DIFFRACTION OF ELECTRONS BY GAS MOLECULES*

L. O. Brockway, California, USA

1. Introduction

The study of the phenomena that occur in collisions of electrons with matter was begun by Lenard^1 ** more than 40 years ago. However, the application of experiments of this kind to the investigation of the structure of matter became possible only after three events that took place 30 years later. The first of these events was the observations of Davisson and Kunsman^2 on the scattering of electrons by the surface of polycrystalline nickel. The results of these observations, together with the results of the later experiments of Davisson and Germer^3 on the scattering of electrons by nickel single crystals, made it possible to discover the interference effects that occur in electron scattering. The second event was the theory proposed by de Broglie^4, which connected the motion of a material particle with the propagation of a certain characteristic wave, whose frequency is determined by the mass and velocity of the particle. The third major event of this period was the creation by Heisenberg and Schrödinger of modern quantum mechanics, on the basis of which it became possible to give a more or less complete theoretical derivation of the relation between the intensity of an electron beam and the angle of its scattering by an atom or molecule.

Soon after the work of Davisson and de Broglie, Thomson^5 and Kikuchi^6 attempted to apply electron diffraction to the study of crystalline structure. In his experiments Thomson directed, onto foils of gold, silver, aluminum, and other substances, a beam of electrons having high velocities (from 30 to 50 kV), and found that the “wavelength” of his electron beams was determined by the de Broglie relation and that the diameters of the rings in the Debye electron-diffraction patterns of various substances that he obtained could be explained on the basis of structures previously determined by X-ray methods. Kikuchi carried out analogous experiments with thin plates of mica.

In 1929 and 1930, Mott^7 derived, in an easily applicable form, a theoretical expression for the interaction of fast electrons with atoms. Thomson^8 and Mark and Wierl^9 at the same time studied the “atomic factor” for electrons by measuring the relati-

* Reviews of Modern Physics, 8, 231 (1936), translated by E. G. Ananyashvili.
** The literature will be given at the end of the article.

of significant ring intensity in electronograms obtained from thin metal films. In 1931 Wierl^10 published the results of an investigation of the molecular structure of 20 different chemical compounds, carried out by observing the diffraction of electrons by the molecules of their vapors. Since that time the method of electron diffraction by gaseous molecules has gained universal recognition as a powerful means for investigating molecular structure.

The three hundred and fifty papers that have appeared in the field of electron diffraction since 1930 fall into four groups.

The first group, comprising approximately 50 papers, contains investigations of the structure of individual molecules. The experiments in them are carried out on a substance in the gaseous state, with the aid of fast electrons; the conclusions of these works concern the arrangement of atoms and the distances between individual atoms within the molecule. The influence of the electronic structure may here be neglected. This special field of electron diffraction is the subject of the present review.

The second group, consisting of approximately 65 papers, comprises investigations of monatomic gases by slow electrons, usually at accelerating-field voltages below 1000 V. This includes the well-known Ramsauer effect. Here the distribution of electrons in the atom already plays an important role, and the object of investigation is the transverse sections of atoms; conclusions about them are drawn on the basis of the angular distribution of intensity, on the one hand, and on the basis of excitation and ionization of atoms by the bombarding electrons, on the other. Some monatomic gases^11 have also been bombarded by slow electrons. Since the atomic factors for slow electrons are not very well known, the positions of the atomic centers (nuclei) in the molecule cannot be determined by means of slow-electron diffraction. The nuclei prove to be more effectively shielded by the electron shell in the case of slow electrons than in the case of fast ones; both elastic and inelastic scattering here, to a greater extent than in the case of fast-electron diffraction, are determined by the electronic structure of the molecules. A complete theoretical interpretation of scattering exists only for the case of the H₂ molecule.^12 We do not touch in the present work on this field of electron diffraction by gaseous molecules, since it is far removed from the subject that concerns us.

The third group of experiments on electron diffraction, comprising at present about 150 publications, concerns the diffraction of fast electrons by solid bodies, including crystalline powders, single crystals of metals and minerals, crystalline and amorphous organic substances, and surface layers. In this field some new types of diffraction phenomena^13 have been observed; the main interest here lies in the distribution of atoms; new results have been obtained in the important field of surface structures. The lesser penetrating ability

electrons, as compared with X-rays, makes the former especially convenient for the investigation of surface properties. The “internal potentials” of crystals were also studied, causing differences in the velocities of electrons outside and inside the crystal.

The fourth group includes investigations of a mixed type, including certain theoretical aspects of the question not touched upon in the preceding groups, theoretical and experimental work on the polarization of electrons, the determination of physical constants, the verification of the de Broglie relation for very high velocities, etc.

Before proceeding to a detailed discussion of the diffraction of electrons by gas molecules, it is necessary to touch upon the interests that, in a broad sense, motivate workers in this field and determine the content of the present work. The development of this field of physical experiment was conditioned by the need to solve numerous problems of structural chemistry, to which, as it turned out, it could give a direct answer. Examples of such problems are the geometrical configuration of isomers, the tetrahedral arrangement of bonds in organic compounds of the aliphatic series, the planar form of the benzene ring, the angles between the valences of atoms of different elements, and the latest considerations concerning the interdependence of the internuclear distance of chemically bonded atoms and other properties of the bond, such as, for example, electronic structure, dissociation energy, electric moment, etc. As a result, the initial use of this method was reduced to the determination of the structure of molecules.

The features of the experiment on the basis of which it was possible to construct a method for determining interatomic distances—one giving sufficiently accurate results and at the same time sufficiently simple to be applied in practice to a large number of compounds—were subjected to thorough development. Owing to these two requirements imposed on the experiment—accuracy of the results and simplicity of application—a complete quantitative comparison of theoretical and observed scattering intensities could not be carried out.

Only in the case of benzene¹ was a comparison of relative intensities carried out for a continuous series of reflection angles. In Section IV, where the method of interpreting electron diffraction patterns is considered, it will be shown that a photograph which makes it possible to judge the accuracy of the theoretical formulas for the distribution of the intensity of reflected electrons is inapplicable for determining interatomic distances.

The interests we have outlined determined the direction of the work carried out in this field. The following exposition has in view equally both future workers whose interests coincide with those just described, and all those wishing to obtain a general idea of the subject or to form an independent judgment about the value of the results obtained. With the exception of the diffraction of slow electrons, we have not omitted in our survey a single case of electron scattering by gas molecules.

II. Theory

Elastic Scattering by Atoms

It is required to derive a theoretical expression for the interference effect of electrons scattered by various atoms in a molecule. For this purpose we shall first consider the case of elastic scattering of electrons, and then the combined effect of all the different kinds of inelastic scattering. In the case of elastic scattering we shall consider first a single atom, and then a group of atoms with fixed relative positions. The solution given below for the problem of scattering by a single atom is presented in the form in which it was first given by Mott^7.

Let there be a homogeneous electron beam moving in a definite direction in a field-free space. The electrons are scattered by an atom represented as a small region within which there is a central force field; the distribution of the scattered electrons is observed at a large distance from the atom. If by \(r\) we denote the distance from the point of observation to the atom, by \(\theta\) the angle between the directions of the incident and reflected electron, and by \(N\) the number of electrons in the incident beam passing through a unit area of the beam cross-section per unit time, then the number of electrons falling on an elementary area \(dS\) per unit time is expressed by the formula:

\[ N I(\theta)\frac{dS}{r^2} \equiv N I(\theta)d\omega, \]

in which \(I(\theta)\) is the unknown function that has to be determined. To solve the problem one resorts to the Schrödinger amplitude equation:

\[ \nabla^2 \psi(x,y,z)+\frac{8\pi^2 m}{h^2}\,[W-V(x,y,z)]\psi(x,y,z)=0. \tag{1} \]

Here the symbol \(\nabla^2\) denotes the operator

\[ \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2} \]

in Cartesian coordinates. The function \(\psi\), which is the solution of the problem, is expressed in the coordinates \(x\), \(y\), and \(z\) with the origin at the center of the atom, and \(|\psi|^2\) represents the distribution of electrons both in the incident and in the reflected beams. \(W\) denotes the kinetic energy of the electrons of the beam and remains unchanged in magnitude during the collision. The value \(W\) is determined by the potential applied in producing the beam to accelerate the electrons. \(V\) represents the potential energy of interaction of the electron with the charged particles constituting the atom, and by assumption is a function only of \(r\), i.e. the atom is assumed to possess spherical symmetry.

The amplitude equation, whose solutions are independent of time, is used here because we are interested in \(|\psi|^2\) for the steady state, or, in other words, because \(W\) remains

constant throughout the entire scattering process. The use of the factor \(e^{\frac{2\pi i}{h}Wt}\) in combination with the solutions we obtain, as is known, leads to satisfactory solutions of Schrödinger’s wave equation that also take time into account; but, since we are interested only in the probable distribution of electrons, we shall neglect the time factor.

For large \(r\) the solution of equation (1) must have the following form:

\[ \psi \sim e^{ikz}+\frac{e^{ikr}}{r} f(\vartheta). \tag{2} \]

The first term here represents the incident plane wave propagating along the \(z\)-axis, and is a solution of the equation for a free electron

\[ \nabla^2\psi+k^2\psi=0. \tag{3} \]

Equation (3) has the form of the classical wave equation, and the constant \(k^2\), which is a function of \(W\), can be related to the wavelength of the ray by the following relation:

\[ k^2=\frac{8\pi^2 mW}{h^2}=\frac{4\pi^2}{\lambda^2}. \]

Expressing the kinetic energy in terms of velocity and mass, we obtain the de Broglie relation:

\[ \lambda=\frac{h}{mv}. \]

The second term of equation (2) represents the reflected wave and, consequently,

\[ I(\vartheta)=|f(\vartheta)|^2. \]

If equation (1) is rewritten in the form

\[ \nabla^2\psi+k^2\psi=\frac{8\pi^2m}{h^2}V\psi, \tag{4} \]

then the most general solution may be written in the following form\({}^{15}\):

\[ \psi_0=\psi_0-\frac{1}{4\pi}\int \frac{e^{ik|r-r'|}}{|r-r'|}\frac{8\pi^2m}{h^2}V'\psi'\,d\tau', \tag{5} \]

where \(\psi_0\) is the general solution of equation (3).

If one assumes that the integral in equation (5) represents the reflected wave, then the quantities under the integral sign—

of the integral, admit the following interpretation. The vector \(\mathbf r\) propagates to the volume element \(d\tau'\) containing the scattering substance. \(V'\) and \(\psi'\) denote, respectively, \(V(x',y',z')\) and \(\psi(x',y',z')\). Then the integrand represents the amplitude and phase at the point \(\mathbf r\) of an elementary wave that has undergone scattering in \(d\tau\) and has, at a distance from the atom equal to unity, the amplitude

\[ \frac{2\pi m}{h^2}\,V'\psi'\,d\tau'. \]

The integral represents the result of summing the elementary waves scattered by all the scattering substance, and propagates over the whole volume in which \(V(x',y',z')\) is different from zero. Thus, the primed coordinates in equation (5) refer to the atom, while the unprimed coordinates refer to the scattered wave. Fig. 1 shows the relation of the coordinates.

Fig. 1. Coordinates in atomic scattering

The first term of equation (5) must represent the incident wave. This is compatible with the definition of \(\psi_0\), i.e. with the solution of equation (3), and we may choose an infinite plane wave propagating along the \(z\)-axis, so that

\[ \psi_0=e^{ikz}. \]

The condition for the acceptability of equation (5) as a solution is that the integral on the right-hand side must have the corresponding asymptotic form, as in the second part of equation (2). Since the observation point is always assumed to be at a great distance from the atom, \(r>r'\), and we may write

\[ |\mathbf r-\mathbf r'|\sim r-\frac{\mathbf r}{r}\mathbf r', \]

the integral then becomes

\[ -\frac{2\pi m}{h^2}\frac{e^{ikr}}{r}\int e^{-ik\frac{\mathbf r}{r}\mathbf r'}\,V'\psi'\,d\tau'. \tag{6} \]

Accordingly, the asymptotic form of equation (5) for large \(r\) has the form of equation (2), in which

\[ f(\vartheta)=-\frac{2\pi m}{h^2}\int e^{-ik\frac{\mathbf r}{r}\mathbf r'}\,V'\psi'\,d\tau'. \tag{7} \]

This integral is already a function of only one \(\vartheta\), since the unprimed coordinates appear in it only in the form \(\frac{\mathbf r}{r}\) of a unit

DIFFRACTION OF ELECTRONS BY GAS MOLECULES

of the vector making an angle \(\vartheta\) with the direction of the incident beam. To compute integral (7) it is now necessary to invoke the Born approximation; \(\psi'\) consists of two parts: \(\psi'_0\) and the integral corresponding to the integral of equation (5). The Born approximation consists in neglecting the integral expression, i.e. taking

\[ \psi' \sim \psi'_0 = e^{ikz'} . \tag{8} \]

This is equivalent to the assertion that inside the atom the amplitude of the incident wave is considerably greater than the amplitude of the scattered wave, or that a wave scattered by one part of the atom cannot be greater than one scattered by another part of it. Further on it is assumed that no phase shift occurs upon scattering. The limitations imposed on the final result by such an approximation are considered below.

If we take \(z'=\mathbf{n}_0\mathbf{r}'\), where \(\mathbf{n}_0\) is a unit vector coinciding with the \(z\)-axis, and if \(\mathbf{n}\) denotes the unit vector \(\frac{\mathbf{r}}{r}\), then substitution of (8) into (7) gives

\[ f(\vartheta)=\frac{2\pi m}{h^2}\int e^{ik(\mathbf{n}_0-\mathbf{n})\mathbf{r}'} V' d\tau' . \tag{9} \]

In order to carry out the integration, we first rewrite the exponent. Since \(\vartheta\) is the angle between \(\mathbf{n}_0\) and \(\mathbf{n}\), the absolute value of \(\mathbf{n}_0-\mathbf{n}\) is equal to \(2\sin\frac{\vartheta}{2}\), and

\[ k(\mathbf{n}_0-\mathbf{n})\mathbf{r}'=2k\sin\frac{\vartheta}{2}\, r'\cos\alpha', \]

where \(\alpha'\) is the angle between \(\mathbf{r}'\) and the vector \((\mathbf{n}_0-\mathbf{n})\). Taking this vector as the polar axis and omitting the primes, we integrate expression (9):

\[ f(\vartheta)=\frac{2\pi m}{h^2} \int_0^{2\pi} d\beta \int_0^\pi \sin\chi\,d\chi \int_0^\infty e^{i2k\sin\frac{\vartheta}{2}\,r\cos\alpha} V(r)r^2dr = \]

\[ =\frac{8\pi^2m}{h^2}\int_0^\infty \frac{\sin sr}{sr}\,V(r)r^2dr, \tag{10} \]

where

\[ s=2k\sin\frac{\vartheta}{2}=4\pi\sin\frac{\vartheta}{2}\cdot\frac{1}{\lambda}. \]

Since \(V(r)\) represents the potential of the field created by the atom, it can be expressed in terms of the charge density in the atom:

\[ V(r)=-\frac{Ze^2}{r}+e^2\int \frac{|\chi(\mathbf{r}')|^2}{|\mathbf{r}-\mathbf{r}'|}\,d\tau' . \tag{11} \]

The first term in the expression obtained owes its origin to the nucleus with charge \(Ze\); the second depends on the distribution of electrons in the atom; \(\varphi(r')\) is the solution of the Schrödinger equation for an atom with a spherical charge distribution.

Expression (11) can be substituted into (10), after which the final result is obtained by two integrations by parts. Another method, leading to the same result, is based on the use of the following formula of Bethe\({}^{16}\):

\[ \int \frac{e^{ik(\mathbf{n}_0-\mathbf{n})\mathbf{r}}}{|\mathbf{r}-\mathbf{r}'|}\,d\tau = \frac{4\pi e^{ik(\mathbf{n}_0-\mathbf{n})\mathbf{r}'}}{k^2|\mathbf{n}_0-\mathbf{n}|^2}. \tag{12} \]

Substituting expression (11) into (9) and simplifying with the aid of formula (12), we obtain\({}^{17}\):

\[ f(\vartheta)= \frac{8\pi^2me^2}{h^2}\frac{1}{s^2} \left[ Z-4\pi\int_0^\infty |\varphi(r)|^2 \frac{\sin sr}{sr}\,r^2dr \right] = \]

\[ = \frac{8\pi^2me^2}{h^2}\frac{Z-F(\vartheta)}{s^2}; \]

\[ s=4\pi\frac{\sin \dfrac{\vartheta}{2}}{\lambda}, \tag{13} \]

where

\[ F(\vartheta)=4\pi\int_0^\infty |\varphi(r)|^2 \frac{\sin sr}{sr}\,r^2dr. \]

The complete solution of the problem has the form

\[ \psi=e^{ikz}+\frac{8\pi^2me^2}{h^2}\cdot\frac{e^{ikr}}{r}\cdot\frac{Z-F}{s^2}. \tag{14} \]

\(F\) is identical with the atomic factor for X-rays; its values have been calculated for all atoms\({}^{18}\). Owing to the fact that the properties of the function \(F\) have already been studied previously, equation (13) is a particularly convenient form of the theoretical expression of the effect of scattering of electrons by an atom.

\(F(\vartheta)\) tends to zero as \(s\) increases, so that for small wavelengths (electrons with high velocities) and large angles scattering occurs predominantly by nuclei; in other words, those of the electrons penetrating into the atom which fly closer to the nucleus are scattered through large angles.

At \(\vartheta=0\), \(F(\vartheta)\) becomes equal to \(Z\), but \(f(\vartheta)\) remains finite, which can be verified by means of the analytical expression for \(\varphi(r)\) for hydrogen-like atoms with screening constants\({}^{18b}\).

Applicability of the Born Approximation

In order to derive theoretically the conditions for applicability of the Born approximation, it is necessary to obtain an exact solution of the problem and compare it with the solution obtained by the approximate method. For our present purpose it will be sufficient to refer to the book by Mott and Massey,^17 where in Chapter II a complete solution of the problem is given, and in Chapter IV the results of the exact and approximate methods are compared. The condition derived there is that

\[ \frac{4\pi^2 m}{h^2}\int V(r)\,[j_{n+\frac12}(kr)]^2\,r\,dr < 1 \quad \text{for any } n. \tag{14a} \]

Here \(V(r)\) is the potential of the atomic field, and \(j_{n+\frac12}(kr)\) is the Bessel function obtained in expanding \(\dfrac{\sin sr}{sr}\). The index \(n\) denotes the successive terms of the infinite series entering into the exact expression for \(f(\vartheta)\). Mott and Massey point out that condition (14a) often proves to be too strict, since in some cases its violation for the first few values of \(n\) introduces no appreciable discrepancy between the results of the exact formula and of the Born approximation.

Since \(V\) is regarded as a small perturbing potential, one should expect the Born approximation to be more accurate for high values of the energies of the bombarding electrons. A quantitative calculation^19 shows that, for the case of scattering by light atoms, condition (14a) is satisfied quite well by electron energies of several hundred volts. For atomic numbers of about 50 and higher, energies of the order of 10,000 V are already required. Since almost all investigations of molecular structure by electron diffraction have been carried out with fields of 40–50 kV, the Born approximation creates no difficulties here; therefore the formula obtained with its aid has an obvious advantage over the exact formula given in the preceding section, whose application to the problems of interest to us would require an extremely large amount of labor.

Elastic Scattering by Molecules

Observation of electron diffraction by gas molecules is carried out under the following experimental conditions: the primary beam falls upon a large number of molecules whose mutual orientation is entirely arbitrary. The concentration of gas molecules is sufficiently small that each molecule can scatter independently of the others. In the theoretical treatment of scattering, one first calculates the effect for a single molecule with a definite fixed

orientation and then take the average result for all possible orientations. The motion of the molecule during the scattering process may be neglected.

Each molecule contains \(m\) atoms that scatter electrons according to equation (14). For this to be possible, it is necessary to make the assumption that the function \(V\) for each atom is independent of the other atoms and that it remains spherically symmetric. In reality, the presence of chemical bonds makes these assumptions incorrect, but in the scattering of fast electrons the electronic structure of the molecule, or of the atoms composing it, plays only a very insignificant role.

According to (14), the amplitude and phase of the scattered wave are given by the expression \(\dfrac{e^{ikr}}{r} f(\vartheta)\). If the origin of coordinates is transferred into the molecule (in the general case, it does not coincide with the center of any atom) and the \(i\)-th atom is denoted by the index \(i\), then the wave scattered by the \(i\)-th atom will be

\[ \psi_i = e^{\frac{ik|r-r_i|}{|r-r_i|}} e^{ikz_1} f_i(\vartheta), \]

where \(\mathbf r\) is the radius vector of the observation point, and \(\mathbf r_i\) is the radius vector of the center of the \(i\)-th atom. The factor \(e^{ikz_1}\) is necessary, since the phase of the scattered wave depends on the phase of the incident wave (Fig. 2).

As before, \(r > r_i\), so that

\[ |\mathbf r-\mathbf r_i| \sim r-\mathbf n\cdot \mathbf r_i, \]

where \(\mathbf n\) is the unit vector in the direction \(\mathbf r\). Then

\[ \psi_i = \frac{e^{ikr}}{r} e^{ik z_i-\mathbf n\mathbf r_i} f_i(\vartheta) = \frac{e^{ikr}}{r} e^{ik(\mathbf n_0-\mathbf n)\mathbf r_i} f_i(\vartheta). \]

Summing the waves scattered by the individual atoms, we obtain the wave scattered by the molecule:

\[ \Psi=\sum_i \psi_i = \frac{e^{ikr}}{r} \sum_i e^{ik(\mathbf n_0-\mathbf n)\mathbf r_i} f_i(\vartheta). \tag{15} \]

Since the phases of the waves scattered by individual molecules are in a completely arbitrary relation, in order to obtain the average result for all possible orientations, it is necessary to

take the mean-square amplitude of the individual molecules. Accordingly, first of all we obtain the expression for

\[ I=\Psi\Psi^*=\frac{1}{r^2} \left[ \sum_i e^{ik(\mathbf n_0-\mathbf n)\mathbf r_i} f_i(\vartheta) \right]^2 = \frac{1}{r^2}\sum_i\sum_j f_i f_j e^{ik(\mathbf n_0-\mathbf n)\mathbf r_{ij}}, \tag{16} \]

where \(\mathbf r_{ij}=\mathbf r_i-\mathbf r_j\) is the distance between the \(i\)-th and \(j\)-th atoms in the molecule.

Fig. 2. Coordinates in molecular scattering

The orientation of the molecule may be specified by any of the vectors \(\mathbf r_{ij}\). The mean result is obtained by integrating each term of the double sum over the variable angle in spherical polar coordinates, taking the vector \((\mathbf n_0-\mathbf n)\) as the polar axis:

\[ I(\vartheta)= \frac{1}{r^2}\sum_i\sum_j f_i f_j \frac{1}{4\pi} \int_0^{2\pi} d\psi_{ij} \int_0^\pi d\alpha_{ij}\, e^{is r_{ij}\cos\alpha_{ij}} = \]

\[ = \frac{1}{r^2}\sum_i\sum_j f_i f_j \frac{\sin s r_{ij}}{s r_{ij}}, \tag{17} \]

where

\[ f_i= \left(\frac{8\pi^2 m e^2}{h^2}\right) \cdot \left(\frac{(Z-F)_i}{s^2}\right), \]

\[ s= \frac{4\pi\left(\sin\frac{\vartheta}{2}\right)}{\lambda}, \]

and \(r_{ij}\) is the distance between the \(i\)-th and \(j\)-th atoms.

This formula (with the exception of the factor \(f_i\)) was derived by Debye\({}^{20}\) and, independently of him, by Ehrenfest\({}^{21}\) in 1915 for the case of the scattering of X-rays by molecules.

It follows from equation (17) that \(I(\theta)\) is a function of the structure of the molecule. The summation extends over all atoms, so that to the distance between any two nuclei there corresponds a definite term of the sum. The terms in which \(i=j\) reduce to \(f_i^2\).

Inelastic Scattering

Inelastic scattering may occur in several ways: the molecule may be raised to one of the levels corresponding to an excited electronic state (which may lead to dissociation of the molecule, but not necessarily); it may, having lost an electron, become ionized, or, conversely, in the case of bombardment by very slow electrons, even acquire an electron; finally, the molecule may undergo a change in vibrational or rotational energy. The question of interest to us is the behavior of the electron beam in scattering processes accompanied by such energy changes. Of particular interest in the study of molecular structure by the electron-diffraction method is the question of the coherence of the scattered waves, or, in other words, of the appearance, in inelastic scattering, of interference effects that depend on the structure of the molecule.

The answer to this question already follows from the circumstance that the effect observed experimentally is the combined effect of all inelastic processes. Interference effects occur\({}^{22}\) when all molecules are excited to one definite state and the wavelength of all scattered electrons undergoes the same change. However, taking into account that in reality the excitation of molecules occurs to all possible levels above the normal one, the combined scattering effect must lead to incoherence of the scattered beam. Excitation of molecules to states corresponding to adjacent levels of vibrational and rotational energy without disturbance of the electronic structure may occur if the energy of the bombarding electrons is less than a certain minimum necessary to produce electronic changes. In the case of fast electrons (i.e., electrons with an energy exceeding the largest ionization potential of the heaviest atom in the molecule), it is already impossible to speak of raising molecules only to a small number of definite energy levels. Therefore, under these conditions inelastic scattering will be incoherent, and no diffraction effects associated with the structure of molecules will be observed.

Nevertheless, it would be desirable to obtain an expression for the angular distribution of inelastically scattered electrons, since on the electron diffraction pattern they produce a background that must be taken into account.

in the interpretation, along with the diffraction pattern of elastically scattered electrons. A complete calculation for the total effect of inelastic scattering by molecules is not yet available. It is probable, however, that by considering the scattering from each atom separately and then summing over all the atoms in the molecule, one may obtain a result that does not differ greatly from reality. Such a solution was given by Morse \(^{23}\), who showed that the expression for incoherent scattering for X-rays, derived by Heisenberg \(^{24}\), is applicable also to fast electrons if it is multiplied by the scattering factor from a single electron,

\[ \left(\sin \frac{\vartheta}{2}\right)^{-4}. \]

The generalized expression for the intensity of the scattered beam, taking into account the change in wavelength, has the form

\[ I(\vartheta)=\sum_l A^2\frac{k_l}{k_0} \left[ \int \Phi_l^*(\mathbf r)\Phi_0(\mathbf r)V(\mathbf r,\mathbf r') e^{\,i(k_l\mathbf n-k_0\mathbf n_0)\mathbf r'}\,d\mathbf r\,d\mathbf r' \right]^2 . \tag{18} \]

The coordinates \(\mathbf r\) and \(\mathbf r'\) are respectively the coordinates of the atomic and the bombarding electrons; \(k_0\) and \(k_l\) denote the wave numbers, \(\frac{2\pi}{\lambda_0}\) and \(\frac{2\pi}{\lambda_l}\), before and after the collision; \(\Phi_l^*(\mathbf r)\) is the wave function describing the state of the atom after the collision, while \(\Phi_0(\mathbf r)\) is the same function before the collision with the electron. \(V\) represents the potential of the bombarding electron in the field of the atom; \(\mathbf n\) and \(\mathbf n_0\), as before, are unit vectors in the directions of the incident and reflected beams. The indices \(l\) denote the various excited states; the summation extends over all possible states. The index \(0\) refers to the initial state, which we may regard as the normal state of the molecule. This expression also contains elastic scattering, since for \(l=0\) it reduces to the formula derived above.

Equation (18) is readily derived from the Schrödinger equation for the complete system, i.e., the wave equation containing explicitly the coordinates of both the atomic and the incident electrons, and in which the term representing the energy is the sum of the energy of the atom in its normal state and the kinetic energy of the incident electron. It is assumed that the solutions include the functions of the incident electron and the other functions for the atom. Expressing the general solution in “integral” form and using the Born approximation, one can obtain equation (18).

Morse derived equation (18) for the case of fast electrons, when \(k_l\) may be taken equal to \(k_0\). The result has the form

\[ I(\vartheta)= \left(\frac{8\pi^2me^2}{h^2}\right)^2 \left\{ \frac{(Z-F)^2}{s^4}+\frac{S(\vartheta)}{s^4} \right\}. \tag{19} \]

The first term agrees exactly with the expression obtained above for the intensity of the elastically scattered wave (cf. equation

(14), while the second term contains \(S(\omega)\), the incoherent scattering function for X-rays, derived by Heisenberg. Bewilogua\(^{25}\) calculated the values \(\left(\dfrac{S}{Z}\right)\), regarding the latter as a function of \(v=sb\), where \(b=\dfrac{0.176}{Z^{\frac13}}\). \(S\) tends to zero together with \(\vartheta\) and increases to the value \(Z\) at large angles. \(\dfrac{S}{s^4}\), on the contrary, reaches a very large finite value at \(\vartheta=0\) and decreases rapidly as the angle increases. The behavior of both terms of (19) is thus very similar.

The complete expression for the effect of diffraction of fast electrons by gas molecules has the form

\[ I(\vartheta)=I_0\left(\frac{8\pi^2me^2}{h^2}\right)^2 \left\{ \sum_i \sum_j f_i f_j \frac{\sin s r_{ij}}{s r_{ij}} + \sum_i \frac{S_i}{s^4} \right\}, \tag{20} \]

where \(I(\vartheta)d\omega\) is equal to the number of electrons scattered within the solid angle \(d\omega\) per unit time; \(I_0\) is equal to the number of electrons passing through unit area of the cross section of the incident beam per unit time; \(f_i=\dfrac{(Z-F)_i}{s^2}\) is the atomic factor for electrons; \(Z\) is the atomic number;

\[ F=4\pi\int_0^\infty |\varphi(r)|^2 \left(\frac{\sin sr}{sr}\right) r^2\,dr, \]

the atomic factor for X-rays\(^{18}\); \(s=\dfrac{4\pi\sin \dfrac{\vartheta}{2}}{\lambda}\); \(r_{ij}\) is the distance between the \(i\)-th and \(j\)-th atoms of the molecule; \(S\) is the tabulated value of the inelastic-scattering function\(^{25}\). The application of this formula to the study of molecular structure is considered in Section IV.

Temperature Effect

The effect of thermal vibrations of atoms in the scattering of X-rays by gas molecules was investigated by James\(^{26}\). The results of his theoretical considerations are also applicable to the diffraction of electrons. James shows that the change, introduced by thermal vibrations, in both the coherent and incoherent intensities can be expressed by a temperature factor introduced into the coherent terms of formula (20). This factor is equal to \(e^{-A}\), where

\[ A=8\pi^2\overline{\delta r_{ij}^2} \left(\frac{\sin \dfrac{\vartheta}{2}}{\lambda}\right)^2 . \]

The mean-square change of the distance \(r_{ij}\) between two atoms \(\overline{\delta r_{ij}^{2}}\) is a function of temperature. When \(i=j\), \(\delta r_{ij}=0\) and the temperature factor becomes equal to unity. For values of \(i\) and \(j\) different from one another, \(e^{-A}\) is less than unity and decreases with increasing angle. The formula for the intensity, taking the temperature factor into account, has the form:

\[ I=K\left\{\sum_i f_i^2+\sum_i\sum_j' f_i f_j \frac{\sin sr_{ij}}{sr_{ij}}e^{-A}+\sum_i \frac{S_i}{s^4}\right\}. \]

The influence of the temperature factor affects only a relatively small part of the total intensity, namely those terms of the double sum with a prime in the formula given, in which \(i\ne j\).

James calculated \(\overline{\delta r_{ij}^{2}}\) from the coordinates of atoms in diatomic molecules and in tetrahedral molecules of the type \(XY_4\). Using frequencies obtained from spectroscopic observations, he calculated \(\overline{\delta r_{ij}^{2}}\) at various temperatures for the molecules \(\mathrm{CCl_4}\) and \(\mathrm{SiCl_4}\). Since the calculated change in intensity was very small, James was unable to detect the temperature effect on x-ray diffraction photographs of \(\mathrm{SiCl_4}\) vapors taken at \(100\) and \(300^\circ\mathrm{C}\). Analogous experiments with \(\mathrm{CCl_4}\), carried out by van der Grinten\(^{27}\), likewise gave a negative result.

It is evident that in the diffraction of electrons by gas molecules the temperature effect must also be insignificant. Degard, Pierard, and van der Grinten state in a recently published note\(^{28}\) that, in interpreting an electron diffraction pattern of \(\mathrm{CCl_4}\), a correction for the temperature factor is necessary in order to explain the intensities in the outer part of the electron diffraction pattern. In any case, when measuring interatomic distances by means of electron diffraction, the temperature factor has no influence on the result.

One further circumstance should be noted: equation (20) applies to fast electrons, but contains no relativistic corrections. Electrons accelerated in fields with voltages from 30 to 60 kV fall between the upper and lower limits of equation (20). When calculating the wavelength by means of the de Broglie relation, one should use the relativistic expression for the mass of the electron.

III. Experimental Method

The observation of electron diffraction by gas molecules in principle presents no special difficulties. The electron beam is made to pass through a jet of gas and is caught by some registering device. In practice, in order to obtain sharp electron diffraction patterns, a number of various precautions must be observed. The velocity (i.e., wavelength) of the electrons of the beam must fluctuate only within very small limits, and the beam must

must be well collimated. The volume of intersection of the beam with the gas jet must be as small as possible, and for this it is necessary, in order to avoid scattering of electrons by gas molecules, to carry out the experiment throughout the apparatus in high vacuum. In view of the fact that the entire exposure must be made within fractions of a second, the recording device is almost inevitably reduced to a photographic emulsion. The emulsion must be protected from X-rays and from other kinds of radiation.

The principle of the electron-diffraction apparatus used at the present time goes back to the design proposed by Wierl[^10]. The instrument consists of two parts: a discharge tube and a diffraction chamber, including the cassette for the photographic plate. The discharge tube serves as the source of electrons and communicates with the diffraction chamber only through a narrow opening in the anode, which serves as a diaphragm for the electron beam. The jet of the gas under investigation moves in a direction perpendicular to the direction of the beam, and is partly condensed on the cold walls, while the remainder is removed by a high-vacuum pump connected with the diffraction chamber. The cassette is arranged in such a way that a fluorescent screen and several plates can be successively brought under the electron beam.

In order better to illustrate the experimental technique, we shall give a description of the instrument used in our laboratory, accompanying it with a brief explanation of all those parts for which we had to choose among different designs.

As the source of electrons, both cold and hot cathodes have been used. In a tube with a cold cathode, the latter consists of a plate with a flat or slightly concave surface. Focusing of the electron beam on the anode surface occurs partly owing to the curvature of the cathode surface. In that part of the apparatus where both electrodes are located, a residual gas pressure of \(10^{-2}\) mm of mercury is maintained. When a high voltage is applied, ions and electrons arise as a result of the gas discharge, the latter acquiring an acceleration directed toward the anode. The advantages usually ascribed to discharge tubes with a cold cathode consist in the simplicity of their construction, in the absence of a cathode-heating circuit, and in the automatic focusing of the electron beam owing to the space charges in the tube. Most investigators working in this field have used discharge tubes of this type.

The author’s experience permits one to assert that a tube with a hot cathode allows better focusing of the beam, permits the voltage and current to be controlled quite independently, and gives a stable discharge without the need to manipulate gas valves. In this case the power losses at the anode amount to approximately one hundredth of the corresponding losses in tubes with a cold cathode. Finally, when working with a hot cathode it is permissible to use unrectified voltages. The discharge tube described below has several special features.

The cathode proper (Fig. 3, A) consists of a tungsten filament with a cross-section of about 0.01 mm, bent in the form of a hairpin. The ends of the filament are clamped in two holders, one of which is fastened to the end of a 3 mm rod of Monel metal, and the other to the end of a similar 10 mm tube. The rod passes inside the tube and is insulated from the latter by a glass tube on the cathode side and by mica spacers on the outside, the joint here being sealed with vacuum cement (Glyptal lacquer). The tube and the rod serve to supply the heating current to the filament. The tube is fastened by an insulating joint in plate C. In plate D

Fig. 3. Cathode assembly

a stainless-steel tube 25 mm in diameter is fastened, carrying at its inner end a cylindrical cap B surrounding the heated filament. This cap, provided with a hole whose diameter is 6 mm and depth 9 mm, serves to focus the electron beam on the anode, located at a distance of approximately 25 mm from the cathode. Owing to the insulating connection at C, an additional voltage of negative sign relative to the filament can be applied to cylinder B, the magnitude of which is usually taken to be about 200 V. The additional voltage and the depth of placement of the filament inside the cylinder are chosen so that the diameter of the focal spot on the anode is no greater than 1 mm.

The position of the filament inside the cylinder has a very strong influence on the behavior of the electron beam. For convenience of adjustment, plate C, connected to D by a flexible metal bellows, is attached to the latter at three points: at one by a ball-and-socket joint, and at the other two by rigid springs, which are stretched by thrust screws. With the aid of long insulating handles, these screws

can be rotated when the filament is already under high tension, observing the effect of the adjustment on the fluorescent screen in the camera. Plate \(D\) is fastened to \(E\) in exactly the same way, and a new pair of adjusting screws makes it possible, by moving the entire cathode relative to the anode, to achieve coincidence of the focal spot with the opening in the latter. Plate \(E\) is fastened by four short screws to \(F\), the joint being sealed with a gasket of special rubber. Thanks to this last connection, the entire cathode assembly can easily be removed from the apparatus, and the filament can be replaced within a few minutes.

To obtain a sharp diffraction pattern it is necessary that all electrons of the beam have identical or, at any rate, very nearly identical velocities. This can be achieved in several

Fig. 4. Connection diagram of the electronograph

different ways. The most convenient source of high voltage at the present time is a transformer, especially in view of the availability on the market of inexpensive medical X-ray installations. The tube described above can be connected directly into the secondary circuit of such a transformer (Fig. 4). The voltage in the secondary circuit of the transformer is approximately sinusoidal, but the discharge tube, owing to its rectifying action, extinguishes half of the wave, so that current passes only during one half of each period. The electrons emitted by the filament during this working half-period would acquire all possible accelerations—from zero up to the maximum value—were it not for an additional potential of opposite sign applied to the focusing cylinder. This reverse voltage does not allow the electrons to leave the cathode until the potential difference between the filament and the anode exceeds a certain minimum value. When working with the described arrangement of the additional

potential of 400 V proved sufficient to extinguish the beam completely at a voltage of 50 kV. During operation the additional voltage is lowered as much as possible without particular harm to the homogeneity of the electron ray. The check can be made by observing, on a fluorescent screen, the effect produced by a magnetic field, or else by the sharpness of the rings of the electron diffraction pattern of a gold film. The current can be regulated without changing the additional voltage, simply by changing the filament heating. If the high voltage varies sinusoidally, then current can pass during six hundredths of each half-period, and the voltage oscillation will not exceed \( \frac{1}{2}\% \). The apparatus described gives good control of the electron beam and requires only a high-voltage installation, which is available in almost every laboratory.

In another method, the electron beam that has passed through the aperture in the anode is passed through a magnetic field. At that place of the magnetic spectrum which corresponds to the desired electron velocity, the beam is cut out by a new diaphragm. This method is less convenient than the preceding one, and with it the useful current constitutes only a small part of the total current.

It is best to use high-voltage installations that give rectified current. Rectifiers and capacitors are commercially available, and although the cost of the equipment must also be taken into account, direct current nevertheless has two great advantages that justify the expense. The first advantage is that voltage oscillations can be smoothed out to a considerable degree; if the primary voltage is stable, a simple filtering system makes it possible to keep them at a value not greater than \(0.1\%\). The second advantage lies in the possibility of directly measuring, with high accuracy, the voltage of the field determining the wavelength of the electron ray at the moment of exposure; for this purpose a voltmeter with very high resistance is used.

Rheostats made of high-quality resistance wire, of approximately 50 megohms, are connected in parallel with the discharge tube. The potential difference at the poles of the discharge tube is judged from the current in the rheostats, or else by measuring the voltage drop over a small section of the parallel resistance. A voltmeter of this kind is especially convenient for observing fluctuations in the power of the high-voltage installation. The pulsation of the rectified voltage, whose frequency is equal to, or twice as great as, the frequency of the alternating current feeding the transformer, can be smoothed by introducing a sufficient filtering capacitance, but oscillations in the primary circuit must be observed on the voltmeter. These oscillations are compensated manually or with the aid of a vacuum regulator, the voltage for which is taken from the parallel resistance. Since the absolute value of the voltage determines the wavelength of the electron ray used in calculating the electron diffraction pattern, it is necessary to carry out a calibration in wavelengths. The voltage reading may be made by means of the high-resistance voltmeter described above, and then

assuming that the accelerating potential imparts to all electrons of the beam the equivalent kinetic energy, the wavelength is calculated from the de Broglie relation in relativistic form. The formula has the following form:

\[ \lambda = \left( \frac{150}{V} \right)^{\frac12} \frac{h}{(e m_0)^{\frac12}} \left\{ 1+\frac{eV}{600m_0c^2} \right\}^{-\frac12} \ \text{cm}, \]

\[ = \left( \frac{150}{V} \right)^{\frac12} \left\{ 1+9.834\cdot 10^{-7}V \right\}^{-\frac12} \ \text{\AA}, \]

where \(V\) is the accelerating potential in volts.

The calibration of the measuring instrument directly in wavelengths can be carried out by another method, by taking electronograms of gold \(^{5,9}\). A gold film, sufficiently thin to be penetrable for electrons, is placed in the path of the beam and, owing to the microcrystalline structure of gold, an electronogram of a powder is obtained, the arrangement of whose rings is identical with the arrangement of the rings on an ordinary Debye X-ray photograph. Bragg’s equation for cubic crystals

\[ \lambda = \frac{\left( 2a_0 \sin \frac{\vartheta}{2} \right)} {(h^2+k^2+l^2)^{\frac12}} . \]

makes it possible to calculate the value of the wavelength corresponding to any ring; \(a_0\) here denotes the edge of the elementary cube and is equal \(^{29}\) to \(4.070\) Å; \(\theta\) is the angle of deviation of the reflected ray from the incident one, denoted in X-ray diffraction by \(2\vartheta\), and \(h, k, l\) are the Miller indices of the reflecting planes. For the face-centered lattice of gold, \(h, k, l\) can be only either all even or all odd. Table 1 shows how the indexing and the calculation of the wavelength are performed for an electronogram of a gold leaf taken at an accelerating voltage of 39,600 V. For overlapping rings the mean value of \((h^2+k^2+l^2)^{\frac12}\) is taken, in which each form enters proportionally to the number of its planes. Calibration in wavelengths with the aid of an electronogram is carried out for several voltages and is repeated at intervals of several months in order to check the voltmeter.

To obtain a sharp diffraction pattern it is necessary, as indicated above, that the volume of intersection of the electron beam with the jet of gas be small. In view of the intensity of the interaction of electrons with matter, it proves impossible to surround the gas jet even with the thinnest walls; on the other hand, this same intensity of interaction makes it possible to be limited to such small exposures that the maintenance of the jet of gas during so short a pro-

Table 1

Calculation of the wavelength of electrons from the electron diffraction pattern of gold

\[ L = 12.91\ \text{cm},\quad a_0 = 4.070\ \text{Å} \]

interval of time presents no difficulties. In this case the principal task consists in preventing the spread of gas into the diffraction chamber, i.e., in maintaining a high vacuum (\(10^{-4}\) mm or higher) in the space into which the gas is introduced. The chamber, of course, is continuously evacuated by sufficiently powerful diffusion pumps, but, besides the pumps, it is also necessary to provide some device for condensing or removing the introduced gas. When using a discharge tube with a hot cathode, there arises, in addition, the further necessity of protecting the discharge tube from gas that may penetrate from the diffraction chamber.

Fig. 5 shows the construction used in our laboratory. The apparatus, built from a three-inch brass tube, is divided into sections joined by means of screwed-on and soldered flanges; the flanges are held together by a device of six pushing and six pulling screws, by means of which it is not difficult to achieve coincidence of the axes of the individual sections. The joints are sealed with rubber gaskets and permit replacement of the sections. Besides the three sections shown in the figure, the apparatus also contains diffraction and photographic chambers for photographing at wide

corners and two additional sections, serving to increase the distance from the specimen to the photographic plate.

The electrons emitted by the filament strike the anode \(A\). At the center of the anode there is a hole of diameter \(0.2\ \text{mm}\), which is the entrance to a channel of the same diameter, whose length is \(15\ \text{cm}\). This long channel, together with the hole \(B\) of diameter \(0.4\ \text{mm}\) and the hole \(C\) of diameter \(0.8\ \text{mm}\), constitutes the diaphragm system. The beam is cut out by the long and narrow channel, while the wide openings retain the electrons diaphragmed by the edges of the inner aperture of the diaphragm. It has been found that, for the same electron current and the same area of the focal spot, a diaphragm in the form of a long channel passes from 10 to 20 times more electrons than two thin platinum plates with holes of the same diameter, placed from one another at a distance equal to the length of the channel. \(D\) is a rod passed through a small metallic bellows and provided on the outside with a lever. It serves as a shutter capable of closing and opening the diaphragm aperture.

Fig. 5. Diaphragm, diffraction and photographic cameras

The first section of the brass tube in Fig. 5 (between the flanges \(E\) and \(F\)) is essentially part of the discharge tube. The flanges \(E\) and \(F\) carry the diaphragm system, which, if necessary, can be removed as a whole from the cathode side. The hole in \(F\) has a conical shape, so that the diaphragm always occupies a strictly definite position and at the same time closes the communication between the diffraction camera and the discharge tube. The two cameras communicate with one another solely through the holes \(B\) and \(C\). The section \(EF\) is connected to a diffusion pump, which evacuates the discharge tube, by means of a two-inch hose; the position of the evacuating aperture is such that the small quantity of gas entering the discharge tube is removed before it

will have time to affect the discharge. This can be verified from the constancy of the electron current at the moment when gas is admitted into the diffraction chamber. Plate \(E\), carrying such an insulating porcelain tube, is provided with four openings for evacuation. Plate \(F\) is lined with lead, serving to protect the photographic camera from X-rays.

A tube is introduced into the second section of the apparatus, through which the gas jet is admitted. The nozzle shown in the figure is used when working with substances having a vapor pressure considerably below \(40^\circ\text{C}\). Through plug \(G\), seated in a conical ground joint, there passes a tube in the closed upper end of which an opening \(0.3\ \text{mm}\) in diameter is drilled, serving for admitting the gas jet. The electron beam meets the jet approximately \(2\ \text{mm}\) above the end of the tube. Outside, the tube is bent through \(90^\circ\) and is connected, by means of a ground joint through a glass stopcock, to the vessel containing the substance under investigation.

The gas jet flowing out of the nozzle is directed into the opening of tube \(H\), which communicates with a vacuum pump having a high pumping speed. The upper end of the nozzle enters into the tube by approximately \(1''\), in such a way that the gap between them does not exceed \(0.5\ \text{mm}\). For passage of the electron beam, two openings are drilled in tube \(H\), the diameter of which is chosen with allowance only for the maximum scattering angle. Owing to the fact that the opening of the evacuating tube \((2\ \text{dm})\) is considerably larger than the openings leading into the body of the apparatus, the greater part of the gas is removed so rapidly that the latter interacts nowhere except in the diffraction space with the electron beam. A separate pump is connected to this section of the apparatus (the place of connection is indicated in the figure by a dotted circumference), serving to evacuate the diffraction and photographic chambers.

The device described proved to be the most convenient for eliminating diffuse scattering from gas molecules distributed in the body of the apparatus. Several other devices were also tried, in which the gas was condensed on a surface cooled to the temperature of liquid air. To improve condensation, we covered the surface with a layer of wood charcoal, but for substances with very low melting points, such as \(\mathrm{CF}_4\), this proved insufficient. Another difficulty consists in the fact that it is impossible to surround the gas jet with a cooled surface without at the same time cooling the tube introducing the gas below the melting point of most of the substances investigated and, consequently, without clogging it with solid substance.

The film holder is a vertical disk with six three-inch cutouts, into which a fluorescent screen and five films are placed. Rotation of the disk is carried out from outside by means of a small magnet attracting pieces of iron fixed around the circumference of the disk. The film holder is placed between two thick brass plates, the edges of which are joined by a rim and through the center of which a screw is passed, serv-

serving also as an axis for the film holder. The rear plate is provided with an opening serving simultaneously for loading the films and for observing the fluorescent screen. This opening is closed with glass and sealed with rubber gaskets.

The electronograms of the gas were taken on X-ray films, although, generally speaking, it is possible to use any other emulsion possessing approximately the same qualities. The emulsion must be characterized by low contrast and a long scale, i.e. it must be sensitive over wide limits of intensity. Eastman “Par Speed Portrait” film proved quite convenient in the work; however, it is apparently somewhat less sensitive (although we did not make quantitative measurements) than X-ray films. For electronograms of gold, characterized by sharp rings, “Process” films were used, giving a high contrast between the diffraction maxima and the background.

In order that it be possible to maintain a high vacuum in the apparatus, despite the continuous inflow of gas from the vessel, the diffusion pumps must have a high pumping speed and be connected by means of short connections of large diameter. The discharge tube and the diffraction chamber must be evacuated by separate pumps. For evacuating the above-described apparatus, four metal diffusion pumps of the type described by Sloan, Thornton, and Jenkins^30 were used. The pumps operated on Apiezon oil “B,” and one of them, with a housing diameter of 45 mm and a nozzle clearance of 6 mm, was connected ahead of the other three, which had a clearance of 3 mm, creating the first stage of rarefaction. One of these three high-vacuum pumps was connected to the discharge tube, another to the diffraction chamber, and the third to tube \(H\), Fig. 5, into which the gas jet entering the apparatus was directed. The first two operated quite satisfactorily without liquid-air traps, although the latter, according to the latest investigations of Hickman^31 concerning the design of high-vacuum oil pumps, raise the limit of attainable vacuum. A liquid-air trap was, however, introduced into the circuit of the third pump for the purpose of trapping the vapors of the substance under investigation which were being pumped out by the pump. The trap was constructed in such a way that the glass vessel containing the condensed vapors of the substance could be removed from the apparatus without disturbing the low temperature. In this way the oil in the pump was protected from contamination; vapors not trapped by the trap do not exert any noticeable effect on it. The fore-vacuum for the diffusion pumps was produced by a mechanical “Cenco Megavac” pump.

The exposure is carried out in the following manner. If the substance under investigation condenses readily, it is placed in a small glass vessel with a stopcock, connected to the inlet tube by means of a ground joint. After this the cassette is loaded and the pumps are started. Half an hour after the pumps have warmed up, the vacuum may be considered sufficiently high. The high voltage is adjusted, setting it at some previously determined value of about 40,000 V.

with the aid of a secondary-voltage voltmeter. The current in the discharge tube is set from 0.05 to 0.20 mA. Observing the central spot on the fluorescent screen, by regulating the additional voltage and the position of the filament they obtain the maximum intensity of the spot. The adjustment of the beam is performed once, and subsequently, over the course of several series of electronograms, requires only slight readjustment. After this, the aperture of the diaphragm is closed with a shutter and the first film is brought into the position for exposure. Heating the vessel with the substance under investigation in a water bath, they bring

Fig. 6. Electronogram of As₄ vapors taken by Dr. L. R. Maxwell of the U.S. Department of Agriculture

it to such a temperature that the vapor pressure of the substance lies between 100–200 mm of mercury. The exposure is made by briefly moving aside the diaphragm shutter and simultaneously opening the valve of the vessel with the substance under investigation. For this purpose the valve is turned through 180° in approximately 1 sec. At the same time access of the gas to the apparatus remains open for approximately \(1/5\) sec., and at this very moment the diaphragm shutter is opened. In some cases three or four such exposures were made on one and the same film. The resulting photograph shows a series of maxima and minima, which appear somewhat blurred in comparison with electronograms of crystalline powders. Fig. 6 reproduces an electronogram of As₄ vapors.

If the substance under investigation has a very low boiling point, the sample vessel is filled with vapor at a pressure of about

200 mm. If the substance has a very low vapor pressure at temperatures below 50° C, then the inlet-tube device shown in Fig. 5 is no longer suitable, since the gas, heated above this temperature, will condense in the cold tube before it reaches the apparatus.

De Laszlo[^32] described a successful design of an apparatus with a gas tube as the electron source, used in several European laboratories. In this apparatus the inlet tube is heated, which, according to the author, makes it possible to bring the temperature up to 1000° C. In this apparatus the electron beam crosses the jet of vapor emerging from an aperture in the removable cover of a small furnace mounted directly in front of the diaphragm aperture. Other types of gas nozzle, described by various investigators, coincide in general outline with this design. The disadvantage of devices of this kind is that, owing to the continuous admission of gas, excessively large amounts of substance enter the apparatus, naturally impairing the quality of the photograph. Certain modifications improving this design are being developed in our laboratory. These modifications amount to a valve regulating the admission of vapor into the apparatus, placed in the furnace cover and opened only at the moment of exposure. This device was successfully used at 175° C and is now being tested for higher temperatures.

Kossel’s apparatus[^33] differs somewhat from De Laszlo’s design; in it the conditions for vapor condensation are improved. Kossel makes the gas flow through a narrow tube, in whose walls there are two openings for the passage of the electron beam. Thus the volume over which the beam intersects the gas jet is limited. On its way the gas meets a surface cooled with liquid air, located at the very end of the tube. If the gas-supply tube is connected to a pump, the possibility of gas penetrating into the diffraction and photographic chambers will be still smaller.

Zeeman, in his laboratory in Freiburg, constructed an electronograph for work with solids[^34], later adapted, according to Wierl’s instructions, for the study of substances in the gaseous state. Still later Zeeman replaced the gas discharge in his apparatus by an incandescent cathode enclosed in a focusing cylinder. The apparatus and the entire installation are very compact and can readily be used as a Lenard or X-ray tube. Its use as an apparatus for electron diffraction by gaseous substances was described by Grether[^35].

(To be continued in the next issue)