HEAVY ELECTRON

D. Ivanenko, Tomsk

1. Blackett’s Experiments

The existence of a new elementary particle—the heavy electron, toward which only a few months ago it was still permissible to adopt a cautious attitude—may at the present time (May 1938) be considered definitively established. Indeed, the initial observations of Street–Stevenson and of Neddermeyer–Anderson have recently been confirmed, along with those of other authors, by careful measurements by Blackett himself, who had earlier supported a different interpretation of the nature of the hard component of cosmic rays. On the other hand, a number of reasonable theoretical hypotheses have also appeared, making it possible to describe the motion of the new particle and to connect it with nuclear phenomena.

Let us briefly recall here the characteristic situation in cosmic-ray physics (for details see our preceding article),¹ which led in 1937 to the discovery of the heavy electron.

The very successful division of cosmic rays into soft and hard components enabled Heitler and other authors in 1937 to construct a theory of the passage of soft rays through matter, proceeding from the assumption of the unrestricted applicability of quantum theory to electrons—positrons, which constitute the soft component. At energies of tens, hundreds, and thousands of electron-volts, the ordinary processes of ionizing collisions play no role for electrons, and energy loss occurs exclusively through the emission of hard photons in collisions with nuclei; moreover, according to quantum electrodynamics, the energy loss in such bremsstrahlung must increase as the energy grows. This consequence of quantum theory was subjected to fundamental doubt as a result of the experiments of Anderson and Blackett with collaborators: the energy loss did in fact increase with energy up to 150–200 million eV, and then clearly fell sharply. All attempts to modify quantum theory in order to take account of the (supposed) “catastrophe” in the region of ultra-high energies led to nothing. Heitler’s theory so successfully explained the formation of “cascade” showers, the form of the cosmic-ray absorption curve in the atmosphere (the Regener curve), transition effects, and in general all the basic facts associated with the soft component, whose passage through matter is characterized

with the formation of showers, so that at present there is no doubt as to the correctness of the quantum theory of electrons up to the very highest energies accessible to measurement. Thereby the hypothesis of the electron–positron nature of the soft cosmic rays is confirmed. Thus, we are compelled to ascribe to the hard penetrating rays another, non-electronic nature. The candidacy of protons must here be rejected, if only because their tracks are not found in Wilson cosmic-ray photographs.

In the decisive 1937 experiments of Neddermeyer and Anderson on the loss of energy by cosmic particles in passing through plates inserted into a Wilson chamber, a separation of all tracks into two groups was carried out: shower particles and single particles. The tracks of particles associated with showers or pairs, as was to be expected, showed, in complete agreement with theory, a continuous increase of energy loss. The single hard particles (obviously not electrons or positrons) showed a considerably smaller energy loss—the tracks of these particles, previously quite unlawfully considered together with electrons, had given grounds for the erroneous judgment that the formulas of bremsstrahlung radiation, and therefore quantum theory in general, were inapplicable. Finally, Street and Stevenson, Anderson and Neddermeyer take the last step (which now, in its naturalness, seems hardly less than obvious) and, from the range in the chamber, the thickness of the track, and the magnitude of the energy loss, ascribe a number of tracks of single hard rays to new particles with a mass intermediate between that of the electron and the proton. The small number of measurements and the boldness of the hypothesis of the existence of a new fundamental particle—the heavy electron—made one await with the keenest interest further work by Blackett, the foremost authority in this field, who in 1937 held the opinion that quantum theory broke down in the region of energies above \(2 \cdot 10^9\ \mathrm{eV}\). As we have indicated earlier, Blackett did not separate particle tracks into shower and single tracks, so that his former measurements could not serve as an argument against the heavy electron. Blackett’s two most recent works put an end to all doubts about the existence of the new particle.^2

Blackett not only reanalyzes his old photographs from the point of view of the criterion of single versus shower tracks, but also carries out, with great care, new experiments on absorption in lead and gold plates up to particle energies of \(2 \cdot 10^9\ \mathrm{eV}\) (in Anderson–Neddermeyer, up to \(5 \cdot 10^8\ \mathrm{eV}\)). The examination of about 150 tracks (in Anderson–Neddermeyer, 55 tracks) fully confirmed Anderson’s conclusion that the energy loss of soft particles associated with showers increases with increasing energy in exact agreement with theory, while single tracks give a small energy loss. With the courage characteristic of a great physicist, without any ambiguity, Blackett acknowledged the incorrectness of his earlier ideas about the limit of applicability of quantum theory near \(2 \cdot 10^8\ \mathrm{eV}\).

In another work, together with Wilson, Blackett investigates the angles

scattering of cosmic rays in passing through plates. The measurement of 170 tracks up to energies of \(9\cdot 10^9\ \mathrm{eV}\) confirmed Williams’ formula, based on the quantum theory of the passage of particles through matter. According to Williams, the mean angle of scattering at velocities of the order of the velocity of light does not depend on the mass of the particles, i.e. it must be the same for ordinary and heavy electrons. If, besides a greater mass, we were to ascribe to the heavy electron some other value of the charge, then the scattering would also have to change. Thus, experiments on energy loss require, for hard particles, which constitute practically all particles with energies greater than \(2\)—\(3\cdot 10^8\ \mathrm{eV}\), a greater mass of the order of 100–200 electron masses; while experiments on scattering require the ordinary electron charge and do not contradict a semiheavy mass.

In a very recently published paper, Corson and Brode\(^3\), on the basis of measurements of ionization, range, and curvature of tracks in a magnetic field (excellent photographs obtained with a delayed exposure of 0.5 sec. to facilitate counting the decayed ions), find for the mass of the new semiheavy particle a value of \(200\pm 50\) electron masses \(m\). Their analysis of the measurements of Anderson–Neddermeyer, Street–Stevenson, Nishina, and Rouliga-Crain showed that the results of these authors in general do not contradict (taking into account the large error of the measurements of small curvature) the indicated value. Williams and Pickup also independently obtained a mean value of \(200\,m\); the experimental data obtained generally lie within the limits of 100–500 electron masses. It is clear that the discovery of the new particle could have occurred earlier, through a very painstaking examination of already existing collections of cosmic-ray photographs, quite analogously to the discovery of the positron.

The theoretical considerations set forth below also speak in favor of the value 100–200 \(m\).

2. Choice of the equation for the heavy electron

After the magnitude of the rest mass and charge, as is known, the most important role in the characterization of a particle is played by the value of its intrinsic angular momentum (spin) and the type of statistics to which the particles obey. These two features, however, are connected with each other, since the presence of an integral number in fractions of \(\frac{h}{2\pi}\) of spin always determines symmetric or Bose statistics, while half-integral spin leads to antisymmetric or Fermi–Dirac statistics. In particular, an ensemble of photons (spin zero) obeys Bose statistics, whereas electrons, possessing spin one-half, obey Fermi statistics.\(^\circ\) Despite the fundamental significance of spin and statistics, the experimental determination of the latter proves to be a very difficult matter and, of course, for the heavy electron is still quite premature. The absence of spin also, in particular, increases the penetrating power, but to an insignificant degree in comparison with the influence of the hundredfold heavy mass. Let us note that concerning the half-integral spin of the posi-

the positron we infer on the basis of conservation laws in reactions of its creation and annihilation, while the statistics of positrons—on the basis of spin, which is, of course, a much more indirect argument than, for example, the obvious proof of Fermi statistics for electrons in a metal. Of course, the theory of the positron is so analogous to the theory of the electron that there is no doubt about such a description of it.

Let us emphasize that, according to modern ideas about elementary particles, only the four or three properties indicated are fundamental: mass, charge, spin (statistics); all the others, for example magnetic and electric moments, etc., are derived from the preceding ones. Moreover, the equations of motion of particles unambiguously give spin and statistics, but are themselves not determined exactly by these properties. Thus, in order to go further, we must, on the basis of reasonable hypotheses, ascribe to the heavy electron one or another spin, and along with it statistics as well. The choice of spin, as we shall see, will almost unambiguously determine the form of the equation to which the new particles are subject.

If we allow a half-integral spin for the heavy electron, we shall obtain, evidently, for it the usual equation of the Dirac type, but with a mass of the order of \(100\text{--}200\,m\). In nuclear reactions involving the heavy electron, neutrinos will necessarily have to participate, the emission of which may be approximately described by Fermi’s theory of \(\beta\)-decay. Since the theory of \(\beta\)-decay and of the neutrino is still far from complete, it seems desirable, as far as possible, to separate the range of problems of the new particle from the particular difficulties of the theory of \(\beta\)-decay and to try another path, i.e. to ascribe to the heavy electron an integer spin, say the value 0 or 1, and thereby Bose statistics. Thus, in contrast to nuclear reactions with emission of an electron, where conservation of spin requires the participation of a neutrino \((\nu)\), for example,

\[ {}^{16}_{7}\mathrm{N}\to e_{-}+\nu+{}^{16}_{8}\mathrm{O} \quad(\text{i.e. } \mathrm{n}\to e_{-}+\nu+\mathrm{p}), \tag{1} \]

we admit the following simple reactions of the new particle \(a\) \([a\)—Anderson\(^1\)] with a proton \(p\) or neutron \(n\)

\[ \mathrm{p}\rightleftarrows \mathrm{n}+a_{+};\qquad \mathrm{n}\rightleftarrows \mathrm{p}+a_{-}. \tag{2} \]

The following reactions with a nucleus of mass \(M\) and charge \(Z\) can probably also occur:

\[ \begin{aligned} a_{\pm}+M_{Z\pm1}&\to M_{Z\pm1}+h\nu, &&\tag{3a}\\ a_{\pm}+M_{Z\pm1}&\to M_{Z\pm1}+(a_{+}+a_{-}), &&\tag{3b}\\ h\nu+M_{Z\pm1}&\to M_{Z}+a_{+}+a_{-}. &&\tag{3c} \end{aligned} \]

To what equation of motion, then, are the new particles subject? The question is, of course, one of a quantum relativistic wave equation,

\(^1\) The designation “anderons” would be, incidentally, a convenient name for particles of this kind.

since the nonrelativistic Schrödinger equation (or, even more so, the classical equation of motion) is too crude to describe the difference in behavior for one or another spin or statistics. As the starting point of the reasoning it is best to take the well-known scalar relativistic equation of second order (the so-called Klein–Gordon equation)

\[ \left(\frac{E^2}{c^2}-p_x^2-p_y^2-p_z^2+m_0^2c^2\right)\psi=0; \]

\[ p_x=\frac{h}{i}\frac{\partial}{\partial x};\qquad E=-\frac{h}{i}\frac{\partial}{\partial t}\quad \text{etc.} \tag{4} \]

or

\[ (\square-k_0^2)\psi=0 \]

(\(h\) is everywhere Dirac’s constant, not Planck’s), which represents the operator translation of the relation between energy and momentum according to the theory of relativity.

For mass equal to zero, we have the simple d’Alembert equation: \(\square\psi=0\), to which, for example, the four components of the vector potential, playing the role of the wave function of the electromagnetic field, are subject. It is essential, however, that all quantities describing the electromagnetic field are real, whereas the \(\psi\)-functions of charged particles are necessarily complex. The presence of a rest mass, in a certain sense, does not at all change the character of the equation as radically as complexity does, although it of course changes the particular solutions. Indeed, at high energies \((E>m_0c^2)\) the term with the rest mass may be neglected, while the character of the statistics, spin, etc. should not change because of this. The reality of the d’Alembert equation for the components of the potential leads to the fact that the general solution in the form of a sum of plane waves is written as follows:

\[ \varphi=L^{-\frac{3}{2}}\sum\left(a_k e^{-ickt+i(\mathbf{kr})}+a_k^* e^{+ickt-i(\mathbf{kr})}\right) \tag{5} \]

(\(L\) is the length of the periodicity cube, \(k_x, k_y, k_z\) are wave numbers,

\[ k=\sqrt{k_x^2+k_y^2+k_z^2}, \]

\(a_k\) are arbitrary Fourier coefficients), where the second term, describing waves traveling in the opposite direction, or referring to absorption, is the complex conjugate of the first, referring to emission, so that the whole sum will be real.

The general solution of equation (4),

\[ \psi=L^{-\frac{3}{2}}\sum\left(a_k e^{-ickt+i(\mathbf{kr})}+b_k^* e^{+ickt-i(\mathbf{kr})}\right) \tag{6} \]

even for \(|k|^2\gg \frac{m_0^2c^2}{h^2}\) will nevertheless be complex, i.e. the arbitrary coefficient \(b^*\) describes not only the motion of particles

in the opposite direction (which, of course, is immaterial), but is independent of \(a\).

As is known, \(|\psi|^2\) gives the probability density for finding particles in one state or another. If we interpret \(a_k^* a_k\) as the number of particles of a given kind, for example electrons, in the state \(k\), then the quantity \(b_k^* b_k\) will already denote the number of certain other independent particles in the \(k\)-th state. Since the coefficient \(b_k^*\) has, in the energy exponent, the opposite sign from that for \(a_k\), it is clear that the \(b_k\) are in some way connected either with particles of negative energy or with holes in states of negative energy, i.e. with antiparticles. The theory of secondary quantization proves that the number of antiparticles in the state \(k\) (for example, positrons, if the \(a_k\) refer to electrons) will be equal to \(b_k^* b_k\) (hence it is clear why we prudently wrote the coefficient \(b_k^*\), and not \(b_k\)).

Since the newly discovered semi-heavy particles may be of two signs, i.e., in all probability, appear in pairs, we must take a wave equation with a complex \(\psi\)-function. Pauli and Weisskopf showed that equation (4) describes spinless particles obeying Bose statistics. This assertion seems quite plausible even without any proof, if one takes into account that spinless photons, which certainly obey Bose statistics, satisfy an equation very close to d’Alembert’s equation, while the presence of a rest mass and the complex character of \(\psi\) cannot change either the kind of statistics or the value of the spin. In 1935, even before the discovery of the semi-heavy particle, Yukawa proposed the scalar equation (4) for particles, then still hypothetical, with mass \(100\text{–}200\,m\). However, a scalar particle cannot be directly emitted or absorbed by other particles. The interaction energy of the emitting particle with the field of the emitted particles must have covariance similar, for example, to the coupling of a charged particle with the scalar or vector potential of the photon

\[ U_0=eA_0,\qquad U=\frac{e}{c}(\mathbf v\mathbf A), \tag{7} \]

where \(A_0\) and \(\mathbf A\), obviously, cannot be a scalar or a semivector. Further development of the scalar theory also leads to a whole series of contradictions with experiment (in the theory of the deuteron), and it must therefore be abandoned. Thus, we must take at least a vector complex function \(\psi\), satisfying the second-order equation (4), and connect its components by some invariant supplementary condition. The simplest is to take a supplementary condition of the Lorentz-equation type for the potentials. In this way we shall finally obtain a natural generalization of Maxwell’s equations to the case of a nonzero rest mass and a complex field, i.e. equations describing particles of two signs of charge. Similar equations (close, in essence, to those considered earlier by Frenkel) were specially investigated by the French theorist Proca two years ago.^5 Proca at first err-

THE HEAVY ELECTRON

usually assumed that his equations can describe the ordinary electron, eliminating a number of difficulties of the Dirac theory of the vacuum. As was shown, however, by the Tomsk theorists Durandin and Ershov ^8, Proca particles must obey Bose statistics and possess an integral spin, i.e. they are not suitable for the normal electron. After, for a number of reasons, the idea arose of the integral spin of the new particle, in Proca’s equations there was found a natural apparatus for describing the motion of the heavy electron. These equations are also a special case of the equations for particles with spin greater than \(1/2\), proposed in 1936 by Dirac “just in case” for future, as yet unknown, particles. The first results obtained with Proca’s equation for the new particle prove to be very promising. Let us note that de Broglie’s idea of the analogy between light and particles finds its most complete expression precisely in the new, most “Maxwell-like” equations of his pupil Proca.

Proca’s equations have the following form:

\[ \mathbf{F}=-\frac{1}{c}\frac{\partial \varphi}{\partial t}-\operatorname{grad}\varphi_0, \qquad \mathbf{G}=\operatorname{rot}\varphi, \tag{8a} \]

\[ \frac{1}{c}\frac{\partial \mathbf{F}}{\partial t}-\operatorname{rot}\mathbf{G}=k_0^2\varphi, \qquad \operatorname{div}\mathbf{F}=-k_0^2\varphi_0, \tag{8b} \]

where \(k_0=\dfrac{m_a c}{h}\), \((\varphi_0,\ \varphi)\) is the wave \(\psi\)-function of the new particle (scalar and vector potential), and \(\mathbf{F}\) and \(\mathbf{G}\) are two vectors analogous to the “electric” and “magnetic” fields, obtained from \(\varphi\) by differentiation (8a). The quantities thus defined also agree in dimension respectively with the potential and the field of the electromagnetic case. For \(m_a\), or \(k_0\), equal to zero, and for real \(\varphi\), system (8) is identical with Maxwell’s equations. Iteration gives, for all components \(\varphi_0, \varphi, \mathbf{F}, \mathbf{G}\), equations of type (4)

\[ \left(\Delta-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-k_0^2\right)\varphi=0. \tag{9a} \]

The Lorentz condition for the “potentials” is here a consequence of the basic equations (8)

\[ \frac{1}{c}\frac{\partial \varphi_0}{\partial t}+\operatorname{div}\varphi=0. \tag{9b} \]

As usual, the second group of equations is a consequence of the definition of the field (8a)

\[ \frac{1}{c}\frac{\partial \mathbf{G}}{\partial t}+\operatorname{rot}\mathbf{F}=0,\qquad \operatorname{div}\mathbf{G}=0. \tag{9c} \]

The secondary quantization of Proca’s equations, carried out in our Tomsk work, and also by Yukawa and collaborators ^7 and by Heitler—Kemmer—Fröhlich ^9, showed that, by analogy with the electromagnetic field, Proca particles, obeying Bose statistics, possess spin equal to unity and can be found

in two states of transverse polarization. Moreover, unlike the photon field, there may exist longitudinal waves and the corresponding heavy electrons of both signs, whereas in Maxwell’s theory the longitudinal part in vacuum is zero, and in the presence of charges reduces to the interaction energy between them.

With the appearance of the Dirac theory of the electron, many unsuccessful attempts were made at a “Maxwellization” of the Dirac equations, which indeed resemble Proca’s or Maxwell’s equations. Now the deep and subtle difference between them is clear to us, namely: the Dirac equation, which is the symbolic square root of the operator (4), describes, by means of four complex functions (constituting not a vector, but a spinor, more precisely a bi-spinor), particles with half-integral (and not integral) spin, obeying Fermi and not Bose statistics.

Thus the Dirac and Proca equations are different ways of linearizing the Klein–Gordon equation of the form (4), which is satisfied by each component of the Dirac or Proca field. According to Kemmer, other equations for particles with spin greater than \(1/2\), considered by Dirac, apart from the Proca system, do not lead to reasonable results.

The harmonious collaboration of experimental and theoretical physics again celebrates a victory, and the particle just discovered immediately finds for itself a ready-made equation among those prepared “in advance” by the powerful apparatus of quantum mechanics.

3. The Problem of the Hard Component

The development of the theory of the semi-heavy particle is proceeding in three directions.

A. The problem of cosmic rays. B. Questions of nuclear physics. C. The theory of elementary particles.

First of all it is necessary to describe as accurately as possible the passage of the new particle through matter. Shall we be able in this way to describe all the phenomena connected with the hard component of cosmic rays? In the present article, devoted chiefly to the inclusion of the heavy electron in the family of elementary particles, we shall confine ourselves to a few remarks on this question. The large mass of the heavy electron determines its weak interaction with the electromagnetic field, i.e. the small probability of bremsstrahlung and, consequently, of the formation of cascade showers. At the same time, the existence of a geomagnetic effect for showers has been experimentally proved, as well as the existence of showers at great depth under water, evidently connected with hard and not soft rays. In the work of Bothe and Schmeiser\(^9\) in 1938, special “hard” showers were studied in detail, the probability of whose formation is proportional to \(Z\), and not to the square of the atomic number \(Z^2\), as in ordinary Heitler cascade showers; in view of this, light elements also give rise to these showers with comparatively high intensity. Bothe’s hard showers have a very small angle of divergence and a considerably greater range than cas-

cascade showers from the soft component. It is possible that the denser core of the shower noticeable in some photographs is connected precisely with Bothe showers. The principal maximum of Rossi’s curve, observed for lead at a thickness of about 5 cm, is caused by cascade showers; the hard showers give the weaker maximum, discovered by Hummel, at 17 cm. One may suppose that the known secondary, much weaker, maxima of Regener’s curve are also partly due to hard showers. Whether Bothe showers coincide with all the showers discovered at great depths under water, or whether the phenomena connected with the hard component are still more complicated, is still premature to judge. Reactions of a semiheavy particle with nuclei of type (3) allow one to expect both a certain number of cascade showers connected with the heavy electron (the photon gives an electron–positron pair, etc.) and burst showers of the Heisenberg or nonlinear type according to reaction (3c): a heavy electron produces other heavy electrons (nonlinear mechanism), which in turn produce new semiheavy particles, etc. At high energy such explosive production of many particles in a single act may be just as probable as the emission of a single pair. It is necessary, of course, to connect showers from the heavy electron with Bothe showers.

Another fundamental question connected with the passage of hard cosmic rays through matter, i.e. with the problem of the interaction of a semiheavy particle with atomic nuclei (since at the present time we may provisionally identify the entire hard component with semiheavy particles, until the possible discovery of protons or other particles in its composition), is the behavior of the new particle when its energy is decreased. On the basis of his most recent measurements, Blackett believes that at energies below \(2 \cdot 10^8\) eV, the particles of the hard group are indistinguishable from soft rays. The impression is created that the new particle proves to be unstable and, at energies of the order of \(2 \cdot 10^8\) eV, changes into an ordinary electron or positron. To conserve angular momentum, a neutrino must then also be emitted, i.e. a reaction of the following kind (in the case of an integral spin of the heavy electron)

\[ a_{\pm} \to e_{\pm} + \nu . \tag{10} \]

To express more definite conjectures about the fate of the heavy electron upon decrease of its energy seems at present risky[^13].

4. The heavy electron and the theory of the atomic nucleus

The question naturally arises: can heavy electrons arise under terrestrial conditions, i.e. in some nuclear reactions, or are they all of cosmic origin? In view of the large value of the rest mass, of the order of \(50—100 \cdot 10^6\) eV, in normal nuclear reactions, which have an energy yield of at most \(10—15 \cdot 10^6\) eV, heavy electrons cannot arise. The maxi-

The maximal energy to which protons or $\alpha$-particles can today be accelerated also lies, at best, within these limits, so that new heavy electrons can arise only as a result of bombarding nuclei with photons or with other particles of the same cosmic energies, for example according to scheme (3). On the other hand, there are no empirical grounds for assuming the presence of ready-made semi-heavy particles in atomic nuclei. Let us recall that the contemporary nuclear model does not so much assert the existence of one or another heavy particle in nuclei as it unconditionally denies the presence in them of any kinds of light particles: electrons, positrons, neutrinos. Light particles must therefore be born during emission, just as photons are born during radiation from an atom.

A strong argument against the presence of electrons in the nucleus is the consideration that, being confined in so small a volume $\left(\Delta q \sim \dfrac{e^2}{mc^2}\right)$, an electron would have to possess an enormous momentum $\left(\Delta p \sim \dfrac{hc}{e^2} mc\right)$ and energy $\left(E \sim c\Delta p \sim \dfrac{hc}{e^2} mc^2 \sim 137\,mc^2\right)$, many times exceeding its own mass. Moreover, such energy values are nowhere observed in nuclear processes. The proton and neutron, however, because of their large mass, will possess energies of the order of those observed in nuclei. Applying this argument to the heavy electron, we see that its energy, when confined in so small a volume, will be of the order of its own energy; therefore this point of view also argues against the possibility of the existence of new semi-heavy particles in atomic nuclei in ready-made form.

Nevertheless, the heavy electron may play an extremely significant role in nuclear processes, carrying the interaction between neutron and proton. The problem of finding intranuclear forces is so essential for all nuclear physics, and the prospects in this direction that have arisen in connection with the discovery of the new particle are so tempting, that we shall dwell somewhat more fully on this point. According to the ideas of quantum electrodynamics, the electromagnetic interaction between two charged particles, for example a proton and an electron, should be represented as realized by means of the following peculiar two-step process: first the first particle (interacting with the photon field) emits a photon (a first-order process), which is then absorbed by the second particle (a second-order process). The emission and absorption of photons occur here virtually, i.e. the photon is by no means emitted “in reality.”

The latter is also clear from the fact that the electromagnetic field exists outside the charge in the form of transverse waves or photons in states of transverse polarization, so that photons of the longitudinal field, which in this case realize the transfer of interaction between charges, are in fact not emitted at all. The process described will become clearer from the following scheme, where $r_1$ and $r_2$ denote the coordinates of the 1st and 2nd particles:

\[ \begin{array}{lll} 0. & e_1(r_1) & e_2(r_2)\\ 1. & e_1(r_1) \searrow\, h\nu & e_2(r_2)\\ 2. & e_1(r_1) & \searrow e_2(r_2) \end{array} \]

Of course, at the same time there occurs the inverse process of emission of the photon by the second particle and its absorption by the first. Thus, as a result, both particles turn out to be connected by two “channels,” i.e. they are in interaction with one another, whereas initially each of them interacted only with the electromagnetic field. Such a picture of the origin of interaction, requiring the formal apparatus of quantum electrodynamics, possesses a peculiar visual clarity not inferior to the model of interaction according to the classical theory of Maxwell.

Moreover, it can be shown that in the case where interaction is transferred by unit quanta-particles, the quantum theory gives a result coinciding with the classical one. Namely, the interaction potential is given by the Green’s function of the equation describing the field of the particles carrying the interaction1.

The field of longitudinal photons is described by Laplace’s equation

\[ \Delta A_0 = 0, \]

whose solution gives the Green’s function \(G\)

\[ G = A_0 = -\frac{1}{4\pi |r_1-r_2|} = \frac{1}{4\pi r} \]

and, at the same time, the interaction according to Coulomb’s law

\[ V = e_1 e_2 G = \frac{e_1 e_2}{4\pi r}. \]

When the vector potential of the field, i.e. transverse photons, is taken into account, terms are added to the Coulomb interaction energy that depend not only on the coordinates but also on the velocities and spins of the interacting particles. The expression for the mutual potential energy in this case is given by the Breit—Møller formulas, which represent a regular approximation to the complete relativistic solution.

The interaction between nuclear heavy particles (the neutron and the proton) obviously cannot have an electromagnetic character, owing to the absence of charge in the neutron. What other particles, then, can the neutron emit, even if only virtually, in order to be able to exchange them with other neutrons and protons? Naturally, the first thought that comes to mind is of electrons, which, after all, are emitted not only virtually but also really in \(\beta\)-decay. In that case, it would seem, the interaction potential between two heavy particles \(V\) would be given by the Green’s function \(G\) of the wave equation for the field

electrons, i.e. the quantized Dirac equation, or, more crudely, the relativistic equation of the second order (4), i.e.

\[ V=\mathrm{const}\cdot \frac{e^{-k_0 r}}{r};\qquad k_0=\frac{mc}{h}. \tag{11} \]

This formula, attractive in its simplicity, was, however, rejected by the present writer in 1933 for the following reasons. The principal merit of a potential of the form (11) is that it gives “short-range” forces, in contrast to the Coulomb energy, which falls off very slowly with distance. Indeed, at distances \(r\) greater than the Compton wavelength of the electron \(\frac{h}{mc}\), i.e. \(r>10^{-11}\ \mathrm{cm}\), the potential \(V\) practically vanishes, thereby reflecting an essential aspect of nuclear forces. Unfortunately, \(10^{-11}\ \mathrm{cm}\) is too large a distance, since nuclear forces extend no farther than \(10^{-13}\)—\(10^{-12}\ \mathrm{cm}\). Substitution in (11) of the Compton wavelength for a heavy particle, on the contrary, gave too small a range, of the order \(10^{-14}\)—\(10^{-15}\ \mathrm{cm}\). A second negative argument was the essential observation that a single electron by itself can never be emitted or absorbed by a neutron, which follows first of all from the impossibility of constructing from a single spinor Dirac wave function of the electron \(\psi_i\) the fourth component of a vector and, consequently, an expression for the energy of interaction with the electron field. Moreover, in the emission by a neutron of only one electron the law of conservation of spin would be violated, since all three particles \((n,p,e)\) have spin equal to \(\frac{1}{2}\) in units of \(\frac{h}{2\pi}\). Hence we see the necessity that, simultaneously with the electron, another particle be produced, likewise possessing spin \(\frac{1}{2}\) and described by a spinor wave function, but obviously without charge; that is, we arrive at the hypothesis of the neutrino, which historically arose by another route, from considerations of the conservation of energy in \(\beta\)-decay. Now from two spinor functions one may simply compose the energy of interaction of a heavy particle with the field of pairs of light particles: electrons and neutrinos, in the form

\[ U=-g\psi_i^{\,e}\psi_i^{\,\nu}, \tag{12} \]

where \(g\) is a new constant of dimension \(\mathrm{erg}\cdot \mathrm{cm}^3\), playing the role of the “charge” of the heavy particle.

On the basis of similar ideas, Fermi in 1934 made expression (12) the foundation of his theory of \(\beta\)-decay, which was the first, and so far the only, coherent picture of this most subtle nuclear process. Despite its always clearly recognized preliminary character, Fermi’s theory undoubtedly contains a significant share of truth. It made it possible to decipher the extremely tangled material of \(\beta\)-spectra, to derive cor—

obviously Sargent’s law (i.e., an analogue of the Geiger—Nuttall law for $\beta$-decay), explained the general characteristic shape of the $\beta$-spectrum curve, made it possible to draw conclusions about the mass of the neutrino and, finally, stimulated the development of the general theory of the interaction of nuclear particles and their magnetic moments[^10].

If we stand firmly on the ground of the hypothesis that the interaction between neutrons and protons is carried by some particles of the electron type, then one can proceed further in two directions:

1) either retain the ordinary electron with the addition of the neutrino, which together will carry the interaction, or 2) introduce a new particle of the electron type, but in any case possessing integral spin, so that the interaction could be carried by single new particles.

A calculation of nuclear forces on the basis of the first possibility, using formula (12), at the suggestion of Tamm and of the present author, indeed gives an interaction energy between heavy particles that decreases very rapidly with distance, has the required exchange character, and is equal in magnitude to

\[ V=\frac{g^2}{hcr^5}. \tag{13} \]

However, substituting the empirical value of the constant $g$, determined by comparing the theory with the data of $\beta$-decay, gives, at the distances of interest to us $r\sim 10^{-13}\ \mathrm{cm}$, a negligibly small result: $V\sim 10^{-6}\ \mathrm{eV}$, instead of the expected $10^{+6}\ \mathrm{eV}$. An equally large discrepancy is given by the calculation of the magnetic moment of the neutron. Although the corrections introduced by Konopinski and Uhlenbeck into the theory of $\beta$-decay [the introduction of derivatives of the $\psi$-functions into formula (12)] improve not only the agreement of Fermi’s theory with experiment, but also reduce the discrepancy in the theory of nuclear $\beta$-forces, the indicated scheme of the neutron—proton interaction could not be brought to a completed form free of shortcomings. The physical basis for the small magnitude of the $\beta$-forces lies in the very small probability of $\beta$-decay, so that the mean lifetimes of nuclei excited before $\beta$-emission (or, roughly speaking, of the protons and neutrons excited in nuclei) are extraordinarily, so to speak “geologically,” large in comparison with the periods of all possible nuclear processes and oscillations of particles in the nucleus. Undoubtedly, the theory of nuclear $\beta$-forces (which are apparently just as insufficient for explaining the enormous binding energies in the nucleus as are gravitational or electromagnetic forces, for example, between two protons) has taught us much: for the first time, by means of quantum theory, information was obtained on the possibility of carrying interaction not by single particles, but by pairs: an electron and a neutrino, or a positron and a neutrino, something that has no analogue in classical theory. Further, the fundamental possibility was clarified of constructing exchange forces of various types: with exchange of coordinates (Majorana forces), or of coordinates and spins

(Heisenberg case). Such a representation of the interaction between \(n\) and \(p\), caused by the virtual emission by a neutron of an electron and a neutrino \((n \to p + e_- + \nu)\) (a first-order process), with the transformation of \(n\) into \(p\), and the subsequent absorption of this pair by a proton \((p + e_- + \nu \to n)\) (a second-order process), with the transformation of \(p\) into \(n\), also made it possible to raise the question of the virtual emission and absorption of a pair of light particles by one and the same heavy particle, i.e., of the self-interaction and magnetic moment of the heavy particles. The general scheme of exchange by an electron and a neutrino has the following form:

\[ \begin{array}{lll} 0. & n(r_1) \searrow & p(r_2)\\ 1. & p(r_1),\ e_- + \nu, & p(r_2)\\ 2. & p(r_1) \searrow & n(r_2) \end{array} \]

From this it is clear that, as a result, the neutron and the proton have exchanged the coordinates \(r_1\) and \(r_2\), i.e., the forces have an exchange character. Of course, at the same time the interaction will be transferred through the positron and the neutrino emitted by the proton. Leaving aside other proposed hypotheses going in the same direction (the transfer of the interaction by a pair: electron + positron, or electron and three neutrinos, etc.), let us turn to the second of the indicated possibilities, connected with the discovery of the heavy electron. In reality a heavy electron may also not be emitted by nuclei, but nevertheless, according to the ideas of quantum electrodynamics, it will with some probability be virtually produced by one nuclear heavy particle and absorbed by another, i.e., it will be capable of carrying the interaction inside the nucleus. Thus, let us admit the following second-order process: suppose that at the beginning there is a neutron at the point \(r_1\) and a proton at \(r_2\), each interacting only with the field of heavy electrons, but not with each other (just as the electron must interact with the field of unborn photons in order to be able to emit the latter).

Let now the neutron at the point \(r_1\) emit a negatively charged heavy electron and turn into a proton; in view of the integral spin of the heavy electron \(a\) and the vector character of its wave function, no neutrino is needed here. Thus, as a result of the first-order process \((n \to p + a_-)\) we shall have two protons and a heavy electron.

Finally, the initial proton at the point \(r_2\) will absorb the heavy electron and turn into a neutron at the point \(r_2\) \((p + a_- \to n)\), and as a result of the second-order process we shall again obtain a neutron and a proton, but now interacting with one another. The interaction will again have an exchange character, which is best seen from the following scheme:

\[ \begin{array}{lllllll} 0. & n(r_1) \searrow & p(r_2) & \quad & n(r_1) & \swarrow p(r_2)\\ 1. & p(r_1) & a_- & p(r_2) & \quad & n(r_1) & a_+ \quad n(r_2)\\ 2. & p(r_1) & \searrow n(r_2) & \quad & p(r_1) \swarrow & n(r_2) \end{array} \]

At the same time the interaction will be carried by the positive semiheavy particle \(a_{+}\), emitted by the proton. Hence it is clear that \(n\) and \(p\) have exchanged coordinates. The energy of the exchange interaction between the neutron and the proton can again be calculated by quantum electrodynamics, starting from the simplest expression for the interaction of each heavy particle with the field of unit semiheavy particles of the form (7)

\[ U_1 = g \varphi_0, \tag{14} \]

where \(\varphi_0\) is the fourth component of the wave \(\psi\)-function of the semiheavy particle, now replacing the ordinary electromagnetic scalar potential \(\varphi_0\), and \(g\) is a new constant—the “charge” of the heavy particle, coinciding in dimension with the ordinary electric charge. It is possible, however, on the basis of the theorem mentioned above, to write at once the desired interaction energy \(V_1\) as the Green’s function \(G\) of the equation describing the static field of semiheavy particles, i.e. equation (4) with the time dependence (11) omitted,

\[ \left. \begin{aligned} \Delta \varphi_0 - \frac{m_a^2 c^2}{\hbar^2}\varphi_0 &= 0,\\ V_1 = \mathrm{const}\cdot G &= \frac{g^2 e^{-k_0 r}}{r}, \end{aligned} \right\} \tag{15} \]

where

\[ k_0=\frac{m_a c}{\hbar}. \tag{16} \]

In view of the large mass of the semiheavy particle \(m_a\) in comparison with the mass of the electron \(m\), the corresponding Compton wavelength

\[ \lambda_a=\frac{1}{k_0}=\frac{h}{m_a c} \]

will be \(100\)—\(200\) times smaller than the ordinary Compton wavelength

\[ \lambda_e=\frac{h}{mc}. \]

Thus, in view of the rapid damping of the factor \(e^{-k_0 r}\) for

\[ r>\frac{1}{k_0}, \]

it is immediately clear that the nuclear forces obtained will be of “short” range, extending over distances not greater than \(\lambda_a\), which gives precisely the correct order for the nuclear radius, \(10^{-13}\ \text{cm}\),

\[ \left(\lambda_a \simeq \frac{\lambda_e}{100}\sim \frac{10^{-11}}{100}=10^{-13}\right). \]

Conversely, of course, from the value of the nuclear radius one can determine the order of the semiheavy mass \(m_a=m\cdot 100\). This fundamental result (which we could not previously obtain by substituting the electron mass \(m\) into equation (4) and formula (11)) compels us to pay attention to this variant of the model of nuclear forces and justifies attempts at a further refinement of the theory, to a brief exposition of which we shall now turn. Of course, the coupling of the heavy particle with the field of the semiheavy particles of the form \(U_1=g\varphi_0\) is only

part of the general interaction, which, along with the 4th component of the potential \(\varphi_0\), must also include terms with the vector potential \(\varphi_i\), and also, possibly, with the six-vector \(F, G\) of the Proca field. As always, relativistic considerations lead directly to reasonable results, which we shall make clear by comparison with the corresponding expressions for the interaction of an ordinary charged particle, for example an electron, with an electromagnetic field described by scalar and vector potentials.

A priori, in the electron equation, besides the coupling with the field potentials, there could also enter a term of the Pauli type, describing the interaction of the magnetic moment with the magnetic field: \(U_2=-\mu H\). As is known, for the electron this is superfluous, since the corresponding addition is automatically taken into account by the Dirac equation. The possibility of such an addition to the equations for a neutral neutron possessing magnetic moments was specially discussed by Oppenheimer and Tamm. In the present case we have no prior arguments against including such terms; moreover, they prove to be essential for a correct theory of the deuteron. Ultimately we shall obtain the following expressions for the interaction with the field:

Charged particle of “charge” \(e\) in an electromagnetic field

\[ U_1=e\left\{\varphi_0+\frac{1}{c}(vA)\right\} \]

\[ U_1\to e\varphi_0\left(\text{for }\frac{v}{c}\ll 1\right) \]

Heavy particle of “charge” \(g\) in a semi-heavy field,

\[ U_1=g\left\{\varphi_0+\frac{1}{c}(v\varphi)\right\} \tag{17} \]

\[ U_2=\frac{f}{k_0}\left\{\rho_2(sF)+\rho_3(\sigma G)\right\} \tag{18} \]

Full interaction:

\[ U=U_1+U_2 \]

in the nonrelativistic approximation becomes

\[ U\to g\varphi_0+\frac{f}{k_0}(\sigma G), \tag{19} \]

where \(v\) is the velocity of the particle, which in the Dirac equation is replaced by the matrix \(c\alpha\), \(f\) is a new constant, a priori independent of \(g\), but of the same dimension as the ordinary charge. Comparison with experiment (see below) permits one tentatively to set \(f\simeq g\).

The table also contains approximate nonrelativistic expressions for the interaction, applicable at small velocities of heavy particles, to the consideration of which we shall restrict ourselves. From the latter it is seen that \(U_1=U_{\parallel}\) will refer to the interaction with the longitudinal part of the field, and \(U_2=U_{\perp}\) to that with the transverse part. Ve-

the quantity \(\dfrac{f}{k_0}\sigma\), where \(\sigma\) are the three Dirac spin matrices, plays the role of the intrinsic quasi-magnetic moment of the heavy particle, \(G\) is the magnetic part of the six-vector of the Proca field. It is needless to emphasize that all these quantities are not electromagnetic in nature; for example, the uncharged neutron possesses a quasi-charge \(g\), etc.

More precisely, the binding energy of each heavy particle with the field of half-heavy ones should be written in the form

\[ U_1=g\left(Q\varphi_0+Q^*\varphi_0^*\right),\qquad U_2=\frac{f}{k_0}\{Q(\sigma G)+\text{complex conjugate}\}, \tag{20} \]

where \(Q\) is the Heisenberg operator for the transformation of a neutron into a proton, \(Q^*\) the operator of the reverse transformation. The proton—neutron interaction energy will then contain a coefficient of the form

\[ P_{\mathrm H}=\frac{Q^*(1)Q(2)+Q(1)Q^*(2)}{2}, \tag{21} \]

indicating the exchange character of the forces (“either the 1st particle is transformed into a neutron and the 2nd into a proton, or vice versa”). The longitudinal part of the interaction \(V_1\) (16) obtained above must also be multiplied by the coefficient \(P_{\mathrm H}\).

A calculation by quantum electrodynamics gives the following expression for the transverse part of the neutron—proton interaction energy:

\[ V_2=P_{\mathrm H}f^2\frac{e^{-k_0r}}{r} \left\{ (\sigma_1\sigma_2)\left(1+\frac{1}{k_0r}+\frac{1}{k_0^2r^2}\right) - \frac{(\sigma_1r)(\sigma_2r)}{r^2} \left(1+\frac{3}{k_0r}+\frac{3}{k_0^2r^2}\right) \right\}, \tag{22} \]

where \(\sigma_1,\sigma_2\) denote the spins of the two heavy particles, and \(r=|r_1-r_2|\) is the distance between them. The total interaction will be the sum of the transverse and longitudinal,

\[ V=V_1+V_2. \]

The exact solution of the Schrödinger equation for the deuteron with potential \(V\) is very complicated, owing to the close coupling of the orbital motion with the spins. We shall not, however, make a large error if we take the wave function in the form of a product of an orbital and a spin function and perform an averaging over the spins. Heitler—Kemmer—Fröhlich and Yukawa with collaborators, restricting themselves to consideration of \(S\)-terms, obtain the proton—neutron interaction in the form of a combination of Heisenberg and Majorana terms

\[ V= \left\{ P_{\mathrm H}\left(g^2-\frac{2}{3}f^2\right) + P_{\mathrm M}\frac{4f^2}{3} \right\} \frac{e^{-k_0r}}{r}, \tag{23} \]

or, for \(g=f\),

\[ V=\frac{g^2}{3}\left(P_{\mathrm H}+4P_{\mathrm M}\right)\frac{e^{-k_0r}}{r} \left(P_{\mathrm H}=P_{\mathrm H}\frac{1+\sigma_1\sigma_2}{2}\right). \tag{23a} \]

It is precisely such a relation between the principal Majorana forces and the weaker Heisenberg forces that is required by the existing semi-empirical theory of the deuteron. For the fundamental triplet and the second singlet states of the deuteron one obtains

\[ {}^{3}S: V=-\frac{e^{-k_{0}r}}{r}\left(g^{2}+\frac{2}{3}f^{2}\right)\to -\frac{5}{3}g^{2}\frac{e^{-k_{0}r}}{r}, \tag{24a} \]

\[ {}^{1}S: V=-\frac{e^{-k_{0}r}}{r}\left(2f^{2}-g^{2}\right)\to -g^{2}\frac{e^{-k_{0}r}}{r}. \tag{24b} \]

It is also not difficult to obtain an analogue of the Møller–Breit formula, when the interaction between heavy particles depends on their velocities. From comparison with the empirical value of the depth of the potential well in the deuteron, equal to \(25\cdot 10^{6}\ \mathrm{eV}\) for the ground state, substituting \(r=3\cdot 10^{-13}\ \mathrm{cm}\), we shall find for the new constants values of the order of several electron charges.

It is interesting to note that purely theoretical considerations, even without introducing empirical data on the deuteron, make it possible to indicate reasonable bounds for the new constants \(g\) and \(m_a\), or \(k_0=\frac{m_a c}{h}\).

The upper bound for \(g\) is determined from the requirement that the dimensionless quantity \(\frac{g^{2}}{hc}\) (an analogue of the fine-structure constant), in powers of which the expansion would proceed in perturbation calculations, should be less than unity. Hence, with \(\frac{e^{2}}{hc}=\frac{1}{137}\), we obtain

\[ \frac{g^{2}}{hc}<1,\quad g<\sqrt{137}\,e;\quad \text{i.e.}\quad g<11e. \tag{25} \]

The best value at present, equal to \(5e\), gives \(\frac{g^{2}}{hc}\sim \frac{1}{6}\). Applying next to the potential \(V=\frac{g^{2}e^{-k_{0}r}}{r}\) the virial theorem and the simplest Bohr quantization, we obtain for determining the radii of stable orbits the equation

\[ \beta e^{\alpha}=a^{2}+\alpha,\quad (\alpha=k_{0}r), \tag{26} \]

where \(\alpha\) must be less than unity. Hence, as the condition for solvability of the transcendental equation, we have \(\beta<\frac{0.7}{2}=0.35\), and for the ratio of the masses of the heavy electron and the proton

\[ \frac{m_a}{M}<0.35\,\frac{g^{2}}{hc};\quad \frac{m_a}{M}<0.3, \]

where \(\beta=\frac{m_a}{M}\frac{hc}{g^{2}}\) of the proton or neutron, whence we obtain the upper bound for the mass of the heavy electron

\[ m_a<0.3M,\quad m_a<500m. \tag{27} \]

Thus, if the heavy electron is at all the transmitter of nuclear forces, then its mass cannot exceed 500 electron masses,

and the new charge of the heavy particles cannot be greater than \(11e\). It is very satisfactory that the extreme experimental values do not go beyond these limits, while the best data (see above the results of the analysis of Corson—Brode and of Heitler’s theory) are equal to one half of the indicated upper limits.

We see that the application of the heavy electron to the model of the nucleus makes it possible, though for the time being not very accurately, to determine theoretically, without any measurements with cosmic rays, the value of the mass of the new particle and the charge of the heavy particles. This theory of nuclear forces in a simpler form was proposed by Yukawa, after the small value of the energy given by \(\beta\)-rays had become clear. After the discovery of the heavy electron, Yukawa’s hypothesis naturally attracted great attention. At the present time an intensive development is under way of Proca’s equations, which apparently are quite capable of satisfactorily describing the properties of the new particle, and, in particular, the fruitfulness of the scheme of nuclear forces under consideration is being clarified. Proca’s Maxwell-like equations (8) describe the motion of a free semi-heavy particle. In the presence, however, of an external electromagnetic field specified by the potentials \(A_0, A\), differentiation with respect to time and coordinates, or the corresponding momenta, must be supplemented by terms with the potentials, as in the completely analogous case of the Schrödinger or Dirac equations

\[ \frac{1}{c}\frac{\partial}{\partial t} \to \frac{1}{c}\frac{\partial}{\partial t} + \frac{ie}{hc}A_0; \qquad \operatorname{grad} \to \operatorname{grad} - \frac{ie}{hc}A . \]

Thus equations (8) will take the form

\[ \left. \begin{aligned} \left(\frac{1}{c}\frac{\partial}{\partial t}+\frac{ie}{hc}A_0\right)F - \left[\operatorname{grad}-\frac{ie}{hc}A\,G\right] -k_0^{\,2}\varphi &= 0,\\ \left(\operatorname{grad}-\frac{ie}{hc}A\right)F +k_0^{\,2}\varphi &= 0, \end{aligned} \right\} \tag{28} \]

where \([\ ]\) denotes the vector product, and \(\varphi_0,\ \varphi\) are the four components of the heavy-electron function.

Hence, after iteration and passage to the nonrelativistic case [replace \(\dfrac{h}{i}\dfrac{\partial}{\partial t}\varphi\) by \((E+m_a c^2)\varphi\); discard terms with \(E^2\) and divide the whole equation by \(2m_a c^2\)], which is an exact repetition of the corresponding operations with the Dirac equation (see, for example, Prof. Fock’s Principles of Quantum Mechanics), we finally obtain the Schrödinger wave equation for a heavy electron of charge \(e\) in an electromagnetic field

\[ \left\{ -E+eA_0 - \frac{h^2}{2m_a} \left( \operatorname{grad}-\frac{ie}{hc}A \right)^2 \right\}\varphi - \frac{ieh}{2m_a c}[\mathbf{H}\varphi] =0 . \tag{29} \]

The additional term

\[ \frac{ieh}{2m_a c}(\mathbf{H}\varphi) \]

can be rewritten in the form

\[ \mu_a(\sigma' H)\zeta, \]

where

\[ \mu_a=\frac{eh}{2m_a c},\quad \sigma'_x= \begin{pmatrix} 0&0&0\\ 0&0&1\\ 0&-1&0 \end{pmatrix},\quad \sigma'_y= \begin{pmatrix} 0&0&-1\\ 0&0&0\\ 1&0&0 \end{pmatrix}, \]

\[ \sigma'_z= \begin{pmatrix} 0&1&0\\ -1&0&0\\ 0&0&0 \end{pmatrix},\quad \varphi= \begin{vmatrix} \varphi_1\\ \varphi_2\\ \varphi_3 \end{vmatrix} \]

From this it is evident that the quantity \(\mu_a\) plays the role of the intrinsic magnetic moment of the heavy electron. Since the matrices possess the eigenvalues \(\pm 1,0\), the heavy electron may have a vanishing moment. In this way the laws of interaction of the new particle with neutrons and protons [see equations (19)] and with the electromagnetic field are established. The value of the magnetic moment of the heavy electron can be used to explain the magnetic moments of heavy particles. Just as the new theory corrects the small value of the binding energy obtained from the \(\beta\)-force model, the value of the magnetic moment of the neutron will now also have the correct order of magnitude.

According to Wick’s hypothesis, the magnetic moment of the neutron will be due to the magnetic moment \(\mu_a\) of the virtually emitted particle, i.e. in the present case a heavy electron of negative charge, and will be given by the value \(\mu_a\), multiplied by the relative time spent by the neutron in the virtually split state: neutron \(\to\) proton \(+\) negative heavy electron. The relative lifetime of the neutron in the split state is approximately

\[ \tau \simeq \frac{g^2}{hc}, \]

where \(g\) is the charge of the heavy particle introduced above. Indeed, the probability of splitting, or the square of the ratio of the perturbed and unperturbed wave functions, must be a dimensionless number proportional to the square of the perturbation parameter, i.e. to the charge \(g\). Hence the negative magnetic moment of the neutron, in agreement with empirical data, proves to be equal to

\[ \mu_n \sim \frac{g^2}{hc}\,\mu_a \sim \frac{g^2}{hc}\,\frac{eh}{2m_a c} \sim \frac{eh}{2mc}\cdot\frac{m}{M}\cdot\frac{M}{m_a}\cdot\frac{1}{6} \sim 1.5\,\mu_0, \]

where \(\mu_0\) denotes the nuclear magneton, equal to

\[ \frac{1}{1840}\cdot\frac{eh}{2mc}, \]

i.e. \(\frac{1}{1840}\) of the Bohr magneton. Similarly one determines the additional magnetic moment of the proton, due to the virtual splitting of the latter into a neutron and a heavy positron. Comparison of the obtained values of the magnetic moment of the neutron and proton with experimental data provides a method for independently determining the mass of the new particle, given the charge \(g\). Hence we find, following Heitler: \(m_a \sim 180m\). Having at our disposal the law of forces between the neutron and the proton, we shall finally be able to construct the theory of the nucleus on a rational basis, although, of course, in view of the known extreme crudeness of the one-body model, it will not be possible here to speak of the immediate construction of a complete periodic system of nuclei; i.e., relatively speaking, in comparison with atomic theory, knowledge of the law of forces between

nuclear particles gives less than the value of Coulomb’s law in the atom. In view of the, at present, not very precise value of the mass of the new particle and of the charge \(g\), the calculation of semi-heavy forces between two protons or two neutrons, requiring the inclusion not of the 2nd but of the 4th approximation, is of an even more preliminary character. According to the interpretation of the experiments of Tuve and his collaborators (Breit and others), between two protons, in addition to Coulomb repulsion, there act attractive forces of the nuclear type, approximately equal to the forces between neutron and proton. It is still difficult to say whether, in order to explain these forces, it will be necessary to introduce the additional hypothesis of the existence of a “neutretto”—neutral semi-heavy particles \(a_0\), which two neutrons or two protons could exchange directly in the 2nd, and not in the 4th, approximation (compare with the exchange of neutral photons). For explanation we give the scheme of exchange by charged (on the left) and neutral (on the right) particles (the symbols 1 and 2 indicate the coordinates of the particles \(r_1\) and \(r_2\), and, for simplicity, we do not take spin into account)

0.  n₁                         n₂      0.  n₁          n₂
1.  p₁, a₁ ——;                 n₂      1.  n₁   a₀     n₂
2.  p₁, a₁ … ; a₂ —, p₂                3.  n₁          n₂
3.  p₁, a₂ …—                 n₂
4.  n₁                         n₂

The calculation of Heitler—Fröhlich—Kemmer gives for the forces between identical heavy particles in the 4th approximation always repulsion, whereas the exchange of neutral heavy electrons will, of course, give attraction, which may reach the magnitude of the neutron—proton interaction. Let us note also that the introduction of a new particle into nuclear physics opens paths also for refining the theory of \(\beta\)-decay. It is possible that heavy particles interact directly only with semi-heavy ones, and the latter both with heavy ones and with the field of pairs of light particles: electrons and neutrinos. A heavy electron, during its journey from proton to neutron, may with some probability give rise to an electron and a neutrino. Since such inclusion of an intermediate agent does not alter the essential aspects of \(\beta\)-decay, the general picture of Fermi’s theory will remain unchanged. The small probability of \(\beta\)-decay will be connected with the fact that semi-heavy particles interact much more intensely with heavy ones than with light ones.

Finally, let us emphasize that the discovery of the heavy electron, besides deciphering the nature and basic properties of the hitherto completely incomprehensible hard component of cosmic rays, besides a very substantial refinement of the model of nuclear forces, will undoubtedly also contribute considerably to the construction of a general theory of elementary particles and, in particular, to elucidating the nature of proper mass. If one assumes that the mass of the electron and positron is electromagnetic in nature, then will the mass of the neutron-proton be mainly of a semi-heavy character, since the interaction

TABLE 1

Table of elementary particles, 1938

heavy particles, and consequently even part, at least, of the self-action is effected through semipositive particles? What, then, is the nature of the mass of the heaviest electron itself? Perhaps it is due precisely to \(\zeta\)-forces, since the new particles can emit an electron and a neutrino or decay into a pair of these particles? In connection with all this, the problem of introducing into the theory a new constant, say, the radius of the electron (or proton, or neutron, or heavy electron), is a very delicate question, going beyond the scope of the present note.

In conclusion of the first survey of the new particle we shall give a table of the entire family of elementary particles known today, entering in their “passports” the most fundamental properties. The general picture of the structure of matter now appears in the following form: cosmic rays consist of electrons, positrons, and (secondary) photons (the soft component); hard cosmic rays, for the most part, probably consist of charged heavy electrons of both signs. Under terrestrial conditions matter consists of electromagnetic fields (photons) and atoms, composed of negative electrons and positive nuclei. Nuclei consist of heavy particles—neutrons and protons. We leave all gravitational properties aside. The most doubtful experimental and theoretical data are marked with a question mark.

In this table the first three rows are the best known and sufficiently guaranteed. By no means are all cells of the table independent; for example: spin and statistics mutually determine one another, and both these characteristics are uniquely connected with the type of equation. The magnetic moment of the electron, which is the most important of the secondary properties, as is known, is also completely determined by the Dirac equation. Obviously, a similar reduction of the table in the horizontal directions will take place, apart from electrons and photons, also for other particles, for which the equations of motion are not yet so reliable.

The most difficult point of the future theory of elementary particles will apparently be the establishment of the connection between mass and charge and the type of equation. At the same time there must be a considerable reduction of the table also vertically, along with the possible addition of new particles, such as antiprotons (?), neutral heavy electrons (?), etc. (particles which at the present moment we can neither convincingly predict nor forbid). A general theory for electron–positron pairs, which frequently arise in pairs, is the beginning of such a reduction. In the order of vertical contraction, the proton and the neutron will probably turn out to be two states of one and the same heavy particle; further, according to the neutrino theory of light, the photon should be regarded as a certain combination of a pair of neutrinos, which gives another example of the probable bringing together of particles (12); finally, the new semiheavy particle may possibly be reduced to the electron and the neutrino. It may be that, in general, all elementary particles will turn out to be different states of one fun-

fundamental particle, so that the different mass values future theory will calculate just as naturally as a series of term values in the spectrum of hydrogen.

If we recall that the greater part of the table dates from 1928 (Dirac’s equation), and that especially intensive filling-in of it began only 5–6 years ago (the neutron, the nuclear model, the discovery of the positron and of the heavy electron, the neutrino hypothesis), then the feeling of optimism characteristic of contemporary physics will seem to us entirely justified, despite the great gaps in our understanding of the connections between the basic particles of matter.

REFERENCES

1. Ivanenko, Uspekhi fizich. nauk, 19, 322, 1938; on the mathematical2 side of shower theory see Ivanenko and Sokolov, ZhETF (in press).
2. Blackett, Proc. Roy. Soc., 165, 11, 1938; Blackett and Wilson, ibid., 165, 209, 1938.
3. Corson and Brode, Phys. Rev., 53, 773, 1938.
4. Iwanenko and Sokolow, Sow. Phys., 11, 590, 1937; Trudy SibFTI, 5, 32, 1937; Weizkopf, Uspekhi fizich. nauk, 16, 293, 1936.
5. Proca, Journ. de Phys., 7, 347, 1936; 8, 23, 1937.
6. Durand and Ershov, Sow. Phys., 12, 466, 1937.
7. Yukawa, Sakata, Taketani, Proc. Phys.-Math. Soc. Japan, 20, No. 4, 1938.
8. Kemmer, Proc. Roy. Soc., 166, 127, 1938; Fröhlich, Heitler and Kemmer, ibid., 166, 154, 1938.
9. Bethe and Schmeiser, Ann. d. Phys., 32, 161, 1938.
10. Heisenberg, Uspekhi fizich. nauk, 16, 1, 1936.
11. Ivanenko, ZhETF, 8, 260, 1938.
12. Sokolov, ZhETF (a series of articles in 1937–1938).
13. See the works of I. Tamm, developing the hypothesis of the half-integer spin of the new particle.

1. • *

   ↩

2. Readers are requested to correct, in connection with this article, the errors that slipped into the equations given in the notes to our previous review in Uspekhi fizich. nauk. ↩