WAVE VIEWS ON THE NATURE OF MATTER AND EXPERIMENT.

P. S. Tartakovsky, Leningrad.

§ 1. The crisis of the old quantum theory, which proved incapable of giving a complete, quantitatively correct description of a number of phenomena in atomic physics, led, through a series of new ideas of de Broglie, which seemed very “radical,” and through the restoration of a forgotten analogy between the basic principles of classical mechanics and geometrical optics, to the establishment of a new “wave” mechanics, with its basic Schrödinger equation, already famous despite the short time of its existence, written in its simplest form as follows:

\[ \nabla^{2}\Psi+\frac{8\pi^{2}m}{h^{2}}(E-U)\Psi=0. \]

The solution of a problem, for example, of the motion of an electron inside an atom, is replaced by the solution of a wave equation, the problem of waves of a certain function \(\Psi\)—a function of coordinates and time. Together with the natural requirements that the solutions be finite and continuous, integration of the wave equation in some cases leads to the same results as the old quantum theory with its quantum conditions “imposed from outside”; in other cases it leads to new results that describe, very exactly and in detail, the actual phenomena of atomic physics—phenomena which the old theory could not describe.

But the derivation of the fundamental equation of wave mechanics—in its simplest form or in one of its more complicated ...

and general—always contains within itself a certain arbitrariness, something hypothetical, and therefore every new success in describing atomic phenomena according to the new theory is above all a new triumph of Schrödinger’s equation. In the present case, a paraphrase of Hertz’s old statement concerning Maxwell’s theory proves very apt: Schrödinger’s theory is Schrödinger’s equation1. A cautious theoretician is often inclined to consider only the formal apparatus of a theory, since the more or less concrete physical ideas that led to the establishment of the theory often turn out to contain internal contradictions.

But in this article we shall be interested chiefly in precisely this aspect of the question.

In formulating his theory, de Broglie proposed2 that every particle is connected with some periodic process, and every moving particle—whether a quantum of light or an electron—is associated with special “phase waves,” whose frequency is determined by the energy of the particle, and whose wavelength is most conveniently determined from the condition that the quantity of motion of the particle \(mv\) is equal to the quantity of motion of some “quantum” corresponding to these waves:

\[ mv=\frac{h\nu}{c}=\frac{h}{\lambda}. \tag{1} \]

Whence

\[ \lambda=\frac{h}{mv}. \tag{2} \]

The velocity of propagation of these phase waves is determined by the formula

\[ w=\frac{c^{2}}{v}=\frac{c}{\beta}, \tag{3} \]

therefore, it is always greater than the speed of light, and only the group velocity of the phase waves \(w'\) turns out to be equal to the velocity of the particle itself:

\[ w' = v. \tag{4} \]

According to de Broglie, the phase waves associated, for example, with a moving electron are approximately just as real as ordinary light waves “associated” with quanta of light, observed by us in all phenomena of absorption or of any other action of light.

A number of ideas in de Broglie’s theory concerning phase waves were wholly borrowed by Schrödinger in the construction of his theory1. The same formulas (3) and (4) for the phase and group velocities also hold in Schrödinger’s theory. For the simplest case of an individual electron (a particle with three degrees of freedom), in the first approximation, the wavelength of the phase wave is determined by formula (2). Only in the case of more complex systems (with many degrees of freedom) do Schrödinger’s phase waves prove to be waves propagating in a multidimensional configuration space, and not in that physical space in which the particles move.

The physical picture of Schrödinger’s theory, however, differs essentially from de Broglie’s theory. In Schrödinger’s theory the waves are not connected with particles, but replace them. In one of his first papers Schrödinger2 attempted to represent a particle as a “wave” packet—that is, as a group of waves occupying a certain part of space and moving as a whole. This attempt was unsuccessful—the wave packet always dispersed. Another physical formulation of Schrödinger’s was connected with the question of the meaning of the function \(\Psi\). Schrödinger proposed that the quantity \(|\Psi|^2\) be regarded as determining the density of charges in space. According to this conception of Schrödinger, the charge of the electron is not concentrated in a small volume, but is “smeared out” over all space; only in some places can the charge density \(|\Psi|^2\) have max-

simum. This conception is very close to replacing the consideration of the motion of the electron by the consideration of the waves of the function $\Psi$ in all space.

Finally, according to the views of Born, now supported by the majority of theorists, the whole meaning of the theory is statistical. One must not consider the motion of an individual electron; the equation of wave mechanics describes the motion of a multitude of electrons, and the quantity $|\Psi|^2$ gives only the probability of finding the electron on one or another soil at a definite moment. Here the wave character of the theory disappears to a certain degree. “Waves” exist only because the “random” equation determining the function $\Psi$ has a wave form.

For convenience of exposition we shall adopt a definite wave point of view, for example the point of view of de Broglie, and see what experimental facts speak in favor of the existence of phase waves. In other words, we shall consider those phenomena from the field of electronic phenomena which are poorly, or cannot at all be, explained from the standpoint of ordinary conceptions, and are well explained if it is admitted that a “phase wave” is associated with the electron, as is required, for example, by de Broglie’s theory.

§ 2. The class of phenomena in which, apparently, the action of phase waves is revealed may be called by the general name “scattering of electrons.” In scattering phenomena, electrons, according to the old terminology, collide with atoms of matter, entering into interaction either with the outer electron shell of the atom or with its nucleus. From the standpoint of phase waves it is natural to expect that, for a definite relation between the length of the phase wave and certain atomic or interatomic distances, diffraction of phase waves will occur, and this in turn will affect the character of the observed phenomenon of electron scattering. The experimental material which can be interpreted from this point of view belongs in part to an earlier period and was treated from the standpoint of phase waves, so to speak, accidentally. Naturally, here we do not have an especially vivid, striking picture. Much more valuable...

are those experiments that were carried out with the special aim of detecting the action of phase waves.

In experimental setups we always classify electrons according to their velocities, assigning the latter by the accelerating field \(V\) through which the electron has passed. It is essential to know the length of the phase wave associated with an electron of a definite velocity, expressed in waves. We have the expression for the kinetic energy of the electron

\[ \frac{1}{2}mv^2=eV. \tag{5} \]

Calculating from this \(v\) (the linear velocity) and substituting in (2), we obtain

\[ \lambda=\frac{h}{\sqrt{2emV}}. \tag{6} \]

If \(V\) is expressed in volts, and \(\lambda\) in ångströms, then this formula takes the form

\[ \lambda=\sqrt{\frac{150}{V}}=\frac{12.25}{\sqrt{V}}. \tag{7} \]

Consequently, electrons at 150 volts have a phase-wave length of \(1\ \mathring{A}\). Slow electrons with velocities of several tens of volts have a phase-wave length of several ångströms; electrons with velocities of the order of a thousand volts correspond to \(\lambda\) in tenths of an ångström; with velocities of tens of thousands of volts—to phase waves equivalent to the hardest X-rays or even to \(\gamma\)-rays.

§ 3. The Ramsauer Effect\(^1\). Ramsauer, and after him a number of other researchers, studied the scattering of slow electrons as they passed through various gases.

\(^1\) For the latest experimental data, as well as a complete summary of the literature, see the articles by Broche (E. Brüche). Ann. d. Phys. 81, 537, 1926; 83, 1065, 1927; 84, 279, 1927; also Handbuch d. Physik, Vol. XXIV, pp. 46–51, Berlin, Springer, 1927.

WAVE VIEWS ON THE NATURE OF MATTER AND EXPERIMENT

The effect observed was, so to speak, an integral one. The essence of Ramsauer’s method (the so-called “two-receiver method,” Zweikäfigmethode) can be understood from Fig. 1. Electrons, knocked out by light from the shield plate \(Z\), are drawn by a small accelerating field toward the first diaphragm \(B_1\), and then by a magnetic field are turned through a series of diaphragms \(B_2, B_3, B_4, B_5\), arranged along a circle of definite radius. In this way a strictly monochromatic beam of electrons (of one velocity) is selected. Through the diaphragm \(B_6\) these electrons enter the receiver \(A_1\), and then some of them, through the diaphragms \(B_7\) and \(B_8\), can enter the receiver \(A_2\). By connecting \(A_1\) and \(A_2\) together and measuring the current with an electrometer, we find the total number of electrons of the given velocity that have passed through the diaphragm \(B_5\). The current in the receiver \(A_2\) gives the number of electrons which have not undergone scattering on the path \(B_6, B_7\). From this one can determine also the number of electrons scattered per unit path. This number depends both on the external conditions (the temperature and pressure of the gas) and on the properties of its atoms or molecules. On the basis of comparatively simple considerations, from the magnitude of the scattering one can calculate the “effective cross-section” of the molecule: the larger this quantity, the greater the scattering. It is quite natural that the effective cross-section is a function of the velocity of the electrons.

Fig. 1.

Indeed, fast electrons (for example, the \(\beta\)-rays of radium) are scattered mainly as a result of interaction with the nucleus of the atom, whereas slow electrons are acted upon by the outer shells.

A remarkable phenomenon, first discovered by Ramsauer in noble gases, consists in the following (Fig. 2): as the velocity of the electrons is lowered, the effective cross section increases, reaches a maximum, and, with further lowering of the velocity, again decreases strongly, attaining values even smaller than those computed from the kinetic theory of gases. Thus, quite unexpectedly, the atoms of the noble gases (it was later shown that

Fig. 2. + Effective cross section A. × Effective cross section. K. λ Effective cross section, X. O. absorbing cross section H₂.

such maxima are observed in all gases) are extraordinarily “transparent” to electrons of very small velocities1.

The interpretation of this phenomenon—the maximum of the effective cross section—presents considerable difficulties if one attempts to explain it from the standpoint of the old views—

classical or quantum (such interpretations had been proposed1). Elsasser2 was the first to propose interpreting the Ramsauer effect, as well as a number of other phenomena of which we shall speak below, from the standpoint of diffraction of phase waves. In fact, at velocities corresponding to the maxima, e.g. for A and X (Fig. 2), the wavelengths of the phase waves turn out to be equal to 2.4 Å (A) and 5.1 Å (X). It is possible that here the conditions for diffraction of phase waves at atoms are exactly satisfied and, consequently, according to de Broglie, we also have a maximum in the scattering of electrons. On the curve this is manifested by an “apparent” increase in the effective cross-section of the atom. Elsasser pointed out that the Ramsauer curves are similar in appearance to the curves of light scattering by colloidal particles. And this phenomenon is typically diffractional. We see that the explanation of the Ramsauer phenomenon from the standpoint of phase waves is of a purely qualitative character.

§ 4. In ordinary phenomena of the diffraction of light, a very characteristic feature is a definite spatial distribution of intensity maxima, depending both on the wavelengths and on the properties of the “instrument” producing the diffraction. In the phenomena of scattering of electrons by helium atoms, observed by Dymond3, such spatial intensity maxima are present—maxima in the numbers of electrons scattered in known directions. This work, too, gives only qualitative indications of the validity of the conception of phase waves: the scattering conditions in this case are so complicated that it does not seem possible to make any quantitative calculations. A diagram of Dymond’s apparatus is given in Fig. 3.

Inside a vessel filled with helium at a known (very low) pressure there was placed, rotating on a ground joint, an “electron gun” \(A\), through whose slit \(S_1\) electrons could emerge at various angles to the axis of the system of slits \(S_2 S_3\).

Electrons were scattered by the gas in a small space in front of the slit \(S_2\) and could enter the box \(D\) through the slits \(S_2, S_3\). In this way electrons scattered at a definite angle to the primary beam were selected. In the box \(D\) they could be subjected to a velocity analysis by the usual magnetic method (by turning the magnetic field toward the Faraday cylinder \(E\)). Thus the apparatus made it possible to study both the distribution of electrons by velocities at a given scattering angle and initial velocity, and the distribution

Fig. 3.

of scattered electrons by angles (at a definite initial velocity).

The study of the velocity distribution showed that near the initial velocity of the electrons there is a very sharp maximum, i.e., the enormous majority of scattered electrons have almost the initial velocity. But, in addition, there is a sharp maximum lying approximately 20 volts from the initial velocity. Consequently, there are collisions accompanied by precisely such a loss of energy of the electron. This aspect of the investigation, however, is not essential for us.

Fig. 4 shows the obtained angular distribution of the electrons. Nos. 1–4 refer to electrons that have lost 20.5 volts, No. 6 to electrons whose velocity is equal to the initial velocity. In each drawing there is an indication of the initial velocity. In all the drawings one can see characteristic maxima in the numbers of electrons scattered in certain directions. The distribution of these maxima depends on the velocity, i.e.

Fig. 4.

on the length of the phase wave. An explanation of such a strange character of electron scattering, at least an unforced qualitative one, is possible only from the point of view of diffraction of phase waves. Quantitative calculations, as has already been indicated, are difficult.

§ 5. We now pass to the consideration of a series of works whose aim, at least in the final stage, was the direct establishment of the “wave nature” of the electron by establishing a direct analogy between certain diffraction and interference phenomena

in the region of X-rays and in the region of electron scattering, where, according to the assumption, phase waves act. All these works concern the scattering of electrons by solids.

Already comparatively long ago, Davisson and his collaborators were engaged in the scattering of electrons upon reflection from a solid surface. A paper on the scattering of low-velocity electrons by platinum and magnesium was published as early as 19231. Since then Davisson has continuously improved the method of investigation, and by 1927, when the investigations that yielded quite quantitative results were published, he had brought the experimental technique to an exceptionally high degree of perfection.

Even the work of Davisson and Kunsman showed that, in investigating the scattering of electrons by metals, an essential condition for success is a very high vacuum and careful degassing of all the metallic parts of the apparatus. For this purpose, in Davisson’s experiments the pumping is carried out by a powerful three-stage pump, with prolonged heating of the entire apparatus and subsequent absorption of the remaining gas by coconut charcoal cooled with liquid air. The most critical metallic parts of the apparatus were heated by cathode bombardment; for this purpose additional hot cathodes were placed in various parts of the apparatus. The vacuum attained by Davisson in his experiments is estimated by him at \(10^{-8}\) mm, and the apparatus can be sealed off from the pump. Such a perfect vacuum, according to Davisson, is necessary in order to make it possible to observe scattered electrons with velocities differing little from the initial ones. The point is that the overwhelming majority of scattered electrons turn out to be “secondary” and have velocities considerably smaller than the initial one. The very detection of the insignificant number of “primary” scattered electrons is possible only under an extreme vacuum. Already in the work of Davisson and Kunsman, measures were taken so that the measur—

the distribution over angles of electrons whose velocities differed from the initial one by no more than 10%. We shall not dwell on the details of this apparatus, since the most recent apparatus was an improvement on the original one. Already in this work, in which electrons with velocities up to 1500 volts were studied, maxima—though rather blurred—were obtained in the study of the angular distribution. Elsasser (loc. cit.) proposed interpreting them from the standpoint of the diffraction of phase waves. It is curious to note that the problem which Davisson set himself when beginning the investigation of electron scattering had nothing in common with the circle of ideas that interests us.

We now pass to the consideration of the investigation by Davisson and Germer1, which is the fundamental, so to speak classical, one for the whole field under consideration.

If one regards electron beams of different velocities as something “wave-like” (phase waves are associated with electrons), then the natural thought arises of establishing an analogy between electron phenomena and certain diffraction phenomena of X-rays of approximately the same wavelength as the corresponding phase waves. When X-rays are reflected from a single crystal, we have, as is well known, a diffraction pattern corresponding to a Laue radiogram (white radiation): a series of spots arranged in a definite way (depending on the structure of the crystal). Approximately the same should be expected in the scattering of electrons by a single crystal. By varying the velocities of the electrons and studying the number of electrons scattered in various directions in space, we obtain a series of maxima corresponding to the spots of a Laue radiogram. Along this path proceeded the investigation of Davisson and Germer. Chance, according to the authors’ statement, forced them to undertake this investigation. Namely, in working with a nickel plate a definite spatial distribution of electrons was observed. An accidental mishap made it necessary to heat the plate again in order to degas it.

At the same time crystallization occurred; large crystals appeared in the plate, and the pattern of distribution of the scattered electrons changed sharply: a number of different maxima appeared.

The apparatus constructed for the special study of the scattering of electrons by nickel single crystals is shown in Fig. 5. Electrons from the “gun” \(G\), closed on all sides, after passing through a series of narrow diaphragms and acquiring the corresponding velocity, fell normally upon the nickel single-crystal plate \(T\), placed on a special rotating table. The scattered electrons could enter the double Faraday

Fig. 5.

cylinder \(C\). The receiver \(C\) is provided with special lugs fastened to an axle passing through the plane of the plate; consequently, the receiver can rotate about this axle. The inner part of the receiver, by means of a well-insulated lead-in, was connected to a sensitive galvanometer. Such an arrangement made it possible, by simply tilting the apparatus, to change the angle “of latitude” at which the scattering was observed (the collector is suspended like a pendulum). This angle \(\theta\) was varied within the limits from \(15^\circ\) to \(85^\circ\), measured from the initial direction of the electrons. The angle was read by means of an index and a limb placed outside the casing that enclosed the entire space where the scattering took place. The outer part of the gun, the plate, the outer part of the receiver, and the casing—all

was at one and the same potential; consequently, the scattering took place in field-free space. By rotating the apparatus about an axis normal to the surface of the nickel plate, it was possible to vary the azimuth in which the scattering was observed. The axis of the stand could rotate, and to it was attached a heavy pendulum \(P\) with an index, which made it possible to read the azimuth on a specially graduated circle. All the dimensions of the apparatus were very small. For example, the distance from the aperture

Fig. 6.

of the gun to the crystal surface was \(7\) mm, and from the crystal to the receiver \(11\) mm. The entire apparatus was enclosed in a shell of “Pyrex” glass and evacuated with all the precautions discussed above. The vacuum was of the order of \(10^{-8}\) mm. The crystal was cut parallel to the plane \((111)\); the cut was etched and polished. Microphotography established the presence of large crystals. The nickel plate could also be heated by electron bombardment, since an additional cathode was placed behind it. By means of special contacts it could be connected to a galvanometer.

Consideration of the arrangement of nickel atoms in planes parallel to the surface (111) leads to the conclusion that, when the azimuth is changed, the scattering pattern must repeat with a period of \(\frac{2\pi}{3}\) (there are, for example, three directions (111) in this plane at angles of \(120^\circ\)).

A special investigation of the velocities made it possible, from the total mass of scattered electrons, to single out the electrons of “full velocity”

Fig. 7.
Vertical axis: collector current.
Horizontal axis: electron velocity, \(V\).
Curves marked: \(35^\circ\), \(40^\circ\), \(45^\circ\), \(50^\circ\), \(55^\circ\), \(60^\circ\), \(65^\circ\), \(70^\circ\).

and thus to separate the inevitable “background.” In all investigations of the angular distribution of electrons, the potential of the inner part of the receiver was selected so as to deal only with electrons of full velocity. The “bombardment” current was of the order of \(10^{-6}\) A, while the current into the receiver was of the order of \(10^{-10}\) and was measured by a sensitive galvanometer.

Figs. 6, 7, 8 convey the general character of the results of the study of the spatial distribution. In Fig. 6

it is evident that there is a gradual increase of the maximum corresponding to 54 volts, with a gradual change of the angle \(\theta\) (latitude).

Fig. 8.

This maximum proves to be most pronounced at \(\theta = 50^\circ\) [each curve corresponds to a definite angle \(\theta\); the azimuth is the direction \((111)\)]. Fig. 10 gives the dependence on azi-

...mutually, and a clear periodicity of \(\frac{2\pi}{3}\) is visible. The curves correspond to maxima at 65 and 54 volts.

The investigation of the aggregate of these most developed maxima in connection with their spatial arrangement gives us an “electrogram” of the crystal [the lengths of the phase waves are calculated by formula (7)].

For a quantitative comparison of the results with the theory, it is necessary to calculate, from the arrangement of the atoms in the crystal, in what directions and at what wavelengths interference maxima should be obtained, using the same rules of calculation as for X-rays. One could, for example, calculate the complete Laue radiogram for a nickel crystal. Davisson uses a simpler and more visual method.

The surface of a crystal with the atoms situated on it, as is easy to see, may be regarded as a collection of several plane diffraction gratings; the rows of atoms represent the rulings of the grating (a monoatomic layer). Depending on the direction in which the rays fall, we shall have several gratings with different constants \(d\), which are easy to calculate. Diffraction is determined by the usual equation of a plane grating:

\[ n\lambda=d\sin\theta . \tag{8} \]

This manner of consideration has great advantages in the case of electrons of low velocities, which are strongly absorbed by the metal, so that it makes sense to speak of reflection in a layer one atom thick. This applies especially to beams falling at a large angle \(\theta\) (close to grazing incidence).

Indeed, a series of beams of this kind was found. The criterion that a beam has been reflected from a plane grating is the following circumstance. By (7), \(\lambda\) is inversely proportional to \(\sqrt{V}\); consequently, for constant \(n\) and \(d\) (a given order of reflection and a given grating), by (8) there must be

\[ \sqrt{V}\cdot\sin\theta=\mathrm{const}. \tag{9} \]

Condition (9) proved to be valid for some beams, for example “54,” within considerable limits; for a number of others it proved to be valid only within very narrow limits and, consequently, these beams must be regarded as reflected from a space lattice.

A space lattice may be regarded as a series of plane lattices superposed one upon another. A ray reflected from a space lattice is obtained by summing rays reflected from the individual plane lattices (strictly speaking, one should speak of amplitudes). In the general case we shall thereby obtain a weakening of the intensity, practically to zero, since we shall have a completely random distribution of phases in the reflected rays and they will extinguish one another. Only under certain strictly defined conditions will the rays be in identical phases, and we shall obtain an increase in intensity. Therefore, on the straight line expressing the dependence of \(\lambda\) on \(\sin \theta\) [for given \(n\) and \(d\), see formula (8)] for “space” rays, we shall obtain only several discrete points (see Fig. 9). The electron beams actually observed are marked by black dots.

Fig. 9.

The general character of their arrangement agrees with the theory, but they are all displaced to the left and downward; the displacement to the left can probably be explained by an incorrect reading of the angles. The downward displacement, however, must correspond to a decrease of the lattice constant in a certain respect. Bethe1 attempted to explain the apparent decrease of the lattice constant by refraction of the phase waves in the crystal, calculating, on the basis of Davisson’s data and simple considerations,

associated with Schrödinger’s theory, of the change of the refractive index with wavelength. However, as Davisson points out, in the displacements of the black spots of Fig. 9 relative to the white ones there is no particular regularity, and this question cannot be regarded as settled. It should be noted that, although the general theoretical character of the pattern does correspond to what is observed experimentally, a number of beams which theoretically ought to exist have not been found in experiment. In addition, there turned out to be superfluous beams, apparently explained by the presence of gas on the surface. In general, the role of the gas in these phenomena, as follows from Davisson’s experiments, is enormous. Only with careful degassing of the plate do the beams stand out quite sharply. In the presence of gas they are greatly weakened and sometimes even disappear altogether. On the other hand, traces of gas cause the appearance of new beams (superfluous ones), which disappear when the plate is degassed.

The general impression from Davisson’s work is a strong confirmation of the point of view of phase waves.

§ 6. Davisson’s investigations have provided the electron analogue of Laue’s roentgenogram; the works which we shall consider in this paragraph give electron analogues of Debye’s roentgenogram.

It is known that the latter are obtained when monochromatic X-ray light falls on a system of randomly oriented crystals (for example, in a crystalline powder or in a polycrystalline piece of metal, etc.); on the roentgenogram we obtain a series of discrete continuous circles. Each circle corresponds to reflection from a definite crystallographic plane, the law of reflection being determined by the well-known Bragg condition for space lattices:

\[ n\lambda = 2d\sin\theta, \tag{10} \]

where \(d\) is the lattice constant, a quantity determined by the edge of the elementary cube (in the general case, parallelepiped) of the lattice and by the totality of the indices \(h, k, l\) of the given

crystallographic plane. Thus the number of rings will be determined by the number of possible reflections from different crystallographic planes of the given kind of crystals. From the size (diameter) of the rings, knowing at what distance from the scattering aggregate of crystals the plate is placed, it is easy to calculate \(d\), if we know \(\lambda\).

Similar phenomena should be expected also in the scattering of electrons by a polycrystalline aggregate. Of the works that established this analogy, we shall first of all dwell on Thomson’s investigation1. This investigator worked chiefly with very fast electrons, i.e. very short phase waves (in contrast to Davisson, who worked with slow electrons). For example, electrons at \(25\,000\) volts correspond to a wavelength of

\(0.075\ \mathring{A}\). These are phase waves corresponding to very short X-ray waves or even to \(\gamma\)-rays. In general, the voltages used by Thomson lie between \(17\,500\) volts and \(56\,500\) volts. To calculate the wavelength in the case of such fast electrons one should use the relativistic formula:

\[ \lambda \ \frac{h\sqrt{1-\frac{v^{2}}{c^{2}}}}{m_{0}v} \]

instead of the former formula (2). The correction, however, amounts to only \(3\%\).

Thomson’s apparatus is shown in Fig. 10. Between the cathode, situated in part of the tube \(A\), and the slit \(B\) (grounded), a high voltage was applied. After passing through slit \(B\) (6 cm long), a narrow beam of electrons fell on a thin foil fastened in holder \(C\). The electrons scattered in the foil traveled a distance of \(32.5\) cm and fell either on a willemite screen \(E\), which served for visual observa-

suspension, or onto the photographic plate \(D\) lowered from above.

Fig. 10.

The foil used was very thin (of the order of \(10^{-5}\) cm) and was obtained by etching a thicker foil. Although fast electrons could pass even through thicker foils, at Thomson’s suggestion, when scattered by a significant number of layers the result would have been so complicated a pattern that nothing could have been made out on the plate. Aluminum, gold, celluloid, and a certain unknown substance \(X\), accidentally obtained when washing the plate with aqua regia and at first taken for platinum, were investigated. Films of metals were deposited, in some experiments on celluloid. In all cases distinct additional rings were observed. Fig. 11 reproduces the photograph for gold. The number of rings in almost all cases agreed with theoretical expectations. The requirement following from the theory

Fig. 11.

\(D\sqrt{V}=\mathrm{const}\), where \(D\) is the diameter of the circle, was also well satisfied. In computing from the observed diameters of the circles the constants of the lattices, in all cases they came out 5 percent smaller than the values found with the aid of X-rays. For example, in the case of gold \(a=3.8\ \text{Å}\) instead of \(a=4.065\ \text{Å}\). The discrepancy is \(6\frac{1}{2}\%\). It is possible that the cause of the discrepancy is a systematic error in the determination of the voltage.

Similar experiments with slow electrons, in which photographs with circles were likewise obtained, are being carried out by Rupp in Göttingen1.

The author2 of this article investigated, by an electrical method, the scattering of electrons on passing through a very thin aluminum foil, placing behind the foil a receiver with a ring circuit. The angle of scattering always remained constant \([\theta=4^\circ45' \text{ in } \varphi\text{-le }(10)]\). Several current maxima were obtained at definite voltages (for different values of \(d\)).

For example, for the plane (111) a maximum was obtained at \(V=1100\) volts, i.e. \(\lambda=0.37\ \text{Å}\), whereas under these conditions Bragg’s law gives \(\lambda=0.38\ \text{Å}\) or \(V=1040\) volts.

As we see, this class of work too leads to a complete confirmation of the wave point of view.

§ 7. The experimental material we have considered, confirming the wave point of view on the nature of matter, belongs wholly to the field of diffraction phenomena.

Among the further tasks in this field, the first is the search for further analogies in the field of diffraction phenomena, for example the detection of diffracti—

... maxima upon reflection from an ordinary grating.

Next, it is necessary to seek electronic analogues of more complex optical phenomena and, finally, such complex and fundamental questions arise as the question of the coherence of the phase waves of electrons.

The posing of these questions obviously makes sense if one considers that the available experimental material proves the “reality” of phase waves.

1. According to information kindly communicated by A. K. Arsen’eva. (Note during proofreading. Rupp’s work has now been published. Ann. d. Phys. 1928, No. 8, p. 981). ↩↩↩↩↩↩↩↩↩

2. P. Tartakovskii. Reports of the Academy of Sciences of the USSR, 1928. ↩↩↩↩

3. Dymond. Phys. Rev. 29, 433, 1927. ↩