GENERATION OF COSMIC-RAY $\pi$-MESONS BY NUCLEAR-ACTIVE PARTICLES OF MODERATE ENERGIES

G. B. Zhdanov

INTRODUCTION

The purpose of the present review is to analyze certain properties of the elementary act of generation of $\pi$-mesons in the energy interval of primary*) nuclear-active particles $10^9$—$10^{10}$ eV.

These mesons have an average energy approximately an order of magnitude lower than that of the primary particle, and from an instrumental point of view a considerable fraction of them should be assigned to the category of slow mesons, i.e., particles having a range in matter comparable with the dimensions of the apparatus and readily deflected in a magnetic field.

The energy region of nuclear-active particles indicated above is of considerable interest from various points of view. First, it is precisely to this energy region that the principal mass of particles of the primary cosmic radiation belongs which, in the process of nuclear interaction with the matter of the atmosphere, create all the main components of the secondary radiation. Second, in this energy region the generation of new particles is, for the most part, limited to $\pi$-mesons, and the role of heavier mesons is still small here. Naturally, it is more difficult to study processes in which several kinds of particles participate, especially since the identification of the particles produced becomes increasingly difficult as their mean energy increases. Third, an energy of the order of $3 \div 5$ Bev constitutes the threshold from which the effective cross section of the process of multiple meson generation

*) Here and below we shall use the terms “primary” and “secondary” particles, referring them only to the act of nuclear interaction (shower), which is not related to the division of cosmic radiation into primary component and secondary radiation.

on nuclei (i.e., the generation of electron-nuclear showers, in the usual sense of this term) approaches the geometrical cross section of the nucleus. It should be expected that it is precisely under these conditions that the specific features of nuclear interactions, due to the presence of a strong coupling of nucleons with the meson field, begin to manifest themselves. Finally, the discovery of at least qualitative features of the phenomena occurring at this energy of the incident nucleons should be of considerable interest to physicists working with accelerator apparatus, since the “mastering” of the energy range \(10^9\)—\(10^{10}\) eV is, in this field, a matter of the nearest future.

What can be said at present about the basic characteristics of the elementary act of \(\pi\)-meson generation? First of all, too little is generally known about the truly elementary act, i.e., about the interaction of two free nucleons in cosmic rays, owing to the great experimental difficulties accompanying work with liquid hydrogen, or the use of differential effects.

As will be shown below, for complex nuclei, especially light ones, the existing experimental material is very incomplete and is often distorted by various kinds of instrumental effects. As for theoretical ideas on the question of interest to us, there do not yet exist any sufficiently definite, even qualitative, considerations that would make it possible to approach the derivation of the basic regularities of the process. In any case, in the energy region of the order of \(10^9 \div 10^{10}\) eV the situation is considerably worse than in the study of processes occurring at still higher energies, where the essential qualitative features of the phenomenon can be understood on the basis of purely statistical and even thermodynamic laws.

1. EXPERIMENTAL METHOD

The principal methods for studying the properties of nuclear interaction with the formation of mesons may be classified as follows:

a) photographic plates;
b) Wilson chamber;
c) ionization apparatus (counters, ionization chambers).

All these instrumental possibilities can be used both for direct and for indirect study of elementary processes. By indirect study we mean the analysis of the so-called “integral” characteristics of one or another component of cosmic radiation in the atmosphere or in dense matter (intensity and its dependence on depth or geographic latitude, spectrum, angular distribution). As a rule,

GENERATION OF RADIATION BY NUCLEAR-ACTIVE PARTICLES

even the most comprehensive study of integral characteristics cannot give an unambiguous conclusion about the properties of the corresponding elementary processes; however, in many cases it makes it possible to carry out a quantitative check of various data obtained by direct methods. Moreover, in some cases the analysis of integral characteristics makes it possible to establish directly certain features of the nuclear interaction that escape direct observation.

One example of such an analysis is the fact established by Grigorov¹ of the concentration of a large fraction (on the average 70%) of the energy on one secondary nucleon, as will be discussed in more detail below.

The most graphic and complete picture of an elementary act can, in principle, be obtained by the photographic-plate method. In a Wilson chamber, observation is usually made difficult both by the necessity of using interactions occurring in the bulk of a solid substance*) and by the distortions introduced into the phenomenon under study by the control system of counters. Similar difficulties also arise when using counters in the form of hodoscopic systems, for which the situation is made still worse because of the poor spatial separation of the individual particles, although on the other hand a sufficiently accurate resolution of them in time is provided.

Alongside this, one may also point to a whole series of shortcomings that hamper the study of the elementary act by the method of photoemulsions. Among the principal shortcomings, which to some extent force one to resort also to other, more “coarse” methods of studying the elementary act, are:

a) difficulties in obtaining data relating to nuclei of definite composition, and moreover with as low an atomic number as possible;

b) difficulties in determining the sign of the charge of particles;

c) low “luminosity,” connected both with the great laboriousness of processing the experimental material and with the small “working volume”**).

In view of the indicated difficulties, it appears rational also to use results obtained with the aid of chamber and ionization techniques; as will be clear from the further exposition, the experimental information on the elementary act obtained by the first and especially by the second of these methods requires the introduction of a whole series of instrumental corrections, thanks to which

*) This difficulty may be overcome after the development of liquid chambers instead of ordinary condensation chambers.

**) Great prospects for overcoming the latter difficulty are offered by the use of stacks of unsupported photoemulsions.

uncertainty and ambiguity of the results go substantially beyond the limits associated with the statistical errors of the experiment, and thereby also with the effective “luminosity” of the instrument. However, the basic question, on the solution of which the reliability of the results and the validity of the corresponding conclusions depend, reduces to the problem of identifying the particles under study and the elementary processes.

In assessing the effectiveness of using various methods, one should bear in mind that a further step forward can be made only through reliable determination of the nature and energy of the particles causing the elementary processes observed in the experiment in each individual case. It is precisely this entirely realistic condition—which up to the present time, as a rule, has not been fulfilled even in work with photographic plates—that will make it possible to bring the value of information obtained for cosmic radiation closer to the corresponding results of accelerator experiments, while at the same time retaining a practically unlimited advantage in the range of primary energies accessible to the experimenter.*)

2. SOME DATA ON THE NATURE AND INTENSITY OF THE COMPONENT GENERATING SLOW π-MESONS IN DENSE MEDIA AND IN THE ATMOSPHERE

In the overwhelming majority of studies of meson-production processes in cosmic rays, experimenters have been deprived of direct data on the nature and energy of the primary particle. Only in work ² was it possible, in a number of cases, to measure directly the momentum of the primary charged particle by the scattering method; however, even in this case, for all momenta \(p > 1.25\,\mathrm{Bev}/c\) there remained an uncertainty associated with the impossibility of distinguishing protons from π-mesons.**) Indirect estimates of the primary energy are based on the use of one of the following three methods:

a) differential measurements at different latitudes in the stratosphere;

b) determination of the energy “threshold” of the apparatus for registering processes of the given type;

c) use of the relation between the primary energy and the angular distribution of secondary particles.

The first method requires no special explanation; obviously, it is suitable only for measurements near the boundary of the Earth’s atmo—

*) A second advantage of studies carried out with cosmic radiation is, evidently, the serious saving in material resources.

**) The question of an “admixture” of π-mesons becomes very important when, under the conditions of the experiment, a considerable layer of dense matter is located above the place where the shower is produced.

spheres, where the influence of the “geomagnetic threshold” is still sufficiently strong.

To use the second method it is necessary to know, first of all, the energy spectrum and the absolute intensity of the nucleon component at various depths in the atmosphere, and also the magnitude and energy dependence \(E\) of the effective cross section \(\sigma\) for the process of meson generation by nucleons. This section will be devoted to these questions; at its end we shall also give an estimate of the role that may be played by processes not connected with the nuclear interaction of nucleons.

In order to obtain the approximate behavior of the function \(\sigma(E)\), averaged over various nuclei, one may use three groups of data:

a) the dependence, on the energy \(E\), of the relative yield of \(\pi\)-mesons in accelerator experiments (see, for example, \(^{3}\));

b) the dependence, on \(E\), of the relative number of so-called inelastic interactions according to photographic-plate data \(^{2}\), and also according to experiments on the “cosmotron” \(^{4}\);

c) the value of the “effective threshold” for the generation of electron–nuclear showers*) according to data on the latitude effect (see, for example, \(^{3}\) and \(^{6}\)).

Fig. 1. Approximate behavior with energy \(E_p\) of the effective cross section \(\sigma\) for the generation of \(\pi\)-mesons by protons (according to data \(^{18,19}\)). \(\sigma_0\) is the geometrical cross section.

The approximate dependence \(\sigma(E)\) obtained from all the indicated data is shown in Fig. 1; here the “geometrical” cross section \(\sigma_0\) in the first approximation corresponds to the radius of various nuclei

\[ R = 1.4 \cdot 10^{-13} A^{1/3} \]

(see, for example, \(^{7}\)).

A large number of works have been devoted to the study of the spectrum and the absolute intensity of the nucleon (mainly proton) component in the atmosphere; these may be divided into the following

*) By this “threshold” is understood that value of the primary energy \((E_c)\) at which either the cross section for multiple generation of mesons, or the cross section for the formation of fast mesons and new nuclear-active particles, approaches the geometrical cross section of the nucleus \(\sigma_0\).

groups (in accordance with the method used by the authors for identifying the nucleon and measuring its energy)*:

a) determination of the momentum and of some other parameter (range, ionization) that makes it possible to distinguish single protons against the background of the other charged particles;

b) determination of the momentum using a subsequent nuclear interaction as the criterion for a nuclear-active particle;

c) estimation of the energies of nuclear-active particles from the character of the nuclear disintegration or shower produced by them;

d) separation of protons from the μ-mesons of the hard component by means of either the ratio between the numbers of positive and negative particles, or one of the characteristic properties of the μ-meson, in particular decay.

Obviously, the first method is suitable in a very limited interval of comparatively low energies and can give only a normalization of the differential proton spectrum, if the form of this spectrum is known. Conversely, the second method is good over a very wide range of momenta, with the exception of comparatively slow protons, for which it is difficult to estimate the probability of recording the interaction. Let us note in passing that the low-energy region in the proton spectrum, on the one hand, is difficult for measurements because of the decrease in the “luminosity” of the instruments in magnetic analysis of momenta and, on the other hand, is distorted, in comparison with the corresponding neutron spectrum, by the influence of ionization losses (Fig. 2).

Fig. 2. Integral momentum spectra of neutrons and protons in the atmosphere:
a — spectrum at the point of generation of the form \(p^{-1.5}\); b — spectrum at the point of generation of the form \(E^{-1.5}\); circles — proton spectrum according to data² (altitude 3.4 km).
It is assumed that the difference in the spectra of protons and neutrons is due only to ionization losses.

The third method is good in that it can also be applied to the study of the neutron component. Indeed, knowing, for example,

* The same classification was used in Tables I and II, in the column “method.”

probabilities of formation of showers with different numbers of relativistic particles \(n_s\) as a function of proton energy, it is possible, using the same photodisintegration data, to determine, generally speaking, both the form of the spectrum and the neutron flux at the given altitude. Such work was carried out by the author of the article using data of the Bristol group\(^8\), and the corresponding results are reflected—

Fig. 3. Excitation functions of “stars” of types \(1_n\) and \(2_n\) by different portions of the spectrum of cosmic-ray neutrons (at an altitude of 3.3 km). The dashed lines show the boundaries of the energy intervals responsible for the generation of 75% of “stars” of type \(1_n\) and 70% of “stars” of type \(2_n\).

Table I

Summary of experimental data on the energy spectra of the (differential) nucleonic component in the atmosphere

*) In this case the method of indirect data processing described above was used.

—are set forth in Tables I and II together with other data. In passing, the excitations of “stars” of types \(1_n\) and \(2_n\) by neutrons of different energies, shown in Fig. 3, were obtained.

After these remarks let us consider a brief summary of the principal experimental results on the study of the spectrum (Table I) and flux (Table II) of the nucleon component at various depths in the atmosphere. A small number of data pertaining to small depths were obtained at high latitudes; for conversion to low latitudes the corresponding latitude dependences should be taken into account here (see, for example, \(^{1,9}\) and \(^{10}\)).

For the convenience of comparing the data of Table II, which refer to various minimum energies \(E_0\), we have recalculated them to one and the same energy \(E_0 = 2\) Bev (Fig. 4), using here

Fig. 4. Various data on the intensity of the nucleon component in the atmosphere, recalculated from Table IV to the minimum nucleon energy 2 Bev: circles—protons, vertical intensity, \(\mathrm{cm}^{-2}\cdot\mathrm{hour}^{-1}\cdot\mathrm{sterad}^{-1}\); triangles—protons, integral intensity, \(\mathrm{cm}^{-2}\cdot\mathrm{hour}^{-1}\); crosses—neutrons, integral intensity, \(\mathrm{cm}^{-2}\cdot\mathrm{hour}^{-1}\).

an energy spectrum of the form \(E^{-2.5}dE\) and taking into account the influence of ionization losses for protons in accordance with Fig. 2. The most reliable should be considered the data \(^{11-13}\), obtained by the method of magnetic analysis of momenta, but even they differ among themselves within a factor of 2, even if one takes into account a certain uncertainty in the knowledge of the shape of the spectrum. In what follows, as a measure of the flux of the nuclear-active component in the atmosphere, we shall use the values \(I_v\) lying on the dashed straight line of Fig. 4.

Let us now estimate the possible contribution of other processes of \(\pi\)-meson generation (besides direct generation by nucleons on nuclei). Such additional processes are:

a) generation by photons on nuclei;
b) generation by mesons on nuclei;
c) decay of heavy mesons in the atmosphere.

To take account of process (a), it is necessary to know the intensity of the “active” photon component (with energy above \(E_0=200\) MeV) and the effective cross section of the process \(\sigma_\phi\). Using experimental data on the electron component in the atmosphere (see, for example, \(^{20}\)), as well as theoretical data on the spectra of electrons and photons in the equilibrium soft component (see \(^{21}\)), we estimated the flux of “active” photons at sea level as \(2\ \text{cm}^{-2}\cdot\text{hr}^{-1}\cdot\text{sterad}^{-1}\). To estimate the effective cross section one may take the limiting value

Table II

Summary of experimental data on the integral intensity of the nucleon component in the atmosphere (\(I_{\mathrm{v}}\) — vertical, \(I_{\mathrm{gl}}\) — global intensity)

* See the footnote to Table I.

...the value $\sigma_\phi$, which is reached already at $h\nu \simeq 300$ MeV and, according to accelerator data (see, for example, $^{22}$), amounts to $3 \div 5 \times 10^{-28}\ \text{cm}^2$ for carbon nuclei, with a geometrical cross section $\sigma_0 \sim 3 \times 10^{-25}\ \text{cm}^2$. In this case $\sigma_\phi$ varies with the atomic number according to the law $A^{1/3}$ (see, for example, $^{23}$), i.e., in the same way as for the generation of mesons by nucleons. Comparing the indicated estimates with the fluxes (see Fig. 4) and cross sections ($\sigma \simeq \sigma_0$) for the nucleon component in the atmosphere, we see that even in the region of the very lowest pion energies the relative contribution of photoproduction should amount to a quantity of the order of several percent. Approximately the same ratio between nucleons and photons remains also in extensive atmospheric showers.

The question of the generation of mesons by mesons themselves will be considered below (Section 4) in connection with the generation spectra of pions in the atmosphere; moreover, it will be shown that in the nuclear-cascade process pions cannot play the determining role.

Finally, the question of the influence of decay processes in the formation of pions may be of interest from the point of view of a possible discrepancy between the generation spectra of single $\mu$-mesons (through pions) in the atmosphere and the generation spectra of pions in dense media. To resolve this question one should use the existing estimates of the relative frequency of generation of pions and heavier mesons in high-energy nuclear interactions. The data, obtained by the photographic-plate method and the Wilson chamber, on the frequency of formation of charged heavy mesons ($K$- and $\tau$-), are admittedly not very definite (see, for example, $^{24-26}$), but give a relative contribution of these processes of no more than $1$–$2\%$ for moderate energies ($10^9$–$10^{10}$ eV). A contribution several times larger (up to $5\%$) may be made by neutral particles according to Wilson-chamber data (see, for example, $^{27}$).

Thus, it should be concluded that the principal part of the slow pions is formed in nuclear interactions of nucleons.

3. MULTIPLICITY OF THE GENERATION PROCESS AND THE RATIO IN THE NUMBER OF SLOW $\pi$-MESONS OF DIFFERENT SIGN

The multiplicity $n$ and “charge asymmetry” $s$*) are very essential characteristics of the process of meson generation, closely connected (along with the angular and energy distribution of the particles) with the mechanism of the process. In addition,

*) In what follows we shall use the “charge asymmetry,” i.e., the quantity $s = n^+/n^-$ (the ratio of the numbers of produced mesons of different sign), instead of the usually adopted “positive excess,” which is defined through

$$ \delta = 2\,\frac{n^+ - n^-}{n^+ + n^-}. $$

knowledge of the quantities \(n\) and \(s\), and especially of their dependence on the primary energy \(E\) and mass number \(A\), is of great methodological interest, since it makes it possible to take into account the influence of instrumental effects and of the control system in studying the properties of the elementary event and to establish the range of primary energies effectively selected by a given installation.

At present one cannot indicate a method that would make it possible to determine the total multiplicity of \(\pi\)-meson generation (over the entire energy range). For the subsequent analysis of the experimental data we shall be interested in two energy regions, partially complementing each other. In the first case very slow mesons stopping in the emulsion were studied, for which there could be no doubt as to the identification of the particles, but the upper limit of the recorded energies was quite small. The second, more extensive group of data, obtained both by the photographic-emulsion and by the chamber method, belongs to the category of “shower” (in the case of photographic emulsions), or penetrating (in the case of a Wilson chamber) particles, and it is characterized by a definite lower limitation on the energies.

The most interesting systematic information on slow mesons was obtained in two works with photographic plates. In the first of them\(^{28}\) the probability \(p_{\pi}\) of emission in “stars” of a very slow meson \((E<40\ \mathrm{Mev})\) was determined as a function of the number of heavy particles \(N_h\), and it turned out that \(p_{\pi}\sim N_h^2\), and consequently \(p_{\pi}\) is approximately proportional to the square of the excitation energy of the nucleus. If one uses the distribution in the number \(N_h\) for “stars” with different numbers of shower particles \(n_s\), given for an altitude of \(3.3\ \mathrm{km}\) in work\(^{8}\), it follows that \(p_{\pi}\) is a monotonically increasing function of \(n_s\) (see Table III), and hence also of the primary energy of the incident nucleon. At the same time, the parallel behavior of the number of the indicated mesons and of the strongly ionizing particles in the “star” compels one to suppose a common origin—

Table III

Relative probabilities (\(p_{\pi}\)) of emission of very slow \((E<40\ \mathrm{Mev})\) \(\pi\)-mesons (mainly negative) as a function of the number of shower particles in the “star” \((n_s)\) for heavy nuclei of photographic emulsion \((N_h>8)\), according to data\(^{28}\) and \(^{8}\)

behavior for particles of both types, attributing it either to the “evaporation” of an excited nucleus, or to an intranuclear cascade process with the participation of secondary slow nucleons and mesons. Obviously (and this circumstance will be analyzed in detail below), for the emission of mesons of higher energy, no longer associated with the “evaporation” of the nucleus, the dependence on the primary energy may turn out to be substantially different.

The dependence of the probability of formation \(p_{\pi}\) on the atomic number of the nucleus was studied in work \(^{29}\), in which photographic plates were surrounded by thick layers of various substances. The effective energy interval of the recorded mesons in this case is not very well defined, since it is connected with the form of the spectrum at the point of generation; however, the mean energy is apparently noticeably greater than in the first case. The dependence \(p_{\pi}\) (in relative units) on the nature of the nucleus, obtained by the authors \(^{29}\), does not look very convincing quantitatively, since the results may be distorted by the change in the form of the spectrum from nucleus to nucleus, and the statistical errors of the measurements are very large. The qualitative conclusion is that for the heaviest nuclei (Pb) there is no increase in the probability \(p_{\pi}\), as compared with light nuclei, and, on the contrary, there is a tendency toward some decrease. Considerably more definite data on the dependence of \(p_{\pi}\) on the atomic weight \(A\) were obtained with accelerators (see \(^{30}\) and \(^{31}\)) for \(\pi^{+}\)-mesons with energies up to 40 MeV. In this case the primary energies were so small (240–340 MeV) that the effect of elastic collisions of protons in the nucleus must substantially affect the degradation of their energy. Considering, along with this degradation, also the absorption of mesons in nuclear matter and their reflection from the potential barrier at the boundary of the nucleus, the author \(^{32}\) was able to give a qualitative explanation of the dependence on \(A\) for the relative yield of \(\pi^{+}\)-mesons with energies from 13 to 40 MeV.

Experimental works devoted to the study of the multiplicity of generation of fast mesons may be divided into 3 groups, depending on the method used:

a) determination of the number of shower particles \((n_s)\) in a photographic emulsion with direct measurement of the energy (more precisely, momentum) of the primary particle by the scattering method;

b) determination of the number \(n_s\) in a photographic emulsion with an estimate of the primary energy from the angular distribution or from the energies of secondary particles in large showers;

c) determination of the number of penetrating particles in a Wilson chamber with an estimate of the shower energy by various methods, including by multiplication of the electron–photon component, by penetrating power, by scattering in plates, and by the angles of emission of the penetrating shower particles.

Before giving a complete summary of the results obtained by various methods (see Table IV and Fig. 5), we shall make a few methodological remarks.

First, both the “shower” particles in a photographic emulsion and the penetrating particles in a Wilson chamber contain a rather appreciable admixture of protons. For photographic emulsions, according to data\(^{33}\), the average fraction of mesons among shower particles is \(79\% \pm 6\%\), and the variation of this fraction with the primary energy and with the atomic number of the nucleus has not been specially studied by anyone. For the Wilson chamber, according to data\(^{16,34}\) and others, the fraction of mesons among penetrating particles emerging from lead varies approximately from 0.3 to 0.8, depending on the controlling system of counters and on the shower energy. The dependence on the atomic number has not been studied by anyone.

Second, even a direct measurement of the energy of the primary (charged) particle still does not give a complete picture of the character of the observed interaction; in particular, the question of the possible fraction of mesons in the generation of the observed showers remains unclear. According to data\(^{2}\), for photographic plates exposed in the stratosphere, \(\pi\)-mesons constitute about 15% of the entire nuclear-active component; however, with a change in the experimental conditions this fraction may prove to be substantially larger.

Finally, when comparing the number of secondary particles \((n_s)\) with the primary energy \(E\), an ambiguity is always present. For small values of \(n_s\), the fluctuations about the mean value \(\overline{n_s}(E)\) agree with Poisson’s law, although calculations of the corresponding statistical weights, carried out by Fermi in the simplest version of his theory\(^{35}\), lead to weaker fluctuations. For large \(n_s\), the fluctuations considerably exceed those that follow from Poisson’s law, and the very character of these fluctuations changes.

In studying the dependence of \(n_s\) on one of the parameters characterizing the angular distribution of secondary particles (for example, on the emission angle of half the particles \(\theta_{1/2}\)), some authors found that this dependence has the form of separate branches. Attempts were made to associate each branch of the curve with a definite number of successive collisions of the incident nucleon with nucleons of the nucleus (see, for example,\(^{36,37}\)), and as an experimental criterion for the number of these collisions it was proposed to count the number of heavy particles in the shower. However, as shown in Ref.\(^{38}\), the idea of successive collisions at large primary energies loses its meaning. More adequate, apparently, is Cocconi’s proposed\(^{39}\) classification of interactions according to the number of nucleons of the nucleus simultaneously participating in them, although the author’s attempt to give a quantitative interpretation of the phenomena within the framework of Fermi’s theory already raises doubts for that reason ...

that the conclusions of this theory do not agree with the data on extensive atmospheric showers (see ^35).

After these remarks let us consider the data of various authors on the mean multiplicity \(\overline{n_s}(E)\) for the case of heavy nuclei (Pb, Au, photomulsion), after which we shall turn to a comparison of the results for nuclei with different atomic numbers.

Table IV

Mean multiplicity \(n_s\) of the formation of “shower” and penetrating particles in heavy nuclei at various primary energies \(E\)

The data of Table IV are repeated, for clarity, in Fig. 5, where the dashed line shows the dependence \(n_s \propto E^{1/4}\), predicted by the theories of Fermi^35 and Landau^42 for sufficiently large primary energies.

To clarify how the multiplicity of the meson-generation process depends on the number of nucleons in the nucleus, the greatest inte-

represent experiments with hydrogen, which were carried out both by the counter method\(^{6,43}\) and with the aid of Wilson chambers\(^{44,45}\). Despite the paucity of the material obtained in the cited works, it may be asserted that multiple production of fast particles in the interaction of two free nucleons does indeed occur; moreover, the intensity of multiple production is smaller than for complex but light nuclei, apparently,

Fig. 5. The absorption law of the integral radiation intensity (see formula (3)) for “true” absorption of the form \(\exp\left(-\dfrac{x}{\lambda_{\Pi}}\right)\). The exponential \(\exp\left(-\dfrac{x}{0.89\lambda_{\Pi}}\right)\) is shown by the dashed line.

at the expense of the higher energy “threshold” of the process; less probable is the assumption that the cross section is several times smaller than the geometrical one (even at sufficiently high energies).

More extensive experimental material has been obtained by the Wilson-chamber method (see, for example,\(^{46,47}\) and \(^{48}\)) in comparing showers produced in light (Be, C) and heavy (Pb) nuclei. Unfortunately, the discrimination introduced by the control system into the distribution of showers according to the number of penetrating particles \((n_s)\) is so large that it is precisely this which determines the mean value of \(n_s\), and the distributions of showers with respect to \(n_s\) in different installations differ much more than in one and the same installation, but for shower production in different substances.

Nevertheless, it may be regarded as established that installations with a sufficiently “hard” control system*) register substantially more large showers \((n_s > 6 \div 8)\) from heavy nuclei than from light ones.

*) By a “hard” control system is usually meant one which selects not fewer than two particles with energies of several hundred MeV each.

Experimental studies of the “charge asymmetry” of mesons may be divided into 3 groups:

a) experiments on the generation of π-mesons on various nuclei in the energy range attainable with accelerators;

b) data on the generation of π-mesons of different signs by comparatively slow nucleons in cosmic rays;

c) studies of single μ-mesons of various energies (also in cosmic rays).

The results obtained with accelerators (see, in particular, works \(^{49-51}\)) give a quite definite picture. At primary nucleon energies of \(\sim 400\) MeV, the charge asymmetry

\[ s = n^+/n^- \]

averaged over the entire meson spectrum differs very strongly from 1, reaching, in the case of light nuclei, approximately 10 for protons and up to \(\frac{1}{10}\) for neutrons. For heavy nuclei (Pb), the quantity \(s\), on the contrary, differs comparatively little from 1. A decrease of \(s\) is also observed for light nuclei as the primary energy of the nucleons increases \(^{52}\). Qualitative confirmation of the same results may also be seen in experiments \(^{28,29,53}\), carried out with photographic plates when observing π-mesons stopped in photoemulsions, whose origin is due mainly to comparatively slow neutrons.

In exactly the same way, a significant charge asymmetry for the momentum interval \(80 \div 340\) MeV/\(c\) was found in experiments \(^{54}\) with Alikhanian’s magnetic mass spectrometer, when mesons with a short range, born in a lead block by neutrons having, on the average, likewise comparatively low energies, were selected. The qualitative interpretation of the facts indicated here, given in work \(^{55}\), is connected with the use of the Pauli principle in counting the statistically possible number of final states of a system consisting of nucleons and one produced meson.

Experimental data on the charge asymmetry of mesons in electron-nuclear showers are almost absent. In particular, in work with a Wilson chamber \(^{56}\), where much attention was paid to the distribution of shower particles by momenta and signs of charge, all the observed charge asymmetry was attributed by the authors to protons; however, the generating component consisted, apparently, to an equal degree of protons and neutrons, and therefore the result is not indicative.

There is, however, an indirect, although not quite unambiguous, method of studying charge asymmetry for light nuclei, connected with the analysis of single μ-mesons. Numerous works on this question were carried out with the aid of: a) Wilson chambers in a magnetic field (see, for example, \(^{57}\), and also the summary of data given in \(^{58}\)); b) magnetic hodoscopic-type analyzers (see \(^{59,60,61}\), etc.); c) the counter method

with delayed coincidences (see ^62); d) the method of counters in the form of a telescope with a so-called magnetic lens, i.e. a magnetized iron filter (see, for example, ^63). A summary of the results concerning mesons of different energies observed at low altitudes is presented in Fig. 6, from which there follows a quite clear increase of the quantity \(s\) with the energy of \(\mu\)-mesons, with a subsequent tendency toward a decrease of \(s\) at energies of the order of \(10^{10}\) eV.

Fig. 6. Dependence between the charge asymmetry
\(s=\dfrac{n^+}{n^-}\) and the energy \(E_\mu\) of single
\(\mu\)-mesons at sea level according to various
data.

An analogous picture is obtained if, using various data, one traces the change with altitude of the quantity \(s\) for slow mesons (see, for example, ^64), since with increasing altitude of the observation point the mean energy of the mesons at the point of their generation decreases monotonically*).

A qualitative interpretation of the indicated dependence of \(s\) on the energy of the mesons themselves and on the associated energy of the primary nucleon is as follows: mesons of low energy originate mainly from nucleons of low energy, for which a large number of intermediate nuclear interactions strongly smooths out the charge asymmetry of the primary cosmic radiation and even shifts it below unity owing to the ionization braking of protons. Conversely, for mesons of high energy, collected from the entire thickness of the atmosphere, the contribution of the primary radiation becomes much larger, and if one denotes by \(\nu(E)\) the mean multiplicity of meson production per primary proton, then the relation must hold

\[ s(E)=\frac{\nu(E)+1}{\nu(E)}. \tag{1} \]

*) In other words, the effective level of generation approaches the registering apparatus.

It follows from (1) that in the region of maximum values of \(s\) (\(s = 1.25—1.3\)) the multiplicity \(\nu\) is \(3.5—4\), while with further increase of \(E\) the quantity \(s\) must decrease owing to the increase in the multiplicity of the process of meson generation. Thus, it may be assumed that, in contrast to the region of small energies, the charge asymmetry of mesons is determined, as it were, by a uniform “spraying” of the charge of the generating particle among all secondaries, which is ultimately connected with the charge symmetry of nuclear interactions of nucleons.

4. ANGULAR DISTRIBUTION OF \(\pi\)-MESONS IN THE ELEMENTARY ACT OF GENERATION

For a quantitative test of one or another conception of the mechanism of the elementary act of meson generation, it is necessary to know not only the total but also the differential cross sections of the process, i.e., the distribution of the emitted mesons over angles and energies. There is every reason to believe that in the laboratory coordinate system these distributions cannot be considered independently of one another, and that simpler regularities will take place in the center-of-mass system of the two colliding systems (nucleon and nucleus). Since we are interested in the region of not too small primary energies \(E\), when, on the one hand, the wavelength of the incident nucleon \(\lambda\) is small in comparison with the range of nuclear forces \(r_0\)), and on the other hand, the energy itself is large in comparison with the binding energy of the nucleon in the nucleus, it makes sense first of all to dwell on the simplest model of the pair interaction of free nucleons and to consider the angular and energy distributions of mesons in the center-of-mass system) of two nucleons. True, there are a number of theoretical considerations in favor of the fact that this simplest model**) is by no means the only possible one, especially at high energies: \(\sim 10^{11}\) eV and above (see, for example, \(^{38}\)), and that the interaction of a nucleon with a nucleus cannot be reduced to successive pair collisions of nucleons.

Let us write down (see, for example, \(^{20}\) and \(^{65}\)) the relations (2)—(6), which connect with one another, on the one hand, the energy \((E = \gamma Mc^2)\) or momentum \((p)\) of the incident nucleon with the velocity \((\beta_c)\) or the corresponding quantity \(\gamma_c\) for the center-of-mass system, and, on the other hand, the angles, energies, and momenta of any particles

) The condition \(\lambda \ll r_0\) is satisfied for \(E \gg 300\) MeV.
) In what follows we shall everywhere use the abbreviation: c.m.s.
**) We shall call it, for brevity, the pair-nucleon model.

in the c.m. system \((\theta_0, \varepsilon_0\) and \(p_0)\) and in the laboratory system \((\theta, \varepsilon, p)\):

\[ \frac{\beta_c}{1-\beta_c^2}=\frac{p}{2Mc}, \tag{2} \]

\[ \gamma_c=\sqrt{\frac{\gamma+1}{2}}, \tag{3} \]

\[ \operatorname{ctg}\theta = \operatorname{ctg}\theta_0 \frac{1}{\sqrt{1-\beta_c^2}} + \frac{1}{\beta_0\sin\theta_0}\cdot \frac{\beta_c}{\sqrt{1-\beta_c^2}}, \tag{4} \]

\[ \varepsilon = \varepsilon_0 \frac{1}{\sqrt{1-\beta_c^2}} + p_0\cos\theta_0 \frac{\beta_c}{\sqrt{1-\beta_c^2}}, \tag{5} \]

\[ p\cos\theta = p_0\cos\theta_0 \frac{1}{\sqrt{1-\beta_c^2}} + \varepsilon_0 \frac{\beta_c}{\sqrt{1-\beta_c^2}}. \tag{6} \]

In relations (2)—(6), \(\theta, \theta_0\) denote the angle with the direction of motion of the primary nucleon, while the energies \(\varepsilon, \varepsilon_0\) and momenta \(p, p_0\) are expressed in units of \(mc^2\) and \(mc\), respectively.

The dependence of \(\beta_c\) on \(E\), calculated with the aid of (2), is presented in Fig. 7.

Fig. 7. Dependence between the kinetic energy of the incident nucleon \((E_k)\) and the velocity \((v_c=\beta_c c)\) of the center-of-inertia system for the case of collision with one stationary nucleon.

Experimental data that would have a direct bearing on the collision of two free nucleons (protons) have been obtained only in accelerator experiments (see \(^{66,77}\)); moreover, for the c.m. system an angular distribution of the form \(\cos^2\theta_0\) was found (the primary energy was \(340\ \text{MeV}\)).

The theoretical considerations developed by Fermi \(^{67}\) show that at higher energies one may expect an even higher degree of anisotropy due to collisions of the noncentral type.

The principal experimental data on cosmic rays relating to complex nuclei at primary energies of \(10^9\)—\(10^{10}\) ev and higher were obtained by two methods—with photographic plates and a Wilson chamber.

In experiments with photographic plates it is difficult to obtain statistically substantiated conclusions directly concerning nuclei of a definite charge number; however, it is possible to approach the study of pair-nucleon interactions by selecting showers with a small number of slow particles (“black” and “gray” tracks). Of special interest are the data on the so-called “jets,” which, together with a small number of heavy particles, contain a large number of shower particles, which facilitates the analysis of individual cases. It cannot be said that the criterion of a small number, or even the complete absence, of slow protons is indisputable. Thus, for example, in paper \(^{68}\) cases are presented in which the assumption of the pair-nucleon character of the interaction (without the participation of slow particles) contradicts the law of conservation of charge. The analysis of the angular distribution of particles in “jets,” carried out in works \(^{39}\) and \(^{69}\), also deserves attention. If, within the framework of the assumption of pair-nucleon interaction, one estimates the value \(\gamma_c\) from the angular distribution of shower particles, it turns out that between \(\gamma_c\) and the multiplicity of production of fast particles \((n_s)\) no unambiguous dependence is obtained; as a result, in particular, Cocconi \(^{39}\) was forced to invoke ideas of interaction with a group of nucleons of the nucleus.

If for the time being one sets aside doubts about the validity of the pair-nucleon model, then the main question arising in the study of the angular distribution of particles is the question of the character of the angular distribution in the c.m.s. and, in particular, of the degree of possible anisotropy of this distribution. Here direct data on the primary energy are usually absent, and therefore, in order to determine the value of \(\gamma_c\), one must again use one or another parameter of the angular distribution in the laboratory system. There are several ways of calculating \(\gamma_c\) from the angular distribution (see, in particular, \(^{69}\) and \(^{70}\)), in each of which either isotropy in the c.m.s. is assumed, or only symmetry of the angular distribution with respect to the plane perpendicular to the direction of motion of the primary nucleon:

a) comparison of the angles \(\theta_f\) and \(\theta_{1-f}\), within which fractions \(f\) and \((1-f)\) of all shower particles are registered (a special case is the estimate of \(\gamma_c\) from the “half-spread” angle \(\theta_{1/2}\)); in the presence of isotropy in the c.m.s. and the additional condition \(\beta_0/\beta_c = 1\), one obtains

especially simple relation

\[ (\gamma_c \operatorname{tg}\theta)^2=\frac{f(\theta)}{1-f(\theta)}; \tag{7} \]

b) determination of the angle \(\theta_{\max}\) (the maximum angle in the laboratory system) for a known value of the mean energy of mesons in the c.m.s.;

c) determination of the value \(\gamma_c\) for which the angular distribution observed experimentally (in an individual shower) proves to be the most probable.

The use of all three methods as applied to the data \(^{69}\) gave mutually consistent values of \(\gamma_c\) (within the range from 20 to 1000 Bev), after which recalculation of the observed angular distribution to the c.m.s. showed that, if anisotropy exists, it is small (approximately such as for the function \(\cos\theta\)).

On the other hand, various authors have observed a number of cases, sometimes relating to still higher energies (up to \(10^{13}\) ev), when the sharply expressed separation of all shower particles into a narrow and a wide cone unambiguously indicates a large anisotropy in the c.m.s. (see, for example, \(^{71}\)).

An entirely different method of testing the assumption of isotropy in the c.m.s. was used in the work \(^{72}\). The authors \(^{72}\) first select a certain analytic form of the spectrum in the c.m.s., namely:

\[ f(\varepsilon_0)\,d\varepsilon_0=\frac{p_0^2\,dp_0}{(\varepsilon_0)^4}, \tag{8} \]

which agrees with the experimentally measured spectrum (in the laboratory system) for showers with a known (from direct measurements) value of \(\gamma_c\).

After this, the total fraction of the backward flux \(B\) in the laboratory system (i.e. the flux of shower particles at angles \(\theta>90^\circ\)) is calculated and compared with the corresponding experimental data \(^{2}\), shown in Fig. 8. The calculated value of \(B\) in this case turns out to be clearly underestimated, which the authors themselves are inclined to attribute not so much to the absence of isotropy in the c.m.s. as to the influence of the intranuclear cascade process in heavy nuclei.

Fig. 8. Backward flux of shower particles as a function of the total number of these particles \((n_s)\), according to data \(^{19}\).

For a qualitative clarification of the question of the influence of the intranuclear cascade process, one may use data by the same authors\(^2\) on the dependence between the angular distribution (in projection) and the number of heavy \((N_h)\) or shower \((n_s)\) particles in a shower.

Fig. 9. Angular distributions of shower particles for various showers at an altitude of 20.5 km according to data\(^ {19}\). Along the abscissa is plotted the angle \(\theta\) (in projection) relative to the direction of motion of the primary particle.
a) Showers with different numbers of relativistic particles \(n_s\): \(1\)—\(n_s = 1 \div 3\), \(2\)—\(n_s = 4 \div 7\), \(3\)—\(n_s \geqslant 8\).
b) Showers with different numbers of heavy particles \(N_h\): \(\circ\ N_h \leqslant 6,\ \times\ N_h \geqslant 7\).

(Fig. 9,a and 9,b). Apparently, some influence of the intranuclear cascade can indeed be detected from the data of Fig. 9,a; however, it is difficult to say to what extent the initial assumption of isotropy in the c.m.s. will be violated by this factor—

difficult. Incidentally, we note that the distributions shown in Figs. 9a and 9b do not give a correct idea of the mean angles of meson deflection relative to the direction of the primary particles. A more correct picture is given by the distribution, taken from work \(^{78}\), presented in Fig. 10, where the angles are counted not in projection but in space, and the ordinates are referred to equal intervals of solid, rather than plane, angles.

Fig. 10. Mean angular distribution of shower particles at an altitude of 3.3 km according to data \(^{31}\). Along the abscissa is plotted the angle \(\vartheta\) (in space) relative to the direction of the primary particle; along the ordinate, the number of cases referred to a unit solid angle.

Let us dwell briefly on the results obtained with a Wilson chamber. First of all, it must be borne in mind that the angular distributions obtained by this method differ rather strongly both for experiments with different installations and for data obtained with photographic plates. Both circumstances are connected with the considerable discrimination introduced into the phenomenon under study by the control system of counters and by the geometrical conditions of the experiment, as well as with the difficulty of separating comparatively slow protons according to their ionizing power. For these reasons the results obtained for a given substance on different installations (see, for example, the data of Chang \(^{47}\) and Green \(^{73}\)) differ from one another considerably more than the data for different substances on one and the same installation.

On the basis of the majority of measurements carried out with light (Be, C) and heavy (Pb) substances under comparable conditions, it may be considered that the mean angles of meson emission differ only slightly. The most interesting information was obtained in one of the recent works \(^{74}\), the authors of which specially investigated the question of the isotropy of the angular distribution in the c.m.s. Using relation (7), they combined showers from C and Pb into groups*) and, for each group, plotted (on a double-logarithmic scale) the dependence between the quantities \(\operatorname{tg}\theta\) and \(\dfrac{f(\theta)}{1-f(\theta)}\). The slight deviation of the obtained curves from straight lines with a slope close to two served as a criterion for the correctness of the initial assumption of isotropy in the c.m.s. (the authors specially veri-

*) Each group included showers with approximately identical values of the mean angles of particle emission.

differed, since a distribution of the form \(\cos^2\theta_0\) in the c.m.s. would lead to a strong deviation from the indicated straight lines). Determining (from the position of the straight lines) the values of \(\gamma_c\), the authors became convinced that for C and Pb these values, measured in tens of BeV, differ little (approximately by a factor of 1.5 at values \(\bar\theta < 25^\circ\)), which indicates the weak influence of the nuclear-cascade process in the Pb nucleus.

In conclusion, let us dwell on the angular distribution of the meson component in the atmosphere. If, at some level \(x_0\), one observes the intensity of the flux of \(\mu\)-mesons with energy above a specified value \((E_0)\) at a specified angle to the vertical \(\varphi\), then it will be determined by the following factors:

a) the absorption law of the nuclear-active component as the depth increases;

b) the angular scattering of \(\pi\)-mesons at the point of their generation;

c) the angular scattering of \(\mu\)-mesons in the decay of \(\pi\)-mesons;

d) multiple angular scattering of \(\mu\)-mesons along their path in the atmosphere;

e) the decay and ionization braking of \(\mu\)-mesons (along the same path), and also their spectrum at the point of generation.

It is not difficult to see that the principal factors are precisely the first and the last, especially at sufficiently high energies \(E_0\). Moreover, in contrast to the case of the angular distribution of the nuclear-active component, absorption of this component over the first few mean free paths of nuclear interaction in the atmosphere may prove essential, since the role of the heavy nuclei of the primary radiation is quite significant there. A calculation carried out by us without taking account of factors b), c), and d), and under certain simplest assumptions concerning the absorption law of the primary radiation in the atmosphere and concerning the meson generation spectrum \(G(E)\), led to the following results:

1. If the usual absorption curve of nuclear-active particles

\[ \exp\left(-\frac{x}{125\ \mathrm{g/cm^2}}\right) \]

is extrapolated all the way to the boundary of the atmosphere, then, under all reasonable assumptions about the form of the function \(G(E)\), the angular distribution of mesons obtained is considerably steeper than that given by experiment (\(\cos^5\varphi\) instead of \(\cos^2\varphi\)), beginning with energy \(E_0 = 2\) BeV and higher.

1. No agreement of the calculated distribution with the experimental one can be obtained even if it is assumed that the energy contained in the heavy nuclei and equal to approximately one half of the entire energy of the primary radiation is absorbed with depth at first considerably more rapidly than for protons, and then just as rapidly, and in both cases the meson spectrum is the same as for protons.

2. The fact that, at the very largest energies \(E\) used in the calculation, the discrepancy with the experimental angular distribution

is not smoothed out indicates that the source of this discrepancy cannot be considered to be neglect of factors b), c), and d).

Apparently, the most plausible explanation of the indicated discrepancy between calculation and experiment is that the meson generation spectrum \(G(E)\), obtained by the photographic-plate method \(^{78}\), differs from the air spectrum more strongly than is usually assumed on the basis of analysis of the altitude dependence of mesons.

5. ENERGY SPECTRUM OF MESONS

The existing data on the spectra of \(\pi\)-mesons arising in nuclear interactions can be divided into three main groups:

a) direct data for interactions with a definite value of the primary energy, obtained with accelerators;

b) direct data obtained with various apparatus in the study of meson generation by the entire nuclear-active component of cosmic rays;

c) indirect data obtained mainly by analyzing the spectrum and absorption curves of single mesons in the atmosphere and in dense matter.

The data obtained with accelerators make it possible not only to determine the meson spectrum at a given primary energy, but also to ascertain the dependence of this spectrum on the target material and on the angle of meson emission. The corresponding results, known from the literature \(^{49,50,75,76}\) mainly for protons with energies \(340 \div 380\,M_{\text{э}}\)*), are presented in Fig. 11. In going from hydrogen to heavier targets, the most probable value of the \(\pi\)-meson energy at first shifts sharply, and then more slowly, toward lower energies. At the same time the position of the hydrogen maximum is determined by the large cross section of the reaction \(p+p \to d+\pi^{+}\), while in the other cases the comparatively small influence of the atomic number is apparently due to secondary interactions of nucleons and \(\pi\)-mesons inside the nucleus, as was already mentioned in connection with the question of the multiplicity of generation.

In going from small angles of emission to large ones (in the laboratory system), the most probable and mean energies of the mesons also decrease, which is connected, ultimately, with the law of conservation of momentum. At the same time, as shown in \(^{77}\), the momentum distribution of nucleons in the nucleus affects the spectra for large angles in a very substantial way, so that analysis of the experimental—

* Recent experiments with the cosmotron \(^{4,52}\) have not yet yielded any detailed information on meson spectra.

Fig. 11. Energy spectra of $\pi$-mesons produced by protons with energies of $340 \div 380$ MeV on various nuclei and for various angles $\theta$ (according to data of $^{49,50,75,76}$):
$a$ — spectra for hydrogen $(\pi^+)$, $\theta = 0$ and $\theta = 60^\circ$;
$b$ — spectra for carbon $(\pi^+, \pi^-)$, $\theta = 0^\circ$ and $\theta = 90^\circ$;
$c$ — spectra for lead $(\pi^+, \pi^-)$, $\theta = 90^\circ$.
$E_\pi$ — kinetic energy of the $\pi$-meson.

GENERATON OF RADIATION BY NUCLEAR-ACTIVE PARTICLES

tal data even makes it possible to test one or another assumption about the form of the momentum distribution of nucleons in nuclei*).

The most detailed information about the energy relations pertaining to the process of meson production in cosmic rays has been obtained by the photographic-plate method, which makes it possible to calculate, within known limits, the energy released in an elementary act by all charged particles, down to the slowest. It turns out that for “stars” produced by protons with energies from 0.5 to 10 Bev, the energy balance is as follows. At an average primary-particle energy of \(2.7 \pm 0.15\) Bev, particles with “black” tracks (with ionization \(I > 7 I_{\min}\)) carry away only about 0.2 Bev; moreover, neutrons with energy \(E < 2.5\) Mev are also included in this group. Particles with “gray” tracks \((1.4 I_{\min} < I < 7 I_{\min})\) and neutrons of the proper energy carry away 0.85 Bev (of this, only 10% falls to \(\pi\)-mesons), and, finally, shower particles \((I < 1.4 I_{\min})\) carry away 1.6 Bev, i.e., 60% of the total energy. Here the main share of the energy of the shower particles is associated with \(\pi\)-mesons, while high-energy protons \((E > 1.6\) Bev) together with the corresponding fraction of neutrons emerge approximately only in 6% of the cases. Meanwhile, for light nuclei the distribution of energy between secondary nucleons and mesons changes substantially in favor of the nucleons. The basis for this statement is provided by indirect data obtained by Vernov and his collaborators (see 5, 9, and 1) from an analysis of the energy relations between the primary and various components of secondary cosmic radiation, studied at different latitudes and through the entire depth of the atmosphere**). The main results of this analysis reduce to the following:

1. Primary protons with an average energy of 3 Bev produce mesons, as a rule, only in the first act of nuclear interaction, and about 30% of all the energy is carried away by mesons in this act (of this, about one half by charged mesons).

2. Protons of higher energy (in any case, starting from 7 Bev and, apparently, up to 1000 Bev) produce mesons already in several successive nuclear interactions; however, in each of these acts no more than 30% of the energy still goes to mesons; about 70% remains with one secondary nucleon.

After these general relations, let us consider the direct and indirect data concerning the form of the \(\pi\)-meson spectrum, obtained by various methods and under several different observational conditions.

*) The author of 77 comes to the conclusion that this distribution is close to Gaussian.

**) Essential for the results presented below are practically depths limited by the stratosphere.

Using photographic plates exposed at an altitude of 20.5 km, Camerini and co-workers \(^{78}\) determined the spectrum of \(\pi\)-mesons born in “stars” of various types. This spectrum, shown in Fig. 12, was determined directly in the energy interval \(30 \div 1500\) MeV. The data for lower energies are not very indicative, since they pertain only to \(\pi\)-mesons stopped in the emulsion, and, for example, in work \(^{28}\) the same portion of the spectrum

Fig. 12. Differential energy spectra of \(\pi\)-mesons and protons formed in “stars” in photographic emulsions. (The proton spectrum is based on data \(^{78}\), the \(\pi\)-meson spectrum on data \(^{28}\) and \(^{78}\).) Along the abscissa axis the kinetic energies are plotted (in MeV), below for protons and above for mesons.

looks quite different (see the dashed curve (1) in Fig. 12)*. For higher energies the falloff of the spectrum apparently becomes steeper (according to the law \(E_n^{-2.5}\) instead of \(E_n^{-1.5}\), where \(E_n\) is the total energy of the \(\pi\)-meson), although this result was obtained only indirectly (extrapolation of the proton spectrum and calculation of the total number of shower particles). By dividing the “stars” into groups with different numbers of heavy particles, the authors \(^{78}\) attempted to investigate the dependence of the form of the spectrum on the atomic number, but they did not find any noticeable dependence by this method.

* The experiments \(^{28}\) were carried out at a greater altitude (\(>26\) km), where both the geomagnetic “cutoff” of the primary-radiation spectrum and the presence of heavy nuclei could have an effect.

The study of the $\pi$-meson spectrum for nuclear interactions at, on the average, higher energy (in electron-nuclear showers) was carried out by the Wilson chamber method[^34][^55][^79] and even with a hodoscope[^80]. In most cases the momenta of the particles were determined by magnetic analysis (in the hodoscope the angle of deflection in magnetized iron was measured); in work[^79] the method of multiple scattering in plates, developed quantitatively in work[^81], was used. The data on the meson spectrum obtained in the works listed are less reliable than the photographic-plate data, both because of the difficulties of separating mesons and protons and because of various apparatus effects that distort the form of the spectrum, especially at low energies. Nevertheless, the qualitative change in the general character of the spectrum, consisting in an increase of both the mean and the most probable energy of $\pi$-mesons by several times in comparison with Cameron’s spectrum, should be regarded as a quite real fact. This principal difference is evidently connected with the presence of a certain, rather high-energy “threshold” for the generating component, set by the controlling system of counters. Still more scanty are the data relating to the comparison of spectra for heavy (Pb) and light (CH$_2$) substances; however, attention should nevertheless be paid to them, since different substances were investigated in one and the same setup[^56]. An indication was obtained that, in passing from a heavy substance to a light one, there is some increase in the mean energy of the mesons. This result can be explained quite naturally by a decrease in the role of secondary interactions inside the nucleus.

As is clear from the foregoing, the principal problems in the information obtained by direct methods on the spectra of generation of $\pi$-mesons are, first, the relatively small range of energies studied; second, the almost complete absence of results relating to light nuclei; and, finally, the absence of sufficiently definite information on the dependence of the form of the spectrum on the energy of the generating particle and on the angle of emission of the meson.

In order to compensate for these gaps to some extent, it makes sense to draw upon indirect, although not always unambiguous, data connected with the study of the properties and behavior of the meson component in the atmosphere and in dense media.

One such investigation is represented by experiments on the study of the so-called transition effects of slow mesons, i.e., phenomena connected with the change in the role of decay processes in the transition of cosmic radiation from the atmosphere into a dense medium. The experiments were set up by one of the following methods:

1) delayed coincidences with counters[^82][^83][^62];

2) a hodoscope with a magnetic field (Alikhanian’s mass spectrometer[^84]);

3) photographic plates[^85][^86].

Experiments carried out as early as 1947–1949 by the author and his collaborators (the delayed-coincidence method) made it possible to establish qualitatively the presence of a transition effect for stopped μ-mesons, especially strong for positive mesons. The interpretation of these results was that there must exist a source of local generation of slow μ-mesons at mountain altitudes, associated with the decay of other nuclear-active mesons (π-mesons), which have an average energy of the order of 100 MeV and are generated, in turn, mainly by neutrons of comparatively low energies (up to 1 BeV).

The results obtained later with a mass spectrometer^84 already make it possible to carry out a quantitative check of one or another assumption about the generation spectrum of slow π-mesons of different signs, which was done with the aid of the corresponding calculations by Saakyan^87. Still more accurate and detailed information about the transition effect of stopped mesons (for the transition from the atmosphere into ice) was obtained in experiments with photographic plates^85,86, and these data make it possible not only to check the form of the spectrum, but also to draw certain conclusions about the ratio of the direct and reverse fluxes of slow π^±-mesons.

The principal result of all the works listed on the transition effect is as follows: for the most diverse substances the generation spectrum of mesons of the entire nuclear-active component of cosmic radiation (at mountain altitudes) in the region of low energies differs little from the spectrum obtained by Camerini^78.

A second method for studying the generation spectra of π-mesons, suitable only for light nuclei (air), but over a considerably wider energy range, is the study of the spectra of the μ-meson component at various altitudes. A serious advantage of this method is the considerably more definite relation between the range and the energy of mesons (the question of taking into account nuclear interactions of π-mesons is eliminated), but, on the other hand, it is necessary to achieve a considerably higher statistical accuracy in all measurements, since the relative role of local generation for fast mesons is much smaller than for slow ones.

The problem of determining the generation spectrum from the altitude variation of the μ-meson component was solved by two different methods. Sands^88 used an integral equation of the form:

\[ F(R,x)=\int_{0}^{x} e^{-\frac{s}{\lambda_{\Pi}}}\,G(R+x-s)\,w(R,x,s)\,ds, \tag{9} \]

where \(F(R,x)\) is the range spectrum at depth \(x\), \(\lambda_{\Pi}\) is the absorption range of the nuclear-active component in the atmosphere, \(G(R)\) is the sought generation spectrum (Sands found it by “fitting” to the experimental values of the spectra \(F(R,x)\)), \(w\) is the inte-

gradient probability that a μ-meson traverses a given segment of its path without decaying.

Garibyan and Goldman \(^{89}\) proceeded from the differential kinetic equation in the variables \(p\) (momentum) and \(t\) (time) instead of \(R\) (range) and \(s\) (thickness of the air layer):

\[ G_\mu [p, x(t)] = -v \frac{\partial F}{\partial x} - \frac{\partial}{\partial p} \left[\frac{dE}{dx} F\right] + \frac{\sqrt{1-\beta^2}}{\tau_0} F . \tag{10} \]

The physical meaning of the individual terms on the right-hand side consists, respectively, in taking account of the effects of the change with time of the meson flux due to their simple displacement, ionization slowing-down, and decay*).

After recalculating the obtained spectrum \(G_\mu\) into the spectrum of generation of \(\pi\)-mesons (in the variables chosen by the authors these spectra differ greatly), one obtains, just as in Sands, a spectrum that practically does not differ from Camerini’s spectrum; however, according to the data of \(^{89}\), the normalization constant is obtained at mountain altitudes to be approximately twice as large as in Sands.

Knowing the normalized spectrum of \(\pi\)-meson generation at a certain level, one can compare it with the corresponding spectrum of the nuclear-active component. Such a comparison is carried out in Fig. 13 for integral energy spectra and for the level \(x = 670 \text{ g}/\text{cm}^2\), the normalization constant of the meson spectrum being referred to the thickness of the nuclear range (65 \(\text{g}/\text{cm}^2\) of air) and to the mean multiplicities of generation \(k = 1\) and \(k = 2\) in each elementary act of generation. The multiplicity \(k = 2\) corresponds more closely to the experimental data, since, according to data \(^{88}\), for example, the mean multiplicity of meson generation in the whole atmosphere is 5.7 (per particle of primary radiation), while according to data \(^{1}\) the mean number of generation acts in the atmosphere is 3.

Fig. 13. Comparison of integral energy spectra for the generation of charged \(\pi\)-mesons (according to data \(^{15}\)) and for the nuclear-active component at a depth of 670 \(\text{g}/\text{cm}^2\) (according to the data of Section 3). \(\pi_1\) — integral spectrum of mesons, normalized to the thickness of the generating layer 65 \(\text{g}/\text{cm}^2\) for process multiplicity \(n = 1\); \(\pi_2\) — the same, but for multiplicity \(n = 2\); \(n\) — integral spectrum of the nuclear-active component. The arrows show the comparison of portions of the spectrum of mesons and nucleons with equal intensity and equal energy.

\[ \frac{N_k}{N_n}=0.18, \qquad \frac{E_m}{E_n}=0.7 \]

*) A specially conducted investigation showed that the influence of Coulomb scattering may be neglected for momenta above \(10^8\) eV/c.

Comparing the meson and nucleon spectra constructed in the indicated way can be done in two ways. First, one may compare the energies of spectral portions of equal intensity, and then (for \(k=2\)) it is seen that the average energy of the mesons produced in each act is approximately 10 times less than the energy of the nucleon, while the total transfer of energy to charged mesons amounts to about \(20\%\)*). Second, one may compare the intensities of spectral portions of equal energy, and then it is seen that \(\pi\)-mesons constitute about \(15\%\) of the flux of nucleons of the corresponding energy.

Such a ratio is already quite sufficient to change substantially (by approximately a factor of 1.5) the range for absorption of the nuclear-active component in dense matter as compared with air, where the energy released into mesons leaves the sphere of the nuclear-cascade process irreversibly.

Summing up the works devoted to the comparison of meson spectra for heavy and light nuclei, one may assert that no substantial difference has been directly found; however, the comparison was usually carried out under different experimental conditions that are difficult to compare.

CITED LITERATURE

1. N. L. Grigorov, DAN 94, 835 (1954).
2. U. Camerini, J. H. Davies, P. H. Fowler et al., Phil. Mag. 42, 1241 (1951).
3. S. B. Jones, R. S. White, Phys. Rev. 78, 12 (1950).
4. W. B. Fowler, R. P. Schutt et al., Phys. Rev. 91, 758 (1953).
5. S. N. Vernov, A. M. Kulikov, A. N. Charakhchyan, Izv. AN SSSR, ser. fiz. 17, 13 (1953).
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*) The last result no longer depends on the choice of the value of \(k\).

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