New Experimental Studies of Electron Waves

V. L. Granovskii, Moscow.

§ 1. The conception of wave laws governing the motion of elementary particles of matter was created by de Broglie in 1924.[^1] The basic idea of this doctrine consists in the fact that, in order to determine the motion of a stream of particles, we must consider the motion of the waves associated with them. If the mass of each particle is \(m\), and all of them move with velocity \(v\), then associated with them is a wave whose length is

\[ \lambda = \frac{h}{mv}, \tag{1} \]

and whose velocity of propagation (phase velocity) is

\[ w = \frac{c^2}{v}. \tag{2} \]

In these formulas \(h\) denotes Planck’s constant, and \(c\) the velocity of light in vacuum.

Developing these ideas further, Schrödinger arrived at the equation

\[ \Delta \psi + \frac{8\pi^2 m}{h^2}(E-U)\psi = 0, \tag{3} \]

where \(E\) is the total energy, and \(U\) the potential energy of the particle at the given point.

The variable quantity \(\psi\), determined from this equation as a function of position, has the meaning that the square of its absolute value—\(|\psi|^2\)—gives us a measure of the probability

finding the particle at the given point. It is not difficult to see that this equation is satisfied by a wave whose frequency is

\[ \nu=\frac{E}{h}, \tag{4} \]

and whose velocity of propagation is

\[ w=\frac{E}{\sqrt{2m(E-U)}} . \tag{5} \]

Readers of Advances in the Physical Sciences are familiar with these ideas from a number of articles²; this enables us not to enter here into a more detailed exposition.

§ 2. The simplest and most convenient object for testing this theory, which has received the name “wave mechanics,” is a beam of electrons moving with equal and parallel velocities. It is practically not difficult to obtain such a beam; from the theoretical side, it corresponds to a plane monochromatic wave, whose behavior is easiest of all to analyze.

The experimental study of the wave properties of matter in general is greatly facilitated by the fact that the questions of propagation of wave-like motion have long been well developed. Reflection, refraction, diffraction, and interference of waves are well known to us at least from the examples of sound and light waves. Therefore physicists who undertook the study of “matter waves” had the possibility, from the large number of wave phenomena, to choose those in which the specifically wave properties of matter would appear most clearly. The most favorable conditions in this respect are presented by the passage of a wave through a medium possessing properties that vary periodically in space, or by the reflection of a wave from such a medium. The ordinary diffraction grating used in optics is precisely such a medium, the properties of which—for example, the coefficient of reflection—vary periodically in one direction. This direction, perpendicular to the rulings of the grating, we shall call \(X\). Theory shows that if a plane wave falls on such a grating,

wave, the normal to which forms an angle \(a_0\) with the \(x\)-axis (Fig. 1), then a whole series of diffracted waves will be obtained in directions \(a\), determined by the condition:

\[ (\cos a_0-\cos a)\cdot d=n\lambda, \tag{6} \]

where \(d\) is the linear period of the grating, \(\lambda\) is the wavelength, \(n=0, 1, 2, 3\ldots\)

Equation (6) means, as is known, that the elementary waves issuing from neighboring scratches give an interference maximum in that direction \(a'\) in which the phase difference between them will be equal to \(2\pi n\).

Let us note that not all waves which are given to us by equation (6) will always actually exist; the number of really observed waves and their relative intensity are determined by the character of the grating.

Fig. 1.

The properties of the grating may be a periodic function of two directions (\(X\) and \(Y\)). We obtain such a case, for example, if we scratch a glass plate with two rows of parallel lines forming a certain angle between themselves. In this case it turns out that the directions of the diffracted waves will be described by two equations:

\[ \left. \begin{aligned} (\cos a_0-\cos a)\cdot d_1&=n_1\lambda,\\ (\cos \beta_0-\cos \beta)\cdot d_2&=n_2\lambda, \end{aligned} \right\} \tag{6'} \]

where \(a_0\) and \(\beta_0\) are the angles formed by the incident ray with the axes \(X\) and \(Y\), and \(a\) and \(\beta\) are the angles formed by the diffracted ray with the same axes; \(d_1\) and \(d_2\) are the linear periods of the gratings in the directions \(X\) and \(Y\); \(n_1\) and \(n_2\) independently run through the values of the integers \(0, 1, 2, 3\).

Each of equations (6′) determines a family of cones with a common axis: the first—with axis \(X\), the second—with axis \(Y\). A screen placed parallel to the grating at some distance from it will cut these cones in hyperbolas. The hyperbolas obtained in this way will intersect one another; the points of intersection will represent possible interference maxima.

In both cases considered, the discussion concerned a plane grating; after passing through it (or being reflected from it), the waves propagated in a homogeneous medium. In a mathematical investigation of the problem we would have to write the differential equation for the propagation of the wave in a homogeneous medium; the grating specifies only the boundary conditions. What is characteristic here is that, for any incidence of a wave of any length, we obtain a definite diffraction pattern.

We encounter a different case in a three-dimensional, i.e. spatial, grating. Here the very propagation of the wave takes place in an inhomogeneous medium. Therefore, in a strict treatment, the differential equation itself will be different here. The result of the calculations in the first approximation has a form analogous to the preceding one; namely, the direction of the diffraction maxima is determined by three equations:

\[ \left. \begin{aligned} (\cos \alpha_0-\cos \alpha)d_1&=n_1\lambda,\\ (\cos \beta_0-\cos \beta)d_2&=n_2\lambda,\\ (\cos \gamma_0-\cos \gamma)d_3&=n_3\lambda. \end{aligned} \right\} \tag{6″} \]

Here \(\gamma_0\) and \(\gamma\) are the angles formed by the incident and diffracted beams of rays with the axis \(z\) (the direction of the third periodicity); the remaining notation is clear. For the quantities \(\alpha,\ \beta\), and \(\gamma\) we have here three equations; moreover, they are connected by the well-known relation of analytic geometry. Under these circumstances the system of equations (6″) generally has no solution. In other words, distinct diffraction phenomena are not obtained for every incidence of a monochromatic wave on a three-dimensional grating. In order for them to occur, it is necessary that the equation relating \(\alpha,\ \beta\), and \(\gamma\) be satisfied identically, if

the equations \((6'')\) are satisfied. This will be so only for a definite relation among \(\alpha_0,\ \beta_0,\ \gamma_0,\) and \(\lambda\).

Let us take the particular case in which the directions \(X,\ Y,\) and \(Z\) are mutually perpendicular, and \(d_1=d_2=d_3=d\). Then the condition for the possibility of diffraction will have the form

\[ n_1\cos\alpha_0+n_2\cos\beta_0+n_3\cos\gamma_0 = \frac{\lambda}{2d}\left(n_1^2+n_2^2+n_3^2\right). \tag{7} \]

If it is satisfied, then the formation of a diffraction image is possible; to each combination of the numbers \(n_1,\ n_2,\ n_3\) for which (7) is satisfied there corresponds one direction of a maximum. If, however, (7) cannot be satisfied for any combinations of \(n_1,\ n_2,\ n_3\), then no diffraction pattern with distinct maxima will be obtained at all.

Another method of presenting the same results was proposed by Bragg. Put \(n_1=n_2=0\). Then the first two equations \((6'')\) give

\[ \cos\alpha=\cos\alpha_0,\quad \cos\beta=\cos\beta_0. \]

From the equality

\[ \cos^2\alpha+\cos^2\beta+\cos^2\gamma = \cos^2\alpha_0+\cos^2\beta_0+\cos^2\gamma_0=1 \]

we find that

\[ \cos\gamma=\pm\cos\gamma_0. \]

The case \(\cos\gamma=\cos\gamma_0\) gives the undeviated wave (all three direction cosines have remained unchanged). The case \(\cos\gamma=-\cos\gamma_0\) gives us ordinary reflection from the plane \(xy\). In this case the third of equations \((6'')\) reduces to the following:

\[ \cos\gamma=n_3\frac{\lambda}{2d_3}. \tag{7'} \]

It is evident that analogous conditions may be derived for reflection from the two other planes \((ZX\) and \(YZ)\), only instead of the index \((3)\) on \(n\) and \(d\) there will stand the index corresponding to the given plane. Thus, from each plane in our lattice one can obtain reflection only if the angle of incidence upon it assumes certain values; at other angles of incidence there is no reflection.

The preceding considerations imply that the wave does not undergo any appreciable attenuation as it propagates in the crystal. In other words, the energy of the waves diffracted by a single plane should not be a significant fraction of the energy incident on the crystal. If this is not the case, then the amplitudes of the waves reflected from successive lattice planes will not be equal; then, even with phases differing by \(180^\circ\), we do not obtain their complete cancellation by one another. With strong absorption of the rays, the restricting action of the third of conditions \((6'')\) will be insignificant. We shall see all beams satisfying the first two conditions \((6'')\), i.e. the effect of two-dimensional diffraction; the effect of the third dimension will be manifested only in the fact that the rays which also satisfy the third condition will be somewhat stronger than the others.

It is evident that an analogous result will also be obtained when the wave is weakly absorbed, provided only that the number of parallel layers (the thickness of the lattice) is small. Then the diffraction pattern will approach more and more closely a two-dimensional one as the thickness of the lattice decreases.

§ 3. From the foregoing it is clear that the study of the wavelength incident on the lattice reduces to measuring the angles at which the wave can give diffraction maxima. In order to measure these angles accurately, they must not be very small. Condition (6) and the subsequent ones say that the linear period of the lattice must be of the same order as the wavelength.

The de Broglie formula (1) for the wave accompanying a beam of electrons can easily be rewritten in the form

\[ \lambda=\frac{12.25}{\sqrt{v}}\,\text{\AA}, \tag{1'} \]

where \(v\) is the energy of the electron expressed in volts. For an electron which has traversed on its path a potential difference of \(150\ \mathrm{V}\), we obtain \(\lambda=1.00\ \text{\AA}\); if \(v=1000\ \mathrm{V}\), \(\lambda=0.39\ \text{\AA}\), and so on. This range of wavelengths in the domain of electromagnetic radiation belongs to X-rays, and therefore

it was natural to transfer to electron waves the well-developed methods of X-ray spectroscopy.

As gratings, X-ray spectroscopists use chiefly crystals, which from this point of view are spatial gratings. The distance between the planes—the grating period—has just the required dimensions: several Å. From the experimental point of view, the methods of obtaining diffraction patterns may be divided into three main groups.

In working by the Laue method, a beam of X-rays is directed at a crystal parallel to one of its axes. On a screen placed behind the crystal we see a group of spots arranged symmetrically with respect to the incident ray; the symmetry of the image corresponds to the symmetry of the axis parallel to which the rays traveled. With a strictly monochromatic beam of rays it may turn out that the equalities (6″) are not satisfied; then no image will be obtained at all. But usually a beam of X-rays is non-uniform in spectral composition. Therefore there will always be rays for which the equalities (6″) are approximately satisfied; these rays will produce one or several diffraction spots.

The Bragg method is based on the selective reflection from the planes of a crystal lattice discussed in the preceding paragraph. From a beam of X-rays incident on a crystal, at a given angle of incidence $\gamma_0$, only the ray possessing a suitable wavelength is reflected. By rotating the crystal with respect to the incident rays, we cause all the rays to be reflected in turn, and on the screen before us the whole spectrum of the incident rays will gradually pass by.

If X-ray waves fall not on a whole crystal, but on a crystalline powder or a medium possessing a microcrystalline structure, then among the disorderly oriented crystallites there will be some in which the given crystallographic plane forms, with the direction of the incident rays, the angles required by Bragg’s formula (7). From all such crystallites the rays will be reflected, and it is easy to see that the reflected rays form a cone whose axis will be the direction of the incident rays. On

on the screen we obtain a ring. There will be several such rings, since each face of the crystal for which \(d > 2\lambda\) gives its own ring. This method was developed by Debye and Scherrer in Germany and Hull in America.

The application of crystals to “electron spectroscopy” is possible if the crystal is indeed a grating for electron waves. We shall be convinced that this is so if we recall that the potential \(U\) inside the crystal is a periodic function of all three coordinates. Therefore, according to formula (5), the velocity of propagation, and consequently also the refractive index of the electron waves, must likewise vary periodically. In this case the theory of three-dimensional gratings, discussed in § 2, can be transferred wholly to the present case. And indeed, all the methods of X-ray spectroscopy applied to electron waves have given a positive result.

Davisson and Germer\(^4\), studying the selective reflection of a beam of electrons from a nickel crystal, used Bragg’s method. G. P. Thomson\(^5\) and E. Rupp\(^6\) observed Debye—Scherrer rings when cathode rays passed through metallic films. A description of these works was given in the article by P. S. Tartakovsky and in Davisson’s article “Are Electrons Waves?”\(^7\).

The Japanese physicist Kikuchi\(^8\) used Laue’s method to detect the diffraction of cathode rays. His work, as it were, concludes this series of applications of the classical methods of X-ray analysis to electrons. The large number of phenomena observed by him and the purity of the results obtained give us reason to dwell in detail on his work.

§ 4. The apparatus used by Kikuchi is shown in Fig. 2. The discharge tube \(A\) served as the source of electrons. Through the channel \(K\), drilled in the anode, they passed into chamber \(B\), where, under the action of a magnetic field produced by a special solenoid, they were resolved into a magnetic spectrum according to velocities. Two diaphragms \(S_1\) and \(S_2\), 0.5 mm thick, selected a narrow monochromatic beam of electrons, which then fell upon a small sheet

\(^5\) Voprosy filosofskikh nauk, vol. IX, issue 3.

of mica \(T\). In this sheet the electrons underwent diffraction. After passing through it, the diffracted rays produced images on the photographic plate \(T'\). The path of the electrons between the diaphragms was \(7\ \mathrm{cm}\); between the mica and the photographic plate, \(14.9\ \mathrm{cm}\); from external electrostatic actions it was protected by a metallic shield. To avoid scattering of the electrons by gas molecules, chambers \(B\) and \(C\) were evacuated to a high vacuum by a three-stage mercury pump. The discharge in tube \(A\) was supplied by a transformer; the pressure in it was maintained by a palladium tube \(E\), which admitted hydrogen when heated. The velocities of the electrons were varied within the limits from \(10{,}000\) to \(85{,}000\ \mathrm{V}\); these velocities correspond to short waves: \(\lambda = 0.12\ \text{Å}\) down to \(0.0423\ \text{Å}\). The wavelengths calculated in this way served only as preliminary data; more exact determinations were made with the aid of a magnetic field. From the formula expressing the action of a magnetic field on an electron:

Fig. 2.

\[ F=\frac{mv^{2}}{R}=evH, \]

where \(R\) is the radius of curvature of the electronic trajectory, \(H\) is the intensity of the magnetic field, and \(e\) is the charge of the electron, we find:

\[ mv=eHR. \]

Substituting into the de Broglie formula (1), we obtain:

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

The magnetic field produced by the solenoid is proportional to the current flowing in it. Let \(H=Ki\) (\(K\) is a constant); then we finally find

\[ \lambda i=\frac{h}{eKR}=\mathrm{const}. \]

Using this formula, Kikuchi counted off the wavelength directly from the strength of the current in the solenoid. In order to determine once and for all the value of \(\lambda i\) for electrons passing through both diaphragms, Kikuchi placed a thin aluminum plate in place of the mica. On the photographic plate rings were obtained similar to those observed by G. P. Thomson; from the radii of these rings and the constants of the crystal lattice of aluminum it was possible to calculate the wavelength. It proved indeed to decrease inversely proportional to the current in the solenoid, and the value of \(\lambda\cdot i\) was found to be equal to 0.310 for \(i\), expressed in amperes, and \(\lambda\) in Å.

Fig. 3.

As has already been indicated, the diffraction phenomena were produced by sheets of mica. Kikuchi took a colorless specimen of muscovite \(KH_2Al_3(SiO_2)_3\). Its crystalline structure had previously been studied with the aid of X-rays. It turned out that in the cleavage planes the molecules are arranged in the form of a rectangular net, at the center of each cell of which there is also one molecule. The sides of these cells are: \(a=5.17\,\text{Å}\), \(b=8.96\,\text{Å}\). This net may also be considered otherwise: as a net of equilateral (approximately) triangles, whose sides are \(5.17\,\text{Å}\) (Fig. 3). The third axis of the crystal makes with the plane of the first two axes an angle close to a right angle \((84^\circ10')\); the distance between the cleavage planes is \(c=20.05\,\text{Å}\).

To obtain good photographs, Kikuchi used thin mica layers \(10^{-4}\)—\(10^{-5}\) cm thick. The thinnest of these sheets did not at all produce the phenomena of thin-plate colors; working with them was very difficult, since it was not easy to secure them without bending them. However, it was precisely these thinnest sheets that gave very curious diffraction patterns. In Fig. 4 two photographs obtained by Kikuchi are reproduced. In them we see lattices similar to that described above, i.e. as if formed by a system of regular triangles.

Fig. 4.

All the points of such a lattice cannot satisfy the three conditions \((6'')\), and therefore these photographs are not a Laue diffraction pattern. But they are easy to understand and interpret as the result of electron diffraction by a two-dimensional lattice; such a lattice is formed by the molecules of the crystal lying in one plane, for example in the cleavage plane. Indeed, we have seen that the spots obtained from diffraction by a two-dimensional lattice must lie on two mutually intersecting families of hyperbolas. When \(\lambda \ll d, -a\), in our case this is exactly what occurs,—these hyperbolas are almost indistinguishable from straight lines, and in the photograph we should see a lattice formed as it were by the intersection of two bundles of parallel straight lines, which is what is observed. By measuring the distance between the points in the photograph and knowing \(\lambda\), one can calculate the distance between the molecules \(d\). Kikuchi made such measurements for different wavelengths; the results are given in Table I. (See the table on p. 319.)

The mean value is \(d = 5.18\text{ Å} \pm 0.013\text{ Å}\), which is in excellent agreement with the value \(d = 5.17\text{ Å}\), measured by means of X-rays; the error is \(0.2\%\).

NEW INVESTIGATIONS OF ELECTRON WAVES

Thus, in these photographs we are undoubtedly dealing with a two-dimensional effect. The Laue effect (three-dimensional) is only barely indicated here; we recognize it in those hints of circles along which spots of equal intensity are grouped. In the mica sheets with which such photographs were obtained, the number of molecular layers was of the order of 50 (\(1000\) Å—the thickness, and \(20\) Å—the distance between layers). Kikuchi indicates that this number of layers is insufficient to make all the spots that do not satisfy equation \((6')\) disappear.

Fig. 5.

On passing to thicker sheets of mica, the two-dimensional pattern disappeared, yielding place to the ordinary Laue pattern. It appears in fully finished form when the thickness of the sheet is \(10^{-4}\) cm; a characteristic example is given in Fig. 5. Here we see the arrangement of spots typical of a Laue pattern, characterized by rotational symmetry with respect to the primary beam. The plane parallelogrammatic lattice, which stands out so clearly in Fig. 4, is completely absent here. But in this photograph we observe yet another curious-

...a phenomenon: pairs of dark and light lines. To each distinctly visible dark line there corresponds a light line parallel to it. This phenomenon is analogous to that observed by Rutherford and Andrade when a divergent beam of γ-rays was passed through a rock-salt crystal. Among the rays incident on the crystal at various angles there will necessarily be some which, with a given crystallographic plane, form the Bragg angles of selective reflection (Fig. 6). These rays, on being reflected, will give on the plate a reflection (dark) at the points \(B\) and \(B_1\); while in the direction of their original propagation less energy will pass than in neighboring directions, and we shall obtain on the plates a shadow (light). From the distances between the image and the shadow and from the plate to the crystal we can calculate the angle of reflection; and knowing the constants of the crystal lattice, the wavelength. Kikuchi distinguished in his photographs reflections (and shadows) from more than 30 different planes in mica. Starting from the wavelength of the electrons as given, he determined the constants for these faces. We cite part of his results. (See the table on p. 321.)

Fig. 6.

The striking agreement with the results of measurements by means of X-rays leaves no doubt as to the correctness of the interpretation set forth above. However, in order for such an effect to occur, a strongly divergent beam of rays is needed, whereas in Kikuchi’s experiments the divergence of the incident beam was insignificant. Kikuchi considers that the beam becomes divergent owing to scattering of the electrons as they move in the crystal. The circumstance that these pairs of lines were obtained with comparatively thick sheets of mica confirms this idea. Kikuchi also compares the relative intensities of various...

Table II.

orders for the cases of electron and X-ray waves; without reproducing his table, we note only that the qualitative agreement is quite satisfactory.

Nishikawa and Kikuchi^9 repeated these experiments, besides mica, also with calcite, topaz, and zinc blende. In all these crystals the same phenomena were observed as in mica; the coincidence of the measurement results with the data of X-ray analysis also proved complete.

Thus it may now be said that, with a beam of electrons in crystals, all those phenomena have been obtained which we have hitherto observed with X-rays.

Moreover, thanks to the development and simplification of the methods of these investigations, we can now say that in some cases, for the purposes of structural analysis, cathode rays are no less, if not more, suitable than X-rays. The much stronger absorption of cathode rays in comparison with X-rays of the same wavelength makes it possible to obtain distinct “Debyegrams” by means of electrons from such thin layers of substance as could scarcely be well analyzed by X-rays. In this respect, the works of

—M. Pont, published in the spring of this year,^11 analyzed by means of a beam of cathode rays thin layers of oxides of metals of the 2nd group, deposited on the metallic part, or edge, of the diaphragm. ZnO, MgO, and CdO were investigated; the electron velocities varied from 7 to 18 kV. From the diameters of the diffraction rings and the electron velocities, Pont calculated the constant \(d\) for various planes. We give the results for ZnO (hexagonal system):

Table III.

The agreement of the results is complete; the mean error does not exceed 1%. In his second work, Pont further improved the accuracy, reducing the magnitude of the deviations from the mean to 0.3%. At the same time the investigator emphasizes the experimental simplicity and comparative economy of the work: the exposure did not exceed \(1\)—\(1\frac{1}{2}\) hours, and the power of the tube was 35 W at 17 kV. A curious result of one of Pont’s experiments especially vividly emphasizes the analytical possibilities of cathode rays. Namely, by passing electrons through a thin sheet of gold, Pont obtained a diffraction ring corresponding to \(\lambda = 4.20\ \text{Å}\). Gold has no such constant. It is, however, observed in the X-ray analysis of paraffin; and thus the cathode rays made it possible to detect a thin film of paraffin that had settled in the vessel on the gold sheet.

In the new work of Germer, cathode waves likewise no longer serve as the goal, but as the instrument of investigation. Namely, using them, Germer studies the structure of a layer of gas adsorbed by the surface of a metal. As is known, such

layers have an extremely strong influence on the clarity of the diffraction pattern obtained from a metal grating. The exposition of this work, however, lies beyond the scope of the present article.

§ 5. In all the experiments described above, crystals served as the diffraction grating for electrons. It is known that, in order to obtain the phenomenon of diffraction of X-rays, it is not at all necessary to make use of the natural gratings that we have in crystals. Compton and Doan[^11] showed that one can obtain a diffraction spectrum of X-rays from an ordinary optical grating ruled on a piece of metal or glass. For this, the rays must fall on the grating at very small grazing angles—smaller than the limiting angle of total internal reflection.

E. Rupp[^12] used this method for measuring the lengths of electron waves. Condition (6) for very small angles takes the form:

\[ (a^2-a_0^2)\frac{d}{2}=n\lambda \tag{8} \]

or, introducing the angle \(\delta=a-a_0\) (the angle between the reflected and diffracted rays),

\[ \frac{d}{2}\delta(\delta+2a_0)=n\lambda . \tag{8'} \]

From formulas (8) and (8′) it is clear that, when \(\frac{\lambda}{d}\) is small, it is advantageous for \(a_0\) to be as small as possible. Therefore Rupp used grazing incidence: \(a_0\) in his experiments did not exceed several minutes. Under such circumstances it was possible to work with a grating of average quality. Rupp used a reflecting grating of specular metal with 1300 lines per 1 cm \((d=7.70\times10^{-4})\). The angle \(\delta\) could be determined by measuring on the photograph the distance between the images of the reflected and diffracted electron beams. As for \(a_0\), it proved impossible to measure it accurately, and therefore in equation (8) there were two unknown quantities: \(a_0\) and \(\lambda\). If only one diffraction image \((n=1)\) was obtained on the photographs, then

verification of the de Broglie formula could be carried out only indirectly. Namely, \(\lambda\) was calculated in advance by formula (1′); varying the voltage, Rupp changed \(\lambda\) and \(\delta\), but \(a_0\) had to remain constant, since the position of the grating relative to the incident electron beam did not change. Consequently, the constancy of the value \(a_0\), calculated from the experimental data by formula (8′), served as the criterion for the correctness of the de Broglie formula. In those cases when more than one diffraction image was obtained on the plate, Rupp had more than one equation for determining \(a_0\) and \(\lambda\), and formula (1) could be measured directly.

Fig. 7.

Rupp’s apparatus is shown schematically in Fig. 7. A beam of electrons, emitted by the heated cathode \(k\), moved under the action of the electric field toward the second cathode \(g\); between this latter and the first diaphragm \(b_1\) the main voltage was applied, forcing the electrons to rush toward \(b_1\).* Having then passed through two more diaphragms \(b_2\) and \(b_3\), the electron beam struck the diffraction grating \(R\). At a distance of 38.5 cm from it there was placed a photographic plate, sensitized for electrons with oil. The diaphragms were made in the form of slits 8 mm high and 0.1–0.2 mm wide. Experiment showed, however, that this was insufficient for obtaining a sharp spot on the plate. Therefore a solenoid \(M\) was placed above diaphragm \(b_3\), playing the role of an “electron lens,” i.e. collecting the beam of electro-

* It was impossible to heat cathode \(g\) directly, since the light emitted by it would have caused the photographic plate to blacken.

anew “in focus” on a photographic plate, Rupp carried out his experiments with electrons that had passed through potential differences of 70, 150, and 310 V; the experiments performed at 40 V gave no results. We shall give only the results obtained by Rupp in processing photographs containing spectra of three orders and where, consequently, \(a_0\) and \(\lambda\) could be calculated and checked (Table IV).

Table IV.
\(V = 150\ \mathrm{V}\) \((\lambda = 1.00\ \text{Å})\).

The value obtained agrees completely with that calculated theoretically for 150 V; the probable error of the result amounts to only 2% (in the second case, 3%).

Describing his experiments, Rupp notes that the difference between these experiments and all the preceding ones consists in the fact that the latter revealed diffraction of electrons by a potential lattice, whereas in his case there was a grating scratched mechanically. It seems to us that such an opposition is incorrect here. In both cases the “cause” of the diffraction is the periodic course of the potential: in the first case, in the bulk of the substance, and in the second, on its surface.

The latter circumstance led R. Gilsch and R. V. Polya\(^{22}\) to the idea of detecting electron diffraction by means of a microscopic model of a potential lattice. For this purpose they stretched a row of wires at equal distances from one another; the wires were charged alternately positive and negative. Such a grating possesses a field of electric potential that varies periodically in a direction perpendicular to the wires and lying in their plane. The thickness of the wires was \(2r = 0.1\) mm, the distance

between them was \(d=0.4\ \text{mm}\); consequently, the period of the potential grating was \(D=2d=0.8\ \text{mm}\). The potential difference between neighboring wires \(P\) was varied between \(42\)—\(440\) volts; the initial velocity of the electrons was \(E=670\) volts. If we use formula (6) to determine the first angle of diffraction of an electron wave by such a grating, we obtain \(\alpha=0.59\cdot 10^{-7}\); such a negligible angle is understandable if one recalls that the constant of this wire grating differs by millions of times from the constant of a crystal. However, Hilsch and Pohl observed on a screen at a distance of \(15\ \text{cm}\) from the grating a displacement of the electron beam by \(34\ \text{mm}\), i.e. \(\alpha=0.26\). Obviously this phenomenon should have no relation whatever to the wave properties of electricity. And indeed, Bethe \((^{23})\) succeeded in giving a natural derivation of this phenomenon from purely classical considerations. Namely, having calculated the field of this grating by Laplace’s formula and the motion in it of electrons as charged particles without any wave properties, simply according to the laws of Newtonian mechanics, Bethe found that a narrow beam of electrons, having passed through such a grating, must split into two approximately equally narrow beams, deflected from the original direction by the angles

\[ \alpha=\pm \frac{\pi P}{4E\lg \frac{a}{\pi r}}. \]

Substitution of the above data into this formula gives \(\alpha=0.315\).

The agreement is not entirely satisfactory quantitatively, but there is no doubt that qualitatively we have an explanation of the Hilsch and Pohl phenomenon. Thus, for example, it follows from Bethe’s formula that the angle of deflection should not depend on the linear dimensions of the grating, but only on its structure, i.e. on the ratio of the gaps between the wires to their thickness. If this ratio is constant, then whether the gaps themselves are of the order of a centimeter or a micron is immaterial; the angle of deflection will be the same. Precisely such a property was noted by Hilsch and Pohl for the phenomenon they discovered, though, of course, within narrower limits. Another

a characteristic feature of it is the dependence of the angle of deflection on the amplitude of the oscillations of the potential, which was likewise observed by the experimenters. Both these properties are sharply different from those which we ascribe to the diffraction of electron waves. Here the angle of deflection depends strongly on the linear dimensions of the grating and only very little on the amplitude of the oscillations of the potential. We have seen that, in the case of a macroscopic grating, the wave effect plays no role whatever. On the contrary, when electrons move through an atomic lattice it plays the dominant role, whereas the Ramsauer effect, in turn, is blurred out owing to the comparatively small oscillations of the potential.

§ 6. The success that crowned the search for diffraction and reflection of electrons naturally led one to think of detecting other wave phenomena with cathode rays. Directly from Schrödinger’s equation (5) it follows that there must exist refraction of cathode waves; they are refracted every time they enter a medium with a different potential. If the energy of the electrons in the first medium was \(E\), and the potential of the second medium differs from the potential of the first by \(V_0\), then the index of refraction is

\[ \mu=\frac{w_1}{w_2}=\sqrt{\frac{E+V_0}{E}} \tag{9}. \]

In itself, the fact of a change in the direction of a beam of electrons passing into a region of another potential represents nothing specifically wave-like. From optics one may recall that the corpuscular theory of light explained the phenomena of the refraction of light just as effortlessly as the wave theory. However, the refraction of waves, besides a change in the direction of the front, is also connected with a change in wavelength: if in the first medium the wavelength is \(\lambda\), then in the second medium it is

\[ \lambda_1=\frac{\lambda}{\mu}. \]

Verification of this relation is of undoubted interest for the wave theory. On the other hand, by measuring \(\mu\), we can calculate \(V_0\); this greatly enriches our information about the internal structure of the crystal, since it makes it possible to draw a number of important conclusions, for example, about the number of conduction electrons.

And in this question we may draw a parallel with the rays of Röntgen. The discovery of the refraction of the latter resulted from the work of Stenström, who noticed systematic deviations of the actually observed reflection angles of X-rays from the values given by Bragg’s formula (7′). These deviations prove especially significant for the first spectra and decrease with increasing order of the spectrum. A simple explanation of this was given by Ewald. In deriving formula (7′) it is assumed that the optical path difference of the rays emerging from two neighboring lattice planes is equal to the geometrical path difference \((2d\cos\gamma)\); in other words, that the velocity of the wave in the crystal is the same as in a vacuum. The deviations from Bragg’s formula directly indicate the incorrectness of this hypothesis and lead us to the idea of the refraction of X-rays.

Davisson and Germer, studying the angles of selective reflection of electrons, likewise established systematic deviations from the expected values. They interpret them as the result of the refraction of electron waves. The dependence of the refractive index on the velocity of the electrons proves to be just what theory would lead one to expect: as the velocity increases, \(\mu\) decreases and, already at about 600 volts, becomes indistinguishable from unity.

E. Rupp, in the work already mentioned on the scattering of electrons by thin metallic films, found that the radii of the Debye–Scherrer rings observed by him agree with those calculated theoretically only on the condition that we take into account the refraction of electrons in the metal. From his observations he derives values of the refractive index for different electron velocities and of the potential difference between the interior of the crystal lattice and the external space \(V_0\) (the so-called internal potential of the lattice). We give some of his results. (See table, p. 329.)

For Ni, Davisson and Germer found, as an average,

\(V_0 = 15\) volts.

Rupp’s results were criticized by G. P. Thomson \(^{24}\) in the following respect. Rupp, turn—

RECENT INVESTIGATIONS OF ELECTRON WAVES

Table 4.

In processing his results, he did not take into account the refraction of the electron waves as they emerged from the film, but introduced the refractive index only as a factor shortening the wavelength in the diffraction process. In Thomson’s opinion, this is inconsistent. Treating the metallic film as a plane-parallel plate, he derived a formula that gave numerical results indistinguishable (within the experimental error) from those calculated without taking refraction into account at all. This undermined the significance of Rupp’s experiments as evidence for the refraction of cathode waves by metals. Thomson’s criticism, of course, did not affect the results of Davisson and Germer. However, in later work Rupp \(^{25}\) found that the thin metal layers which Thomson and he had used to obtain diffraction patterns do not possess the properties of plane-parallel plates. Indeed, when waves fall obliquely on such a plate, they are refracted not only upon leaving it, but also upon entering it. In that case, when the angle of incidence is changed, all the diffraction angles must be shifted in a definite way. Rupp carried out such experiments; the result proved negative, i.e. no regular refraction at the surfaces of the metallic film was found. This result could be interpreted

in two ways: as evidence of the absence of a regular surface in thin metallic layers, or as evidence of the absence of electron refraction. Rupp adopted the first point of view and tried, in his new, more accurate observations—carried out no longer by the photographic method but by an electrical method (in the variant proposed by P. S. Tartakovsky^27)—to determine the internal potential of the metal. The results obtained were ambiguous: in some cases no refraction was observed at all; in others an internal potential of the order of 12–17 volts was obtained, but without any confidence in the reality of this number. Thus, from the controversy between Rupp and Thomson there emerged the indisputable conclusion that, for the study of the refraction of electron waves, polycrystalline metal films are unsuitable because of the uncertainty of their structure. As an illustration of this one may cite the results of the experiments of M. V. Deryagin^28, who studied the diffraction of electrons in a layer of cobalt and found an obviously incorrect value for the internal potential, between 4 and 5 volts. Therefore Rupp acted quite correctly in abandoning this path and turning, in his next work (26), to reflection from single-crystal metals, with which Davisson and Germer had worked from the very beginning.

The results of these experiments are given in extract in the following table (see p. 331).

These results may be regarded as reliable to within $\pm 3$ volts.

Turning to their interpretation, it should be noted that the introduction of the refractive index into the calculation of the angles of selective reflection constitutes a second, more accurate approximation to the true picture of the motion of electrons in a crystal. Strutt^20 and Bethe^21 treated more rigorously the problem of the propagation of an electron wave in a crystalline medium. The solution is given by Schrödinger’s equation, (3), in which the potential $V$ must be expressed exactly as a function of the coordinates. This can be done by means of Poisson’s equation: $\Delta V = 4\pi\rho$. But for this it is necessary to know how there are arranged: 1) the charges within the atoms and 2) the atoms themselves relative to one another.

Table 5.

of the other. The latter is known from the structure of the crystal. As regards the former, Bethe makes use of the results of wave mechanics and, in addition, introduces two simplifying hypotheses: a) that the bound electrons are arranged in each atom with spherical symmetry, and b) that the conduction electrons are uniformly distributed throughout the whole crystal. On this basis Bethe calculates the mean value of the internal potential \(V_0\) and its oscillations, and from this the motion of the electron wave. It turns out that the angles of reflection calculated in this way are still somewhat displaced from those which are obtained when refraction is taken into account; moreover, reflection occurs not exactly at a definite angle, but over a certain interval of angles. This means that the reflected beam has a certain angular thickness, and its center is displaced relative to the angle previously calculated. From the angle of reflection we determine the wavelength; depending on the degree of approximation we shall obtain different results. Bethe gives the following calculation. In one of the experiments of Davisson and Germer, at an electron velocity of \(160\ \mathrm{V}\), the diffraction pattern (622) was obtained. The wave number (the number of waves fitting into a length equal to \(2\pi\)), calculated by de Broglie’s formula, proved to be \(6.80\ \text{\AA}^{-1}\). If the refractive index is taken into account, it decreased by \(0.32\ \text{\AA}^{-1}\), i.e. by \(5\%\); the next approximation gave a further decrease by \(0.05\ \text{\AA}^{-1}\), i.e. by \(0.8\%\). As for the width of the beam, expressed in wave numbers, it proved to be equal to \(0.064\ \text{\AA}^{-1}\); consequently, the resolving power of the crystal for these waves is equal to \(\frac{6.80}{0.064}\sim 100\). Bethe finds the value of \(V_0\) itself to be \(18.9\ \mathrm{V}\); under the simplifying assumptions on which the derivation is based, the agreement with the results of Davisson and Germer and of Rupp must be regarded as encouraging.

It should be noted that in the experiments of Thomson, Kikuchi, and Ponte the influence of refraction and of higher effects was not noticeable, as was to be expected at the high velocities with which these investigators worked.

No influence of the internal potential of the metal was noticeable also in Rupp’s experiments with a reflecting grating; this is understandable, since the diffraction of electrons here depended on the periodicity of the surface properties of the metal.

§ 7. Even more interesting is the question of the possibility of detecting the polarization of electron waves. From the original conception of the wave mechanics of de Broglie and Schrödinger, the necessity for the existence of polarization does not follow. In Schrödinger’s equation the function \(\psi\) is a scalar; consequently, all directions are indifferent to it. However, the further development of quantum mechanics led to the description of the motion of electrons by a vector field. The new formulation of this theory, given by Dirac,^14 leads directly to the conclusion that the electron, in addition to charge, also possesses a magnetic moment. This idea, expressed even before the advent of quantum mechanics, gave, in application to questions of the classification of spectra, a number of good results. But in such a case a beam of electrons whose magnetic moments are not distributed chaotically in all directions, but are oriented in certain preferred directions, may possess different properties in different directions, i.e. be more or less polarized.

The simplest method of detecting this effect is suggested by analogy with optics. A beam of rays of natural, unpolarized light, after being reflected from some surface, turns out to be partially (sometimes completely) polarized. If the reflected rays are made to be reflected a second time from another mirror, then, rotating the latter about the incident beam, without changing the angle between them, we shall observe periodic changes in the intensity of the reflected rays; during one complete revolution it will pass twice through a maximum and twice through a minimum. This experiment was carried out with an electron beam by a number of authors.

Davisson and Germer^15 made a beam of electrons fall at an angle of \(45^\circ\) on the face \((111)\) of a nickel crystal. The electrons reflected from it fell at the same angle on a sec-

…crystal. The secondarily reflected rays were collected in a Faraday cylinder, and measures were taken to ensure that electrons whose velocities differed appreciably from the velocity of the original beam did not enter it, since such electrons could be of secondary origin. Therefore, of the \(10^8\) electrons incident on the first crystal, on average only 2.5 electrons entered the Faraday cylinder. The velocity of the electrons was varied from 10 to 150 V; selective reflection was obtained at 20, 55, 75, 103, and 120 V. Under these conditions the ratio of the current in the collector \((5\cdot 10^{-12}\ \mathrm{A})\) to the current on the first crystal \((2\cdot 10^{-4}\ \mathrm{A})\) did not change even by

\[ \frac{1}{200} \]

of its value. In other words, the polarization of the electron flux, if it exists at all, did not manifest itself in these experiments.

A. F. Ioffe and A. N. Arsen’eva[^16] carried out the same experiments with mirrors made of glass, brass, and steel—media that are not monocrystalline. Consequently, in contrast to Davisson and Germer, these researchers were not working with selective reflection. In the collector there were gathered not only electrons possessing a velocity equal to the initial one, but all electrons moving at an angle of reflection equal to the angle of incidence. The velocities of the electrons were varied within the range from 80 to 6400 V. Ioffe and Arsen’eva likewise arrived at a negative conclusion.

Cox, McIlraith, and Karelmeier[^29] studied the polarization of a beam of \(\beta\)-rays emitted by a radium preparation. The polarizer and analyzer were mirrors of polycrystalline gold; the collector was a Geiger counter, in which each incident electron caused a change in the discharge current. These authors found a certain polarization effect exceeding the experimental errors. However, the sources of error themselves are quite serious: 1) insufficient shielding from \(\gamma\)-rays emitted by the same preparation, 2) strong instability of the discharge characteristic in the counter; both these sources of error, in the opinion of Davisson and Germer, make the results of the named three authors unreliable.

F. Wolf1 used, as a “polarizer” for a beam of electrons, not a mirror but a magnetic field. Quantum mechanics requires that, when electrons move in a magnetic field, their magnetic moment be oriented either along the lines of force or against the lines of force; an orientation inclined or perpendicular to them is impossible. Wolf therefore made the beam of electrons move in a magnetic field, which at the same time served as the means for their magnetic splitting, and made the part that had passed through three diaphragms fall at an angle of 45° upon a reflecting surface. The reflecting cylinder, with a polished mirror surface, could rotate about an axis coinciding with the direction of the incident beam; the materials used for it were brass, a single crystal of copper, and PS. The ratio was measured of the number of electrons received by the cylinder to the number entering the chamber \(D\) (this ratio is evidently additional, up to unity, to the coefficient of reflection of the electrons). No periodic dependence on the angle of rotation of the cylinder was obtained.

Finally, Rupp2 combined all these experiments, testing the influence of three factors: 1) simple and selective reflection, 2) the absence and presence of a magnetic field between the mirrors, 3) the direction of the lines of force of the magnetic field (parallel and perpendicular to the path of the electrons). Only in one of these cases was there obtained a certain dependence on the rotation of the analyzer that could be taken for the expected effect; but precisely in this case (selective reflection at an angle of 11°—third order for 150 V) the difficulty of accurately centering the entire apparatus was so great that errors arising from defects of the latter could quite cover the observed effect.

As the general result of these investigations, it must be acknowledged that the reflection of electrons from a solid body gives us no possibility of judging the polarization of electrons. Ya. I. Frenkel3 pointed out that in this case it would be unfounded to expect an analogy with optics. For all that is common between the propagation of light and the motion of electrons, in the phenomena of reflection there is also a moment of essential differ-

tion—these are boundary conditions. Whereas for electromagnetic waves at the boundary of two media a discontinuity of the normal component of the electric vector is observed, for electron waves quantum mechanics requires that all the functions determining the motion of the electrons remain continuous. The mathematical calculation carried out in this way shows, indeed, that the amplitude of the reflected wave should not depend on the orientation of the vector of the incident wave; in other words, that the reflection coefficient does not depend on the polarization of the incident wave.

Darwin also arrived at an analogous conclusion \((^{30})\). Namely, he showed that, in contrast to optics, an unpolarized beam of electrons incident on the surface of a solid is also reflected unpolarized. From these two theoretical works it follows that, for electrons, the surface of a solid can serve neither as a polarizer nor as an analyzer.

Fig. 8.

§ 8. The preceding account has covered works that studied a number of characteristic wave phenomena: reflection, refraction, diffraction, and polarization. This, however, does not exhaust everything that has been done to reveal the wave properties of an electron beam. New interesting data were obtained by Rupp in the spring of this year \((^{31})\). The absorption of electrons by various metals at different electron velocities was investigated, and was then compared with their reflection at the same velocities.

Fig. 8 gives us an idea of the course of these phenomena. Along the ordinate axis in the upper curves is plotted \(A = 1 - \frac{J}{J_0}\), where \(J_0\) is the number of electrons that reached the plate; \(J\) is the num-

number of electrons that have passed through the plate without loss of velocity; in the lower curves is plotted \(R = \frac{J'}{J_0'}\), where \(J'\) is the number of electrons reflected from the plate without loss of velocity. Along the abscissa axis the electron velocity is everywhere plotted in volts. We note that the absorption and reflection of electrons have an entirely identical course as a function of velocity; the maxima of both coefficients coincide for all the metals studied. This fact corresponds to the law known in optics which establishes a connection between selective absorption and reflection; thus, for example, the maximum of reflection for infrared rays (“residual rays,” Reststrahlen) is at the same time the maximum of their absorption.

This preliminary communication by Rupp adds a new link to the chain of those proofs by which modern experimental physics has supported the ideas of de Broglie and Schrödinger.

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