Average Characteristics of the Interaction Event of Primary Cosmic Particles of Different Energies (2–1000 Bev) with Light Atomic Nuclei

N. L. Grigorov

INTRODUCTION

In the last five years, substantial changes have occurred in the understanding of the basic processes that determine the passage of cosmic rays through matter. The nucleonic nature of primary cosmic rays, with an absolute predominance of protons, has been definitively established1,2. Study of the mechanism of formation of secondary cosmic radiation, varied in its nature, has revealed the decisive role of nuclear interactions3.

It has been established that \(\pi^{\pm}\)-mesons4, which give rise to the penetrating component, and \(\pi^0\)-mesons5, which give rise to the electron-photon component of cosmic rays, are generated in events of nuclear interaction of high-energy nucleons with the atomic nuclei of the matter of the atmosphere. It has been established that, in the total flux of cosmic radiation in the atmosphere, an essential role is played by the component producing “stars”—disintegrations with comparatively small energy release6,7. New elementary particles have been discovered. Finally, the study of extensive atmospheric showers has shown that in them as well the determining mechanism is the mechanism of nuclear interaction of nuclear-active particles of ultrahigh energies8.

Thus, the totality of investigations carried out in recent years has led to substantial changes in views on the role of nuclear and electromagnetic interactions in cosmic rays. At the present time it may be asserted that the science of cosmic rays represents one of the branches of nuclear physics, namely that part of it which deals with particles of high and ultrahigh energy. In this connection the main direction of cosmic-ray investigations has also become defined—the study of the processes of nuclear interaction of high-energy particles (the study of the elementary event

N. L. GRIGOROV

collisions nucleon—nucleon, nucleon—nucleus, and the study of the formation of new particles and of their nature).

Great successes in the study of nuclear interactions of cosmic-ray particles were achieved, chiefly, thanks to the broad use of the Wilson chamber and photographic plates, which make it possible to observe directly the picture of an elementary act of nuclear interaction of high-energy particles. However, despite the great visual clarity of the results thereby obtained, it has so far not been possible simultaneously to measure the energy of the primary particle causing the observed interaction and the energy of all secondary particles; therefore the energy regularities of the basic processes of nuclear interactions and their dependence on the energy of the primary particles have until recently remained little studied.

Progress in this direction was made possible by using methods for measuring the integral fluxes of various secondary components of cosmic radiation. The present article is devoted to the presentation of the results of these measurements and of the conclusions drawn from them.

The basic considerations underlying the work we carried out are as follows.

The Earth’s atmosphere is a filter upon which there falls a flux of primary cosmic particles with a known energy distribution. As a result of interaction with atomic nuclei, the flux of primary particles is absorbed in this filter. The energy of the absorbed primary particles is transferred to other components of cosmic radiation: \(\pi\)-mesons and heavy products of nuclear disintegrations—“stars.” If one measures the energy obtained by these secondary components in different layers of the filter, i.e. at different depths of the atmosphere, then one can form an idea of the characteristics of the interaction of primary particles with the atomic nuclei of the substance of the filter (the atmosphere).

All measurements of this kind, performed with thin filters (filter thickness \(x \ll L_{\mathrm{int}}\), where \(L_{\mathrm{int}}\) is the mean free path for the nuclear interaction of primary particles), will obviously give information on what energy is transferred on average to secondary particles in a single act of interaction of a primary particle with an atomic nucleus.

If such measurements are made at different geomagnetic latitudes, then we obtain information on the energy transferred to the secondary components by primary particles possessing energy greater than a certain value \(E_c\), which depends on the geomagnetic latitude \(\lambda\). Therefore measurements carried out at two latitudes \(\lambda_1\) and \(\lambda_2\) give us information on the characteristics of nuclear interactions for which the primary particles lying in the energy interval from \(E_c(\lambda_1)\) to \(E_c(\lambda_2)\) are responsible (the values of the critical energies \(E_c(\lambda)\) are taken from the theory of geomagnetic effects of cosmic rays\(^{9}\)).

It should be noted that the terrestrial atmosphere, as an absorber of primary radiation, has a feature that cannot be imitated under laboratory conditions: a great extent at a low density of matter. Owing to this feature, all charged \(\pi^\pm\)-mesons, if in the upper layers of the atmosphere they have energies less than \(10^{11}\) eV, have time to decay into \(\mu^\pm\)-mesons and neutrinos before they interact with atomic nuclei. Thus, the nuclear-active component in the atmosphere will in practice consist only of nucleons; the energy of the \(\pi^\pm\)-mesons will pass to the nuclear-passive particles—the electron-photon and \(\mu\)-meson components of cosmic rays. Therefore, only the nucleon component of cosmic rays is responsible for the processes of nuclear interactions occurring in the atmosphere.

I. FORMATION OF \(\pi\)-MESONS BY PRIMARY PARTICLES OF DIFFERENT ENERGIES

1.1. Method of measurements

The soft component of cosmic rays in the atmosphere consists predominantly of relativistic electrons, which are produced by two principal processes: the decay of \(\pi^0\)-mesons into two \(\gamma\)-quanta with subsequent conversion into electrons, and by the \(\pi^\pm \to \mu^\pm \to e^\pm\) decays. Thus, at the basis of both sources of formation of the electron-photon component of cosmic rays lie the processes of formation of neutral \((\pi^0)\) and charged \((\pi^\pm)\) mesons.

The energy \(E_{\pi^0}\) transferred to \(\pi^0\)-mesons passes completely into the energy of the soft component. Of the energy \(E_{\pi^\pm}\) transferred to charged \(\pi^\pm\)-mesons, only a small part passes into the energy of the soft component. In fact, from the total energy \(E_{\pi^\pm}\) of the \(\pi^\pm\)-mesons, the part

\[ E_\mu=\frac{m_\mu}{m_\pi}E_{\pi^\pm}\simeq \frac{3}{4}E_{\pi^\pm} \]

passes to the \(\mu^\pm\)-mesons.

In the decay of \(\mu^\pm\)-mesons, on average \(1/3\) of the total energy of the \(\mu\)-mesons passes into the energy of the soft component. During passage through the atmosphere, part of the meson energy is spent on ionization, \(E_{\mu\text{ion}}\). Therefore, from the total energy \(E_\mu\) of the \(\mu\)-mesons obtained by them in the decay of \(\pi\)-mesons, the decay electrons receive an energy equal to:

\[ \frac{1}{3}\left(E_\mu-E_{\mu\text{ion}}\right) = \frac{1}{3}\cdot\frac{3}{4}E_{\pi^\pm} - \frac{1}{3}E_{\mu\text{ion}} = \frac{1}{4}E_{\pi^\pm} - \frac{1}{3}E_{\mu\text{ion}}. \]

Thus, the energy transferred to the soft component is equal to

\[ E_{\text{s.c.}}=E_{\pi^0}+\frac{1}{4}E_{\pi^\pm}-\frac{1}{3}E_{\mu\text{ion}}. \tag{1} \]

It is known that in the atmosphere the ionization produced by \(\mu\)-mesons (the hard component) constitutes only a small fraction of the ionization produced by the soft component. Therefore the term \(\frac{1}{3}E_{\mu\text{ ion}}\) in equality (1) constitutes a very small correction (known, incidentally, from experimental data \(^{10}\)).

The quantity \(E_{\text{m.c.}}\) can be measured simply. The soft component consists of relativistic electrons and photons (the energy of the latter sooner or later is transferred to electrons and is expended by them on ionization of the atmosphere). The ionization losses \(\beta\) per \(1\ \text{g}\) of matter for electrons are known; therefore, if the global flux of electrons \(N_{\text{m}}(p)\) at all depths \(p\) of the atmosphere is known, then the expression

\[ \beta \int_{0}^{\infty} N_{\text{m}}(p)\,dp = E_{\text{m.c.}} \tag{2} \]

gives the total energy transferred to the soft component in the whole atmosphere.

If we assume that \(E_{\pi^\pm}=kE_{\pi^0}\), then from equalities (1) and (2) one easily obtains the values of \(E_{\pi^0}\) and \(E_\pi=E_{\pi^\pm}+E_{\pi^0}=(1+k)E_{\pi^0}\):

\[ E_{\pi^0}= \frac{E_{\text{m.c.}}+\frac{1}{3}E_{\mu\text{ ion}}} {1+\frac{k}{4}}, \tag{3} \]

\[ E_\pi= \frac{1+k}{1+\frac{k}{4}} \left(E_{\text{m.c.}}+\frac{1}{3}E_{\mu\text{ ion}}\right). \tag{3a} \]

Thus, in order to answer the question of how the energy transferred to \(\pi\)-mesons by primary cosmic particles upon their complete absorption in the atmosphere depends on the energy of the primary particles themselves, it is sufficient to measure the global flux of particles of the soft component at different heights in the atmosphere and at different geomagnetic latitudes.

It is considerably more difficult to determine the energy transferred to \(\pi\)-mesons by primary particles in a thin layer of matter. In this case we must proceed from equality (1), understanding by the quantities entering into it those energies which have been released in the layer under consideration of thickness \(p\ \text{g}/\text{cm}^2\) (as applied to the atmosphere this means that the point of observation is at an altitude where the pressure of the overlying part of the atmosphere is equal to \(p\ \text{g}/\text{cm}^2\)). However, in this case the quantity \(E_{\text{m.c.}}(p)\) will no longer be determined as simply as for an infinitely thick filter \((p\to\infty)\). The point is that in a thin

layer \(p\ \mathrm{g}/\mathrm{cm}^{2}\), only part of the energy transferred in this layer to the electron–photon component will be lost to ionization. Some part of the energy (and possibly a considerable part) will leave this layer in the form of an energy flux \(S^{r}(p)\), carried by photons and electrons.

Thus, for the given case, \(E_{\mathrm{m.c.}}(p)\) will be determined by the equality

\[ E_{\mathrm{m.c.}}(p)=\beta \int_{0}^{p} N_{\mathrm{m}}(x)\,dx+S^{r}(p). \tag{4} \]

If all the quantities entering into (4) are known, then, using expression (3) or one similar to it (from the standpoint of a more exact allowance for the fraction of energy transferred from mesons to decay electrons), one can find the energy \(E_{\pi}(p)\) transferred by the primary particles to \(\pi\)-mesons in an atmospheric layer of thickness \(p\ \mathrm{g}/\mathrm{cm}^{2}\). Taking the energy flux brought by all primary radiation to the boundary of the atmosphere to be known (this is \(\bar E_{0}N_{0}\), where \(N_{0}\) is the number of primary particles and \(\bar E_{0}\) is the mean energy of one particle), and also taking as known the mean free path for interaction of primary particles in matter (the atmosphere), \(L_{\mathrm{int}}\), it is easy to determine the fraction of energy transferred to \(\pi\)-mesons in one interaction. In fact, the number of primary particles that have undergone an interaction in a layer \(p\) is equal to

\[ \Delta N_{\mathrm{prim}}(p)=N_{0}\left[1-e^{-\frac{p}{L_{\mathrm{int}}}}\left\{1-\frac{p}{L_{\mathrm{int}}}-\left(\frac{p}{L_{\mathrm{int}}}\right)^{2}\operatorname{Ei}\left(\frac{p}{L_{\mathrm{int}}}\right)\right\}\right] = \]

\[ = N_{0}F\left(\frac{p}{L_{\mathrm{int}}}\right). \tag{5} \]

The energy \(\varepsilon\) transferred to \(\pi\)-mesons by one primary particle that has undergone an interaction will be equal to

\[ \varepsilon=\lim_{p\to 0}\frac{E_{\pi}(p)}{\Delta N_{\mathrm{prim}}(p)} =\lim_{p\to 0}\frac{E_{\pi}(p)}{N_{0}F\left(\frac{p}{L_{\mathrm{int}}}\right)}, \]

and the fraction of energy transferred to \(\pi\)-mesons in one interaction will be

\[ \frac{\varepsilon}{\bar E_{0}} =\lim_{p\to 0}\frac{E_{\pi}(p)}{\bar E_{0}N_{0}F\left(\frac{p}{L_{\mathrm{int}}}\right)} =\frac{1}{\bar E_{0}N_{0}}\lim_{p\to 0}\frac{E_{\pi}(p)}{F\left(\frac{p}{L_{\mathrm{int}}}\right)}. \tag{5a} \]

Thus, the question of the average fraction of energy transferred to \(\pi\)-mesons in one act can be solved if \(E_{\mathrm{m.c.}}(p)\) is known. And for this it is necessary to be able to measure \(S^{r}(p)\)—the energy flux of elec-

electron-photon component carried through a horizontal surface at an altitude where the pressure is equal to \(p\ \mathrm{g/cm^2}\).

To measure the energy flux of the soft component passing through a horizontal surface, one may make use of the fact that the soft component is strongly absorbed in lead filters. If, at a given point of the atmosphere, one records the absorption curve of particles of the soft component in lead, i.e., the dependence of the number of particles \(N_{\mathrm{Pb}}(x,p)\) on the thickness \(x\) of the lead filter, then

\[ \beta_{\mathrm{Pb}}\int_0^\infty N_{\mathrm{Pb}}(x,p)\,dx=S(p) \tag{6} \]

(practically, for complete absorption of the soft component a lead thickness of \(8\text{--}10\ \mathrm{cm}\) is sufficient). If the lead filters are horizontal plates covering a solid angle \(2\pi\), then \(S(p)=S^{r}(p)\). If, however, the lead filters are spherical in shape, then the quantity \(S(p)\), determined by formula (6), will give the global energy flux \(S^{\mathrm{glob}}(p)\). As shown in \({}^{6}\), in the absence of particle scattering there is a definite relation between \(S^{\mathrm{glob}}(p)\) and \(S^{r}(p)\):

\[ S^{r}(p)=S^{\mathrm{glob}}(p)- \int_0^{\frac{\pi}{2}} S^{\mathrm{glob}}\left(\frac{p}{\cos\theta}\right)\cos\theta\sin\theta\,d\theta . \tag{7} \]

For the real conditions existing in the stratosphere, formula (7) gives an error of no more than \(10\%\) \({}^{11}\).

It should be noted that determination of the energy flux of the electron-photon component from absorption in thick lead filters is methodologically irreproachable only at such altitudes where one may neglect the formation of an electron-photon component by primary cosmic particles (through the formation of \(\pi^0\)-mesons in the very thickness of the filter). Experimental results (see Fig. 6, p. 612) show that the indicated condition holds at altitudes below \(15\ \mathrm{km}\) (\(p>100\ \mathrm{g/cm^2}\)). To determine \(S^{r}(p)\) at greater altitudes (\(p\ll100\ \mathrm{g/cm^2}\)) one may make use of the features of the cascade multiplication of the electron-photon component and of certain relations between

\[ \int_0^\infty N_{\mathrm{Pb}}(x,p)\,dx \]

and the height of the maximum of the transition curve—the absorption curve in lead of particles of the electron-photon component (for more on this see § 3).

Thus, the solution of the problem posed reduces to measuring the intensity of the soft component under various thicknesses of a lead filter, i.e., to obtaining air–lead transition curves at different altitudes in the atmosphere.

1.2. Apparatus and experimental results

It is known that in an electron–photon shower produced by a photon or an electron, there is a considerable number of electrons whose energy is below the critical energy\({}^{12}\). In view of the fact that for lead the critical energy has the value \(6.4\) MeV, one should expect that in the electron spectrum there will be a significant fraction of particles possessing very small energies. Therefore, in order to measure the transition curve air–lead, as experiments\({}^{13}\) have shown, one should use thin-walled recording instruments (counters and ionization chambers) and surround these instruments on all sides with lead filters (in order to eliminate the effect of electron scattering in the lead).

Under measurement conditions in the stratosphere, the use of flat lead filters is practically excluded. It is most expedient to use spherical lead filters, spherical ionization chambers, and “spherically symmetric” counters with a diameter 2–3 times smaller than the length.

In the process of measurements under stratospheric conditions, the possibility of changes in the sensitivity of the recording instrument is not excluded. To prevent uncontrolled changes in sensitivity from distorting the measured transition effect, our measurements were carried out alternately: for 2 minutes the intensity of cosmic radiation in the atmosphere was measured, and for 2 minutes the intensity of the radiation under the lead filter was measured. This measurement technique made it possible to obtain correct relative data even if the sensitivity of the apparatus changed during the flight.

Since for our purposes it is necessary to know the total flux of energy brought to a given geomagnetic latitude by the primary radiation, and this flux is determined by the quantity \(32.5 \displaystyle\int\limits_0^\infty I(p)\,dp\) eV, where \(I(p)\) is the ionization produced by cosmic rays in the atmosphere at a depth \(p\ \text{g}/\text{cm}^2\), it was necessary, in addition to determining the number of particles of the soft component \(N_m(p)\), also to measure the ionization \(I(p)\). Moreover, the use of an ionization chamber in measuring the transition effect air–lead made it possible to isolate strongly ionizing particles arising as a result of nuclear disintegrations.

In order to be able reliably to compare the results of measurements of ionization and of the number of particles, and to exclude possible errors in determining the altitude, in each instrument the ionization and the number of particles were measured simultaneously.

The instruments developed by us contained one charged-particle counter and one ionization chamber, by means of which the number of particles and the ionization in the atmosphere and under lead were measured successively.

filter (Fig. 1). In each instrument there was a lead filter of spherical shape. Filters of different thicknesses—1, 2, and 4 cm—were used in different instruments. The inner cavity of all filters had a diameter of 9 cm. The instruments weighed from 17 to 27 kg and were raised into the stratosphere on pilot balloons. All measurement results during the flight were transmitted by radio and, at the receiving station, were recorded by continuous photographing on cine film of the coded radio signals fed from the receiver to the screen of a cathode oscillograph.

Fig. 1. Schematic representation of the apparatus for studying the transition effect (plan). In position 1 the number of particles under the lead filter and the ionization in the atmosphere are measured; in position 2 the number of particles in the atmosphere and the ionization under the filter are measured. The dotted line shows the displacement of parts of the filter when the position of the chamber (counter) is changed.

radio signals fed from the receiver to the screen of a cathode oscillograph.

The counters were made of glass and had walls 0.2–0.3 mm thick. The diameter of the counters was 8–9 mm; the length of the working part varied in different specimens from 1.5 to 2.5 cm. The counters were filled with a mixture of argon (80%) and ethylene (20%) to a total pressure of 130 mm Hg.

In order to obtain information on the absolute intensity of cosmic rays in the atmosphere, it was necessary to know the effective length of the counters used and their efficiency. For this purpose X-ray photographs were taken of all the counters, from which the geometrical dimensions of the working region were determined—the inner diameter of the cathode and the distance between the guard tubes \(l_{\text{geom}}\). In addition, for seven counters the effective length was determined by “transilluminating” them with electrons from RaE. It was established that for all specimens \(l_{\text{eff}} = 0.77\, l_{\text{geom}}\). The efficiency of the counters used was determined; it proved to be \(87 \pm 6\%\).

Ionization chambers were made of aluminum 1 mm thick. The inner diameter of the chamber was 80 mm. The diameter of the collecting electrode was 20 mm. The chambers were filled with spectrally pure argon to a pressure of about 2.9 atm. Since electrons in spectrally pure argon have high mobility and do not attach to neutral atoms, one should expect very little recombination of ions even in the track columns produced by α-particles. Special control experiments showed that, at an argon pressure in the chamber of 2.4 atm, saturation of the ionization current caused by the α-particles of Po occurs[^11] at a potential difference between the chamber electrodes of 30 V. Under operating conditions a potential difference of 32–35 V was applied between the chamber electrodes.

Measurement of the ionization current was carried out by the method of charge flow onto the collecting electrode of the chamber over a known time interval \(\Delta t\). In our experiments this interval was about 2 minutes.

The direct measurement of the charge formed on the collecting electrode was carried out by the method proposed by A. E. Chudakov. The essence of this method is as follows. After the time \(\Delta t\) has elapsed, the collecting electrode of the chamber is connected to the grid of an amplifying vacuum tube. The grid potential \(V_g\) changes abruptly by an amount \(\Delta V_g\), proportional to the amount of charge \(\Delta q\) collected on the electrode during the time \(\Delta t\). This voltage pulse is then amplified and transformed into a square pulse whose duration \(T\) is proportional to the amplitude of the input signal; the square signal is modulated by an audio frequency \((1000 \div 2000\ \text{Hz})\) and transmitted over the air. The radio circuit of the apparatus and a sample of the signals are shown in Figs. 2 and 3.

Fig. 2. Image on motion-picture film of radio signals sent over the air by the apparatus for studying the transition effect.

Labels in the figure:

• pulse from the counter
• end of the measurement cycle
• pulse from the chamber
• signals indicating the direction of motion of the counter (chamber)
• beginning of a new measurement cycle
• barographic signals

Measurements were made at geomagnetic latitudes \(31^\circ\) and \(51^\circ\). Six apparatuses were released at geomagnetic latitude \(31^\circ\); measurements

measurements were carried out in the altitude interval \(5\)–\(30\) km. At geomagnetic latitude \(51^\circ\), the measurements were carried out in the altitude interval \(5\)–\(24\) km (in the 1947–1948 experiments\(^{14}\), performed at geomagnetic latitude \(51^\circ\), altitudes up to \(28\) km were reached). In all flights there was good reproducibility of the results of the direct measurements.

The results obtained are shown in Figs. 4 and 5. As is seen from these figures, surrounding the counter or ionization chamber with a lead filter of thickness 1 and 2 cm leads to a strong increase in the number of particles (ionization) under the lead filters. With a further increase in the thickness of the lead filter to 4 and 8 cm, a decrease in the radiation intensity is observed.

In order to separate the soft component from the total flux of charged particles registered by the counter, it is necessary to know the flux of particles of the hard component—particles not absorbed by a lead filter of thickness 10–15 cm (\(\mu\)-mesons, high-energy protons). At geomagnetic latitude \(51^\circ\), in work\(^{15}\) the angular distribution of particles of the hard component was measured. From these data it is easy to obtain the global flux \(N_{\mathrm{h}}(p)\) of particles of the hard component

\[ N_{\mathrm{h}}(p)=2\pi \int_{0}^{\frac{\pi}{2}} N_{\mathrm{h}}(p,\theta)\sin\theta\,d\theta \tag{8} \]

and to find the magnitude of the flux of particles of the soft component

\[ N_{\mathrm{s.c.}}(p,x_{\mathrm{Pb}})=N(p,x_{\mathrm{Pb}})-N_{\mathrm{h}}(p), \]

where \(N(p,x_{\mathrm{Pb}})\) is the number of particles measured by the counter under a lead filter of thickness \(x_{\mathrm{Pb}}\) at that depth of the atmosphere where the pressure is equal to \(p\ \mathrm{g/cm^2}\).

At geomagnetic latitude \(31^\circ\), the angular distribution of particles of the hard component was not measured. Therefore, assuming that the absorption of these particles is determined mainly by the amount of atmospheric matter traversed by them, one may put \(N_{\mathrm{h}}(p,\theta)\simeq N_{\mathrm{h}}(p/\cos\theta,0)\), and, under these assumptions, compute from formula (8) the global intensity of particles of the hard component, using the dependence—known from work\(^{16}\)—of the flux of particles of the hard component for the vertical direction, \(N_{\mathrm{h}}(p,0)\), on the depth \(p\).

In view of the fact that the fraction of particles of the hard component in the stratosphere at latitude \(31^\circ\) in the total flux of all particles is very small, possible deviations of \(N_{\mathrm{h}}(p,\theta)\) from \(N_{\mathrm{h}}\left(\dfrac{p}{\cos\theta},0\right)\) will not lead to any appreciable errors in determining the intensity of the soft component.

Fig. 4. Results of measurements of the number of particles (a) and ionization (b) at latitude \(51^\circ\). 1 — with a filter of thickness 1 and 2 cm; 2 — with a filter of thickness 4 cm; 3 — without a filter; 4 — with a filter of thickness 8 cm; 5 — global flux of the hard component according to measurements \(^{15}\).

Fig. 5. Results of measurements of the number of particles (a) and ionization (b) at latitude \(31^\circ\). \(1\)—with a filter of thickness 1 and 2 cm; \(2\)—with a filter of thickness 4 cm; \(3\)—without a filter; \(4\)—global flux of the hard component, obtained under the assumption that

\[ N_{\mathrm{h}}(p,\theta)=N_{\mathrm{h}}\left(\frac{p}{\cos\theta},0\right). \]

To determine the ionization produced by the hard component, it is sufficient to multiply the value \(N_{\text{h}}(p)\) by the ionizing power of relativistic mesons, measured in papers 11, 17.

1.3. Generation of \(\pi\)-Mesons by Primary Particles of Different Energies

In setting forth in the introduction the essence of the method for studying the average characteristics of the interaction of primary particles with the nuclei of atoms of the atmosphere, we indicated that it is necessary to measure the energy flux \(S(p)\) contained in the electron-photon component at different altitudes. We shall show that the necessary values of \(S(p)\) can be obtained from the results of the measurements carried out.

If, from the intensity of cosmic radiation measured by a counter (chamber) under a lead filter of a given thickness \(x_{\text{Pb}}\), one subtracts the intensity of the hard component, then we obtain the dependence of the intensity of the soft component of cosmic rays on the thickness

Fig. 6. Transition curves of the soft component of cosmic rays according to measurements of the number of particles at different depths at geomagnetic latitude \(51^\circ\). Along the abscissa axis—the mean thickness of the lead filter; along the ordinate axis—the intensity of the number of particles of the soft component under a filter of the given thickness. The intensity of the soft component in the atmosphere at the given atmospheric depth \(p\) is taken as unity.

of the lead filter. A graphical representation of this dependence is given in Fig. 6. As is seen from this figure, all curves corresponding to different altitudes are similar to one another, while a certain increase in the number of particles under filters of large thickness, observed at high altitudes, is due, as was noted earlier, to the generation of \(\pi^0\)-mesons by primary particles in the body of the filter itself.

Let us consider the features of the curves obtained.

1. At all altitudes (for different values of \(p\)) the curves have a sharply pronounced maximum, as should be the case if the soft component is of electron-photon nature.

2. Beyond the maximum the intensity decreases exponentially with an absorption coefficient of \(0.20 \div 0.02\) \(t\)-units (a \(t\)-unit is a shower unit of length, equal for lead to \(\sim 0.5\) cm). According to the cascade theory describing the absorption of high-energy electrons and photons in lead, the decrease of the curve beyond the maximum should be of exponential type with an absorption coefficient of about \(0.22\) \(t\)-units (the minimum absorption coefficient of photons in lead).

3. From a comparison of the increase in ionization and the number of particles under lead filters one can determine the mean ionizing power of those particles which arose in the filter and caused the presence of the transition effect.

The experimental results obtained show that the particles arising in lead filters of thickness 1 and 2 cm possess a specific ionizing power in air of

\[ 87 \pm 9 \frac{\text{ion pairs}}{\text{cm}\cdot\text{atm}} . \]

This value differs little from the ionizing power of the electrons of the soft component of cosmic rays at sea level, equal to

\[ 75 \pm 5 \frac{\text{ion pairs}}{\text{cm}\cdot\text{atm}} \,^{17}. \]

Thus, these particles have relativistic velocities, as should also be the case if the observed transition effect is due to the electron-photon component.

The listed features of the curves characterizing the transition effect of the soft component from air into lead show that this effect is due to multiplication in lead of the electron-photon component of cosmic rays. If this assertion is correct, then for each of the transition curves there must be a definite relation between the height of the maximum, its position, and the area bounded by the curve and the coordinate axes. This relation is established by cascade theory.

If an electron (photon) with energy \(E_0\) falls on lead, then the number of particles created as a result of the process of cascade multiplication (emission by electrons of bremsstrahlung quanta and formation by the latter of electron-positron pairs), at a certain depth \(t_{\max}\), reaches a maximum value \(N_{\max}\). According to \(^{12}\)

\[ N_{\max}=k_1\left(\frac{E_0}{\beta}\right)\frac{\frac{E_0}{\beta}}{\sqrt{\ln \frac{E_0}{\beta}}} \quad \text{and} \quad t_{\max}=k_2\left(\frac{E_0}{\beta}\right)\ln\frac{E_0}{\beta}, \tag{9} \]

where \(E_0\) is the energy of the electron incident on lead, \(\beta\) is the critical energy in lead, equal to 6.4 MeV. But the area \(\sigma\) under

of the cascade curve \(N(t)\), which determines the number of electrons under a layer of thickness \(t\), in accordance with the law of conservation of energy must be equal to \(\dfrac{E_0}{\beta}\). Since

\[ E_0=\beta\int_0^\infty N(t)\,dt=\beta\sigma,\quad \text{i.e.}\quad \sigma=\frac{E_0}{\beta}, \]

then

\[ N_{\max}=\frac{k_1(\sigma)\,\sigma}{\sqrt{\ln \sigma}};\qquad t_{\max}=k_2(\sigma)\ln \sigma. \tag{9a} \]

The quantity \(\dfrac{k_1(\sigma)}{\sqrt{\ln \sigma}}=\gamma(\sigma)\) can be calculated according to cascade theory. It turns out that \(\gamma(\sigma)\) changes extremely weakly with increasing \(\sigma\) (or \(E_0\)). Thus, when \(E_0\) changes from \(20\ \text{MeV}\) to \(10\,000\ \text{MeV}\), \(\gamma(\sigma)\) changes from 0.145 to 0.08.

At a depth \(p=300\ \text{g}/\text{cm}^2\), the area under the transition curve is \(\sigma=20.4\pm1\) \(t\)-units. Using formulas (9a), one can determine that \(\sigma=20\) corresponds to \(\gamma(20)=0.1\); consequently, it should be \(N_{\max}=2.0\) and \(t_{\max}=4.3\) \(t\)-units \(\approx 2.0\ \text{cm}\).

As is seen from Fig. 6, the experiment is in complete agreement with the calculated values: the experimental values are

\[ N_{\max}=2.0\pm0.1,\quad t_{\max}\approx 2\ \text{cm}. \]

Thus, the features of cascade multiplication, owing to which there is a one-to-one relation between the maximum of the transition curve and the area under it, make it possible to determine the energy flux of the electron-photon component not from the area under the entire absorption curve of the soft component in lead, but from the height of the maximum of the transition curve. This method is especially necessary at great altitudes, where the transition curve under large thicknesses of lead is distorted by the formation of \(\pi^0\)-mesons in the lead filter itself by primary particles.

According to (9a), \(N_{\max}=\gamma(\sigma)\sigma\); \(N_{\max}\) varies at all altitudes from 2 to 3, whence the value \(\gamma(\sigma)\) remains practically constant and equal to 0.096. Thus,

\[ \sigma=\frac{N_{\max}}{\gamma}=\frac{\overline{E}}{\beta} \quad \text{and} \quad \overline{E}=\frac{\beta}{\gamma}N_{\max}. \]

\(\overline{E}\) is the mean energy falling on one charged particle of the soft component. The total energy flux of the electron-photon component

\[ S=\overline{E}N_0=\frac{\beta}{\gamma}N_0N_{\max}. \]

But

\[ N_{\max}=\frac{N(t_{\max},p)}{N_0}, \]

whence

\[ S^{\text{deep}}(p)=\frac{\beta}{\gamma}\,N(t_{\max},p). \tag{10} \]

Taking into account that up to 15% of the particles emerging from the lead filters could be absorbed in the walls of the counters used, we finally obtain:

\[ S^{\text{deep}}(p)=76N(t_{\max},p)\ \text{Mev}\cdot\text{cm}^{-2}\text{sec}^{-1}. \tag{10a} \]

The experimental data give \(N(t_{\max},p)\) (the intensity of the soft component under 1 or 2 cm of lead). Using formulas (10a) and (7), we obtain \(S^r(p)\). Then, from the experimental data on the number of particles of the soft component in the atmosphere \(N_{\text{m}}(p)\), we determine

\[ \beta\int_0^p N_{\text{m}}(p)\,dp \]

and, using formula (4), obtain \(E_{\text{m.k}}(p)\).

If the quantity \(E_{\text{m.k}}(p)\) is divided by the fraction of primary particles that have interacted in the layer \(p\),

\[ \frac{\Delta N_{\text{prim}}}{N_0}, \]

determined by formula (5), then we obtain the dependence

\[ N_0\varepsilon(p)= \frac{E_{\text{m.k}}(p)} {\left(\dfrac{\Delta N_{\text{prim}}}{N_0}\right)} \]

on the thickness of the atmospheric layer \(p\), in which the energy \(\varepsilon(p)\) has been transferred to the electron-photon component.

To determine the value

\[ \frac{\Delta N_{\text{prim}}(p)}{N_0}, \]

one must know the interaction mean free path of the primary particles in the atmosphere. From experiments\({}^{18}\) it follows that the interaction mean free path in carbon of protons with energy \(\sim 10^{10}\) ev is \(70\pm 7\ \text{g}/\text{cm}^2\), i.e., the effective interaction cross section leading to the formation of secondary particles is approximately equal to the geometrical cross section of the carbon nucleus. Taking for the interaction mean free path of primary particles in air the value \(60\text{--}70\ \text{g}/\text{cm}^2\), we obtain the dependence

\[ \frac{E_{\text{m.k}}(p)} {\left(\dfrac{\Delta N_{\text{prim}}}{N_0}\right)} \]

on \(p\), shown in Fig. 7. The upper two curves refer to the data obtained at geomagnetic latitude \(31^\circ\) (the primary particles have energies greater than \(7\ \text{Bev}\), i.e. \(\bar E_0=20\ \text{Bev}\)), and the lower curve refers to the difference of the data obtained at geomagnetic latitudes \(51^\circ\) and \(31^\circ\) (the primary particles have energies lying in the interval \(1.5\ \text{Bev}\leq E_0\leq 7\ \text{Bev}\), i.e. \(\bar E_0=3.3\ \text{Bev}\)),

Before analyzing these curves, let us note one feature of meson generation by primary particles with the indicated mean energies.

Fig. 7. Dependence of
\[ \frac{E_{\mathrm{m.c}}(p)} {\left(\dfrac{\Delta N_{\mathrm{prim}}(p)}{N_0}\right)} \]
on the atmospheric depth \(p\).

Curves 1 were obtained for latitude \(31^\circ\) for two values of the geomagnetic cutoff of the primary particles: crosses—\(L_{B3}=70\ \mathrm{g/cm^2}\); dots—\(L_{B3}=60\ \mathrm{g/cm^2}\). Curve 2 is for the latitude difference \(51^\circ\) and \(31^\circ\), \(L_{B3}=60\ \mathrm{g/cm^2}\).

For \(p=1000\ \mathrm{g/cm^2}\), the energy transferred in the whole atmosphere to the soft component is equal to:

at latitude \(31^\circ\)
\[ E_{\mathrm{m.c}}^{31^\circ}(1000\ \mathrm{g/cm^2}) =(7.20\pm0.14)\cdot10^8\ \mathrm{eV\ cm^{-2}\ sec^{-1}}, \]

at latitude \(51^\circ\)
\[ E_{\mathrm{m.c}}^{51^\circ}(1000\ \mathrm{g/cm^2}) =(9.80\pm0.22)\cdot10^8\ \mathrm{eV\ cm^{-2}\ sec^{-1}}. \]

Thus, by primary particles with energy \(\overline{E_0}\simeq 3\ \mathrm{BeV}\), the following energy is transferred to the soft component throughout the whole atmosphere:
\[ E_{\mathrm{m.c}}^{51^\circ-31^\circ}(1000\ \mathrm{g/cm^2}) =(2.6\pm0.26)\cdot10^8\ \mathrm{eV\ cm^{-2}\ sec^{-1}}. \]

At the same latitudes, the energy released in the atmosphere (by ionization of the atmosphere) is
\[ E_{\mathrm{ion}}^{31^\circ} =32.5\ \mathrm{eV}\int_0^{1000} I^{31^\circ}(p)\,dp =(10.8\pm0.18)\cdot10^8\ \mathrm{eV\ cm^{-2}\ sec^{-1}}, \]
\[ E_{\mathrm{ion}}^{51^\circ} =32.5\ \mathrm{eV}\int_0^{1000} I^{51^\circ}(p)\,dp =(19.2\pm0.21)\cdot10^8\ \mathrm{eV\ cm^{-2}\ sec^{-1}}, \]

and

\[ E_{\mathrm{ion}}^{51-31^\circ}=(8.4\pm0.28)\cdot 10^8\ \text{eV}\cdot\text{cm}^{-2}\cdot\text{sec}^{-1}. \]

The ratio of the energy transferred to the soft component to the total energy expended on ionization of the atmosphere by primary particles with \(\bar E_0=20\ \text{BeV}\) and \(\bar E_0=3\ \text{BeV}\) will be, respectively,

\[ \frac{E_{\mathrm{s.c.}}^{31^\circ}}{E_{\mathrm{ion}}^{31^\circ}}=67\pm1.4\% \qquad\text{and}\qquad \frac{E_{\mathrm{s.c.}}^{51-31^\circ}}{E_{\mathrm{ion}}^{51-31^\circ}}=31\pm3\%. \]

Since the energy transferred in the whole atmosphere to the soft component is approximately proportional to the energy transferred by primary particles to all \(\pi\)-mesons, two conclusions follow from the comparison given above:

a) With complete absorption in matter of small atomic number, primary particles with \(\bar E_0\simeq3\ \text{BeV}\) transfer to mesons approximately half as large a fraction of their energy as do primary particles with \(\bar E_0=20\ \text{BeV}\).

b) Energy losses occur that compete with the losses of energy for meson production. These losses play a substantially greater role for particles of low energies (these losses will be considered below, in part II).

This is the situation with the formation of \(\pi\)-mesons when the primary particles are completely absorbed in the atmosphere.

To obtain the fraction of the energy transferred to \(\pi\)-mesons by primary particles in the first interaction with the nuclei of N and O atoms, it is necessary, in accordance with formula (5a), to find

\[ \frac{1}{E_0N_0}\lim_{p\to0}\frac{E_\pi(p)}{F(p/L_{\mathrm{int}})}; \]

however, what we measure directly is \(E_{\mathrm{s.c.}}(p)\). Let us find the relation between \(E_\pi(p)\) and \(E_{\mathrm{s.c.}}(p)\).

If the \(\mu\)-mesons did not have time to decay, then all the energy \(E_{\mathrm{s.c.}}(p)\) of the soft component produced in the layer \(p\) would owe its origin to the decay of \(\pi^0\)-mesons. However, some fraction of the \(\mu\)-mesons in the layer \(p\) does have time to decay and thereby transfers part of its energy to decay electrons, i.e. makes a certain contribution to \(E_{\mathrm{s.c.}}(p)\).

If the energy of the \(\mu\)-mesons is sufficiently large, so that \(E_\mu\gg\beta p\), and if the depth of the observation point is small, so that \(p/L_\pi\ll1\), where \(L_\pi\) is the range for absorption of the nuclear-active component, then it can be shown\({}^{11}\) that the fraction of the energy transferred by \(\mu\)-mesons to decay electrons will not depend on \(p\). If the mean energies of the \(\mu\)-mesons are taken into account, then for latitude \(31^\circ\) we obtain that at depths \(p\lesssim200\ \text{g}/\text{cm}^2\) the fraction of the energy passing from \(\pi^\pm\)-mesons to electrons amounts to \(10\%\) of \(E_{\pi^\pm}\), so that

\[ E_{\mathrm{s.c.}}^{31^\circ}(p)=E_{\pi^0}^{31^\circ}(p)+0.1\,E_{\pi^\pm}^{31^\circ}(p). \]

If we assume that \(E_{\pi^\pm}^{31^\circ}=kE_{\pi^0}^{31^\circ}\), then

\[ E_{\text{m. k.}}^{31^\circ}(p)=(1+0{,}1k)E_{\pi^0}^{31^\circ};\qquad E_{\pi^0}^{31^\circ}(p)=\frac{E_{\text{m. k.}}^{31^\circ}(p)}{1+0{,}1k} \]

and

\[ E_{\pi}^{31^\circ}(p)=(1+k)E_{\pi^0}(p) =\frac{1+k}{1+0{,}1k}E_{\text{m. k.}}^{31^\circ}(p). \tag{11} \]

For the latitude difference \(51-31^\circ\), the average energy of the \(\mu\)-mesons is appreciably less than \(10^9\) eV (see \(^{16}\)), and therefore the contribution of the \(\pi^\pm\)-mesons to the energy of the soft component amounts to \(\sim 0.25\) of \(E_{\pi^\pm}\). Thus,

\[ E_{\pi^0}^{51-31^\circ} = \frac{E_{\text{m. k.}}^{51-31^\circ}(p)}{1+0{,}25k} \]

and

\[ E_{\pi}^{51-31^\circ}(p) = \frac{1+k}{1+0{,}25k}\, E_{\text{m. k.}}^{51-31^\circ}(p). \tag{11a} \]

Consequently,

\[ \lim_{p\to 0}\frac{E_{\pi}(p)}{F(p/L_{\text{вз}})} = \frac{1+k}{1+ak} \lim_{p\to 0} \frac{E_{\text{m. k.}}(p)}{F(p/L_{\text{вз}})}, \]

where \(a=0.1\) and \(0.25\), respectively, for latitude \(31^\circ\) and for the latitude difference \(51-31^\circ\).

To obtain

\[ \lim_{p\to 0}\frac{E_{\text{m. k.}}(p)}{F(p/L_{\text{вз}})}, \]

it is necessary to extrapolate the curves presented in Fig. 7 to their intersection with the ordinate axis \((p=0)\). In doing so, we obtain the following data:

for latitude \(31^\circ\):

\[ \lim_{p\to 0} \frac{E_{\text{m. k.}}(p)}{F(p/L_{\text{вз}})} = (2.04\pm0.2)\cdot 10^8\ \mathrm{eV}\cdot \mathrm{cm}^{-2}\cdot \mathrm{sec}^{-1}, \]

for the latitude difference \(51-31^\circ\):

\[ \lim_{p\to 0} \frac{E_{\text{m. k.}}(p)}{F(p/L_{\text{вз}})} = (1.26\pm0.28)\cdot 10^8\ \mathrm{eV}\cdot \mathrm{cm}^{-2}\cdot \mathrm{sec}^{-1}, \]

and in accordance with (11) and (11a)

\[ \lim_{p\to 0} \frac{E_{\pi}^{31^\circ}(p)}{F(p/L_{\text{вз}})} = \frac{1+k}{1+0{,}1k} (2.04\pm0.2)\cdot 10^8\ \mathrm{eV}\cdot \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}, \]

\[ \lim_{p\to 0} \frac{E_{\pi}^{51-31^\circ}(p)}{F(p/L_{\text{вз}})} = \frac{1+k}{1+0{,}25k} (1.26\pm0.28)\cdot 10^8\ \mathrm{eV}\cdot \mathrm{cm}^{-2}\mathrm{sec}^{-1}. \]

Now it remains to determine the energy flux that is brought in by the primary radiation. The difficulties that arise here are as follows:

1) a part (and the main part) of the energy transferred to the \(\pi^\pm\)-mesons goes into neutrinos and is not released in the atmosphere;

2) a part of the energy (although a small one) associated with the binding energy in the nucleus of the products of nuclear splittings escapes registration.

The energy carried away by neutrinos can be determined from the following considerations.

In \(\pi \to \mu\) decay, \(\frac{1}{4}E_{\pi^\pm}\) will go to the neutrino, and \(\frac{3}{4}E_{\pi^\pm}\) will go to the \(\mu^\pm\)-mesons. Of this energy the \(\mu\)-mesons will expend \(E_{\mu\text{ ion}}\) on ionization, and the remainder \(\frac{3}{4}E_{\pi^\pm} - E_{\mu\text{ ion}}\) will go to the decay products, with \(^{2}/_{3}\) of this energy going to neutrinos. Thus,

\[ E_{\text{neutrino}} = \frac{1}{4}E_{\pi^\pm} + \frac{2}{3}\left(\frac{3}{4}E_{\pi^\pm}-E_{\mu\text{ ion}}\right) = \frac{3}{4}E_{\pi^\pm} - \frac{2}{3}E_{\mu\text{ ion}} . \tag{12} \]

If we use formula (3) and the fact that \(E_{\pi^\pm}=kE_{\pi^0}\), and substitute in (12), we obtain:

\[ E_{\text{neutrino}} = \frac{ 3kE_{\mathrm{m.k}}-\dfrac{8-k}{3}E_{\mu\text{ ion}} }{ 4+k }. \tag{12a} \]

The quantities entering into (12a) are measured experimentally. If one sets \(k=2\), in accordance with the majority of experimental data, then we obtain the following values of the energy fluxes of the primary particles:

for latitude \(31^\circ\)

\[ N_0\overline{E}_0^{31^\circ} = (18.3\pm0.24)\cdot10^8\ \text{eV cm}^{-2}\text{ sec}^{-1}; \]

for the latitude difference \(51-31^\circ\)

\[ N_0\overline{E}_0^{51^\circ-31^\circ} = (12.3\pm0.4)\cdot10^8\ \text{eV cm}^{-2}\text{ sec}^{-1}. \]

Now we have all the data that enter into formula (5a), which determines the average fraction of the energy transferred to \(\pi\)-mesons in the act of the first interaction of primary particles with light atomic nuclei.

For \(k=2\) we obtain for latitude \(31^\circ\):

\[ \frac{\varepsilon_\pi}{\overline{E}_0} = \frac{3}{1.2}\, \frac{(2.04\pm0.2)\cdot10^8\ \text{eV cm}^{-2}\text{sec}^{-1}} {(18.3\pm0.24)\cdot10^8\ \text{eV cm}^{-2}\text{sec}^{-1}} = 28\pm3\%, \]

and for the latitude difference \(51—31^\circ\):

\[ \frac{\varepsilon_\pi}{\bar E_0} = \frac{3}{1.5}\, \frac{(1.26\pm0.28)\cdot10^8\ \text{ev cm}^{-2}\text{sec}^{-1}} {(12.3\pm0.4)\cdot10^8\ \text{ev cm}^{-2}\text{sec}^{-1}} = 21\pm4.5\%. \]

Although we did not make direct measurements of the energy fluxes of the soft component at geomagnetic latitude \(0^\circ\), it is nevertheless possible to determine, with a sufficient degree of accuracy, the fraction of energy transferred to mesons by primary particles with \(\bar E_0=40\) Bev. This estimate is based on the following considerations. The mean energy of primary particles at the geomagnetic equator is \(\sim 40\) Bev. The mean energy of primary particles added in going from latitude \(0^\circ\) to latitude \(31^\circ\) is approximately \(\sim 10\) Bev. The flux of primary particles at latitude \(31^\circ\) is twice the flux of primaries at latitude \(0^\circ\). Thus, the energy flux of primary particles at latitude \(31^\circ\) is due, to the extent of \(3/4\), to particles with \(\bar E_0=40\) Bev.

Let primary particles with \(\bar E_1=10\) Bev transfer in the first act of interaction to mesons a fraction \(\delta_1\), and primaries with energy \(\bar E_2=40\) Bev a fraction \(\delta_2\) of their energy. Then

\[ 28\pm3\%= \frac{\delta_1 N_1\bar E_1+\delta_2 N_2\bar E_2} {N_1\bar E_1+N_2\bar E_2} = \frac{\delta_1\dfrac{N_1\bar E_1}{N_2\bar E_2}+\delta_2} {1+\dfrac{N_1\bar E_1}{N_2\bar E_2}}. \]

But \(\delta_1\) is obviously no less than for primary particles with \(\bar E_0=3\) Bev, i.e. \(\delta_1>20\%\); therefore

\[ \delta_2<30\pm4\%. \]

On the other hand, apparently \(\delta_1\leqslant\delta_2\); therefore, if \(\delta_1=\delta_2\), then \(\delta_2=28\%\), and consequently

\[ 28\pm3\%\leqslant\delta_2<30\pm4\%. \]

Thus, for primary particles with \(\bar E_0=40\) Bev the fraction of energy lost to the production of mesons in the first act of interaction with a light nucleus is equal to \(\sim 30\%\) of the energy of the primary particle.

It is possible to determine the value \(\dfrac{\varepsilon_\pi}{\bar E_0}\), assuming various values of \(k\). Such an assumption is, in essence, equivalent to the assumption of the production, besides \(\pi\)-mesons, of other particles which

then decay into \(\pi\)- or \(\mu\)-mesons. The results of such a calculation are given in Table 1.

Table 1

Dependence of \(\dfrac{\varepsilon_\pi}{\bar E_0}\) on the value of \(k\)

\(k\)	\(E_{\text{neutrino}}\), \(10^8\ \text{eV cm}^{-2}\text{s}^{-1}\) 31°	\(E_{\text{neutrino}}\), \(10^8\ \text{eV cm}^{-2}\text{s}^{-1}\) 51–31°	\(N\bar E_0\), \(10^8\ \text{eV cm}^{-2}\text{s}^{-1}\) 31°	\(N\bar E_0\), \(10^8\ \text{eV cm}^{-2}\text{s}^{-1}\) 51–31°	\(E_\pi\) in the first act of interaction, \(10^8\ \text{eV cm}^{-2}\text{s}^{-1}\) 31°	\(E_\pi\) in the first act of interaction, \(10^8\ \text{eV cm}^{-2}\text{s}^{-1}\) 51–31°	\(\dfrac{\varepsilon_\pi}{E_0}\) 31°	\(\dfrac{\varepsilon_\pi}{E_0}\) 51–31°
1	3.5	1.30	15.4	11.2	3.7	2.0	24%	18%
2	6.4	2.40	18.3	12.3	5.1	2.5	28%	21%
3	5.8	3.20	20.7	13.1	6.3	2.9	30%	22%
4	10.5	3.8	22.4	13.7	7.3	3.1	32.6%	23%

The table shows that the obtained average fraction of energy transferred to mesons in the first collision is extremely insensitive to the presence, among the generated particles, of still heavier mesons that decay into \(\pi\)- or \(\mu\)-mesons.

We have seen that in the entire atmosphere the energy transferred to mesons is considerably greater than in the first act of interaction of the primary particles with the nuclei of atmospheric atoms. Since in previous estimates the energy carried away by neutrinos and the energy passing into the hard (\(\mu\)-meson) component of cosmic rays were not taken into account, we can now include these two quantities in order to obtain a more correct value of the total energy transferred to \(\pi\)-mesons in the entire atmosphere. Obviously,

\[ \left. \begin{aligned} E_\pi &= E_{\text{m. k}} + E_{\text{neutrino}} + E_{\mu\ \text{ion}},\\ E_{\mu\ \text{ion}}^{31^\circ} &\approx 1.7\cdot 10^8\ \text{eV cm}^{-2}\text{s}^{-1},\\ E_{\mu\ \text{ion}}^{51^\circ-31^\circ} &\approx 0.6\cdot 10^8\ \text{eV cm}^{-2}\text{s}^{-1}. \end{aligned} \right\} \tag{13} \]

The values of \(E_{\text{neutrino}}\) are given in Table 1. Using relation (13) and all the numerical values given, we obtain that for latitude \(31^\circ\) (primary particles with energy \(\bar E_0=20\ \text{BeV}\)),

\[ \frac{E_\pi}{\bar E_0}=84\pm 2\%, \]

and for the difference of latitudes \(51-31^\circ\) (primary particles with energy \(\bar E_0\approx 3\ \text{BeV}\)),

\[ \frac{E_\pi}{\bar E_0}=45\pm 3\%. \]

Comparing these quantities with the fractions of energy transferred to mesons in a single act, we see that primary particles with $\bar E_0 \simeq 20\ \mathrm{Bev}$ in the atmosphere transfer to mesons three times more energy than in the first act of interaction. Consequently, as a result of the first interaction there remain nuclear-active particles (nucleons), which in subsequent acts of nuclear interactions generate $\pi$-mesons. Thus, we see that the primary protons are not absorbed in the first interaction. They are absorbed as a result of several collisions with atomic nuclei. It is interesting to note that the absence of single-act absorption occurs not only for particles of high energy ($\bar E_0 \simeq 20\ \mathrm{Bev}$), but also for particles of substantially lower energies ($\bar E_0 \simeq 3\ \mathrm{Bev}$). The multiple-act character of the absorption of primary particles was first established in the work$^{19}$.

For a complete characterization of the elementary act of interaction of primary particles of the indicated mean energies (3–40 $\mathrm{Bev}$) with nuclei of atoms of the atmosphere, it is necessary to determine how many secondary nucleons are formed on the average, which carry away the energy not expended by the primary particle in the first collision, and what energy is spent on processes competing with meson formation. These questions will be considered in Part II.

1.4. Generation of $\pi$-Mesons by Primary Particles with Energies $10^{11}$–$10^{12}$ ev

In order to consider the processes of meson production by primary particles with energies $10^{11}$–$10^{12}$ ev, it is necessary to use the spectrum of $\mu$-mesons in the high-energy region, obtained as a result of measurements of the intensity of the hard component at various depths underground$^{20}$, and the spectrum of primary particles in the energy region up to $10^{12}$ ev$^{21}$.

In Fig. 8 are presented the integral spectra of $\mu$-mesons and primary particles. A direct comparison of these spectra shows that from one primary particle with energy $\gg 10^{11}$ ev there are formed approximately $\dfrac{1}{40}=0.025$ $\mu$-mesons with energy $\gg 10^{11}$ ev. If we assume that all $\mu$-mesons are formed from the decay of $\pi^\pm$-mesons, then from this we immediately obtain that the probability of transferring to a $\pi^\pm$-meson energy of the order of the energy of the primary proton is very small.

To obtain quantitative characteristics of the probability of formation of $\pi^\pm$-mesons with a given energy in the interaction of high-energy primary particles, $10^{11}$–$10^{12}$ ev, it is necessary to take into account that by no means all $\pi^\pm$-mesons of such high energy have time to decay in the atmosphere and produce a high-energy $\mu$-meson. The process competing with decay is the nuclear interaction

mesons. If the decay path becomes significantly greater than the path for interaction, then most of the high-energy \(\pi\)-mesons will not decay, but will be knocked out of the particle flux because of nuclear interaction.

In our estimates we shall assume that the interaction path of high-energy \(\pi\)-mesons in air is \(\sim 65\ \mathrm{g/cm^2}\), i.e., that the effective interaction cross section is equal to the geometrical cross section of the nucleus, and that as a result of the interaction of a \(\pi\)-meson with a nucleus all the energy of the incident meson is transferred to secondary mesons with energy substantially less than that of the incident meson, i.e., regeneration of mesons with energy of the same order as that of the incident meson does not occur.

Fig. 8. Integral spectra: 1 — of primary particles; 2 — of \(\pi\)-mesons generated in the atmosphere; 3 — of \(\mu\)-mesons at sea level.

We shall assume that all \(\mu\)-mesons observed at sea level are products of the decay of \(\pi^\pm\)-mesons.

The assumptions listed ensure the maximum number of \(\pi\)-mesons generated by high-energy particles.

The equation describing the absorption of \(\pi\)-mesons in the atmosphere has the form:

\[ \frac{\partial N_\pi(x,E_\pi)}{\partial x} = -\frac{N_\pi(x,E_\pi)}{L_{\mathrm{вз}}} -\frac{A}{x}N_\pi(x,E_\pi) + \frac{N_0(E>E_\pi)\,\nu(E_\pi)e^{-x/L_\pi}}{L_{\mathrm{вз}}}, \tag{14} \]

Here \(N_\pi(x,E_\pi)\) is the flux of \(\pi^\pm\)-mesons with energy \(E_\pi\) at atmospheric depth \(x\ \mathrm{g/cm^2}\). The first term of this equation takes into account the nuclear absorption of mesons, the second—their decay, and the third—the generation of \(\pi\)-mesons by primary particles with energy \(E \geq E_\pi\), whose intensity at the point \(x\) is equal to \(N_0(E \geq E_\pi)e^{-x/L_\Pi}\) (where \(L_\Pi\) is the absorption path in the atmosphere of the generating particles). \(\nu(E_\pi)\) is the mean number of \(\pi\)-mesons with energy \(E_\pi\) generated by all primary particles with energy \(>E_\pi\).

The solution of equation (14) has the form:

\[ N_\pi(x,E_\pi)=\frac{N_0(E\geq E_\pi)\nu(E_\pi)}{L_{\mathrm{вз}}} \int_0^x \left(\frac{\xi}{x}\right)^A e^{-\xi/L_\Pi}e^{-(x-\xi)/L_{\mathrm{вз}}}\,d\xi \tag{14a} \]

(here, as in what follows, the energy losses to ionization of the atmosphere by \(\pi\)- and \(\mu\)-mesons are not taken into account, in view of their smallness in comparison with the energy of the particles themselves).

In the case when

\[ A=\frac{p_0m_\pi c^2}{\rho_0 c\tau_\pi E_\pi} =\frac{1.8\cdot 10^{11}\ \mathrm{eV}}{E_\pi}\gg 1, \]

the expression

\[ \left(\frac{\xi}{x}\right)^A \]

differs from 0 only at points \(\xi \simeq x\) (since \(\xi/x\leq 1\)), where

\[ e^{-(x-\xi)/L_{\mathrm{вз}}}\simeq 1. \]

Consequently, in this case

\[ N_\pi(x,E_\pi)= \frac{N_0(E\geq E_\pi)\nu(E_\pi)}{L_{\mathrm{вз}}} e^{-x/L_\Pi}\frac{x}{A+1}. \tag{15} \]

If the number of \(\pi\)-mesons at each point of the atmosphere is known, then it is easy to determine how many \(\mu\)-mesons are formed in a layer of thickness \(p\ \mathrm{g/cm^2}\):

\[ N_\mu(\bar E_\mu)=\int_0^p N_\pi(x,E_\pi)\,w(x,E_\pi)\,dx, \tag{16} \]

where \(w(x,E_\pi)\) is the probability of decay of a \(\pi\)-meson possessing energy \(E_\pi\).

In calculating the form of the spectrum of \(\mu\)-mesons, as noted in paper \(^{22}\), in a first approximation one may neglect the spread of the momenta of \(\mu\)-mesons arising from the decay of \(\pi\)-mesons with a given momentum.

\[ w(x,E_\pi)=\frac{A}{x}, \]

therefore, for \(A\gg 1\),

\[ N_\mu(\bar E_\mu)= \frac{N_0(E\geq E_\pi)\nu(E_\pi)}{L_{\mathrm{вз}}} \frac{A}{A+1}\int_0^p e^{-x/L_\Pi}\,dx= \]

\[ =\frac{L_\Pi N_0(E\geq E_\pi)\nu(E_\pi)}{L_{\mathrm{вз}}} \left(1-e^{-x/L_\Pi}\right). \tag{17} \]

i.e., in the case when \(E_{\pi} \ll 1.8\cdot 10^{11}\) eV, the number of \(\mu\)-mesons with mean energy \(\overline{E}_{\mu}=\dfrac{1}{1.3}E_{\pi}\), formed from \(\pi\)-mesons with energy \(E_{\pi}\), will be the same as if these \(\mu\)-mesons arose directly in the interactions of the primary particles.

At high \(\pi\)-meson energies another case is realized: \(A \lesssim 1\). To analyze it, we replace, in expression (14a), \(\dfrac{\xi}{x}\) by \(t\). Then

\[ N_{\pi}(x,E_{\pi})= \frac{N_{0}(E\geqslant E_{\pi})\,\nu(E_{\pi})}{L_{\mathrm{int}}}\, x\int_{0}^{1} e^{-\frac{xt}{L_{\mathrm{p}}}}\, t^{A}\, e^{-\frac{x}{L_{\mathrm{int}}}(1-t)}\,dt, \]

and the number of \(\mu\)-mesons formed in an atmospheric layer \(p\ \mathrm{g/cm^{2}}\) will be, according to (16),

\[ N_{\mu}(\overline{E}_{\mu},p)= \]

\[ =\frac{N_{0}(E\geqslant E_{\pi})\,\nu(E_{\pi})}{L_{\mathrm{int}}}\, A\int_{0}^{p} dx \int_{0}^{1} e^{-\frac{x}{L_{\mathrm{p}}}t}\, t^{A}\, e^{-\frac{x}{L_{\mathrm{int}}}(1-t)}\,dt . \]

Experiment gives\(^{23,24}\) that up to \(E\sim 10^{12}\) eV \(L_{\mathrm{p}}\simeq 120\ \mathrm{g/cm^{2}}\). Taking into account that \(\dfrac{L_{\mathrm{p}}}{L_{\mathrm{int}}}\simeq 2\), we obtain:

\[ \int_{0}^{p} e^{-\frac{xt}{L_{\mathrm{p}}}}\, e^{-\frac{x}{L_{\mathrm{int}}}(1-t)}\, \frac{dx}{L_{\mathrm{int}}} = \int_{0}^{p} e^{-\frac{x}{L_{\mathrm{int}}}\left(1-t+\frac{L_{\mathrm{int}}}{L_{\mathrm{p}}}t\right)} \frac{dx}{L_{\mathrm{int}}} = \]

\[ = \frac{2}{2-t} \left[ 1-e^{-\frac{p}{L_{\mathrm{int}}}\left(1-\frac{t}{2}\right)} \right]. \]

We are interested in the number of \(\mu\)-mesons at sea level, i.e., at \(p=1000\ \mathrm{g/cm^{2}}\); consequently, for all values of \(t\) lying in the interval from 0 to 1,

\[ e^{-\frac{1000}{60}\left(1-\frac{t}{2}\right)}\ll 1 \]

and

\[ N_{\mu}(\overline{E}_{\mu},p=1000\ \mathrm{g/cm^{2}}) = N_{0}(E\geqslant E_{\pi})\,\nu(E_{\pi})\,A \int_{0}^{1}\frac{2t^{A}}{2-t}\,dt. \]

The number of \(\pi\)-mesons with energy \(E_{\pi}\) produced in the entire atmosphere is equal to

\[ N_{\pi}^{0}(E_{\pi})= \int_{0}^{1000} N_{0}(E\geqslant E_{\pi})\,\nu(E_{\pi})\, e^{-\frac{x}{L_{\mathrm{p}}}}\frac{dx}{L_{\mathrm{int}}} = 2N_{0}(E\geqslant E_{\pi})\,\nu(E_{\pi}). \]

Therefore the ratio of the number of μ-mesons with energy \(E_\mu=\dfrac{E_\pi}{1.3}\) to the number of \(\pi\)-mesons born in the atmosphere with energy \(E_\pi\) is equal to

\[ \frac{N_\mu\left(E_\mu=\dfrac{E_\pi}{1.3}\right)} {N_\pi^0(E_\pi)} = A\int_0^1 \frac{t^A\,dt}{2-t} = f(A); \]

whence

\[ N_\pi^0(E_\pi)= \frac{N_\mu\left(E_\mu=\dfrac{E_\pi}{1.3}\right)} {f(A)}. \tag{18} \]

The expansion of the function \(f(A)\) can be obtained easily by integrating by parts the expression

\[ \int_0^1 \frac{t^A}{2-t}\,dt: \]

\[ f(A)=\frac{A}{A+1}\left[ 1-\frac{1!}{A+2} +\frac{2!}{(A+2)(A+3)} - \frac{3!}{(A+2)(A+3)(A+4)} +\cdots \right]. \]

Using formula (18) and the integral spectrum of μ-mesons presented in Fig. 8, one can obtain the total number of \(\pi\)-mesons with energy \(>E_\pi\) born in the entire atmosphere. The result of this calculation is shown in Fig. 8 by the dashed line.

To estimate the upper limit of the number of \(\pi\)-mesons of a given energy produced by primary particles of one or another energy, let us assume that all \(\pi\)-mesons arising in the atmosphere with energy \(>E_\pi\) were generated by primary particles with energy \(E_0>E_1>E_\pi\) (obviously, such an assumption overestimates the sought number of \(\pi\)-mesons, since part of the \(\pi\)-mesons with \(E>E_\pi\) could have been generated by primary particles with energies from \(E_\pi\) to \(E_1\)). Then

\[ N_\pi^0(E>E_\pi)>2N_0(E_0>E_1)\,\nu(E>E_\pi) \]

and hence

\[ \nu(E>E_\pi)< \frac{N_\pi^0(E>E_\pi)} {2N_0(E>E_1)}. \tag{19} \]

Assigning values to the energy of the primary particles \(E_1\) and to the energy of the generated \(\pi\)-mesons \(E_\pi\), and using the data given in Fig. 8, one can determine from formula (19) the upper limit of the number of \(\pi\)-mesons of different energies that can be born in a single interaction.

with the light nucleus of the primary particle with energy \(> E_1\). The results of such calculations are given in Table II.

Table II

Number of \(\pi\)-mesons \(\nu\,(E \geq E_\pi)\) produced with energy \(E \geq E_\pi\) by primary particles whose energy \(E_{\mathrm{prim}} > E_1\)

| \multicolumn{2}{c}{\(E_1 = 50\) Bev} | \multicolumn{2}{c}{\(E_1 = 300\) Bev} | \multicolumn{2}{c}{\(E_1 = 1000\) Bev} |
|---:|---:|---:|---:|---:|---:|
| \(E_\pi\), Bev | \(\nu(E \geq E_\pi)\) | \(E_\pi\), Bev | \(\nu(E \geq E_\pi)\) | \(E_\pi\), Bev | \(\nu(E \geq E_\pi)\) |
| 6 | \(< 3.6\) | 16 | \(< 7.4\) | 42 | \(< 7.5\) |
| 9 | \(< 1.2\) | 29 | \(< 2.3\) | 68 | \(< 3.1\) |
| 16 | \(< 0.34\) | 42 | \(< 1.0\) | 130 | \(< 1.3\) |
| 29 | \(< 0.10\) | 68 | \(< 0.44\) | 260 | \(< 0.44\) |
| 42 | \(< 0.048\) | 130 | \(< 0.17\) | 390 | \(< 0.24\) |
| 50 | \(< 0.032\) | 300 | \(< 0.05\) | 520 | \(< 0.15\) |
| | | | | 650 | \(< 0.10\) |
| | | | | 1000 | \(< 0.053\) |

Two conclusions follow from Table II:

1) \(\pi\)-mesons with energy equal to the energy of the primary particle are generated with probability not exceeding \(5\%\).

2) A \(\pi\)-meson with a given energy \(E_\pi\) is generated mainly by a primary particle possessing an energy approximately 8 times greater than \(E_\pi\). Therefore, in a flux of nuclear-active particles with energy \(\sim 10^{10}\) ev, observed at any depth in the atmosphere, \(\pi\)-mesons will constitute a negligible fraction (\(\sim 5\%\) of the nucleon fraction), even if the energies of the \(\pi\)-mesons are large enough that their decay path is much greater than the interaction path.

Information on the mean energy losses of nucleons in their collisions with light nuclei in the energy range \(10^{11}\)—\(10^{12}\) ev can be obtained from consideration of the absorption of high-energy particles in the atmosphere (see Part III).

II. FORMATION OF NUCLEAR DISINTEGRATIONS IN THE ATMOSPHERE BY PRIMARY PARTICLES OF DIFFERENT ENERGIES

2.1. Method of study

In the experiments of 1947—1948\({}^{25,26}\), carried out at geomagnetic latitude \(51^\circ\), it was found that the ionization increases from sea level to the stratosphere by a factor of 150, whereas the number of particles increases by a factor of 90. Consequently, in the stratosphere the mean specific ionizing

the ionizing capacity of cosmic particles is approximately 70% higher than the average ionizing capacity of relativistic particles present at sea level.

The significant increase in the average specific ionizing capacity of cosmic particles in the stratosphere, as compared with sea level, indicates the appearance in the stratosphere of slow, strongly ionizing particles.

It is natural to associate these strongly ionizing particles with products of nuclear disintegrations—protons, deuterons, tritons, $\alpha$-particles, flying out of the disintegrating nuclei. If such an interpretation of the nature of the “excess” ionization $I_{\mathrm{s.i.}}(p)$ (i.e., ionization which cannot be explained by relativistic particles, but is produced by strongly ionizing particles) is correct, then this suggests a method for detecting it: it must be produced in the ionization chamber in the form of ionization bursts of greater or lesser magnitude, i.e., instantaneous increases of ionization above some mean level, caused by the passage through the chamber of a large number of relativistic particles.

If the “excess ionization” is produced by products of nuclear disintegrations, then the quantity

$$ 32.5 \int_{0}^{\infty} I_{\mathrm{s.i.}}^{\lambda}(p)\,dp $$

determines the energy of all strongly ionizing particles produced in the atmosphere as a result of the disintegration of nuclei of air atoms. If to this quantity one further adds the binding energy of these particles in the nucleus $E_{\mathrm{bind}}^{\lambda}$, then we obtain the total energy $E_{\mathrm{n.d.}}^{\lambda}$, transferred in the whole atmosphere to nuclear disintegrations by primary particles at geomagnetic latitude $\lambda$:

$$ E_{\mathrm{n.d.}}^{\lambda} = 32.5\,\text{eV} \int_{0}^{1000} I_{\mathrm{s.i.}}^{\lambda}(p)\,dp + E_{\mathrm{bind}}^{\lambda}. \tag{20} $$

Dividing $E_{\mathrm{n.d.}}^{\lambda}$ by the number of primary particles $N_{0}^{\lambda}$ falling on the boundary of the atmosphere, we obtain $\varepsilon_{\mathrm{n.d.}}^{\lambda}$—the amount of energy lost to nuclear disintegrations in the atmosphere by one primary particle at geomagnetic latitude $\lambda$, i.e.

$$ \varepsilon_{\mathrm{n.d.}}^{\lambda} = \frac{E_{\mathrm{n.d.}}^{\lambda}}{N_{0}^{\lambda}}. $$

Knowing $\varepsilon_{\mathrm{n.d.}}^{\lambda}$ and the character of the energy losses for the formation of mesons by primary particles of different energies, one can determine the average multiplicity of high-energy secondary nucleons arising in the act of the first interaction of the primary particles (see 2.5).

2.2. Apparatus and Measurement Results

The study of the formation in the atmosphere of heavy particles—products of nuclear disintegrations—requires the measurement not only of the total number of bursts \(N_{\tau}\), but also of their distribution by magnitude, i.e.

\[ \frac{\partial N_{\tau}(p,J_{\tau})}{\partial J_{\tau}}. \]

Knowing \(\frac{\partial N_{\tau}}{\partial J_{\tau}}\), one can readily determine the ionization \(I_{\tau}^{\lambda}\) produced by the products of nuclear disintegrations:

\[ I_{\tau}^{\lambda}(p)=\int_{J_{\min}}^{\infty} J_{\tau}\frac{\partial N_{\tau}^{\lambda}}{\partial J_{\tau}}\,dJ_{\tau}. \tag{21} \]

Here \(J_{\tau}\) is the magnitude of the ionization burst, expressed as the number of ion pairs.

We developed two types of instruments which were used in the study of bursts. One version of the instrument consisted of a pulse ionization chamber and an amplifier. The second version of the instrument, in addition to the chamber and amplifier, contained 25 counters surrounding the chamber and connected into 14 hodoscope cells (Fig. 9). The control signal was generated by an ionization burst exceeding a certain minimum threshold \(J_{\min}\).

The method for transmitting the amplitudes of ionization bursts, proposed by A. E. Chudakov, was the same in both instruments and amounted to the following: the amplified signal from the ionization chamber was transformed into a rectangular pulse, which was modulated at an audio frequency and transmitted over the air in the same way as was done in the previously described instruments for studying the transition effect. Following the transmission of the pulse from the chamber, data on the operation of the hodoscope counters were transmitted over the air. The “polling” of the triggering of the hodoscope cells was carried out with the aid of a mechanical commutator and a continuously rotating scanner, which successively connected the hodoscope cells to the output of the radio circuit\({}^{44}\) (see Fig. 9). The radio signals received on the ground were fed to the input of a cathode oscilloscope and photographed on continuously moving film.

In instruments of both types, identical ionization chambers were used. They were aluminum spheres with wall thickness \(1.0\text{–}1.5\ \mathrm{mm}\) and diameter \(15\ \mathrm{cm}\). The diameter of the spherical collecting electrode was \(2\ \mathrm{cm}\). The chambers were filled with spectrally pure argon to a pressure of \(4.0\text{–}4.5\ \mathrm{atm}\) and operated in electron-pulse mode. A high voltage of \(1000\text{–}1100\ \mathrm{V}\) was applied to the collecting electrode of the chamber through \(100\ \mathrm{M\Omega}\).

A brass tube was introduced into the chamber; in it there was a Po α-particle source, used for calibrating the chamber and for monitoring the purity of the gas filling the chamber. The α-particle source was introduced into the working volume of the chamber by means of a small solenoid fitted onto the tube. Through this same tube the chamber was evacuated and filled with argon.

Fig. 9. Radio circuit of the apparatus for studying ionization bursts using a chamber surrounded by Geiger counters.

To monitor the radio-engineering amplification factor during the flight of the instrument, the amplifier was provided with a circuit that periodically (6–8 times per minute) applied to the amplifier input a calibration signal of standard magnitude.

The constructed apparatus made it possible to obtain results with an error not exceeding 10% over a wide range of measured bursts. The measurement results showed that the discrepancy between the data from different flights did not exceed 10%.

The resolving time of the hodoscope cell, equal to \((4.3 \pm 0.9)\times 10^{-5}\) sec, ensured relatively small distortions in the operation of the counters due to accidental coincidences of a shower in the chamber and a discharge in one of the counters.

The number of accidental coincidences was checked in flight by cases of coincidence of a discharge in any of the counters with a calibration signal (such coincidences, by their nature, could only be accidental).

Before flight the apparatus was carefully calibrated, i.e., an unambiguous relation was established between the magnitude of the ionization in the chamber, the magnitude of the input signal, and the duration of the output signal[^11].

The special feature of our measurements, in comparison with what had been done earlier, is that we succeeded in measuring showers beginning with a very small value, \(10^4—2.5\cdot 10^4\) ion pairs, and in using hodoscope counters, which made it possible to determine the nature of the recorded showers.

The study of ionization showers was carried out at geomagnetic latitudes \(0;\ 31,\ 38\) and \(51^\circ\), in the altitude range from 5 to \(20—28\) km. At geomagnetic latitudes 31 and \(51^\circ\), apparatus of the two types indicated above was used; therefore the most detailed information on the nature of the showers was obtained at these two latitudes. The data obtained on the spectra of showers at different altitudes are presented in Fig. 10, borrowed from work[^27].

To clarify the nature of the showers, let us consider the data (Table III) on the operation of the hodoscope counters surrounding the ionization chamber. The data were obtained at latitude \(51^\circ\) and refer to showers with ionization \(>27000\) ion pairs.

Analogous results[^27] were also obtained at latitude \(31^\circ\).

Calculations[^27] show that, for an isotropic angular distribution of particles, the probability that a particle enters the chamber while missing the hodoscope counters is 0.12. Therefore, if all showers were caused by charged particles, then showers without operation of the counters could account for no more than 12% of the total number of showers, and with operation of the counters, 88%. In reality, as is seen from Table III, the cases in which showers are accompanied by operation of the counters amount to 33%. Let us suppose that all these cases are caused by a charged particle arriving from the air. This assumption obviously overestimates the number of showers produced by the charged component of cosmic rays. However, even under this assumption we find that, at altitudes of \(11—21\) km, at least 62% of all showers are caused by the nonionizing component.

It is clear that these showers cannot be showers of relativistic particles, since the threshold for triggering of the apparatus corresponded to an ionization of 27000 ion pairs, which is equivalent to the passage through the chamber, along the mean chord, of 5.6 relativistic particles.

Fig. 10. Integral spectra of ionization bursts at different altitudes, obtained at geomagnetic latitudes 0, 31, and 51°. The abscissa shows the burst magnitude (number of ion pairs), and the ordinate shows the number of bursts in 1 minute.

It is quite incredible that at such a minimal shower density not a single one of the counters, covering about 90% of the total solid angle, should have fired. Consequently, these pulses represent cases of the occurrence within the chamber volume of heavy, strongly ionizing particles incapable of passing through the chamber wall and the counter wall, and produced by a neutral particle, evidently a neutron.

Table III

Height	Number of counters that fired	Number of pulses in the chamber with the given number of counters that fired	%	After allowing for random coincidences
\(H = 21\ \mathrm{km}\)	0	347	54%	62%
\(H = 21\ \mathrm{km}\)	1	183	28.8%	24.8%
\(H = 21\ \mathrm{km}\)	2	64	10%	6%
\(H = 21\ \mathrm{km}\)	\(>3\)	46	7.2%	7.2%
\(H = 21\ \mathrm{km}\)	Total...	640	100%	100%
\(H = 16\ \mathrm{km}\)	0	534	59%	67%
\(H = 16\ \mathrm{km}\)	1	227	25%	21%
\(H = 16\ \mathrm{km}\)	2	94	10.6%	6.7%
\(H = 16\ \mathrm{km}\)	\(>3\)	48	5.4%	5.3%
\(H = 16\ \mathrm{km}\)	Total...	903	100%	100%
\(H = 11\ \mathrm{km}\)	0	381	64.7%	68.7%
\(H = 11\ \mathrm{km}\)	1	152	25.6%	23.6%
\(H = 11\ \mathrm{km}\)	2	36	6.1%	4.1%
\(H = 11\ \mathrm{km}\)	\(>3\)	21	3.6%	3.6%
\(H = 11\ \mathrm{km}\)	Total...	590	100%	100%

Let us now consider what is represented by the pulses accompanied by the operation of counters. To do this, let us turn our attention to the distribution of counter firings in different ranges of the magnitude of the recorded pulses. These data, relating to an altitude of \(16\ \mathrm{km}\) at latitude \(51^\circ\), are presented in Table IV (at other altitudes the distribution of counter firings as a function of pulse magnitude has the same ...

the same character as that given in Table IV. Therefore the data presented should be regarded as typical).

Table IV

Ionization interval in the chamber, in \(10^3\) ion pairs	\(27.0\text{–}37.4\)	\(37.4\text{–}54.0\)	\(54.0\text{–}69.2\)	\(69.2\text{–}84.6\)	\(84.6\text{–}100\)	\(>100\)	Total
Total number of pulses of this magnitude	114	244	138	87	34	286	903
Number of cases of operation of one counter	30	62	36	27	10	62	227
Number of cases of operation of any two counters	14	28	12	6	3	31	94
Number of cases of operation of any three or more counters	10	12	4	3	3	16	48

If it is assumed that all pulses accompanied by the operation of three or more counters are caused by showers from relativistic particles (such an assumption obviously overestimates the role of showers), then even then we find that showers produce no more than 5% of all pulses. Thus, no less than 95% of all pulses are caused by strongly ionizing particles. Most of them arise in the walls of the chamber and in the gas filling the chamber, and have ranges less than \(1.1\ \mathrm{g/cm^2}\) Al (the total thickness of the chamber wall and the counter wall). As is seen from Table IV, about 30% of all pulses are accompanied by the operation of one and two counters. A detailed analysis of the operation of these counters for pulses producing more than 27,000 ion pairs in the chamber shows\(^{11}\) that single protons entering the chamber from the air can account for no more than 40% of the pulses accompanied by the operation of one counter (i.e., no more than \(0.4 \cdot 25\% = 10\%\) of all pulses) and no more than 15% of all pulses accompanied by the operation of two counters (i.e., no more than 1% of all pulses). The remaining 20% of all pulses, which are accompanied by the operation of one and two counters, cannot be explained either by the fact that they are produced by strongly ionizing single particles (mainly protons) entering the chamber from the air, or by showers from relativistic-

...particles (since the mean momentum in these showers corresponds to the ionization produced by 15 relativistic particles, while the number of counters triggered is 1–2).

We arrive at the conclusion that 20% of all showers (accompanied by the operation of 1–2 counters) are also produced by strongly ionizing particles arising as a result of nuclear disintegrations occurring in the walls and gas of the chamber and in the walls of the counters.

Thus, it may be asserted that at altitudes up to 21 km ($p = 47\ \mathrm{g/cm^2}$) no fewer than 80% of all showers are caused by nuclear disintegrations occurring in the walls and gas of the ionization chamber, and no fewer than $3/4$ of all disintegrations are generated by the neutral component of cosmic rays—neutrons. Experiments$^{28}$ carried out with photographic plates at geomagnetic latitudes $54^\circ$ and $28^\circ$ show that 72% of all disintegrations are produced by neutrons.

An analogous result was also obtained at geomagnetic latitude $31^\circ$, where no fewer than 75% of all disintegrations are caused by neutrons.

In experiments carried out at latitude $31^\circ$, it was possible to record ionization showers with an ionization threshold in the chamber greater than 10,000 ion pairs. It was found that, in showers with ionization less than 27,000 ion pairs, a large role is played by single strongly ionizing particles entering the chamber from the air. From the measured ionization and the minimum range that a particle producing a shower accompanied by the firing of two diametrically opposite counters must have, in these measurements it was possible to estimate the mass of the particles$^{27}$. It turned out to be close to the mass of the proton. In this way it was shown that the single strongly ionizing particles causing showers in the chamber and arriving from the air are, for the most part, protons.

If one takes into account that strongly ionizing particles arriving from the air, before entering the chamber, must have a minimum range of not less than $1.1\ \mathrm{g/cm^2}$ Al, then, according to the terminology adopted in work with photographic plates$^{4}$, they must be assigned to particles forming “gray” tracks. In work$^{4}$ it is shown that the principal fraction of “gray” tracks arising in nuclear disintegrations of the nuclei of the photoemulsion consists of protons.

Thus, the study of showers shows that:

1) part of the showers is produced by strongly ionizing single particles arriving from the air; these particles have the same nature as the heavy particles arising in the “stars”;

2) the remaining showers are produced by strongly ionizing particles arising as a result of disintegrations of the atomic nuclei of the material of the walls and gas of the chamber.

We have already shown earlier that the greater part of the particles generating showers in the chamber and “stars” in photographic plates are neutrons.

If the bursts and “stars” are caused by one and the same component, then the altitude dependence of these phenomena must be the same. Indeed, Figs. 7 and 8 in paper\(^6\) show that bursts and “stars” have the same altitude dependence and latitude effect; consequently, one may write that the number of bursts \(N_{\mathrm{t}}\) is proportional to the number of “stars,” \(N_{\mathrm{z}}\):

\[ N_{\mathrm{t}}^{\lambda}(p)=\varkappa N_{\mathrm{z}}^{\lambda}(p). \tag{22} \]

All the features of the recorded bursts prove that they are products of nuclear disintegrations. Therefore, if we measured all the ionization produced by all bursts, we could determine \(E_{\text{n.d.}}\). In reality, however, we record bursts beginning from a certain threshold—from some minimum value \(J_{\min}\). Therefore the quantity

\[ I_{\mathrm{t}}^{\lambda}(p)= \int\limits_{J_{\min}}^{\infty} J_{\mathrm{t}}\, \frac{\partial N^{\lambda}(J_{\mathrm{t}},p)}{\partial J_{\mathrm{t}}}\, dJ_{\mathrm{t}} \]

does not give the ionization produced by all strongly ionizing particles arising in nuclear disintegrations. It may be expected, however, that as the threshold of the recorded bursts \(J_{\min}\) is decreased, the quantity \(I_{\mathrm{t}}^{\lambda}(p)\) will tend to \(I_{\text{s.i.}}^{\lambda}(p)\). The results of measurements carried out at latitudes \(51\) and \(31^\circ\) show that the ratio

\[ \frac{I_{\text{s.i.}}^{\lambda}(p)}{I_{\mathrm{t}}^{\lambda}(p)} \]

does not depend, within the errors, either on the atmospheric depth \(p\) or on the geomagnetic latitude \(\lambda\), i.e.,

\[ \frac{I_{\text{s.i.}}^{\lambda}(p)}{I_{\mathrm{t}}^{\lambda}(p)} =\mathrm{const}=\alpha, \tag{23} \]

where \(\alpha=2.6\) for \(J_{\min}=27000\) ion pairs. If the threshold of the recorded bursts is lowered to \(J_{\min}=10000\) ion pairs, then \(\alpha\) decreases to \(1.6\). Consequently, there is reason to suppose that the quantity \(I_{\text{s.i.}}^{\lambda}(p)\) is determined mainly by the ionization produced in the atmosphere by all strongly ionizing products of nuclear disintegrations.

Thus, at those latitudes where we have measured the total ionization \(I^{\lambda}(p)\) and the number of particles \(N^{\lambda}(p)\), the value

\[ I_{\text{s.i.}}^{\lambda}(p)= \bigl(I^{\lambda}(p)-83\,N^{\lambda}(p)\bigr) \frac{\text{ion pairs}}{\text{cm}^{3}} \]

can be determined. At the same latitudes (\(0\) and \(38^\circ\)), where these data are lacking, we shall determine \(I_{\text{s.i.}}^{\lambda}(p)\) using relation (23), taking \(\alpha=2.6\) for \(J_{\min}=27000\) ion pairs.

2.3. Formation, by Primary Particles of Different Energies, of Secondary Particles Causing Nuclear Disintegrations (“Stars”)

It is easy to show that at an altitude of 20 km, in the flux of particles generating “stars,” primary particles make up no more than 15%, and, consequently, about 85% of the particles are of secondary origin.

We may write that the entire flux of particles of the star-generating component \(S^\lambda(p)\) at depth \(p\) and geomagnetic latitude \(\lambda\) consists of \(N_{\text{prim}}^\lambda(p)\)—primary particles that have reached the given depth without interaction—and \(N_{\text{sec}}^\lambda(p)\)—secondary particles that have arisen in the layer \(p\) as a result of the interaction of primary particles:

\[ S^\lambda(p)=N_{\text{prim}}^\lambda(p)+N_{\text{sec}}^\lambda(p). \]

Experiment shows (Fig. 11) that the latitude effect of the number of prongs is the same as the latitude effect of the number of primary cosmic particles \(N_{\text{prim}}^\lambda(0)\), i.e.,

\[ \frac{S^\lambda(p)}{N_{\text{prim}}^\lambda(0)}=\text{const}=a. \]

Consequently,

\[ a=\frac{N_{\text{prim}}^\lambda(p)}{N_{\text{prim}}^\lambda(0)} +\frac{N_{\text{sec}}^\lambda(p)}{N_{\text{prim}}^\lambda(0)}. \]

But at \(p \simeq 50\ \text{g}/\text{cm}^2\) (altitude about 20 km above sea level)

\[ \frac{N_{\text{prim}}^\lambda(p)}{N_{\text{prim}}^\lambda(0)}\simeq 0.15, \]

therefore

\[ \frac{N_{\text{sec}}^\lambda(p)}{N_{\text{prim}}^\lambda(0)} = a-0.15 = \]

\[ = \text{const}\,(a \geqslant 1). \]

Fig. 11. Latitude effect at a depth \(p=40\ \text{g}/\text{cm}^2\) of the number of prongs (open circles) and of the ionization produced by these prongs (filled circles). The solid curve shows the latitude effect of primary particles, according to measurements\[^{16}\]. Along the ordinate is plotted the ratio of the measured quantity at latitude \(\lambda\) to its value at latitude \(0^\circ\).

We have found that in a thin layer of the atmosphere at different geomagnetic latitudes, one primary particle produces the same number of secondary particles that subsequently give nuclear disintegrations (“stars”). In other words, \(\nu\)—the mean number of star-generating particles—

... generating component arising in the interaction of primary cosmic particles with a light nucleus—turns out to be independent of the energy of these primary particles in the range of their mean energies \(3\)—\(40\) Bev.

The value \(\bar{\nu}\) can be determined in the following way. Suppose that at geomagnetic latitude \(\lambda\), in an atmospheric column of cross section \(1\ \text{cm}^2\), \(n_3^\lambda\) “stars” are formed in one second. Dividing \(n_3^\lambda\) by the number \(N_{\text{prim}}^\lambda(0)\) of primary particles incident on the boundary of the atmosphere, we obtain the number of nuclear disintegrations \(n_{\text{dis}}^\lambda\) which one primary particle produces in the whole atmosphere. How do these \(n_{\text{dis}}^\lambda\) disintegrations arise?

From the character of the collisions leading to the formation of mesons, clarified in Part I, one may assert that after the first collision a considerable part of the energy is retained in secondary nucleons, which in turn, upon collision with nuclei, are capable of generating mesons, etc.; i.e., the primary particles and their nuclear-active progeny of high energy undergo several interactions in the atmosphere, \(n_{\text{int}}^\lambda\). Naturally, at each such interaction there will arise the disintegration of that nucleus with which the interaction has occurred. For this reason alone \(n_{\text{int}}^\lambda\) disintegrations will already be produced in the atmosphere (per one primary particle). However, at each such interaction secondary particles of the star-generating component will also be produced, each of which can then cause the disintegration of the nucleus with which it collides.

We have seen that the number of secondary particles \(\bar{\nu}\) arising in an interaction does not change on the average, at least as long as the energy of the nucleon causing the interaction is not less than \(1.5\) Bev (the minimum energy of primary particles at latitude \(51^\circ\)).

Therefore, if we consider interactions in the atmosphere of particles with energy greater than \(3\) Bev (which certainly ensures for us the independence of \(\bar{\nu}\) from the energy of the incident particle), then the total number of disintegrations that will be produced in the atmosphere by one primary particle will be equal to

\[ n_{\text{dis}}^\lambda = n_{\text{int}}^\lambda = n_{\text{int}}^\lambda + \bar{\nu} n_{\text{int}}^\lambda = (1+\bar{\nu}) n_{\text{int}}^\lambda . \]

Hence

\[ \bar{\nu} = \frac{n_{\text{dis}}^\lambda}{n_{\text{int}}^\lambda} - 1 . \tag{24} \]

Incidentally, it should be noted that if \(\bar{\nu}\) does not depend on energy, then its value should not depend on the geomagnetic latitude \(\lambda\).

Let us consider the experimental data. In order to compute \(n_{\mathrm{split}}^\lambda\), it is first necessary to find

\[ n_3^\lambda=\int_0^{1000} N_3^\lambda(p)\,dp. \]

But

\[ N_3^\lambda(p)=\frac{1}{\chi}\,N_T^\lambda(p). \]

Therefore

\[ n_3^\lambda=\frac{1}{\chi}\int_0^{1000} N_T^\lambda(p)\,dp. \]

The coefficient \(\frac{1}{\chi}\) can be determined if one compares the number of “stars” in \(1\ \mathrm{cm}^3\) of emulsion and the number of bursts at some one altitude. According to the data of \(^{28}\), at latitude \(54^\circ\), at altitudes where the pressure is \(15\text{--}50\ \mathrm{g}/\mathrm{cm}^2\), the number of “stars” in nuclear emulsion is \(2400\ \mathrm{cm}^{-3}\ \mathrm{day}^{-1}\) (in this case “stars” with number of rays \(N_h \geq 3\) were recorded). If, using the data of \(^{29}\), one estimates the possible number of “stars” with \(N_h \geq 0\), one obtains \(4500\) “stars” per \(\mathrm{cm}^{-3}\ \mathrm{day}^{-1}\). At latitude \(51^\circ\) and altitude \(20\ \mathrm{km}\) we record \(300\) bursts per minute. From these data one can find \(\chi\).

The result of calculating \(n_3^\lambda\) at different geomagnetic latitudes is given in Table V in the second row, while in the fourth row of the same table the value of \(n_{\mathrm{split}}^\lambda\) is given. As is seen from the table, \(n_{\mathrm{split}}^\lambda\) increases sharply with increasing energy of the primary particle. Thus, for primary particles with \(\bar E_0=3.3\ \mathrm{Bev}\), \(n_{\mathrm{split}}\simeq 5.6\), while for primary particles with \(\bar E_0=40\ \mathrm{Bev}\), \(n_{\mathrm{split}}=17\).

It is obvious that the increase in the number of splittings per primary particle occurs because, with increasing \(\bar E_0\), the number of interactions in the atmosphere undergone by high-energy nuclear-active nucleons increases. That this is so is evident not only from the fact that, with increasing \(\bar E_0\), the number of acts in which mesons are generated in the atmosphere increases (see Part I), but also from the altitude dependence of the number of protons with energy greater than \(\sim 3\ \mathrm{Bev}\), obtained in \(^{16}\) at different geomagnetic latitudes.

To obtain the total number of interactions \(n_{\mathrm{int}}^\lambda\) undergone in the atmosphere by primary and secondary high-energy nucleons, it is necessary to know how their flux changes with atmospheric depth. If \(S^\lambda(p)\) is the flux of all nuclear-active nucleons at depth \(p\ \mathrm{g}/\mathrm{cm}^2\), then

\[ n_{\mathrm{int}}^\lambda= \frac{1}{N_{\mathrm{prim}}^\lambda(0)\,L_{\mathrm{int}}} \int_0^\infty S^\lambda(p)\,dp. \tag{25} \]

The experiments\({}^{16}\) give only the change with altitude

\[ N^{\lambda}_{\text{prim}}(p)+N^{\lambda}_{\text{sec. prot}}(p)=P^{\lambda}(p), \]

i.e., the sums of the fluxes of primary and secondary protons (high-energy neutrons were not registered in experiments\({}^{16}\)). However, with a high degree of probability it may be assumed that high-energy secondary neutrons are formed in the same way as high-energy secondary protons are formed, i.e.,

Table V

Dependence of \(n^{\lambda}_{\text{spall}},\ n^{\lambda}_{\text{int}}\), and \(\gamma\) on the energy of the primary particles

Geomagnetic latitude \(\lambda\)	\(51^\circ\)	\(31^\circ\)	\(0^\circ\)	\(51{-}31^\circ\)	\(31{-}0^\circ\)	\(0^\circ\)
Energy of the primary particle, Bev	\(>1.5\)	\(>7\)	\(>14\)	3.3	9.2	40
Number of primary particles at the boundary of the atmosphere, \(\text{cm}^{-2}\text{s}^{-1}\)	0.47	0.094	0.047	0.376	0.047	0.047
Number of spallations \(n^{\lambda}_{3}\) in the atmospheric column, \(\text{cm}^{-2}\text{s}^{-1}\)	3.5	1.4	0.8	2.1	0.6	0.8
Number of spallations per one primary particle, \(n^{\lambda}_{\text{spall}}\)	7.5	15.0	17.0	5.6	13	17
Number of interactions \(n^{\lambda}_{\text{int}}\) per one primary particle	2	4	5	\(\sim 1.5\)	3	5
\(\dfrac{n^{\lambda}_{\text{spall}}}{n^{\lambda}_{\text{int}}}=\gamma+1\)	—	—	—	3.7	4.3	3.4

\[ N^{\lambda}_{\text{sec. neutr}}(p)=kN^{\lambda}_{\text{sec. prot}}(p). \]

At great depths in the atmosphere, where \(N_{\mathrm{prim}}^\lambda(p) \simeq 0\), it is known that \(N_{\mathrm{sec.\,neutr}}^\lambda(p)=N_{\mathrm{sec.\,prot}}^\lambda(p)\), i.e. \(k=1\). Thus,

\[ S^\lambda(p)=N_{\mathrm{prim.\,prot}}^\lambda(p)+N_{\mathrm{sec.\,prot}}^\lambda(p)+N_{\mathrm{sec.\,neutr}}^\lambda(p) \]

\[ = N_{\mathrm{prim}}^\lambda(p)+2N_{\mathrm{sec.\,prot}}^\lambda(p). \]

From experiment we know the quantity \(N_{\mathrm{prim}}^\lambda(p)+N_{\mathrm{sec.\,prot}}^\lambda(p)=P^\lambda(p)\); consequently,

\[ S^\lambda(p)=2P^\lambda(p)-N_{\mathrm{prim}}^\lambda(p). \]

Since

\[ N_{\mathrm{prim}}^\lambda(p)=N_{\mathrm{prim}}^\lambda(0)e^{-\frac{p}{L_{\mathrm{int}}}}, \]

then finally

\[ \bar n_{\mathrm{int}}^\lambda = \frac{2}{N_{\mathrm{prim}}^\lambda(0)L_{\mathrm{int}}} \int_0^{1000} P^\lambda(p)\,dp - 1. \tag{25a} \]

The number \(\bar n_{\mathrm{int}}^\lambda\), calculated on the basis of the data of \({}^{16}\), is presented in Table V in the 6th row, and in the 7th row the values of \(\dfrac{n_{\mathrm{calc}}^\lambda}{n_{\mathrm{int}}^\lambda}\) are given. As is evident from this table, with sufficient accuracy one may consider the ratio to remain constant and equal to \(3.8\pm0.4\) for all energies of the primary particles in the interval \(3\text{--}40\) Bev.

Using the data of Table V and formula (24), we obtain that \(\bar \nu\) indeed does not depend on the energy of the primary particles and is equal to \(2.8\pm0.4\).

From the method of determining the quantity \(\bar \nu\) it follows that the number found gives the mean number of particles of the star-producing component formed (neutrons with energies of tens to hundreds of Mev) in one interaction of a high-energy nucleon with a light nucleus (if each such neutron on average produces one disintegration). More precisely, the quantity \(\bar \nu\) gives the mean number of secondary disintegrations per one interaction of a high-energy nucleon.

2.4. Energy losses of primary particles due to nuclear disintegrations

As the energy of the primary particles increases, the total number of interactions undergone in the atmosphere by secondary high-energy nucleons increases. Correspondingly, one may expect an increase in the energy that will be transferred in the atmosphere

primary particles and their descendants, the products of nuclear disintegrations.

To clarify this question, one should determine the energy transferred to nuclear disintegrations in the column of the whole atmosphere, in accordance with formula (20); for this it is necessary to estimate the energy lost in the disintegration of nuclei in overcoming the binding of nucleons in the nucleus. Such an estimate\({}^{11}\) gives that \(E_{\text{bind}}^\lambda \simeq 0.3 E_{\text{kin}}\), i.e., is equal to \(0.3 E_{\text{s.i.}}^\lambda\). An estimate of the energy expended on ionization of the atmosphere by protons with ionization close to relativistic shows\({}^{11}\) that this energy is about \(0.1 E_{\text{s.i.}}^\lambda\).

Thus,

\[ E_{\text{n.d.}}^\lambda \simeq 1.4 E_{\text{s.i.}}^\lambda = 1.4 \cdot 32.5 \int_{0}^{1000} I_{\text{s.i.}}^\lambda(p)\, dp \ \text{eV}. \]

\(E_{\text{n.d.}}^\lambda\) can be calculated both by using data on the quantity \(I_{\text{s.i.}}^\lambda(p)\), obtained from measurements with an integrating chamber and a counter, and by using data from measurements of the spectrum of bursts.

The results of both methods of determining \(E_{\text{n.d.}}^\lambda\) are given in Table VI.

The results presented in Table VI show that protons with energy \(\sim 3\) Bev, when completely absorbed in a light substance, lose about 50% of their energy to the formation of nuclear disintegrations.

Primary protons with energy \(\sim 20\) Bev, when completely absorbed in a light substance, lose only 15% of their energy to the formation of nuclear disintegrations, while protons with \(\bar{E}_0 = 40\) Bev lose about 10% of their energy.

Table VI shows that, with increasing energy of the primary particle, an increase in the energy lost to nuclear disintegrations is indeed observed, but this increase is very weak. Thus, when \(\bar{E}_0\) increases from 3.3 Bev to 20 Bev, i.e., by a factor of 6, \(\varepsilon_{\text{n.d.}}\) increases from 1.8 Bev to 2.8 Bev, i.e., by only a factor of 1.6. The energy losses for the formation of \(\pi\)-mesons increase in the same range of primary particles from 1.5 Bev to 17 Bev, i.e., by more than a factor of 11. To understand the reason for the weak dependence of \(\varepsilon_{\text{n.d.}}\) on the energy of the primary particle \(\bar{E}_0\), it is necessary to find out what energy is transferred to the products of nuclear disintegrations in the first interaction of the primary particle with the nucleus and how this energy depends on the energy of the primary particle.

To answer the question posed, one may use the following feature of strongly ionizing particles arising in nuclear disintegrations. These particles, by virtue of ionization ...

Table VI

Geomagnetic latitude \(\lambda\)	\(51^\circ\)	\(31^\circ\)	\(0^\circ\)	\(51^\circ–31^\circ\)	\(31^\circ–0^\circ\)
Mean energy \(E_{\mathrm{Be}}^{\lambda}\) of primary particles, BeV	6.5	19.5	40	3.3	9.2
Value of \(E_{\mathrm{n},p}^{\lambda}\), obtained from the values of \(r_c\) and \((p)\), \(10^8\ \mathrm{eV\ cm^{-2}\ sec^{-1}}\)	\((9.25 \pm 0.55)\)	\((2.66 \pm 0.34)\)	—	\((6.6 \pm 0.6)\)	—
Value of \(E_{\mathrm{n},p}^{\lambda}\), obtained from measurements of the burst spectrum, i.e. \(I_T^{\lambda}(p)\), \(\mathrm{eV\ cm^{-2}\ sec^{-1}}\)	\((9.16 \pm 0.62)\)	\((2.95 \pm 0.23)\)	\((1.68 \pm 0.14)\)	\((6.2 \pm 0.65)\)	\((1.27 \pm 0.27)\)
Energy \(E_{\mathrm{n},p}^{\lambda}\) lost to nuclear disintegrations throughout the entire atmosphere by one primary particle, BeV	\(2.0 \pm 0.12\)	\(2.84 \pm 0.36\)	\(3.7 \pm 0.3\)	\(1.75 \pm 0.18\)	\(2.6 \pm 0.5\)
Fraction of energy lost to nuclear disintegrations, \(\dfrac{E_{\mathrm{n},p}^{\lambda}}{E_0}\)	31%	15%	9%	\(53 \pm 5\%\)	\(28 \pm 6\%\)

losses have a considerably smaller range than the range of those particles which produce them, i.e., a range smaller than the range of the component generating the “stars.” Therefore there will be an energy “equilibrium” between the strongly ionizing particles and the component generating them, i.e., the energy lost to ionization in 1 g of atmospheric matter by strongly ionizing particles will be equal to the energy which the star-producing component transfers in 1 g to the strongly ionizing particles.

The energy of strongly ionizing particles at a given depth \(p\) will be replenished through nuclear disintegrations produced directly by the primary particles at the place of observation and through the energy of those secondary particles of the star-producing component which arose above the level \(p\). If the altitude is sufficiently great (\(p\) is small), then the secondary particles of the star-producing component are, for the most part, particles which arose in the acts of the first interaction of the primary particles with atmospheric nuclei. These first-generation particles are precisely the carriers of that energy \(\varepsilon\) which is expended on the disintegration of nuclei by the primary particles. Therefore the quantity

\[ \int_{p_{\min}}^{p_{\max}} I_{\mathrm{s.i}}(p)\,dp \]

is a measure of the energy transferred by the primary particle to the products of nuclear disintegration in the act of its interaction with the nucleus.

The value \(p_{\min}\) is chosen from the condition that in the quantity \(I_{\mathrm{s.i}}(p)\) the primary multiply charged particles should play a small role (i.e., \(p_{\min} \geq 40\ \mathrm{g/cm^2}\), since at \(p = 40\ \mathrm{g/cm^2}\) multiply charged particles produce no more than 10% of \(I_{\mathrm{s.i}}^\lambda\)). \(p_{\max}\) is chosen from the condition that, in the global flux of the star-producing component, particles of the second and subsequent generations should play a small role (i.e., \(p_{\max} \lesssim 100\ \mathrm{g/cm^2}\)).

Since

\[ \frac{I_{\mathrm{s.i}}^\lambda(p)}{N_{\mathrm{prim}}^\lambda} \simeq \mathrm{const} \]

(see Fig. 11) for values

\[ p \lesssim 150\ \mathrm{g/cm^2}, \]

then, when choosing \(p_{\max} < 100\ \mathrm{g/cm^2}\), we obtain that

\[ \varepsilon \sim \int_{p_{\min}}^{p_{\max}} \frac{I_{\mathrm{s.i}}^\lambda(p)\,dp}{N_{\mathrm{prim}}^\lambda} = \mathrm{const}; \]

or the energy transferred to the products of nuclear disintegrations, in the first approximation, does not depend on geomagnetic latitude, i.e., on the energy of the primary particles.

It is easy to estimate the upper limit of the quantity \(\varepsilon\). For this it is sufficient to divide the total energy released in the entire atmosphere in nuclear disintegrations by the number of interactions undergone throughout the atmosphere by nuclear-active particles. From such an estimate we obtain that \(\varepsilon < 700\) MeV.

Consequently, the picture of energy losses in the atmosphere by a primary particle due to meson production and nuclear disintegrations appears as follows: a primary particle of high energy, interacting with a nucleus, transfers only a small part of its energy to mesons; less than 700 MeV on average is transferred to heavy particles—the products of nuclear disintegration. Thus, most of the energy of the primary particle is carried away by some number of secondary nucleons of high energy. These nucleons will subsequently interact with nuclei, and each of the high-energy nucleons will repeat everything that the primary proton did in its first interaction. Gradually the energy of the nuclear-active nucleons will decrease and will reach such a value \(\varepsilon_1\) at which the process of meson production will take away a negligible part of the nucleon energy. At this nucleon energy, the energy losses will be completely determined by the process of nuclear disintegrations and ionization braking, and it may be assumed that almost all the energy \(\varepsilon_1\) will pass to strongly ionizing particles. When the particle energy falls below 2–3 GeV, then, on the one hand, they will no longer be registered as nuclear-active particles, and, on the other hand, they will lose the greater part of their energy to nuclear disintegrations. By not taking this circumstance into account, we have substantially overestimated the value of \(\varepsilon\).

2.5. Multiplicity of formation of secondary high-energy nucleons and the picture of the elementary act of interaction of nucleons of different energies with light nuclei

After the general outlines of the picture of energy losses by primary particles into two principal processes—meson production and nuclear disintegrations—have been clarified, it is possible to determine the mean number \(\overline{m}\) of nuclear-active nucleons carrying away the energy not lost in the first interaction.

To estimate the quantity \(\overline{m}\), let us assume that in each collision \(\overline{m}\) nuclear-active nucleons are produced. If before the collision the primary particle had energy \(E_0\), then after the collision \(\overline{m}\) nucleons will have the total energy \((1-\alpha)E_0\), where \(\alpha\) is the fraction of energy transferred to \(\pi\)-mesons. As a result of the next collision of these \(\overline{m}\) nucleons, \(\overline{m}^{\,2}\) nucleons will arise, whose total energy will be \((1-\alpha)^2E_0\), and so on.

Let, as a result of \(k\) cascades, the last, \(k\)-th generation of nucleons consist of nucleons with mean energy \(\overline{E}_k = 3\) Bev. But particles with energy \(\sim 3\) Bev in the atmosphere lose \(1.75\) Bev to nuclear disintegrations. Consequently, the resulting \(\overline{m}^k\) nucleons with \(\overline{E} = 3\) Bev will all together lose in the atmosphere, to nuclear disintegrations, an energy no greater than that transmitted in the atmosphere to this process by the primary particle with energy \(\overline{E}_0\). If \(\overline{E}_0 = 20\) Bev, then \(\overline{m}^k(1.75 \pm 0.18)\) Bev \(< 2.84 \pm 0.36\) Bev, and \(\overline{m}^k < 1.62 \pm 0.25\).

But before \(\overline{m}^k\) nucleons with mean energy \(3\) Bev were formed, the preceding generations produced \(n_{\mathrm{int}}^{\lambda}\) interactions, which we call interactions of nuclear-active particles. For primary particles with \(\overline{E}_0 = 20\) Bev, \(n_{\mathrm{int}} = 4\); thus,

\[ n_{\mathrm{int}} = 1 + \overline{m} + \overline{m}^{2} + \ldots + \overline{m}^{k-1} = \frac{1 - \overline{m}^{k}}{1 - \overline{m}} = 4, \]

i.e. \(\overline{m}^{k} = 1 - 4(1 - \overline{m}) < (1.62 \pm 0.25)\), whence \(m < 1.16 \pm 0.06\).

One may also reason as follows. All nucleons of the \(k\)-th generation, with mean energy \(3\) Bev, will possess a total energy

\[ (1 - \alpha)^k \overline{E}_0 = 3\overline{m}^{k}\ \text{Bev}; \]

therefore, \(\overline{m}_k = \dfrac{20}{3}(1-\alpha)^k\). But \(\alpha \approx 0.3\), and \(\overline{m}^{k} < 1.62 \pm 0.25\), i.e. \(6.7\alpha^k < 1.62 \pm 0.25\) and \(k > 3.7\).

Hence

\[ \overline{m} < (1.62 \pm 0.25)^{\frac{1}{3.7}} = 1.13 \pm 0.04 . \]

For primary particles with \(\overline{E}_0 = 40\) Bev we obtain:

\[ k > 5.2 \quad \text{and} \quad m < 1.15 \pm 0.04 . \]

On the other hand, since after the first interaction \(70\%\) of the energy of the primary proton is carried away by nucleons, \(\overline{m}\) cannot be appreciably less than 1. Consequently, we must conclude that, as a result of the collision of a high-energy nucleon with a light nucleus, on average one secondary nucleon of high energy carries away the unspent energy.

This conclusion is valid in the region of mean energies of primary particles \(3\)—\(40\) Bev.

Primary particles with \(E > 1.5\) Bev, when interacting with light nuclei, on average produce identical nuclear disintegrations. This

means that, as a nucleon passes through the atmosphere, the average loss of energy to nuclear disintegrations in each collision of the nucleon with a nucleus will remain a constant quantity equal to $\varepsilon$, while the total energy loss in the atmosphere will increase as the number of interactions increases.

This will obviously be true as long as the energy of the nucleon is greater than $1.5$ Bev. However, already at $E = 1.5$ Bev a very small fraction of the nucleon energy is lost in the process of meson production. As the nucleon energy decreases, for example when $\overline E_0 < \varepsilon_1$, one may assume that practically all the nucleon energy $E_0$ goes into the production of strongly ionizing particles.

Since the average multiplicity of production of high-energy nucleons is not greater than unity, on the average one nucleon with energy $\varepsilon_1$ is produced in its shower by one primary proton.

Therefore we may write that the energy $E_{\text{n.d.}}^\lambda$ lost to nuclear disintegrations in the entire atmosphere by one primary particle at geomagnetic latitude $\lambda$ is equal to

\[ E_{\text{n.d.}}^\lambda = \varepsilon n_{\text{int}}^\lambda + \varepsilon_1. \]

We can write such equalities for three different latitudes, i.e. we shall have three independent equations for the two quantities $\varepsilon$ and $\varepsilon_1$. It turns out that these three equations are satisfied by the values

\[ \varepsilon = 440 \pm 160 \ \text{Mev}, \]

\[ \varepsilon_1 = 1100 \pm 280 \ \text{Mev}. \]

Primary particles with $\overline E_0 = 20$ Bev in the first interaction transfer $28 \pm 3\%$ of their energy to $\pi$-mesons. They lose $440$ Mev to nuclear disintegration of the target nucleus, i.e. $2\%$ of $E_0$.

Thus, the total energy loss in the first interaction is $30 \pm 3\%$ of $\overline E_0$ (the same is true also for primary particles with $\overline E_0 = 40$ Bev).

Primary particles with an average energy of $3.3$ Bev transfer $21 \pm 4.5\%$ of their energy to $\pi$-mesons. They lose $440 \pm 160$ Mev to nuclear disintegration of the target nucleus, i.e. $13 \pm 5\%$ of their energy. Thus, the total energy loss in the first interaction is $34 \pm 7\%$ of $\overline E_0$. Therefore it may be assumed that the average energy loss by primary particles in the first act of their interaction with a light nucleus is $\sim 30\%$ of $\overline E_0$ and does not depend on $\overline E_0$, at least in the range of average energies $3$–$40$ Bev.

In the energy region of primary particles $10^{11}$—$10^{12}$ ev, direct data are lacking. However, the considerations discussed above

on the production of high-energy π-mesons and the altitude variation of nucleons of such high energies, in comparison with certain calculations (see part III), lead to the conclusion that in the energy region \(10^{11}—10^{12}\) eV the average characteristics of interaction with light nuclei apparently differ little from those found for primary particles with energies \(3—40\) BeV.

If the collisions of nucleons with a light nucleus are interpreted as a series of successive independent nucleon-nucleon collisions, then we must arrive at the following conclusion about the character of nucleon-nucleon interactions at energies of the incident nucleon \(10^{10}—10^{12}\) eV. In a light nucleus with atomic weight 14—16, the incident nucleon will on average undergo about two collisions; therefore, in each collision it will lose on average 15% of its energy to the generation of π-mesons. In a nucleon-nucleon collision there will arise a recoil nucleon having, on average, a kinetic energy of about 200 MeV. The energy of these recoil nucleons is ultimately expended in the form of the energy of nuclear disintegrations, both of the nuclei in which these recoil nucleons were produced and of other nuclei. (These considerations on the importance of the energy of the recoil nucleon are valid, at least, for average energies of the incident nucleon up to 40 BeV.)

If, in a light nucleus, on average only one nucleon-nucleon collision occurs, then in it an energy \(\sim 30\%\) of \(E_0\) is transferred to π-mesons, while an energy \(\sim 400\) MeV is transferred to the recoil nucleon.

One may consider\({}^{11}\) the kinematics of a nucleon-nucleon collision.

In this case, from the specified values of the average energy loss to meson production and of the average energy of the recoil nucleon, one can determine the average angle of emission \(\bar{\varphi}\) of the nucleons after their collision in the system of their center of mass.

Thus, assuming that in a light nucleus, on average, one nucleon-nucleon collision occurs, we obtain from our experimental data

\[ \bar{\varphi}=51^\circ \pm 11^\circ \quad \text{for} \quad \bar{E}_0=3\ \text{BeV} \]

and

\[ \bar{\varphi}=20^\circ{}^{+10^\circ}_{-20^\circ} \quad \text{for} \quad \bar{E}_0=20\ \text{BeV}. \]

If, however, one assumes that in a light nucleus, on average, two nucleon-nucleon collisions occur, then we obtain:

\[ \bar{\varphi}\approx 30^\circ \quad \text{for} \quad \bar{E}_0=3\ \text{BeV} \]

and

\[ \bar{\varphi}\approx 10^\circ \quad \text{for} \quad \bar{E}_0=20\ \text{BeV}. \]

Thus, the data we have obtained, under the interpretation indicated above, lead to the conclusion that the scattering anisotropy in the center-of-mass system of the colliding nucleons exists, and that the scattering anisotropy should increase with increasing energy of the incident particle.

Our data can be compared with the results obtained for energies of \(1\)--\(2\) BeV at the Cosmotron\({}^{42}\). In collisions of neutrons with energy \(E = 1.8\) BeV with protons, the average energy loss for meson production is \(44 \pm 11\%\) (averaged over all types of reactions with meson production). However, for comparison with our results, the data obtained at the Cosmotron must be referred to all kinds of collisions. For proton energies of \(1.0\) BeV the fraction of inelastic collisions is about \(60\%\) of all \((p-p)\)-collisions\({}^{43}\). If the dependence

\[ \frac{\sigma_{\mathrm{inel}}(E)}{\sigma_{\mathrm{tot}}(E)} \]

is extrapolated to \(E = 1.8\) BeV, we obtain that

\[ \frac{\sigma_{\mathrm{inel}}}{\sigma_{\mathrm{tot}}} \simeq 0.7 \]

at \(E = 1.8\) BeV. Then, taking elastic collisions in the \(n-p\) reaction into account, the average energy loss for meson production will be \(31 \pm 8\%\). The mean scattering angle of the nucleons in the center-of-mass system (averaged over all types of reactions) will be \(42 \pm 4^\circ\).

The conclusions we have obtained concerning the character of nucleon-nucleon collisions agree better with the results obtained at accelerators if it is assumed that in a light nucleus the incident nucleon undergoes one nucleon-nucleon collision.

III. PASSAGE OF NUCLEONS OF VARIOUS ENERGIES THROUGH THE ATMOSPHERE

3.1. Method of calculation

The average characteristics of the interaction of nucleons with light nuclei were obtained mainly from an analysis of the formation of various secondary components in thin layers of atmospheric matter, i.e., from the results of measurements performed at great altitudes. The processes that take place when nucleons pass through large thicknesses of matter did not enter directly into the results of our analysis; therefore, a priori, it does not follow that the characteristics we have found can be reconciled with what is known about the nucleon component of cosmic rays at great depths in the atmosphere, i.e., under large thicknesses of matter.

It is necessary, on the basis of the average characteristics found for the elementary event, to consider the passage of nucleons through large thicknesses of matter and to compare the results of the consideration with experimental data.

The equations describing the change in the fluxes of protons \(P(x,E)\) and neutrons \(N(x,E)\) with energy \(E\) as the depth of the atmo-

spheres \(x\), can be written in the form

\[ \left. \begin{aligned} \frac{\partial P(x,E)}{\partial x} &=-a(E)P(x,E)+\int_E^\infty P(x,E')a(E')W_1(E,E')\,dE' + \\ &\quad +\int_E^\infty N(x,E')b(E')W_2(E,E')\,dE', \\[6pt] \frac{\partial N(x,E)}{\partial x} &=-b(E)N(x,E)+\int_E^\infty P(x,E')a(E')W_3(E,E')\,dE' + \\ &\quad +\int_E^\infty N(x,E')b(E')W_4(E,E')\,dE'. \end{aligned} \right\} \tag{26} \]

Here \(W_1\) and \(W_3\) are, respectively, the numbers of protons and neutrons with energy \(E\), produced by a proton with energy \(E'\); \(W_2\) and \(W_4\) are the numbers of protons and neutrons with energy \(E\), produced by a neutron with energy \(E'\); \(\frac{1}{a(E)}\) and \(\frac{1}{b(E)}\) are, respectively, the interaction ranges of protons and neutrons with energy \(E\).

As has been shown\(^{30,31,11}\), equations (26) can be solved in general form if the form of the functions \(W_1, W_2, W_3, W_4, a(E)\), and \(b(E)\) is known. The solution of equations (26) has the form

\[ \left. \begin{aligned} P(x,E)&=e^{-x}\sum_{n=0}^{\infty}\frac{x^n}{n!}\Phi_n(E), \\ N(x,E)&=e^{-x}\sum_{n=0}^{\infty}\frac{x^n}{n!}F_n(E), \end{aligned} \right\} \tag{27} \]

where

\[ \Phi_{n+1}(E)=[1-a(E)]\Phi_n(E)+ \]

\[ +\int_E^\infty \Phi_n(E')a(E')W_1(E,E')\,dE' +\int_E^\infty F_n(E')b(E')W_2(E,E')\,dE', \]

\[ F_{n+1}(E)=[1-b(E)]F_n(E)+ \]

\[ +\int_0^\infty \Phi_n(E')a(E')W_3(E,E')\,dE' +\int_E^\infty F_n(E')b(E')W_4(E,E')\,dE'. \]

However, the form of the functions \(W_1, W_2, W_3, W_4, a(E)\), and \(b(E)\) is not known to us. Therefore we shall assume that

1) \(W_1=W_2=W_3=W_4=W(E,E')\),

2) \(a(E)=b(E)=1\), if \(x\) is measured in mean free paths for interactions corresponding to the geometrical cross section of nuclei,

3)
\[ W(E,E')\,dE = W\!\left(\frac{E}{E'}\right)\frac{dE}{E'} = W(u)\,du, \quad \text{where } u=\frac{E}{E'}. \]

In order to satisfy the established mean characteristics of the elementary interaction of nucleons with light nuclei, one must set

\[ \int_0^1 W(u)\,du=1; \]

\[ \int_0^1 uW(u)\,du=a\approx 0.7. \]

After these assumptions, system (26) will take the form:

\[ \left. \begin{aligned} \frac{\partial P(x,E)}{\partial x} &=-P(x,E) +\frac{1}{2}\int_0^1 \left[ P\!\left(x,\frac{E}{u}\right) + N\!\left(x,\frac{E}{u}\right) \right]W(u)\,\frac{du}{u}, \\[6pt] \frac{\partial N(x,E)}{\partial x} &=-N(x,E) +\frac{1}{2}\int_0^1 \left[ P\!\left(x,\frac{E}{u}\right) + N\!\left(x,\frac{E}{u}\right) \right]W(u)\,\frac{du}{u}. \end{aligned} \right\} \tag{28} \]

When a flux of primary particles with a purely power-law spectrum falls on the boundary of the atmosphere,

\[ P(0,E)=\frac{c}{E^\gamma}, \]

then, in accordance with formulas (27), we obtain:

\[ \left. \begin{aligned} P(x,E)&=\frac{c}{E^\gamma}e^{-\mu x}; \\ N(x,E)&=\frac{c}{E^\gamma}\left[e^{-\mu x}-e^{-x}\right]. \end{aligned} \right\} \tag{29} \]

where

\[ \mu = 1 - \int\limits_{0}^{1} u^{\gamma - 1} W(u)\,du . \tag{30} \]

Consequently, in the case of a purely power-law primary spectrum, the nucleon cascade will be absorbed in the atmosphere according to an exponential law with absorption coefficient \(\mu\), depending on the spectral index, while the nucleon spectrum will remain, at all depths, a power-law with the same index \(\gamma\).

In reality, the flux of particles incident on the boundary of the atmosphere does not have a purely power-law spectrum. At high latitudes the spectral index \(\gamma\) itself is a function of the energy, changing from 2 in the energy region \(10^{9}\)—\(10^{10}\) ev to 2.5 in the energy region \(E > 10^{11}\) ev. In addition, if we are interested in the passage through the atmosphere of nucleons of different energies at different geomagnetic latitudes, where the primary spectrum is cut off on the side of low energies, then in this case there is a sharp deviation from the conditions of a purely power-law primary spectrum. In this case, for nucleon energies \(E > E_c\) (\(E_c\) is the cutoff energy in latitude), the solutions obtained (29), (30) will be valid (under the condition that \(\gamma\) is constant), but for \(E < E_c\) the solutions (29) and (30) will be knowingly incorrect. Taking both these circumstances into account, it is necessary to find general solutions of equations (28).

To solve equations (28) one may use the method developed in electromagnetic cascade theory\(^{12}\). Using this method, we obtained a solution for the boundary conditions in which one proton of a given energy \(E_0\) falls on the boundary of the atmosphere. Knowing these solutions \(P(x,E,E_0)\) and \(N(x,E,E_0)\), it is easy to generalize them to the case of an arbitrary spectrum of primary particles \(\dfrac{dF(E_0)}{dE_0}\). Indeed, the desired solution will be

\[ \left. \begin{aligned} P(x,E) &= \int\limits_{E_c}^{\infty} P(x,E,E_0)\,\frac{dF}{dE_0}\,dE_0, \\[6pt] N(x,E) &= \int\limits_{E_c}^{\infty} N(x,E,E_0)\,\frac{dF}{dE_0}\,dE_0 . \end{aligned} \right\} \tag{31} \]

Here \(E_c\) is the energy of geomagnetic cutoff. If \(E > E_c\), then the lower limit of integration will be \(E\). If (31) is integrated with respect to \(E\), then we obtain the integral spectra of protons and neutrons at depth \(x\).

We give formulas that provide the solution of problem 11:

\[ \left. \begin{aligned} P(x, > E, E_0) &= \frac{1}{2}\left\{ \frac{e^{\lambda(s)x+ys}}{\sqrt{2\pi [1+s^2 x\lambda''(s)]}}+e^{-x} \right\},\\[4pt] N(x, > E, E_0) &= \frac{1}{2}\left\{ \frac{e^{\lambda(s)x+ys}}{\sqrt{2\pi [1+s^2 x\lambda''(s)]}}-e^{-x} \right\}, \end{aligned} \right\} \tag{32} \]

where

\[ y=\ln \frac{E_0}{E}, \]

\[ \lambda(s)=\int_0^1 u^s W(u)\,du-1, \]

and the relation between the parameter \(s\) and \(x\) is given by the expression

\[ x=-\frac{1}{\lambda'(s)}\left[y-\frac{1}{s}\right]. \]

The primary spectrum is taken in the form proposed in work \(21\):

\[ \frac{dF}{dE_0}= \frac{B}{E_0^{2/3}\left[1+0.09E_0^{4/3}\right]^{3/2}}, \]

where \(E_0\) is measured in Bev.

A complete calculation of \(P(x, > E)\) was carried out for the functions:

\[ W(u)=2u \]

and

\[ W(u)=20u^3(1-u), \]

which are characterized by the fact that for them

\[ \int_0^1 W(u)\,du=1 \]

and

\[ \int_0^1 uW(u)\,du=\frac{2}{3}, \]

which means that, on the average, as a result of interaction with a nucleus there remains one nuclear-active nucleon carrying \(2/3\) of the energy of the incident nucleon.

In Fig. 12 are presented the results of calculating proton spectra at various heights at three geomagnetic latitudes: 51, 31, and \(0^\circ\), for the two indicated forms of the function \(W(u)\).

One way of checking the correctness of the assumptions on which the calculation is based is to compare the absolute intensities of protons obtained for sea level (depth \(1000\ \mathrm{g/cm^2}\)) by calculation with experimental data. It is clear that for such

Fig. 12. Results of calculating integral proton spectra at various depths of the atmosphere \(x\), measured in nuclear interaction mean free paths, for three geomagnetic latitudes. Solid curves show spectra computed for the function \(W(u)=2u\); dashed curves, for the function \(W(u)=20(u^3)(1-u)\).

a great depth even a small deviation of the absorption mean free path from the true value, with an exponential law of absorption of nucleons in the atmosphere, will produce a large discrepancy between experiment and calculation.

We shall compare the results of the calculation with the experimental data \({}^{32}\). For the number of protons with energy \(1.5\ \mathrm{Bev}\) the experiment gives an intensity \((2.0\pm0.4)\cdot 10^{-8}\ \mathrm{cm^{-2}\ sec^{-1}\ sterad^{-1}\ Mev^{-1}}\), while our calculations \({}^{11}\), in which the scattering of protons of this energy is roughly taken into account, give \(3.4\cdot 10^{-8}\ \mathrm{cm^{-2}\ sec^{-1}\ sterad^{-1}\ Mev^{-1}}\) (for the function \(W(u)=20u^3(1-u)\)). Thus, with a change in the vertical flux

of protons between the boundary of the atmosphere and sea level by more than \(10^4\) times, the calculation gives the correct value of the intensity at sea level with an accuracy up to a factor of \(\sim 2\).

From the calculated spectra presented in Fig. 12, it is easy to construct the altitude dependence of the number of protons with energy above a given value and to compare it with experimental results. The most detailed in this respect are the results of measuring the relative intensity of protons with energy \(E>3\) Bev (see \(^{16}\) and \(^{33}\)). A comparison of these experimental data with the results of our calculation is shown in Fig. 13. As is seen from the figure, the calculation agrees well with the experimental data. The experimental data give, with sufficient accuracy, the altitude variation of protons

Fig. 13. Altitude variation of protons with energy \(E>3\) Bev (solid curves) and the altitude variation of electron-nuclear showers according to the data of work\(^{16}\) (circles and crosses) and work\(^{33}\) (triangles). \(1\)—at latitude \(51^\circ\), \(2\)—at latitude \(31^\circ\), and \(3\)—at latitude \(0^\circ\). Along the abscissa axis—the atmospheric depth \(x\) in mean free paths for nuclear interaction (\(L_{\text{int}}\) taken as \(60\ \mathrm{g/cm^2}\)); along the ordinate axis—the number of protons (the experimental points are normalized to the theoretical curves).

in the stratosphere from depths of \(30\ \mathrm{g/cm^2}\) to depths of \(300\text{–}400\ \mathrm{g/cm^2}\). The altitude variation at great depths in these experiments was obtained with low statistical accuracy. However, there is a large number of works in which the altitude variation of the nucleon component generating electron-nuclear showers was measured between depths of \(1000\text{–}300\ \mathrm{g/cm^2}\). The results of some of them are summarized in Table VII.

As the calculations carried out show, at mountain depths and lower, the spectrum of the nucleon component is purely power-law, with exponent

Table VII

Author	Altitude interval	Absorption range \(L_p\), g/cm\(^2\)	Method of measurement
Freter \(^{34}\)	Sea level—mountains	\(123 \pm 10\)	Electron-nuclear showers in a Wilson chamber
Birger, Wexler et al. \(^{35}\)	Sea level—mountains	\(\approx 120\)	Electron-nuclear showers (counters)
Welsh, Piccioni et al. \(^{36}\)	Sea level—10 km	\(112 \pm 2\)	Electron-nuclear showers (counters)
Christy et al. \(^{23}\)	5 km—10 km	\(\approx 130\)	Pulses in an ionization chamber, \(E > 10^{11}\) eV
Tinlot \(^{37}\)	Sea level—10 km	\(118 \pm 2\)	Electron-nuclear showers (counters)
Milroy \(^{32}\)	Sea level—10 km	\(140 \pm 10\)	For the vertical flux of particles
Rosen \(^{38}\)	Sea level—mountains	\(136^{+13}_{-8}\)	For the vertical flux of particles

degree \(\gamma = 2.6\). Therefore, to calculate the absorption path length \(L_{\text{п}}\), one may use formula (30). However, this formula gives the absorption of a parallel beam of particles. In reality, at great depths in the atmosphere, because of exponential absorption, the flux of protons (neutrons), although it will be very collimated in the vertical direction, will nevertheless not be strictly parallel. As experiments show \(^{39,40}\), the angular distribution of high-energy protons at mountain altitudes is \(P(\theta) \sim P(0)\cos^n \theta\), where \(n = 6\text{--}7\).

Owing to the sharp collimation of the radiation in the vertical direction, an experimental setup measuring the altitude dependence of electron-nuclear cascades measures a flux more or less close to the global one; hence the absorption path length \(L_{\text{п}}\) measured at great depths is close to the absorption path length of the global flux of nuclear-active particles.

Table VIII

Type of function \(W(u)\)	Absorption path length of the vertical particle flux, \(г/см^2\)	Absorption path length of the global particle flux, \(г/см^2\)	Mean fraction of energy lost in a collision: \(\displaystyle \int_0^1 u W(u)\,du\)
\(W(u)=2u\)	135	118	\(1/3\)
\(W(u)=20u^3(1-u)\)	133	116	\(1/3\)
\(W(u)=\delta\!\left(\dfrac{2}{3}-u\right)\)	125	110	\(1/3\)
\(W(u)=1\)	98	88	\(1/2\)
\(W(u)=6u(1-u)\)	94	85	\(1/2\)
\(W(u)=\delta\!\left(\dfrac{1}{2}-u\right)\)	90	82	\(1/2\)

Table VIII gives the values for the absorption path lengths of nuclear-active nucleons between atmospheric depths of \(600\text{--}1000\ г/см^2\), calculated by formula (30) for various forms of \(W(u)\) for the global and vertical particle fluxes. In this calculation, the path length for interaction in air was taken to be \(60\ г\cdot см^{-2}\).

Table VIII shows that the obtained mean characteristics are in good quantitative agreement with the experimental value of the range for absorption of nuclear-active nucleons, whereas the assumption of an average loss of 50% of the energy in one collision event leads to excessively strong absorption of the nucleon component, incompatible with the experimental data.

The assumption that the unspent energy is carried away by two nucleons with approximately equal energies substantially reduces the absorption range of the nucleon component[^11]. Therefore, if experiments confirm that for particles with energy \(10^{11} \div 10^{12}\) eV the interaction range in air is \(\sim 60 \text{ g}/\text{cm}^{2}\), then from the fact that the range for their absorption is \(120 \text{ g}/\text{cm}^{2}\) it will follow that also at energies \(\sim 10^{12}\) eV the average energy loss to mesons in one interaction event is about 30%, while 70% of the unspent energy is carried away by one particle, most likely a nucleon.

To check how correctly the calculations performed reflect the change with depth of the flux of protons and neutrons of different energies, one may compare the results of the calculations with the data[^4] obtained by the photographic-plate method at depths of \(60 \text{ g}/\text{cm}^{2}\) and \(700 \text{ g}/\text{cm}^{2}\).

In making such a comparison we assume that the “stars” with a given number of shower particles \(n_s\) correspond to generating particles with mean energy \(\overline{E}(n_s)\).

The results of the calculation and the experimental data are given in Tables IX and X. These tables show that the calculation correctly describes the relative change in the intensity of protons and neutrons of different energies over a large range of atmospheric depths.

It should be especially emphasized that the calculation and experiment agree with respect to the fraction of neutrons and protons of a given energy at high altitudes \((x=1)\).

From the fact that for \(x=1\) the calculation gives

\[ \frac{N(1;>E)}{P(1;>E)} = 0.55, \]

while the experiment gives

\[ \frac{n_n(1;>E)}{n_p(1;>E)} = 0.61 \pm 0.03, \]

there follows a justification of the assumption underlying the calculation (the conditions \(W_1 = W_2\) and \(W_3 = W_4\)).

As is seen from Tables IX and X, the calculation correctly reflects the change in the range for absorption of particles with change in their energy. It should be noted here that the observed decrease of \(L_n\) with increasing \(E\) occurs not because the mechanism of interaction of particles changes as their energy increases, but because with increasing \(E\) we pass to different parts of the energy spectra. The particle spectrum itself changes with depth (see Fig. 12).

Summarizing the results of comparing the calculations with the experimental data on high-energy nucleons (greater than \(2\text{–}3\) Bev), one may conclude that the calculation, based on the characteristics of an elementary event obtained in our experiments,

Table IX

Variation of the global flux of protons of different energies at latitude 51° with depth

Proton energy \(E\), Bev	Calculation: proton flux \(P(x, > E)\), \(\mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\), \(x = 1\) \((60\ \mathrm{g/cm^2})\)	Calculation: proton flux \(P(x, > E)\), \(\mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\), \(x = 11.7\) \((700\ \mathrm{g/cm^2})\)	Calculation: \(\dfrac{P(1, > E)}{P(11.7, > E)}\) and \(L_{\mathrm p}\)	Experiment: number of proton “stars” \(n_{\mathrm p}(x, > E)\), \(\mathrm{cm}^{-3}\ \mathrm{day}^{-1}\), \(x = 1\) \((60\ \mathrm{g/cm^2})\)	Experiment: number of proton “stars” \(n_{\mathrm p}(x, > E)\), \(\mathrm{cm}^{-3}\ \mathrm{day}^{-1}\), \(x = 11.7\) \((700\ \mathrm{g/cm^2})\)	Experiment: \(\dfrac{n_{\mathrm p}(1, > E)}{n_{\mathrm p}(11.7, > E)}\) and \(L_{\mathrm p}\)
\(> 4\)	\(1.26 \cdot 10^{-1}\)	\(3.3 \cdot 10^{-4}\)	\(380\) \(L_{\mathrm p} = 107\ \mathrm{g/cm^2}\)	\(79 \pm 2.4\)	\((2.6 \pm 0.24)\cdot 10^{-1}\)	\(300 \pm 30\) \(L_{\mathrm p} = 112 \pm 2\ \mathrm{g/cm^2}\)
\(> 10\)	\(3.7 \cdot 10^{-2}\)	\(8.1 \cdot 10^{-5}\)	\(460\) \(L_{\mathrm p} = 104\ \mathrm{g/cm^2}\)	\(21 \pm 1.2\)	\((5.3 \pm 1.1)\cdot 10^{-2}\)	\(400 \pm 80\) \(L_{\mathrm p} = 106 \pm 4\ \mathrm{g/cm^2}\)
\(> 40\)	\(7.4 \cdot 10^{-3}\)	\(4.7 \cdot 10^{-6}\)	\(1600\) \(L_{\mathrm p} = 86\ \mathrm{g/cm^2}\)	\(4.0 \pm 0.5\)	\((2.1 \pm 0.7)\cdot 10^{-3}\)	\(1900 \pm 700\) \(L_{\mathrm p} = 84 \pm 4\ \mathrm{g/cm^2}\)

Table X

Change in the global flux of neutrons of different energies at latitude \(51^\circ\) with depth

Neutron energy \(E\), neutrons, BeV	Calculation: neutron flux \(N(x, > E)\), \(\mathrm{cm}^{-2}\,\mathrm{sec}^{-1}\), \(x = 1\)	Calculation: neutron flux \(N(x, > E)\), \(\mathrm{cm}^{-2}\,\mathrm{sec}^{-1}\), \(x = 11.7\)	Calculation: \(\dfrac{N(1; > E)}{N(11.7; > E)}\)	Experiment: number of neutron “stars” \(n_{\mathrm{n}}(x, > E)\), \(\mathrm{cm}^{-3}\,\mathrm{day}^{-1}\), \(x = 1\)	Experiment: number of neutron “stars” \(n_{\mathrm{n}}(x, > E)\), \(\mathrm{cm}^{-3}\,\mathrm{day}^{-1}\), \(x = 11.7\)	Experiment: \(\dfrac{n_{\mathrm{n}}(1; > E)}{n_{\mathrm{n}}(11.7; > E)}\)
\(> 4\)	\(6.9 \cdot 10^{-2}\)	\(3.3 \cdot 10^{-4}\)	\(207\)	\(48 \pm 1.9\)	\((2.2 \pm 0.2)\cdot 10^{-1}\)	\(215 \pm 22\)
\(> 10\)	\(1.9 \cdot 10^{-2}\)	\(8.1 \cdot 10^{-5}\)	\(234\)	\(6.4 \pm 0.7\)	\((3.1 \pm 0.8)\cdot 10^{-3}\)	\(206 \pm 58\)
\(> 40\)	\(3.7 \cdot 10^{-3}\)	\(4.7 \cdot 10^{-6}\)	\(790\)	\(1.0 \pm 0.3\)	\((1.8 \pm 0.6)\cdot 10^{-3}\)	\(570 \pm 270\)

with the addition of the assumption of equality of the probabilities of recharging of a proton into a neutron and of a neutron into a proton, gives good agreement with the following set of experimental data:

1) with the altitude variation of the nucleon component over the entire range of depths in the atmosphere from \(p = 30 \text{ g}/\text{cm}^2\) to \(p = 1000 \text{ g}/\text{cm}^2\);

2) with the altitude variation of protons in the stratosphere at different geomagnetic latitudes;

3) with the altitude variation of neutrons of different energies, from several Bev to several tens of Bev, between depths in the atmosphere \(p = 60 \text{ g}/\text{cm}^2\) and \(p = 700 \text{ g}/\text{cm}^2\).

The calculation gives:

4) the correct change of the proton spectrum with depth between the atmospheric depths \(p = 60 \text{ g}/\text{cm}^2\) and \(p = 700 \text{ g}/\text{cm}^2\);

5) the correct value (to within a factor of \(\sim 2\)) of the absolute flux of high-energy protons at a depth of \(1000 \text{ g}/\text{cm}^2\) (with the intensity of protons changing from the boundary of the atmosphere to sea level by 10,000 times);

6) the correct ratio between the fluxes of nuclear-active protons and neutrons in the stratosphere and at large depths in the atmosphere.

3.2. Calculation of the formation in the atmosphere of the star-generating component at different geomagnetic latitudes

Experimental data on the altitude and latitude dependence of bursts and “stars” show that the star-generating component is not identical with the component generating electron-nuclear showers, although it is genetically related to it.

Our experiments and the data of other authors\(^{4}\) show that the secondary particles generating “stars” without shower particles are mainly neutrons. Analysis of the angular distribution of secondary protons with energies \(50\text{--}100\) Mev, produced in air, has shown\(^{27}\) that they are generated predominantly in the direction of the primary particles, i.e., of the particles that produced the disintegration in which the indicated protons arose. This circumstance gives grounds for assuming that neutrons with energies \(\sim 100\) Mev and higher are also emitted in the disintegration of light nuclei predominantly in the direction of the “primary” particles. (In this paragraph the term “primary” particles will refer to nuclear-active nucleons with energy \(> 1.5\) Bev, whose intensity and spectrum were calculated in the preceding paragraph and whose collision with a nucleus leads to the formation of neutrons of the star-generating component. The boundary \(1.5\) Bev is, of course, very conventional.) Since a large part of these neutrons apparently has energies \(\sim 100\) Mev, their range for interaction in air should be\(^{41}\) about \(120 \text{ g}/\text{cm}^2\), and in interaction with a nucleus a neutron causes disintegration and, as a rule, is absorbed.

In the case of substantially higher energies, where the path length for interaction approaches \(60\text{--}70\ \mathrm{g/cm^2}\), the probability of an inelastic interaction increases substantially, which leads to an increase in the path length for absorption as compared with \(60\text{--}70\ \mathrm{g/cm^2}\).

Thus, it may be assumed that: 1) the absorption path length of particles of the star-producing component is equal to \(\sim 120\ \mathrm{g/cm^2}\) for a very wide range of energies of these particles; 2) secondary neutrons arising in the act of interaction with a nucleus of high-energy particles retain the direction of motion of the primary particle. The change in the intensity of the parallel neutron flux \(n(x)\) with depth \(x\) may be written as follows:

\[ \frac{dn}{dx}=-\frac{1}{2}\,n(x)+\bar{\nu}\,[P(x)+N(x)], \]

where \(P(x)+N(x)\) is the total flux of protons and neutrons with \(E>1.5\ \mathrm{Bev}\).

The solution of this equation is elementary:

\[ n(x)=e^{-\frac{x}{2}}\nu\int_{0}^{x} e^{\frac{t}{2}}\,[P(t)+N(t)]\,dt. \]

Since \(P(x)+N(x)\) is known to us at all geomagnetic latitudes, the quantity \(n(x)\) can also be readily calculated for any value of \(x\) and any geomagnetic latitude. When comparing with experiments, it must be kept in mind that nuclear disintegrations are caused not only by neutrons of moderate energies \((n(x))\), but also by nuclear-active particles with energy \(E>1.5\ \mathrm{Bev}\). Therefore, the total flux \(\Pi(x)\) of the component producing disintegrations (“stars,” prongs) is equal to

\[ \Pi(x)=n(x)+P(x)+N(x). \]

Since in all experiments performed up to the present time the global flux of particles producing disintegrations was recorded, it is necessary to calculate \(\Pi^{\mathrm{glob}}(x)\).

Within the framework of the assumptions made,

\[ \Pi^{\mathrm{glob}}(x)=2\pi x\int_{x}^{\infty}\Pi(t)\,\frac{dt}{t^{2}}. \]

In the quantity \(\Pi^{\mathrm{glob}}(x)\) there is one as yet undetermined quantity, \(\bar{\nu}\). We shall determine it on the basis of experimental data, and then compare the results of the calculation with data from various experiments.

It was shown earlier that for one interaction event of a nuclear-active particle there are \(2.8\pm0.4\) secondary disintegrations; therefore \(\bar{\nu}=2.8\pm0.4\).

We carried out a calculation of \(\Pi^{\mathrm{glob}}(x)\) for the value \(\bar{\nu}=2.6\). Considering that the secondary particles forming “stars” are mainly neu-

…trons, one can determine the ratio of protons to the total intensity of the star-generating component, i.e. the ratio \(\dfrac{P}{P+N+n}\). (As protons in this component one takes particles with \(E \geq 1.5\,B_{\mathrm{Б}}\).) It is clear that the ratio \(\dfrac{P}{P+N+n}\) at great altitudes will be very sensitive to the value of the quantity \(\bar{\nu}\). In Fig. 14 the experimental and calculated ratio \(\dfrac{P}{P+N+n}\) is presented at two geomagnetic latitudes. As is seen from this figure,

Fig. 14. Dependence of the fraction of proton stars on atmospheric depth. Along the ordinate is plotted the ratio of the number of stars caused by protons to the number of all stars. The solid curves correspond to \(\bar{\nu}=2.6\), the dashed curve to \(\bar{\nu}=2.3\); points are the experimental results of work\(^{28}\), the cross is the results of work\(^{4}\).

the adopted value \(\bar{\nu}=2.6\) is in good agreement with experiment.

In Fig. 15 are presented the results of a calculation of the altitude dependence of the star-generating component \(\Pi_{\mathrm{глоб}}(x)\) at different geomagnetic latitudes, for the adopted value \(\bar{\nu}=2.6\). As is seen from the figure, the calculated altitude dependence of the flux of the star-generating component agrees well with the experimental values of the number of bursts over a wide range of depths and at different geomagnetic latitudes.

Moreover, the adopted calculation scheme with the value \(\bar{\nu}=2.6\) also gives the correct value of the absolute flux of the star-generating component. The calculation results give the flux of the star-generating component for geomagnetic latitude \(51^\circ\) and \(p=15\ \mathrm{g}/\mathrm{cm}^2\):

\[ \Pi_{\mathrm{глоб}}(p=15)=1.38\ \frac{\text{particles}}{\mathrm{cm}^2\,\mathrm{sec}}; \]

The experiment, corrected for the number of stars with \(N_h < 3\), gives a flux of the star-generating component at geomagnetic latitude \(54^\circ\) and \(p = 15\ \mathrm{g/cm^2}\) equal to \(1.20\ \dfrac{\text{particles}}{\mathrm{cm^2\,sec}}\).

Comparison of the calculations with the experimental data makes it possible to draw the following conclusions:

1. A wide range of experimental data on the component generating “stars” is in good agreement with the idea that, in the act of interaction with a light nucleus of a high-energy nucleon, on the average 2.6 secondary neutrons arise, with low energies of the order of \(100\ \mathrm{Mev}\), which subsequently generate “stars.”

Fig. 15. Dependence of the global flux of the star-generating component on atmospheric depth for latitudes \(51^\circ\) (curve 1), \(31^\circ\) (curve 2), and \(0^\circ\) (curve 3). Along the ordinate are plotted the values of the flux of the star-generating component. The experimental points are the numbers of bursts, normalized to the theoretical curves at the point \(x = 4\).

1. The possibility of correctly explaining, by a single mechanism, the altitude dependence of the star-generating component at different geomagnetic latitudes and its latitude effect confirms the correctness of the assertion made earlier that the number of secondary star-generating particles born in the act of interaction with the nucleus of a high-energy nucleon does not depend on its energy.

The analysis of the experimental material, set forth very briefly in the present article, makes it possible to draw the following main conclusions:

1) at primary cosmic-particle energies of several billion electron-volts, the absorption of primary particles is due to two principal processes: nuclear disintegrations and generation of \(\pi\)-mesons.

The role of both processes in the energy losses under complete absorption of primary particles depends substantially on their energy. Thus, at a mean energy of primary particles \(\overline{E}_0=3\) Bev, the latter lose \(53\pm5\%\) of their energy in the formation of heavy particles in nuclear disintegrations; whereas primary particles with a mean energy \(\overline{E}_0=20\) Bev lose only \(15\pm2\%\) of their energy in this same process. Primary particles with mean energy \(3\) Bev lose \(45\pm3\%\) of their energy to the generation of \(\pi\)-mesons, while primary particles with mean energy \(20\) Bev lose \(84\pm2\%\) of their energy.

2) The process of \(\pi\)-meson generation is not a single-act process. In a collision with a light nucleus, primary particles in one act transfer to \(\pi\)-mesons a comparatively small fraction of their energy. In the first act of collision with a light nucleus, primary particles with mean energy \(3\) Bev transfer \(21\%\) of their energy to \(\pi\)-mesons, while primary particles with mean energies \(20\) Bev and \(40\) Bev transfer \(28\text{--}30\%\) of their energy to \(\pi\)-mesons. Primary particles with energies \(100\text{--}1000\) Bev transfer to mesons on the order of \(30\%\), and certainly less than \(50\%\), of their energy.

3) The energy not expended on the formation of \(\pi\)-mesons after collision with a light nucleus is carried away, on the average, by one nucleon. For particles with energy \(10^{11}\text{--}10^{12}\) ev, after the collision the unexpended energy (on the order of \(70\%\) of the primary energy) is also carried away, on the average, by one particle, most probably a nucleon.

4) The main features of the mechanism of formation of the star-generating component in the atmosphere are as follows. In the collision with a light nucleus of a high-energy particle (several Bev and above), one and the same energy

\[ \bar{\varepsilon}=440\pm160\ \text{Mev}, \]

independent of the energy of the incident nucleon (at least in the range of mean energies \(3\text{--}40\) Bev), is expended on the disintegration of the nucleus on the average. An energy of \(\sim 200\) Mev is transferred in this disintegration to charged heavy particles, and the remaining energy, \(\sim 200\) Mev, is carried away on the average by \(2.8\pm0.4\) neutrons, which subsequently expend the energy received in secondary nuclear disintegrations. The mean number of secondary neutrons flying out of the disintegrated nucleus likewise does not depend on the energy of the incident particle.

5) The experimentally found mean characteristics of the act of interaction of high-energy particles with a light nucleus quantitatively describe well the passage of the nucleon component of cosmic rays through the atmosphere.

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