Theoretical and Experimental Data on Particle Production at High Energies

I. L. Rozental and D. S. Chernavskii

Introduction

In the present article we attempt to give a survey of the experimental and theoretical data relating to the production of particles at high \((>10^{10}\ \text{eV})\) and especially ultrahigh \((>10^{12}\ \text{eV up to }10^{18}\ \text{eV})\) energies.

It should be noted that the experimental study of processes at such high energies began comparatively recently (with the development of the photographic-emulsion technique, on the one hand, and the detailed study of extensive atmospheric showers, on the other). Nevertheless, by the present time a number of very interesting results have already been obtained in this field. The main one is that, in the collision of a nucleon of such high energy with a nucleon (or nucleus), many particles at once are produced (of the order of 10 or more). Processes in which many particles are produced in the collision of two nucleons have been called multiple production. It must be said that physics had not previously encountered processes of this type.

Indeed, in electrodynamics, as is well known, multiple generation of particles is not observed, since, owing to the smallness of the coupling constant \(\left(\dfrac{e^2}{\hbar c}\sim\dfrac{1}{137}\right)\), the probability of producing \(n\) particles is, in order of magnitude, \((137)^{n-1}\) times smaller than the probability of producing one photon (or one electron pair).

In nuclear reactions occurring at lower energies \((\sim 300\ \text{MeV})\), multiple generation of particles was likewise not observed. Here, to be sure, this is explained by energetic considerations (there is not always enough energy for the production of several particles). There were also theoretical considerations that at high energies multiple production would be hindered by the so-called damping phenomenon². In one way or another, some physicists

multiple production seemed to be an unexpected process, and they sought to explain the experimental facts testifying in favor of multiple production on the basis of the concept of so-called repeated production. Repeated production is the process in which a nucleon, passing through a nucleus, produces many particles (for example, \(\pi\)-mesons), one in each act of interaction with the nucleons of the nucleus (see § 2).

On the other hand, theoretical indications of the possibility of multiple generation appeared comparatively long ago, even before the basic experimental investigations in this field. Indeed, the coupling constant of nucleons with the meson field

\[ \left(\frac{g^2}{\hbar c}\right) \]

is much larger than in electrodynamics. It is very likely that

\[ \frac{g^2}{\hbar c}\sim 1. \]

If, in addition, this coupling has a vector character (for which there are now also certain indications \(^{1}\)), then it must increase with energy, which at high energies should increase the probability of processes of higher order (in the sense of perturbation theory), i.e. lead to multiple production.

Furthermore, as Heisenberg \(^{3}\) noted, one may expect that at high energies nonlinear terms in the Hamiltonian of the interaction (present, for example, in the theory of \(\beta\)-decay) will play a role; these too will lead to multiple production.

Despite the fact that all these circumstances have been known for a comparatively long time, there is still not a single theoretical work that would be based entirely on the initial premises of modern field theory and would consistently describe multiple processes. This is explained by the fundamental difficulties of modern theory associated with the infinite self-energy of particles.

All theoretical works are based on some hypothesis that makes it possible to circumvent these difficulties; the validity of the hypothesis is established only as a result of comparison with experimental data. Thus the leading role in this field now belongs to experiment.

The most plausible hypothesis concerning the establishment of statistical equilibrium among many simultaneously formed particles in a small volume, in which the colliding particles are located, was put forward by Fermi \(^{4}\). However, the theory contained certain inconsistencies \(^{5,6}\). The consistent development of Fermi’s hypothesis, carried out by L. D. Landau \(^{7}\), leads to the necessity of considering the aggregate of particles at the moment of their formation as a kind of liquid, i.e. it requires consideration of this aggregate on the basis of hydrodynamics.

One of the tasks of the present article is to compare the various directions of theoretical investigations of multiple processes (see § 1). Another purpose of the review is to systematize the experimental material and to compare theoretical and experimental data on interactions at high energies (see § 4).

All experimental material may be divided into two groups:

a) Observations of individual acts of multiple production.

This includes data obtained by means of photographic plates, and the results of experiments with counters. The energy of the events recorded by such methods does not exceed \(10^{13}\) eV, which is the principal shortcoming of these methods.

The results obtained by means of counters can show whether multiple processes exist at all. In order to obtain information about other characteristics of an elementary act, it is expedient to use photographic plates; however, even in this case the analysis of the results is complicated by the presence of intranuclear acts of transformation and multiplication (see § 2).

b) Data on extensive atmospheric showers.

They are necessarily averaged, since the observed result is the result of many collisions at once, and this constitutes a shortcoming of the method. In addition, when comparing these data with theoretical ones, it should be borne in mind that showers develop as a result of collisions of nucleons with light nuclei, and not on the basis of nucleon–nucleon collisions. On the other hand, extensive atmospheric showers are the only source of information about elementary acts at superhigh energies (up to \(10^{18}\) eV), and in this lies their indisputable advantage over all other methods of investigating the elementary act.

Here it should be noted that the principal works elucidating the role of nuclear interaction in the development of extensive atmospheric showers were carried out by Soviet physicists (see \(^{8}\) and review \(^{9}\)).

Let us dwell on the significance of each of these methods. The role played in the development of the concept of multiple production by acts of multiple production recorded in photographic plates is well known. Schein’s “star,” recorded in emulsion, was, as is known, the first touchstone of Fermi’s theory. However, at present data on extensive atmospheric showers are beginning to play an ever greater role. At the present time the most definite information for testing one or another hypothesis or theory of multiple production is given to us precisely by extensive atmospheric showers. This is explained by the fact that extensive atmospheric showers provide information about processes at extremely high energies \((\sim 10^{14} \div 10^{18}\ \text{eV})\), when all the characteristic features of multiple production are expressed especially vividly (see §§ 3 and 4).

Before proceeding to a detailed analysis of the question of interest to us, let us give the system of units, the basic notation, and formulas.

In what follows, as a rule, we use a system of units in which

\[ \hbar=c=\mu=1 \]

(\(\mu\) is the nucleon mass).

In this system of units the mass of the \(\pi\)-meson is \(\mu \sim \frac{1}{6}\); the unit of energy is equal to \(0.9\) Bev; the range of action of the nuclear forces is \(R \sim \frac{1}{\mu}\); \(\bar E_0\) is the energy of the incident particle in the coordinate system connected with the center of gravity (\(C\)-system); \(\bar E\) and \(\bar\vartheta\) are the energy of the secondary particle and the angle between the directions of motion of the incident and secondary particles in the \(C\)-system. \(E\), \(\vartheta\) are the same quantities in the laboratory coordinate system (\(L\)-system); \(E_0\) is the energy of the incident particle in the \(L\)-system. (As applied to extensive atmospheric showers, \(E_0\) is the energy of the primary particle.) \(V\) is the velocity of the \(C\)-system relative to the \(L\)-system; \(\gamma=\frac{1}{\sqrt{1-V^2}}\). In the case of a collision of a nucleon with a nucleon, \(\gamma\) is also the energy of the nucleons in the \(C\)-system (since \(\mu=1\)).

In the case of the collision of two particles of equal mass, in the extreme relativistic case one may write the following relations, which we shall use extensively below:

\[ E_0=\frac{\bar E_0^{\,2}}{2}; \tag{A} \]

\[ \gamma=\sqrt{\frac{E_0}{2}}; \tag{Б} \]

For the transformation of angles in the extreme relativistic case one may use the formula

\[ \vartheta=\frac{\operatorname{tg}\frac{\bar\vartheta}{2}}{\gamma}. \tag{В} \]

1. THEORETICAL CONCEPTIONS OF THE INTERACTION OF A NUCLEON WITH A NUCLEON AT HIGH ENERGY

Theoretical investigations of multiple production began even before the appearance of the basic experimental indications of the existence of such processes. At the present time, on the basis of comparison with experimental data, it is already sufficiently clear that the conclusions of a number of theoretical constructions do not correspond to reality.

Still, despite this, we consider it not superfluous to survey all the theoretical works in historical perspective; this is important also because some ideas, although they lead, because of the underdevelopment of field theory, to incorrect results, nevertheless may in themselves play an important role in the creation of a correct consistent theory.

Heisenberg was the first to draw attention to the possibility of the production of a large number of particles in the collision of two nucleons.^3 It was he who noted that the theory of β-decay, owing to the nonlinear term in the Hamiltonian, contains the possibility of the simultaneous production, at high energies, of several electron-positron pairs and neutrinos. Therefore, turning to meson physics, he attached special importance to nonlinear theories and, as an example, analyzed^10 the equation obtained from a Lagrangian function analogous to that proposed by Born,

\[ L=l^{-4}\sqrt{1+l^{4}\left[\left(\frac{\partial\varphi}{\partial\tau}\right)^{2}-(\operatorname{grad}\varphi)^{2}\right]},\qquad \tau=t, \tag{1} \]

where \(l\) is a certain universal constant having the dimension of length. Subsequently Heisenberg takes it to be equal to

\[ l=\frac{1}{\mu}\quad(\mu\text{—the mass of the }\pi\text{-meson}). \]

This Lagrangian function is characterized by the fact that, in processes occurring at low energy \(\left(E_{0}\ll \frac{1}{l}\right)\), it can be expanded in a series in powers of \(l\), and then, in the first approximation, the expression goes over (up to a constant term) into the usual Lagrangian of the linear theory. The nonlinear terms begin to make themselves felt only at energies exceeding \(\frac{1}{l}\): \(E_{0}\gg \frac{1}{l}\), i.e. \(E_{0}\gg \mu\). Indeed, already from energy considerations it is clear that multiple production can occur only in the indicated energy region.

To determine the number of particles produced, Heisenberg finds their energy spectrum*). In doing so he regards the field \(\varphi\) as classical, i.e. neglects the commutation relation of the function \(\varphi\) with its conjugate. In other words, he considers the two colliding nucleons as certain bodies whose dimensions in the direction perpendicular to the motion are equal to \(l\). In the direction of motion these bodies are Lorentz-contracted, and their dimensions in the \(C\)-system are equal to \(\frac{2l}{E_{0}}\). In the case

*) More or less consistently, the entire calculation was carried out only in his last work.^11

of very high energy \(E_0 \gg 1\); the Lorentz contraction is very large, and one dimension of the system may be neglected in comparison with the others. The whole system may be represented as a certain very thin sheet. From symmetry considerations it is clear that at the initial moment of the process the velocities of the particles, their gradients, etc., will have only one direction—along the \(x\)-axis, which coincides with the direction of motion of the primary nucleons. Thus, consideration of the first stage of the process leads to a one-dimensional problem.

The equation for the function \(\varphi\), obtained from the Lagrangian (1), in this case has the form:

\[ \varphi \frac{d}{ds}\left(s\varphi'\right)+\mu^{2}\varphi = 8l^{4}s(\varphi')^{2} \frac{\varphi' + \mu^{2}\varphi}{1+l^{4}\mu^{2}\varphi^{2}}, \tag{2} \]

where \(s\) is the interval, equal to \(\sqrt{x^{2}-\tau^{2}}\).

Solving this equation, Heisenberg obtains the spectrum of particles; moreover, it turns out that the average energy of the particles is, in order of magnitude, equal to \(M\). Hence the number of particles \(n\) is found by dividing the entire energy \(\bar E_0\) in the \(C\)-system by \(\mu\): \(n \sim \dfrac{\bar E_0}{\mu}\)*). It should be said that Heisenberg’s main results are determined by the presence of the nonlinear term and by the strong interaction among all particles connected with it. Moreover, the result \(n \sim \dfrac{\bar E_0}{\mu}\) does not depend even on the form of the nonlinear term, but follows exclusively from the assumption of strong interaction. The angular distribution of the particles produced is not specially investigated by the author, but he assumes that, in view of the strong interaction of the mesons with one another, it should be isotropic in the \(C\)-system.

In a number of works, especially in the latest articles\(^{11,12}\), Heisenberg gives a new, somewhat different interpretation of his theory. In the first works Heisenberg’s theory had a purely field-theoretical character. In the later works he is inclined to regard the whole system of nucleons and mesons as a certain liquid, whose behavior is described by a hydrodynamic equation. For this purpose, in order to justify the spectrum of the particles produced, he even invokes results of work on the statistical theory of turbulence\(^{13}\). It should be said that the use of these data is completely unjustified, since work\(^{13}\) was carried out in the nonrelativistic approximation, whereas the process of multiple production under consideration is essentially relativistic. In addition, Heisenberg’s theory—

*) Let us note that, on the basis of the laws of conservation of energy and momentum, such a number of particles is the maximum possible. Indeed, if all \(\pi\)-mesons are produced at rest, which is possible, because of the law of conservation of momentum, only in the center-of-mass system, then

\[ n = \frac{\bar E_0}{\mu}. \]

Berg’s cannot in general correspond to ordinary classical relativistic hydrodynamics. Indeed, in the relativistic region the Reynolds number is very large, and consequently the forces of interaction between the particles of the fluid are smaller than the inertial forces. Therefore, in a relativistic hydrodynamic equation, as a rule, the term describing viscosity may be neglected. In that case, however, the process cannot be accompanied by dissipation of energy, and the total entropy of the system must remain constant. In the processes described by equation (2), however, the dissipation of energy is very considerable, and the entropy of the system constantly increases.

Thus, if it were possible to give Heisenberg’s theory a hydrodynamic form, it would have to be the hydrodynamics of an entirely special fluid, in which the forces of interaction between particles are so large that they remain greater than the inertial forces even in the relativistic region. Therefore the hydrodynamic terminology appearing in Heisenberg’s articles is essentially conventional in character, since formally any field described by a nonlinear equation may be regarded as a certain continuous medium.

Thus, Heisenberg’s theory is an internally consistent and coherent field theory of multiple production. Formally, it may perhaps also be regarded as a hydrodynamic theory. However, a consistent construction of such a relativistic hydrodynamics with anomalously large viscosity on the basis of a nonlinear wave equation has not yet been carried out.

It must be said that all of the above applies to that part of Heisenberg’s work in which he considers a head-on collision of two nucleons. However, in his last work, written already after it had become clear that the main conclusions of the theory do not agree with the experimental data, Heisenberg, in order to reduce the discrepancy, turned to consideration of peripheral collisions between nucleons. In this part of the work the author allowed a number of inaccuracies, so that the result as a whole is incorrect. Let us discuss this in more detail.

Heisenberg considers the collision of ultrafast nucleons in the \(L\)-system (Fig. 1). In this system their meson fields are strongly flattened in the direction of motion, while in the perpendicular direction \(y\) they decrease according to the law \(e^{-y/R}\), where \(R = \frac{1}{\mu}\). Heisenberg assumes that, when two nucleons pass one another at a distance \(b\) (the impact parameter), the nucleons transfer to each other a momentum

Fig. 1. Scheme of a noncentral collision of two nucleons.

\(\sim \bar p=\bar p_0 e^{-b/R}\), where \(\bar p_0\) is the total momentum of the particle. Such a result is obtained if one assumes that the intensity of the interaction is proportional to the magnitude of the mutual overlap of the meson clouds. Heisenberg does not restrict himself to cases of small values of \(b\) and applies the same formula for \(b \gg R\). Owing to this, the number of particles produced in the collision is obtained by him to be smaller than in a central collision by a factor \(e^{-b/R}\):

\[ n \sim \frac{\bar E_0}{\mu}\, e^{-b/R}. \tag{3} \]

The author considers the maximum possible impact parameter \(b\) to be that at which two mesons can still be produced. The maximum impact parameter \(b_{\max}\) determined in this way depends on the energy

\[ \left( b_{\max} \sim R \ln \frac{\bar E_0}{\mu} \right) \]

and, consequently, the cross section for the interaction of two nucleons should also increase with energy according to the law

\[ \sigma \sim \pi (b_{\max})^2 \sim \pi R^2 \left( \ln \frac{\bar E_0}{\mu} \right)^2 . \tag{4} \]

However, this conclusion is incorrect, since for \(b \gg R\) one cannot regard the nucleon fields as classical\(^*\). Indeed, the classical approximation is valid only when the momentum transfer \(\bar p \gg \Delta \bar p\), where \(\Delta \bar p\) is the uncertainty of the momentum. In accordance with the uncertainty relation \(\Delta \bar p \sim 1/\Delta l\), where \(\Delta l = 1/(\mu p_0)\) is the thickness of the nucleon field. Consequently, the condition for applicability of the classical approximation is expressed by the inequality

\[ \bar p_0 e^{-b/R} \gg \bar p_0 \mu \quad \text{or} \quad e^{-b/R} \gg \mu . \]

For \(b \gg R\) this condition is violated and the process is essentially quantum-mechanical.

Another aspect of the same phenomenon is also possible. In considering the collision of two very energetic nucleons with impact parameter \(b\), one can expand the meson field of one of them at the point \(b\) into meson plane waves and find the number of mesons exchanged by the nucleons (in the spirit of the Weizsäcker-Williams method \(^{15,16}\)). For \(b \lesssim R\) (head-on collisions and those close to them) this number turns out to be large and the process may be considered classical. For \(b \gg R\) this number is less than unity and, consequently, here one can speak only of calculating the probability of exchange of a single meson. A classical treatment in this region (similar to that carried out by Heisenberg) would correspond to the assumption of an exchange by fractions

\(^*\) For more details on this, see \(^{14}\).

meson, which, of course, is impossible. A successive quantum treatment of peripheral collisions does not lead to the conclusion that the cross section grows with energy, leaving it, in order of magnitude, equal to the geometrical one. The numbers of particles produced in a head-on and in a peripheral collision of two nucleons also turn out to be close, and differences appear only in the angular distributions of the particles (see^6).

Recently, the field theory based on the assumption of nonlinearity of the interaction of nucleons and the meson field has been developed by D. D. Ivanenko and V. V. Lebedev^17. In these works it is assumed that the binding energy of the nucleons and the meson field has the form

\[ U = g_1 \Phi + g_2 \Phi^2 + \cdots + g_n \Phi^n, \tag{5} \]

where \(\Phi\) is the wave function of the mesons and \(g_i\) are coupling constants. However, the theory gives neither the number of such constants nor their magnitudes. At any rate, at present one cannot be guided by the expectation that experimental data make it possible to determine unambiguously the magnitudes of these constants and their number. Therefore we shall not dwell on this theory in more detail.

Another approach to the problem of multiple production was developed in the work of Lewis, Oppenheimer, and Wouthuysen^18. These authors proceeded from the ordinary theory of the interaction of the meson field with nucleons and mainly analyzed a symmetric pseudoscalar meson field with vector coupling. The authors explained their choice by the fact that, in the classical approximation, this variant provides a strong increase of the meson field near the nucleon (according to a law \(\sim 1/r^3\)). Such a dependence of the field strength on distance reflects, to some extent, the dependence of the strength of the interaction of two nucleons on energy. Indeed, as the energy and momenta of the colliding nucleons increase, ever smaller distances begin to play a role. Therefore, in the collision of two nucleons of high energy, such a form of the meson field should lead to multiple generation of mesons.

For calculation of the process the authors used the Bloch–Nordsieck method^19, i.e. they regarded the spin vectors \((\sigma)\) and isotopic spin \((\tau)\) in the expression for the Hamiltonian of the system as classical unit vectors, neglecting their commutation properties. The latter circumstance makes it possible to carry out a canonical transformation of the original Hamiltonian, to find the wave function of the system, the dependence of the cross section of the process on the interaction potential between the nucleons \(V(r)\), the number of particles produced, etc.

Concerning the interaction potential, the authors made the following assumptions: 1) it is sufficiently smooth near zero (falls off more slowly than \(1/r^3\)) and, consequently, its components

Fourier components corresponding to very large momenta (\(\gg \mu\)) tend to zero; 2) the matrix element of the potential depends only on the nucleons and does not depend on the states of the meson field, i.e., on the number \(n\) of nucleons produced. It should be noted that the latter assumptions are not consistent with those made earlier. Indeed, it is known that the pseudoscalar variant with vector coupling of the meson field to the nucleons leads precisely to a very singular interaction potential of the type \(\dfrac{1}{r^3}\). Moreover, in the calculation the authors make a number of further simplifications: they do not take into account the recoil of the nucleons in their interaction with the meson field, which, in the production of mesons of such high energy, cannot be considered justified; they use perturbation theory to estimate cross sections, although they themselves note its inadmissibility, etc. Thus, this work is inconsistent even within the framework of the assumptions made. As a result, they obtain the cross section for the formation of \(n\) particles in the form

\[ \sigma(n)=\left(\frac{g^2}{\pi}\right) \frac{\bar E_{\max}^{\,2n}}{(n!)^3} \left(\frac{g_1+g_2}{f_1^{\,f_1} f_2^{\,f_2}}\right)^n, \tag{6} \]

where \(\bar E_{\max}\) is the maximum value of the energy transferred in the \(C\)-system, \(g\) and \(f\) are quantities of order 1 depending on the character of the field and of the coupling (for example, for a neutral pseudoscalar field \(f=\dfrac{1}{2}\)); the indices 1 and 2 refer to the first and second nucleons.

The maximum value of the transferred energy \(\bar E_{\max}\) depends on the character of the collision. In any collision in the \(C\)-system the nucleons transfer to each other almost all the kinetic energy (i.e., all the energy except the rest mass). Therefore \(\bar E_{\max}=\gamma-1\), where \(\gamma\) is the energy of the nucleons in the \(C\)-system. At high energies \(\gamma \gg 1\), and with sufficient accuracy one may put \(\bar E_{\max}=\gamma\).

As a rule, the process whose probability (or the quantity proportional to it—the cross section of the process) is close to the maximum will be realized. The number of particles \(\bar n\) for which \(\sigma(n)\) reaches its maximum is equal to:

\[ \bar n= \left(\frac{g^2\gamma^2}{\pi}\right)^{1/3} \left(\frac{g_1+g_2}{f_1^{\,f_1} f_2^{\,f_2}}\right)^{1/2}. \tag{7} \]

Thus, the number of particles produced depends in the \(C\)-system on the energy as \(E_0^{2/3}\), or, passing to the \(L\)-system, we obtain:

\[ \bar n \sim E_0^{1/3}. \tag{8} \]

This result is characteristic of the theory being described. Investigating the angular distribution of the particles produced, the authors came to the conclusion that in the center-of-mass system it must be isotropic. Subsequently their ideas were developed in papers \(^{20-23}\).

In papers \(^{20,21}\) a relativistic generalization of the Bloch–Nordsieck method and other refinements of the calculation were carried out. However, the main characteristic features of the theory were not changed by this: the dependence of the number of particles produced on the energy still retained the form (8), and the angular distribution of the particles in the \(L\)-system remained isotropic. Quite recently, when it became clear that both conclusions are difficult to reconcile with the experimental data, an attempt was made \(^{22}\) to find a factor reducing the multiplicity. The author assumed that not all the kinetic energy of the colliding nucleons passes into the meson field (as in paper \(^{17}\)), but only a small part of it \(\left(\sim \frac{1}{10}\right)\). However, in paper \(^{22}\) no arguments are given in favor of such an assumption.

In a recent paper by Lewis \(^{23}\) the same idea is pursued, but a concrete mechanism is indicated by which not all the energy of the nucleons, but only part of it, can be transferred to the particles produced. This mechanism takes place only in peripheral collisions of two nucleons and is analogous to that considered earlier in paper \(^{6}\). It consists in the fact that the nucleons, flying past one another at a distance \(R\), exchange mesons. Owing to this, in the \(L\)-system the distribution is not isotropic, as before, but of the characteristic two-cone type. In all other respects, however, such a process differs only slightly from the process of a central collision. Indeed, if in a central impact the nucleons transfer to each other the momentum \(\bar p_0\), then in meson exchange the transferred momentum cannot be less than \(2p\bar p_0\). Thus, taking these corrections into account does not change the remaining conclusions of paper \(^{18}\).

It should be noted that in articles \(^{22,23}\) a somewhat different interpretation of paper \(^{18}\) is given, close in spirit to the ideas of Fermi \(^{4}\). It is useful to dwell on this in more detail below.

Fermi \(^{4}\) approached the solution of the problem of multiple production in an entirely different way. In doing so he proceeded from ideas that he had successfully applied in the theory of \(\beta\)-decay. The essence of it is as follows.

It is well known that the probability \(W\) of a transition of a system from one state \(A\) to another \(B\) is equal to:

\[ W = 2\pi |H_{AB}|^2 \rho(\bar E_0), \tag{9} \]

where \(H_{AB}\) is the matrix element of the transition, and \(\rho(\bar E_0)\) is the density of levels of the final state.

In multiple-production processes the factor \(\rho_n(\overline{E}_0)\) is the relative probability that in some volume \(\Omega\) there will be \(n\) particles, each of which has its own momentum \(\overline{p}_i\) \((i=1,2,\ldots,n)\), so that the total energy of the whole system is equal to \(\overline{E}_0\).

For \(n \gg 1\), \(\rho_n(\overline{E}_0)\) is a very sharp function of \(n\), possessing a pronounced and narrow maximum. Therefore the maximum of the probability \(^{20}\) must to a considerable extent be determined by the behavior of the function \(\rho_n(\overline{E}_0)\). The matrix element \(|H_{AB}|^2\) can substantially affect the probability \(W\) only in the case where it is a function of \(n\) at least as sharp as \(\rho_n(\overline{E}_0)\), which is, in general, hard to expect. Fermi’s basic idea consists in assuming sufficient smoothness of the function representing the matrix element. In this case the result of the process depends very little on it and is determined entirely by the statistical weight \(\rho_n(\overline{E}_0)\).

The question of the volume \(\Omega\) in Fermi’s theory has not been sufficiently clarified. Fermi takes it in the \(C\)-system to be equal to

\[ \Omega=\frac{\Omega_0}{\gamma}=\Omega_0\frac{2}{E_0}, \tag{10} \]

where \(\Omega_0=\dfrac{4}{3}\pi R^3\) is the volume of one nucleon in its rest system.

Such a choice of volume in work \(^{4}\) is justified insufficiently clearly, and it is precisely this point that has repeatedly been questioned.

Here there is indeed an element of arbitrariness, connected, however, with all of Fermi’s other assumptions. Nevertheless, the assumption that formula (10) may be used is apparently still correct; the weightiest arguments for this will be given below.

Thus, predominantly those states will be realized for which the probability \(W\) (and, consequently, the quantity \(\rho(\overline{E}_0)\)) is maximal. In this way the problem reduces to finding the maximum of the expression \(\rho(\overline{E}_0)\); moreover, since this function possesses a sharp maximum, only the states corresponding to its largest value will be realized.

However, the problem of determining the maximum of the weight function at a given energy coincides exactly with the problem solved in classical statistical physics for determining the equilibrium state corresponding to maximum entropy*).

We arrive at the classical thermodynamic problem of finding the equilibrium state of a system with a large number

* Strictly speaking, there is also contained here the assumption that equilibrium is established sufficiently rapidly. For this it is necessary only that all particles undergo several collisions with one another, i.e., that the mean free path be sufficiently small.

of degrees of freedom, contained in the volume \(\Omega\) and possessing energy \(\bar E_0\).

Next Fermi assumes that the energy of interaction between the particles may be neglected. This means that the kinetic energy of the particles is much greater than their interaction energy and corresponds to the previously made assumptions about the independence of the process from the matrix element. Indeed, if the interaction energy between particles were large, of the order of or greater than their kinetic energy, this would have to be reflected in the matrix element, and it would be impossible to neglect its influence. The assumption made permits one to use the formulas of the thermodynamics of a relativistic ideal gas and to find the number of particles \(n\) produced at a given energy \(\bar E_0\). Indeed, the energy density, according to Stefan’s law, is \(\sim (kT)^4\). Consequently, the energy

\[ \bar E_0 \sim (kT^4)\Omega = \frac{2}{E_0}\Omega_0(kT)^4. \]

The particle density in this case is

\[ \Delta n \sim (kT)^3 \sim \bar E_0^{3/2} \]

and the total number of particles

\[ n \sim \Delta n \Omega \sim \bar E_0^{1/2}. \]

Consequently,

\[ n = k\bar E_0^{1/2}. \tag{11} \]

The proportionality coefficient \(k\), generally speaking, depends on the kind of newly produced particles (\(\pi\)-mesons, nucleon—antinucleon, etc.) and, in order of magnitude, is equal to unity.

Let us turn to the question of the distribution of the produced particles over angles and energies. It is considered in another paper by Fermi\(^{24}\), which is much less substantiated than the first; in this paper there are a number of inconsistencies, and it has been criticized many times\(^{6,7}\), as a result of which it became clear that it is in fact incorrect. Nevertheless, the results of the named paper are widely used, and we consider it not superfluous to give briefly its main conclusions. Here Fermi does not go beyond thermodynamics. He considers noncentral collisions of two nucleons and tries to find the angular distribution of particles, using in this case the law of conservation of angular momentum.

Assuming that only \(\pi\)-mesons are produced, the distribution of particles over angles and momenta, with allowance for the conservation of energy and momentum, may be written in integral form:

\[ dn = \frac{A}{\bar E_0}\, g\, \bar p^{\,2}\, d\bar p\, d\bar\eta \int_{-1}^{+1}\frac{(1-\zeta^2)}{e^{\bar p}-1}\,d\xi, \tag{12} \]

where

\[ A=\frac{1}{2\pi \mu^3}, \qquad g=3 \text{ (statistical factor)}, \qquad \bar{\eta}=\cos\vartheta,\quad \zeta= \]

\[ =\nu\rho(1-\rho\bar{\eta}), \]

\(\nu\) and \(\rho\) are quantities determined from the conservation laws. Their physical meaning is as follows: \(\rho\) reflects the magnitude of the impact parameter and varies from \(0\) (head-on collision) to \(1\) (the most peripheral collision); \(\nu=\dfrac{1}{kT}\) is the quantity reciprocal to the temperature.

Integration over \(\bar p\) leads to the distribution of particles over angles in the form

\[ dn=\frac{a f(\rho\bar{\eta})}{2\pi\mu^3 E_0\vartheta}, \tag{13} \]

where

\[ a=2\sum_{r=1}^{\infty}\frac{1}{r^3}=2.413 \]

and

\[ f(\rho\bar{\eta})=\frac{2}{(\rho\bar{\eta})^3\left[1-(\rho\bar{\eta})^2\right]}-\frac{1}{(\rho\bar{\eta})^3}\ln\frac{1+\rho\bar{\eta}}{1-\rho\bar{\eta}}. \]

Let us now dwell on the shortcomings of work \(^{24}\).

First, Fermi assumes that in a comparatively distant collision with an impact parameter of the order of the nucleon radius (as in the case of a head-on collision) one aggregate excited state is formed. However, for this it is necessary that, during the collision time, equal in the \(L\)-system to

\[ \frac{2}{\mu E}, \]

the disturbance should propagate in the direction perpendicular (to the motion) over distances of order \(R\), which, if one does not abandon the theory of relativity, is difficult to imagine (since this would require velocities greater than the velocity of light).

Second, the interaction of the particles after they have flown out of the initial Lorentz-contracted volume is completely ignored.

However, the most important shortcoming lies in the incorrect approach to the solution of the question. Indeed, in order to obtain such “averaged” information as the total entropy of the system, the number of particles, etc., one may use the formulas of thermodynamics (the hydrodynamical consideration will introduce nothing new in this respect—see below). But in order to obtain information concerning the angular and energy distribution of the particles, it is necessary, as was indicated by I. Ya. Pomeranchuk \(^{5}\) and L. D. Landau \(^{7}\), to take into account the interaction of the particles during their expansion, i.e., it is necessary to use the equations of hydrodynamics. Indeed, the domains of applicability of thermodynamics and hydrodynamics coincide. For both the one and the other it is necessary

DATA ON PARTICLE PRODUCTION AT HIGH ENERGIES

and it is sufficient that the number of particles be large, while the mean free path is small in comparison with the dimensions of the system. Hydrodynamics and thermodynamics are not different approaches, but constitute only two aspects of one and the same statistical approach to the phenomenon. And if, for the description of one side of the phenomenon—the description of stationary states—thermodynamics is used, then for the description of the other side—the development of the process in space and time—it is necessary to apply the equations of hydrodynamics. In the case of high energies, in accordance with all the assumptions made by Fermi in his first paper, it is necessary to use the hydrodynamics of an ideal, nonviscous relativistic fluid. This was done in L. D. Landau’s paper^7.

Thus, Landau’s work is a continuation and development of Fermi’s first paper^4. At the same time, it sharply diverges from Fermi’s second paper^24, which, as was indicated above, is inconsistent.

In what follows, when the theories of Fermi and Landau are compared, it should be borne in mind that what is meant is Fermi’s second paper^24.

Landau’s consideration concerns the stage of expansion of the newly formed particles. As the initial state he takes that state of the system which in Fermi’s first paper was regarded as final, i.e., he assumes that in the \(L\)-system, in the volume \(\Omega\) (see (10)), the energy \(\bar{E}_0\) is contained. As for the stage of formation of this state, it too could have been considered hydrodynamically in the spirit of Fermi’s ideas; however, as we shall see, such a consideration does not lead to new results. Indeed, from the assumption that the results of any process are determined by the statistical weight of the various states of the nucleon, it follows that the nucleon may be regarded as a certain “continuous” gaseous or liquid body of volume \(\Omega_0\). In the collision of two such Lorentz-contracted bodies, at the instant of impact a shock wave arises, propagating inside the nucleons in both directions with the velocity of light.

Fig. 2. Scheme of the central collision of two nucleons according to Landau: dashed line—the shock-wave front; solid line—the boundaries of the nucleons.

During the propagation of the shock wave, dissipation of energy and an increase of entropy occur in the system. This stage ends when the fronts of the shock waves reach the boundaries of the nucleons. Since the velocities of both boundaries are the same and approximately equal to the velocity of light, their meeting with the boundary of the incident nucleon occurs at the moment when the wave has traversed half the diameter of the nucleon (Fig. 2). The volume of the entire system by the end of this stage becomes equal to \(\Omega=\Omega_0 \cdot \dfrac{2}{\bar{E}_0}\), which fully corresponds to Fermi’s work. Then, since the waves

spread further in vacuum, and not in nuclear matter, they no longer have a collisional character, no energy dissipation occurs, the entropy does not increase, and the stage of expansion considered by Landau begins. Meanwhile, the number of newly formed particles in the relativistic case is proportional precisely to the total entropy of the \(S\)-system\(^7\)

\[ n \sim S. \]

Consequently, as in Fermi’s first paper, it will be equal to:

\[ n_1 = k_1 E_0^{1/4}, \tag{14} \]

where \(k_1\) is a constant, and \(E_0\) is the energy in the \(L\)-system. To obtain the angular and energy distribution, however, it is necessary to solve the equations of relativistic hydrodynamics for an ideal fluid. In view of the strong relativistic contraction, the initial state may be regarded as plane, and therefore at the first stage of expansion one solves the one-dimensional system of equations of relativistic hydrodynamics.

As a result of solving this problem Landau obtained that the principal fraction of the system’s energy is concentrated at the very front of the wave. The principal fraction of the particles being produced, however, is contained in the region behind the wave front, no longer so rich in energy. In the energy distribution of the particles this circumstance is expressed in the fact that a comparatively small fraction of the particles carries a significant fraction of the energy. However, the initial assumption that the system is plane and that the motion is quasi-one-dimensional must sooner or later break down. Indeed, in the final analysis the longitudinal dimensions will become of the same order as the transverse ones. Landau showed that by this moment, i.e. at the stage of “three-dimensional expansion,” the interaction of the various parts of the system with one another will be so small that they will already be moving rectilinearly. The distribution of the particles over angles at this stage, therefore, can no longer be distorted and remains the same as at the end of the first stage. Therefore the ratio of the transverse components of the velocity to the longitudinal ones (directed along the axis of motion) in the second stage will remain the same as at the end of the first stage, i.e. small. In view of the smallness of the transverse components, one may base oneself on the solution for the one-dimensional problem, and take the lateral components into account as corrections. Using this circumstance, Landau obtained the angular distribution of the particles in the \(L\)-system. It is given in parametric form:

\[ dn \sim e^{\sqrt{L^2-\lambda^2}}\,d\lambda, \tag{15} \]

where

\[ \lambda = -\ln \operatorname{tg}\frac{\vartheta}{2}; \qquad L = \ln \frac{\overline{E_0}}{2}. \]

In the case of sufficiently large energy, as a rule, only values \(\lambda \ll L\) play an essential role. Then

\[ dn \sim e^{-\frac{\lambda^2}{2L}}\,d\lambda, \]

and here one may already regard \(\lambda\) as

changes from 0 to \(\infty\). It should be noted that, under such a consideration, each element of the “gas-like nuclear substance” possesses a quite definite energy and a definite direction of motion.*)

Therefore the angular and energy distributions in Landau’s scheme are rigidly connected with one another, i.e., particles of quite definite energy are emitted at a definite angle. The dependence of the energy on the angle of emission (or on the parameter \(\lambda\)) is given by the expression

\[ \bar E \sim e^{-\frac{L}{6}+\lambda+\frac{1}{3}\sqrt{L^2-\lambda^2}} . \tag{16} \]

Expression (15) gives the distribution of particles in the \(Ц\)-system. One may pass to the \(Л\)-system by the aberration formulas, since the particles both in the \(Ц\)-system and in the \(Л\)-system move with velocities very close to the speed of light. This transition leads to the distribution:

\[ dn=2\sqrt{\frac{L}{2\pi}}\left(\frac{E_0}{2}\right)^{-\frac{\delta}{4}+\frac{1}{2}\sqrt{1-\delta^2}}\,d\delta, \tag{17} \]

\[ -1<\delta<1,\qquad \delta=\frac{\lambda}{L}, \]

and to the particle energies

\[ E=C\left(\frac{E_0}{2}\right)^{\frac{5}{12}\delta+\frac{1}{2}+\frac{1}{6}\sqrt{1-\delta^2}}, \tag{18} \]

where \(C\) is determined from the condition \(\int E\,dn=E_0\).

The angle of emission

\[ \vartheta=\left(\frac{E_0}{2}\right)^{-\frac{1+\delta}{2}} . \tag{19} \]

The question of the composition of the particles produced is treated in Landau’s work differently than in Fermi’s. In Fermi’s work it was assumed that all particles are formed at once, and the ratio between the various kinds of particles (mesons and nucleons) is determined not by the temperature (since \(kT\gg 1\) and \(kT\gg \mu\)), but by the number of possible spin and charge states of particles of each kind. As a result, the ratio of the number of nucleons—antinucleons to mesons in Fermi’s theory is equal to \(8/3\). In Landau’s theory it is taken into account that at the first stage of the collision one cannot judge the composition of the particles (since

*) It is necessary to note that the hydrodynamic calculation of an elementary act is extremely complicated. Therefore Landau resorted to a number of simplifications. Thus, for example, he used the equations of state of an ultrarelativistic fluid and at the same time noted that the majority of particles have energy \(\sim 1\). However, the accuracy of the calculations fully corresponds to the accuracy of the experimental data on an elementary act available at the present time.

as a result of interaction it changes all the time), and only the total entropy of the system has meaning. One can speak of the composition only when the temperature in the system has fallen so much that the interaction will be weak and the composition will be approximately constant. Landau takes the temperature corresponding to such a moment to be equal to \(kT \sim \mu\). It follows from this that the main part of the newly born particles must consist of \(\pi\)-mesons, while the fractions of any other particles of mass \(x > \mu\) will be of order \(e^{-x/\mu}\), i.e., very small.

However, this conclusion is valid only in the case when equilibrium has time to be established in the system at every instant of time. This assumption at low temperatures \((kT \sim \mu)\) may be violated because the interaction between \(\pi\)-mesons at small energies is weak. If one adopts the point of view that the interaction between \(\pi\)-mesons is due to virtual nucleon–antinucleon pairs, then one should expect that this interaction will be effective only at energies \(\sim 1\). Equilibrium in the system will be established sufficiently rapidly at temperature \(kT \sim 1\), and at lower temperatures the equilibrium corresponding to \(kT \sim 1\) will be preserved in a “frozen” form.

In this case, generally speaking, one may expect that among the newly born particles there will be present, in appreciable quantity, also particles heavier than \(\pi\)-mesons; nucleon pairs may also be formed.

Thus, the question of the composition of the shower of newly born particles is closely connected with the question of the interaction of \(\pi\)-mesons with one another at small (from \(\mu\) to 1) energies. Further investigations are necessary in order to answer these questions.

Such are the characteristic features and the principal results of the three directions of the theory. Despite the difference in their starting assumptions, there is something common among these theories. Thus, in work\(^{22}\) the idea is developed that results formally coinciding with the results of the theory of Lewis, Oppenheimer, and Wouthuysen can be obtained on the basis of Fermi’s ideas. For example, one may assume that statistical equilibrium is established not in the volume \(\dfrac{2}{E_0}\Omega_0\), but in a volume whose transverse dimensions are \(\sim \dfrac{1}{\mu}\), while the longitudinal dimensions are, in order of magnitude, equal to the wavelength of the emitted mesons. In this case the dependence of the number of particles on the energy also has the form (8).

Heisenberg’s theory, despite the fact that it is based on assumptions different from those of the Fermi and Landau theories, can also be correlated with a certain hydrodynamical picture (as was already mentioned above). But at the same time, as was предпolo-

The assertion of the independence of the probability of a process from the matrix element leads to the equation of a classical relativistic ideal fluid (without viscosity); the assumption of a strong dependence on the matrix element and an extremely strong interaction of the produced particles can lead only to hydrodynamics with extremely strong viscosity and dissipation.

Purely speculatively it is difficult to say which of the theories corresponds more closely to reality. As already mentioned, the theory of Oppenheimer et al. is inconsistent and internally contradictory. The other two theories (Fermi–Landau and Heisenberg), within the framework of the assumptions made, are consistent and internally coherent, although Heisenberg’s theory has not been developed to the same extent as Landau’s theory. Which of them is closer to reality will be shown only by experiment, and, as will be seen from what follows, experiment can already give an unambiguous answer to this question.

In conclusion, let us note that the principal theories of multiple production are phenomenological in character and are not directly connected with the fundamental problems of the theory of elementary particles (for example, the problem of intrinsic mass). Therefore, at first sight one might get the completely false impression that there can be no such connection at all, i.e., that neither experimental nor theoretical investigations of multiple production can advance us in the study of the fundamental properties of elementary particles. This, however, is not true, and a good example of this is provided by Heisenberg’s theory.

Indeed, it will be shown below that Heisenberg’s theory of multiple production disagrees with experiment and that, consequently, Lagrangian (1), at least at very high energies, does not correspond to reality. Therefore it makes no sense to use it without modifications in problems of the intrinsic energy of elementary particles, and it must at least be modernized by requiring that at high energies such a Lagrangian should not lead to too high a multiplicity.

Thus, on the basis of the results on multiple production one can formulate a restriction, or requirement, imposed on any theory claiming to describe the fundamental properties of elementary particles: in the region of high energies the theory must give the correct dependence of multiplicity on energy: \(n \sim E^{1/4}\). The fact that not every theory satisfies this requirement is evident at least from the example of Heisenberg’s theory.

On the other hand, as was indicated above, the question of the composition of the particles produced is closely connected with the question of the interaction between them and of its dependence on energy; therefore the study of the composition can, for example, shed light on the problem of the interaction of two mesons.

2. INTERACTION OF HIGH-ENERGY NUCLEONS WITH COMPLEX NUCLEI

In the preceding section we considered theoretical ideas about the elementary act, i.e. about the interaction of a nucleon with a nucleon. In reality, however, the experiment usually yields the result of the interaction of fast nucleons with complex nuclei.

The question of whether such a process will be essentially analogous to an elementary interaction, or whether intranuclear interactions will strongly distort it, and how the result of the process will depend on the atomic number of the nucleus \(A\), is a very topical one. Comparatively recently a number of works appeared\(^{25,26}\), carried out with a Wilson chamber, in which, on the basis of the coincidence of the characteristics (multiplicity and angular distribution) of electron-nuclear showers formed in heavy (Pb) and light (C, Be) elements, a conclusion is drawn in favor of the first supposition. However, in these works the selecting system recorded showers containing comparatively many particles moving at a small angle. Therefore, whatever the substance in which the showers were formed, the registered showers were characterized by the same properties, reflecting mainly the features of the apparatus. Thus these experiments cannot be regarded as decisive. On the contrary, there are now arguments in favor of the supposition that intranuclear processes accompanying the passage of a nucleon through a nucleus will lead to consequences different from those of elementary collisions. In our opinion, the most convincing arguments in favor of the existence of intranuclear phenomena (in any case at energies \(\sim 10(10^{10}\ \text{eV})\)) are provided by considering the dependence of the number of fast relativistic particles formed in showers \((n_s)\) and their angular distribution on the number of slow charged particles \(N_h^*)\).

In work \(^{28}\) all showers were divided, according to the number of slow particles, into two classes. To the first class were assigned showers with \(N_h > 8\). Since the slow particles formed in disintegrations are usually nucleons, showers with such a large value of \(N_h\) could arise only in collisions with heavy elements (Ag, Br) which are included in the composition of the photographic emulsion. Showers of the second class, \(N_h \leqslant 8\), could be formed either in collisions with light nuclei of the emulsion (O, N, C, H), or in disintegrations of heavy nuclei in which, because of the features of the process (small energies of the primary particles, peripheral collisions), a small energy is transferred to the nucleus.

It was observed that in “stars” with \(N_h > 8\) the mean number of relativistic particles is \(n_s = n_s' = 0.16 \pm 0.02\), while in “stars” with \(N_h \leqslant 8\)

*) We use the designations introduced by Powell and co-workers\(^{27}\) for the characteristics of showers observed in photographic emulsion.

$n_s = n'_s = 0.23 \pm 0.03.$ It also turned out that the mean angles $\vartheta'$ of the shower-particle spread for $N_h > 6$ are larger than the mean spread angle $\vartheta''$ for $N_h \leqslant 6$. These results show an explicit dependence of the characteristics of showers (and, consequently, of the interaction process) on atomic weight.

Indeed, under the opposite assumption the cause of the observed difference could only be the difference in the energies of the particles producing showers of the two classes. However, as has already been noted, showers of the second class (small values of $N_h$) are on the average characterized by lower energy than showers of the first class (large values of $N_h$). Therefore, in contradiction to the observational results, the relations $n'_s > n''_s$ and $\vartheta' < \vartheta''$ would have to hold.

Against the conclusion drawn, it might seem possible to raise the objection that stars with small $N_h$ arise as a result of peripheral collisions of the nucleon with heavy nuclei. However, such a conclusion means that the character of the collision depends essentially on the path traversed by the nucleon in the nuclear matter, i.e., that the size of the nucleus affects the character of the interaction of the nucleon with the nucleus. Thus, intranuclear phenomena noticeably influence the results of the collision. Therefore it becomes necessary to generalize the picture of the elementary act and to consider separately the case of the passage of fast nucleons through a nucleus.

We shall proceed from the fact that the cross section for the interaction of two nucleons with energy greater than 1–10 ($10^9$–$10^{10}$ eV) is equal to the geometrical one ($\sigma = \pi R^2$). Although there is no direct experimental proof of the validity of this assertion, the circumstance that the interaction cross section of nucleons of such energies is close to the geometrical one for a very wide range of atomic numbers29 (from C to Pb) makes it very plausible.

Since nucleons in the nucleus are at distances of order $R$ from one another, the nucleus is “opaque” in this energy region. In other words, it is difficult to assume that a primary particle, flying through a nucleus along a chord passing through several nucleons, will interact with only one of them. More likely, it will interact with all these nucleons, or at least with a large fraction of them. Starting from such an assumption, one can develop two extreme points of view on the process of the passage of nucleons through a nucleus. According to the first of them, the nucleus is treated as a certain volume filled with continuous, structureless nuclear matter. We shall call such a picture scheme (a). According to the second point of view, the process of the passage of a nucleon is regarded as a series of successive elementary acts independent of one another (scheme (b)).

The most general criterion for choosing between the two schemes follows from the relation between the time $\tau_B$ between interactions

and durations \(\tau_A\) of a single act*). Scheme (b) can be definitely accepted only in the case where \(\tau_A \ll \tau_B\). In the opposite case (\(\tau_A \gg \tau_B\)) it is impossible to consider the individual acts independently. In the case where \(\tau_A \sim \tau_B\), nothing can be said in advance, and if there is no convincing picture of the elementary act, the decisive word belongs to experiment.

The criterion for choosing one or the other scheme can also be considered from another point of view. It is obvious that collisions may be regarded as independent if one can neglect the influence of the particles formed in the collision with the \(r\)-th nucleon on the particles produced in the preceding \(r-1\) collisions. However, such neglect is possible only if the distance between the particles is much greater than the radius of the forces between them; otherwise the interaction between the particles is so large that it becomes impossible to separate the acts.

In the very important intermediate case, no definite conclusions can be drawn a priori. In this case the result of the collision of a nucleon with a nucleus will depend on the character of the elementary act. From the work of I. Ya. Pomeranchuk and E. L. Feinberg \(^{30}\) it follows that if in a nucleon—nucleon collision a small fraction of the energy \(E_0\) is transferred

\[ \left(\sim \frac{1}{5}\right), \]

then the effective distance at which the interaction takes place considerably exceeds the geometrical diameter of the nucleons. Therefore, in the case of small energy losses, the interaction time is very large and the relation \(\tau_A \gg \tau_B\) is satisfied. If, at certain energies, collisions predominate in which the primary particles lose a small fraction of their energy, then in considering the passage of a nucleon through a nucleus it is necessary to use scheme (a). In the case where, at moderate energies (when a small number of particles is produced), the principal role is played by elementary collisions with large energy losses, one may expect that the application of scheme (b) will give a satisfactory picture of the passage of a nucleon.

The choice between the two possibilities belongs to experiment. According to N. L. Grigorov and V. S. Murzin \(^{31}\), nucleons with energy of the order of \(10^{10}\) ev lose on the average in each collision 30% of their initial energy.

These experimental data indicate that, on the average, the nucleon transfers to the nucleus a small fraction of its energy and that long-duration collisions predominate. Thus, in order to obtain characteristics averaged over a large number of collisions, it is apparently impossible to apply the model of an intranuclear cascade (scheme (b)).

*) It is necessary to emphasize the difference between the time during which an act proceeds and the time of passage of one nucleon past another. This difference may, in particular, be due to the interaction of secondary particles, which is a necessary element of the act.

However, if one considers specially selected cases of intense intranuclear interaction at the same energy (\(10^{10}\) eV), for example many-prong stars recorded in photographic plates, then in these cases the mean impact parameter will be small and the duration of the collision of a nucleon with a nucleon will be of the same order as the time between collisions. It is possible that the intranuclear-cascade model will be applicable to the description of such processes. To settle this question definitively, and to determine the limits of applicability of either scheme at energies of \(10^{10}\) eV, additional experimental and theoretical investigations are necessary.

In the limiting case, when the energy \(E_0\) is so small that it is transferred completely to the meson\(^*\), \(\tau_A \ll \tau_B\). In this case the time of interaction may be taken equal to the interval between the emission of the meson by one nucleon and its absorption by another:

\[ \tau_A = \sqrt{(\Delta t)^2 + (\Delta x)^2}, \]

where \(\Delta x \sim R\), \(\Delta t = \dfrac{R}{v}\), and \(v\) is the velocity of the meson. Therefore

\[ \tau_A \sim \frac{R}{\sqrt{1-\dfrac{M^2}{E_0^2}}}\,\frac{M}{E_0}. \]

The interval between interactions is:

\[ \tau_B \sim \frac{R}{\sqrt{1-\dfrac{1}{E_0^2}}}\,\frac{1}{E_0} \gg \tau_A . \]

Thus, it is quite possible that at low energies (when few particles are produced) the process represents a sequence of acts.

The theory of multiple processes was developed by Janossy, Heitler, Mészáros, and others.\(^{32-35}\) In this theory it is assumed that when a nucleon collides with the nucleons of the nucleus lying in its path, one meson is produced in each such collision\(^ {**}\). The basic premise underlying the calculations is the assumption of the statistical independence of meson emission in collisions. However, the consistent implementation of this idea even in its simplest form proves difficult. Indeed,

\(^*\) Naturally, in this extreme case the cascade process in the nucleus, if it develops at all, will do so only very weakly.

\(^ {**}\) The assumption that no more than one meson is produced in each nucleon–nucleon collision is based on the theory of radiation damping (see 2).

even in the simplest form of the Heitler–Janossy conception (mesons and secondary nucleons do not interact), the production of mesons occurs if the energy of the nucleons is above a certain critical value. In a more complicated modification (secondary $\delta$-nucleons also cause the formation of mesons), the number of mesons formed in the $r$-th collision depends essentially on $r$.

In such a treatment, taking into account the interaction of nucleons and mesons plays a very fundamental role. Mesons, in passing through the nucleus, transfer part of their energy to nucleons and can also produce new mesons. Calculations which, in a definite scheme, take this interaction into account have been carried out*). However, in them only the interaction of the particles of the shower with those nucleons through which it passes was taken into consideration, in a completely inconsistent manner, and the interaction of the particles of the shower with one another was not taken into account (and, in particular, the reverse action of the nucleons on the act of generation). This simplification, as was mentioned earlier, is completely unjustified in the case of the formation of a large number of particles. In the case when the number of particles formed is large, the description of the passage of nucleons through the nucleus as a series of successive collisions is internally contradictory; in this case it is obviously necessary to apply scheme (a), the concrete expression of which is the hydrodynamical picture proposed by Landau.

The final choice between the two schemes should be made on the basis of experimental data. At the present time there are no data that would make such a choice possible at energies of the order of $10^{10}$ eV (see above); however, at higher energies, $10^{11}$–$10^{12}$ eV, there are indications that the hydrodynamical theory, and hence scheme (a), as was to be expected (see above), reflects reality better. This energy region plays a special role for a number of reasons. First, only at energies $\lesssim 10^{11}\div 10^{12}$ eV can one still expect cascade intranuclear processes; second, for these energies one can obtain data with sufficient statistical accuracy; third, nucleons of such energy, in not very peripheral collisions, must produce—

*) In the first variants of the calculations^33 it was assumed that mesons do not interact with nucleons. In the subsequent article by Heitler and Janossy^34, and then also in Messel, p. 35, an attempt was made to take the interaction of mesons into account. However, since for moderate energies $10$–$10^2$ ($10^{10}$–$10^{11}$ eV) no substantiated theory of the interaction of mesons with nucleons exists, the authors have to introduce additional assumptions about the character of such an interaction. The results obtained with the aid of such a model are still worse in agreement with experimental data than the conclusions from a model in which the interaction of $\pi$-mesons is not taken into account. Therefore Messel, referring to Heitler, expressed, in our view, the completely unfounded point of view that the cross section for the interaction of $\pi$-mesons falls with energy. Such a conclusion contradicts the experimental data on the interaction of $\pi$-mesons at energies of the order of $10^{10}$ eV.

to cause a practically complete breakup of the nucleus, which will make it possible (from the magnitude of the total charge of the slow particles) to judge more definitely with which particular nucleus the collision occurred; and, finally, fourth, the secondary particles in this case possess sufficiently high energy to give rise again to mesons. This circumstance makes it possible, in estimating the value of the total multiplicity \(n\) of secondary particles according to scheme (b), to neglect the effect of the energy threshold for meson production and, consequently, gives this estimate greater generality. All these circumstances create favorable conditions for choosing between the two schemes. A convenient possibility here is provided by comparing the dependence of the multiplicity \(n\) on the atomic weight \(A\) of the nucleus with which the collision occurred. Indeed, for collisions at high energy this dependence will be entirely different in the two indicated schemes. Let us estimate the ratio \(\dfrac{n_{\mathrm{h}}}{n_{\mathrm{l}}}\) of the numbers of particles formed in heavy and light elements.

It is easy to see that in scheme (b) the lower bound of the required ratio is obtained when the secondary particles interact neither with one another nor with the nucleons of the nucleus. In this limiting case, which is certainly not realized, the number of particles formed is proportional to the path length of the nucleon in the nucleus, i.e., to the nuclear radius. Consequently, in this case

\[ \frac{n_{\mathrm{h}}}{n_{\mathrm{l}}} = \left(\frac{A_{\mathrm{h}}}{A_{\mathrm{l}}}\right)^{1/3}. \]

At high energies any mechanism that assumes interaction of the secondary particles will only increase this ratio. Thus, if one assumes that only nucleons interact (the primary nucleon and \(\delta\)-nucleons), then

\[ \frac{n_{\mathrm{h}}}{n_{\mathrm{l}}} \sim 2^{\frac{A_{\mathrm{h}}^{1/3}-A_{\mathrm{l}}^{1/3}}{k}}, \]

where \(k \sim 1.5\) is a geometric factor.

Since for the group of heavy and the group of light elements in a photoemulsion \(A_{\mathrm{h}}\sim 90\), \(A_{\mathrm{l}}\sim 13\), it follows that, in scheme (b), for the photoemulsion

\[ \frac{n_{\mathrm{h}}}{n_{\mathrm{l}}}\sim 1.9 \]

(the secondary particles do not interact) and

\[ \frac{n_{\mathrm{h}}}{n_{\mathrm{l}}}\sim 3 \]

(only nucleons interact).

Taking the interaction of mesons into account will increase the ratio still more.

More detailed calculations, based on the Heitler–Janossy model \(^{35}\), give, under the assumptions known for this ratio (for \(E_0 \sim 500,\ n_\pi \sim 10\)), a value equal to 2.5.

A completely different dependence should be expected if scheme (a) is analyzed (more precisely, its concrete expression—the hydrodynamic theory of Landau). The corresponding calculation can be simplified as follows. In the \(C\)-system the nucleon and the nucleus, owing to relativistic contraction, are strongly flattened. Therefore the collision time in this system is very small; during this time the disturbance caused by the nucleons cannot propagate very far in the transverse direction. The nucleon interacts only with that nuclear matter which lies directly in its path, i.e., it as it were cuts out in the nucleus a “tube” and interacts with it as with a whole, forming a combined excited state of the matter which, upon expansion, behaves as an ideal liquid. In the following calculation it is assumed that the expansion in the \(C\)-system occurs symmetrically with respect to the plane perpendicular to the direction of motion of the primary nucleon. This assumption, quite natural when considering a nucleon–nucleon collision, in the case of a nucleon–nucleus collision requires additional justification.

According to Landau (see § 1),

\[ n \sim S \sim E_0^{3/4}\Omega_T^{1/4}, \]

but

\[ \Omega_T=\Omega_T^{(0)}\frac{2M_1}{E_0}, \]

where \(M_1\) and \(\Omega_T^{(0)}\) are the mass and volume of the “tube” in its rest frame.

Since \(\Omega_T^{(0)}\) and \(M_1 \sim A^{1/3}\), we have \(n \sim \bar E_0^{1/2} A^{1/8}\). For the energy in the \(L\)-system we have \(E_0=\dfrac{\bar E_0^2}{2M_1}\). Therefore

\[ n \sim E_0^{1/4} A^{1/4}. \tag{20} \]

In this case, consequently,

\[ \frac{n_T}{n_L}=\left(\frac{A_T}{A_L}\right)^{1/4}\sim 1.5. \tag{20a} \]

Thus, in scheme (a) the number of particles formed when a nucleon enters a nucleus depends much more weakly on the atomic number than in scheme (b). This circumstance makes it possible to hope that the corresponding experiments will lead to a final choice between the two schemes. So far there are only few indications obtained on the basis of experiments in which a selection was made of large-energy showers formed in photoemulsion \(^{36}\). In this case it is natural to suppose that an almost complete destruction of the nucleus has occurred; therefore the nucleus can

characterized by the total charge \(N_h\) carried by relativistic particles.

It turned out that, of the 16 showers of greatest energy, in 6 showers \(N_h < 10\), while in the remaining ten \(N_h > 15\). The absence of showers with \(10 < N_h < 15\) suggests that the first 6 showers were formed in collisions with light nuclei, and the other 10 in collisions with heavy ones. The author found that for the first group of showers, apparently formed in light nuclei, the mean number of relativistic particles \(\bar n_s^{(l)} = 9.1\), and for showers of the second group (heavy nuclei), \(\bar n_s^{(t)} = 12.6\).

Thus,

\[ \frac{\bar n^{(t)}}{\bar n^{(l)}} \simeq 1.4, \]

which is in good agreement with formula (20a) and disagrees with the predictions of the theory of successive collisions (scheme (b)), which gives for this ratio the value \(2 \div 3\).

Thus, at sufficiently high energies (\(10^{11} — 10^{12}\) eV), both experimental and theoretical data indicate that the model of a one-act interaction of a nucleon with a nucleus (more precisely, with a nuclear “tube”—scheme (a)) reflects reality better than the scheme of successive collisions.

At lower energies the question of the applicability of one or the other scheme has not been finally clarified. It is possible that both are applicable, but to describe different acts. However, “on average” the predominant acts are those for whose description scheme (b) is inapplicable*).

3. ON THE EXISTENCE OF MULTIPLE PROCESSES

Let us begin the comparison of the theories with experiment by proving the multiple character of particle generation at high energies. Consideration of this question is expedient because the very concept of multiple production has not yet received general recognition, and works still appear which attempt to explain all observed facts by multiple processes.

a) Counter method

An unquestionable proof of the multiple character of the process would be the production of many particles at once on a hydrogen nucleus. However, with counters it is impossible to carry out experiments with pure hydrogen, since the presence of a “vessel” is necessary, the mass of which

*) Biswas et al. \(^{37}\), analyzing the angular distribution of particles recorded in photographic plates, come to the conclusion that it agrees with the idea of multi-step processes. However, the considerable arbitrariness in the interpretation of such a complex scheme greatly reduces the value of this conclusion.

always comparable with the mass of hydrogen. Therefore they study the difference effect, i.e., they measure the number of particles formed in the “vessel” with hydrogen and subtract the number of particles formed in the empty “vessel.” This method was applied by N. A. Dobrotin, B. N. Verkhovskii, I. I. Levintov, and G. N. Khodakov1, who compared the absorption cross section of electron-nuclear showers in paraffin and graphite.

In this case the role of the “vessel” was played by graphite. The authors studied the absorption of electron-nuclear showers formed in lead in blocks of paraffin and carbon (equal in mass). It turned out that hydrogen absorbs the particles of the generating component in the same way as the nucleons of graphite nuclei. Since, in the authors’ opinion, absorption is not caused by nucleonic δ-processes, they regard equal absorption as evidence in favor of multiple particle production.

A similar idea was used in another variant in work2. However, in this case it was not absorption that was compared, but the generation of electron-nuclear showers in layers of graphite and paraffin containing the same number of carbon nuclei. The frequency of fourfold, fivefold, and sixfold coincidences was measured. The measurement results are given in Table I.

Table I

Multiplicity	Without filter (0)	Graphite (C)	Paraffin (Par)	C — 0	Par — 0	Par — C
4	120.9±1	152.9±2.0	162.9±2	32±2.3	42.0±2.3	10.0±2.8
5	65.8±0.9	74.6±1.4	81.1±1.4	8.8±1.7	15.3±1.7	6.5±2
6	30.3±0.7	35.3±1	80.8±1	5.0±1.3	10.5±1.3	5.5±1.8*

Analyzing the data they obtained (the Par—C column), the authors concluded that, when nuclear-active particles collide with hydrogen, electron-nuclear showers are formed which cause fourfold, fivefold, and sixfold coincidences.

On the basis of comparing the frequencies of fivefold and sixfold coincidences (in the Par—C column), an estimate was obtained for the multiplicity of such processes, which turned out to be \(\gg 10\).

Thus, the results of this work also, and more definitely, testify in favor of the concept of multiple production. However, in this work too there are a number of unclear points that cast doubt on its main result. Thus, comparison of the figures in the last two rows of the table leads to the conclusion that more particles are produced on the hydrogen nucleus than on the graphite nucleus. This circumstance is not discussed in the article. The observed effect

* In the article another, incorrect value of the error is given (1.4).

can, for example, also be ascribed to the influence of double showers in lead, produced by the primary nucleon and by the hydrogen nucleon that collided with it. Owing to the greater probability of registration, such double showers may imitate an increase in the number of interactions in paraffin as compared with the number of interactions in carbon.

Thus, although experiments based on comparing the characteristics of electron–nuclear showers formed in graphite and in paraffin definitely testify in favor of the existence of multiple processes, this conclusion cannot be regarded as final.

A widely known attempt to detect multiple generation of showers in a nucleon–nucleon collision was undertaken by Schein and co-workers.^40 In these experiments a thin-walled copper Dewar vessel (total wall thickness \(\sim 1.5\) mm), filled with liquid hydrogen, was raised to an altitude of 27 km (corresponding depth \(20\ \text{g}\cdot\text{cm}^{-2}\)). The hydrogen (in an amount of \(1.6\ \text{g}\cdot\text{cm}^{-2}\)) remained at this altitude for 130 min., after which 85% of the hydrogen was released into the air. The vessel, together with the remaining hydrogen, stayed at the maximum altitude for 110 min.

With the aid of a small hodoscope, electron–nuclear showers containing two or more particles were recorded. The results obtained by the authors are summarized in Table II.

Table II

Number of triggered hodoscope counters	Number of showers with hydrogen	Number of showers without hydrogen
2	\(43 \pm 5\)	\(33 \pm 4\)
3	\(24 \pm 3\)	\(12 \pm 3\)
4	\(15 \pm 3\)	\(17 \pm 3\)

From the data presented in Table II, the authors conclude that the role of multiple processes in elementary acts is very great.

Assuming that the showers formed in hydrogen (the difference between the quantities placed in columns 3 and 2 of Table II) are due to nucleon–nucleon collisions, the authors, comparing the angular distribution of showers formed with hydrogen and without hydrogen, drew the following conclusions:

1) At high energies the cross section of the nucleon–nucleon interaction is equal to the geometrical one.

2) Protons with energies \(\sim 10^{11}\ \text{eV}\) produce more than four charged mesons.

It should be noted, however, that the authors’ basic assumption that the difference effect can be attributed only to nucleon–nucleon collisions is not sufficiently convincing; in particular, it may be caused by the presence in the primary radiation flux of \(\alpha\)-particles, which disintegrate upon collision with hydrogen nuclei.

Indeed, it is well known that at the altitudes where the measurements were made, the flux of \(\alpha\)-particles \((P_{\mathrm{He}})\) amounts to \(\sim 0.2\) of the total flux, which in the following calculation we shall take to be equal to 1.

Assuming that the cross section for the interaction of \(\alpha\)-particles with hydrogen is equal to the geometrical cross section of an \(\alpha\)-particle,* and taking into account that the ratio \(K\) of the masses of hydrogen and of the walls is equal to 1.2, one can obtain the ratio of the number of collisions of \(\alpha\)-particles with hydrogen to the number of collisions of all particles with copper nuclei. The number of collisions is proportional to the flux, the magnitude of the geometrical cross section, and the number of atoms in the layer of target material.

Therefore the required ratio is equal to:

\[ K A_{\mathrm{Cu}}^{1/3} A_{\mathrm{He}}^{1/3} P_{\mathrm{He}} \sim 2.5. \]

Thus the entire effect observed by Shain et al. with an excess can be attributed to the interaction of \(\alpha\)-particles. Such an explanation, naturally, agrees with the fact that for \(n \gg 4\) no difference in the number of showers with hydrogen and without hydrogen was observed.

The excess of the calculated value of the effect over the observed one may be attributed to the difference in the probabilities of registering He–H and H–Cu collisions. Naturally, for this reason Shain’s experiments in no way testify either for or against the existence of multiple processes. The experiments prove nothing and disprove nothing.

b) The photographic-plate method

Since intranuclear interactions exist (see § 2), the appearance of “stars” with many relativistic particles does not in itself prove the existence of multiple processes. Therefore special attention was paid to “stars” that could be interpreted as the result of a single (or at least a small number of) collision(s) of nucleons. Most suited to such an interpretation were “stars” containing very few (or no) slow particles (“black” and

\[ \text{* In reality it is somewhat larger (see, for example, }^{41}\text{).} \]

“gray” tracks). Recently, several dozen “stars” have been observed in which the ratio of the numbers of slow and fast particles is \(\ll 1/10\). Some of them have played an important role in the development of our ideas about multiple processes (thus, the star observed by Schein et al.\(^{42}\) was the first touchstone of Fermi’s theory).

However, not so many such stars have been observed after all, and all of them were obtained by means of a very “rigid” selection. In addition, the interpretation of such showers as the consequence of a nucleon—nucleon collision often comes into contradiction with the law of charge conservation\(^{41}\). Therefore attempts arise to explain them on the basis of multiple production on a nucleus. In this respect characteristic is the recently published article by Messel et al.\(^{35}\), which is a continuation of Heitler and Janossy’s work. Its authors believe that, with improvement of the theory of multiple production, on its basis it will be possible to explain the principal features of showers observed in photographic plates.

The success of such attempts is very doubtful; nevertheless they show that the results of photographic-plate work are not an unappealable proof of the multiple character of production, although the assumption of such processes naturally explains many features of showers observed with the aid of photographic plates.

c) Wide atmospheric showers

As was indicated by G. T. Zatsepin\(^{8}\), the development of wide atmospheric showers is determined mainly by nuclear interaction. Therefore, by comparing the observed and calculated characteristics of wide atmospheric showers, one can obtain definite information about the nature of the elementary act. In this section we shall apply this approach to proving the existence of multiple processes. In particular, G. T. Zatsepin\(^{44}\) proposed the following method for determining the multiplicity. It is known\(^{43}\) that the observed maximum of registered wide showers (with energy \(10^4—10^5\) \((10^{13}—10^{14}\ \mathrm{eV})\)) is located at a depth of \(350\ \mathrm{g\cdot cm^{-2}}\), i.e., higher than was predicted by the electromagnetic cascade theory. From this one may conclude that the multiplication of particles, caused by the combined influence of nuclear and electromagnetic processes, occurs faster than multiplication in electron–photon showers. If the atmospheric depth \(l\) is expressed in units equal to the nuclear mean free path, then the growth of the number of particles in electron–photon showers (at energies \(\sim 10^4—10^5\)) at the beginning of the cascade curve must have the character of the exponential \(e^{\lambda}\), where \(\lambda\sim 5\). Hence, assuming that multiplication due to electromagnetic processes plays a small role, one may conclude that the value \(\lambda=5\) is a lower limit for the multiplicity. However, since

the possibility of neglecting electromagnetic processes in this case is unclear, we have carried out a determination of the value of the multiplicity by comparing the calculated and experimental altitude variations of extensive showers, based on a somewhat different idea.

This idea consists in comparing the experimental and calculated positions of the maximum of extensive showers. The calculation takes into account the influence of electromagnetic processes, and the calculation itself is based on assumptions that either are founded on experimental data or contain an arbitrariness that can only lower the resulting multiplicity, and therefore are consistent with the problem posed of determining a lower limit of the multiplicity.

The calculations are presented in Appendix I. In the present section we shall confine ourselves to a discussion of the results.

If one assumes that the value of the multiplicity is \(n=5\), then the maximum of the curve describing the altitude variation of electrons produced by a primary particle with energy \(E_0=10^4—10^5\), must lie at a depth exceeding

\[ t=a\frac{\ln \dfrac{E_0}{E_{\mathrm{кр}}}}{\ln n}\sim 14 \]

(formula (15,1); see curve \(II\) in Fig. 6).

This depth is greater than the observed one, which, as was mentioned earlier, is of order 10.

An even greater average multiplicity is obtained if one analyzes the altitude variation of showers of high energies (according to approximate estimates \(10^7—10^8\) (\(10^{16}—10^{17}\) eV)). It is known that the number of such showers decreases with height (between sea level and an altitude of 4 km) very strongly—approximately in the same way as showers of low energy (\(10^4—10^5\)). Therefore one may expect that the maximum of high-energy showers lies considerably above 4 km. However, if one assumes that in each act 10 particles are born, then a shower of energy \(10^7—10^8\) would reach its maximum at a distance of 17–20 units (4–5 km).

Hence it may be concluded that at very high energies the multiplicity exceeds 10. If one also takes into account that the development of extensive showers is determined by secondary particles of relatively high energies*), then the true multiplicity (i.e. the total number of particles) at high energies turns out to be greater than 20.

Such a large value of the multiplicity of particles produced in a collision with a light nucleus cannot be reconciled with the concept of multiple production. Indeed, from elementary

*) The main share of the energy is carried away by particles moving in the \(Ц\)-system at an angle less than \(90^\circ\) relative to the direction of motion of the primary particle. The number of such particles is approximately equal to one half of the total number.

considerations it follows that, since on average in a collision with a light nucleus a nucleon collides with only two particles of the nucleus (assuming that in each collision a meson and a \(\delta\)-nucleon are produced), the total number of particles is less than 10.

The already mentioned fact of the practical independence of the height of development of broad atmospheric showers on energy (in the interval \(10^4\)—\(10^7\)) likewise cannot be reconciled with the assumption of the absence of multiple processes. Indeed, it follows from this that the number of particles produced increases with energy, which also cannot be explained on the basis of the concept of multiple production \(^{44}\).

Thus, from the totality of the data one may conclude that at high energies (\(\gtrsim 10^4\), i.e. \(10^{13}\) eV) multiple processes exist. The experimental data at lower energies are not so definite, although in this region too such processes apparently take place.

4. COMPARISON OF EXPERIMENTAL AND THEORETICAL DATA ON THE INTERACTION OF HIGH-ENERGY PARTICLES

The experimental material already accumulated in past years has made it possible to evaluate theories and to recognize some of them with great confidence as untenable. Thus, from the data obtained in the study of broad atmospheric showers it follows that the multiplicity can depend on energy no more strongly than

\[ n \sim E_0^\gamma,\quad \text{where } \gamma < \frac{1}{3}^{46}. \]

Hence it follows that Heisenberg’s conclusion \(^{12}\) that this dependence at very high energies is represented by the function \(n \sim E_0^{1/2}\) does not correspond to reality. The theory of Oppenheimer et al. \(^{17}\), which predicts an isotropic distribution of the particles produced in the center-of-mass system, is not consistent with the data on broad atmospheric showers or with observations in photographic plates.

Indeed, as was noted by Zatsepin \(^{44}\), who proposed, in comparing the characteristics of an elementary act and of broad showers, using the angular distribution of energy fluxes, in the case of an isotropic distribution (in the \(C\)-system) the energy in the \(L\)-system will be distributed approximately uniformly within an angle

\[ \frac{1}{2}\sqrt{\frac{2}{E_0}}. \]

For energies \(\sim 10^{14}\) eV this angle is \(\sim 2\cdot 10^{-3}\). Such a large value leads to the fact that the width of the “plateau” (i.e. the region of approximately uniform particle density near the axis) in the curve of the spatial distribution of particles should be \(\sim 20\) m. This contradicts experimental data \(^{47,48}\) on the width of the plateau, which, if it exists, is no wider than \(1\)—\(2\) m \(^{45,47,48}\). (For more detail on the width of the “plateau,” see below.)

The assumption of isotropy in the \(L\)-system also contradicts the observed angular distribution of particles in a number of “stars” of high energy. Thus, Shain and collaborators\({}^{42}\) observed a “star” with 15 tracks of relativistic particles and 2 tracks of slow particles. Seven particles moved inside a narrow cone with half-angle at the vertex \(\vartheta_1 \sim 0.003\); the remaining 8 moved within a broad cone with half-angle \(\vartheta_2 \sim 0.13\).

Assuming that a nucleon–nucleon collision occurred, it is easy, using formula (B), to estimate the value of the half-angle \(\vartheta\) in the \(L\)-system:

\[ \left. \begin{aligned} \vartheta_1 &\sim \frac{\operatorname{tg}\dfrac{\overline{\vartheta}_1}{2}}{\gamma},\\[4pt] \vartheta_2 &\sim \frac{\operatorname{tg}\overline{\vartheta}_2}{\gamma} \end{aligned} \right\} \qquad \gamma=\sqrt{\frac{\overline{E}_0}{2M}}. \]

Since \(\overline{\vartheta}_1 = 180^\circ - \overline{\vartheta}_2\) (equality of the particles),

\[ \operatorname{tg}^2 \frac{\overline{\vartheta}}{2} \sim \frac{\vartheta_1}{\vartheta_2}. \]

If Shain’s data are used, we obtain \(\overline{\vartheta}\sim 20^\circ\)\({}^{*}\). Thus, if the assumption of an elementary collision of two nucleons is correct, then the distribution in the \(L\)-system must be anisotropic.

Let us turn to other theories. Fukuya’s assumption that a considerable \(\left(\sim \dfrac{5}{6}\right)\) fraction of the energy remains with the primary particle contradicts the data obtained in the study of extensive atmospheric showers.

Thus, from this assumption it follows that the maximum number of showers should lie near sea level (i.e. considerably below that observed), and the range of the cascade should be \(\sim 1500\ \text{g}\cdot\text{cm}^{-2}\), in contrast to the experimentally observed value \(\sim 200\ \text{g}\cdot\text{cm}^{-2}\).

The recently published new work of Heisenberg\({}^{11}\) predicts (as was shown in § 1, incorrectly) an increase of the interaction cross section with energy, which is not consistent with the experimental data on a high-energy star\({}^{49}\) \((\sim 10^{14}\ \text{eV})\), where it was found that at energies of the order of \(10^{12}\ \text{eV}\) the interaction cross section is close to the geometrical one, whereas according to Heisenberg it should have been an order of magnitude larger \((\sim 10\sigma_{\text{geom}})\).

The same is evidenced by the “stars” recorded in photographic plates. Among them there are a number of “stars” interpreted by the authors as nucleon–nucleon collisions. The number of particles in most such “stars” agrees well both with the theory

\({}^{*}\) It is necessary to note that the value obtained is somewhat smaller than follows from Fermi’s theory, which gives for the angle \(\overline{\vartheta}\sim 40^\circ\). This circumstance was not noted by the authors of work\({}^{42}\).

Fermi’s, as well as Landau’s theory, and is in rather sharp contradiction with the other theories*).

Thus it remains to consider only Fermi’s theory and Landau’s theory**). In order to choose between them, a more detailed analysis of the experimental data is necessary. It is necessary to include data on collisions of nucleons with nuclei and to extend the range of applicability of the theory in this direction. In the case of high energy this is easy to do if one assumes that the primary particle collides only with those nucleons of the nucleus that lie in its path.

The first stage of the collision of a nucleon with a nucleus ends with the formation, in the \(L\)-system of the nucleon and the tube, of an excited clump of nuclear matter, the volume and width of which are determined by the dimensions of the “tube.” Then begins the stage of expansion of the matter found in such an excited state. We applied Landau’s theory to this stage and assumed that the expansion would occur symmetrically with respect to the plane perpendicular to the direction of motion. Generally speaking, this assumption may also fail to hold, since the problem is not initially symmetric (for example, the expansion may begin from the end into which the nucleon flew, before the shock wave reaches the other end). Moreover, such an asymmetry should even be expected in the case of a central collision with a heavy nucleus (a long “tube”). However, in cases of collision with a small “tube” (containing 2–3 nucleons) such a model still seems applicable.

Let us compare Landau’s theory, modified in this way, with the experimental data obtained by Kaplan et al.\(^{51,52}\).

The authors used a new technique based on the use of the so-called “emulsion chamber,” which consisted of a stack of photographic plates interleaved with sheets of copper. Collisions of particles with copper nuclei were investigated. Not only the photographic plates in the immediate vicinity of the place where the shower originated were examined, but also all 15–20 plates traversed by the shower. The authors were able to study in detail the angular distribution of the particles. In processing each star they plotted the dependence of \(n_{\vartheta}\) on \(\ln \vartheta\), where \(n_{\vartheta}\) is the number of particles emitted at angle \(\vartheta\) (in the \(L\)-system).

*) The exception is a “star” observed by Teucher\(^{50}\), in which more particles were recorded than should be according to the Fermi–Landau theory. However, it is possible that this “star” is a collision of a nucleon with a nucleus, since the criterion for nucleon–nucleon collision content, as indicated above, is ambiguous.

**) Let us note once again that by Fermi’s theory we mean here not Fermi’s basic paper\(^{4}\), on which Landau’s is based, but his second paper\(^{24}\), devoted to the angular distribution.

A typical graph is shown in Fig. 3.

The authors found that in every case one can find on the curve a point, corresponding to an angle \(\vartheta_0\), such that the curve is symmetric with respect to this point. It follows from this that one can always find such a moving coordinate system in which the angle \(\vartheta_0\) will have the value \(90^\circ\), and the distribution of particles will be symmetric with respect to \(90^\circ\). Such a distinguished system is naturally to be regarded as the system of the center of gravity\(^*\). Thus, in the \(C\)-system the angular distribution is not isotropic, but has a “two-cone” character, i.e. is symmetric with respect to \(90^\circ\).

Fig. 3. Integral angular distribution of particles in the \(C\)-system, obtained in \(^{52}\).

In papers \(^{51,52}\) the authors give the following experimental quantities: the number of charged secondary particles \(n\), the quantity

\[ \gamma=\frac{2}{1-v^2} \]

(\(v\) is the velocity of the \(C\)-system\(^ {**}\)), and also, characterizing the angular distribution, the quantity \(X=\frac{x}{x_1}\), where

\[ x=\frac{\vartheta_{3/4}}{\vartheta_{1/4}}, \]

and \(x_1\) is the same quantity corresponding to an isotropic distribution in the \(C\)-system; \(\vartheta_{3/4}\) and \(\vartheta_{1/4}\) are the angles in the \(L\)-system within which \(3/4\) and \(1/4\) of all particles are emitted (for a large anisotropy in the \(C\)-system, \(X\) is large; in the case of an isotropic distribution, \(X=1\)).

Using these data, one can process the experimental material of Kaplon et al. in accordance with Landau’s theory, generalizing it to the case of a collision of a nucleon with a nucleus; that is, using the quantities \(\gamma\) and \(X\), one can find the velocity of the \(C\)-system, as well as the length and mass of the tube with which the nucleon collides in each case. On the basis of these quantities one can compute the number of particles \(n\) which should be produced in each case, and compare it with the experimental value \(n_{\mathrm e}\).

The results of such a calculation are given in Table III. It is evident from the table that in those cases where the nucleon collides with a not very long tube (containing 2–3 nucleons), the values calculated according to

\(^*\) The authors’ conclusion that there is symmetry in the angular distribution in the \(C\)-system is very important, since from the theoretical point of view, as indicated above, in collisions of a nucleon with a large nucleus one should expect an asymmetric distribution. Unfortunately, in papers \(^{51,52}\) the authors give only one curve \(n_\vartheta-\ln\vartheta\), on the basis of which, of course, it is impossible to judge all stars.

\(^ {**}\) It must be said that \(\gamma\) coincides with the energy of the primary particle in the \(L\)-system only in the case of nucleon–nucleon collision, which is not always the case.

Landau’s values \(n_{\mathrm{L}}\) agree well with the experimental ones. In cases of collision with a long tube a noticeable discrepancy is observed; however, as was indicated above, in this case the method of calculation is inapplicable.

Table III

Star No.	\(\gamma \cdot 10^{-3}\)	\(X\)	Tube length \(\dfrac{l}{R}\)	\(n_{\mathrm{L}}\) (according to Landau)	\(n_{\mathrm{F}}\) (according to Fermi)	\(n_{\Gamma}\) (according to Heisenberg)	\(n_{\mathrm{e}}\) (experim.)
104p	1.3	1.33	4	30	11.7	702	20
67pn	4.5	1.33	7.6	78*)	16	2430	9
60pn	19.5	1.33	15.8	234*)	23	10530	26
91pn	1.3	1.7	2.65	20	10.8	456	16
224p	1.8	1.9	2.54	21	11.6	514	24
63K	6.1	2.5	2.64	29	12	984	19
77	4	1.17	9	90*)	16.5	106	24
17p	5.1	3.7	—	—	7.9	321	24**)
59pn	8.9	4	—	—	8.8	2714	24**)

) In these cases the method of calculation is inapplicable (see text).
*) These stars are the result of a nucleon–nucleon collision.

In the last two columns of the table are given the values \(n_{\mathrm{F}}\)—the number of particles calculated according to Fermi’s theory by the authors themselves, and \(n_{\Gamma}\)—the number of particles calculated by us according to Heisenberg’s theory.*) It is evident that the numbers \(n_{\mathrm{F}}\) do not differ very sharply from the experimental data, but in any case they agree with experiment worse than \(n_{\mathrm{L}}\). The value \(n_{\Gamma}\), however, is greater than the experimental values by at least an order of magnitude, which once again proves the inadequacy of Heisenberg’s theory.

In a number of cases it turned out that the mass of nuclear matter with which the primary particle should have collided, according to the calculation given above, is less than one nucleon. This means that a collision occurred with one of the outer nucleons of the nucleus, with a large impact parameter. For such processes, as was indicated above, a quantum treatment is necessary, which reduces to the need to consider meson exchange (one, two, etc.).

*) In the calculation according to Heisenberg’s theory we assumed that a head-on collision of a nucleon with the tube occurred, since, quite apart from the fact that his theory of peripheral collisions is not correct (see § 1), in the present case the angular distribution clearly indicates that the impacts are close to central. We note that, since Heisenberg did not consider the question of the angular distribution in essence, the calculation cannot be made with any precision. Therefore the figures given in the column \(n_{\Gamma}\) are correct only as to order of magnitude.

The process of collision of a nucleon with mesons was here considered by analogy with the process of collision of two nucleons, i.e., Landau’s theory was applied directly to this case.

The results of the calculation are given in Table IV. From the table it is seen that the numbers of produced particles \(n_\pi\) calculated in this way agree well with the observed values. A calculation according to Fermi’s theory (performed by the authors themselves \(^{48,49}\)) gives values that are too small and do not agree with experiment.

Table IV

Star No.	\(\gamma \cdot 10^{-3}\)	\(X\)	Calculated quantities: number of exchanged mesons \(r\)	Calculated quantities: \(X\)	Calculated quantities: \(n_\pi\) (according to Landau)	Calculated quantities: \(n_\Phi\) (according to Fermi)	\(n_e\) (experim.)
72K	0,5	4	2	3,8	7	4,5	17
55K	1,29	3,3	2	4,5	9	6	18
222pn	1	6	2	4,3	8,4	4,2	16
66pn	0,64	6,67	2	4	7,5	2,7	11
T(p)	19,4	6,67	2	7	18	6,3	36
58K (pn)	30	8,7	2	7,5	19	8	15
179(pn)	0,8	20	1	6	6	2	24
67K (pn)	0,8	20	1	6	6	2	15
Zpn	50	24	1	11	17	3,4	15
184pn	5	33	1	8	9,4	2	10
45K	7,2	66	1	8	10,3	2	14
Sp	32	—	1	9,8	15	6	15
GL(p)	23	9	1	9,5	14	6	18

Attention should be drawn to the fact that in cases of exchange by one meson the calculated values \(X\) differ from the measured ones. Unfortunately, the authors \(^{51,52}\) do not give the errors in measuring \(X\), and in the case of large values of \(X\) (which corresponds to very narrow cones) these errors should precisely be very large. Thus, this cannot be regarded as a sharp discrepancy.

Thus, the result of comparing the experimental data of Kaplan et al. with the theoretical data is as follows: Landau’s theory, extended both to the case of collision with a nucleus and to the case of distant peripheral collisions, gives in all cases where it is applicable better agreement with experiment than Fermi’s theory, not to mention other theories (Heisenberg’s, etc.), which sharply contradict the experimental data.

Finally, in the “star” caused \(^{49}\) by collision with a nuclear emulsion nucleus of a nucleus with energy \(\sim 10^{14}\) eV, the angular distribution of the particles agrees only with Landau’s theory, although the authors themselves erroneously believe that agreement with Fermi’s theory takes place.

Since in the present case a collision of a nucleus with a nucleus takes place, Fermi’s theory cannot be applied directly. If, nevertheless, one attempts to use Fermi’s theory, then one should rather expect an angular distribution corresponding to a central collision (i.e., isotropy in the \(L\)-system), which is in complete disagreement with what is observed experimentally.

Very important information about the elementary interaction is provided by extensive atmospheric showers, and certain characteristics of showers can also be used to analyze the theories of Fermi and Landau. In particular, for this purpose it is expedient to investigate the spatial distribution of particles near the shower axis and the altitude dependence.

Let us turn to the analysis of the spatial distribution. The region near the shower core is of special interest, since it reflects more than anything else the angular distribution of energy fluxes in the primary act. Thus, for example, it has already been pointed out that if the distribution of energy in the first act of collision in the center-of-gravity system is isotropic, then the spatial distribution of particles in showers with energy \(\sim 10^5\) should vary little within 20 m from the axis. The existence of such a large region of a “quasi-plateau” contradicts the experimental data. The spatial distribution of particles near the core was studied by Hazen \(^{48}\). He indicates that showers with energy \(10^5\) are characterized at an altitude of 3 km by a “plateau” of width \(\sim 1\) m. Therefore any theory of the elementary act must predict an angular distribution of energy fluxes that ensures the absence of a “plateau” at distances greater than \(1\text{--}2\) m. However, Hazen’s conclusion requires further clarification, since he determined the shower core by the presence of high-energy electrons. In reality, because of the nuclear-cascade process, such electrons can be removed to considerable distances from the core. If this effect plays an important role (there is no reliable experimental information about it), then the data obtained by Hazen on the spatial distribution characterize not the core, but other regions. In the following paper \(^{53}\) Hazen et al. compared their data on the width of the core with Fermi’s theory. Although Hazen comes to the conclusion that there is a contradiction between the values of the shower energy calculated, on the one hand, from the number of particles in it, and on the other hand, from the width of the “plateau” on the basis of Fermi’s theory (the energy calculated by the second method exceeded by 1–2 orders of magnitude the value determined by the first method), this conclusion does not seem indisputable, since certain inaccuracies were allowed in the calculation (for example, what was used in essence was the distribution in particle number, and not the angular distribution of energy fluxes in the elementary act).

In addition, in \(^{53}\) it is assumed that the collision takes place, on average, at the mean nucleon–nucleon collision parameter.

In reality, the nucleon collides with a nucleus of air, and therefore the mean parameter in the present case has a different value (more on this below).

In other works (G. T. Zatsepin 44; Green and Messel 54) it was also pointed out that Fermi’s theory incorrectly predicts the angular distribution of the particle.

We have compared the experimental data 47 with the predictions of the Fermi and Landau theories, using as the characteristic of the elementary act the angle at which one half of the energy is emitted.

The angle \(\vartheta_{1/2}\), in accordance with Fermi’s theory, is determined by equations (11, II) and (12, II), which are derived in Appendix II.

From these equations it follows that the magnitude \(\vartheta_{1/2}\) depends on the parameter \(\rho\), which is determined by the impact parameter. Therefore, in the case when an averaged result of many collisions is being studied (for example, the characteristics of broad showers), it is necessary to use the mean value of the parameter \(\rho\). In the case of a nucleon–nucleon collision, the mean value of \(\rho\) is determined without difficulty.

Indeed, let us define the parameter \(r_{1/2}\) so that the probability of collisions with \(r < r_{1/2}\) is equal to the number of collisions with \(r > r_{1/2}\). Consequently,

\[ \frac{\displaystyle \int_{0}^{r_{1/2}} r\,dr}{\displaystyle \int_{0}^{R} r\,dr} = \frac{1}{2}; \]

whence \(r_{1/2} \sim 0.7\).

To this value of \(r_{1/2}\) there corresponds the value \(\rho = 0.959\), for which \(\vartheta_{1/2} \sim 0.3\).

The generalization of Fermi’s theory (and consequently also the calculation of \(r_{1/2}\)) to the case of a collision of a nucleon with a nucleus is considerably more complicated. If one adopts the most plausible model (collision of a nucleon with particles located in a “tube”), then most collisions will be central, and consequently the angular distribution in the \(L\)-system will be isotropic, which, as noted above, contradicts the experimental data. Only in the case of collision with the peripheral nucleons of the nucleus is an anisotropic collision possible according to Fermi. The introduction of \(r_{1/2}\) (even if the nucleon collides with a light nucleus) is meaningless, since in this case \(r_{1/2} > 1\). It is expedient to calculate the fraction \(\delta\) of cases of collisions with peripheral nucleons when \(R_{\text{n}} - r < 0.3\,(R_{\text{n}} = R A^{1/3})\). If we take \(A = 15\), then \(\delta \sim 0.25\).

Consequently, only in 25% of cases should, if one follows Fermi, a strong anisotropy be observed. In the remaining 75% of cases the distribution in the \(L\)-system will be close to isotropic.

Therefore the calculation of the angular distribution in the case of a nucleon–nucleus collision (and, consequently, also of the spatial distribution of particles in showers) for the parameter value \(r_{1/2}=0.7\) gives only a lower bound for the angles of deviation. However, since there is no consistent generalization of Fermi’s theory to the case of collision with a nucleus, we have carried out the calculation with this value of the parameter. Thus, the results of the calculation presented can give only a lower bound for the width of the “plateau” corresponding to Fermi’s theory. A more consistent application of Fermi’s theory should lead to a more isotropic distribution (in the \(L\)-system), and, consequently, to a broader “plateau.” As we shall see below, even this smallest value of the width of the “plateau” is much larger than the experimental one.

Table V

Energy in eV	\(\bar{\vartheta}_{1/2}\) according to Fermi	\(\vartheta_{1/2}\) according to Fermi	\(\bar{\vartheta}_{1/2}\) according to Landau	\(\vartheta_{1/2}\) according to Landau
\(10^{14}\)	0.31	\(6.7\cdot 10^{-4}\)	0.10	\(1.3\cdot 10^{-4}\)
\(10^{15}\)	—	\(2.1\cdot 10^{-4}\)	0.054	\(2.2\cdot 10^{-5}\)
\(10^{16}\)	—	\(6.7\cdot 10^{-5}\)	0.031	\(3.8\cdot 10^{-6}\)

In the second and third columns of Table V are given the values of \(\vartheta_{1/2}\), calculated from formulas (11, II), (12, II) for \(\rho=0.959\left(\dfrac{r}{R}=0.7\right)\).

Let us proceed next to the calculation of \(\vartheta_{1/2}\) in accordance with Landau’s theory*). For this purpose we first calculate the quantity \(\delta_{1/2}\), determined by the relation

\[ \frac{\displaystyle \int_{\vartheta_{1/2}}^{1} E\,dn} {\displaystyle \int_{-1}^{+1} E\,dn} = \frac{\displaystyle \int_{\vartheta_{1/2}}^{1} \left(\frac{E_0}{2}\right)^{\frac{1}{6}+\frac{\delta}{2}+\frac{2}{3}\sqrt{1-\varepsilon^2}}\,d\vartheta} {E_0} = \frac{1}{2}, \tag{21} \]

*) It is necessary to note that, since Landau considers central collisions, in calculating \(\vartheta_{1/2}\) difficulties do not arise with the determination of the mean parameter in the collision of a nucleon with a nucleus. Landau’s theory makes it possible to describe more consistently the picture of this complex process.

where

\[ \delta_{1/2}=\frac{\lambda_{1/2}}{L};\qquad \lambda_{1/2}=\left|\ln \tg \frac{\vartheta_{1/2}}{2}\right| \]

(thus, \(\delta_{1/2}\) is the parameter corresponding to the angle into which one half of the energy is emitted).

In the energy interval \(E_0=10^5—10^7\), \(\delta_{1/2}=0.54—0.55\), whence

\[ \vartheta_{1/2}\sim \left(\frac{E_0}{2}\right)^{-\frac{1+\delta_{1/2}}{2}} \sim \left(\frac{E_0}{2}\right)^{-0.77}. \tag{22} \]

\(\overline{\vartheta}_{1/2}\) can be determined from (B) and (22).

Table V gives the values of \(\vartheta_{1/2}\) and \(\overline{\vartheta}_{1/2}\), calculated in accordance with formula (22), for different energies. It is seen from the table that the distribution of energy fluxes is considerably “narrower” than according to Fermi, and that this difference increases with energy.

Another feature of the angles calculated in accordance with Landau’s theory is the dependence of \(\vartheta_{1/2}\) on the energy (in contrast to the predictions of Fermi’s theory).

In order to calculate the averaged plateau width (i.e., the width of the region of uniform density under the assumption that, within the angle \(\vartheta_{1/2}\), the energy is distributed uniformly), we assumed that the showers originate at a depth \(\sim 100\ \mathrm{g\cdot cm^{-2}}\), which corresponds to a distance of \(\sim 15\ \mathrm{km}\) from the altitude at which Hazen and Nikol’skii, Vavilov, and Tukish carried out their measurements.

Table VI

Energy in ev	Experimental value of the “plateau”	Lower limit of the averaged plateau width according to Fermi (in m)	Averaged plateau width according to Landau (in m)
\(10^{14}\)	\(\lesssim 1—2\ \mathrm{m}\)	10	3
\(2\cdot 10^{14}\)	\(\lesssim 1—2\ \mathrm{m}\)	7	2

Table VI summarizes the calculated values of the mean “plateau” width for the energy values that were registered in both works. Since an exact calculation of the energy is not possible at the present time, we have also given the “plateau” values for an energy twice as large (relative to the authors’ estimate).

It is seen from Table VI that the lower limit of the “plateau” width according to Fermi considerably exceeds the experimentally observed upper limit. The plateau width calculated in accordance with Landau’s theory agrees with the observational results much better.

Even more important than the mean values, as a characteristic of the elementary act, is the angular distribution of the energy fluxes \(E_\vartheta\). For Fermi’s theory it can be calculated by formula (13, II); in accordance with Landau’s theory (see (17)—(19)) it is determined by the relation

\[ E_\vartheta = \frac{2k}{\sqrt{2\pi \ln \frac{E_0}{2}}}\, \frac{1}{\vartheta} \left(\frac{E_0}{2}\right)^{ \frac{1}{6}+\frac{\delta}{2}+\frac{2}{3}\sqrt{1-\delta^2} } \,d\vartheta, \tag{23} \]

where

\[ \delta=-1-2\,\frac{\ln \vartheta}{\ln \frac{E_0}{2}}. \]

Figure 4 gives the angular distributions of the energy fluxes, calculated for the value \(E_0=10^5\) \((10^{14}\ \mathrm{eV})\) in accordance

Fig. 4. Angular distribution of energy fluxes: curve \(I\)—distribution according to Fermi, curve \(II\)—according to Landau. These curves also describe the spatial distribution of electrons near the axis, caused by the first act. In this case, along the abscissa axis are plotted \(M\) (lower scale); the quantities on the ordinate are proportional to the energy flux of the particle fluxes at a distance from the axis. In converting the scales it was assumed that the first collision occurred at a distance of \(15\ \mathrm{km}\) from the observation point.

with both theories. The spatial distribution of particles in the shower, determined only by the first act, is represented, with the corresponding change of scale, by the same curves.

The character of the curves shows that, whereas according to Landau one should expect a very sharp increase in the particle density as the distance decreases, formula (13, II) predicts the presence of a “plateau” in the region \(1\)—\(6\) m. Experimental data show that there is no “plateau” in this region \(^{47,48}\).

Unfortunately, the accuracy of the calculations carried out by Landau is such that one cannot judge the spatial distribution of particles in showers with an energy \(10^{15}\) at distances less than 1 m. Essentially the same situation holds in Fermi’s theory. Therefore Fig. 5 does not give the distribution for such small distances.

Thus, in accordance with Landau’s theory, right up to distances of \(\sim 1\) m, strictly speaking, there should be no regions of constant density.

Thus, all the experimental data set forth above on interaction at high energy agree with Landau’s theory. And if the experiments of Kaplon et al. \(^{51,52}\) testify to this partly indirectly (additional assumptions were made in the treatment of these experimental data), then the data on wide atmospheric showers directly confirm the angular distribution predicted by Landau’s theory. Further investigation both of stars in photographic plates and of wide atmospheric showers apparently cannot substantially alter this conclusion, although refinements in this direction are highly desirable.

However, the experimental study of the elementary act at high energies can by no means be considered complete. Many unresolved questions remain and, in particular, the very important question of the composition of showers at high energies. Some light may be shed on this problem by a detailed investigation of the core of a wide atmospheric shower, in particular the study of the composition of the core, the fraction of the penetrating component, etc.

On the other hand, the experimental study of electron–nuclear showers is very promising. In this direction, the observation of showers of high energy \((E \sim 10^{13}\ \text{eV})\), formed in various elements (both heavy and light), would be of great interest. With the aid of such experiments it would be possible to check in detail not only the theory of the elementary act, but also the assumptions set forth above concerning the character of the nucleon–nucleus collision.

CONCLUSIONS

1. On the basis of the totality of all experimental data, one may conclude that multiple processes exist at high energies.

2. The theories of Heisenberg, Oppenheimer et al., and Fukuda contradict the experimental data on interaction at superhigh energy.

1. The theories of Fermi and Landau give the same dependence of the multiplicity on the energy of the primary particle, but substantially different angular and energy distributions of the secondary particles.

2. The angular distribution of particles given by Fermi’s theory cannot be reconciled with the spatial distribution of particles in broad atmospheric showers and with the data obtained by means of photographic plates.

3. Landau’s theory agrees, within the limits of its applicability, with all experimental data available at the present time.

In conclusion, the authors express their deep gratitude to E. L. Feinberg, whose advice was widely used in the course of the work. The authors also thank Yu. A. Smorodin for discussion of the experiments of Schein et al.^40

ADDENDUM I

On the maximum of the curve describing the altitude variation of the electronic component of broad atmospheric showers

Let us first carry out the calculation of the altitude dependence of the nuclear-active component of broad showers under the following assumptions:

1) In each collision of a nuclear-active particle whose energy is greater than the critical energy*), \(n\) (\(n=\mathrm{const}\)) particles are always produced.

2) The energy of the incident particle is divided equally among the secondary particles.

3) The fraction of energy \(\Delta\) transferred to neutral \(\pi\)-mesons (and consequently to the electron-photon component) does not depend on the energy of the colliding particles.

4) All secondary particles interact with the nucleus with the same cross section, equal to the geometrical cross section.

It is easy to see that the assumptions we have made are to a considerable extent based on experimental facts. To the extent that these assumptions are arbitrary, they have been chosen so that they can lead only to an underestimate of the multiplicity. Meanwhile, our problem is precisely to determine the lower limit of the multiplicity.

Thus, for example, the assumption that the energy of the incident particle is divided uniformly among the secondary particles is equivalent

*) The critical energy \(E_k\) is the energy at which the generation of secondary relativistic particles ceases. According to data^27 \(E_k \sim 5\cdot 10^9 — 10^{10}\) eV.

to a decrease of the multiplicity and, consequently, leads to a “stretching” of the shower line*).

A decrease of the effective cross section also leads to a lowering of the position of the maximum. An increase of the cross section cannot be excluded a priori. However, there are experimental indications\(^{49}\) that, up to energies \(\sim 10^{13}\), the size of the cross sections for secondary particles is of the order of the geometrical one. It is natural to expect that at higher energies as well it remains constant.

Let us proceed to the calculations.

From the scheme we have adopted there follows the expediency of introducing the concept of a “generation.” We shall define it as follows: all particles of the \(i\)-th generation have the same energy \(E_i\) and are formed in collisions with air nuclei of particles of the \(i-1\)-st generation.

Then, taking as the unit of length the mean free path \(L\) traversed between two successive collisions, one can write the following system of equations describing the distribution \(P_i(l)\) of particles of different generations with depth:

\[ \frac{dP_i(l)}{dl}=-P_i(l)+n(1-\Delta)P_{i-1}(l), \tag{1,I} \]

where \(i=1,2,\ldots,r\); \(r\) is determined from the equation \(E_0\left(\frac{1}{n}\right)^r\sim E_{\mathrm{cr}}\) (\(E_0\) is the energy of the primary particle). This relation has a simple physical meaning: the energy of the \(r\)-th generation is equal to the critical energy.

Taking into account that \(P_0(l)=e^{-l}\), we obtain:

\[ P_i(l)=\frac{[n(1-\Delta)]^i e^{-l}l^i}{i!}, \tag{2,I} \]

The total number of particles is determined by the sum**)

\[ N(l)=\sum_{i=0}^{r}P_i(l) =e^{-l}\sum_{i=0}^{r}\frac{[n(1-\Delta)]^i e^{-l}l^i}{i!}. \tag{3,I} \]

Taking into account that \(\Delta \lesssim 0.1 \div 0.2^{55}\), and also that near the maximum of the cascade curve describing the altitude dependence of the nuclear-active component the determining contribution is made by particles of the \(r\)-th generation, we obtain that the corresponding maximum depth is

\[ l_{\max}\sim r. \]

\[ \underline{\hspace{5cm}} \]

*) We note that the assumption of a uniform distribution of energy is based on the existence of a maximum in the energy distribution of secondary particles in statistical-hydrodynamical theories.

**) The equations written down do not take into account the decay of charged particles into nuclear-passive ones. There are the following grounds justifying, for the problem posed, this assumption (although a rigorous proof cannot yet be given): 1) we are interested in the main range of high energies; 2) the total energy of the meson component constitutes, apparently, only a small part of the energy \(E_0\).

We next proceed to calculate the depth \(l_{\max}^{(e)}\) corresponding to the maximum of the curve describing the altitude dependence of the electron-photon component. First let us calculate the number of electrons \(P_i^{(e)}\) formed in the avalanche multiplication of photons arising from the decay of \(\pi^0\)-mesons of the \((i-1)\)-st generation.

Passing to ordinary radiation \(t\)-units of length and taking into account that \(\pi^0\)-mesons decay into two photons, we obtain:

\[ P_i^{(e)}(t)= \]

\[ = \frac{2n\lambda [n(1-\Delta)]^{i-1}}{(i-1)!\,a^i} \int_0^t e^{-\frac{(t-t_1)}{a}} (t-t_1)^{i-1} \Pi\!\left[\frac{E_i}{2},\,t_1\right]\,dt, \tag{4,I} \]

where \(a=\dfrac{t}{l}\sim 2.2\text{—}2.4\), \(\Pi\!\left[\dfrac{E_i}{2},\,t_1\right]\) is the number of electrons in an avalanche formed by a photon with energy \(\dfrac{E_i}{2}\), if the avalanche has traversed a path \(t_1\).

Using the representation \(^{56,57}\)

\[ \Pi\!\left[\frac{E_i}{2},\,t_1\right] = e^{\varphi_i^{(m)}-\frac12 b_i(t_1-t_m)^2}, \tag{5,I} \]

where

\[ e^{\varphi_m}=\Pi_i^{(m)},\qquad b_i=2\pi\left[\frac{2\beta\Pi_i^{(m)}}{E_i}\right]^2 \sim \frac{0.6}{t_i^{(m)}};\! \]

\(\Pi_i^{(m)}\) is the number of electrons of the \(i\)-th generation at the maximum of the avalanche; \(t_i^{(m)}\) is the corresponding depth of the maximum. Equation (4,I) can be written in the form

\[ P_i(t)= \frac{2n\lambda [n(1-\Delta)]^{i-1}}{(i-1)!\,a^i} e^{-\frac{t}{a}+\varphi_i^{(m)}} \sum_{k=0}^{i-1} (-1)^k C_{i-1}^k t^{\,i-k-1} \times \]

\[ \times \int_0^t e^{\frac{t_1}{a}}t_1^k e^{-\frac12 b_i\left(t_i^{(m)}-t_1\right)^2} \,dt_1, \tag{6,I} \]

or

\[ P_i(t)= \frac{2n\lambda [n(1-\Delta)]^{i-1}}{(i-1)!\,a^i} \Pi_i^{(m)}e^{-\frac{t}{a}} \sum_{k=0}^{i-1} (-1)^k C_{i-1}^k t^{\,i-k-1} \times \]

\[ \times \frac{ e^{-\frac12 b_i[t_i^{(m)}]^2+c_i} }{ \left(\sqrt{\frac{b_i}{2}}\right)^{k+1} } \int_{-\sqrt{c_i}}^{\sqrt{\frac{b_i}{2}}\,t-\sqrt{c_i}} e^{-x^2}(x+\sqrt{c_i})^k\,dx, \tag{7,I} \]

where

\[ C_i=\frac{\left(\frac{1}{a}+b_i t_i^{(m)}\right)^2}{2b_i}\sim \frac{1}{2b_i}. \]

Let us find \(\dfrac{dP_i(t)}{dt}\):

\[ \frac{dP_i(t)}{dt} = \frac{2n\Delta [n(1-\Delta)]^{i-1}}{(i-1)!\,a^i}\, \pi_m e^{-\frac{1}{2}b_i\left[t_i^{(m)}\right]^2+C_i-\frac{t}{a}} \times \]

\[ \times \sum_{k=0}^{i-1} \frac{C_i^k{}_{-1}}{\left(\sqrt{\frac{b_i}{2}}\right)^{k+1}} \,t^{\,i-k-2} \times \]

\[ \times \left\{ \left[-\frac{1}{a}t+(i-k-1)\right] \int_{-\sqrt{C_i}}^{\sqrt{\frac{b_i}{2}}\,t-\sqrt{C_i}} e^{-x^2}(x+\sqrt{C_i})^k\,dx \right\}. \tag{8,I} \]

As we shall see below [see (10,I), (11,I)], in the region of the maximum of the electron curve (when \(\dfrac{dP_i}{dt}=0\)) \(\sqrt{\dfrac{b_i}{2}}\,t>\sqrt{C_i}\); therefore we study the region \(t>\sqrt{\dfrac{2C_i}{b_i}}\).

Then

\[ \int_{-\sqrt{C_i}}^{\sqrt{\frac{b_i}{2}}\,t-\sqrt{C_i}} e^{-x^2}(x+\sqrt{C_i})^k\,dx \simeq \frac{\sqrt{\pi}}{2}\,(\sqrt{C_i})^k . \]

The sum in (8,I) is simplified, and

\[ \frac{dP_i(t)}{dt} \sim i-1-\frac{t-\sqrt{\frac{2C_i}{b_i}}}{a} = i-1-\frac{t-1.6t_i^{(m)}}{a}. \tag{9,I} \]

Consequently, the maximum \(t_i^{(M)}\) of the cascade curve describing the behavior of electrons of the \(i\)-th generation (under the assumption that the electrons are produced through the nuclear-active component) is located at

\[ t=t_i^{(M)}=a\left(i-1+\frac{1.6t_i^{(m)}}{a}\right).^{*} \tag{10,I} \]

\[ \text{}^{*}\ \text{We recall that } t_i^{(m)} \text{ is the maximum of the pure electron-photon avalanche with energy } \frac{E_i}{2}. \]

Since the energies of the particles of the \(i\)-th generation are \(E_i=\dfrac{E_0}{n^i}\), then

\[ t_i^{(\mathrm{M})}=a\left[i-1+1.6\,\frac{\ln \dfrac{E_0}{2}-i\ln n-\ln \beta}{a}\right]. \tag{11,I} \]

Regarding the index \(i\) as a continuous function, we find

\[ \frac{dt_i^{(\mathrm{M})}}{di} = a\left[1-\frac{1.6\ln n}{a}\right]. \tag{12,I} \]

It follows from (12,I) that if \(\dfrac{1.6\ln n}{a}<1\) (small multiplicity, \(n\leqslant 4\)), then \(t_i^{(\mathrm{M})}\) is an increasing function of \(i\), and therefore the smallest value is \(t_i^{(\mathrm{M})}=t_1^{(\mathrm{M})}\), and, consequently, the depth \(t^{(\mathrm{M})}\) corresponding to the maximum of the total number of electrons satisfies \(t^{(\mathrm{M})}\gtrsim t_i^{(\mathrm{M})}\). In the case when \(\dfrac{1.6\ln n}{a}>1\), the smallest value is \(t_i^{(\mathrm{M})}=t_r^{(\mathrm{M})}\); in this case \(t^{(\mathrm{M})}\gtrsim t_r^{(\mathrm{M})}\).

Let us examine both cases:

1) \[ \frac{1.6\ln n}{a}<1, \]

\[ t_1^{(\mathrm{M})}=1.6\ln \frac{E_0}{2\beta n}. \tag{13,I} \]

Consequently, for very small values of \(n\) the position of the maximum is approximately 1.5 times lower than the position of the maximum of the curve describing the longitudinal development of an electron-photon shower produced by an electron with energy \(E_0\).

2) \[ \frac{1.6\ln n}{a}>1, \]

\[ t_r^{(\mathrm{M})} = a\left[2-1+1.6\,\frac{\ln \dfrac{E_0}{2\beta n^r}}{a}\right] > ar, \tag{14,I} \]

i.e., in this case the maximum of the number of electrons always lies lower than the maximum of the number of nuclear-active particles.

Since \(r\sim \dfrac{\ln \dfrac{E_0}{E_{\mathrm{cr}}}}{\ln n}\), then

\[ t_r^{(\mathrm{M})} > a\,\frac{\ln \dfrac{E_0}{E_{\mathrm{cr}}}}{\ln n}. \tag{15,I} \]

In Fig. 5 are shown curves describing the longitudinal development of the nuclear-active and electronic components for two values of \(n\):

\[ n=3 \quad \text{and} \quad n=5 \qquad [E_0=10^5]. \]

As is seen from Fig. 5, for \(n=3\) (curves \(I\) and \(II\)) \(t^{(M)}\sim 1.5\ln \dfrac{E_0}{\beta}\);

Fig. 5. Cascade curves. Curves \(I\) and \(II\) describe, respectively, the longitudinal development of the electronic and nuclear-active components if \(n=3\); curves \(III\) and \(IV\)—if \(n=5\). The ordinates of curve \(V\) are proportional to the experimental values \(^{13}\). On the ordinate axis the left-hand scale corresponds to the number of electrons, and the right-hand scale to the number of nuclear-active particles.

for \(n=5\) (curves \(III\) and \(IV\)) the depth at which the maximum of the curve describing the longitudinal development of the electrons is located lies considerably below the maximum of the nuclear-active component.

APPENDIX II

Calculation of the angle \(\vartheta_{1/2}\) in accordance with Fermi’s theory

We first calculate the angle \(\vartheta_{1/2}\) (in the \(L\)-system) at which the secondary particles carry away one half of the energy of the primary particle. In calculating \(\vartheta_{1/2}\) in accordance with Fermi’s theory we use the variant in which the formation of nucleon–antinucleon pairs is not assumed. As shown by Hazen et al. \(^{53}\) (which, incidentally, follows naturally from the theory itself), taking into account the possibility of nucleon production does not change the result*). According to Fermi, the number of \(\pi\)-mesons in an element of phase volume of the \(L\)-system is equal to:

\[ dn=\frac{3A}{E_0}\frac{1}{\left[e^{\zeta}-1\right]}(1-\xi^2)\,\bar p^{\,2}\,d\bar p\,d\eta\,d\xi . \tag{1,II} \]

*) The generalization of the conclusions to the case of nucleon production is carried out without difficulty.

(On the physical meaning of the quantities entering into (1, II), see § 1.) This expression is conveniently represented in the form

\[ dn=\frac{3A}{E_0}\sum_{r=1}^{\infty} e^{-r\xi}(1-\xi^2)\,\overline{p}^{\,2}\,d\overline{p}\,d\overline{\eta}\,d\xi . \tag{2,II} \]

Let us calculate the energy carried away by particles with momentum \(\overline{p}\), moving at an angle \(\arccos \overline{\eta}\):

\[ \overline{E}_{\overline{p},\,\overline{\eta}} = \frac{3A}{E_0}\,\overline{p}^{\,3}\,d\overline{p}\,d\overline{\eta} \sum_{r=1}^{\infty}\int_{-1}^{1} e^{-r\xi}(1-\xi^2)\,d\xi = \]

\[ = \frac{6A}{E_0(\gamma p\eta)^2}\,\overline{p} \sum_{r=1}^{\infty}\frac{1}{r^2} \left[ e^{-r\gamma\overline{p}(1+p\eta)} \left(1+\frac{1}{r\gamma p\overline{p}\eta}\right) + \right. \]

\[ \left. + e^{-r\gamma\overline{p}(1-p\eta)} \left(1-\frac{1}{r\gamma p\overline{p}\eta}\right) \right]. \tag{3,II} \]

The total energy \(\overline{E}_{\overline{\eta}}\), carried away by particles moving within the angular interval \((\arccos \overline{\eta},\,\arccos(\overline{\eta}+d\overline{\eta}))\), is equal to

\[ \overline{E}_{\overline{\eta}}\,d\overline{\eta} = \frac{6A}{E_0(\gamma p\eta)^2}\,d\overline{\eta} \sum_{r=1}^{\infty}\frac{1}{r^2} \int_{0}^{\infty}\overline{p} \left[ e^{-r\gamma\overline{p}(1+p\eta)} \left(1+\frac{1}{r\gamma p\overline{p}\eta}\right) + \right. \]

\[ \left. + e^{-r\gamma\overline{p}(1-p\eta)} \left(1-\frac{1}{r\gamma p\overline{p}\eta}\right) \right]d\overline{p} = \]

\[ = \frac{6AB}{E_0(\gamma^2 p\eta)^2}\,d\overline{\eta} \left[ \frac{1}{(1-p\eta)^2} - \frac{1}{p\eta(1-p\eta)} + \frac{1}{(1+p\eta)^2} + \frac{1}{p\eta(1+p\eta)} \right], \tag{4,II} \]

\[ B=\sum_{r=1}^{\infty}\frac{1}{r^4}=1.08. \]

Integrating with respect to \(\overline{\eta}\) from \(\overline{\eta}_m\) to \(1\), we obtain the total flux \(\overline{E}_{>\overline{\eta}_m}\) of energy carried away by particles moving at angles \(\vartheta<\arccos \overline{\eta}\):

\[ \overline{E}_{>\overline{\eta}_m} = \frac{6AB}{E_0\gamma^4 p} \left[ \frac{2p}{1-p^2} +\ln\frac{1+p}{1-p} - \frac{2p\overline{\eta}}{1-p^2\overline{\eta}^{\,2}} - \ln\frac{1+p\overline{\eta}}{1-p\overline{\eta}} \right], \tag{5,II} \]

and the total energy

\[ \overline{E}_0= \frac{12AB}{E_0\gamma^4 p} \left[ \frac{2p}{1-p^2} +\ln\frac{1+p}{1-p} \right]. \tag{6,II} \]

Equation (6,II) can serve to determine the magnitude \(v\). In order to determine \(\bar{\vartheta}_{1/2}\), we shall use the relation

\[ E=\gamma\left(\bar E+p\sqrt{\bar\eta}\right), \tag{7,II} \]

where \(\gamma=\dfrac{\bar E_0}{2M}=\dfrac{1}{\sqrt{1-v^2}}\) and \(E\) is the energy in the \(L\)-system.

Taking into account that relativistic particles are being considered, one may put

\[ E=\gamma \bar E(1+\bar\eta). \tag{8,II} \]

The angle \(\bar{\vartheta}_{1/2}\), within which the particles carry away half the energy in the \(C\)-system, is determined from the condition:

\[ \gamma\int_{\bar\eta_{1/2}}^{1}\bar E(\bar\eta)(1+\bar\eta)\,d\bar\eta=\frac{E_0}{2}. \tag{9,II} \]

Since in the cases of interest to us \(\bar\eta_{1/2}\sim 1\) (see § 4), (9,II) may be written in the form

\[ \int_{\bar\eta_{1/2}}^{1}\bar E(\bar\eta)\,d\bar\eta=\frac{E_0}{4\gamma}=\frac{\bar E_0}{4}. \tag{10,II} \]

From (5,II), (6,II), (10,II) it follows that \(\bar\eta_{1/2}\) is a root of the equation

\[ \frac{\rho}{1-\rho^2}+\frac{1}{2}\ln\frac{1+\rho}{1-\rho} = \frac{2\rho\,\bar\eta_{1/2}}{1-(\rho\bar\eta_{1/2})^2} +\ln\frac{1+\rho\bar\eta_{1/2}}{1-\rho\bar\eta_{1/2}}. \tag{11,II} \]

Passing to the \(L\)-system, we obtain:

\[ \vartheta_{1/2}=\frac{1}{\gamma}\operatorname{tg}\frac{\bar{\vartheta}_{1/2}}{2}. \tag{12,II} \]

Equation (4,II) describes the angular distribution of the energy fluxes in the \(C\)-system. To obtain the corresponding angular distribution in the \(L\)-system, we use the fact that the region where \(\bar\eta\sim 1\) is of interest to us. Then from (B) and (7,II) it follows:

\[ 2\gamma\,d\vartheta=d\bar\vartheta \quad\text{and}\quad E_{\vartheta}=2\gamma \bar E_{\bar\vartheta}. \]

Hence the magnitude of the energy \(E\) emitted in the interval of angles \(\vartheta\), \(\vartheta+d\vartheta\), is equal to:

\[ E_{\vartheta}\,d\vartheta\sim \frac{48\gamma^3 AB}{E_0(\psi^2-\rho\eta)^2}\, \frac{1}{(1-\rho\eta)^2}\, \vartheta\,d\vartheta, \tag{13,II} \]

where \(\bar\eta=\bar\eta(\vartheta)\) is determined from relation (B).

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