Abstract
A paper read at a meeting of the Franklin Institute in Philadelphia on March 21, 1928.
Full Text
ARE ELECTRONS WAVES?
K. Davisson, New York1
The very title I have chosen for today’s communication—“Are Electrons Waves?”—shows that certain doubts have arisen concerning the nature of electrons. And indeed, phenomena have been discovered in which electrons behave not at all like particles, but rather like waves.
As an example of such behavior I shall describe the simplest type of those experiments which Dr. Germer and I have been carrying out over the past several months. We directed a narrow beam of electrons onto a face of a nickel crystal and found that, under certain conditions, a sharply defined beam of electrons begins to emerge from the crystal in the direction of regular reflection, i.e. in accordance with the law “the angle of reflection is equal to the angle of incidence.”
At first glance it may seem that there is nothing strange in this. Indeed, why should electrons not undergo regular reflection from the surface of a metal? We know that Newton and other supporters of the corpuscular theory of light were not at all troubled by the fact of the regular reflection of light from a plane mirror. This phenomenon can be easily explained. It is well known that, in an elastic impact between a particle and a plane surface, the particle rebounds according to the law of regular reflection,—
hence, according to the corpuscular theory, the regular reflection of light followed. Why, then, can the regular reflection of electrons not be explained in the same way?
The point here is that, in giving such a description of reflection, the adherents of the corpuscular theory were in more favorable circumstances than we are now. They were not at all bound by ideas about the dimensions of corpuscles and knew nothing about the structure of the surface of a metal. We, however, have grounds for believing that the diameter of the electron is a quantity of the order of \(10^{-13}\) cm. We know, furthermore, that the order of magnitude of the diameter of an atom is \(10^{-8}\) cm, and that the smallest distance between atoms in a nickel crystal is \(2.48 \cdot 10^{-8}\) cm. If \(10^{-13}\) cm is taken as the unit of length, then the diameter of the electron will be equal to 1, the diameter of the nickel atom to 100,000, and the smallest distance between atoms in a nickel crystal to approximately 250,000.
It is clear that to give an explanation of the regular reflection of such small particles as electrons from a surface formed by such relatively large bodies as atoms is very difficult. If we were to fire shot at a pyramid made of cannonballs, then, of course, one could not expect the regular reflection of the shot from the surface of the pyramid. A surface formed by cannonballs is much too coarsely rough to reflect regularly such small particles as pellets.
The analogy just given is not a good one, because one cannot, of course, liken the rebound of an electron from the surface of an atom to the rebound of a pellet from a cannonball. We are accustomed to regard the atom rather as a solar system: a massive “sun” nucleus, surrounded by planetary electrons moving in closed orbits. From the standpoint of these ideas, the electron must be likened to a comet entering a region rather densely filled with solar systems.
It might seem that an electron, having struck an atom near the surface of the metal, turns around it like a comet and is thrown back from the metal without loss of energy. In
this, the direction of departure of the electron ought, it would seem, to be a matter of chance and to depend only on the collision of the electron with the given atom; at the very least it is quite unclear what role neighboring atoms could play here. Meanwhile it turns out that fast-moving scattered electrons move preferentially in the direction of regular reflection from the plane forming the surface of the metal. The position of this plane is determined by at least three atoms; consequently, the direction of departure of the electron depends not on one, but at least on three atoms.
It may be said without exaggeration that, within the framework of the usual conceptions of the interaction of atoms and electrons, the regular reflection of electrons from the surface of a metal is inexplicable.
Of course, if electrons were waves, then all these difficulties would disappear. We can, after all, explain the regular reflection of light and X-rays—in exactly the same way one could explain the regular reflection of electrons, if they were not particles but waves. But although this observation is true, it is hardly especially valuable. It is like someone who, seeing a rabbit climbing a tree, were to say: “True, for a rabbit this is somewhat strange, but in the end there is nothing to be surprised at. Cats can climb trees; so if the rabbit were a cat, we would understand its behavior perfectly.” The only way out here remains to admit that what we have hitherto taken for a rabbit is in fact not a rabbit at all, but a cat. But is it possible that we are making a similar mistake with respect to the electron? Is it possible that we have been mistaken all along in considering electrons to be particles, whereas in reality they are waves? There is, of course, no need for me to repeat here before you all those arguments that allow us to think—or, more sharply, to assert—that electrons really are particles.
These arguments are quite numerous; let us note that among them is the very method by which we ob-
...discovered the correctly reflected beam. The correctly reflected beam is detected in the following way: a small cylinder is moved in front of the crystal, and it turns out that when this cylinder coincides with the direction of correct reflection, the greatest number of electrons enters it.
The diagram of the experimental apparatus is shown in Fig. 1.
A filament placed in a metal vessel constitutes an “electron gun,” delivering a constant stream of electrons. The velocity of the electrons is known, and it can
Fig. 1. Experimental apparatus for investigating the scattering of electrons by a crystal and a typical curve showing the beam of correctly reflected electrons.—Bombarding potential 83 volts; angle of incidence 30 degrees.
be given any value depending on the potential difference between the filament and the walls of the vessel. This stream is directed onto the crystal, and electrons with various velocities rebound from the bombarded surface.
In order to investigate how they are distributed in different directions, we move the collector, i.e. our cylinder, and observe how many electrons enter it at its various positions. To enter the inner vessel, the electrons must pass through an aperture in the outer vessel. If they succeed in doing so, they enter the galvanometer, whose deflection indicates the number of electrons that have entered the vessel.
The same method, in essence, could have been applied in measuring the scattering of shot by a pyramid of cannonballs. It is fundamentally different from the methods used in studying the scattering of X-rays.
In the observations, the collector is moved in front of the crystal and the curve of the current in the collector is plotted as a function of the angle. On the right-hand side of Fig. 1 such a curve is shown for an angle of incidence of 30° and a bombarding potential of 83 V. On this curve a sharp hump is clearly visible at the point corresponding to the direction of regular reflection. The question arises about the electrons that fly out of the crystal in other directions. It turns out that almost all of them belong to the class of slow secondary electrons, whereas the majority of the regularly reflected electrons have the same velocities as the incident electrons.
There is no doubt that the incident electrons, in accordance with the position of the surface of the crystal, prefer to move in the direction of regular reflection.
The following experiment, which I shall briefly describe, is even simpler than the first. We directed a beam of electrons onto a sheet of ordinary nickel—that is, not onto one large crystal, but onto a multitude of small crystallites. Under these circumstances, never, under any conditions, was even a trace of regular reflection observed. Thus, from a sheet of ordinary polycrystalline nickel, the electrons are reflected irregularly.
This fact—that electrons are regularly reflected only from the face of a crystal—is very curious. The same is true of X-rays. X-rays undergo regular reflection only from the face of a crystal, and not from a polycrystalline mirror. It is known that this difference in the behavior of ordinary light and X-rays is explained by the difference in their wavelengths. The wavelengths of light are large in comparison with the distance between atoms in solids, whereas the wavelengths of X-rays are comparable with these distances.
Both of these results—the regular reflection of electrons from the surface of a crystal and the absence of such reflection
... from a polycrystalline surface—would be understandable if the electrons were waves whose length was comparable with the distances between atoms in solids.
Let us recall that the reflection of X-rays is characterized by a very sharp selectivity. When a beam of monochromatic X-rays is directed onto the surface of a crystal, the intensity of the beam reflected in a given direction is almost always equal to zero, unless the wavelength of the incident wave lies very close to one of a series of definite discrete values. The situation here is as though we had a mirror that reflected red light of a definite wavelength and blue light of a definite wavelength, but did not reflect light of any of the intermediate wavelengths.
This suggests one interesting experiment. Electrons resemble X-rays in the sense that they are regularly reflected from the face of a crystal and irregularly reflected from a polycrystalline surface. Perhaps they resemble X-rays also in the selectivity of reflection? One might expect, for example, that if reflection in electrons proceeds in fact in the same way as in X-rays, then it should possess the property of selectivity with respect to the speed of bombardment. Remarkably, this proves to be precisely the case. Measuring the intensity of the reflected beam as a function of the speed of bombardment, we found that, as this speed increased, the intensity passed through a series of maxima. The curve characterizing this behavior is shown in Fig. 2.
Along its ordinates are plotted the intensities of the reflected beam, and along the abscissae—the square roots of the bombarding potential, proportional to the velocities of the electrons in the incident beam. The observations were made for an angle of incidence of \(10^\circ\); the curve on the left side of the drawing gives the reflected beam at the second maximum of the intensity curve.
The phenomenon of selective reflection of X-rays has been very thoroughly investigated and explained. In explaining it, one must constantly make use of the fact of wav—
new nature of X-rays. It is precisely this phenomenon that gives us the most convenient means for measuring the wavelength of X-rays. Therefore it is very significant that, in this respect as well, the reflection of electrons recalls the reflection of X-rays.
I shall try, in a few minutes, briefly to set forth the theory of the reflection of X-rays from a crystal, which explains its selective character.
When, as shown on the left-hand side of Fig. 3, a beam of X-rays falls on a single layer of atoms, it passes through this layer with only slightly weakened intensity. In doing so, forced oscillations arise in the atoms, and they begin to emit spherical waves that stand in definite phase relations to one another. The result of their superposition is a wave moving away from the crystal in the direction of regular reflection. Thus the reflection of X-rays from a single layer of atoms is not selective.
Fig. 2. Selective character of the reflection of electrons. Angle of incidence 10 degrees.
Selectivity appears only in reflection from an entire series of parallel layers of atoms, as we have it in a crystal. This case is illustrated on the right-hand side of the figure. The reflected beams arising in the various layers are superposed upon one another, and the intensity of the resulting beam has a sharp maximum when these elementary waves emerge from the crystal in phase, as is shown in the figure. Obviously, such a maximum is attain-
...occurs in the case when the path difference from plane \(AA\) to plane \(BB\) for different layers of atoms differs by an integral number of wavelengths. This path difference is equal to twice the product of the distance between two neighboring layers of atoms by the cosine of the angle of incidence. Consequently, the intensity of the reflected beam has a maximum when
\[ 2d \cos \theta = n\lambda . \]
Fig. 3. Diagram illustrating the selective reflection of X-rays from a crystal.
This condition may be expressed in words as follows: the intensity of the reflected beam has a maximum when the wavelength of the incident beam takes one of the values
\[ \lambda = \frac{1}{n} 2d \cos \theta \]
or when its reciprocal has one of the values
\[ \frac{1}{\lambda} = n \frac{1}{2d} \frac{1}{\cos \theta}. \]
Thus the dependence of the intensity of the reflected beam on the quantity \(\frac{1}{\lambda}\) must be expressed by a curve with maxima equally spaced from one another. A curve of this kind is given in the upper half of Fig. 4.
In the lower half of the figure, for comparison, a curve is given for the intensity of the reflected beam of electrons as a function of \(V^{1/2}\), i.e. of the square root of the bombarding potential.
As we see, the maxima of this curve are separated by almost equal intervals. On this basis one may say,
Fig. 4. Selective reflection of X-rays and selective reflection of electrons.
that the phenomena connected with the reflection of electrons, including selectivity, would be explained with sufficient accuracy if it were possible to regard electrons as waves whose wavelength is inversely proportional to the square root of the bombarding potential, i.e. inversely proportional to the velocity of the electrons. Evidently, such an explanation would not be absolutely exact, since the intervals between the maxima on the electron curve are only approximately equal to one another. Nevertheless it would explain the experiments quite well.
Thus we have almost reached the point of beginning to calculate the wavelengths of electrons—knowing perfectly well that electrons are particles. Our exposition has reached the point where one must adopt a definite standpoint on this question, and for the time being I propose not to give up the fact known to us, that electrons are particles, but simply to accept that they behave as if they were waves—more precisely, that we can describe the phenomena we observe by taking electrons to be waves, and that we do not know how to do this if we take them to be particles. Let us, for the time being, adopt this point of view and see how long it can still be maintained.
In the scattering of X-rays by a crystal, the reflected beams arise not only in the direction of regular reflection, but also in other directions. This becomes understandable if one imagines that the atoms in the crystal may be regarded as arranged not only in planes parallel to the surface of the crystal, but also in planes not parallel to this surface; moreover, with respect to the X-rays all these rows of atomic planes are equivalent (see Fig. 5).
If a beam of X-rays falls on the surface of a crystal at an angle \(\theta_1\), then the regularly reflected beam appears when the wavelength has one of the values
\[ \lambda=\frac{1}{n}\,2d_1\cos\theta_1. \]
This beam is due to regular reflection from atomic planes parallel to the surface of the crystal and is usually called the Bragg reflected beam. But, as shown in the figure, the atoms may be imagined as arranged also in other planes, which likewise give reflected beams. For example, we shall obtain a beam regularly reflected from the atomic planes shown in Fig. 5 at the upper right, when the wavelength has one of the values
\[ \lambda=\frac{1}{n}\,2d_2\cos\theta_2. \]
Such beams are usually called Laue diffraction beams.
We have seen that electrons resemble X-rays in that they are regularly reflected from the surface of a crystal and, in doing so, exhibit the property of selectivity. The question arises: does this similarity extend to the formation of electron diffraction beams? Indeed, thanks to
Fig. 5. Reflection of X-rays from various planes in a crystal: illustration of the formation of Laue diffraction beams.
a peculiar accident, we observed these diffraction beams about a year before we began to look for regularly reflected beams.
These diffraction beams are not as easy to describe as the reflected beams. There is a substantial difference between the behavior of electron beams and the behavior of X-rays, so that at first glance it may seem that electrons do not behave all that well from the wave point of view. It turns out, however, that this difference can be explained and that the diffraction data
lead to definite numerical values of the electronic wavelengths. I shall first describe the conditions under which the observations were made, and the experimental results which would have been expected if we had been dealing with X-rays, and then pass on to the results obtained in the observation of electrons.
Let us first recall how the atoms are arranged in a nickel crystal. Nickel forms crystals of the face-centered cubic type. The basis of its structure is a cube—with edge \(3.51\ \mathring{\mathrm A}\)—in which there is one atom at each vertex and at the center of each face. The large cube shown
Fig. 6. Schematic representation of a nickel crystal: a cube with face-centered faces.
on the left side of Fig. 6 consists of 27 such small cubes. In the figure only the atoms situated on the surface of this large cube are shown. In what follows, this large cube will represent in the drawings the nickel crystal with which we began our experiments. The top of the crystal was cut off at right angles to one of its diagonals, so that a triangular surface was formed, shown in the center of the figure. Onto this surface a beam of electrons was directed perpendicular to it (see the right-hand part of the figure), and the number of electrons emerging from the crystal was measured as a function of the direction and of the velocity of bombardment. A diagram of the experimental arrangement itself is given in Fig. 7. The collector could be moved in one plane—the plane of the drawing—and the crystal rotated
tate about the vertical axis, so that any azimuth of the crystal could be brought into the plane of rotation of the collector.
It is clear that the crystal has symmetry of the 3rd order. If, for example, we measure the beam emerging from the crystal when one of the vertices of the triangle coincides with the plane of the collector, then, naturally, one should expect to find the same beam when the crystal is rotated by \(120^\circ\), so that
Fig. 7. Schematic representation of the experimental setup for studying electron diffraction.
another of its vertices falls into the plane of the collector, or when it is rotated by \(240^\circ\). We shall call the azimuths of the crystal that include one of the vertices of the triangle azimuths A; azimuths that include one of the midpoints of its sides, azimuths B; and, finally, azimuths parallel to one of the sides of the triangle, azimuths C.
Figure 8 shows a section of the crystal by the plane of the A- and B-azimuths. The circles denote chains of atoms of the crystal perpendicular to the plane of the drawing. The crystal may be regarded as built up of atomic planes parallel to its surface. The distance between these planes is equal to \(2.03\ \text{Å}\); the distance between atomic chains
in each plane is \(2.15\ \text{Å}\). It should be noted that the chains of atoms in a given plane are not situated directly beneath the corresponding chains in the preceding plane, but are displaced to the right with respect to the latter by a distance equal to approximately \(1/3\) of the distance between the chains.
After this brief description of the crystal, we may now proceed to the calculation of the wavelengths and the positions of those diffracted beams of X-rays which may appear in the planes of the A- and B-azimuths. The atoms may be regarded as arranged in the planes shown on the left side of Fig. 9. The distance between these planes is \(1.24\ \text{Å}\). The angle of incidence is \(35^\circ\); consequently, the diffracted beam of X-rays will appear at an angle \(\theta' = 70^\circ\), if the wavelength has one of the values
Fig. 8. Section of a nickel crystal by the plane of the A- and B-azimuths.
\[ \frac{1}{n}2d\cos\theta = \frac{1}{n}\,2\cdot 1.24\cos 35^\circ = \frac{2.87}{n}\ \text{Å}. \]
The three beams in the A-azimuth shown in the upper part of the figure have the greatest values of the quantity \(2d\cos\theta\), i.e. the greatest wavelengths. In the lower part of the figure are shown the beams in the B-azimuths with the greatest wavelengths. From this figure it is seen that, if the wavelength of the incident beam of X-rays is gradually decreased, the diffracted beams will appear in the following order: first at \(70^\circ\) in azimuth A, then at \(59^\circ\) in azimuth B, then at \(44^\circ\) in azimuth A, and, finally, at \(39^\circ\) in azimuth B.
Thus would the picture be if we were dealing with X-rays. Judging by the regular reflection of elec-
Fig. 9. Principal diffracted beams that should appear in the A- and B-azimuths if the incident beam were an X-ray beam.
trons from atomic planes parallel to the surface of the crystal, one might expect that, at certain
at bombardment velocities, and the electrons will give diffraction beams in the directions just indicated. However, although at certain critical bombardment velocities
Fig. 10. Curves showing the growth and disappearance of the “54-volt” diffraction beam of electrons in azimuth A. The surface of the crystal is clean.
the diffraction beams of electrons do indeed begin to appear from the crystal in the principal azimuthal planes, the direction of these beams coincides with none of the directions of the Laue beams. They seem not to undergo regular reflection from any of the principal atomic planes. As the bombardment velocity is increased from zero, the first of these beams appears in the A-azimuth, but
Fig. 11. The “54-volt” diffraction beam, weakened as a result of contamination of the crystal surface. The curve below shows maxima in the A azimuths.
not at 70° or at 44°, but at 50°; the second appears in the B-azimuth, but not at 59° or 39°, rather at 44°. The curves for the first of these beams are shown in Fig. 10. The beam first appears at 40 V, disappears at 70 V, and reaches its greatest intensity at a bombarding potential of 54 V. Fig. 11 shows the same beam, somewhat weakened owing to adsorption of gas by the surface of the crystal. The curve lying lower is obtained if the collector is set along the axis of the beam at its maximum and the current in it is measured while rotating the crystal about the vertical axis. From it it is evident that, as was to be expected, a sharp maximum appears in each of the A-azimuths.
Fig. 12. Growth and disappearance of the “65 volt” beam in the B-azimuth. The surface is contaminated with gas. The lower curve shows maxima in the B-azimuths.
The corresponding curves for the first beam in the B-azimuth are shown in Fig. 12. The maximum intensity is reached at 65 V in the direction at 44°. There is an interesting possibility of explaining this discrepancy between the directions of the diffraction beams of electrons and of X-rays. It is based on the conception of the crystal as a refracting medium for electrons, i.e. a medium with an index of refraction different from unity.
I shall try briefly to develop this idea. Let us imagine a beam of radiation falling normally on the surface of a crystal with refractive index \(\mu\). The wavelength of the beam outside the crystal is \(\lambda\), inside it \(\lambda'=\dfrac{\lambda}{\mu}\). On entering the crystal, the beam does not change its direction and encounters a certain series of atomic planes at an angle \(\theta\) (Fig. 13). If
\[ \lambda'=\frac{1}{n}\,2d\cos\theta, \]
then a regularly reflected beam begins to move from these planes. It falls on the surface of the crystal at an angle \(2\theta\), but, passing through it, is refracted and emerges from the crystal in the direction \(\theta'\). Thus the beam emerging from the crystal does not appear to have been reflected from any of the principal atomic planes.
\[ \mu=\frac{\lambda}{\lambda'}=\frac{\sin\theta'}{\sin 2\theta} \]
\[ \lambda'=\lambda\,\frac{\sin 2\theta}{\sin\theta'}=\frac{1}{n}(2d\cos\theta) \]
\[ \lambda=\frac{1}{n}\left(\frac{2d\cos\theta}{\sin 2\theta}\right)\sin\theta' \]
\[ \lambda=\frac{1}{n}\left(\frac{2d\cos\theta}{2\sin\theta\cos\theta}\right)\sin\theta' \]
\[ \lambda=\frac{1}{n}\left(\frac{d}{\sin\theta}\right)\sin\theta'. \]
But
\[ \frac{d}{\sin\theta}=D. \]
Hence
\[ \lambda=\frac{1}{n}D\sin\theta' \]
Fig. 13. Diffraction by a refracting crystal.
Now I ask you to follow the brief mathematical derivation on the left side of Fig. 13, which leads to an interesting and important relation. We shall regard electron radiation as light and write:
\[ \mu=\frac{\lambda}{\lambda'}=\frac{\sin\theta'}{\sin 2\theta}. \]
Solving this with respect to \(\lambda'\), we have \(\lambda'=\lambda \dfrac{\sin 2\theta}{\sin \theta'}\). But according to Bragg’s law,
\[ \lambda'=\frac{2d\cos\theta}{n}. \]
Equating these two expressions for \(\lambda'\) and solving for \(\lambda\), we obtain:
\[ \lambda=\frac{1}{n}\frac{2d\cos\theta}{\sin 2\theta}\sin\theta'. \]
Replacing \(\sin 2\theta\) by \(2\sin\theta\cos\theta\) and cancelling by \(2\cos\theta\), we have from this
\[ \lambda=\frac{1}{n}\frac{d}{\sin\theta}\sin\theta'. \]
But from the construction in Fig. 13 it is evident that \(d/\sin\theta=D\)—the distance between two neighboring chains of atoms on the surface of the crystal1. Hence
\[ \lambda=\frac{1}{n}D\sin\theta'. \]
This relation is useful because, from it, knowing the distance between the chains of atoms on the surface of the crystal and the angle at which the given diffracted beam emerges, one can calculate its wavelength (if the order \(n\) is known). In doing so it is not necessary to know either the atomic plane from which the beam is reflected or the refractive index of the crystal.
The idea of treating a crystal as a refracting medium for electrons belongs to Dr. Eckart of the California Institute of Technology, although we had already earlier applied the formula written above for calculating the wavelengths of diffracted beams.
Let us substitute into this formula the data for electron diffraction beams. The distance between chains of atoms normal to the A and B azimuths is \(2.15\) Å. The beam, na—
observed in the A-azimuth with an accelerating potential of 54 V, is directed at \(\theta' = 50^\circ\). Consequently, the wavelength of the “54-volt” electron is
\[ \lambda = \frac{2.15}{n}\sin 50^\circ = \frac{1.65}{n}. \]
Since this beam is the first in azimuth A, for it, of course, \(n = 1\) and \(\lambda = 1.65\ \text{Å}\). An analogous calculation for the 65-volt beam in azimuth B gives \(\lambda = 1.50\ \text{Å}\). The fact that it is possible by such a simple and direct method to calculate the wavelength of an electron stream seems, of course, very remarkable. At the present time, however, it surprises us less than it would have surprised us, say, five years ago. During the last 4–5 years the view has rapidly developed that the principles of mechanics known to us up to now, in their various formulations, serve only, so to speak, as a first approximation to the truth. For a whole range of fields this approximation is astonishingly accurate and, for example, in practical mechanics and astronomy it will hardly ever need to be replaced by anything else. Yet we are now nevertheless convinced that classical mechanics is, as it were, a degenerate form of the true mechanics and has a limited field of application. It serves as a first approximation to the true mechanics and is applicable only to those systems in which the products of momenta by linear dimensions are large in comparison with Planck’s constant of action \(h\), i.e. in comparison with \(10^{-26}\ \text{erg}/\text{sec}\). Therefore, when dealing with the solar system, one may without hesitation use classical mechanics, since there the linear dimensions are quantities of the order of the major axes of planetary orbits, the momenta are the quantities of motion of the planets, and consequently the products of these quantities are enormous in comparison with \(h\). But, on the other hand, one must not think that the approximate form of the true mechanics applicable to the solar system is also applicable to a system of the Bohr atom type. Here the products of linear dimensions by momenta are not large in comparison with \(h\); they are of the same order as \(h\), and,
therefore the laws of mechanics in the form known to us up to now are not applicable. The domain of their application is limited, and the present case does not fall under it.
This conviction grew out of the increasing dissatisfaction with, and artificiality of, Bohr’s model of the atom as a means for describing and generalizing the facts of spectroscopy. It led to various attempts to create a new system of mechanics, which would turn into the ordinary one for the case of large systems and at the same time would be applicable to systems containing atoms and electrons. One of these attempts was made by L. de Broglie, who about 3 years ago advanced the idea that every mechanical phenomenon is, in a certain sense, a wave phenomenon; that, consequently, every problem of mechanics is a kind of problem of optics; and that, in the rigorous solution of all problems of this type, the propagation and interference of waves must be taken into account. This idea was received with great enthusiasm by theorists—especially by Schrödinger—and underwent a rapid and remarkable development. It is this idea that constitutes the essence of the new wave mechanics, which perhaps is the true mechanics that has been awaited for so long. For the time being it is still in a state of development, and it is unknown what its final form will be. Its form and interpretation are constantly changing; however, de Broglie’s fundamental idea—that with every freely moving particle having quantity of motion \(mv\) there is associated a train of waves of length \(\frac{h}{mv}\)—has remained in force up to the present.
Whether the particle itself exists only in the form of this group of waves, whether the waves themselves have a real physical existence—as, for example, light waves do—or whether they represent only a mathematical apparatus, cannot yet be said.
Until now this theory has developed only in the direction of a new model of the atom and of a new, less arbitrary establishment of rules for describing and interpreting spectroscopic data. To the periodic phenomena with which we are concerned in the present case, considerably
devoted considerably less attention. And yet, several months before the beginning of our experiments, and more than a year before the first of the results I have set forth was obtained, the German physicist Elsasser predicted that confirmation of wave mechanics would be found in the interaction of a stream of electrons with a crystal.
Most of you probably know that the values of the wavelengths of electrons obtained from our measurements agree well with the values
\[ \frac{h}{mv}, \]
given by the new mechanics. We have:
\[ \lambda=\frac{h}{mv}, \]
and for electrons with not very large velocities (such as we used)
\[ \frac{mv^{2}}{2}=\frac{Ve}{300}, \]
where \(e\) is the charge of the electron in electrostatic units, and \(V\) is the bombarding potential in volts. Eliminating the velocity \(v\) from these equations, we obtain:
\[ \lambda=\left(\frac{h^{2}}{me}\right)^{1/2} \left(\frac{150}{V}\right)^{1/2}\ \text{cm}. \]
The quantity
\[ \left(\frac{h^{2}}{me}\right)^{1/2} \]
differs from \(10^{-8}\), approximately, by 2 thousandths; consequently, with high accuracy one may write:
\[ \lambda=\left(\frac{150}{V}\right)^{1/2}10^{-8}\ \text{cm} =\frac{150^{1/2}}{V^{1/2}}\ \text{\AA} =\frac{12.25}{V^{1/2}}\ \text{\AA}. \]
The value of
\[ \frac{h}{mv} \]
for electrons situated in a field with a potential difference of \(54\ V\) is
\[ \frac{12.25}{54^{1/2}}\ \text{\AA}=1.67\ \text{\AA}, \]
which closely agrees with the value \(1.65\) Å found from our measurements. For a “65-volt” electron the theoretical wavelength is \(1.52\) Å, the measured one \(1.50\) Å.
In all, we determined more than twenty electron wavelengths. All these values are shown in Fig. 14 as a function of the reciprocal value of the square root of the accelerating or bombarding potential. The values which we have reason to regard as the most reliable are placed in circles or squares. The straight line passing through the origin is the graph of the equation
Fig. 14. Graphical comparison of all experimentally found values of the electron wavelength with the quantities calculated from the relation
\[
\lambda=\frac{h}{mv}.
\]
\[ \lambda=\frac{12.25}{V^{1/2}}. \]
If the observed values of the wavelengths coincided exactly with the quantities \(\frac{h}{mv}\), then all the points would lie on this
straight line. The resulting deviations are no greater than those which would have to be expected by virtue of the inaccuracy of the measurements. Consequently, one may say that, in certain cases, a stream of electrons of velocity \(v\) behaves like a beam of waves of wavelength \(\dfrac{h}{mv}\), in accordance with the postulates of wave mechanics.
Let us return for a few minutes again to the regularly reflected beam.
I have already indicated that if one plots the intensity curve of the reflected electron beam at an angle of incidence of \(10^\circ\) as a function of the square root of the bombarding potential, then we obtain a series of maxima at intervals equally spaced from one another.
Let us calculate what the position of these maxima ought to be if the refractive index of nickel with respect to electrons were equal to 1. We have:
\[ \frac{1}{\lambda}=\frac{mv}{h}=\frac{V^{1/2}}{12.25}=n\,\frac{1}{2d\cos\theta} \]
(we take the wavelength of the electron to be equal to \(\dfrac{h}{mv}=\dfrac{12.25}{V^{1/2}}\)
and then use Bragg’s formula to find those values of \(V^{1/2}\) at which the intensity of the reflected beam has a maximum). Substituting for \(d\) and \(\theta\) the values \(2.03\) Å and \(10^\circ\), we find:
\[ V^{1/2}=n\cdot 3.06. \]
These are the values of \(V^{1/2}\) at which the maxima would be found if the refractive index were equal to 1. They are shown in Fig. 15.
The observed maxima lie to the left of the calculated ones, and this displacement decreases with increasing order. This is precisely the kind of displacement that should be expected if the refractive index of nickel with respect to electrons were
greater than 1. Therefore these displacements can be used to determine the refractive index. A more general form of Bragg’s formula, applicable also to the case of \(\mu\) different from 1, is:
\[ U^{1/2}=n\,\frac{12.25}{2d(\mu^2-\sin^2 b)^{1/2}}. \]
Applying this formula to the available data, we obtain values of the refractive index for different electron velocities, plotted on the lower diagram of Fig. 15. The values marked by circles were taken from observations at an angle of incidence of \(10^\circ\); the remaining values were taken from other observations.
Fig. 15. The upper curve shows the positions of the maxima calculated on the assumption that the refractive index is equal to unity. The lower curve depicts the dependence of the “refractive index” of nickel for electrons on their velocity.
We see that, as the velocity of the electrons increases, the refractive index approaches 1. Thus, at high velocities, the observed purely geometrical difference in the reflection and diffraction of X-rays,
on the one hand, and of electrons—on the other, disappears. The validity of such a conclusion is confirmed by the experiments of Prof. J. P. Thomson at the University of Aberdeen, who studied the diffraction of beams of fast-flying electrons (from 15,000 to 60,000 V) by very thin metallic sheets. Prof. Thomson found that electrons of such velocities are diffracted by a polycrystalline metal in exactly the same way as X-rays in the “powder method,” discovered by Hull simultaneously with Debye and Scherrer. The cross section of the transmitted beam consists of a central spot surrounded by concentric rings. These rings coincide exactly with those that would be observed if, instead of electrons, we had a beam of X-rays of wavelength \(\frac{h}{mv}\). Hence it is clear that, for such fast electrons, the refractive index in the metal is close to unity.
Are electrons waves? The simplest way to answer this question is to pose another question: are X-rays waves? If they are waves, then electrons too are waves. But now we are far less certain than before that X-rays are waves. The Compton and photoelectric effects are most easily described if we regard X-rays, in a certain sense, as particles.
All this is somewhat paradoxical and unclear. It turns out that, in some respects, not only are rabbits cats, but cats are rabbits.
I do not wish, however, to say that this is unclear to everyone. There are many theorists for whom it is not at all unclear. In his latest paper Prof. Darwin writes on this subject:
“The central difficulty of quantum theory has always been the conflict between waves and particles. On the one hand, we have the laws of conservation of matter, energy, etc., which tell us that matter is conserved, and which endow each particle of energy with individuality, so that we can trace its history. On the other hand, we have the theorems of the interference of light, and now, if you like, of mate-
ries, which also tell us quite definitely that those things which we formerly regarded as particles must become dispersed, losing their individuality in the process. Bohr’s latest work shows how, at least in general outline, this contradiction can be removed. These two kinds of representations do not contradict, but complement one another. In order to verify the conservation laws we must necessarily have a closed system, and this excludes the possibility of observing what is being done inside it. If, however, nothing is observed, then one may say that nothing is happening: the system is in a stationary state outside time and space, outside our intuition. If, on the other hand, we wish to find out what is happening in it, it will be necessary to make an opening in the closed vessel that contains it and to see what happens. But by this very act we shall destroy the conservation laws and shall obtain, in their stead, interference phenomena, which will entail the introduction of geometry and a connection with space and time. This, of course, imperfect reasoning shows that, in speaking of events occurring in space and in time, we may freely make use of the wave theory without being troubled by difficulties connected with the conservation laws, since in reality these difficulties do not exist.”
As you see, all this is very simple.