl, and the third column gives values of the moments calculated from X-ray data.

In 1932 Richter$^{15}$ investigated trimethylamine (CH$_3$)$_3$N, for the molecule of which he adopted, in the calculation, three models: a tetrahedron, a “flat” pyramid, and a triangle. The distance between N and CH$_3$ was assumed constant and equal, according to crystallographic data, to 1.5 Å. In the case of the tetrahedron the three CH$_3$ groups are situated at the vertices of an equilateral triangle with sides CH$_3$—CH$_3$ = 1.5 Å, and the nitrogen atom is at the vertex of the tetrahedron. If now the N atom, descending vertically downward, reaches the center of gravity of the tetrahedron, then it forms with the three CH$_3$ groups a “flat” pyramid with sides: $l_1 =$ CH$_3$—CH$_3 = 2.45$ Å, $l_2 =$ N—CH$_3 = 1.5$ Å and height $h = 0.73$ Å. With further displacement of the nitrogen atom downward, until it reaches the center of gravity of the equilateral triangle, the third model (the triangle) is obtained, for which $l =$ CH$_3$—CH$_3 = 2.6$ Å.

The calculation of the intensity was carried out first for all three models without taking account of the hydrogen atom, i.e., instead of (CH$_3$)$_3$N the formula C$_3$N was adopted. In Fig. 16 the results of the calculation are presented in the form of three curves, of which: the dotted one is for the tetrahedron, the solid one for the “flat” pyramid, and the curve with mixed dots for the triangle.

All three curves give satisfactory agreement at large angles; at small angles, however, they diverge noticeably. If they are compared with the experimental curve shown in Fig. 17, it can be observed that over a fairly large interval it agrees well with the theoretical curves for the pyramid and the triangle, and differs considerably from the curve calculated for the tetrahedron. It is true that on the experimental curve a certain wav—

Fig. 16.

Fig. 18.

Fig. 17.

Fig. 19.
Legend: solid line — theor.; dashed line — exp.; “flat” pyramid; \(+5\%\); \(-5\%\).
Lower plot legend: solid line — theor.; dashed line — exp.; triang.

intensity that is not present on the theoretical curves. The reason for this, apparently, is that the scattering by hydrogen atoms has not been taken into account. If this is allowed for, a considerably better agreement is obtained (Fig. 18).

To decide which of the three proposed models should be assigned to the molecule \((\mathrm{CH}_3)_3\mathrm{N}\), it is sufficient to plot the curves on an enlarged scale along the ordinate axis (Fig. 19).

It is easy to see that the model of the “flat” pyramid fits best. The discrepancy observed near the first maximum (marked with a cross) is less than \(\pm 5\%\).

The presence in a molecule of even only two heavy atoms is quite sufficient for obtaining interference, on the basis of which quite definite conclusions can be drawn about the structure of molecules (Bewilogua \(^{9}\)).

However, in some cases one can make accessible the study of the characteristic features of a molecule even without replacing light atoms by heavy ones. The point is that, using the conclusions of Debye’s theory, one can calculate not only the positions of maxima and minima on the X-ray photograph of a molecule, but also determine the general course of the intensity of the scattered X-rays. From the general course of the intensity, even in the case when there are no sharply expressed maxima, one can, as Gaevskii \(^{16}\) showed, determine at least approximately the structure of molecules.

Gaevskii carried out experiments with nitrogen and oxygen and found that on the intensity curve for nitrogen, on which, according to calculations by the point theory, a maximum is obtained at \(s = 1.74\), no such maximum is observed experimentally. For the distance between atoms he took \(1.10\ \text{Å}\)—the value found by Rasetti from the moment of inertia of Raman spectra. If the size of the atom (atomic factor) is taken into account, using the characteristic radius

\[ a=\frac{0.47}{Z^{1/3}}=0.246\ \text{Å}, \]

and corrections for incoherent radiation, calculated according to Heisenberg \(^{12}\), are introduced, then the intensity curve obtained in this way agrees with the experimental one.

For \(\mathrm{O}_2\), according to the point theory, taking, following Rasetti, \(l = 1.21\ \text{Å}\), a maximum is obtained at \(\vartheta\) approximately equal to \(105^\circ\), which, when the atomic factor is taken into account, is smoothed out, but does not disappear completely, as in nitrogen. In Fig. 20 \(a\) and \(b\), the curves for nitrogen \((a)\) and oxygen \((b)\) are given for comparison.

An interesting application of the method of X-ray interference is measurements on isomers.

Thus, for example, Ehrhardt \(^{17}\), who investigated by this method the molecules of the derivatives ethane and ethylene (dichloroethane \(\mathrm{C}_2\mathrm{H}_4\mathrm{Cl}_2\) and dichloroethylene \(\mathrm{C}_2\mathrm{H}_2\mathrm{Cl}_2\)), arrived at very important conclusions concerning their structure.

Proceeding from the tetrahedral model for the indicated molecules, he obtained for the intensity of the scattered X-rays, at

on the basis of the Debye formula (5) given above, the following expression:

\[ I \sim \psi_{\mathrm{Cl}}^{2}\left\{1+\frac{\sin ksl_{\mathrm{Cl-Cl}}}{ksl_{\mathrm{Cl-Cl}}}\right\} +0.706\,\psi_{\mathrm{C}}\psi_{\mathrm{Cl}} \left\{ \frac{\sin ksl_{\mathrm{C_1-Cl}}}{ksl_{\mathrm{C_1-Cl}}} + \frac{\sin ksl_{\mathrm{C_2-Cl}}}{ksl_{\mathrm{C_2-Cl}}} \right\} +0.125\,\psi_{\mathrm{C}}^{2} \left\{1+\frac{\sin ksl_{\mathrm{C-C}}}{ksl_{\mathrm{C-C}}}\right\}. \]

The second term in this formula has only two terms, since the distances C—Cl are always pairwise equal.

Fig. 20a.

Fig. 20b.

In processing the results, polarization was taken into account, as well as incoherent radiation, calculated from Bevilogue’s data1. The hydrogen atoms were not taken into account.

Let us now consider the results obtained by Ergardt, for example, for the dichloroethylene molecule.

According to the theory of van ’t Hoff, at a double bond the two carbon tetrahedra touch one another by edges (Fig. 21). Therefore both chlorine atoms of dichloroethylene may occupy two positions distinct from one another: 1) if they are located adjacent to one another on the same side of the molecule, the cis-isomer is obtained; 2) if the two chlorine atoms are distant from one another, such a position corresponds to the trans-isomer (Fig. 22).

A similar distinction in structure is also given by X-ray analysis, as is seen from Fig. 23. For cis-dichloroethylene two maxima are obtained, whereas for trans-dichloroethylene there is clearly visible also

third maximum. The first maximum for the trans-isomer lies at a smaller angle than for the cis-isomer. From this one may conclude that the distance between the chlorine atoms for the trans-isomer must be greater than for the cis-isomer.

Comparison of the experimental curves for both isomers with the theoretical ones, presented in Fig. 24, gives good agreement

Fig. 21.

Fig. 22.          Fig. 23.

between theory and experiment. In this case, for the cis-isomer the distance between the chlorine atoms is \(3.7\ \text{\AA}\), and for the trans-isomer \(4.7\ \text{\AA}\).

These conclusions agree well with the data obtained from measurements of electric moments.

Similar results with X-ray measurements on the isomers of para- and ortho-dichlorobenzene were obtained by Pierce\(^{18,19}\).

Up to the present time the structure of the benzene ring cannot be considered established. Two types of its structure have been proposed: a planar one (corresponding to graphite), and a second according to which the carbon atoms are assumed to be arranged as in diamond. X-ray investigations carried out on solid benzene confirm the first structure\(^{20}\). The planar structure is also supported by the fact that it was found for hexachlorobenzene and hexamethyl-

benzene.^21 In 1935, Kaiser^22 investigated benzene and hexachlorobenzene in the gaseous state by X-ray diffraction. It turned out that in this case too the planar structure is confirmed; at least, it is supported by the general course of the intensity on the experimental scattering curve for benzene. For the distances between atoms the following values were obtained: \(l_{\mathrm{C}-\mathrm{C}} = 1.42 \pm 0.03\ \text{Å}\) (for benzene and hexachlorobenzene) and \(l_{\mathrm{Cl}-\mathrm{Cl}} = 3.35 \pm 0.05\ \text{Å}\) (for hexachlorobenzene).

In discussing the results of the investigation of the \(\mathrm{CCl}_4\) molecule, it was pointed out that discrepancies are observed on the experimental and theoretical curves in the magnitudes of the maxima and minima, and that a possible cause of these discrepancies would naturally be ascribed to the thermal motion of the atoms in the molecule.

However, James,^23 in his very thorough work, showed that this is not quite so. In any case, explaining the deviations of the experimental curves from the theoretical ones by vibrations of the atoms in the molecule presents great difficulties.

In order to find out whether the thermal vibrations of atoms really affect the picture of X-ray scattering, it is necessary, for the given molecule, to find the difference between two curves obtained at different temperatures, then to calculate the temperature effect theoretically from the constants known for this molecule, and to compare the results of theory with the experimental data.

Fig. 24.

For this purpose James carried out experiments with \(\mathrm{SiCl}_4\) at \(100^\circ\) and \(300^\circ\). The technique of his experiments essentially did not differ at all from Bevilogue’s method; only the chamber, the windows, and also the gas inlet and outlet tubes were made not of metal but of quartz. The aim in this was, first, to get rid of moisture, in the presence of which \(\mathrm{SiCl}_4\) decomposes easily, and, second, to ensure that, on heating, all parts underwent the same expansion; otherwise bending or breakage of the windows might occur, which would lead to corresponding changes in the photographs having nothing in common with the thermal effect.

atom. The photographs were taken in copper rays with the use of a nickel filter.

In Fig. 25 are shown the results of three photographs, two at \(100^\circ\) and one at \(300^\circ\), obtained by James under entirely identical conditions. The photographs at \(100^\circ\) are marked by circles, and those at \(300^\circ\) by crosses. Comparing the curves, one can notice a difference between them near the first maximum, i.e., at small angles—a difference that certainly cannot be attributed to a temperature effect subject to measurement. On the other hand, the minimum at \(300^\circ\) is deeper than at \(100^\circ\); in reality, precisely the opposite phenomenon should have been expected. From these data, and also from their theoretical treatment, James comes to the conclusion

Fig. 25.

of the impossibility of detecting the influence of temperature within the limits of errors of measurement, and, consequently, of explaining the observed deviations of the experimental curves from the theoretical ones by oscillations of the atoms in the molecule.

SCATTERING OF X-RAYS IN SOLUTIONS OF “HEAVY” MOLECULES AND THE STRUCTURE OF COMPLEX IONS

What has been set forth above shows that the determination of the structure of molecules in gases, especially at low densities, gives good results. If, however, the gas is sufficiently compressed, then, according to Debye, at small angles a maximum may appear whose position and relative intensity are no longer determined by the structure of the molecule, but depend on the mean intermolecular distances and, in all probability, on that ordering of the molecules which

appears when molecules approach one another at very close distances.

In the scattering of X-rays in liquids, conditions are possible under which what has just been said about gaseous scattering is also realized in liquids. What is meant here is scattering by solutions \(^{24,25,26}\).

Let a beam of X-rays fall on a solution containing molecules I and molecules II. Let the atomic factor of the former (the solvent) be \(\psi\), and of the latter (the dissolved substance) be \(\varphi\). In a rough estimate, in the first approximation, we shall take them as equal to the number of electrons of the atom. Further, let there be, in the illuminated volume, \(N\) atoms (molecules) of the solvent and \(n\) atoms (ions) of the dissolved substance. Then the resulting amplitude of the radiation scattered in a direction making an angle \(\vartheta\) with the incident ray will be expressed as follows:

\[ \begin{aligned} A={}&\psi_1 e^{ik(s,r_1)}+\psi_2 e^{ik(s,r_2)}+\cdots+\psi_N e^{ik(s,r_N)} +\varphi_1 e^{ik(s,r_{N+1})}+{}\\ &+\varphi_2 e^{ik(s,r_{N+2})}+\cdots+\varphi_n e^{ik(s,r_{N+n})}=\\ &=\sum \psi e^{ik(s,r)}+\sum \varphi e^{ik(s,r)}, \end{aligned} \]

where \(s=2\sin \dfrac{\vartheta}{2}\), and \(r\) is the radius vector of the atom (molecule, ion) under consideration.

The intensity will be proportional to the square of the amplitude, i.e.:

\[ \begin{aligned} I\sim A^2 \sim \left[\sum \psi e^{ik(s,r)}+\sum \varphi e^{ik(s,r)}\right]^2 ={}&\\ =\sum_j\sum_i \psi^2 e^{ik(s,r_i-r_j)} +\sum_j\sum_i \varphi^2 e^{i(s,r_i-r_j)} +{}&\\ +\sum_j\sum_i \psi\varphi e^{i(s,r_i-r_j)}. \end{aligned} \]

If we now turn our attention to the three double sums in this expression, we shall see that the first sum determines the intensity in the direction \(\vartheta\) due to scattering by the molecules of the solvent; the second sum is due to the interference of rays scattered by the molecules of the dissolved substance; and, finally, the third sum characterizes the interference of rays proceeding from the molecules of the solvent and the dissolved substance. The first sum contains \(N^2\) terms, the second \(n^2\), and the third \(2Nn\). Each of these sums may give an intensity distribution close to that in liquids.

In a rough estimate, the role of each of the double sums in the formation

of the general pattern of the intensity distribution may be taken as proportional to the quantities:

\[ N^{2}\psi^{2};\quad n^{2}\varphi^{2};\quad 2Nn\psi\varphi . \]

It is obvious that for large \(\varphi\), small \(\psi\), and at a sufficiently high concentration of the solution, the second double sum begins to play an essential role, i.e., under these conditions, at any rate at not very small scattering angles, one may expect the appearance on the X-ray photograph of a maximum due to the structure of the molecules, or ions, of the dissolved substance. This conclusion is supported by the fact that the atomic factor decreases with angle for “heavy” atoms much more slowly than for “light” ones. Therefore, at the angles at which maxima appear that are due to intramolecular scattering, the role of the “heavy” molecules of the dissolved substance in forming the general pattern of the intensity distribution becomes still greater.

TABLE 8

Substance	Distances between like ions	Distances between unlike ions
CdJ\(_2\)	4.15	2.60
CdCl\(_2\)	3.72	2.52
ZnJ\(_2\)	4.14	2.40
ZnCl\(_2\)	3.79	—

Prince \(^{27}\) showed, for a series of aqueous solutions (ZnCl\(_2\), ZnJ\(_2\), CdCl\(_2\), and CdJ\(_2\)), that maxima do indeed occur on the X-ray photographs—at large scattering angles—the positions of which do not depend on the concentration of the solution and agree well with those calculated on the assumption that these maxima are due to intramolecular scattering. Table 8, taken from Prince’s work, is given below; it gives the distances between ions obtained by Prince for various solutions. If one turns to solutions of complex ions, such as, for example, \(\mathrm{HgJ}_{4}^{--}\), then here too, at large scattering angles, one may expect the appearance on the X-ray photograph of intensity maxima due to “intramolecular” diffraction. Naturally, the presence of such maxima may be used in an attempt to determine the structure of the ions of these solutions. It should be noted, however, that the degree of accuracy possible in determining molecular structure in the case of X-ray scattering in gases can, of course, not be attained here. An attempt of this kind was made in the work of Danilov, Finkel-

stein and Sirotenko,\(^{28}\) where they investigated an aqueous solution of \(K_2HgJ_4\). I repeated the experiments with this solution by a somewhat refined method and, in addition, investigated aqueous solutions of \(K_2ZnJ_4\) and \(ZnJ_2\).

Since all these solutions strongly absorb X-rays, it was natural to use photographs from a free surface. The photographs were taken in copper radiation at a tube voltage of 18–20 kV. Monochromatization was achieved with a nickel filter of \(0.02\) mm. A celluloid plate \(0.6\) mm thick was placed between the specimen and the filter. The angles at which the beam of rays fell on the surface of the liquid were from 8 to \(12^\circ\). The solution was placed in a small cup 18 mm in diameter. By raising or

Fig. 26.

lowering the latter with a micrometric screw, it was possible to set the level at the same height. The setting was carried out with the aid of a retractable needle, to the sharp end of which the surface of the solution was brought. If the point of the needle (in the extended position for adjustment) was at the center of the chamber, then the position of the maxima could be calculated from the dimensions of the chamber. This chamber is shown schematically in Fig. 26. Under the same conditions as for the solutions, photographs were taken from a flat polished copper surface, and the chamber was calibrated by the copper lines. Microphotometry was carried out on a Moll microphotometer. No absorption corrections were introduced, since we were interested in the angular interval from 35 to \(70^\circ\), in which, for photographs from a flat surface and a glancing angle of \(12^\circ\) or less, the correction curve runs almost parallel to the abscissa axis.

Figure 27 gives the intensity curves for solutions of \(K_2HgJ_4\) (a) and \(K_2ZnJ_4\) (b). Three maxima appear on both curves. Alongside the experimental curves are also given the theoretical ones, calculated for tetrahedral models of the ions \(HgJ_4^{--}\)

and \(\mathrm{ZnJ}_{4}^{--}\), taking into account the atomic factors. Along the abscissa axis are plotted the values of the quantity

\[ x=\frac{4\pi}{\lambda}a_{ij}\sin\frac{\vartheta}{2}, \]

and along the ordinate axis—the calculated intensities. The calculation was carried out under the assumption of independent (gas) scattering by the ions.

Fig. 27a.

After summation, the following expression for the intensity was obtained:

\[ I=\frac{1+\cos^{2}\vartheta}{2} \left\{ 4\psi_{1}^{2}\left[1+3\frac{\sin x}{x}\right]+ \right. \]

\[ \left. +8\psi_{1}\psi_{2}\frac{\sin x'}{x'}+\psi_{2}^{2} \right\}, \]

where \(x\) corresponds to the distance between atoms \(1—1\), i.e. between atoms \(J—J\), and \(x'\)—between atoms \(1—2\) (\(\mathrm{Hg}—J\) in the case of a solution of \(\mathrm{K}_{2}\mathrm{HgJ}_{4}\) and \(\mathrm{Zn}—J\) for a solution of \(\mathrm{K}_{2}\mathrm{ZnJ}_{4}\)).

The first maxima, both in the case of a solution of \(\mathrm{K}_{2}\mathrm{HgJ}_{4}\) and of \(\mathrm{K}_{2}\mathrm{ZnJ}_{4}\), lie in that range of scattering angles where radiation of considerable intensity, arising from diffraction by solvent molecules as well as by solvent—ion molecules, may be superimposed on the radiation scattered by the \(\mathrm{HgJ}_{4}^{--}\) and \(\mathrm{ZnJ}_{4}^{--}\) ions. In a word, this is precisely the region that had to be left out of consideration in discussing the experimental curves, and in which, obviously, it is not possible to compare the theoretical and experimental curves.

As for the second and third maxima, their relative position and intensity are such that, when comparing the experimental and theoretical curves, an attempt to calculate the interionic distances on the basis of the tetrahedral model and of the positions of the intensity maxima on the experimental curves seems entirely justified. Moreover, if in the expression for \(x = \dfrac{4\pi}{\lambda} a_{j-j}\sin \dfrac{\vartheta}{2}\) the distance between two neighboring ions proves to be the same when it is calculated from the second

Fig. 27c.

and third maxima, then this may be regarded as sufficiently convincing evidence that the ion has a tetrahedral structure.

The values of \(x\) to which, on the tetrahedral curve for \(\mathrm{HgJ}_4^{--}\), the second and third maxima correspond are \(x_2 = 13.3\) and \(x_3 = 21\). On the experimental curves they correspond to angle values of \(39\) and \(62^\circ\). Substituting into the expression

\[ x = \frac{4\pi}{\lambda} a_{j-j}\sin \frac{\vartheta}{2}, \]

we find \(a_{jj} = 4.9\) and \(5.0\) (the mean is \(4.9\ \text{Å}\)). Greater agreement under the conditions of scattering in solution is difficult to expect, since determining the position of the maxima here is associated with considerable difficulties. For the same distance from the curves for \(\mathrm{ZnJ}_4^{--}\) cor-

respectively we find 4.6 and 4.6. Using \(a_{J-J}\), from the relation \(a_{\mathrm{Hg}-J}=\dfrac{1}{2}\sqrt{\dfrac{3}{2}}\,a_{J-J}\) one can calculate the distances between the mercury and iodine ions in the complex ion \(\mathrm{HgJ}_4^{--}\); they are equal to \(a_{\mathrm{Hg}-J}=3.04\,\text{\AA}\). The distance between the zinc ion and iodine in \(\mathrm{ZnJ}_4^{--}\) is equal to

Fig. 28a.

\(a_{\mathrm{Zn}-J}=2.8\,\text{\AA}\). This difference in the distances \(a_{\mathrm{Hg}-J}\) and \(a_{\mathrm{Zn}-J}\) may arise from the different size of the zinc ion and the mercury ion.

A concentrated aqueous solution of \(\mathrm{ZnJ}_2\) gives an X-ray pattern with exactly the same course of intensity as \(\mathrm{K}_2\mathrm{ZnJ}_4\). The slight

Fig. 28b.

difference consists only in the fact that the second and third intensity maxima turn out to be slightly diminished in comparison with the maxima in the X-ray pattern for \(\mathrm{K}_2\mathrm{ZnJ}_4\). The calculation of “gas” scattering by \(\mathrm{ZnJ}_2\) molecules,* carried out according to the formula:

\[ I=\frac{1+\cos^2 \vartheta}{2}\left\{2\psi_1^2+2\psi_1^2\frac{\sin x}{x}+4\psi_1\psi_2\frac{\sin x'}{x'}+\psi_2^2\right\}, \]

* In the calculation the \(\mathrm{ZnJ}_2\) molecule was assumed to be dumbbell-shaped.

obtained after summation from (5), leads to the curve shown in Fig. 28a.

As can be seen from this figure, only one maximum appears on the curve, and it is so weak that it does not seem possible to think that it would appear under conditions of scattering by molecules in solution.

The presence of the second and third maxima on the experimental curve for the ZnJ$_2$ solution (see Fig. 28b), with exactly the same position and intensity as for K$_2$ZnJ$_4$, forces one to think that the scattering centers causing the positions of the maxima are constructed in the same way. There is no reason to suppose that in a K$_2$ZnJ$_4$ solution we have ZnJ$_2$ molecules, if only because there is no correspondence between the experimental curve obtained for this solution and the theoretical curve for ZnJ$_2$. It is more plausible to suppose that in the ZnJ$_2$ solution there are ions of the type ZnJ$_4^{--}$.

Electrochemical studies by a number of authors indicate that in concentrated aqueous solutions of ZnJ$_2$ and K$_2$ZnJ$_4$ there exist ZnJ$_4^{--}$ ions. Indeed, the transport numbers of the anion, measured in concentrated solutions, turn out to be greater than unity. For example, according to Jahn’s work, $^{29}$ at a dilution of 4.02 l, $n_A = 1.003$. According to Hittorf’s old investigations, $^{30}$ in a four-molar solution (our solution contains 3.8 mol/l) the transport number of the anion was found to be $n_A = 1.157$. At the same time the theoretical value, calculated on the assumption of transport of the anion J$_2$, is $n_A = 0.591$. And indeed, for the dilute solution Jahn found $n_A = 0.558$ at a dilution of 300.4 l.

Thus the conclusion that can be drawn on the basis of the results of an X-ray investigation of an aqueous solution of ZnJ$_2$ is confirmed by electrochemical data.

LITERATURE

1. L. Bewilogua, Phys. Z. 32, 714. 1931.
2. L. Bewilogua, Phys. Z., 33, 688, 1932.
3. P. Debye, Phys. Z., 28, 135, 1927.
4. G. G. Harvey, Phys. Rev., 46, 441, 1934.
5. G. G. Harvey, Phys. Rev., 45, 848, 1934.
6. H. S. Gingrich and B. E. Warren, Phys. Rev., 46, 248, 1934.
7. P. Debye, Phys. Z., 31, 419, 1930.
8. L. Bewilogua, Phys. Z. 32, 740, 1931.
9. L. Bewilogua, Phys. Z., 32, 265, 1931.
10. H. Meyer, Ann. d. Phys., 5, 701, 1930.
11. W. Van der Grinten, Phys. Z., 33, 769, 1932.
12. W. Heisenberg, Phys. Z., 32, 737, 1931.
13. P. Debye, L. Bewilogua u. F. Ehrhardt, Phys. Z., 30, 84, 1929.
14. P. Debye, “Polar Molecules,” p. 226, GTTI, 1931.
15. H. Richter, Phys. Z., 36, 85, 1935.
16. H. Gejewski, Phys. Z., 33, 122, 1932.
17. F. Ehrhardt, Phys. Z., 33, 605, 1932.
18. Pierce, Phys. Rev., 43, 145, 1933.

1. Pierce, J. Chem. Phys., 2, 1, 1934.
2. E. G. Cox, Proc. Roy. Soc., A 135, 491, 1932.
3. K. Lonsdale, Proc. Roy. Soc., A 133, 536, 1931.
4. R. Kaiser, Phys. Z., 36, 92, 1935.
5. R. W. James, Phys. Z., 33, 737, 1932.
6. J. A. Prins, Z. Physik, 56, 617, 1929.
7. J. A. Prins, J. Chem. Phys. 3, 72, 1935.
8. V. Danilow, Acta Physicochimica URSS, 3, 725, 1935.
9. J. A. Prins et R. Fonteyne, Physica 11, 1016, 1935.
10. V. Danilow, W. Finkelstein u. D. Sirotenko, Phys. Z. d. Sowjetun. 3, 653, 1933.
11. Jahn. Z. physik. Chem. 37, 673, 1901.
12. Hittorf, Pogg. Ann., 106, 548, 1859.

1. Bevilogue. ↩