Abstract
Lecture delivered at Girton College on March 8, 1928.
Full Text
BEYOND THE ELECTRON1
Sir J. J. Thomson, Cambridge
Not so very long ago the atom was considered the ultimate boundary beyond which the very nature of things does not allow us to pass. The atom was regarded as something indivisible, impenetrable, eternal, inaccessible to the influence of heat, electricity, or any other physical agent. The interior of the atom was declared to be that territory into which the physicist would never succeed in penetrating. But the time came when this holy of holies was subjected to investigation, and it turned out that the atom itself is built of still smaller particles: negatively charged electrons and positively charged protons. Means were found for counting the number of electrons in an atom, and it was established that the atom is by no means a small solid particle, as had formerly been thought, but is a very complex system, comparable in complexity to the solar system. Moreover, it turned out that precisely this complexity, this delicate structure of the atom, imparts to matter its electrical and chemical properties. To explain these properties it is not enough simply to suppose that matter consists of a large number of small particles; this explanation is directly connected with the question of the internal structure of these particles, with their electronic structure.
Experiments give us the possibility of finding the numerical values of the mass of the electron and of its associated charge; they,
however, say nothing about its structure. One cannot, for example, say whether the electron is simply a point charge of negative electricity, or whether it, like the atom, is built up from still smaller particles: sub-electrons and sub-protons. True, we have no evidence that, analogous to the various kinds of atoms, there also exist various types of electrons. But it is possible that, alongside the observed type of electrons, there are others, considerably less stable and therefore almost entirely unobserved. Having no concrete data on this question, it is natural, of course, to make the simplest assumption and to regard the electron as a charged point surrounded by a medium possessing no internal structure. The mathematical analysis under such a hypothesis is considerably simpler than under any other. From this, naturally, nothing follows; for one cannot think that the universe is constructed according to the principle of the greatest convenience for mathematicians. It is quite probable that, in the light of further advances of science, such a point of view regarding the electron will prove just as unable to withstand criticism as the former point of view regarding the atom.
The task of my lecture today is to show that these advances are already at hand, and that the true structure of the electron and of the space surrounding it is entirely unlike the usual conceptions of it.
Some of you may ask: is it necessary to go so far into the depths; would it not be better to put a full stop somewhere? To this I shall answer as follows: the whole charm of physics consists in the fact that it has no rigid and definite boundaries. Every new discovery does not bring us to an end; on the contrary, it opens the way for further investigations; and therefore, so long as science exists, there will always be many new unsolved problems. Physicists need not fear being left without work.
The main reason which, in my opinion, compels us to abandon the former views of the electron consists in the fact that, as has recently been shown, a moving (in particular, a uniformly moving) electron is always accompanied by a train of waves. These waves, as it were, carry with it [[unclear: text obscured by stamp]].
constitute its path. Thus a moving electron is a considerably more complicated thing than a simple point charge.
It seems to me that the most obvious evidence for the existence of the waves surrounding the electron is furnished by the experiments of my son, Prof. G. P. Thomson, who studied the passage of electrons through very thin metallic plates. These plates, considerably thinner than the thinnest gold leaves, are very valuable physical instruments, since they make it possible to determine whether the radiation passing through them is a stream of particles or a train of waves. Indeed, suppose we have a narrow beam of some rays and wish to determine whether it consists of a stream of particles moving in one direction, or of a train of waves. If this beam were to fall directly upon a photographic plate, then in both cases a sharply outlined image would be obtained upon it. Let us now see how the image will change if a thin metallic plate is placed in the path of the beam. Suppose that the beam consists of a stream of particles; these particles will strike the molecules of the plate, changing the direction of their path at each collision. The magnitude of this deflection will be different for different particles, since it is determined to a considerable extent by the laws of chance. Therefore, upon emerging from the plate, the particles will no longer move in one and the same direction, and their stream will assume the form of a cone. The image on the photographic plate will become larger and more diffuse, having turned into a spot without a definite boundary. If, however, instead of a stream of particles we have a train of waves, then the plate, by virtue of the regular arrangement of its molecules, will act like a diffraction grating. From the properties of such gratings it is known that, for wavelengths comparable with the distance between the molecules of the grating, the original spot does not become blurred, but will be surrounded by a series of bright rings with definite ratios of radii.
In Fig. 1 is shown the result obtained by my son in experiments on the passage of a beam of electrons through a plate. The figure shows a series of rings, whose position
coincides exactly with the position of the diffraction rings of light of a definite wavelength that would be obtained if it passed through this same plate. The fact that these rings indicate the directions of motion of the electrons was proved by placing a magnet near the photographic plate: the rings were deflected by the magnet, like the trajectories of electrons (Fig. 2). Hence it is clear that the blackening of the plate is due precisely to electrons, and not to light waves, since the latter are not deflected in a magnetic field. Thus it turns out that, in passing through a metal, electrons change the direction of their motion not as particles, but as waves of a definite wavelength. From this one may conclude that each electron is accompanied by a train of waves and that these waves entirely determine the direction of its motion. The electron is, as it were, compelled to follow these waves.
Fig. 1
Fig. 2
A thin metallic plate not only reveals the presence of waves, but also makes it possible to determine their length. My son did this and obtained a very interesting result. It turned out that electron waves have extraordinarily high frequencies. The smallest of these frequencies is a million times greater than the frequency of visible light and considerably exceeds the frequency both of X-rays and of the most hard-
of all known kinds of high-frequency radiation—the γ-rays of radium. Electron waves constitute an entirely new type of radiation, whose properties may in many respects differ from the properties of all types of radiation known to us hitherto.
Just as the concept of light particles proved insufficient for explaining the properties of light and it became necessary to introduce the concept of light waves, so too the concept of electrical particles proved insufficient for explaining the properties of electrons, and it became necessary to suppose that these particles are accompanied by systems of waves. Wave–particle dualism thus has a place in the most diverse domains of physics and, apparently, is rooted in the very nature of things.
In order to assess the importance of this fact, let us consider briefly how energy passes from one place to another. Suppose, for example, that an electron changes its position—the question arises: by what path does its energy follow it? This question can be formulated more clearly if we return to the old conception of the electron as a little sphere with a radius of \(10^{-13}\) cm. When this little sphere moves, is its energy concentrated within it, or is it distributed throughout the external space and making its way through the ether? If, as I do, we consider, following Faraday and Clerk Maxwell, that the properties of charged bodies are determined by lines of force in the ether surrounding them, then the energy of the electron must be conceived as concentrated not in the small sphere symbolizing the electron, but throughout the external space. According to this view, all energy is concentrated in the ether and is propagated from one place to another by means of ether waves. The fact of the transmission of energy through the ether was first clearly and distinctly formulated by my old friend Prof. John Henry Poynting. His arguments lead to results which at first sight are somewhat strange, although I consider them absolutely correct. For example, I am convinced that most people think that the energy of electric lamps is imparted to them from the power station through the connecting—
BEYOND THE LIMITS OF THE ELECTRON
their copper wires. According to Poynting, the matter stands quite differently, and energy is propagated not along the wire, but through the external space surrounding it. The role of the wire is not that it carries the energy, but rather that it directs its path in the external space. Energy is propagated in the form of waves through the ether outside the wire, with a velocity that does not depend on the dimensions of the wire or on its material.
The fact of the wave-like propagation of electrical energy through the ether is generally recognized; but can one go further and say that every kind of energy is transmitted by the same path? Such an assumption is quite plausible, since it is natural to suppose that there exists energy of only one type, concentrated in the ether and propagated through it. But if one tries to transfer the idea of the wave-like propagation of electrical energy to other kinds of energy, we at once encounter a difficulty which at first glance is insurmountable. The point is that all electromagnetic waves, regardless of their wavelength, are propagated through the ether with one and the same velocity—the velocity of light; the energy that they carry with them must obviously possess the same velocity. But let me, or any one of those present, try to move from one place to another and thus transfer his energy—even the youngest among us will at once fall hopelessly behind a ray of light. In other words, we must reckon with the fact that the velocity of propagation of energy may have the most varied values. It seems to me, however, that this fact does not contradict the fact of the propagation of energy through the ether. Indeed, the propagation of electromagnetic waves of any frequency with the velocity of light does not occur if there are electrons or, in general, charged bodies in the ether. These charged bodies are set in motion by electromagnetic waves and themselves begin to emit them; the secondary waves, combining with the primary ones, change their character, i.e. their wavelength, while leaving the frequency unchanged and, consequently, affect the speed of their propagation.
An example of such an effect on a cosmic scale may be found in the upper parts of the atmosphere, in the region bearing the name of the Heaviside layer. The Heaviside layer, whose existence, among other things, makes long-distance wireless telegraphy possible, is the upper region of the atmosphere in which the atmospheric pressure is very small and the solar radiation is considerably more intense than at the surface of the earth, since it has not yet had time to be absorbed in the atmosphere. The intensity of this radiation there is so great that it dissociates a large number of air molecules into electrons and positive ions. Thus in the Heaviside layer there is a large number of excess positive and negative charges. The behavior of electromagnetic waves in this layer is of an entirely different character than in space near the earth, which is devoid of free charges. In the lower layers of the atmosphere all waves, regardless of their wavelength, travel with the velocity of light; in the Heaviside layer, however, their velocity is always greater than the velocity of light and is different for different wavelengths: the greater the wavelength, the greater the velocity of its propagation. The relation between wavelength and velocity of propagation is shown by the curve \(APL\) in Fig. 3, where the velocity is plotted along the abscissas and the wavelength along the ordinates1. I shall call a medium of this type a superdispersive medium.
Fig. 3.
The vertical dashed straight line in the figure gives the relation between the velocity of propagation and the wavelength in nor-
normal medium. You see that the influence of electric charges reduces to increasing the velocity of propagation of the waves. At first glance it seems that this circumstance cannot in any way help us explain why the energy moves there more slowly than in the pure ether. This, however, is incorrect. The point is that, in speaking of the wave-like propagation of energy, one must distinguish between the velocity of the waves and the velocity of the energy carried by them. The greater the velocity of propagation of the waves, the smaller the velocity of propagation of the energy. This fact is often forgotten, since in the most common examples of wave motion—in light and in sound—all waves propagate with one and the same velocity, and therefore the question of a change in the velocity of the energy does not arise.
Since this point is of fundamental importance, I shall dwell on it somewhat more fully. When we in some way bring about the appearance of waves—for example, by throwing a stone into water or by producing a spark discharge—the initial disturbance arises over a comparatively small area. In this case the waves excited are usually of different wavelengths; in order that only a single wavelength be obtained, the excitation must have an entirely special character and at once spread over a large area. And since in practice the excitation is limited to a small area, there arises not one wave, but a whole group of waves of different wavelengths, propagating in a superdispersive medium with different velocities. If the wavelengths of the excited waves are close to one another, then the initial disturbance, spreading in all directions, will retain its former character, i.e. will be concentrated in a region of approximately the same size as that in which it arose. All the energy will be concentrated in this region and will propagate with the same velocity as the disturbance. This velocity may differ from the velocity of propagation of the waves. The velocity of propagation of the energy is called the group velocity; the velocity of propagation of the waves, the phase velocity. Let us illustrate the difference between them by a simple example: suppose
J. J. Thomson
We have two trains of waves. Let us represent one of them as a row of boys walking in a straight line with constant speed at equal distances from one another. Their speed will correspond to the phase velocity, and the distance between them to the wavelength. The second train we shall represent by a row of girls moving with another speed and separated from one another by another distance. Let us now imagine both trains walking side by side. If the boys denote the maxima of one train of waves, and the girls the maxima of the other train, then at those places where a boy and a girl are side by side, the amplitude of the oscillation and its energy have maximum values. Let us direct our attention to these points and find the speed of their motion. If the observer stands still, sooner or later he will see one of the girls beside a boy; however, since the two rows do not keep step, in the next pair this coincidence will no longer occur, and it may be that he will have to wait a very long time for it. Can the observer shorten this waiting time if he himself begins to move, and what must his speed be in that case? Let us denote this speed by \(W\) and suppose that the observer began walking immediately after the appearance of the first pair: after how much time will the nearest girl overtake him? If \(V\) is the speed of the girls’ step and \(D\) the distance between them, then this time will evidently be equal to
\[ \frac{D}{V-W}. \]
The same time for the boys will be equal to
\[ \frac{d}{v-W}, \]
where \(v\) is the speed of the boys’ step, \(d\) the distance between them. In the case when these quantities are equal to one another, i.e. when
\[ \frac{D}{V-W}=\frac{d}{v-W} \]
or
\[ W=\frac{vD-Vd}{D-d}, \]
the boys will all the time appear simultaneously with the girls, and it will seem to the observer,
that the whole procession consists only of pairs. This velocity \(W\) is, obviously, the velocity of motion of energy. We see that it may be quite different from the phase velocities \(V\) and \(v\); for example, when \(vD=Vd\), the group velocity is equal to zero, whatever the phase velocities may be. From this example, incidentally, it is clear that although the speed of energy may be much smaller in magnitude than the speed of propagation of the waves, the direction of their path is always one and the same. The waves, as it were, direct the path of the energy.
The velocity of energy can easily be determined from a diagram giving the dependence of the phase velocity on the wavelength (Fig. 4). The point \(P\) denotes a wave with velocity \(ON\) and wavelength \(PN\). The velocity of the energy carried by a group of waves of wavelengths close to \(PN\) is equal to \(OM\), where \(M\) is the point of intersection of the tangent to our curve at the point \(P\) with the axis of abscissas. It may be shown that \(OM\cdot ON=c^2\), where \(c\) is the speed of light in a non-dispersive medium. Thus, the greater the phase velocity, the smaller the velocity of propagation of the energy. From the drawing it is evident that \(OM\), depending on the position of the point \(P\), may have any value from zero to the speed of light. Hence it is clear that energy can be transmitted by waves and yet not possess the speed of light. When charged bodies are present in the ether, the velocity of energy, depending on the wavelength, may have any value from \(0\) to \(c\).
Fig. 4.
An example of this difference between the velocity of propagation of energy and the velocity of propagation of waves may be provided by a storm at sea. In deep water the waves, like electromagnetic waves in an ether filled with charges, move with different velocities; the velocity of water waves also increases with wavelength, though not as rapidly as
In electromagnetic waves. A stormy sea is not at all like the beautiful undulating surface in the form in which it is depicted in pictures; it is covered with irregularly scattered white crests, considerably higher than the waves. If you direct your attention to the top of some wave close to one of these crests, you will see that this top moves much faster than the crest. At the place where the crest is situated the greatest amount of energy is concentrated, and observation shows that this energy moves more slowly than the waves accompanying it.
I shall draw your attention to one feature of our drawing which is of fundamental importance. The wavelength \(PN\) is equal to the speed of its propagation \(ON\), multiplied by the period of oscillation; consequently the period of oscillation is equal to the ratio of the segments
\[ \frac{PN}{ON}. \]
But from the drawing it is evident that the magnitude
\[ \frac{PN}{ON} \]
never exceeds a certain limit, since \(P\) always lies below the straight line \(OK\). Consequently the period of oscillation of our waves always remains less than some finite quantity. Our medium transmits only oscillations above a certain frequency and does not transmit lower oscillations.
Thus we see that the ether, which by itself can carry energy only with the speed of light, in consequence of the presence of electric charges passes into a superdispersive state and becomes capable of serving as a transmitter of energy with any speed less than the speed of light. In its motion the energy is accompanied by a train of waves which determine the direction of its motion and which, by themselves, outside the places where the energy carried by them is located, possess only a small amount of energy. Moreover, to each speed of motion of the energy there corresponds a definite wavelength, which is the smaller the faster the energy moves. The speed of the waves themselves is considerably greater than the speed of the energy, and the product of these two quantities is equal to the square of the speed
of the propagation of waves through a pure nondispersing ether.
The experiments described by me above show that precisely this state of affairs obtains in the electron. The energy concentrated in the electron itself moves together with a certain train of waves in the direction determined by them. At the end of the present lecture, as an addendum, there is appended a mathematical investigation of the question of the passage of waves through a superdispersing medium filled with electric charges. It is shown there that between the velocity of the electron \(u\) and the wavelength of the waves accompanying it, \(\lambda\), there must hold the relation
\[ \frac{u\lambda}{\sqrt{1-\frac{u^2}{c^2}}}=C, \]
where \(C\) is a constant, and \(c\) is the velocity of light, i.e. \(3\cdot 10^{10}\ \mathrm{cm/sec}\). This relation was precisely confirmed by the experiments of my son, who, for \(u=10^{10}\ \mathrm{cm/sec}\), found \(C=8.3\), and consequently \(\lambda=7.8\cdot 10^{-10}\ \mathrm{cm}\). Since the frequency of our waves, i.e. the number of their oscillations per second, is equal to
\[ \frac{c^2}{C\sqrt{1-\frac{u^2}{c^2}}}, \]
its smallest value is \(\frac{c^2}{C}\), or, in the present case,
\[ 1.08\cdot 10^{20}. \]
Thus the electron behaves as though it were passing through an atmosphere filled with electric charges.
The view of the electron set forth above attributes to it a dualistic structure. One part of it, in which the energy is concentrated, is built of electric lines of force; while the other “part” is a train of waves in resonance with the electron and determining its path. This conception of the electron coincides strikingly with that conception of the structure of light which I gave
in Philosophical Magazine for October 1924. According to the latter, light, like the electron, also possesses a dualistic structure. On the one hand, it is a closed electric force ring in which its energy is concentrated; on the other hand, it, like the electron, is accompanied by a system of electromagnetic waves which themselves have no energy, but merely determine its path. If it is assumed that these waves are in resonance with the ring and that the energy of the ring is proportional to the frequency of the light, then for this structure one can derive consequences in agreement with Planck’s law of radiation. On the other hand, by adopting such a conception of light, we at once rid ourselves of all the difficulties connected with reconciling the electric properties of light, which for their explanation require a corpuscular theory, with its optical properties—for example, the phenomenon of interference—which require a wave theory. In fact, our waves undergo ordinary interference and direct the energy precisely into the bright parts of the interference pattern.
This dualism is a necessary consequence of the conception of the wave-like propagation of energy through the ether, since one must always distinguish between the transmission of energy and the propagation of waves. In the case of light waves this distinction is obscured by the fact that the velocity of transmission of energy turns out to be equal to the velocity of propagation of the waves. Such a coincidence is, however, pure chance—as we have seen, energy may move much more slowly than the waves; then the dualism becomes quite evident. In optical phenomena the waves play the principal role, and therefore we may quite well confine ourselves to a wave theory; in electrical phenomena, however, it is energy that is at issue, and therefore the corpuscular theory, which concentrates attention on particles of energy, is the most suitable. Speaking of waves, we obtain a wave theory; speaking of energy, a corpuscular theory. Thus each of these theories embraces only part of true reality. Strictly speaking, in all optical phenomena—just as in the motion of electric particles—
cathode, $\alpha$- and $\beta$-rays, it is necessary to take into account both particles and waves.
The behavior of the electron indicates that it moves in a superdispersive medium. The question arises: is this medium concentrated in the immediate vicinity of the electron, as might be the case if the electron, like the atom, consisted of still smaller electrical particles; or does the ether itself have a structure of this kind. Of course, for ordinary light the ether is not a superdispersive medium, but this does not mean that it cannot be such for oscillations many thousands of times more rapid, since dispersion depends on the frequency of the oscillations. Thus, for example, glass has no dispersion with respect to the long electromagnetic waves used in wireless telegraphy, since the period of oscillation of the latter is considerably greater than the period of the natural oscillations of the molecules of glass, and at the same time it has dispersion with respect to visible light, whose period of oscillation differs only slightly from the period of the natural oscillations of these molecules. If the structure of the ether is such that the period of its natural oscillations is considerably smaller than the period of oscillations of light, and yet greater than the period of oscillations of electron waves, then it may be a dispersive medium with respect to electron waves and not be one with respect to light.
Let us see in what way it might be possible to choose between the two possibilities formulated above.
According to the first assumption, the superdispersive region is concentrated near the electron, and its dimensions coincide with the dimensions of the electron. The value usually assigned to the diameter of the electron is $10^{-13}\ \text{cm}$. This quantity is not the result of direct measurements; it is calculated from a formula into which enter the charge and mass of the electron, known to us. Every system possessing an electric charge has, by virtue of this, a certain mass whose value depends on the configuration of the system. If we assume that the electron is a charged
sphere of radius \(a\), and that all its mass is of electromagnetic origin, then we obtain:
\[ m_0 c^2=\frac{2}{3}\frac{e^2}{a}, \]
where \(m_0\) is the mass of the electron at rest, and \(c\) is the velocity of light. Hence, knowing \(m_0\), \(e\), and \(c\), one can calculate \(a\). In this way the value \(10^{-13}\ \mathrm{cm}\) was obtained. We have seen that this quantity is connected with certain ideas about the structure of the electron—in fact, no one has measured the diameter of the electron. If one gives another picture of this structure, for example if one assumes that the electric field of the electron is distributed around it not uniformly, but has “protrusions,” then one can obtain another, higher value for this diameter. From our point of view, the quantity most characteristic of the dimensions of the electron is the size of the superdispersing region surrounding it. This quantity can at least approximately be determined from direct experiments similar to those of G. P. Thomson on obtaining diffraction rings when electrons pass through thin metallic sheets. Knowing the radii of these rings, one can determine the wavelength of the electron waves lying within the superdispersing region, whose interference gives rise to the appearance of the rings. In order for the interference to be noticeable, the electronic ray must contain at least several such waves. According to G. P. Thomson’s measurements, electrons with a velocity of \(10^{10}\ \mathrm{cm/sec}\) correspond to a wavelength close to \(7.8\times10^{-10}\ \mathrm{cm}\). Consequently, the diameter of the superdispersing region must be at the very least \(10^{-9}\ \mathrm{cm}\), i.e. it has a value considerably greater than those \(10^{-13}\ \mathrm{cm}\) given by the usual theory. By using slower electrons, one can obtain greater wavelengths; in this case, however, when the wavelength becomes greater than the diameter of the superdispersing region, the interference pattern should become blurred. Experiment has shown that the sharp outlines of the rings disappear only when the velocity of the electrons becomes considerably less than \(10^{10}\ \mathrm{cm/sec}\), and the length
BEYOND THE ELECTRON
waves is considerably greater than \(10^{-9}\) cm. It is possible, however, that this blurring of the interference is also affected by secondary causes.
Well-defined rings are obtained at wavelengths of \(10^{-9}\) cm; consequently the diameter of the superdispersing region is in any case not less than this quantity. This fact may well be reconciled with the properties of the electron, since one must not forget that the frequency of electron waves is very high and that the dispersing capacity of the electron manifests itself only with respect to very rapid oscillations. With respect to lower frequencies, the superdispersing region may behave like a normal ether and, consequently, not be detected. Thus it is not at all surprising that the study of electron waves gives a higher value for the dimensions of the electron than other methods.
Among the \(\gamma\)-rays emitted by radioactive substances there are some whose wavelengths are comparable with the wavelengths of electron waves; they are not, however, accompanied by electrons. If the superdispersing state exists only in the immediate vicinity of the electron, then the velocity of these \(\gamma\)-rays must be different from the velocity of electron waves; but if the superdispersing capacity is a property of the entire ether, then the velocity of \(\gamma\)-rays must coincide with the velocity of electron waves of the same wavelength and consequently be greater than the velocity of light. The wavelengths of \(\gamma\)-rays can be measured by a method which in principle coincides with the method of measuring electron wavelengths; the frequencies of \(\gamma\)-rays can be determined by knowing the velocity of the \(\beta\)-particles knocked out by them from metals. Assuming that all the energy of the \(\gamma\)-rays is transferred to the \(\beta\)-particles, one can determine this energy \(E\), and with it also the frequency \(\nu\), from the relation \(h\nu = E\). The propagation velocity of the \(\gamma\)-rays that interests us is equal to \(\lambda\nu\). Measurements of \(\lambda\) for very hard \(\gamma\)-rays were carried out by Kovarik, and measurements of \(\nu\) by Ellis and Skinner. These measurements showed that the values of \(\lambda\nu\) are very close to the ordinary value of the velocity of light. Hence one may conclude that the superdispersing state—
caused by the presence of the electron itself; in other words, the electron itself “provides ether for itself.”
One may expect that the effect of collision with an electron will be, for these hard $\gamma$-rays, considerably greater than for light of lower frequency. In fact, with respect to $\gamma$-rays the electron is a superdispersive medium. The refractive index in such a medium is considerably less than unity; for example, for the waves with which G. P. Thomson dealt, it is equal to $\frac{1}{3}$. Thus, on entering the region surrounding the electron, the path of the $\gamma$-rays must be curved as though they were repelled by the electron. Since the jump in the refractive index is rather large, this deflection must be very great. A change in the direction of motion of the $\gamma$-rays is associated with a loss of momentum; moreover, by the law of conservation of momentum, the momentum lost by the rays passes to the electron. The masses of hard $\gamma$-rays are comparable with the mass of the electron; consequently the collision between the rays and the electron may be likened to a collision between two bodies with masses of the same order of magnitude, one of which is initially at rest. In such a collision, when one of the bodies is deflected through a considerable angle, the body at rest receives from the moving one a large amount of energy. Consequently, the energy of hard $\gamma$-rays must, upon their collision with electrons, decrease considerably, and a decrease of energy is equivalent to a decrease of frequency. The change of frequency upon collision is called the Compton effect; from our reasoning it follows that this effect must be especially large for hard $\gamma$-rays.
Since the electron is a complex system, it possesses a whole series of its own periods of oscillation. The number of these periods may be very large; they form a series of discrete values, each of which is separated from the next by a finite interval. The electron can oscillate in various periods, but not in any arbitrarily chosen period. In this case, between the very elec-
trón and the waves accompanying it must be in resonance. The momentum and energy of the electron are connected by simple relations with the frequency of the guiding waves: in Appendix A it is shown that the energy of the electron is proportional to the frequency of the wave and that the product of the electron’s momentum by its wavelength is constant. Hence it follows that the momentum and energy of the electron, like its frequencies and wavelengths, cannot take arbitrary values. The possible values, for example of the momentum, are always separated from one another by finite intervals, so that the increase in the quantity of motion proceeds not continuously, but in jumps. Thus we arrive at a kind of quantization of the quantity of motion. From our point of view, quantization is a consequence and expression of the definite structure of the electron—only those motions are possible, or at least stable, which are in resonance with the intra-electronic oscillations.
Whatever point of view one may take regarding the structure of the electron, we shall almost inevitably arrive at the idea of its own periods of oscillation. Indeed, when the electron is in equilibrium, the distribution of the lines of force of the electric field surrounding it is such that its potential energy has a minimum value. If this distribution is disturbed, for example as a result of the passage through the field of a rapidly flying cathode ray, then the new distribution will no longer be an equilibrium one and will begin to undergo oscillations. The period of these oscillations will be comparable with the time that light spends in traversing a distance equal in magnitude to the linear dimensions of the electron.
Experiment shows that electrons are deflected under the action of electric and magnetic forces. Since the path of electrons is determined by waves, it must be admitted that these forces influence the direction of propagation of the waves. It is known that the paths of light rays always remain straight lines, except in the case of a variable refractive index (for example, in a mirage the bending of rays is explained by the fact that the air near the earth is hotter and therefore refracts less than in the upper layers of the atmosphere). Therefore, since electric—
electric and magnetic forces change the direction of the rays, hence they make the refractive index of the super-dispersive medium surrounding the electron a quantity varying from point to point. From our point of view this is precisely what was to be expected, since the refracting power of a given region depends on the distribution of electric systems within it. If the electron is far from other charged bodies, then the lines of force surrounding it are arranged symmetrically; there are as many of them on one side as on the other, and the refractive index has the same values on both sides. The appearance of a charged body disturbs the symmetry near the electron: the lines of force become denser on one side, leaving the other. The refractive index in the region surrounding the electron depends on the number of lines of force passing through that region. Consequently, owing to the change in the distribution of the lines of force caused by the appearance of the charged body, the refractive index on one side of the electron becomes different from the refractive index on the other side. This in turn causes a deflection of the path of the waves as they pass from one side of the electron to the other; and the deflection of the waves entails a curvature of the electron’s trajectory. The mathematical theory of this effect is given in Appendix B.
In the present lecture I have tried to show how the recently discovered properties of electrons lead us to the conclusion that the electron is not the final stage in the structure of matter; that it itself possesses structure, being built up of still smaller electric charges. With the aid of ideas of this kind it has proved possible to explain recently discovered phenomena. The results of the theory set forth above agree in many respects with the results obtained by the development of the new wave mechanics, whose emergence we owe to Louis de Broglie, Schrödinger, and others. This agreement is all the more remarkable in that the two above-named theories differ sharply from one another in their very foundations. De Broglie’s theory is purely analytical; the theory expounded by me today is of a purely physical character. I have tried to show that the recently discovered proper-
the properties of the electron analogous to what we have in other branches of physics, and at the same time to give a picture of the structure of the electron that would explain these properties.
It is interesting to note that, with such a view of the electron, the application of the methods of classical mechanics leads to results that were formerly regarded as an exceptional peculiarity of quantum mechanics. From this, it seems to me, one may draw the conclusion that the necessity for this new type of mechanics is connected with special, particular conceptions of the nature of the electron.
The experiments described by me, just like the experiments of Davisson and Kunsman and of Davisson and Germer on the reflection of electrons from crystals, open up a field for entirely new investigations. Let us hope that these investigations will help us to resolve the question, of enormous importance, of the nature of the electron.
Appendix A.
Propagation of Waves in a Superdispersive Medium.
The equations for the propagation of waves through a medium containing electric charges are written in the following form:
Let \(X, Y, Z\) be the components of the electric force; \(\alpha, \beta, \gamma\) the components of the magnetic force; \(x_r, y_r, z_r\) the coordinates of an electric charge of mass \(m_r\); \(c\) the speed of light. Then three equations of the following form hold:
\[ \frac{dX}{dt} + 4\pi c^2 \sum e \frac{dx_r}{dt} = c^2 \left( \frac{\partial \beta}{\partial z} - \frac{\partial \gamma}{\partial y} \right), \tag{1} \]
three equations of the form
\[ \frac{\partial X}{\partial z} - \frac{\partial Z}{\partial x} = \frac{\partial \beta}{\partial t}, \tag{2} \]
and three equations for each charge of the form
\[ m_r \frac{d^2 x_r}{dt^2} + n_r^2 x_r = Xe + \left( \beta \frac{dz_r}{dt} - \gamma \frac{dy_r}{dt} \right)e. \tag{3} \]
Hence we have:
\[ \frac{d^2X}{dt^2}+4\pi c^2\sum e\frac{d^2x_r}{dt^2} = c^2\left(\frac{\partial^2X}{\partial x^2}+ \frac{\partial^2X}{\partial y^2}+ \frac{\partial^2X}{\partial z^2}\right) -c^2\frac{d\psi}{dx}, \tag{4} \]
where
\[ \psi=\frac{\partial X}{\partial x}+\frac{\partial Y}{\partial y}+\frac{\partial Z}{\partial z}. \]
For the case of wave motion, when all the equations are linear, the term with \(\psi\) may be discarded.
Suppose that all charged particles have identical masses \(m'\), identical periods of natural oscillations, and identical charges; then, denoting by \(N\) the number of charges per unit volume, we obtain:
\[ 4\pi c^2\sum e\frac{d^2x_r}{dt^2} = 4\pi Nc^2e\frac{d^2x}{dt^2}. \]
In the superdispersing state \(m'\dfrac{d^2}{dt^2}\) is large in comparison with \(n^2\) and with \(\dfrac{\alpha^2+\beta^2+\gamma^2}{m'}e^2\); consequently, according to (3), the right-hand side of the last relation is equal to
\[ 4\pi c^2\frac{Ne^2}{m'}X. \]
Substituting this in (4), we have
\[ \frac{d^2X}{dt^2}+4\pi^2BX = c^2\left(\frac{\partial^2X}{\partial x^2}+ \frac{\partial^2X}{\partial y^2}+ \frac{\partial^2X}{\partial z^2}\right), \tag{5} \]
where
\[ B=\frac{c^2Ne^2}{m'}. \]
This is the equation of propagation of waves in a superdispersing medium.
Let us consider the case of a plane wave, when
\[ X=A\cos\frac{2\pi}{\lambda}(vt-z), \]
where \(v\) is the phase velocity, and \(\lambda\) the wavelength. For this case formula (5) gives
\[ v^{2}=B\lambda^{2}+c^{2}. \tag{6} \]
The group velocity \(u\) is equal to
\[ u=v-\lambda\frac{dv}{d\lambda}=\frac{c^{2}}{v}, \]
whence
\[ uv=c^{2} \]
or, by formula (6),
\[ c^{4}=B\lambda^{2}u^{2}+c^{2}u^{2}, \]
and finally
\[ \frac{\lambda u}{\sqrt{1-\frac{u^{2}}{c^{2}}}}=\frac{c^{2}}{\sqrt{B}}, \tag{7} \]
i.e., \(\dfrac{\lambda u}{\sqrt{1-\frac{u^{2}}{c^{2}}}}\) is constant. This relation between \(\lambda\) and \(u\) is precisely confirmed by the experiments of my son.
The frequency of our waves \(\nu\) is
\[ \nu=\frac{v}{\lambda}, \]
or, since \(v=\dfrac{c^{2}}{u}\),
\[ \nu=\frac{c^{2}}{\lambda u}= \frac{\sqrt{B}}{\sqrt{1-\frac{u^{2}}{c^{2}}}}. \tag{8} \]
Consequently the smallest possible frequency is equal to \(\sqrt{B}\).
My son found that at an electron velocity of \(10^{10}\ \mathrm{cm/sec}\) the wavelength \(\lambda\) is equal to \(7.8\cdot 10^{-10}\ \mathrm{cm}\). Substituting in equation (7) the values \(u=10^{10}\), \(\lambda=7.8\cdot 10^{-10}\), we obtain
\[ \sqrt{B}=1.08\cdot 10^{20}. \]
Thus the smallest frequency of the electron waves is \(1.08\cdot 10^{20}\); it corresponds in air to a wavelength
in \(2.7\cdot 10^{-10}\) cm. This length is considerably less than the wavelength even of hard X-rays and, in general, of all kinds of radiation, with the exception of the hardest \(\gamma\)-rays; it is small also in comparison with the radius of the atom.
Since
\[ \frac{1}{\sqrt{1-\frac{u^2}{c^2}}}=\frac{m}{m_0}, \]
where \(m\) is the mass of an electron moving with velocity \(u\), and \(m_0\) is the mass of an electron at rest, from equation (8) we have:
\[ \nu=mc^2\frac{\sqrt{B}}{m_0c^2}. \tag{9} \]
But \(mc^2\) is the total energy of the electron; consequently the frequency of the waves is equal to the energy of the electron multiplied by a constant. This fact is known as a relation of quantum theory; from our point of view, however, it is not a postulate of quantum mechanics, but a logical consequence of the fact that the energy of the electron moves through a superdispersive medium.
It is also interesting that if, according to the rules of quantum mechanics, one calculates the mass of a light quantum of any frequency, the result will coincide with the magnitude of the mass of the electron which, according to our theory, would accompany electron waves of the corresponding frequency. Further, the refractive index \(\mu\) is equal to
\[ \mu=\frac{c}{v}=\frac{u}{c}. \]
But by formula (8)
\[ \frac{\nu}{\nu_0}=\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}, \]
where \(\nu_0=\sqrt{B}\) is the least possible value of \(\nu\). Hence
\[ \mu=\frac{u}{c}=\sqrt{1-\frac{\nu_0^2}{\nu^2}}. \tag{10} \]
If
\[ X=A\cos(pt-mz), \]
Beyond the Electron
then from equation (2)
\[ \beta=\frac{m}{p}A\cos(pt-mz)=\frac{1}{v}A\cos(pt-mz). \tag{11} \]
The energy per unit volume due to the electric force is
\[ \frac{1}{8\pi c^{2}}A^{2}\cos^{2}(pt-mz). \]
The energy per unit volume due to the magnetic force is
\[ \frac{1}{8\pi}\beta^{2}=\frac{1}{8\pi v^{2}}A^{2}\cos^{2}(pt-mz). \]
Since in a superdispersive medium \(v\) is greater than \(c\), the magnetic energy there is not equal to the electric energy—as in a nondispersive medium—but is always less than it; in particular, it is equal to zero when the frequency of the waves reaches the limiting value \(\nu_{0}\). The mean value of the difference between the electrostatic and magnetic energies is equal to the kinetic energy of those charges whose presence causes the appearance of dispersion. Indeed,
\[ m_{1}\frac{d^{2}x}{dt^{2}}=Xe=eA\cos(pt-mz) \]
\[ m_{1}\frac{dx}{dt}=\frac{eA}{p}\sin(pt-mz), \]
and, consequently, the kinetic energy of our charges
\[ \frac{1}{2}Nm_{1}\left(\frac{dx}{dt}\right)^{2} \]
is equal to
\[ \frac{1}{2}\frac{Ne^{2}}{m'p^{2}}A^{2}\sin^{2}(pt-mz). \]
But from formula (6)
\[ \frac{Ne^{2}}{m'p^{2}}=\frac{1}{4\pi}\left(\frac{1}{c^{2}}-\frac{1}{v^{2}}\right), \]
and, consequently, the kinetic energy is equal to
\[ \frac{1}{8\pi}A^{2}\left(\frac{1}{c^{2}}-\frac{1}{v^{2}}\right)\sin^{2}(pt-mz) \]
and its mean value is
\[ \frac{A^{2}}{16\pi}\left(\frac{1}{c^{2}}-\frac{1}{v^{2}}\right). \]
As we see, it indeed coincides with the mean value of the difference between the electric and magnetic energies. Therefore the mean value of the total energy per unit volume is equal to twice the mean value of the electrostatic energy, i.e. to the quantity
\[ \frac{A^{2}}{8\pi c^{2}}. \]
The flux of energy per unit time through a unit surface is given by the Poynting vector
\[ \frac{1}{4\pi}X\beta=\frac{1}{4\pi}\frac{X^{2}}{v}\quad [\text{by formula (11)}]= \]
\[ =\frac{1}{4\pi}\frac{A^{2}}{v}\cos^{2}(pt-mz). \]
Its mean value is
\[ \frac{A^{2}}{8\pi v}. \]
But if \(u\) is the velocity of energy, then this flux must be equal to the mean energy density multiplied by \(u\). Hence
\[ \frac{A^{2}}{8\pi v}=u\,\frac{A^{2}}{8\pi c^{2}} \]
and \(uv=c^{2}\).
In this way one can obtain the value of the velocity of energy without resorting to the notion of interference between two waves. Since
\[ \beta=\frac{X}{v}=\frac{uX}{c^{2}}, \]
one may say that the magnetic force is proportional to the velocity of energy. Introducing the limiting frequency \(\nu_{0}\), one may write equation (5) in the form
\[ \frac{d^{2}X}{dt^{2}}+4\pi^{2}\nu_{0}^{2}X = c^{2}\left( \frac{d^{2}X}{dx^{2}}+ \frac{d^{2}X}{dy^{2}}+ \frac{d^{2}X}{dz^{2}} \right). \]
On the other hand, if \(\nu\) is the frequency of the waves transmitted, then
\[ \frac{d^2 X}{dt^2}=-4\pi^2\nu^2 X. \]
Consequently, the last equation may be put in the form
\[ 4\pi^2(\nu_0^2-\nu^2)X = c^2\left( \frac{d^2 X}{dx^2} + \frac{d^2 X}{dy^2} + \frac{d^2 X}{dz^2} \right), \]
or, by formula (10):
\[ c^2\left( \frac{d^2 X}{dx^2} + \frac{d^2 X}{dy^2} + \frac{d^2 X}{dz^2} \right) + 4\pi^2\mu^2\nu^2 X = 0. \]
Addition B.
The path of an electron under the action of an external force.
The equations determining the path of a ray of light in a medium with a variable refractive index \(\mu\) have the form:
\[ \frac{d}{ds}\left(\mu\frac{dx}{ds}\right)=\frac{d\mu}{dx}; \qquad \frac{d}{ds}\left(\mu\frac{dy}{ds}\right)=\frac{d\mu}{dy}; \qquad \frac{d}{ds}\left(\mu\frac{dz}{ds}\right)=\frac{d\mu}{dz}. \]
If \(q\) is the velocity of the electron’s energy; \(u, v, w\) are its components, then \(\mu=\frac{q}{c}\); and these equations take the form
\[ \frac{d}{ds}\left(\frac{u}{c}\right)=\frac{\partial\mu}{\partial x}. \]
If \(ds\) is an element of the path of the energy, \(ds=q\,dt\), whence
\[ \frac{du}{dt}=cq\frac{\partial\mu}{\partial x} = \frac{1}{2}c^2\frac{\partial\mu^2}{\partial x} \]
and similarly for \(\frac{dv}{dt}, \frac{dw}{dt}\).
It follows from this that the path of our energy is the same as the path of a mass \(m\) acted upon by a force with components
\[ \frac{1}{2}mc^{2}\frac{d\mu^{2}}{dx},\qquad \frac{1}{2}mc^{2}\frac{d\mu^{2}}{dy},\qquad \frac{1}{2}mc^{2}\frac{d\mu^{2}}{dz}. \]
As we see, this force is derived from the potential \(V\), where in the superdispersive region
\[ \mu^{2}=\mu_{0}^{2}+\frac{2V}{mc^{2}}, \]
where \(\mu_{0}\) is the value that \(\mu\) has in the absence of external forces.