THE MESOTRON THEORY AND NUCLEAR FORCES1

I. E. Tamm, Moscow

As is known, the hypothesis of the existence of mesotrons, as particles with a mass several hundred times greater than the mass of the electron, was first put forward by Yukawa long before their experimental discovery. Yukawa arrived at this hypothesis after the failure of numerous attempts to construct a theory of nuclear forces, i.e., the forces acting between heavy particles (protons, neutrons), in which it was assumed that these forces arise because heavy particles exchange light particles (electrons and positrons). The very fact that the existence of mesotrons had to be postulated even before their discovery testifies to the unusually important role of mesotrons in the theory of nuclear forces.

Furthermore, in the field of cosmic rays a large number of facts have accumulated indicating the essential role played by mesotrons in the processes that occur when cosmic rays pass through matter. For example, such a fundamental fact as the presence at the earth’s surface of cosmic rays of considerable intensity cannot be understood without taking into account the role of mesotrons. In essence, only with the discovery of mesotrons did attempts become possible to construct a coherent picture of the passage of cosmic rays through the atmosphere. It is true that at the present time the construction of this picture is not yet complete; there remain many unclear and unexplored questions. However, the central role of mesotrons in all questions of cosmic-ray physics is already visible with complete clarity.

At present it has been established that the mass of the mesotron is approximately two hundred electron masses; the charge of the mesotron is, in absolute value, equal to the elementary charge and occurs with both signs (positive and negative mesotrons).

Besides charge and mass, the third fundamental quantity characterizing a particle is its spin. Up to the present time it has not yet been possible to determine the spin of the mesotron experimentally.

In Yukawa’s theory it is assumed that the spin of the mesotron is equal to one. This assumption is accepted, as a certain working hypothesis, by almost all investigators, although there is as yet no direct confirmation of it.

The study of the properties of mesotrons in cosmic rays has shown that mesotrons are unstable and decay spontaneously, apparently into an electron and a neutrino (or a positron and a neutrino). The decay time of a mesotron at rest is approximately \(2 \cdot 10^{-6}\) sec.; the decay time of moving mesotrons increases as

\[ \frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}} . \]

The existing experiments lead to considerable difficulty in the question of the decay of mesotrons. Namely, in most of the measurements made up to the present time, the required number of decay electrons has not been found. However, none of the experiments carried out is conclusive, and the question of the mechanism and products of mesotron decay requires further, more precise investigations.

There is no doubt that the question of mesotron decay is one of the most important and urgent questions in the physics of cosmic rays.

From the fact that mesotrons prove to be unstable and decay spontaneously, it follows directly that they cannot arrive at the Earth from outer space, since in that case they would all have time to decay en route. It is therefore necessary to assume that they are formed in the upper layers of the atmosphere by other particles or \(\gamma\)-rays entering into the composition of cosmic radiation. Direct measurements of the latitude dependence of the penetrating component of cosmic rays confirm the assumption of the formation of mesotrons in the upper layers of the atmosphere.

The question of the formation of mesotrons is at present the central problem of cosmic-ray physics. The mechanism of mesotron generation has not yet been investigated. One of the probable processes leading to the formation of mesotrons should apparently be considered a process of the type \(p + h\nu \rightleftarrows n + \mu^{+}\), or of the type \(n + h\nu \rightleftarrows p + \mu^{-}\), where \(h\nu\) is a \(\gamma\)-quantum, \(p\) a proton, \(n\) a neutron, and \(\mu^{+}\) and \(\mu^{-}\), respectively, positive and negative mesotrons.

However, such a mechanism leads to the following difficulty: if, from the known number of photons, protons, and neutrons in the upper layers of the atmosphere and from the number of mesotrons formed in them, one determines the effective cross-section of this process, then, with the aid of the principle of microscopic reversibility, one can also find the cross-section of the inverse process. The inverse process is evidently the capture of mesotrons by heavy particles. In this case it turns out that the effective capture cross-section obtained is too large and would lead to a considerably greater absorption of mesotrons in passing through matter than actually occurs. Therefore one may regard the question of the mechanism of mesotron generation as still entirely open and requiring further study.

The next important problem in cosmic-ray physics is the study of ionization bursts caused by mesotrons, as well as the composition and energy spectrum of cosmic rays at great depths. Of particular interest is the question of the generation of light particles (electrons, positrons) or photons by fast mesotrons.

As theory shows, the probability of the formation of light particles and photons when mesotrons pass through matter depends essentially on the spin of the mesotron. A characteristic feature of particles with spin equal to unity is their ability, in a single act, to transfer a considerable fraction of their energy to light particles.

The existing experimental data seem to favor the supposition that the spin of the mesotron is equal to unity. In particular, there apparently exists a number of processes, such as, for example, Hoffmann ionization bursts, in which the mesotron transfers enormous energy to a light particle, which then forms a large shower containing several hundred particles. Furthermore, the same is apparently indicated by the increase in the percentage of the soft component at great depths. However, the accuracy of the data obtained is still very low, and their experimental refinement is highly desirable.

In nuclear physics the mesotron plays no smaller a role than in cosmic rays. This is connected with the fact that, according to the theory of nuclear forces, first proposed by Yukawa and then developed by the works of Heitler, Kemmer, Bhabha, and a number of others, the interaction of heavy particles in the nucleus is effected by means of a “mesotron field,” just as the electrical (Coulomb) interaction of particles is effected by means of a “photon field.”

According to Yukawa’s theory, a proton can turn into a neutron by emitting a positive or absorbing a negative mesotron, while a neutron can turn into a proton by the inverse process.

The interaction of heavy particles with one another, according to Yukawa, is connected with a number of successive transformations of a proton into a neutron and back: a proton, emitting a positive mesotron, turns into a neutron. The emitted mesotron is absorbed by a neutron, which turns into a proton, and so on. When negative mesotrons are emitted, the transformations proceed in the reverse order.

Since the transfer of mesotrons from one nuclear particle to another is also connected with the transfer of energy, such an exchange of mesotrons leads to an energetic interaction between heavy particles.

This process is formally quite analogous to the process of photon exchange leading to the Coulomb interaction between charged light particles.

As theoretical calculations show, the potential energy of such an interaction in the most general case has the form

\[ U \sim \frac{e^{-\frac{r}{\lambda_0}}}{r^n}, \]

where \(r\) is the distance between the particles, \(n\) an integer power, and \(\lambda_0\) the Compton wavelength of the particle carrying the interaction,

\[ \lambda_0=\frac{\hbar}{m_0 c}, \]

where \(m_0\) is its rest mass.

In particular, for the interaction of two electric charges, effected by the exchange of photons, \(m_0=0\), \(\lambda_0=\infty\), and \(n=1\), so that Coulomb’s law is obtained,

\[ U\sim \frac{1}{r}. \]

For the interaction of two magnetic dipoles, likewise effected by photons, \(m=0\), \(\lambda_0=\infty\), but \(n=3\), etc.

In the case of heavy particles the interaction energy decreases very rapidly with distance. The action of nuclear forces extends to distances of the order of the size of the nucleus \(r_0\), i.e. \(10^{-13}\) cm.

In order that the interaction should break off at a distance of the order of \(r_0\), it is evidently necessary that \(\lambda_0\) be of the order of \(r_0\). In this case, for the mass of the particle carrying the interaction, one obtains the value

\[ \mu \sim 200m, \]

where \(m\) is the mass of the electron.

Such an estimate of the mesotron mass is in good agreement with experiment.

In Yukawa’s theory radioactive \(\beta\)-decay is associated with the decay of mesotrons. \(\beta\)-decay proceeds according to the following scheme: a heavy particle, for example a proton, emits a negative mesotron, thereby becoming a neutron, while the emitted mesotron decays into an electron and a neutrino.

Since all nuclei in \(\beta\)-decay change their spin by an integral number, the spin of the mesotron must be integral.

Further, in order to explain the experimentally established dependence of the proton-neutron nuclear forces on the spins of both particles, it is necessary to assume that the spin of the mesotron is different from zero. Since the spin is integral, it is natural to assume that it is equal to unity (the assumption that the spin is equal to 2 would lead to substantial complications).

Yukawa’s theory, improved by other authors, gives highly valuable qualitative results and correctly depicts the qualitative course of nuclear processes. However, it is already clear that in quantitative respect it by no means agrees with experiment. Even now this theory leads to a number of difficulties which compel one to raise the question of its complete revision. As an example of difficulties of this kind I. E. Tamm gave the following.

The theory cannot explain the existence of proton—proton attractive forces, whose reality has been proved experimentally. To explain proton—proton forces it is necessary to postulate the existence of a new type of particles—neutral mesotrons—a circumstance constituting a substantial shortcoming of the theory.

Further, the interaction of nuclear particles, effected by the exchange of mesotrons with spin one, is analogous to the interaction of magnetic dipoles. Therefore the exponent \(n\) in the expression for the potential energy of nuclear particles is equal to three, so that

\[ U \sim \frac{e^{-\frac{r}{\lambda_\mu}}}{r^3}, \]

where \(\lambda_\mu\) is the Compton wavelength of the mesotron.

However, such a law for the potential energy cannot be valid all the way to \(r = 0\). It can be shown that an increase of the potential energy proportional to \(\frac{1}{r^3}\) as \(r \to 0\) would lead to the falling of heavy particles onto one another. It is therefore necessary to assume that at small distances, for \(r\) less than some critical distance \(r_{cr}\), the present theory is no longer valid and that for \(r < r_{cr}\) the increase of the potential energy ceases.

The quantity \(r_{cr}\) must be introduced into the theory as a certain constant. In addition, the law of interaction of the mesotron with heavy particles contains four more arbitrary constants. The values of these five constants may be chosen so that the theory gives the best explanation of the known experimental facts. However, even this latitude proves insufficient for a consistent explanation of all observed phenomena. For this reason numerous attempts are being made to “correct” the theory. The very character of these “corrections” points to the extremely unfavorable state of the theory. For example, Bethe proposes to explain nuclear forces by means of neutral mesotrons alone, which have nothing in common with the charged mesotrons observed in cosmic rays. In particular, the existence of the magnetic moment of the neutron then remains completely incomprehensible. Heitler proposes to postulate the existence of protons with different charges \(+2\), \(+1\), \(-1\) and different spins: \(^{1}/_{2}\), \(^{3}/_{2}\), \(^{5}/_{2}\), etc. Møller and Rosenfeld propose to admit the existence of two kinds of mesotrons with different equations of motion and different periods of decay, and so on.

But even with the aid of such arbitrary and far-reaching assumptions, it is possible to explain qualitatively only a limited range of facts relating to cosmic rays and nuclear forces.

All these difficulties confronting the mesotron theory have as their source the specific properties of particles with spin 1. In this connection I. E. Tamm investigated the peculiarities of the behavior of a particle with spin equal to one. For this purpose an exact solution was obtained of the equations of motion of a particle with spin 1 (Proca’s equation) in a Coulomb field, i.e., a problem analogous to the problem of the hydrogen atom for the electron was solved.

This investigation led to a completely unexpected result: it turned out that a particle with spin 1 moves in a Coulomb ...

THE THEORY OF THE MESOTRON AND NUCLEAR FORCES

a field with potential \(\dfrac{1}{r}\), just as a particle with spin \(1/2\) or 0 would move in a field with potential \(\dfrac{1}{r^3}\). Namely, in the Coulomb field \(\dfrac{e^2}{r}\) of a particle with spin 1, along with stationary states of the usual type, there also exist “anomalous” solutions of the equations of motion, corresponding either to the “fall” of the particle onto the charge creating the Coulomb field, or to the radiation of particles by this charge.

The reason leading to this difference in the behavior of particles with spin 1 and \(1/2\) in a Coulomb field is as follows.

In Dirac’s theory, at nonrelativistic velocities the electron moves as a particle with charge and its own (“spin”) magnetic moment. However, at high velocities the electron in Dirac’s theory behaves as a point charge without any intrinsic magnetic moment.

Unlike the electron, a particle with spin 1 also in the relativistic region moves as a particle possessing its own magnetic moment. In this sense it may be likened to a true magnetic dipole.

According to the well-known relation of the theory of relativity, a particle with magnetic moment \(\mathbf m\) also acquires, when moving, a certain electric moment \(\mathbf p\), equal to

\[ \mathbf p=\frac{[\mathbf v\cdot \mathbf m]}{c\sqrt{1-\frac{v^2}{c^2}}}, \]

where \(\mathbf v\) is the velocity of the particle.

At high particle velocities, close to the speed of light, its electric moment grows rapidly, and it is attracted by the Coulomb field toward the center up to falling onto the proton (or is repelled by the field, depending on the sign of the magnetic moment and the direction of the velocity \(\mathbf v\)). Since the fall of a mesotron onto a proton is physically absurd, it follows from this that for a charged particle with spin 1 Coulomb’s law must lose its meaning at small distances. Landau and Tamm made use of this specific feature of a particle with spin 1 in order to propose an entirely new theory of nuclear forces.

L. Landau drew attention to the fact that if Coulomb’s law is valid down to very small values of \(r\), then the binding energy of a mesotron on a proton can be very large.

Then the proton and the negative mesotron may form a quasihydrogen atom, with an enormous binding energy.

If Coulomb’s law breaks off at sufficiently small values of the radius, the binding energy \(E\) may reach a value of the order \(\mu c^2\), where \(\mu\) is the mass of the mesotron.

In this case the mass of the quasihydrogen atom will be equal to

\[ M_n \simeq M_p+\mu-\frac{E}{c^2}=M_p, \]

where \(M_p\) is the mass of the proton, i.e. approximately equal to the mass of the proton.

According to the theory proposed by Landau and Tamm, such a quasi-hydrogen atom is identified with the neutron. Thus, the neutron is regarded as a composite formation, which may be represented schematically as

\[ p+\mu^- \to n+E(100\ \mathrm{MeV}). \]

At the same time, in contrast to the former theory of nuclear forces, the Landau–Tamm theory does not postulate the existence of any special forces of interaction between the proton and the mesotron. The sole assumption of the theory is that the spin of the mesotron is equal to unity, and the only unknown parameter is that critical value of the radius at which Coulomb’s law is “cut off.”

At first sight the new hypothesis contradicts the approximate equality in rights of neutrons and protons, manifested, for example, in the approximate independence of the forces of interaction between nuclear particles from the charge of these particles. However, this contradiction is removed if one takes into account relativistic effects—the formation of mesotron pairs and their annihilation, as well as the so-called polarization of the vacuum. Thus, for example, the capture of a negative mesotron by a proton leads to the formation of a neutron and the liberation of energy \(E=\mu c^2\) (100 MeV). It is possible, however, also for a positive mesotron to be captured by a neutron, as a result of which this mesotron annihilates with the negative mesotron present in the neutron. As a result, a proton remains and an energy is liberated equal to the difference between the energy of the annihilated mesotrons \(2\mu c^2\) and the binding energy of the negative mesotron in the neutron \(E=\mu c^2\). Thus, the energy liberated in this process is also equal to \(\mu c^2\):

\[ n+\mu^+ \to p+E(100\ \mathrm{MeV}). \]

Thus, from the energy point of view, the processes of formation of \(p\) from \(n\) and \(\mu^+\), and of formation of \(n\) from \(p\) and \(\mu^-\), are equal in rights. There is reason to expect that the consistent application of relativistic theory will also lead to the equality in rights of neutrons and protons in other respects, for example with regard to the forces of interaction between them.

According to the new hypothesis, in order to explain nuclear forces there is no need to postulate the existence of specific nuclear forces. These forces are a peculiar manifestation of ordinary electromagnetic forces, while their specific features are explained by the distinctive laws of motion of mesotrons. Mesotrons likewise cause the binding of protons in the atomic nucleus, just as electrons cause the binding of atoms in a molecule. In general, one may establish a certain analogy between nuclear forces and ordinary chemical forces.

For example, the attractive forces between a neutron and a proton are analogous to the chemical forces in the system hydrogen atom–proton, while the neutron–neutron forces are analogous to the forces binding two hydrogen atoms into a molecule.

In the theory of Landau and Tamm, the need for assumptions about a new law of interaction, which figured in all previous theories of nuclear forces, disappears. In essence, the only arbitrary assumption made in the new theory is the assumption that the spin of the mesotron is equal to unity.

In the present state of the theory it is still difficult to speak of its quantitative verification. The estimates made by the authors give quite satisfactory results for a number of different quantities, such as, for example, the magnetic moments of heavy particles. However, in order to construct a complete theory that could be directly compared with experimental data, it is necessary to overcome a number of still very considerable difficulties, partly purely computational and partly connected with the fact that in the general relativistic theory of particles with integral spin a number of questions of a general, fundamental character still remain unexamined (for example, the question of the relativistic interaction of identical charged particles with integral spin).

1. Report at the Conference on the Atomic Nucleus organized by the Physics and Mathematics Division of the Academy of Sciences of the USSR in Moscow, November 20–26, 1940 (see in this issue, p. 241). The text is printed from a record of the report made by V. G. Levich and reviewed by the speaker. ↩