EXTENSIVE AIR SHOWERS OF COSMIC RAYS

N. A. Dobrotin, G. T. Zatsepin, I. L. Rozental,
L. I. Sarycheva, G. B. Khristiansen, L. J. Eidus

I. INTRODUCTION

In 1929 D. V. Skobeltsyn[^1], observing the tracks of cosmic-ray particles in a Wilson chamber, discovered the property of these particles to appear in the form of groups or showers, constituting streams of ionizing particles passing simultaneously through the chamber.

In 1938 Auger and his collaborators, as well as Kolhörster with his collaborators, undertook a study of atmospheric showers by means of counters connected in a coincidence circuit. They found that, as the distance between the counters was increased, the number of coincidences at first decreased rapidly and then much more slowly (Fig. 1, a); the number of coincidences remains quite appreciable up to very large distances between the counters (of the order of a hundred meters). In Fig. 1, b is shown the dependence, obtained by Auger[^2], of the number of double coincidences on the distance between counters for an altitude of \(\sim 3500\) m above sea level. Each counter had an area of \(200\ \text{cm}^2\). Auger and Kolhörster correctly explained the presence of coincidences at large distances between the counters by the existence of large atmospheric showers covering areas of thousands of square meters.

Auger succeeded in tracing showers up to distances between counters of 300 m, and in 1946 in the Pamirs, at an altitude of 3860 m, thanks to a new method for registering showers by means of counters, proposed by D. V. Skobeltsyn, giant, super-wide showers were discovered, covering up to \(1\ \text{sq. km}\) and more.

Subsequently, a number of authors investigated the altitude dependence of showers. It turned out that the number of showers recorded by any system of counters increases rapidly with altitude, reaching a maximum at an altitude of about 8–9 km, and then falls toward the boundary of the atmosphere.

Fig. 1. Dependence of the number of double coincidences on the distance between counters: \(N\) is the number of double coincidences per hour; \(D\) is the distance between the counters in meters.

Fig. 2. Photograph of a portion of a broad atmospheric shower in a Wilson chamber.

Besides counters, extensive atmospheric showers can also be studied successfully with the aid of ionization chambers and Wilson chambers. If a shower consisting of a sufficiently large number of particles enters an ionization chamber, an ionization pulse arises in the chamber. By using coincidences in several ionization chambers, or coincidences between a chamber and counters, it is possible to separate ionization pulses caused by showers of weakly ionizing relativistic particles from pulses caused by nuclear disintegrations. This makes it possible to clarify such questions as the distribution of showers with respect to density, the structure of showers, etc.

A Wilson chamber, controlled by sufficiently widely spaced counters, gives a visual picture of the passage of shower particles through the volume of the chamber and helps to clarify their nature and properties. In Fig. 2 a typical photograph is shown of the tracks formed by shower particles in a Wilson chamber.

Finally, the use of various absorbing screens and of a large number of counters connected into a hodoscopic system*) also makes it possible to study the behavior of individual particles in showers, their interaction with matter, and thereby makes it possible to determine the nature of the shower particles.

The flux density of shower particles, i.e. the number of particles arriving per unit area, can be estimated with the aid of ionization chambers or various coincidence systems with counters. By such methods it was established that in the central part of a shower the particle flux density is very large, but it falls off comparatively rapidly toward its periphery. We shall consider this question in more detail below. The total number of charged particles in large showers may reach \(10^8—10^9\). If the mean energy of shower particles is taken to be \(10^8\) eV, and if one takes into account that part of the shower energy is expended in the higher layers of the atmosphere, and also the large number of photons in showers that are not registered by counters, then the energy of the primary particles causing large atmospheric showers should be estimated at \(10^{16}—10^{17}\) and even \(10^{18}\) eV.

The study of the processes of formation of extensive atmospheric showers at present provides the only experimental approach to investigating the properties of particles with such gigantic energy. It is precisely this circumstance that explains the great importance which the study of extensive atmospheric showers has now acquired.

Shortly before the discovery of extensive atmospheric showers, a number of authors\(^{3,4}\) developed the foundations of the cascade, or avalanche, theory describing the process of development of an avalanche of electrons formed when a high-energy electron**) or photon falls upon a layer of matter.

*) That is, into an apparatus in which the operation of each counter causes the lighting of a corresponding neon lamp.

**) We shall, as is customary, understand by the word “electrons” both electrons and positrons.

The form of shower theory developed by L. D. Landau and J. B. Rumer^5 substantially advanced the development of this theory and made it possible to determine the basic characteristics of electron–photon showers. By observing the multiplication of the particles entering a shower in various substances (aluminum, lead) and comparing the data obtained with the conclusions of the theory, it was easy to establish that the shower particles, in every case in the overwhelming majority, are electrons and photons. Direct observation of the multiplication of shower particles in a Wilson chamber placed in a magnetic field likewise confirmed this conclusion.

On this basis, extensive atmospheric showers were interpreted by Euler^6 and others as electron–photon showers developing in the atmosphere when a primary cosmic electron of superhigh energy appears.

Such an interpretation seemed all the more natural because, in that period, the primary radiation arriving in the atmosphere was believed to consist of electrons.

This view of extensive atmospheric showers persisted until the most recent years.

However, the subsequent development of cosmic-ray physics showed it to be erroneous.

On the one hand, an intensive study of the properties of showers, carried out mainly by a group of Soviet investigators under the general direction of D. V. Skobeltsyn, beginning in 1944, on the Pamirs (altitude about 3860 m) and at sea level, showed that the properties of showers do not agree with the more rigorously developed shower theory (chiefly by the Soviet theoreticians L. D. Landau, I. E. Tamm, S. Z. Belenky, and I. Ya. Pomeranchuk).

On the other hand, experiments to study the nature of the particles in showers, likewise carried out to a considerable extent on the Pamirs, established the presence of an appreciable number of penetrating particles of non-electronic nature (see Section III).

Therefore, beginning in 1946–1947, a conception gradually began to take shape according to which extensive atmospheric showers are a flux of genetically related particles of both electron–photon and other nature, and according to which non-electromagnetic processes play a fundamental role in their development; this possibility had already been indicated in 1942 by D. V. Skobeltsyn^7 in an attempt to explain the discrepancy between Auger’s experiments (see below) and shower theory.

Some authors tried to find a way out of the situation that had arisen by revising the cascade electromagnetic theory, proposing its inapplicability in the region of high electron and photon energies. However, the further accumulation of experimental facts led to the resolution of this problem on an entirely different plane.

During 1940–1947 it was established that the primary particles of cosmic rays are not electrons, but protons (and heavier atomic nuclei, as was shown somewhat later).

The discovery of electron-nuclear showers and their detailed study, carried out in the Pamirs (in the first stages under the direction of V. I. Veksler), showed that high-energy nucleons, in collisions with atomic nuclei, effectively produce not only particles of the penetrating type, but also an electron-photon component (this also explains the name “electron-nuclear showers”\(^8\)).

The nucleonic composition of the primary component and the existence of the process by which an electron-photon component is produced by nucleons made it possible for G. T. Zatsepin\(^9\) in 1948 to put forward a new concept of extensive air showers, according to which extensive air showers should be regarded as electron-nuclear showers produced by primary nucleons of ultrahigh energy. This concept was based on the idea that, in electron-nuclear showers generated by high-energy nucleons, new nuclear-active particles are produced, which in turn are capable of generating electron-nuclear showers. This circumstance leads to the emergence of a nuclear-cascade process. According to the new concept, the nuclear-active component forms the basis of an extensive air shower, while the electron-photon component arises as a secondary product of the nuclear-cascade process, and its development can be described by cascade electromagnetic theory without resorting to a revision of the latter.

The fact, discovered in the 1948 investigations carried out in the Pamirs, of the presence of a large number of nuclear-active particles in extensive air showers confirmed this concept. From that time the second stage began in the experimental investigation and construction of the theory of extensive air showers.

Quantitatively, the new theory of extensive air showers has not yet been developed to completion. The absence of information on the elementary cross sections for the interaction of mesons and other particles at high energies has not yet allowed a complete quantitative theory to be created. However, as will be considered below, a qualitative—and in some cases also quantitative—explanation of a number of difficulties that arose in attempts to describe showers with the aid of the ordinary electromagnetic cascade theory shows the correctness of the new concept.

Recently other investigators have also begun to arrive at this same point of view\(^ {10}\).

We shall consider in turn the properties of the individual components of extensive air showers, beginning with the electron-photon component, which characterizes the general picture of the shower that is directly observed experimentally.

N. A. DOBROTIN, G. T. ZATSEPIN, I. L. ROZENTAL, et al.

II. THE ELECTRON–PHOTON COMPONENT OF EXTENSIVE ATMOSPHERIC SHOWERS

§ 1. Principal Conclusions of Shower Theory*)

First of all let us describe the picture of extensive atmospheric showers that was given by electromagnetic shower theory, which regarded them as a purely electron–photon formation.

In the electromagnetic interaction of high-energy electrons (photons) with the nuclei of atmospheric atoms, particles multiply intensively through the production by photons of electron–positron pairs and through the emission by the latter, in turn, of bremsstrahlung quanta. At the same time there occurs a sharp degradation of the mean energy of the particles. The rapid increase in the number of particles continues until the mean energy of the shower particles has fallen to the so-called critical energy \(\beta\) (for air \(\beta\) is of order \(7\cdot 10^{7}\) eV; for lead, \(6.4\cdot 10^{6}\) eV), below which the processes of ionization braking of electrons begin to predominate**). The number of shower particles, having reached a maximum at some depth, then begins gradually (at the end, approximately exponentially) to decrease. It is easy to see that, in order of magnitude, after \(n\) cascades one should obtain \(e^n\) particles. Their number at maximum can be estimated very roughly from the initial energy of the shower \(E_0\) and the critical energy \(\beta\); since the total energy of the shower is equal to \(E_0\), and the mean energy of the particles at maximum is close to \(\beta\), \(E_0=\beta e^{n_{\max}}\). Since the number of cascades on the path to a given depth is approximately proportional to the depth, it follows from this that the depth \(t\) of the layer of matter at which the maximum is obtained increases only weakly (logarithmically) with the initial energy \(E_0\):

\[ n \sim t \approx \ln \frac{E_0}{\beta}. \]

At the same time the mean energy of the particles in the region beyond the shower maximum changes only insignificantly.

In this connection it should be noted that, at those altitudes at which the main investigations of extensive atmospheric showers were carried out (from sea level to altitudes of about 4 km), at the maximum of their development, according to the theory of electron–photon showers, there can be only showers produced by electrons that had, near the boundary of the atmosphere, energies of order \(10^{16}\) eV and higher. The majority of showers, however, must reach these depths of the atmosphere after having passed far beyond the maximum.

*) For a detailed exposition of shower theory, see the monograph by S. Z. Belenkii: Avalanche Processes in Cosmic Rays.\(^{11}\)

**) At \(E=\beta\) the mean ionization losses of an electron per unit path are equal to its mean radiation losses.

In the theory of electron–photon showers it is convenient to choose as the unit of length the so-called \(t\)-unit (“shower unit”), characterizing the path over which the energy of an electron, owing to bremsstrahlung, decreases on average to \(1/e\) of its initial value. This unit is inversely proportional to the density of the substance and to the square of the nuclear charge \(Z^2\) (more precisely \(Z(Z+1)\)). For lead it is \(5.24\ \mathrm{g}/\mathrm{cm}^2\) (or about \(5\ \mathrm{mm}\)); for air, about \(34\ \mathrm{g}/\mathrm{cm}^2\) (at normal temperature and pressure—about \(260\ \mathrm{m}^{*}\)). The Pamir station is located at a depth of about \(20\) \(t\)-units below the boundary of the atmosphere; sea level corresponds to a depth of about \(30\) \(t\)-units.

A shower produced by one electron or photon upon passing through a layer of matter \(t\), on the average consists of \(N(E,t)\) electrons whose energies are greater than \(E\), where

\[ N(E,t)\approx \Phi(s)\left(\frac{E_0}{E}\right)^s \cdot e^{\lambda(s)t}, \tag{1} \]

\(E_0\) is the energy of the primary electron (it is assumed that \(t\gg 1\), \(E_0\gg \beta\), and \(E\gg \beta\), i.e., that the shower consists of a large number of particles). Here \(s\) is a certain slowly increasing function of \(t\) and \(E\); \(\Phi(s)\) and \(\lambda(s)\) are also slowly varying functions. The parameter \(s\) determines the degree of development (“age”) of the shower. It increases as \(t\) increases, and at the shower maximum \(s=1\). Before the maximum \(\lambda(s)>0\), and the shower grows with depth. At the maximum \(\lambda(s)=0\); afterward \(\lambda(s)\) is negative, with the limiting value \(\lambda(\infty)=-0.773\).

When the parameter \(s\) reaches the value \(s=2\), on average a negligible number of particles remains in the shower, and the absorption coefficient for the shower becomes equal to

\[ -\frac{d\ln N}{dt}=-\lambda(s)=0.526. \]

\[ {}^{*})\quad X_0=\frac{1}{4\pi a Z(Z+1) r_0^2 L_i}; \quad \text{here } a=\frac{e^2}{\hbar c}=\frac{1}{137},\quad r_0=\frac{e^2}{mc^2} \]
is the “classical radius” of the electron, and \(n\) is the number of atoms in \(1\ \mathrm{cm}^3\) of substance.

The quantity \(L_i\) can be calculated by using the Thomas–Fermi model for the atomic shell. However, for light elements the use of the Thomas–Fermi model seems unjustified. Therefore A. Kirpichev and I. Pomeranchuk\(^{12}\) proposed determining the quantity \(L_i\) from experiments on the scattering of X-rays. Then \(L_i\) is found to be approximately \(10\%\) larger. Adding unity to \(Z\) in one of the factors in the denominator takes into account that bremsstrahlung and pair production can occur not only on the nucleus but also on atomic electrons. Owing to these two circumstances, the true value of the \(t\)-unit is noticeably smaller than the value

\[ X_0=\frac{1}{4\pi a r_0^2 Z^2 \ln\left(191\cdot Z^{-1/3}\right)}, \]

which is used in many foreign works. Accordingly, in these works the depth of sea level is obtained as 24 units, whereas with the correct value of the \(t\)-unit it is approximately 30 units.

In Fig. 3 the so-called “cascade curves” are shown for showers produced in air by high-energy electrons.

The total number of particles at the shower maximum is equal to

\[ N_{\max}=\frac{0.3}{\sqrt{\ln \frac{E_0}{\beta}}}, \]

and the depth of the maximum is

\[ t_{\max}=1.01\ln \frac{E_0}{\beta}. \]

We shall confine ourselves here to presenting these conclusions of shower theory, and the necessary additional formulas will be given subsequently. Let us note only, moreover, that the mean energy of one particle at the maximum of a shower produced by an electron with energy \(E_0\) is determined by the expression\(^*\)

\[ \bar E \simeq \beta \sqrt{\ln \frac{E_0}{\beta}}. \]

Shower particles also fly apart in directions perpendicular to the direction of motion of the entire shower. Historically, questions connected with the so-called “width” of showers have played an important role. This is explained not only by the fact that the very existence of showers could be discovered only by recording genetically connected particles spatially separated from one another, but also by the considerable development of these questions in the theory, allowing comparison with experimental data.

The deviation of electrons from the direction of motion of their “ancestors” is determined mainly by their Rutherford scattering on the nuclei of air atoms. In comparison with this scattering one may neglect both the Compton scattering of photons and the deviation from the initial direction of the electrons upon the production of pairs and of photons in the emission of bremsstrahlung quanta by electrons. If one is interested in a small resultant deviation, then multiple scattering will be of primary importance. The greater the energies of the particles, the smaller their deviation from the initial direction. For particles of sufficiently high energy the mean

\(^*\) This expression is obtained in the following way: on the path to the maximum, the shower expends on ionization about half of the initial energy \(E_0\). The remaining energy \(E_0/2\) is divided approximately equally between the electrons, the number of which, as stated, is equal to

\[ \frac{0.3}{\sqrt{\ln \frac{E_0}{\beta}}}\cdot \frac{E_0}{\beta}, \]

and photons. Therefore the mean energy of one electron is equal to

\[ \bar E \simeq \frac{E_0}{4N_{\max}} \simeq \beta \sqrt{\ln \frac{E_0}{\beta}}. \]

the angle of multiple scattering is small, in connection with which the entire shower propagates in the direction of motion of the primary particle. Thus, for example, for particles with energy \(E\) sufficiently large in comparison with the critical energy, the mean angle of deviation at the maximum of the avalanche is determined by the expression

\[ \sqrt{\overline{\theta^2}(E)} = 0.8\,\frac{E_k}{E}, \]

where

\[ E_k \simeq 21\ \mathrm{MeV}. \]

In this connection, particles of high energy are concentrated near the axis of the shower, and it is they that determine the density of the particle flux near the axis of the shower and the penetration depth of the given shower. The probability of the total deviation through a large angle as a result of multiple scattering decreases sharply with the growth of this angle. Therefore, when considering deviations through large angles, the principal role is played by single scattering.

Since the mean scattering angle increases rapidly as the energy decreases, the spatial distribution of each given group of particles is determined mainly by the scattering precisely of these particles, and not of their “ancestors,” which possessed substantially greater energy. In other words, scattering effectively occurs over the course of the last cascade before the level of observation. Thus, the spatial distribution of particles in an electron-photon shower is determined mainly not by the “history” of the shower, but by the properties of the particles themselves located at the point of observation. Therefore the deviation of particles from the initial direction will approximately be expressed as follows:

\[ \overline{r^2}(E) \simeq \overline{\theta^2}(E)\cdot X^2, \]

where \(X\) is the length of a \(t\)-unit at the given altitude. Let us note that for particles with energy less than the critical energy, the magnitude of the effective path length must be calculated with allowance for ionization losses, as a result of which \(\overline{r^2}\) decreases.

Fig. 3. Cascade curves for air.

\[ y=\ln \frac{E_0}{\beta}. \]

Cascade theory makes it possible to calculate the root-mean-square radius

\[ R=\sqrt{\overline{r^2}}=\sqrt{\frac{\int r^3\rho(r)\,dr}{\int \rho(r)\,r\,dr}} \]

(\(\rho(r)\) is the flux density of shower particles at a distance \(r\) from the shower axis) of the distribution of electrons of a given energy (and also, in total, of electrons of all energies) for that region of the shower where the spatial distribution is due to multiple scattering*). The value of the root-mean-square radius depends only weakly on the total energy of the shower and increases only very slowly as the shower develops**).

For the maximum of the shower, cascade theory leads to a value of the root-mean-square radius \(R_0\) equal to \(70\ \mathrm{m}^{11}\) at an atmospheric pressure of \(760\ \mathrm{mm}\) Hg.

The dependence of the root-mean-square radius of a shower on the degree of its development, characterized by the parameter \(s\), has been theoretically investigated insufficiently. Under certain special assumptions one can obtain

\[ \sqrt{\overline{r^2}}=R_0\sqrt{\frac{s(s+1)}{2}}. \]

The increase of the shower radius expressed here as the shower develops is a consequence of the softening, with depth in the atmosphere, of the energy spectrum of the shower, as a result of which an ever smaller fraction of the particles travels in its core. Since all the results of the theory are expressed in dimensionless quantities, with the unit of length chosen as the \(t\)-unit, whose length is inversely proportional to the density of the substance, it follows that, if one neglects the weak dependence of \(R\) on the degree of shower development mentioned above, the latter must contract toward the axis in proportion to the geometrical length of the \(t\)-unit (or, what is the same, inversely proportional to the density that the air has at approximately the same depth at which the observation is made).

The distribution of particles in the shower is characterized in more detail by the function of the spatial distribution of particles \(\rho(r)\). Its form has been analyzed by a number of investigators. As I. Ya. Pomeranchuk showed\(^{13}\), near the shower axis, i.e. at distances from the axis \(r \ll R\), the particle flux density \(\rho(r)\) decreases with distance as

\[ \frac{1}{r^{2-s}}, \]

where \(s\) is the above-mentioned parameter of shower theory characterizing the degree of development of the shower. Such a dependence is due to the fact that for \(r \ll R\) the shower density at a distance \(r\) from the axis is determined

*) Sometimes, to characterize the spatial distribution of particles, the concept of a “half” radius is introduced, i.e. the radius of a circle within which half of all particles are contained.

**) Here the difference in atmospheric pressure at different altitudes is not taken into account, and the conclusion about the change in radius refers to an atmosphere whose pressure is assumed identical at all altitudes.

mainly by electrons with energy of order \(E\sim \dfrac{1}{r}\). Consequently, as the shower penetrates into the atmosphere the decrease in density with distance from the axis becomes slower, the “trunk” of the shower gradually spreads out, and the high-energy electrons in it expend their energy, transferring it to particles of low energy, which move away from the axis and are absorbed there. For distances \(r\) close to \(R\) and \(r>R\), unfortunately, even at the present time there is no exact analytical expression for the function \(\rho(r)\). It may be assumed that at such large distances from the shower axis, where the spatial distribution of particles is determined by single scattering, but for which nevertheless \(\sqrt{\overline{\theta^2}}\ll 1\), the density \(\rho(r)\) falls off as \(\dfrac{1}{r^4}\), in accordance with Rutherford’s formula for the angular distribution of scattered particles. (Owing to the finite dimensions of the nucleus, an effect may appear because of which, at high energies, the fall of \(\rho(r)\) becomes still sharper.)

For the special case of showers at the maximum of development, Molière\({}^{14}\) calculated the spatial distribution of particles with allowance for both multiple and single scattering. The calculation was performed by numerical integration and gives a graphical representation of the function \(\rho(r)\) up to \(r\sim 6R_0\). However, even Molière’s function cannot be regarded as correct. In particular, the energy spectrum of particles in the shower used in deriving it (Arley\({}^{15}\)) gives an evidently overestimated number of electrons with energy above the critical energy. The correct form of the electron energy spectrum for the maximum of shower development was calculated by I. E. Tamm and S. Z. Belen’kii\({}^{16}\). The values of the critical energy and of the magnitude of the \(t\)-unit adopted in work\({}^{14}\) are also obsolete.

Thus, up to the present time sufficiently rigorous calculations of the function of the spatial distribution have not been carried out. This applies not only to the general case, characterized by an arbitrary parameter \(s\), but even to a shower at the maximum of its development \((s=1)\). As was already said above, it is known only that near the axis

\[ \rho(r)\sim \frac{1}{r^{2-s}}. \]

§ 2. Spatial structure of extensive atmospheric showers and comparison of theory with experiment

In most cases the results of experiments can be compared with the conclusions of the theory only after an appropriate recalculation, taking into account the degree of selectivity of the installation with respect to the given phenomenon, as well as the imperfection of the apparatus, side effects

and so on. Thus, for example, one of the widespread methods for studying showers is the registration of the simultaneous passage of shower particles through several counters placed at some distance from one another. The number of such coincidences can be calculated, in particular, on the assumption of the electron-photon nature of the shower, if one specifies the energy spectrum of the primary particles at the boundary of the atmosphere. Within the framework of shower theory, the dependence of this number on the area of the counters, on the distance between them, on the form of the energy spectrum of the primary particles and, finally, on the altitude of the place of observation has been calculated in a number of works. These questions were first treated in a sufficiently detailed and mathematically rigorous manner by A. B. Migdal^17.

We shall point here to one essential conclusion of that work, which can be quite simply understood qualitatively. It is evident that the triggering of a given system of counters connected in coincidence is favored by an increase in the particle flux density in the shower above the apparatus, i.e., by an increase in the total energy of the shower \(E_0\). On the other hand, the number of primary particles, and therefore of showers, decreases rapidly with energy. Therefore, other conditions being equal, at a given altitude counters of area \(\sigma\) mainly register showers caused by primary particles with energy lying in a certain interval—namely, as the theory shows, with such an energy for which the mean particle flux density at the given point of the shower is of the order of one particle per counter \((\rho\sigma \sim 1)\).

Let us give an example showing what error may result from neglecting this circumstance. Suppose that in an experiment the number of showers is registered by means of a system of counters. Then all the counters are covered with a very thick layer of lead, so that the shielded counters register only penetrating particles. One might think that the decrease in the number of coincidences would be a measure of the fraction of penetrating particles in the shower. However, this is not so: under the new conditions, showers will be registered with a density of penetrating particles equal to one particle per counter. It is evident that these will be quite different showers from those without lead.

Thus, a system of counters, possessing different sensitivity to different showers depending on the energy of the primary particles (and also on the distance of the apparatus from the point where the shower axis strikes, the depth at which the shower is initiated in the atmosphere, the structure, etc.), introduces a sharp discrimination into the phenomenon itself.

Let us note, in addition, that when experimental data are compared with theory, assumptions are usually made not only about the form of the energy spectrum of the primary particles, but also about their angular distribution, and in some cases also about the form of the function of the spatial distribution of particles \(\rho(\mathbf{r})\), the details of which, as indicated above, theory cannot always provide. The assignment of the func-

WIDE ATMOSPHERIC SHOWERS

of the function \(\rho(r)\), in turn, reduces to a more or less reasonable “stitching together” of solutions valid for individual parts of the shower. Thus, very often the comparison of theory with experiment contains considerable elements of arbitrariness. Taking this circumstance into account is very essential for conclusions about the validity of one or another theory of wide atmospheric showers.

Let us now consider the experimental data on the spatial distribution of shower particles.

Auger carried out his experiments at an altitude of \(3457\ \mathrm{m}\) above sea level and reached a maximum distance \((D)\) between counters of \(\sim 300\ \mathrm{m}\). The method of recording showers used by Auger (separation of two counters) did not make it possible to increase the distance further, since even at \(D = 300\ \mathrm{m}\) random coincidences already constituted half of the entire measured effect.

Møller\(^{14}\) made a theoretical calculation of the dependence of the number of double coincidences \((C_2)\) on the distance \(D\) between counters under the following assumptions: 1) the integral energy spectrum of the primary electrons is expressed by the power-law dependence \(F(>E_0)=\)

\[ =\frac{\mathrm{const}}{E_0^\gamma}, \]

where \(\gamma = 1.8\); 2) all showers originate at the boundary of the atmosphere and fall vertically; 3) all showers have the same spatial distribution, corresponding to the dependence determined by this author for the maximum of shower development (see above). It turned out that the theoretical dependence \(C_2(D)\) agrees with the experimental one in the range of distances from 2 to \(200\ \mathrm{m}\). On this basis Møller stated that there was good agreement between theory and experiment.

Outside the indicated interval, the experimental number of coincidences exceeds the calculated value. Møller attributed the discrepancy to the inapplicability of the calculation method for large distances, to the neglect of inclined showers, and also to the approximate nature of the assumptions made\(*\).

D. V. Skobeltsyn\(^{7}\), who independently calculated the dependence \(C_2(D)\)\(**\), arrived at a different conclusion. He showed that although the general character of this curve (taking for the altitude at which Auger made his measurements \(R = 100\ \mathrm{m}\)) does correspond to the theoretical dependence,

\(*\) We note that Møller’s attempt to explain the discrepancy at very small distances \((D < 2\ \mathrm{m})\) by the influence of low-density showers located far beyond the maximum of development is unconvincing, since, according to the theory, the distribution of particles in such showers is always diffuse \((s > 1)\), which could have caused a discrepancy only of the opposite sign.

\(**\) In this calculation a definite form of the function of the spatial distribution of particles was adopted, namely

\[ \rho(r)=\rho_0 e^{-\frac{\sqrt{2}\,r}{R}}\cdot \frac{1}{r}. \]

However, a systematic deviation of the experimental curve from the theoretical one is nevertheless observed. Thus, at small distances between counters (up to distances of 60–70 m) the number of coincidences decreases with distance more sharply than is predicted by the calculation. Of greatest interest was the discrepancy at large distances (\(D = 300\) m), where the experimental number of coincidences exceeded the calculated value by more than a factor of two. In this work D. V. Skobeltsyn suggested the presence in showers of another mechanism, connected with penetrating particles and leading to a stretching of the curve \(C_2(D)\) at large distances.

From this time there begins an intensive study of the properties of extensive atmospheric showers by Soviet experimentalists, leading to the discovery of the nuclear-cascade process.

To confirm the indicated anomaly in the “width” of showers, D. V. Skobeltsyn proposed a method of registration different from Auger’s. Auger recorded a shower with the aid of only two counters, i.e. he selected cases of the simultaneous passage of two particles at a specified distance from one another. D. V. Skobeltsyn proposed selecting, at each of the sites separated from one another, cases of the passage of a shower, i.e. of more than one particle. In practice this means the simultaneous triggering, for example, of four counters, brought close together pairwise, at different distances \(D\) between the pairs. Such a method made it possible to reduce random coincidences to a minimum and to reach distances considerably exceeding 300 m. In the summer of 1946 on the Pamir the corresponding measurements were carried out \(^{18}\). In Fig. 4 a skeleton diagram of this installation is given. Four groups of counters of large area (\(\sigma = 1840\ \text{cm}^2\) in each group) were separated pairwise to distances of up to 1 km between them.

As is seen from Table I, even at the maximum distance between the counters a noticeable number of coincidences \(C_4\) is observed.

Table I

\(D\) (m)	2	12	30	100	285	585	980
\(C_4\) per hour	\(469 \pm 14\)	\(366 \pm 9\)	\(274 \pm 8\)	\(60.0 \pm 3.5\)	\(4.5 \pm 0.5\)	\(0.60 \pm 0.15\)	\(0.15 \pm 0.09\)

Random coincidences made only an insignificant contribution. Attempts to ascribe the observed effect to the influence of inclined showers \(^{19}\) proved untenable. D. V. Skobeltsyn showed \(^{20}\) that, even taking into account the angular distribution of showers, the calculation, carried out—

... based on a successive application of avalanche theory, does not give any satisfactory agreement with experiment. A discrepancy of tens of times remains, which is not eliminated either by changing (within admissible limits, of course) the form of the energy spectrum of the primary particles, or by changing the magnitude \(R\). Thus, for example, at a distance of \(1\ \mathrm{km}\) this discrepancy between experimental and theoretical data can be eliminated only by changing the exponent \(\gamma\) in the distribution of primary particles over energies from the value \(\gamma=1.8\) (as was assumed in the calculation) to \(\gamma=1.3\), or even less. This value of \(\gamma\) sharply contradicts the value \(1.7\text{--}2.0\), accepted by all authors on the basis of a number of other experiments concerning the determination of the form of the spectrum in the corresponding energy interval. Moreover, special measurements carried out in the Pamirs for showers recorded by counters separated by \(1\ \mathrm{km}\) showed that \(\gamma=2.1\pm0.1\) (see § 3).

Fig. 4. Diagram of the setup for recording broad atmospheric showers: \(S, S_1, S_2\)—amplifiers recording coincidences of pulses from counters 1, 2, 3, 4.

The detection of showers at such large distances between counters is greatly hindered by the smallness of the observed effect. It is enough to say that, for \(D=1000\ \mathrm{m}\), in order to record showers appearing at an altitude of \(3860\ \mathrm{m}\) on average once every 3 hours, it is necessary to use counters with an area of \(0.8\ \mathrm{m}^2\) each (sixfold coincidence). Nevertheless, in recent years, in experiments in the Pamirs at such distances, a considerable number (several hundred) of showers have already been recorded,\(^{21}\) and in this way the anomalous width of the showers has been established with sufficient reliability.

It should be noted, however, that the calculations mentioned above according to avalanche theory\(^{20}\) are very complicated and may depend substantially on possible inaccuracies in the theory underlying the calculations. In particular, the results of the calculation are sensitive to the magnitude \(R\), which determines the form of the function \(\rho(r)\). In this connection, another proof of the anomalous width of the showers is of considerable interest, one that is free, in particular, from the arbitrariness connected with the choice of the magnitude \(R\). As shown by D. V. Skobeltsyn,\(^{22}\) one can establish a simple relation between the dependence \(C_4(D)\) for \(D\gg R\) and the density distribution \(\rho(r)\) in a shower at large distances \(r\) from its axis.

If it is assumed that in the region of the given (large) distances from the axis the density \(\rho(r)\) varies as \(\dfrac{1}{r^n}\), and the function \(C(D)\), at large distances between counters, can be approximated by a power-law dependence \(C(D) \sim \dfrac{1}{D^k}\), then \(n\) and \(k\) are related by the relation \(k = n\gamma - 2\), where \(\gamma\) is the exponent in the expression for the energy spectrum of the primary particles. This relation is obtained with the aid of only one natural assumption: that near the maximum of showers the total number of particles is proportional to the energy of the primary particle. Experiments carried out in the Pamirs\(^{18}\) give a value of \(k\) equal to 2.7. Measurements made by L. H. Eyges et al.\(^{23}\) at sea level lead to approximately the same value of this quantity. If \(\gamma = 1.8\) is taken, then the value \(n \simeq 2.6\), calculated from the formula \(k = n\gamma - 2\), is in sharp contradiction with the spatial distribution of particles at large distances from the axis expected from theory (\(n \geq 4\)); for \(n = 4\) one should have \(k = 5.2\). Thus the existence of an “anomalous width” of showers is confirmed.

In Section V it will be shown that the existence of the “width anomaly” finds its explanation in the scheme of the cascade-nuclear process. However, the “anomalous width” of showers is not caused directly by penetrating particles.

As will be shown below, penetrating particles in a shower constitute, at moderate altitudes, only about \(1/10\) of the total number of particles. Although their fraction increases toward the edges of the shower (see Section III), even at the periphery, at distances of \(100\)—\(400\) m from the axis, strongly absorbed particles predominate. The origin of these particles has not yet been sufficiently studied; however, their appearance at large distances from the shower axis cannot in any case be explained by the trivial processes of formation of \(\delta\)-electrons or electrons from the decay of \(\mu\)-mesons.

At the same time, in recent years it has been found that the spatial distribution of particles in the central regions of showers (\(r < 70\)—\(100\) m) likewise cannot be explained within the framework of the usual electron-photon scheme of showers.

Direct experiments\(^{24}\) to determine the form of the function \(\rho(r)\) in the central regions of showers, carried out with a number of hodoscopes (\(H = 3260\) m above sea level), indicate that at such altitudes the function \(\rho(r)\), in the interval \(2\ \text{m} < r < 200\ \text{m}\), has a somewhat steeper falloff than follows from shower theory for these altitudes. However, the discrepancy with the theory at these altitudes is still very small. This, in particular, explains the existence of the opinion, often expressed to this day by various

...by the authors*), concerning the fact that the main properties of showers, and above all their spatial distribution, are on the whole well described by the usual scheme, according to which the primary electron of ultra-high energy creates a shower near the boundary, which subsequently develops in accordance with the avalanche theory.

In reality, this agreement is observed only with respect to those properties of showers which are not very sensitive to the mechanism of their development, and therefore it still does not make it possible to distinguish electron–photon avalanches from showers whose structure is determined mainly by processes of nuclear interaction.

An investigation of the change in shower properties with altitude in the atmosphere (see § 3) helps to detect the difference between these two mechanisms of shower development.

In particular, from the change in the form of the dependence \(C_4(D)\) with altitude (the corresponding measurements were made in the Pamirs \((3860\ \mathrm{m})^{18,21}\) and at sea level\({}^{23}\)) it follows that the spatial distribution of particles in the central regions of showers changes only weakly with a change in the altitude of the observation site. At the same time, for electron–photon showers it should change substantially, as a result of which the discrepancy between the experimental data and the cascade theory increases on approaching sea level and reaches a large value (Fig. 5).

Fig. 5. Dependence of the number of fourfold coincidences (see Fig. 3) on the distance between pairs of counters (at sea level): I — experimental dependence; II — theoretical dependence (electromagnetic cascade theory).

Thus, if experiments on spreading the counters apart, carried out at an altitude of \(3.5—4\ \mathrm{km}\) above sea level, revealed a discrepancy between the avalanche theory and experiment only at very large (\(\gtrsim 500\ \mathrm{m}\)) distances \(D\) between the counters, while at distances \(D < 100\ \mathrm{m}\) the discrepancy between experiment and theory was small, then with increasing depth in the atmosphere the situation is different, and the discrepancy between this theory and experiment becomes considerable even near the axis.

*) Including by the authors of work \(^{24}\).

§ 3. The density spectrum and altitude dependence of extensive atmospheric showers

If the theory of the spatial distribution of particles in an electron-photon cascade shower has so far been developed rather weakly, and disagreement between experimental data and theory can be reliably established only in the case of a very sharp discrepancy, then the question of the density spectrum of showers and their altitude dependence is a well-developed problem, as a result of which it is possible to carry out a careful comparison of theory with experiment. The analysis of the altitude dependence and of the density spectrum of showers is facilitated, as will be shown below, by the fact that for the theoretical calculation a detailed knowledge of the spatial distribution function is not required.

Over an element of the shower path \(dt\), the \(N(t)\) electrons present in it expend on ionization an energy \(\beta N\,dt\). Therefore the area under the cascade curve, multiplied by the magnitude of the ionization losses in air per unit length, i.e. the total energy released by the shower for ionization, is equal to the total initial energy of the shower:

\[ \beta \int_{0}^{\infty} N(t)\,dt = E_{0}. \tag{1} \]

The cascade character of the development of the shower leads to the fact that the curve of the number of particles \(N(t)\) has a sharp maximum, so that the area under the curve is determined to a considerable extent by the region of the maximum of \(N(t)\). From the cascade character of the development of the shower it also follows that the width of the cascade curve near the maximum depends only weakly on the energy of the primary particle. Hence it follows that the number of particles at the maximum of the shower is approximately proportional to the energy of the primary particle. (In regions far from the maximum, the number of particles depends on the energy of the primary particle in a more complicated way, determined by the character of the cascade process.)

In an electron-photon shower the number of particles \(N\) at depth \(t\) can be approximated over a sufficiently wide interval by a power function \(\sim E_{0}^{s}\), where the cascade parameter \(s\) is a function of \(E_{0}\) and \(t\). For cascade showers of any nature, in particular for real extensive atmospheric showers, the number of particles in the shower can also, over some interval, be approximated by a power function of the energy \(N \sim E_{0}^{sN}\), and therefore the subsequent arguments, based on this relation, have a sufficiently general character.

Let us compute the spectrum of showers with respect to the number of particles, assuming that

\[ N = A E_{0}^{s}. \tag{2} \]

Hence, a shower with number of particles \(N\) corresponds to an energy of the primary particle

\[ E_0=\left(\frac{N}{A}\right)^{\frac{1}{s}}. \]

If the energy spectrum of primary particles is approximated by a power function

\[ F(>E)=B\cdot E_0^{-\gamma}, \tag{3} \]

then we obtain that the number of showers with number of particles greater than \(N\), whose axes fall on a unit area, is equal to:

\[ \Phi(>N)=B\left(\frac{N}{A}\right)^{-\frac{\gamma}{s}}=C\cdot N^{-\frac{\gamma}{s}}, \tag{4} \]

i.e. the shower spectrum by number of particles can in this case be approximated by a power law:

\[ \Phi(>N)=C\cdot N^{-\varkappa}, \quad \text{where } \varkappa=\frac{\gamma}{s}. \]

Thus, having obtained experimentally, at any depth, the spectrum (by number of particles) of showers whose axes pass through a given area, in the case of applicability of the electron-photon scheme we can obtain the energy spectrum of the primary electrons.

If the mechanism of shower development is different, then the number of particles in a shower may vary with the energy of the primary particle according to another law, and the exponent in \(E_0\) in the relation \(N\sim E_0^s\) may differ from \(s\), calculated from the cascade electron-photon theory. Only in the region of the maximum of showers, as shown above, is this exponent practically independent of the type of cascade multiplication and, with great accuracy, equal to unity: \(s_{\max}=1\). Consequently, in order to obtain the energy spectrum of primary particles, it is best to take experimental data obtained at a sufficient altitude, where the showers under investigation are not far beyond the maximum of their development.

Up to the present time the spectrum of showers by number of particles has not been obtained because of the difficulty of the experimental method. However, if it is assumed that the form of the spatial-distribution function depends only insignificantly on the number of particles in the shower, then it can be shown that the exponent of the spectrum of shower densities differs little from the exponent of the distribution of showers by number of particles.

Let the number of axes (passing through a unit area) of showers having a number of particles in the interval \(N, N+dN\) be equal to

\[ \varphi(N)dN=A\cdot N^{-(\varkappa+1)}dN, \tag{5} \]

if the spatial-distribution function does not depend on the number of particles, then the shower density at a distance \(r\) from the shower axis can be written in the form:

\[ \rho(r)=N\cdot u(r). \tag{6} \]

Let us now compute the number of showers passing through a given place with density in the interval \(\rho,\rho+d\rho\), for an arbitrary position of the shower axis.

Showers whose axes pass at a distance \(r\) from the place under consideration with density \(\rho\) have the number of particles

\[ N=\frac{\rho}{u(r)}, \qquad dN=\frac{d\rho}{u(r)} . \]

The number of showers whose axes fall on a unit area, which have a density lying in the interval \(\rho,\rho+d\rho\) at a distance \(r\) from the axis, will therefore be

\[ \begin{aligned} \Phi(\rho,r)\,d\rho &=\varphi\left(\frac{\rho}{u(r)}\right)\frac{d\rho}{u(r)} \\ &=A\left(\frac{\rho}{u(r)}\right)^{-(x+1)}\frac{d\rho}{u(r)} =Au(r)^x\rho^{-(x+1)}\,d\rho . \end{aligned} \tag{7} \]

The total number of showers for an arbitrary position of the axis is equal to

\[ C(\rho)\,d\rho=A\rho^{-(x+1)}d\rho\cdot 2\pi \int_{0}^{\infty} u^x(r)\,r\,dr =\operatorname{const}\rho^{-(x+1)}\,d\rho . \tag{8} \]

If \(x\) and \(U(r)\) are constant or depend only very weakly on \(N\), and consequently on \(\rho\), so that the value of the integral does not change with \(\rho\), then it follows directly from this that the distribution of showers by density repeats the distribution of showers by number of particles. Knowledge of the actual function of the spatial distribution \(U(r)\) is not required.

In the derivation it was obtained that, if the function of the spatial distribution of particles in a shower is \(U(r)\), then the number of axes falling on a unit area decreases with distance from the place of registration as \(U^x(r)\), i.e., much faster than the particle density in the shower (since \(x \geqslant 1.4\)). Calculations carried out by one of the authors (G. Z.) show that more than half of the showers registered by installations measuring the local shower density have axes passing at distances \(\leq 1/3\) of the “radius” of the shower. The question of the density spectrum of showers was considered in detail in the work of A. B. Migdal\({}^{17}\) for showers that are electron-photon avalanches.

Analysis shows that in the lower half of the atmosphere, for electron-photon avalanches, the condition of coincidence of the density spectrum with the shower spectrum by number of particles is fulfilled, and therefore one may assume that

\[ x=\frac{\gamma}{s} \tag{9} \]

to an accuracy of a few percent.

Before presenting the experimental data and their comparison with theory, let us consider the method of multiple coincidences of pulses from counters, which is the most widespread method for determining the density spectrum.

Although all particles in a shower are in one way or another genetically connected with one another, nevertheless, as a result of multiple scattering of trajec-

of particle trajectories prove to be distributed in space practically independently. The validity of the Poisson law for the probability of particles passing through a given area was checked in a number of experiments \(^{76—78}\), which confirmed that, when showers are recorded by open counters, practical independence of the places where particles strike is observed. The following criterion for the independence of trajectories may be proposed. The greatest correlation should be observed between the electrons of pairs produced in the last \(t\)-unit above the apparatus. Independence of passages through the counters will occur if the spatial separation of the electrons of one pair is much greater than the mean distance between particles in the shower. It turns out that this is satisfied in almost all showers, with the exception of the central regions of low-density showers, which under ordinary conditions are practically not recorded.

If the particle trajectories are distributed in space statistically independently, then, when a shower of density \(\rho\) passes through, the probability that at least one particle will strike a counter of area \(\sigma\) is equal to \(1-e^{-\rho\sigma}\). If, however, \(n\) counters are included in the coincidence system, then the probability of its firing will be equal to \((1-e^{-\rho\sigma})^n\). Assuming that the distribution of showers by densities has a power-law form \((\sim \rho^{-\chi})\), we obtain that the total number of coincidences per unit time is equal to

\[ C_n(\sigma)=B\int_0^\infty (1-e^{-\rho\sigma})^n \rho^{-(\chi+1)}\,d\rho, \tag{10} \]

or, after the substitution \(\rho\sigma=x\),

\[ C_n(\sigma)=B\sigma^\chi \int_0^\infty (1-e^{-x})^n \frac{dx}{x^{\chi+1}}=A\sigma^\chi\cdot Y(n,\chi), \tag{11} \]

where \(Y(n,\chi)\) is an easily calculable function.

The value of the exponent \(\chi\) is most often obtained by measuring the values of \(C_n\) at two values of \(\sigma\) for a given \(n\)

\[ \chi=\frac{\ln\dfrac{C_n(\sigma_1)}{C_n(\sigma_2)}}{\ln\dfrac{\sigma_1}{\sigma_2}} \tag{12} \]

(the method of area variation). The value of \(\chi\) can also be obtained from the results of measurements of the number of coincidences at constant \(\sigma\), but variable \(n\) (the method of changing the multiplicity). However, when the method of changing the multiplicity of coincidences is used, the spatial structure of the shower affects the results, as a consequence of which this method gives less reliable results.

In recent years, thanks to the development of hodoscopic techniques for recording showers, the spectrum has been determined by some authors

densities by an improved coincidence method, making it possible to study the density spectrum for an arbitrary dependence \(x(\rho)\).

Fig. 6. Density spectrum of broad atmospheric showers at an altitude of 3860 m.

For this purpose, several groups of counters of different area \(\sigma\) are simultaneously connected to the hodoscope. If, when a shower passes, the number of counters that have fired (coincidences) is, as is the number that have not fired (anticoincidences), sufficiently large, then a sufficiently narrow region is selected from the density spectrum, whereas in the simplest coincidence method this region is considerable and rapid changes of \(x\) with \(\rho\) could not be observed. However, both these methods give the same result, showing a very slow change of \(x(\rho)\): when \(\rho\) is increased by a factor of 1000, \(x\) increases by approximately 25% (Fig. 6).

Table II gives the results of measurements of the density spectrum carried out by the indicated methods. The table does not include data from a number of other authors (for example,\(^{25}\)) obtained in 1947–1949, which are not free from errors of a methodological nature, nor data from older works that were not subsequently confirmed.

Table II

Author	Height above sea level, m	Density interval	Value of \(-x\)
Zatsepin\(^{26}\) (1947) . . .	sea level	20–40	\(1.43 \pm 0.13\)
Zatsepin\(^{26}\) (1947) . . .	3860	5–300	\(1.42 \pm 0.02\)
Zatsepin\(^{26}\) (1947) . . .	4800	5–300	\(1.52 \pm 0.04\)
Cocconi\(^{27}\) (1949) . . .	260	2–300	1.36–1.45
Cocconi\(^{27}\) (1949) . . .	3260	2–1000	1.33–1.55
Eidus\(^{28}\) (1949) . . .	3860	1500–10 000	1.65–1.76
Brodein\(^{29}\) (1950) . . .	sea level	5–500	\(1.425 \pm 0.022\)

The method of varying the area of the counters can also be applied in the case when the counters are arranged at arbitrary distances from one another

from one another, i.e., when the shower density is different at the locations of each of the counters.

Indeed, let us denote the shower density at the location of the \(i\)-th counter by \(\rho_i\), and its distance from the shower axis by \(r_i\). Then \(\rho_i=N\cdot f(r_i)\), where \(N\) is the total number of particles in the shower. Taking into account that the number of showers with a number of particles in the interval \(N, N+dN\), whose axes pass through a unit area, is equal to \(A\cdot N^{-(x+1)}\), we obtain the total number of showers registered by the system

\[ C_n(\sigma)=\iint\limits_{N\,S} A\cdot N^{-(x+1)}\,dN\cdot \prod_{i=1}^{i=n}\left(1-e^{-N\cdot f(r_i)\sigma}\right)\,dS, \tag{13} \]

where \(\int\limits_S\) denotes integration over the plane.

Substituting \(N\sigma=x\), we obtain

\[ C_n(\sigma)=A\sigma^x\iint\limits_{x\,S} x^{-(x+1)}\,dx\cdot \prod\left(1-e^{-x f(r_i)}\right)\,dS, \tag{14} \]

or

\[ C_n=A\cdot \sigma^x\cdot I_S(i), \tag{15} \]

where \(I_S(i)\) is a function depending on the mutual arrangement of the counters.

Consequently, the structure of the shower does not affect the results of determining \(x\) obtained by the method of area variation.

Since, as the distance \((D)\) between counters increases, the number of particles in the selected showers also increases, the method of area variation of counters separated from one another is very effective for obtaining information about the density spectrum of showers in the region of very high energies, where the statistics obtained with the aid of a locally arranged system of counters are extremely small.

This method was applied in the works of 1946–1951 in the Pamirs at distances \(D\) between counters of 100 and 1000 m. It turned out that for \(D=100\ \text{m}\), \(x=1.8\pm0.15\); for \(D=1000\ \text{m}\), \(x=2.1\pm0.1\).

At sea level the data were obtained in 1948–1949\({}^{30}\) at several distances \(D\). With increasing distance between the counters a systematic increase of \(x\) was observed, which to some extent corresponds to the growth of \(x\) with increasing shower density. In addition, a certain role in the increase of \(x(D)\) may be played by the dependence of the function of the spatial distribution of particles on the shower energy.

Measurement of the value of \(x\) at large distances between counters showed quite unambiguously that showers registered at \(D=100\div1000\ \text{m}\) have a density spectrum with exponent \(x=1.8\div2.0\) and, consequently, a slow rate of decrease

the number of showers registered with distance is due not to the small value of \(\varkappa\), but to the slow decrease of the spatial distribution function.

From the data presented it is seen that the exponent \(\varkappa\) at an altitude of \(\sim 3\)–\(4\) thousand meters above sea level slowly increases with shower density.

Calculations carried out by one of the authors (G. Z.) in accordance with electromagnetic cascade theory showed that the value of \(\varkappa\) observed over the entire density range presented leads to a purely power-law form of the energy spectrum of the primary electrons \(E_0^{-\gamma}\), with \(\gamma = 1.80 \pm 0.05\). The increase of \(\varkappa\) with shower density is explained in this case by the decrease of the parameter \(s\).

Using the given value of \(\gamma\), one can calculate the spectrum of shower densities for any altitude above sea level; for this it is only necessary to determine the corresponding parameter \(s\). With increasing depth in the atmosphere, the parameter \(s\) corresponding to the given shower density increases, and, consequently, the exponent \(\varkappa\) must decrease. In particular, for sea level we obtain that, for showers of low density \(\rho \sim 3\) particles/\(\text{m}^2\), according to the calculations \(s = 1.4\) and \(\varkappa = 1.24\), while for showers of 10 times greater density (\(\rho\) about 30 particles/\(\text{m}^2\)) \(s = 1.38\) and \(\varkappa = 1.30\). However, comparison of these conclusions with the table shows that there is a sharp discrepancy between the results of theory and experiment: the values of \(\varkappa\) obtained experimentally, contrary to theory, do not decrease toward sea level. It should be noted that this contradiction cannot be removed by changing the magnitude of the \(t\)-unit or by changing the form of the energy spectrum of the primary electrons.

The contradiction between experiment and theory is revealed still more clearly when comparing the altitude dependence of showers of different densities.

The altitude dependence of showers with a number of particles above a given value is determined entirely by the absorption of particles in an individual shower and by the exponent of the shower spectrum with respect to the number of particles \(\varkappa\). Indeed, if the absorption coefficient of particles in a shower is equal to \(\mu_N\), then

\[ \frac{d \ln N}{dt} = -\mu_N, \tag{16} \]

and the shower spectrum with respect to the number of particles is determined by the function

\[ \Phi(> N) = C \cdot N^{-\varkappa}; \tag{17} \]

then the absorption coefficient of showers with a number of particles above the given value is written in the form

\[ \mu_{\text{shower}} = \frac{d \ln \Phi(>N)}{dt} = -\varkappa \mu_N . \]

As indicated above, cascade theory predicts a weak dependence of the spatial distribution function on the depth in the atmosphere *).

Detailed calculations show that the correction to the value of \(\chi\), arising from the change of the spatial distribution function with atmospheric depth, is very small. We have also calculated the altitude variation of electron–photon showers from an altitude of \(3860\ \text{m}\) down to sea level, with an approximate allowance for the change of the spatial distribution function with depth in the atmosphere—which proves insignificant—and for the angular distribution of showers.

It turns out that, according to cascade theory, the altitude variation of showers with densities of \(10\ \text{particles}/\text{m}^2\) and \(300\ \text{particles}/\text{m}^2\) in this interval of altitudes should differ by a factor of 3, whereas according to experimental data the altitude variation of these showers is practically the same. It differs, in agreement with the theory, for showers of high density (the difference is approximately a factor of 11).

Analogous results, obtained by the method of numerical calculation of the number of coincidences caused by showers, were published in work \({}^{31}\). According to these data, the number of showers with density \(\rho \geq 3\ \text{particles}/\text{m}^2\) at the two indicated altitudes differs only by a factor of 13, whereas in the case of electron showers one should have expected a change with altitude of more than a factor of 50.

It should be noted that this discrepancy between theory and experiment also cannot be corrected by changing the value of the \(t\)-unit (to bring the altitude variation of low-density showers into agreement, it would be necessary to increase the \(t\)-unit by a factor of 1.3, but then we would obtain a discrepancy in the altitude variation for dense showers of almost a factor of 3).

Thus one may state that one of the basic characteristics of a shower—the change of the density spectrum with altitude—cannot be described within the framework of the electron–photon scheme. The practical equality of the absorption coefficients \((\mu_{\text{sh}})\) for showers of different density indicates that the dependence of the absorption coefficients \((\mu_N)\) of the number of particles in showers on the energy of the generating particle, for real showers, is considerably weaker than for electron–photon showers. Therefore, owing to the increase of the index \(\chi\) with shower density, the product
\[ \chi(\rho)\,\mu_N(\rho)=\mu_{\text{sh}} \]
remains constant.

It follows from the electron–photon scheme of a shower that the absorption coefficient of particles in cascades depends very substantially on the number of particles in the cascade (i.e., on the energy of the primary particle producing

*) If the spatial distribution function does not depend on the depth in the atmosphere, then the absorption coefficient of showers with density above a given value coincides with the absorption coefficient of showers with a number of particles above the given value.

showers), whereas in actually observed showers this dependence is considerably weaker.

Budini’s work^32 was also devoted to calculations of the altitude variation and to the analysis of experimental data. In this work, too, a discrepancy is noted between theoretical data and experiment. However, Budini adopted a value of the \(t\)-unit in air 20% larger than follows (but according to work^12), as a result of which the anomalies in the altitude variation, in accordance with what was said above, are shifted into another range of densities; agreement between the experimental and theoretical data is obtained for showers of low density and disagreement for showers of high density.

Thus, the experimental study of the density spectrum of broad atmospheric showers, in connection with a detailed development of the theory of these experiments, has led to conclusions on the existence of serious discrepancies between the observed properties of showers and those properties which are predicted by the electron-photon scheme.

III. PENETRATING PARTICLES OF BROAD ATMOSPHERIC SHOWERS

§ 1. Discovery of the penetrating component of broad atmospheric showers

As is known, initially the entire flux of cosmic rays was phenomenologically divided into two components, assigning to one of them particles penetrating through a layer of lead of thickness on the order of 10 cm (the hard component), and to the other—those absorbed in it (the soft component).

Previously it was usually assumed that the particles of the soft component are electrons (positrons or photons), and the particles of the hard component are mesons and protons. At present, however, such a division appears correct only approximately. Indeed, in a number of works by Soviet authors^33,34, which will be considered in detail below, it was established that electron-photon showers of high energy are capable of penetrating through 12 and even through 16 cm of Pb. Therefore, in what follows, by penetrating particles we shall mean mesons and nucleons, irrespective of their energy, and by particles of the soft component, electrons and photons.

Cosmic-ray particles interacting with atomic nuclei with an effective cross section approaching the “geometrical cross section” of nuclei we shall call “nuclear-active” particles. Thus, nuclear-active particles include protons, neutrons, and \(\pi\)-mesons. Nuclear-active particles form part of the penetrating component. The particles making up the other part of the penetrating component—\(\mu\)-mesons—are known to interact weakly with nuclei and are “nuclear-passive” particles.

Already the first experiments by Auger\(^2\) on determining the penetrating power of particles of atmospheric showers, carried out with apparatus consisting of 2 and 3 counters connected in a coincidence circuit, showed that, when one of the counters is shielded with lead, the number of coincidences drops sharply as the thickness of the lead is increased from 0 to 10 cm; with a further increase in the thickness, however, the number of coincidences changes little. Owing, however, to the undeveloped state of the theory of shower registration by counter systems, the authors obtained, for different experimental configurations, data that were poorly consistent with one another. Subsequently many experiments were set up with the aim of proving the existence of penetrating particles. In particular, Dode\({}^{42}\) studied the composition of showers with the aid of a Wilson chamber with a lead plate 1.5 cm thick. He showed that in showers there exists a certain number of particles passing through the lead plate without multiplication or scattering.

Some data supporting the presence of penetrating particles in atmospheric showers were obtained by Rogozinski\({}^{33}\), and also by Cocconi and collaborators\({}^{36}\). However, the inconsistency of the data obtained in these works, and the incorrectness of the quantitative estimates, led to the result that the penetrating particles were regarded as a secondary and second-order phenomenon accompanying electron–photon showers.

Fig. 7. Layout of counters for studying the penetrating power of particles of extensive atmospheric showers. Counters \(A\), \(B\), \(C\) are shielded with lead. The unshielded counter is used to study the relation between penetrating and electron atmospheric showers.

A detailed investigation of the penetrating power of particles of extensive atmospheric showers was carried out in 1946–1947 in the Pamirs with the aid of the installation shown in Fig. 7. The use of triple coincidences made it possible to eliminate completely accidental coincidences, and careful shielding of the counters—other side effects.

In the first experiments the dependence of the number of coincidences on the thickness \(d\) of lead above the counters was measured, and it was established that the number of coincidences, with an increase in the shield thickness to 20 cm, rapidly decreases, reaching

\[ \frac{1}{200} \]

of the number of coincidences observed at \(d = 0\).

To determine the nature of the particles of extensive atmospheric showers that produce coincidences under considerable thicknesses of lead (over 12 cm), the following experiment was performed. Above the lead shielding each counter, a layer of aluminum 10.5 cm thick was placed. Such a layer, in ionization absorption, is equivalent

4 cm Pb. If, however, particles that have passed through a large thickness of lead are absorbed as a result of radiation losses, then lead and aluminum must be compared in \(t\)-units, and this layer of aluminum is equivalent to only 0.6 cm Pb. The experiment showed that the absorption of the particles by aluminum, to within the statistical errors of the experiment, coincides with the absorption in a layer of lead 0.6 cm thick, and, consequently, that the particles passing through 12 cm Pb are electrons. These experiments were continued in work \(^{34}\), in which the thickness of the lead layer already exceeded 16 cm. The dependence obtained in this work of the number of triple coincidences on the thickness of lead above the counters is shown in Fig. 8. Thus, the available data make it possible to regard as proven that, in an apparatus with several shielded counters, the particles of broad showers capable of passing through 16 cm of lead, but absorbed when the thickness of lead is increased to 20 cm, are in fact, for the overwhelming part, electrons. Such a large penetrating power of the electron component seemed to contradict the cascade theory accepted at that time, in which the decrease in photon absorption at low energies, considerably increasing the penetrating power of shower photons, was not taken into account in the calculations. Thus, from the corresponding formulas it followed that an electron shower passing through 12 cm Pb should be produced by a particle with an energy greater than \(2 \cdot 10^{10}\) eV; however, the assumption of the presence of so large a number of such particles led to a contradiction with other data on the spectrum of particles in the shower and seemed unlikely.

Fig. 8. Dependence of triple coincidences on the thickness of lead above the counters.

Meanwhile, as we have already mentioned, the effective cross section for the absorption of photons in lead decreases when their energy is lowered to energies of the order of the critical energy.

Indeed, the absorption of photons in matter is determined by two processes: the Compton effect and pair production (the ordinary photoelectric effect, which occurs only at very low energies, and the nuclear photoelectric effect, whose probability is comparatively small, we shall not consider). As is known, the effective cross section for

the Compton effect falls with increasing photon energy, while the effective cross section for pair production, beginning with the energy \(1 M_{\text{eV}}\), increases and then reaches a constant value. Theory and experiment show that the total effective cross section for the Compton effect and pair production, which determines the probability of converting photons into electrons, for light elements remains practically constant and does not depend on the photon energy. On the contrary, for heavy elements in the region of energies close to the critical one, the total effective cross section proves to be considerably smaller than for photons of high or low energies. This circumstance, as S. N. Vernov noted, leads to a significant increase in the penetrating power of photons in heavy elements. Calculations made by S. N. Vernov and S. Z. Belenky\(^{11,38}\) made it possible to explain the absorption curve of the electron-photon shower in lead.

Taking account of the above-mentioned features of shower theory for heavy elements proves very essential when considering the penetrating power of particles of extensive atmospheric showers. Neglect of this circumstance, which was often allowed, especially in the first years of the study of extensive atmospheric showers, may lead to serious errors in the interpretation of experimental data. Calculations\(^{37}\), carried out on the basis of the work of S. Z. Belenky\(^{11}\) (Fig. 9) and taking into account the dependence of the photon cross section on their energy, showed that this effect leads to a considerable increase in the penetrating power of the electron component in lead. The energy of an electron (photon) required so that under \(12\) cm of Pb, on the average, one electron remains, is reduced by approximately a factor of 5 in comparison with what was obtained from the approximate form of shower theory. Owing to the small absorption coefficient of shower particles in lead and the relative softness of the spectrum of the electron-photon component in showers, electrons and photons incident on lead with relatively low energy \(10^8—10^9\) eV make a large contribution to the number of coincidences. It should be noted that the underestimation of the penetrating power of the electron-photon shower in lead once led Auger to the construction of an erroneous

Fig. 9. Cascade curves for lead.

hypotheses concerning the existence, in broad showers, of new light particles—\(\lambda\)-mesons\(^{39,40}\)* with a mass several times greater than the electron mass. Underestimation of this circumstance also led Cocconi et al. in 1948 to the erroneous conclusion that the energy spectrum of the mesons comprising atmospheric showers is extraordinarily soft.

Despite the very great penetrating power of the electron-photon shower, from the work carried out in the Pamirs\(^{33,34}\), as well as by certain foreign authors\(^{42,43}\), it nevertheless follows with complete certainty that penetrating particles are indeed present in broad showers and carry a significant fraction of the energy of the entire shower. Indeed, already in the interval \(20\)—\(24\ \mathrm{cm}\ \mathrm{Pb}\) the absorption coefficient of particles is not consistent with the assumption that all particles of broad atmospheric showers that have passed through \(18\)—\(20\ \mathrm{cm}\ \mathrm{Pb}\) are electrons. Direct proof that these particles are particles of the penetrating type was obtained in experiments showing that, when the thickness of lead is increased beyond \(24\ \mathrm{cm}\), the number of coincidences practically does not change\(^{34}\).

Some authors considered it very probable that, alongside broad electron showers, there also exist broad, low-density penetrating showers of a special type\(^{35}\). However, the Pamir studies\(^{34}\), as well as the work of Cocconi et al.\(^{36}\), showed that this point of view is untenable. Indeed, it was established that penetrating particles are, as a rule, accompanied in the air by electron-photon showers of considerable density. Thus a genetic connection was established between broad penetrating and electron-photon showers, and it was shown that penetrating broad showers are not a special type of shower, but one of the components of ordinary atmospheric showers.

§ 2. Origin of the Penetrating Component of Broad Atmospheric Showers

Penetrating particles were discovered in broad showers at a time when the prevailing conviction was that these showers were purely electron-photon cascade formations, and that electromagnetic shower theory described them well. In this connection, the discovery of penetrating particles in showers led to the supposition that the penetrating particles do not enter into the composition of the shower as a necessary element, but arise in the apparatus itself, in the absorber surrounding the counters of the recording setup (under the action, for example, of photons of high energy).

* For criticism of the hypothesis of \(\lambda\)-mesons, see also the article by Cocconi and Greisen\(^{41}\).

Such an assumption is, of course, very inconsistent (since it leaves unclear why the photon forms penetrating particles only in dense material, but not in air). However, it was discussed and tested over a number of years by many authors44, 45, 46.

At the same time it was additionally assumed that photons form penetrating particles with an effective cross section that otherwise depends on the atomic number of the absorber than does the effective cross section for pair production (for example, \(\sigma_{\mathrm{eff}} \sim Z\), and not \(\sigma \sim Z^2\)). If this is so, then replacing the upper layer of absorber—usually Pb, for example—by a layer of Al equivalent to lead in \(t\)-units should lead to a change in the number of registered penetrating particles. However, the authors mentioned above did not succeed in observing such a change. Therefore it proved impossible to decide, on the basis of experiments with replacing the filter substance, whether the penetrating shower particles are formed in the air or in the filter above the apparatus.

The solution of this problem was given in one of the Pamir works in 194747, which used a completely different idea. The authors compared the spatial distribution (i.e. the distribution in the horizontal plane) of penetrating particles and electrons entering into the composition of extensive atmospheric showers. As a result it was found that electrons, and consequently also photons, have a narrower spatial distribution than penetrating particles.

Electrons and high-energy photons, which should have made the principal contribution to the supposed formation of mesons in the filters above the apparatus, are concentrated to an even greater degree near the axis. This proves that penetrating particles arise mainly not under the action of photons (or electrons) in the apparatus, but are formed (at least for the most part) in the air. Subsequently (in 1949) this conclusion was confirmed by an analogous method in the work of Cocconi24 et al.

Thus, in an extensive atmospheric shower, alongside electrons and photons there are also penetrating particles (including, as we shall see below, nuclear-active ones).

If, on the basis of absorption experiments, one assumes that the mean energy of a penetrating particle is equal to \(\simeq 2 \cdot 10^9\) eV, i.e. exceeds by a factor of 20 the mean energy of the electrons (according to cascade theory \(\sim 10^8\) eV), then the detection of even a small fraction of penetrating particles in showers shows that their role in the energy balance of showers is very substantial. This conclusion confirms the incorrectness of the conception according to which extensive atmospheric showers are electron-photon showers formed in accordance with electromagnetic cascade theory.

§ 3. Spectrum of the flux densities of penetrating particles

By studying the distribution of showers according to the number of penetrating particles in them, we can draw conclusions about the properties of the penetrating particles and obtain indications of the nature of the processes in which they are formed. Therefore, experiments on the study of the spectrum of densities of penetrating shower particles are of substantial interest.

The difference between the method used for this purpose in the Pamirs^48 and the corresponding experiments for electrons consists only in covering the counters with lead screens 22 to 32 cm thick. In the method of variation of areas, the dependence of the number of triple coincidences on the counter area \(C_3(\sigma)\) is represented as a curve (Fig. 10), which is approximated by the function \(C_3(\sigma)\sim \sigma^{x_n}\), with the value \(x_n\) equal to \(1.47\pm0.07\).

Fig. 10. Dependence of the number of triple coincidences of discharges in screened counters \(C_3(\sigma)\) on their area.

To determine the quantity \(x_n\), one may also use the method of different multiplicities. Measurement of the ratio of the number of quadruple coincidences to triple coincidences (with counter area \(0.4\ \text{m}^2\)) gave the value

\[ \frac{C_4}{C_3}=0.57\pm0.07, \]

whence we obtain \(x_n=1.60\pm0.15\), which agrees with the value obtained by the method of variation of areas.

Although in these experiments a comparatively small interval of densities of penetrating particles was studied,

\[ \left(2-15\ \frac{\text{particles}}{\text{m}^2}\right), \]

nevertheless they show that the value of the exponent in the density spectrum for penetrating particles coincides, within the experimental errors, with the corresponding value for the density spectrum of shower electrons. From this one may conclude that the flux densities of penetrating particles are approximately proportional to the electron flux density. The same relation over a somewhat larger interval of densities was studied, by another method, by Cocconi and collaborators^65. In this work the authors found a weak decrease of the fraction of penetrating particles with density (as \(\rho^{-0.13}\)). However, the discrepancy between the results of the two works lies, in general, within the limits of statistical errors.

§ 4. Fraction of penetrating particles in showers and its altitude dependence

In determining the percentage content of penetrating particles, two experimental approaches are possible: 1) determination of the percentage content in showers which give, in the apparatus, a specified electron density. In this case showers of different energies are taken into account, whose axes pass at correspondingly different distances from the apparatus; 2) determination of the percentage content in a shower having a definite energy, at different distances from the axis.

The second aspect of the problem posed appears to be the more important, since only it makes it possible to obtain the characteristics necessary for constructing a scheme of the development of an individual shower. (For example, the total number of penetrating particles in a shower of definite energy.) However, at the present time this problem has still not been solved completely.

The first formulation of the question from the experimental side is considerably simpler.

Proceeding from the fact that the differential spectrum of densities both for electrons and for penetrating particles can be represented (see § 3) by a power function with the same exponent

\[ N_p\,d\rho_p \sim \rho_p^{-(\chi_p+1)}\,d\rho_p \]

for penetrating particles, and

\[ N_e\,d\rho_e \sim \rho_e^{-(\chi_e+1)}\,d\rho_e \]

for electrons), one measures the ratio of the number, for example, of triple coincidences of discharges of \(n\) unshielded counters \(C_n^{(e)}\) to the number of coincidences of discharges of the same counters placed under large thicknesses of lead \(C_n^{(\mathrm{p})}\). It is, evidently, equal to

\[ \frac{C_n^{(e)}}{C_n^{(\mathrm{p})}} = \frac{ \displaystyle \int_0^\infty \rho_e^{-(\chi+1)} \left(1-e^{-\rho_e\sigma}\right)^n \,d\rho_e }{ \displaystyle \int_0^\infty \rho_{\mathrm{pr}}^{-(\chi+1)} \left(1-e^{-\rho_{\mathrm{pr}}\sigma}\right)^n \,d\rho_{\mathrm{pr}} }, \tag{18} \]

where \(n\) is the number of counters.

We are interested in the ratio \(\dfrac{\rho_e}{\rho_{\mathrm{pr}}}=k\), which approximately does not depend on the density (see § 3). The integral is readily calculated. Substituting \(\rho_e=k\rho_{\mathrm{pr}}\) in (3) and making a change of variable of integration in the upper integral, we obtain in this case:

\[ \frac{C_n^{(e)}}{C_n^{(\mathrm{pr})}}=k^\chi. \tag{19} \]

From the data obtained in work \(^{34}\), we find that at an altitude of about \(4\ \text{km}\), \(K=70\). From analogous measurements carried out by Cocconi et al. \(^{65}\), it may be concluded that at an altitude of about \(3.2\ \text{km}\), \(K=80\text{--}90\). Consequently, in a broad shower the flux density of penetrating particles at altitudes of \(3\text{--}4\ \text{km}\) is \(1\text{--}1.5\%\) of the density of the electron fluxes.

As follows from what was set forth in Section II, in the case of measurements of this type the effective distance from the shower axis is about \(1/3\) of the cascade radius. Therefore, measurements carried out by the method of multiple coincidences give the fraction of penetrating particles in the central region of the shower (see Section II, § 3).

Let us turn to the second aspect of the problem posed. In addition to the qualitative conclusion already noted earlier, that penetrating particles are distributed more broadly than electrons, some quantitative estimates were made at an altitude of \(3260\ \text{m}\) (Cocconi, Tongiorgi, and Greisen \(^{34}\)) and at sea level (Eidus et al. \(^{50}\)).

In the first work it was found that, when the distance from the shower axis is varied from \(5\) to \(100\ \text{m}\), the fraction of penetrating particles doubles.

Fig. 11. Dependence of the fraction of penetrating particles on the distance to the shower axis. Along the ordinate axis is plotted the value of the percentage of penetrating particles out of the total number of particles at a given distance \((r)\) from the shower axis.

A considerably more complete investigation was carried out in the second work \(^{50}\), in which the total percentage of penetrating particles in broad showers was studied. This quantity at sea level was determined in the following way.

At several points located on the earth’s surface and separated by large (\(\sim 200\ \text{m}\)) distances from one another, the flux density of all shower particles and, separately, of the penetrating particles was determined. Then a certain function of the spatial distribution of particles in showers was assumed (see §§ 1, 2), and by means of calculation the point of passage of the shower axis was determined. Knowing the distance of the shower axis from the detectors of penetrating particles, it was possible to determine the percentage of penetrating particles at different distances from the shower axis. With the aid of such a method the authors were also able to calculate the total number of particles in each of the showers investigated. Figure 11 shows the dependence of the percentage of penetrating particles on the distance to the shower axis, obtained at

On the basis of an analysis of 57 showers recorded by the authors. As is seen from Fig. 11, the percentage of penetrating particles increases strongly with distance from the axis, which agrees well with the results of work \(^{61}\). If the contribution of penetrating particles to the total number of particles in showers is estimated, it turns out that the penetrating particles constitute about \(1/10\) of the total number of particles.

Using these data, one may also approximately estimate the contribution that penetrating particles at sea level make to the total energy of a shower. If one assumes that the mean energy of electrons in the depth of the atmosphere does not exceed the “critical” energy, while the mean energy of penetrating particles is \(\sim 2\cdot 10^9\) eV, then we find that the penetrating component contains more than half of the entire energy of the shower.

In the same work the authors, dividing all the showers recorded by them into two groups (with the number of particles \(<10^7\) and \(>10^7\)), came to the conclusion that penetrating particles make a somewhat smaller contribution, in terms of number of particles, to showers with greater energy (\(N>10^7\)) than to smaller showers (\(N<10^7\)).

Thus, the following conclusions may be drawn regarding the contribution of penetrating particles to the total number of charged particles:

1) Within the central region of showers, the fraction of penetrating particles at an altitude of 3–4 km is 1–1.5%; this fraction increases with decreasing altitude, reaching 2.5–3% at sea level.

2) The total fraction of penetrating particles in a shower (averaged over all distances from the axis) is greater than the indicated value because of the contribution of the peripheral regions; moreover, at sea level the total number of penetrating particles in a shower reaches approximately 10%.

3) At sea level, at the periphery of a shower, the fraction of penetrating particles reaches 60%.

IV. THE NUCLEAR-ACTIVE COMPONENT OF EXTENSIVE ATMOSPHERIC SHOWERS

§ 1. Discovery of the nuclear-active component

Indications of the existence of nuclear processes occurring under the influence of particles of extensive atmospheric showers were obtained as early as 1939 in the work of Auger and collaborators \(^{2}\), using a Wilson chamber placed under 12 cm of lead. In 32 photographs associated with the passage of extensive atmospheric showers, 7 tracks of particles with high ionization density were found, probably belonging to protons.

In 1942, in the work of Auger and Dode \(^{64}\), it was shown that, when extensive atmospheric showers pass through a block of lead, there arise

penetrating particles are produced; however, it was not determined whether this production occurs under the influence of the electron–photon component or under the influence of particles of the penetrating type.

In 1946–1947, the question of the production of penetrating particles in lead was the subject of work by Janossy and collaborators[^66]. In these works it was also shown that, when broad atmospheric showers pass through a block of lead, penetrating particles are produced. However, on the basis of these investigations the authors arrived at the erroneous conclusion that the electron–photon component, and not the component of the nucleon type, is responsible for these processes.

Until very recently (1947–1948), the overwhelming majority of investigators believed that the penetrating particles of broad atmospheric showers are ordinary \(\mu\)-mesons.

Fig. 12. Diagram of the apparatus for comparing the shower-producing ability of “single” and shower penetrating particles.

In order to clarify the nature of the interaction of penetrating particles of broad atmospheric showers with matter, in 1947 in the Pamirs a small hodoscope was used to compare the shower-producing ability of particles of the “single” penetrating component of cosmic rays and penetrating particles entering into broad atmospheric showers. The results of this comparison indicated that penetrating particles of broad atmospheric showers create showers in lead from many particles more often than “single” \(\mu\)-mesons. However, the low statistical accuracy of the observations did not permit final conclusions to be drawn. A more detailed investigation was undertaken in the following year[^53] in connection with the appearance of the hypothesis of the nuclear-cascade character of the development of broad showers.

The scheme of the apparatus intended for the investigation of the nuclear-active component is shown in Fig. 12. The discharges of 72 hodoscope counters, arranged in a lead block, registered showers only in the case when discharges occurred in all counters of the control group. In the study of shower-produc-

...the particle of a single component, the system was controlled by the discharges of three of these counters, forming a vertical “telescope.”

To study the shower-producing ability of penetrating particles included in extensive showers, the circuit was controlled by triple coincidences between pulses from the counters of group \(A\) and unshielded counters located at a distance of \(1\)—\(2\) m from the lead block. The results of the experiments are summarized in Table III.

Table III

Number of particles recorded by the group of counters under the lead	1	2	3	4	5	6	7	8	9	10	11	12
Percentage of such events when single penetrating particles pass . . . . .	95.5	2.5	0.5	0.3	0.1	1.1	0	0	0	0	0	0
Percentage of such events when penetrating particles included in an atmospheric shower pass . . . . .	57.4	18.0	6.2	5.8	3.1	2.3	1.3	1.5	2.3	0.3	0.5	1.6*

In the second row of Table III is presented the distribution of the setup’s operations according to the number of particles under the lead when “single penetrating particles” entered the lead; in the third, the distribution when penetrating particles included in atmospheric showers (“shower” particles) passed through.

The results of this work clearly showed that the penetrating particles of extensive atmospheric showers cannot be \(\mu\)-mesons, since they possess a shower-producing ability far exceeding the shower-producing ability of \(\mu\)-mesons, but characteristic of “nuclear-active” particles.

From this work it followed that from 25 to 50% of the penetrating particles of extensive atmospheric showers recorded by the setup are nuclear-active particles, generating electron-nuclear showers when interacting with matter.

Thus, strong evidence was obtained for the validity of the nuclear-cascade scheme of extensive atmospheric showers.

Later, Cocconi, Greisen, and others came to analogous conclusions in their works.^24 It should be noted that the first work of Cocconi and Greisen,^49 undertaken in this direction, gave a negative result, contradicting the data obtained at the Pamirs. However, Cocconi and Greisen themselves subsequently acknowledged it as erroneous.

§ 2. Spectrum of Flux Densities of Nuclear-Active Particles

The investigation of the spectrum of flux densities of nuclear-active particles is essential for studying the dependence of the fraction of nuclear-active particles on shower energy and is therefore necessary for refining our ideas about the mechanism of its development.

The study of the spectrum of flux densities of nuclear-active particles was carried out by a somewhat indirect method1. Namely, what was studied directly was only the incidence of nuclear-active particles on a certain apparatus (which may be regarded as a detector of such particles) at various flux densities of electrons in the broad showers accompanying this particle, i.e. at various areas \(\sigma\) registering these showers of three unshielded counters (Fig. 13). (The measurements were made in the Pamirs.)

Fig. 13. Schematic of the apparatus for investigating the properties of the nuclear-active component of broad atmospheric showers. Counters \(M\) are actuated by particles of electron-nuclear showers formed in the filter. Block \(B\) serves to determine the density of showers, and the hodoscopic group of counters serves to determine the density of atmospheric showers.

For the interpretation of the dependence obtained (it is presented in Fig. 14), let us assume: 1) that the particle density of the nuclear-active component is in fact proportional to the flux density of charged particles:

\[ \rho_{\mathrm{n}} = k_1 \rho_{\mathrm{e}} \quad [k_1=\mathrm{const}]; \]

2) that the probability \(w\) of recording a nuclear-active particle incident on the upper filter with effective area \(S\) does not depend on the density of the air shower \(\rho_{\mathrm{e}}\).

In that case the probability \(P\) of actuation of the counters \(M\) when a shower of density \(\rho_{\mathrm{e}}\) passes through is equal to:

\[ P = w \cdot k_1 S \cdot \rho_{\mathrm{e}} \tag{20} \]

and, consequently, the number of coincidences of discharges in the counters \(M\) and in three unshielded counters of area \(\sigma\), \(C_4(\sigma, w \cdot k_1 \cdot S)\), may be

write in the form

\[ C_4(\sigma, w k_1 s)=A\int_0^\infty \rho^{-(\chi+1)}(1-e^{-\rho\sigma})^3 w k_1 s \rho\, d\rho =B\sigma^{\chi-1}, \tag{21} \]

where \(A\rho^{-(\chi+1)}\,d\rho\) is the number of showers per hour with density between \(\rho\) and \(\rho+d\rho\);

\[ B=\mathrm{const}. \]

Since \(\chi \simeq 1.4\), the dependence \(C_4(\sigma)\), calculated under the assumptions made, is represented by a function \(B\sigma^{0.4}\) (solid curve, Fig. 14).

Fig. 14. Dependence of fourfold coincidences on the density of fluxes of extensive atmospheric showers. Showers in one of four systems of counters (group \(M\) in Fig. 13) are caused by nuclear-active particles. On the abscissa is plotted the effective area \(\sigma\) of the unshielded counters; on the ordinate—the number of fourfold coincidences per unit time \(C_4(\sigma)\).

As can be seen from the figure, the three values of \(C_4(\sigma)\) determined experimentally lie well on the curve. The discrepancy for the smallest area (i.e., for the densest showers) may be attributed to the simultaneous entry into the apparatus of several nuclear-active particles; in this case the probability of triggering the counters is not proportional to \(S\), and it cannot be written in the form (20).

An approximate estimate of the influence of this effect, made in the Pamir work of 1948–1949 (see \(^{54}\)), gives the dashed curve, which removes the discrepancy. Thus, the assumption

that the flux densities of all charged and nuclear-active particles are proportional to one another is in agreement with experiment. Consequently, the distribution of showers according to the flux densities of nuclear-active particles can be approximated by the same power function, with exponent \(x \simeq 1.4\), as is the case for all components of the shower.

This conclusion agrees with that made by Tonkiyorzhi\(^{55}\), who used, as a detector of electron-nuclear showers, a system of neutron counters immersed in paraffin (see § 5 of this section).

§ 3. Percentage content of nuclear-active particles in extensive showers

The determination of the content of nuclear-active particles in a shower is based on measuring their fraction in the penetrating component. Since we have arrived at the conclusion that the densities of all three components (electronic, penetrating inactive, and nuclear-active) are approximately in a constant ratio (at least in the central regions of the shower), this makes it possible to estimate the fraction of nuclear-active particles in extensive showers.

However, all measurements carried out so far have been made within the framework of the first of the approaches indicated in § 1 of Chapter III, and therefore the numbers obtained are very crude. In particular, the difference in the spatial distributions of the various components is not yet taken into account here.

The first estimate of the quantity of interest to us was made in work\(^{53}\). The measurements were carried out in the Pamirs (3860 m). With the aid of the setup shown earlier (see Fig. 12), the ratio was determined of the number of passages of penetrating particles accompanying showers from the detector itself to the number of passages of penetrating particles through the detector without multiplication (in both cases, of course, accompanied by extensive showers, i.e., by discharges in unshielded counters). If one assumes that all penetrating single particles are inactive, then this ratio, which turned out to be approximately 0.15, gives the fraction of nuclear-active particles in the penetrating component of extensive showers. In reality, this fraction is larger, since by no means every nuclear-active particle entering the setup can be distinguished from a single penetrating particle. In fact, on the one hand, this particle may produce a shower that is not dense enough and that triggers only one counter; on the other hand, the particle has a significant probability of passing through the lead without interacting at all. Taking these factors into account leads to the conclusion that the fraction of nuclear-active particles may be taken to be approximately \(1/3\) of the number of particles of the entire penetrating component of extensive showers.

Greisen et al.56 found, using an analogous method, that at an altitude of 4260 m this fraction is \(0.60 \pm 0.15\), while Sitte57 for an altitude of 3260 m gives the figure \(0.26 \pm 0.03\).* Thus, approximately, one may assume that in the penetrating component included in extensive showers at altitudes of the order of 3–4 km, from one quarter to one half of the particles are nuclear-active. MacCusker58 obtained at sea level, for this ratio, a value \(\simeq 0.35\). It should be noted that the fraction of the nuclear-active component relative to the number of electrons, as recorded by a system of neutron counters (Tongiorgi55, see § 5), changes hardly at all when the altitude is decreased from 4300 to 260 m, and, consequently, the number of nuclear-active particles decreases with altitude approximately in the same way as the number of electrons.

§ 4. Spatial distribution of nuclear-active particles in extensive showers

The first indications of the character of the spatial distribution of nuclear-active particles in an extensive shower were obtained (by means of different techniques) in works53,59.

In the first of these works (the apparatus shown in Fig. 12 was used), the position relative to the shower axis was determined from the energy of the electrons accompanying a given penetrating particle. For this purpose, next to the detector of nuclear-active particles, the absorption of the flux of accompanying electrons in 6 cm of lead was determined. If the number of electrons was not reduced by this layer, this testified in favor of the electron energy being very large and, hence, the measurement taking place near the shower axis. It turned out that single penetrating particles in the detector of nuclear-active particles are accompanied for the most part by electrons absorbed in 6 cm of lead, whereas nuclear-active particles producing showers in the detector are accompanied by electrons passing through lead of such thickness. The result obtained could be interpreted as an indication that nuclear-active particles have a “narrower” spatial distribution than \(\mu\)-mesons. Further, after passing through 6 cm of lead, the shower consisted on average of one particle. It follows from this that the average energy of the electrons accompanying nuclear-active particles is of the order of \(3 \cdot 10^8\) eV. According to cascade theory11, the mean square radius for them at an altitude of 4 km will be equal to 32 m. Consequently, this quantity may be taken

* Naturally, in order to estimate the fraction of the nuclear-active component in the whole shower, it is necessary to take into account the difference in the spatial distribution of nuclear-active particles and \(\mu\)-mesons. As will be shown in the following paragraph, the spatial distribution of nuclear-active particles is somewhat “narrower” than that of \(\mu\)-mesons; therefore the figures given characterize, apparently, the upper limit of the fraction of the nuclear-active component in the whole shower.

as a very rough estimate of the width of the region over which the nuclear-active particles are distributed.

In another paper\(^{59}\) the passage of penetrating particles through lead plates in a controlled Wilson chamber was studied. The control was effected by a system of counters connected in a coincidence circuit and placed under a thick (9 cm) layer of lead. Such a system approximately fixes the point of incidence of very fast electrons, i.e. the shower axis (shower selector). Thus, the principle of fixing distances from the shower axis was essentially the same as in the first paper. Nuclear-active particles were recorded by the secondary effects they produced—by electron-nuclear showers and “stars,” i.e. nuclear disintegrations of small energies.

Fig. 15. Spatial distribution of penetrating and nuclear-active particles obtained with a Wilson chamber. Along the axis of abscissas is plotted the distance between the shower selector and the Wilson chamber, and along the axis of ordinates—the number of passages of penetrating particles (curve A) and cases of nuclear interaction in the chamber (curve B).

In Fig. 15 the dependence of the number of such effects on the distance between the shower selector and the Wilson chamber is plotted (in the interval 5–25 m). For comparison, the same figure also gives an analogous curve for penetrating particles obtained by the same method. Within the limits of statistical errors both curves are parallel to one another. As is known, at small distances from the axis the spatial distributions of the electron and penetrating components coincide\(^{59}\). Therefore one may say that the “total” spatial distribution of particles of the nuclear-active component, including both particles of high energies (forming electron-nuclear showers) and particles of small energies (forming “stars”), coincides at small distances from the shower axis with the spatial distribution of the electron and penetrating components.

The spatial distribution of nuclear-active particles of very high energy (according to approximate estimates, \(\sim 10^{11}\) eV) was studied in paper\(^{54}\) with the aid of two detectors of nuclear-active particles; each of them recorded the passage of an electron-nuclear shower of considerable density. The simultaneous formation of electron-nuclear showers in both detectors was considered. As a result of observations (carried out, however, with low statistical accuracy) it turned out that when the distance between the installations was changed from 2 to 30 m the number of such coincidences changed by \(8^{+6}_{-4}\) times.

This is a certain indication that the flux density of nuclear-active particles decreases with distance from the shower axis more rapidly than the flux density of electrons. Indeed, the number of double coincidences of pulses in unshielded counters, for the same change in distance, changes only by a factor of \(1.9 \pm 0.2\) \(^{60}\).

Thus, one may suppose that nuclear-active particles of comparatively high energy \((\simeq 10^{10}—10^{11}\ \mathrm{eV})\) are distributed in space more concentratedly than electrons, whereas all nuclear-active particles (at any rate within the central region of the shower) have a distribution approximately coinciding with the distribution of electrons.

The results of the experiments of Cocconi and Cocconi-Tongiorgi \(^{61}\) also testify in favor of this conclusion, although the small statistical accuracy does not permit a definitive conclusion to be drawn in this case either.

The apparatus of these authors consisted of a selector of cores, detectors of nuclear-active particles with energy \(>10^9—10^{10}\ \mathrm{eV}\), and a recorder of the density of electron fluxes located in the immediate vicinity of the detector*). By varying the distance from the core selector to the nuclear-active-particle detector in the range 10—95 meters, the authors compared the spatial distribution of electrons and nuclear-active particles.

The results showed that the nuclear-active particles recorded by this apparatus are situated closer to the core than the electrons, and that for showers of lower density (and, consequently, energy) this effect is less pronounced. Thus these experiments confirm the conclusion made above.

§ 5. Composition of the Nuclear-Active Component of Extensive Air Showers

We have already said that nucleons, \(\pi\)-mesons and (if they exist) other nuclear-active mesons should be classed as nuclear-active particles. However, at the present stage experiment does not allow one to identify nuclear-active particles of different nature. Therefore, confining ourselves in this section to only a general consideration, we shall discuss:

1) the existence in extensive showers of nuclear-active particles of low energy (including neutrons); 2) the ratio between the number of charged and neutral particles in the nuclear-active

*) The nuclear-active-particle detector consisted of 16 neutron counters placed in a paraffin block in which there was a lead plate that served as the site of generation of electron-nuclear showers. The counters thus recorded neutrons moving as part of an electron-nuclear shower passing through the detector (or formed in it).

component of showers (the presence of neutral particles as part of the nuclear-active component, not included in showers, was discovered in works \(^{62,63}\)). An indication of the existence in atmospheric showers of nuclear-active particles of low energies was obtained in the experiments of Tonzhiorgi \(^{55}\), who used a system of neutron counters immersed in a paraffin block. The passage of broad atmospheric showers was recorded by three Geiger counters. The registered neutrons were either produced in the material surrounding the neutron counters, or were created in the air. A comparison of the data obtained by Tonzhiorgi shows that, when a lead filter is present inside the paraffin block, neutrons are formed mainly in this filter, whereas in the absence of the lead filter neutrons reach the apparatus chiefly from the air. Thus, slow neutrons are present in atmospheric showers. According to Tonzhiorgi’s estimates, at low altitudes they amount to about \(1\%\) of the total number of charged particles. Let us note that an apparatus of this type is also a good detector of nuclear-active particles of high energies. Indeed, according to the author’s estimates, in lead, in each event, on average about 60 slow neutrons are produced; such a number of neutrons can arise only in a nuclear interaction of high-energy particles.

The ratio between the number of charged and neutral nuclear-active particles was studied by Zitte \(^{57}\) and by Greisen et al. \(^{56}\) at mountain altitudes (\(3\)—\(4\) km) by the hodoscope method. In both works the nuclear-active particles were recorded by the electron-nuclear showers produced by them in lead. Zitte determined the total number of nuclear-active particles (both charged and neutral, \(N_N\)), and also separately the number of charged ones (\(N_p\)), relative to the number of penetrating non-active particles (\(N_\mu\)), and obtained

\[ \frac{N_N}{N_\mu}=0.26\pm 0.03 \]

and

\[ \frac{N_p}{N_\mu}=0.105\pm 0.022. \]

From a comparison of these ratios it follows that neutral particles constitute about \(60\%\) of the total number of nuclear-active particles*).

*) It should, however, be noted that in the apparatus used by Zitte the conditions for recording the total number of nuclear-active particles (neutral and charged) were, because of the “geometry” of the apparatus, better than for recording only charged ones. This circumstance could have led to the percentage of neutral particles actually being somewhat smaller than that indicated by Zitte.

In the work of Greisen et al. \(^{56}\) it was found that neutral particles constitute about \(40\%\) of all nuclear-active particles*). As for the nature of the neutral particles, one may suppose that they are, at least to a considerable extent, neutrons.

Thus, it may be regarded as established that penetrating particles are present in extensive showers; their fraction in the central regions amounts to several percent and increases with distance from the shower core. From one third to one half of the penetrating particles lying within the central region of the shower are nuclear-active, and the particles with the highest energy (by a very rough estimate, of the order of \(10^{11}\) ev) are concentrated near the shower axis, while the bulk of all nuclear-active particles lies within several tens of meters of this axis.

These facts, established in recent years, form yet another group of evidence for the inadequacy of the electron-photon concept of extensive showers.

V. EXTENSIVE ATMOSPHERIC SHOWERS AS A NUCLEAR-CASCADE PROCESS

As follows from the experimental data set forth in the preceding chapters, the totality of the results obtained at the present time shows the inadequacy of the electromagnetic electron-photon scheme for describing extensive atmospheric showers.

The principal contradictions between the experimental data and the electromagnetic scheme of electron-photon showers may be formulated as follows:

1) The shower has a complex composition: in addition to the electron-photon component, the shower includes penetrating particles, both nuclear-active and nuclear-passive (\(\mu\)-mesons). Strongly ionizing particles and slow neutrons are present in the shower. The energy carried by the penetrating particles exceeds the energy of the electron-photon component; the energy carried by the nuclear-active component is at least of the same order as the energy of the electron-photon component.

2) No sharp dependence of the altitude development of showers on their density is observed, contrary to the conclusions from the electron-photon scheme.

3) The spatial distribution of shower particles is considerably broader than cascade theory predicts. At large distances from the core, the decrease in shower density with distance from the shower axis proceeds more slowly than follows from cascade theory.

*) The scheme used by these authors registered charged and neutral nuclear-active particles with equal probability.

Moreover, the purely electron-photon shower scheme probably does not fit the description of showers already because the primary particles entering the atmosphere from cosmic space, at least in the region of moderate energies (\(\sim 10^{10}\)—\(10^{11}\) eV), are not photons or electrons, but protons and, perhaps, heavier nuclei.

The way out of the situation that had arisen, as was already mentioned in the introduction, was found in 1948 in the construction of a new scheme of extensive atmospheric showers, based on the notion of a nuclear-cascade process.\(^9\) According to this scheme, extensive atmospheric showers are electron-nuclear showers formed by primary particles of ultrahigh energy and developing in the air.

Thus, in principle there is no difference between the mechanism of formation of particles of the “single” component and of shower components. The difference lies in the energy of the primary particle. When the energy of the primary particle is small, the number of particles in the resulting “showers” is small and they are observed as “single” particles. When the energy of the primary particle is large, a shower is formed with a number of particles sufficiently large for its observation. Owing to the low density of air and the comparatively large emission angles of the particles being born, as well as their subsequent scattering, the particles diverge over large distances. Thus, all showers in the air, at a sufficient distance from the point of their generation, will be extensive. Hence one may conclude that, in principle, all cosmic-ray particles in the depths of the atmosphere are part of extensive atmospheric showers.

However, when we speak of “extensive atmospheric showers,” we in fact mean only showers consisting of a very large number of particles, i.e., showers generated by primary particles of very high energy. Therefore, it might perhaps have been more correct to call them large atmospheric showers or “high-energy atmospheric showers,” but the term “extensive atmospheric showers” has historically become firmly established in the literature, and we do not consider it expedient to replace it.

After its emergence, the nuclear-cascade scheme of extensive atmospheric showers received numerous weighty confirmations.

First, the nuclear-cascade process was directly observed in photographs obtained with a Wilson chamber\(^{67—69}\); second, all the contradictions of the experimental data with the electron-photon scheme, which by 1948 had only become apparent, were later clearly proved by carrying out a series of detailed experimental investigations and theoretical calculations; third, all the consequences of the nuclear-cascade shower scheme, at least qualitatively, were confirmed experimentally.\(^{18, 47, 50, 53}\) All this led to the fact that the new scheme has now become universally accepted.\(^{43, 55, 70}\)

The nuclear-cascade scheme describes the development of a shower as follows:

1. The beginning of the development of the shower is due to the collision of a primary proton or a heavier nucleus of ultrahigh energy with the atomic nucleus of one of the atoms of the air in the Earth’s atmosphere.

2. In the act of such a collision new nuclear-active particles are born. The subsequent collisions of each of them with atomic nuclei lead to a cascade process of multiplication of nuclear-active particles.

3. The neutral $\pi^0$-mesons born in the processes of nuclear collisions, in their decay, give rise to the electron-photon component of the shower, whose development is described by the electromagnetic avalanche theory.

4. The charged $\pi^\pm$-mesons, in their decay, give the $\mu$-meson component of extensive atmospheric showers.

Such a scheme of shower development makes it possible to divide the consideration of the totality of processes in a shower into two parts:

1) the development of an avalanche of nuclear-active particles, which is, as it were, the skeleton of the shower, and

2) the development of the secondary components of the shower, including the electron-photon one.

In the multiplication of nuclear-active particles, each particle of the next generation has an energy smaller than the energy of the particles of the preceding generation. As the energy of the nuclear-active particles decreases, the relative role of the different processes occurring in nuclear collisions changes.

The energy of a nuclear-active particle in nuclear collisions is transformed not only into the energy of new, likewise nuclear-active particles, but also goes (and, one may suppose, practically irreversibly) into the formation of photons (through $\pi^0$-mesons), $\mu$-mesons (through the decay of $\pi^+$-, $\pi^-$-mesons), and into nucleons of comparatively small energy, arising in the destruction of atomic nuclei. The role of these processes increases sharply as the energy of the nuclear-active particles decreases, so that, apparently, for the nuclear-cascade process too one may introduce the concept of an energy “threshold,” analogous to the concept of “critical energy” in the cascade electron-photon theory. The numerical value of this threshold lies in the interval $10^9$—$10^{10}$ ev.

The existence of each of the processes described may at present already be considered well established by independent experiments (with photographic plates, etc.), so that now the necessity of such a scheme is a simple consequence of the assumption of the existence of nucleons with energies of the order of $10^{14}$—$10^{16}$ ev in the primary component. From this scheme there follows directly the presence of all the known components of the shower:

1) nuclear-active,

2) $\mu$-meson,

3) electron-photon.

It also becomes clear that there are strongly ionizing particles and slow neutrons.

At present, the nuclear-cascade scheme makes it possible to understand the composition of the shower not only qualitatively, but also semiquantitatively. Indeed, in a developing shower the number of nuclear-active particles \(N^{(я)}\) must be (to within a coefficient of order unity) equal to the ratio of the total energy carried by the nuclear-active particles to the “threshold” energy \(\varepsilon_k\). By an analogous ratio of the total energy of the electron-photon component \(E^{(e)}\) to \(\beta\), the number \(N^{(e)}\) of electrons is determined. Therefore the fraction of nuclear-active particles in a shower is determined by the relation

\[ \frac{N^{(я)}}{N^{(e)}} \sim \frac{\dfrac{E^{(я)}}{\varepsilon_k}}{\dfrac{E^{(e)}}{\beta}} . \]

If one assumes that in a developing shower the energy carried by nuclear-active particles is close to the energy of the electron-photon component \(\bigl(E^{(я)} \simeq E^{(e)}\bigr)\), then the fraction of nuclear-active particles must be of order \(\dfrac{\beta}{\varepsilon_k}\). For \(\varepsilon_k \simeq 10^{10}\) eV, as follows from experiments with photographic plates and from the study of electron-nuclear showers, we obtain:

\[ \frac{N^{(я)}}{N^{(e)}} \simeq 10^{-2}, \]

which agrees with experiment.

According to the nuclear-cascade scheme, in a developing shower there should be observed an approximate proportionality between the number of penetrating particles and electrons, with a slow decrease in the fraction of penetrating particles in showers produced by primary particles of high energy.

Let us denote the number of nuclear-active particles born by a primary proton with energy \(E_0\) by \(n\), and suppose that each of them has received the energy

\[ \frac{E_0(1-\varepsilon)}{n}, \]

where \(\varepsilon\) is the fraction of the energy of the primary particle that has passed into the nuclear-passive component (electron-photon). Consequently, one primary particle with energy \(E_0\) is equivalent (in the sense of the subsequent development of the penetrating component of the shower) to \(n\) particles with energy

\[ \frac{E_0(1-\varepsilon)}{n}. \]

The number of penetrating particles at the maximum \(N_{\mathrm{п}}(E_0)\) therefore satisfies the relation

\[ N_{\mathrm{п}}(E_0)=nN_{\mathrm{п}}\left(\frac{E_0(1-\varepsilon)}{n}\right). \tag{22} \]

Approximating \(N_{\pi}\) by a power function \(E_0^k\), we obtain:

\[ k=\frac{\ln n}{\ln n-\ln(1-\varepsilon)} \simeq 1-\frac{\varepsilon}{\ln n}. \tag{23} \]

If \(\varepsilon \ll 1\) and \(n \gg 1\), then, consequently, at the maximum of the shower the number of penetrating particles is proportional to \(E_0\) raised to a power close to unity, but smaller than it: \(k < 1\).

Conversely, the fraction of the energy passing into the electron-photon component, with increasing energy of the primary particle, must increase somewhat, since the number of acts of nuclear collisions occurring before the shower reaches its maximum increases. However, this increase will be slow if \(n\) is large and \(\varepsilon\) is small. Since the number of electrons at the maximum is proportional to the energy transferred to the electron-photon component, the number of electrons at the maximum must also increase only somewhat faster than \(E_0\).

All this should lead to the result that the fraction of penetrating particles at the maximum of showers, in agreement with experiment, should only weakly decrease with an increase in the number of particles in the showers.

At great depths the fraction of penetrating particles should decrease more noticeably with increasing number of particles in the shower, since the number of penetrating particles must decrease with depth very slowly, while the number of electrons in showers of higher energy decreases somewhat more weakly than in showers of lower energy.

Experimental data on this question are still very scanty. At mountain altitudes\(^{49}\) only data have been obtained concerning the change in the ratio of the densities of penetrating particles and electrons. The fraction of penetrating particles in the recorded regions of showers, if it decreases at all, does so approximately only as \(\rho^{-0.13}\). At sea level, preliminary data concerning the fraction\(^{50}\) of penetrating particles confirm a certain decrease of this fraction with the number of particles in the shower; however, the statistical material is still insufficient for quantitative conclusions.

As follows from the arguments presented, the fraction of the energy passing in each act into the nuclear-passive component of the shower must be small, so that it would be possible to explain the approximate independence, with respect to the shower energy, of the ratio of the number of penetrating particles to the number of electrons.

In work\(^{71}\) a proof of this proposition is given from another point of view. At an atmospheric depth of \(600\ \mathrm{g/cm^2}\), which corresponds to approximately 10 free paths of particles interacting with atomic nuclei with an effective cross section equal to the geometrical cross section, the energy of the nuclear-active component is very considerable. If in each collision act a fraction \(\varepsilon\) of the energy of the incident particle goes into the nuclear-passive component, while a fraction \(1-\varepsilon\) remains in the nuclear-active component, then at a depth,

corresponding to 10 nuclear cross sections, only a fraction equal to \((1-\varepsilon)^{10}\) of the energy of the primary particle will remain in the nuclear-active component. Assuming that the nuclear-active component thereafter retains not less than 0.1 of the total shower energy (which is apparently a considerable underestimate), we obtain for the quantity \(\varepsilon\):

\[ \varepsilon = 1 - 10^{-1/10} \simeq 0.2. \]

The nuclear-cascade scheme also makes it possible, even without detailed calculations and an exact model, to give a qualitative explanation of the anomalously slow decrease of the shower density at large distances from the axis. Indeed, if in an electron-photon shower one may neglect (in comparison with Coulomb scattering) the angles of emission of particles (produced pairs and photons emitted by electrons), then in a nuclear-cascade shower it is necessary to take into account the angles at which particles produced in nuclear processes fly out relative to the direction of the generating particle.

The angular distribution of particles generated in the process of nuclear collisions, of course, depends on the mechanism of generation. However, if in the coordinate system of the center of inertia of the colliding nucleons the process of particle production does not possess a sharply pronounced anisotropy (in the variants of the theory of such processes proposed so far one obtains either isotropy or sufficient proximity to it), then in the laboratory system the order of magnitude of the mean angle of divergence of the produced particles is determined by the formula

\[ \theta_{0} = \sqrt{\frac{2mc^{2}}{E}} \]

(where \(m\) is the mass of the nucleon, \(E\) is the energy of the incident nuclear-active particle).

If, as the “threshold” of the nuclear-cascade process, even such a high energy as \(10^{11}\ \text{eV}\) is adopted, then, for a mean path length of the particles of the nuclear-active component in air (determined by the geometrical cross section of the nucleus) of the order of \(1\ \text{km}\), we find that the component generating the electron-nuclear shower will spread out to distances up to \(100\ \text{m}\) from the shower axis.

The electron-photon component (see, for example, \({}^{62}\)) is effectively generated by nucleons with energy greater than \(10^{10}\ \text{eV}\). In this case neutral mesons must fly out at angles of order \(\theta_{0} \simeq 0.4\). With a path length of showers formed by the decay photons of \(\mu\)-mesons of about \(100\text{–}150\ \text{g}/\text{cm}^{2}\), we find that the electron-photon component can spread out to distances as great as \(500\text{–}700\ \text{m}\) from the shower axis.

The widest spatial distribution should belong to the \(\mu\)-mesons, since they can be formed only with relatively low energy \((\lesssim 3 \cdot 10^{10}\ \text{eV})\) (at high energies \(\pi^{\pm}\)-mesons will not have time to decay). Therefore they will arise mainly as a result of events caused by nuclear-active

WIDE ATMOSPHERIC SHOWERS

particles of small energies, as a result of which their angles of production will be large. The weak interaction of $\mu$-mesons with matter and their longer lifetime lead to large ranges; therefore their distance from the shower axis may be very considerable. This is well confirmed by experimental data (see Section III).

Since neither the electron-photon nor, still less, the nuclear-active components of showers can diverge from the axis to very large distances, at the far periphery wide atmospheric showers must consist of $\mu$-mesons alone (and about 30% electrons moving in equilibrium with them).

The nuclear-cascade scheme makes it possible to explain qualitatively the anomalies connected with altitude dependences. For this, however, special assumptions are required concerning the character of the elementary act of particle production in the collision of a high-energy nuclear-active particle with a nucleus.

Indeed, the act of shower formation occurs, on the average, at a depth corresponding to one nuclear cross section, i.e. $\sim 70\ \mathrm{g}/\mathrm{cm}^2$ of air, whereas in the case of the incidence of a primary high-energy electron, beginning at the boundary of the atmosphere, the number of particles grows exponentially as $e^{\lambda t}$, reaching large values (hundreds of particles) at a depth of $70\ \mathrm{g}/\mathrm{cm}^2$, since a high-energy electron emits, in one $t$-unit ($35\ \mathrm{g}/\mathrm{cm}^2$), a large number of photons with energy above the critical energy. Nevertheless, according to experiments, the shower reaches the maximum of its development at atmospheric depths ($\sim 300\ \mathrm{g}/\mathrm{cm}^2$) smaller than the depths corresponding to the maximum of electron-photon avalanches from primary electrons. Hence it is necessary to conclude that, in the collision of high-energy nuclear-active particles with atomic nuclei, multiple production of secondary particles occurs. Owing to this, despite the fact that the nuclear range (the mean free path $\sim 70\ \mathrm{g}/\mathrm{cm}^2$, corresponding to the geometrical cross section of nuclei of air atoms) is twice as large as the $t$-unit in air, multiplication of particles will proceed faster than in an electron-photon avalanche.

The nuclear-cascade scheme makes it possible to explain qualitatively the weaker dependence of the absorption coefficient of shower particles on the energy of the particle that generated it than follows from the scheme of electron-photon showers. Indeed, in the case of electron-photon showers the rate of energy fragmentation does not depend on energy, as a result of which, even at large depths, showers produced by a primary particle of high energy are richer in high-energy electrons than showers arising from particles of low energy. This leads to a strong dependence of the absorption coefficient on the energy of the primary particle.

In nuclear collisions the fragmentation of energy must be very great at high particle energy, i.e. increase with energy,

Then, in a shower originating from a particle with high energy, the particle energy will be fragmented more rapidly and at a sufficient depth. The energy spectrum of showers arising from primary particles of different energies will differ less in showers of different energy than in the case of electron–photon showers. Hence there also follows a weaker dependence of the particle absorption coefficient on the primary energy of the shower.

As was already said above, the absorption of particles of showers of low density proves to be much smaller than follows from calculations carried out for electron–photon showers. Therefore one should conclude that such an altitude dependence is caused by the nuclear-cascade process of shower development. In work \(^{31}\) the assumption is put forward that at great depths in the atmosphere the electron–photon component of showers is in equilibrium with the nuclear-active one.

Thus, qualitatively all the principal experimental data concerning extensive atmospheric showers find a good explanation in the nuclear-cascade scheme, under two additional assumptions which have not yet been fully confirmed in independent experiments (although the corresponding indications undoubtedly already exist).

1) The multiplicity of the production process in the act of a nuclear collision increases with energy.

2) In the collision of nuclear-active particles of high energy, the fraction of energy transferred to nuclear-passive particles is small.

However, the multiplicity of production processes in the collision of a nuclear-active particle with a nucleus cannot be very large, since in that case already after several cascades the mean energy of the particles would have fallen to the “threshold” of the nuclear-cascade process, and with a further increase in depth the number of nuclear-active particles (of any energy) would decrease exponentially, with an absorption coefficient corresponding to the effective interaction cross section of high-energy nuclear-active particles, i.e. \(\mu \simeq \dfrac{1}{80\, g/cm^{2}}\); experimentally, however, an altitude dependence is observed corresponding approximately to \(\mu \simeq \dfrac{1}{200\, g/cm^{2}}\), i.e. to a considerably weaker absorption of particles. If this is due not to particles possessing a correspondingly smaller effective cross section for nuclear interaction, then it testifies to the great role of cascade processes for the nuclear-active component even at great depths in the atmosphere and, consequently, the multiplicity of particle production processes cannot be very large.

From the conservation laws it follows that the energy in the center-of-inertia system of two colliding nucleons in an ultrarelativistic

in this case is proportional to $E_0^{1/2}$, where $E_0$ is the energy of the incident nucleon in the laboratory coordinate system. It follows from this that the multiplicity of particle production cannot grow faster than $E_0^{1/2}$ (in this case the mean energy of the produced particle in the center-of-inertia coordinate system does not depend on $E_0$). At the same time, however, any power exponent smaller than $1/2$ is possible; its value is determined by the specific mechanism of particle production.

Heisenberg $^{72}$ proposed a version of the theory of the elementary act (as less well founded than the others), according to which $N \sim E_0^{1/2}$; Oppenheimer $^{73}$: $N \sim E^{1/2}$. The best founded is the theory of multiple generation at superhigh energies, proposed by Fermi $^{74}$. In this theory $N \sim E^{1/4}$.

Each version of the theory of nuclear processes at high energies leads to a corresponding version of the theory of the development of atmospheric showers. Thus, comparison of experimental results with the results of theoretical calculations makes it possible to choose the version closest to the truth.

The calculation of showers developing according to the scheme of a nuclear-cascade process (for various versions of the model of the elementary act of nuclear collisions) was carried out by I. L. Rozental $^{51}$. He calculated the dependence of the number of particles in the various components of extensive atmospheric showers on altitude.

The calculation is based on the following scheme of the nuclear-cascade process: 1. The primary particle is a nucleon with energy $E_0$. 2. The nuclear-active component consists of nucleons and $\pi$-mesons; the effective interaction cross section of these particles is approximately equal to the geometrical cross section of the nucleus. 3. In the interaction of a nuclear-active particle of energy $E$ with a nucleus, $\left(\dfrac{E}{M}\right)^\nu$ particles are produced ($\nu=\mathrm{const}$; $M$ is the rest energy of the nucleon, which we take as unity). 4. All particles arising as a result of such an interaction have the same energy $E^{(1-\nu)}$. 5. The fraction of energy $b$ transferred in each act to nucleons does not depend on the energy of the incident particle. 6. Charged $\pi$-mesons decay with a lifetime $\sim 1\cdot10^{-8}$ sec, producing $\mu$-mesons in the decay. 7. One third of the $\pi$-mesons are neutral; immediately after arising they decay into two photons, which, multiplying, form the electron-photon component. 8. Nuclear-active particles possessing energy $E < E_c$ ($E_c = 10^{10}—5\cdot10^9$ ev is the critical energy for the nuclear-cascade process), upon collision with the nuclei of air atoms, do not produce $\pi$-mesons and nucleons capable of further nuclear interaction.

Denoting by $N^{(N)}(e)$, $N^{(\pi)}(e)$, and $N^{(\mu)}(e)$ respectively the total number of nucleons, charged $\pi$- and $\mu$-mesons with energy $E_i > E_c$,

\(N^{(e)}(e)\) is the total number of electrons in the shower, and taking as the unit of depth the quantity \(L\), equal to the mean free path of nuclear-active particles between two successive collisions (in accordance with \(^{70}\), \(L=82\ \mathrm{g/cm^2}\)), we obtain results, some of which are presented as curves in Fig. 16. Shown here are the altitude dependences of the various components of extensive atmospheric showers, calculated under the assumption

\[ \nu=\frac{1}{4},\quad E_0=10^{16}\ \mathrm{eV}. \]

In the calculation, ionization losses are not taken into account, nor is the decrease in the number of \(\mu\)-mesons due to their decay in flight.

Fig. 16. The theoretically calculated altitude dependence of the various components of extensive atmospheric showers.

Table IV brings together some experimental data on extensive showers and the results of calculations carried out under two assumptions concerning \(\nu\). In the first row is given the fraction of penetrating particles in the shower; in the second, the fraction of nuclear-active particles relative to the penetrating nuclear-passive ones.

It follows from Table IV that the assumption \(\nu=\frac{1}{4}\) does not contradict the experimental data, whereas the assumption \(\nu=\frac{1}{2}\) cannot be reconciled with them.

The scheme of the nuclear-cascade process described above was used to calculate the spatial distribution of the various components. Under the assumption that the angular distribution of the produced particles in the center-of-mass system of the colliding particles does not depend on energy, as follows from some proposed schemes of the elementary act \(^{74,75}\), values were obtained for the root-mean-square radii of the various components; these are given in Table V.

These estimates, in comparison with those set out above, include an additional assumption about the angular distribution of particles in the elementary act, for which at present there are no sufficiently reliable data. In accordance with the qualitative consideration presented above, it is also found here that the radius of extensive showers, calculated with allowance for the nuclear-cascade process,

Table IV

Quantities of ratios	Experimental values: $H=3\text{–}4\ \mathrm{km}$	Experimental values: sea level	Theoretical values: $\nu=\dfrac{1}{4}$, $H=3.5\ \mathrm{km}$	Theoretical values: $\nu=\dfrac{1}{4}$, sea level	Theoretical values: $\nu=\dfrac{1}{2}$, $H=3.5\ \mathrm{km}$	Theoretical values: $\nu=\dfrac{1}{2}$, sea level
$\dfrac{N^{(H)}+N^{(\pi)}+N^{(\mu)}}{N^{(H)}+N^{(\pi)}+N^{(\mu)}+N^{(e)}}$	$0.01\text{—}0.015$	$0.1^{*}$)	$0.02$	$0.1$	$0.3$	$0.9$
$\dfrac{N^{(H)}+N^{(\pi)}}{N^{(H)}+N^{(\pi)}+N^{(\mu)}}$	$0.3\text{—}0.5$	$0.1$	$0.6$	$0.08$	$0.06$	$0.005$

*) Table IV gives data obtained in work$^{50}$. They correspond to the total fraction of penetrating particles in a shower, whereas the data obtained at high altitude refer only to the central parts of the shower (see Section IV).

Table V

$H$	$R_{\mathrm{e}}$ (cascade theory), m	$R^{(H)}$, m	$R^{(\pi)}$, m	$R^{(\mu)}$, m	$R^{(e)}$, m
$3\ \mathrm{km}$	100	190	80	400	180
Sea level	70	190	90	600	200

significantly exceeds the radius corresponding to the electromagnetic cascade theory, which is also in agreement with the experimental data.

Thus, the calculations have shown that, under reasonable assumptions about the elementary act, the nuclear-cascade scheme of extensive atmospheric showers makes it possible to explain not only qualitatively, but also semi-quantitatively, their principal characteristics: the composition and spatial distribution of the various components. Moreover,

they show that the resulting picture of broad atmospheric showers is very sensitive to the multiplicity of the production process in the elementary act. Therefore, the experiments make it possible to determine this multiplicity approximately.

If the development of a shower with depth is determined primarily by the multiplicity of production in the elementary act, then the spatial distribution (in particular, of the penetrating component) is governed by the angular divergence of the particles at their production. This gives grounds to hope that a detailed study of broad atmospheric showers, together with the corresponding theoretical estimates, will make it possible to obtain the necessary experimental parameters characterizing the production of particles in the elementary act of collision of nuclear-active particles of ultrahigh energy with nucleons.

Comparison of the experimental results of the study of broad atmospheric showers with theoretical estimates and calculations, as carried out, permits the following conclusions to be drawn concerning the processes occurring when nucleons of ultrahigh energy collide with atomic nuclei of the air:

1) The effective cross section of interaction between nucleons and atomic nuclei, up to energies of \(10^{16}—10^{18}\) ev, remains appreciable, close in order of magnitude to the geometrical cross section of the nucleus (mean free path in air \(\ll 100\ \text{g}/\text{cm}^2\)).

2) At very high energy of nuclear-active particles, in the act of collision with a nucleus the overwhelming part of the energy is transferred to nuclear-active particles.

3) The multiplicity of the process of particle production in acts of nuclear collisions increases with energy, but more slowly than \(E^{1/2}\), and, apparently, not faster (or only somewhat faster) than \(E^{1/4}\).

For more precise conclusions, additional experimental data and further, more detailed calculations are needed.

The authors express their gratitude to Academician D. V. Skobeltsyn for valuable advice and comments, which were widely used in writing the article; the authors are also very grateful to E. L. Feinberg for the extensive editorial work he carried out on the manuscript, and also to S. Z. Belenkii and M. I. Podgoretskii for a number of comments made by them when reading the manuscript.

CITED LITERATURE

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1. Reference number as shown in the source. ↩