INTERPRETATION OF PHENOMENA OCCURRING IN COSMIC RAYS

Bruno Rossi*)

FROM THE EDITORS

B. Rossi’s article is a more or less systematic survey, devoted mainly to the results of experimental investigations of the non-electromagnetic interaction of cosmic-ray particles with matter. Whereas the theory of electromagnetic interaction has been well developed**), mainly through the efforts of Soviet physicists, the theory of nuclear forces is still in an extremely unsatisfactory state. There is no doubt that this state of theory is connected with the scarcity and unreliability of the empirical material relating to this field. The data available in the literature are fragmentary, incomplete, and often contradictory. Therefore an attempt to summarize the experimental facts concerning the nuclear interaction of cosmic-ray particles is highly desirable.

A major shortcoming of Rossi’s article is the fact that the author passes over in complete silence the work of Soviet scientists, who have made a very substantial contribution to the science of cosmic rays. Among the works of primary importance should be noted the discovery of varitrons by A. I. Alikhanov and A. I. Alikhanyan, investigations of cosmic rays in the stratosphere carried out by S. N. Vernov, as well as the discovery and study, by the participants of the high-altitude expedition (headed by N. A. Dobrotin), of a fundamental process in cosmic rays—the generation of “special showers.”

In order at least partially to make up for these substantial omissions in the article, the editors considered it advisable to supplement it with indications of the most important works of Soviet scientists relating to the circle of phenomena discussed in the article.

PREFACE

The present article describes the main experiments on the study of cosmic rays and attempts to arrange their results in a logical sequence.

*) Reviews of Modern Physics 20, No. 3 (1948). Translated by P. E. Kunin and I. L. Rosental.

**) See, for example, the books: S. Z. Belen’kii, Cascade Processes in Cosmic Rays, State Publishing House of Technical-Theoretical Literature, 1948; and B. Rossi and K. Greisen, Interaction of Cosmic Rays with Matter, GITTL, 1948.

The article does not consider all phenomena in cosmic rays. For example, questions connected with atmospheric showers are not touched upon, and geomagnetic phenomena are discussed only qualitatively. The historical course of the development of the science of cosmic rays is not covered here. Therefore the author apologizes for the absence of references to the corresponding works.

I. SOME QUANTITATIVE DATA ON COSMIC RAYS

1. Definitions

The cosmic rays observed by us in the atmosphere contain electrons, photons, mesons (possibly of various kinds), protons, neutrons, and heavy nuclear fragments.

The electron and photon components are closely connected with each other, since if electrons are present, then photons are generated by bremsstrahlung, just as photons give rise to electron–positron pairs. Thus, electrons and photons form one component, which in what follows we shall call the electronic component of cosmic rays. It is well known that the curve representing the number of coincidences between two or more Geiger–Müller counters, arranged along a vertical straight line, as a function of the thickness of lead placed between them, undergoes a sharp change in slope at a thickness close to \(10\ \mathrm{cm}\). Being very steep for small thicknesses, it becomes flatter for large ones. This fact is due to the presence in cosmic rays of electrons, which are rapidly absorbed. Independently of its interpretation, it leads to an empirical division of cosmic-ray particles into hard and soft components. The hard component includes all particles capable of passing through a given thickness of lead, while the soft component comprises all particles capable of penetrating the walls of the counter but stopped in the given thickness of lead. The choice of this critical thickness is, of course, to some extent arbitrary, and the subdivision of cosmic-ray particles into hard and soft components is meaningful only because the relative intensity of both components depends little on this choice.

Observations of the passage of particles through lead in a Wilson chamber make it possible to distinguish electrons from particles of greater mass (mesons, protons). Indeed, for an electron the probability of emitting a high-energy photon in lead is considerably greater than for a meson or proton. Therefore electrons undergo greater energy losses and have a greater probability of producing showers. Since in photographs in the chamber it is often difficult to distinguish mesons from protons, it is convenient to combine them under one name, applying to both kinds of particles. In what follows we shall call them penetrating

ing particles (independently of their energy). The division of cosmic-ray particles into electrons and penetrating particles is not identical with the division into the soft and hard components, since although it is true that the hard component contains practically only penetrating particles, the soft component consists not only of electrons, but also of mesons and protons of low energies.

For a quantitative description of the various components of cosmic rays we shall define the following quantities:

a) Directional intensity \(I\). \(I\,d\omega\,d\sigma\,dt\) is the number of particles of the given kind incident on an area element \(d\sigma\), during a time \(dt\), within a solid-angle element \(d\omega\), perpendicular to \(d\sigma\).

\(I\) is measured in \(\mathrm{cm}^{-2}\,\mathrm{sec}^{-1}\,\mathrm{sterad}^{-1}\);

\(I_v\) (vertical intensity) is the value of \(I\) in the vertical direction.

b) Flux \(J_1\). \(J_1\,d\sigma\,dt\) is the number of particles of the given kind crossing a horizontal area element \(d\sigma\) from top to bottom during a time \(dt\). \(J_1\) is related to \(I\) by the equation

\[ J_1=\int J\cos\theta\,d\omega, \tag{1} \]

where \(\theta\) is the angle between the vertical direction and the direction \(d\omega\). The integration extends over the entire upper hemisphere. \(J_1\) is measured in \(\mathrm{cm}^{-2}\,\mathrm{sec}^{-1}\).

c) Integral intensity \(J_2\). \(J_2\) is defined by the equation

\[ J_2=\int I\,d\omega, \tag{2} \]

where the integration extends over all directions. \(J_2\) is measured in \(\mathrm{cm}^{-2}\,\mathrm{sec}^{-1}\).

In this article the following notation is used:

\(M\) — mass of the proton, equal to \(0.935\cdot10^9\,\mathrm{eV}/c^2\);

\(\mu\) — mass of the ordinary meson, equal to \(10^8\,\mathrm{eV}/c^2\);

\(\tau\) — lifetime of the ordinary meson, equal to \(2.15\cdot10^{-6}\,\mathrm{sec}\);

\(c\) — speed of light, equal to \(3\cdot10^{10}\,\mathrm{cm\,sec}^{-1}\);

\(\beta\) — ratio of the velocity of the body to the speed of light;

\(Z\) — atomic number and

\(A\) — atomic weight.

2. Hard and soft components at sea level at geomagnetic latitudes exceeding \(45^\circ\)

Up to the present no change in any effects in cosmic rays has been detected with a change of geomagnetic latitude from \(45^\circ\) to the pole (with the possible exception of the integral intensity at very great altitudes); therefore, in this section we shall consider all experimental data obtained at latitudes greater than \(45^\circ\).

Precise measurements of the intensity of the hard component and of the total intensity of charged particles (the sum of the soft and hard components) were carried out by Greisen \(^{G3}\) at \(50^\circ\) and at an altitude of \(259\ \mathrm{m}\) above sea level (atmospheric depth equal to \(1007\ \mathrm{g\,cm^{-2}}\)). To separate the hard component from the soft one, a layer of lead was used whose thickness was \(167\ \mathrm{g\,cm^{-2}}\).

After introducing corrections for the difference between the geometrical and effective lengths of the Geiger–Müller counters for the hard component, the following results were obtained:

\[ I_\nu = 0.82\cdot 10^{-2}\ \mathrm{cm^{-2}\ sec^{-1}\ sterad^{-1}}, \]

\[ J_1 = 1.26\cdot 10^{-2}\ \mathrm{cm^{-2}\ sec^{-1}}, \]

\[ J_2 = 1.66\cdot 10^{-2}\ \mathrm{cm^{-2}\ sec^{-1}}. \]

In the above estimates, the decrease in the number of coincidences due to scattering of particles and the increase caused by showers were neglected. The errors thereby arising can be estimated by comparing the values just given with the value of this quantity obtained by Greisen \(^{G5}\) in an apparatus in which scattering had practically no significance and a correction for shower production was introduced. This comparison, taking into account the difference in the thicknesses of the absorbers used in the two experiments (\(107\ \mathrm{g\,cm^{-2}}\) instead of \(167\ \mathrm{g\,cm^{-2}}\), which corresponds to a correction of \(4\%\)), shows that, in order to allow for the combined influence of scattering and showers, the above values of \(I_\nu\), \(J_1\), and \(J_2\) must be increased by \(4\%\). To obtain the intensity at sea level (\(1030\ \mathrm{g\,cm^{-2}}\)), a correction of approximately \(3\%\) must be introduced in the opposite direction. With these two corrections, the following values are obtained for the hard component:

\[ I_\nu = 0.83\cdot 10^{-2}\ \mathrm{cm^{-2}\ sec^{-1}\ sterad^{-1}}, \]

\[ J_1 = 1.27\cdot 10^{-2}\ \mathrm{cm^{-2}\ sec^{-1}}, \]

\[ J_2 = 1.68\cdot 10^{-2}\ \mathrm{cm^{-2}\ sec^{-1}}. \tag{3} \]

The statistical errors of the measurements were about \(1\%\). Owing to the uncertainty associated with the various corrections, the systematic errors could have been considerably larger. As regards the angular dependence, Greisen’s results, in agreement with previous data, show that \(I\) varies almost as the square of the cosine of the zenith angle.

The measurement of the absolute value of the total intensity is more difficult than the measurement of the intensity of the hard component. Because of the presence in cosmic rays of a large number of low-energy particles (especially electrons), the number of coincidences in a telescope with no absorber between the counters depends strongly on the thickness of the counter walls. The presence of any covering over the apparatus may also significantly affect the results of the measurements.

In Greisen’s experiments the walls of the counters were equivalent to the presence of \(2.3\ \mathrm{g\,cm^{-2}}\) of brass between the effective volumes

counters. As a result of the first series of measurements, after corrections for the effective length of the counters, the following values were obtained:

\[ I_{\nu}=1.23\cdot 10^{-2}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\ \mathrm{sterad}^{-1}, \]

\[ J_{1}=1.93\cdot 10^{-2}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}, \]

\[ J_{2}=2.60\cdot 10^{-2}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}. \]

In the second series of measurements\({}^{5}\), for \(J_{2}\) the following value was obtained:

\[ J_{2}=2.53\cdot 10^{-2}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}. \]

The difference between the two values is possibly due to the statistical error of the measurement. Otherwise this difference may be explained by the influence of showers, which were taken into account in the second measurement, but not in the first. We shall therefore consider

[In the graph: vertical axis \(I(s)/I(h)\); horizontal axis \(g\,\mathrm{cm}^{-2}\) of brass.]

Fig. 1. Absorption curve of “soft” cosmic-ray particles in brass. Along the abscissa is plotted the minimum range of soft particles, determined by the thickness of the counter walls. Along the ordinate is plotted the ratio of the intensity of the soft component to the hard component at sea level.

the second value of \(J_{2}\) to be closer to the true one, and we shall correct the values \(I_{\nu}\) and \(J_{1}\), obtained in the first series of measurements, in accordance with the value of \(J_{2}\). After introducing the correction for altitude (which for the total intensity amounts to about 5%) we obtain the following values for the total intensity at sea level and for the intensity of the soft component, measured with \(2.3\ g\,\mathrm{cm}^{-2}\) of brass between the effective volumes of the counters:

Total intensity:

\[ \begin{aligned} I_v &= 1.14\cdot 10^{-2}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\ \mathrm{sterad}^{-1},\\ J_1 &= 1.79\cdot 10^{-2}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1},\\ J_2 &= 2.41\cdot 10^{-2}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}. \end{aligned} \tag{4} \]

Soft component (obtained as the difference):

\[ \begin{aligned} I_v &= 0.31\cdot 10^{-2}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\ \mathrm{sterad}^{-1},\\ J_1 &= 0.52\cdot 10^{-2}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1},\\ J_2 &= 0.73\cdot 10^{-2}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}. \end{aligned} \tag{5} \]

The intensity of the soft component as a function of the wall thickness of the brass counters is presented in Fig. 1. This curve was obtained from measurements of absorption in graphite carried out by Greisen G5. The calculation of absorption in brass from the data on absorption in graphite is based on comparison of the mean energy losses of electrons in the two materials. The thickness considered is the mean thickness between the effective volumes of the two outer counters for a disordered distribution of particles and is approximately equal to \((n-1)d/2\), where \(n\) is the number of counters and \(d\) is the wall thickness of each counter.

3. Altitude dependence of the hard and soft components at geomagnetic latitudes greater than 45°

Curve \(H\) in Fig. 2 represents the vertical intensity of the hard component \(I_v\) as a function of depth, reckoned from the boundary of the atmosphere. The lower part of the curve (from 1030 to \(616\ \mathrm{g\ cm^{-2}}\), i.e., from sea level to an altitude of 4300 m) is based on the measurements of Rossi, Hilberry, and Hoag R4, in which the telescope registering cosmic rays was well shielded from lateral showers*). These measurements agree well with the results obtained by other authors G3; B2. The upper part of the curve is based on measurements carried out by Gill, Shein, and Young G1 on radiosondes and airplanes. This curve is normalized to the value of \(I_v\) at sea level given in the preceding section.

Curve \(T\) in Fig. 2 represents the total vertical intensity as a function of depth.

The part of the curve between 616 and \(250\ \mathrm{g\ cm^{-2}}\) is based on measurements carried out by Sands S2 in an airplane. In Sands’s experiments the counter walls were equivalent to \(5\ \mathrm{g\ cm^{-2}}\) of brass placed between the effective volumes of the counters.

It follows from Fig. 1 that the vertical intensity of the soft component at sea level, measured with counters having such walls, is equal to \(0.25\cdot 10^{-2}\ \mathrm{cm^{-2}}\ \mathrm{sec^{-1}}\ \mathrm{sterad^{-1}}\). Sands’s measurements of the total intensity are normalized to the sea-level value equal to

*) See also S. Azimov, V. Veksler, G. Zhdanov, and A. Lyubimov, JETP 17, 87 (1947), (Editor’s note).

\[ (0.83 + 0.25)\cdot 10^{-2} = 1.08\cdot 10^{-2}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\ \mathrm{sterad}^{-1}. \]

Sands did not carry out measurements in the interval from \(1030\) to \(616\ \mathrm{g\,cm}^{-2}\).

Legend in the figure:

• Prottser
• Millikan et al.
• Carmichael et al.
• Normalization point

Axis labels in the figure:

• \(I_V\ \left(\mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\ \mathrm{sterad}^{-1}\right)\)
• Atmospheric depth \(\left(\mathrm{g\,cm}^{-2}\right)\)

Fig. 2. Vertical intensity of the hard component (\(H\)), the soft component (\(S\)), and the total ionizing component (\(T\)), as a function of atmospheric depth at geomagnetic latitudes greater than \(45^\circ\). The minimum range of the soft particles is equal to \(5\ \mathrm{g\,cm}^{-2}\) of brass.

To fill this gap, use was made of the results obtained by Greisen at various altitudes\({}^{63}\). In these experiments-

thus the thickness of the walls was equivalent to \(2.3\ \mathrm{g\,cm^{-2}}\) of brass. In constructing the curve it was assumed that the change in the intensity of the soft component with altitude between \(1030\) and \(616\ \mathrm{g\,cm^{-2}}\) is the same both in measurements with a wall thickness of \(2.3\ \mathrm{g\,cm^{-2}}\) and in measurements with \(5\ \mathrm{g\,cm^{-2}}\) of brass.

Fig. 3. Vertical intensity of the hard component \((H)\), the soft component \((S)\), and the entire ionizing component \((T)\) as a function of atmospheric depth near the geomagnetic equator. The minimum range of the soft particles is \(5\ \mathrm{g\,cm^{-2}}\) of brass.

In measurements with \(5\ \mathrm{g\,cm^{-2}}\) of brass. For depths less than \(250\ \mathrm{g\,cm^{-2}}\), there are measurements by various experimenters carried out on balloon sondes*) \(P^2, M^3, C^1\). Since in some of these measurements the point

*) See also L. Baradzei, S. Vernov, and D. Smorodin, DAN 62, 465 (1948) (Ed. note).

of ground was not precisely determined, then the data obtained with balloon-borne sondes are normalized to the Sandström curve at a depth of \(300\ \mathrm{g\,cm^{-2}}\). As shown in the figure, the results of various experiments differ greatly from one another, and it is not clear whether these discrepancies are a consequence of experimental errors or of fluctuations in the intensity of cosmic rays at great altitudes. Therefore there is great uncertainty in the values of the total intensity near the boundary of the atmosphere. The curve shown in Fig. 2 is, in a certain sense, averaged from data obtained by different experimenters. The curve \(S\) is the difference between the curves \(T\) and \(H\) and represents the vertical intensity of the soft component as a function of depth.

4. Hard and soft components near the equator

The curves \(T\), \(H\), and \(S\) in Fig. 3 represent, respectively, the altitude dependence of the vertical intensity for the total intensity and for the intensities of the hard and soft components near the equator. The intensity at sea level was obtained from measurements at latitudes greater than \(45^\circ\), under the assumption of the existence of a five-percent latitude effect both for the hard and for the soft components, which follows from the recent work of Morris, Swann, and Taylor \(^{\mathrm{M4}}\). The total vertical intensity at great altitudes was obtained from the results of Millikan, Neher, and Pickering \(^{\mathrm{M3}}\). These results are normalized by multiplying all intensities on the Millikan, Neher, and Pickering curve by the appropriate factor, chosen so as to bring the value of the intensity at latitudes greater than \(45^\circ\) and at a depth of \(300\ \mathrm{g\,cm^{-2}}\) into agreement with the values given on the curve in Fig. 2.

The vertical intensity of the hard component at great altitudes, obtained from the results of Gill et al. \(^{\mathrm{G1}}\), is again normalized to the value of the intensity of the hard component at \(300\ \mathrm{g\,cm^{-2}}\) and at latitudes greater than \(45^\circ\). The latitude effect for the hard component at great altitudes was found by the measurements of Gill et al. to be in good agreement with the observations of Morris et al.*) \(^{\mathrm{M4}}\). The curve \(S\) is the difference between the curves \(T\) and \(H\).

5. Momentum spectrum of mesons at sea level

As already noted, by the Wilson chamber method it is easy to distinguish electrons from penetrating particles. Measurements of the curvature of electron tracks in a magnetic field and observations of shower generation in lead plates have shown that the curve of the energy distribution of electrons at sea level falls off very rapidly with increasing

) Concerning the latitude effect in cosmic rays in the stratosphere, see the article by S. N. Vernov in Trudy Fizicheskogo instituta AN SSSR, vol. III, issue 1 (1945) (Editor’s note*).

energy. From observations of the frequency of occurrence of large showers under several centimeters of lead it follows that, at sea level, electrons with energy greater than \(10^{10}\ \mathrm{eV}\) occur less often than 1 in 10,000 particles (see, for example, B7).

Penetrating particles with momenta smaller than \(7\cdot 10^8\ \mathrm{eV}/c\) can be separated into protons and mesons by estimating the specific

Fig. 4. Differential momentum spectrum of mesons at sea level. Wilson’s experimental results W7 are shown by circles.

ionization from the density of their tracks in the Wilson chamber. This method becomes unsuitable if the momenta exceed \(7\cdot 10^8\ \mathrm{eV}/c\). However, as will be shown below, there is evidence for an extreme paucity of high-energy protons at sea level (it is estimated that, of all particles with momenta greater than \(7\cdot 10^8\ \mathrm{eV}/c\), less than 1% are protons; see § 20). Therefore, in determining the momentum spectrum of mesons at sea level from observations with a Wilson chamber placed in a magnetic field, one may regard as mesons all

penetrating particles, not differing appreciably from mesons in specific ionization.

The circles in Fig. 4 show the differential momentum spectrum of mesons at sea level, measured by Wilson^W7, whose results agree well with the earlier determinations of Blackett^B6 and other authors^I6, H5, but extend to the region of smaller momenta.

In this figure the momenta along the abscissa axis are measured in units of \(10^6\ \mathrm{eV}/c\). Since the relative number of particles with momenta exceeding \(2\cdot 10^9\ \mathrm{eV}/c\) is well known from observations in Wilson’s chamber, the normalization factor has been chosen so as to bring the number of particles with momenta greater than \(2\cdot 10^9\ \mathrm{eV}/c\) into agreement with the absolute value of this quantity determined from absorption experiments (see § 6). The curve in Fig. 4 shows the most appropriate estimate of the momentum spectrum that can be made at present, taking into account measurements by deflection in a magnetic field and by absorption (see § 6).

As for the nature of the mesons observed at sea level, it was found that the greater part, if not all, of the mesons with sufficiently small energies occurring at sea level have a mass, determined from experiments in a Wilson chamber or from observations of their passage through matter, equal to 200 electron masses^B5, F1, τ, e, and are ordinary mesons. (There is no direct evidence for the existence of high-energy mesons at sea level.)

Evidence for the existence of mesons of larger masses was obtained in processing photographic emulsion exposed at high altitudes^L1, L2 *). The absence of an appreciable number of such heavy mesons in Wilson chambers at sea level may be explained by the assumption that heavy mesons decay into ordinary ones with a very short half-life, so that they can be observed only near the place of their origin.

6. Distribution of meson ranges at sea level

Emert^E1 and Wilson^W8 measured, respectively, the vertical intensity of cosmic rays under water and under ground by means of telescopes, between whose counters there was no lead (see also the works of Clay^K2, K3). If the data of Emert and Wilson, normalized to one depth, are plotted against the equivalent absorber thicknesses, they fall on a single curve shown in Fig. 5. Absorber thicknesses of different materials are considered equivalent if mesons of the same energies have in them the same

) The first evidence for the existence of mesons of larger masses (varitrons) was obtained by other methods by A. I. Alikhanov, A. I. Alikhanian, and collaborators. See the note to § 17, e. (Ed. note.)*

and the same range. Along the abscissa of Fig. 5 is plotted the thickness of the absorber above the counters, at sea level, in \(g\,cm^{-2}\) of air equivalent. A lead thickness of \(167\ g\,cm^{-2}\) corresponds to an air thickness of \(100\ g\,cm^{-2}\). The vertical scale in Fig. 5 is chosen so that the ordinate

Fig. 5. Integral range spectrum of mesons at sea level. The range is measured in \(g\,cm^{-2}\) of air. The circles show the results of measurements of the momentum spectrum \(W^7\), normalized at \(920\ g\,cm^{-2}\) (double circle).

corresponding to \(100\ g\,cm^{-2}\) is equal to \(0.83\cdot 10^{-2}\) (the value of \(I_v\) for the hard component at sea level).

It may be assumed that Fig. 5 shows the integral range spectrum of mesons at sea level, i.e. that the ordinate at each point gives the number of mesons in \(cm^{-2}\ sec^{-1}\ sterad^{-1}\) with range greater than the corresponding abscissa.

This assumption is justified by the following circumstances: a) the number of protons in the penetrating component is very small; b) the number of electrons capable of penetrating through \(100\ g\,cm^{-2}\) of air is also

very small. Electrons detected under \(100\ \mathrm{g\ cm^{-2}}\) or even greater depths must be generated mainly by mesons as a result of secondary processes. In this case their number should be proportional to the number of mesons at all depths. This point of view agrees with the results of Emert, who found that the percentage decrease in the number of coincidences caused by inserting between

Fig. 6. Differential spectrum of meson ranges at sea level. The range is measured in \(\mathrm{g\ cm^{-2}}\) of air. The circles show results obtained by the following authors: Emert \(E^1\), Kœnig \(K^1\), Nielsen et al. \(N^2\), Rossi et al. \(R^4\), Sands \(S^3\), J. G. Wilson \(W^7\), and V. K. Wilson \(W^8\).

counters \(5\ \mathrm{cm}\) of lead is, within the limits of experimental error, the same at all depths. Fig. 6 shows the differential spectrum of meson ranges at sea level, \(i_\nu\), i.e. the derivative with respect to \(R\) of the curve giving the integral spectrum of ranges. The quantity \(i_\nu(R)\,d\omega\) represents the number of mesons arriving at sea level in one second, within the solid angle \(d\omega\) in the vertical direction, and stopping in \(1\ \mathrm{g}\) of a light absorber after traversing

thicknesses \(R\) of the same absorber. It is measured in \(g^{-1},\ sec^{-1}\times sterad^{-1}\). The experimental data used to determine the differential range spectrum are shown in Fig. 6 by circles. Some of these data were obtained by the coincidence method, i.e., they correspond to the difference in the number of coincidences of the telescope counters for different absorber thicknesses above the counter. The accuracy of this method is limited by fluctuations in the intensity of cosmic rays and by statistical errors; both circumstances have a stronger effect at the lower end of the range spectrum, where one has to operate with small differences in absorber thickness.

The first source of errors is eliminated, and the second is greatly reduced, when the anticoincidence method is used, in which the relative number of particles passing through a given absorber thickness and stopping in an additional (small) thickness is measured directly. Some data obtained by this method are shown in Fig. 6. In determining the absolute value of \(i\) by the anticoincidence method, corrections must be allowed for scattering and for secondary electrons produced as a result of the decay of mesons stopped in the absorber. These corrections are very uncertain. In addition, the coincidence and anticoincidence methods become completely inapplicable for ranges smaller than \(100\ g\,cm^{-2}\) of lead, since in this case electrons and protons begin to play a significant role. In this region the only possible method is the method of delayed coincidences, in which an apparatus is used that is very similar to the apparatus employed for determining the mean lifetime of mesons. In these arrangements, after passing through a certain thickness of lead or another material, the mesons stop in an absorber and subsequently decay into electrons, which are registered by Geiger–Müller counters surrounding the absorber. Some data obtained by this method are shown in Fig. 6.

The delayed-coincidence method unambiguously distinguishes mesons from other particles, and its accuracy in relative measurements is limited only by statistical fluctuations. However, determination of the absolute intensity of the differential range spectrum, even at a single point, requires calculation of the probability of registering a decay electron born in the absorber. The accuracy of these calculations is very doubtful. Moreover, one must also take into account the existing uncertainty in the energy of the decay electrons. The curve in Fig. 6 obtained from the available data represents the best estimate of the differential spectrum of mesons at sea level. In drawing this curve, the data obtained by the delayed-coincidence method were used to determine the slope of the curve between \(10\) and \(200\ g\,cm^{-1}\), whereas the estimate of the absolute intensity is based on results obtained by the coincidence method for thicknesses greater than \(150\ g\,cm^{-2}\).

If mesons lose energy only in collisions, their distribution in ranges can be calculated from the distribution in momenta (see § 5), using the theoretical relation between momentum and range (see Appendix, § 27).

Fig. 7. Differential intensity of slow mesons as a function of atmospheric depth.

Some data obtained by this method (the magnetic method) are plotted in Figs. 5 and 6, after normalizing the total number of mesons with momentum greater than \(2 \cdot 10^9\ \mathrm{eV}/c\) to the intensity of the integral range spectrum corresponding to a range of \(920\ \mathrm{g\,cm^{-2}}\). Very good agreement is obtained in the range interval from \(920\) to \(4000\ \mathrm{g\,cm^{-2}}\) (momentum interval from \(2 \cdot 10\) to \(10^{10}\ \mathrm{eV}/c\)). The discrepancy obtained at \(8400\ \mathrm{g\,cm^{-2}}\) (\(2 \cdot 10^{10}\ \mathrm{eV}/c\)) is probably not significant, if one takes into account the large errors obtained in

measurements of momenta exceeding \(2\cdot 10^{10}\ \mathrm{eV}/c\). It is possible that the discrepancy in the low-energy region of the spectrum is caused, in the experiments with Wilson chambers, by the strong magnetic field, which prevents the registration of some of these particles (although this effect was taken into account and experimentally estimated in the experiments of Blackett and Wilson).

7. Slow mesons at various altitudes

The variation of the number of slow mesons with altitude was measured*) R9, S3 by the delayed-coincidence method (see § 6). Three different experiments were carried out, in which mesons were recorded respectively with ranges between 5 and 22, 13 and 32, and 53 and 83 \(g\,cm^{-1}\) of air equivalent. No substantial differences were found in the results of these experiments, which indicates the absence of a slope (within the experimental errors) in the curve of the differential range spectrum at all altitudes between 5 \(g\,cm^{-2}\) and 80 \(g\,cm^{-2}\). The arrangement of the counters allowed particles to pass within a wide solid angle around the vertical. Consequently, the measured quantities refer rather to the integral intensity of slow mesons than to their vertical intensity. Fig. 7 presents the experimental data, normalized at sea level to the absolute value of the differential range spectrum of mesons at 10 \(g\,cm^{-2}\) (Fig. 6). The curve passing through the experimental points should be interpreted as determining the number of mesons per second per steradian, incident in the vertical direction and stopping in 1 g of air, as a function of altitude. Such an interpretation is based on the assumption of a small dependence of the angular distribution of slow mesons on depth. It should be noted that this assumption has not been checked experimentally.

II. TRANSFORMATIONS OF PARTICLES IN COSMIC RAYS

8. General considerations

As a result of processes in which elementary particles and quanta disappear and are produced, as well as processes in which energy is transferred to electrons and nucleons initially at rest, the structure of cosmic rays gradually changes as they pass through matter. These processes may be divided into three categories: electromagnetic interaction, nuclear interaction, and spontaneous decay. The theory of electromagnetic interaction has been developed in detail on the basis of the general principles of quantum electrodynamics and correctly represents the observed phenomena. For nuclear interaction and spontaneous decay

) See also G. Zhdanov and A. Naumov, DAN, 60, 1519 (1948). (Ed. note.)*

there exists no well-founded theory. The experimental data relating to these two classes of phenomena are described briefly in the following sections.

9. Meson decay

In the case when mesons are stopped in an absorber, delayed emission of charged particles occurs, which are apparently electrons^R1, M1, A4, R8, N1, C6, C7, C8, C9, C10, M2, S8, T3. If the absorber has a large atomic number, then delayed emission of electrons is observed only upon absorption of positive mesons; if, however, the absorber has a small atomic number, then it is observed after the absorption of both positive and negative mesons^C7, C10, V1. The probability that an electron will be emitted with a delay greater than \(t\), after absorption of a positive meson, is represented by the exponential law \(e^{-t/\tau^+}\). Within the limits of experimental errors, \(\tau^+\) has, for all absorbers, the value \(\tau^+ = (2.15 \pm 0.1)\) microseconds.

The emission of electrons that is a consequence of the absorption of negative mesons in light materials obeys the same law \(e^{-t/\tau^-}\). However, the lifetime \(\tau^-\) is different for different absorbers^T4, V2, V3, T5. In Fig. 8 the experimental values of \(\tau^-\) are plotted as a function of the atomic number \(Z\) of the absorber. From Fig. 8 it is seen that \(\tau^-\) practically coincides with \(\tau^+\) for \(Z \leqslant 4\) and rapidly decreases as \(Z\) increases from 4 to approximately 20.

These results can be explained by the assumption of the instability of free mesons with a natural lifetime, with respect to spontaneous decay, equal to

\[ \tau = (2.15 \pm 0.1)\,10^{-6}\ \text{sec}. \tag{6} \]

Because of electrostatic repulsion, a positive meson is stopped in the absorber at a large distance from the atomic nucleus and then decays spontaneously with its natural lifetime. However, a negative meson can fall onto a \(K\)-orbit, and therefore the fact that in elements with intermediate atomic number \(\tau^-\) is less than \(\tau\), while in elements with large atomic number decay is not observed, is attributed to the influence of neighboring nuclei.

It has been pointed out that the strong electric field existing near nuclei can reduce the natural lifetime of mesons^V2; however, the more plausible hypothesis is the capture of mesons by atomic nuclei. The competition between capture and decay reduces the true lifetime of negative mesons to values \(\tau^-\) determined by the equation

\[ \frac{1}{\tau^-} = (1/\tau) + (1/\tau_c), \tag{7} \]

where \(\tau_c\) is the characteristic lifetime for capture processes.

Since the radius of the \(K\)-orbit of mesons is inversely proportional to the atomic number \(Z\), the density of the meson wave function in the nucleus is proportional to \(Z^{3}\). On the other hand, the probability of capture is proportional to the density of the meson wave function in the nucleus and to the number of nucleons in it, which, at least for light elements, is approximately proportional to \(Z\). Consequently, \(\tau_c\) should vary as \(Z^{-4}\) ^{1}. The curves plotted in Fig. 8 represent \(\tau^{-}\) as a function of \(Z\), calculated from equation (7) on the assumption that \(\tau^{-}\) varies respectively as \(Z^{-3}\), \(Z^{-4}\), and \(Z^{-5}\). The curves are fitted to the experimental value at \(Z = 13\). Although the experimental data are consistent with a power law with exponent \(-4\), the accuracy is nevertheless insufficient to prove its validity.

If the capture hypothesis is correct, there should exist a relation connecting the true lifetime of negative mesons \(\tau^{-}\) and the relative number of absorbed and then decayed negative mesons. This quantity is given by the expression

\[ f = \tau^{-}/\tau . \tag{8} \]

Experiments were carried out to determine \(f\) in aluminum. The results obtained do not have sufficient accuracy either to disprove or to prove the capture hypothesis.

It is assumed that moving positive and negative mesons decay with the characteristic lifetime \(\tau\). Apparently this accounts for the anomalous absorption of mesons in the atmosphere; indeed, observation of this phenomenon first indicated radioactive instability

Fig. 8. Observed lifetime of negative mesons as a function of the atomic number of the absorber. The experimental points were obtained from measurements with beryllium (\(Z = 4\)), fluorine sodium (\(Z_{\text{mean}} = 10\)), magnesium (\(Z = 12\)), and aluminum (\(Z = 13\)). The three curves were calculated on the assumption that \(\tau_c\) is proportional respectively to \(Z^{-3}\), \(Z^{-4}\), and \(Z^{-5}\). They are normalized to the experimental value obtained for aluminum (\(Z = 13\)).

ity of mesons. It should be noted, however, that there are certain difficulties in a quantitative explanation of anomalous absorption if one adopts for \(\tau\) the value given by (6) and the most reliable values for the meson mass.

10. Decay products of mesons

At present there are very few data concerning the decay products of mesons. The laws of conservation of energy and momentum require that, in the decay of a meson, at least two particles be produced*). If only two particles are produced, then in the coordinate system in which the meson was at rest they have equal but oppositely directed momenta. One of the two particles, by virtue of the law of conservation of charge, must be charged. Since its mass is smaller than the mass of the meson, it is either an electron or a particle with a mass intermediate between the mass of the electron and the mass of the ordinary meson.

As experimental data show, when a meson is stopped in an absorber only one charged particle is emitted W3, A2, V3. There is also experimental evidence showing that the decay of a meson is usually not accompanied by the emission of photons H4, S4. If each meson decays into an electron and a neutral particle, then in the coordinate system in which the meson is at rest the decay electron must always have one and the same energy. This energy is approximately equal to half the rest energy of the meson, i.e. 50 MeV, in the case where the neutral particle is a neutrino. If the neutral particle is considerably heavier than the electron (a neutretto), then the energy of the decay electron must be correspondingly smaller. If more than one neutral particle is formed, then the energy of the decay electrons has a continuous distribution with a mean energy value smaller than 50 MeV.

So far only two photographs have been published of mesons in a Wilson chamber which stopped in the gas of the chamber and decayed, apparently into electrons, whose momenta were measured from the curvature of the tracks in a magnetic field.

One of the photographs W3 gives a value of the electron energy equal to approximately \(70 \pm 35\) MeV, in good agreement with the hypothesis of meson decay into an electron and a neutrino. In the second photograph A2 the electron energy is 24 MeV, which is more consistent with the hypotheses of decay into an electron and a neutretto or into an electron and several neutrinos**). To the scant information obtained with the aid of the Wilson chamber,

*) New data show that mesons may possibly decay into three particles. On this see G. Zhdanov and V. Khaidarov, DAN 65, 287 (1949). (Editor’s note.)

**) One more doubtful photograph, apparently showing a decay electron with an energy of 24 MeV, was demonstrated by Anderson at the January meeting of the American Physical Society in New York.

one may also add the results of the experiments of Conversi and PiccioniC9 on absorption, showing that the range of the decay electrons cannot be appreciably less than the calculated range of electrons with energy 50 MeV. Hence it follows that, before it is possible to determine reliably the nature of the products of meson decay, a still larger number of experiments will be required.

The problem of the fate of the rest energy of those mesons which stopped but did not decay in elements of medium or large atomic number has not been solved. When a meson is captured by a nucleus, one may expect that the release of the energy of the vanished meson will be followed by the disintegration of the nucleus. So far there are no Wilson-chamber photographs showing nuclear disintegration at the end of a meson’s path, although, as PiccioniP3 pointed out, if this phenomenon occurred it could hardly have remained unnoticed. In photographic emulsion, nuclear disintegrations have been found at the ends of meson tracksP1, O1, L2.

It is probable, however, that the mesons which produced these disintegrations are not ordinary mesons, but rather “heavy” mesons.

Piccioni investigated the possibility that the excitation energy, after absorption of a negative meson by a nucleus, is lost in the form of a $\gamma$-quantum (either directly or with the participation of hypothetical short-lived mesons). His preliminary results show that mesons, after stopping in iron, do not produce high-energy $\gamma$-rays.

11. Interaction of Mesons with Matter*)

Studies of the passage of mesons through matter do not provide sufficient grounds for asserting the existence of a nuclear interaction of moving mesons, i.e., an interaction which cannot be explained by electromagnetic phenomena.

Direct measurements by EhrenfestE2 and WilsonW4, W5 of the energy losses of penetrating particles in the metal plates of a Wilson chamber, as well as the comparison, discussed in § 6, of the momentum spectrum with the range spectrum of penetrating particles at sea level, show that if mesons do experience energy losses in nuclear interactions, the mean value of these losses is considerably smaller than the energy losses due to collisions with electrons.

One may expect that nuclear collisions produce scattering through large angles and reactions in which protons and mesons of high energies appear. As observations at sea level of the passage of several thousand particles through metal plates in a Wilson chamber show, such phenomena occur extremely rarely. For example, WilsonW6 found, in the passage of penetrating particles through a total thickness equivalent to 50 m of lead, only one case of nuclear interaction with the emission of a proton.

) For this chapter, see the review by V. L. Ginzburg, “On the Nuclear Scattering of Mesons,” in the collection Meson, Gostekhizdat, 1947. (Editor’s note*.)

Brode and StarrB11 discovered three such cases during the passage of penetrating particles through a total thickness equivalent to 200 m of lead. FreterF3 did not find a single case of nuclear disintegration during the passage of penetrating particles through 180 m of lead. CoddC4 observed the passage of about 450 penetrating particles through 3.8 cm of tungsten (a total thickness equivalent to 30 m of lead). He did not observe a single nuclear disintegration, but he did find several cases of scattering through large angles, which are difficult to explain by Coulomb interaction. Assuming that the particles undergoing nuclear interaction are mesons, from these scanty experimental data one can estimate that the value of the effective cross section for the meson interaction producing nuclear disintegrations of high energies, or scattering through large angles, lies in the range from 3 to \(7\cdot 10^{-27}\,\text{cm}^2\) per lead nucleus. On the other hand, this effective cross section may be much smaller, or even equal to zero, because, probably, the majority—and possibly all—of the observed cases of nuclear interaction are produced by high-energy protons.

SinhaS9 and ShuttS7 reported experiments showing that the partial scattering of mesons in matter through small angles is caused by a non-Coulomb interaction. However, the grounds for this conclusion do not seem sufficiently convincing. In any case, all these phenomena, interesting in themselves, cannot have a strong effect on the behavior of mesons. Thus, it may be considered justified to neglect the nuclear interaction of ordinary mesons in describing phenomena in cosmic rays.

12. Stars and penetrating showers*)

Observations of certain secondary phenomena in cosmic rays clearly testify to the existence of nuclear interaction. Among such observations are:

a) Photographs of “stars” in the Wilson chamber (see, for example, D1, M2, P4). In photographs in an uncontrolled Wilson chamber, groups of strongly ionizing particles are sometimes encountered, diverging at large angles from a point situated in the gas, the walls of the chamber, or solid material that may be located inside the chamber. Such groups of particles, regardless of their accompaniment by fast, weakly ionizing particles, we shall call “stars.” The tracks of stars can often be identified with the tracks of protons, \(\alpha\)-particles, or heavy nuclear fragments.

The number of particles moving upward is comparable with the number of particles moving downward. The energy of the individual particles of stars has the order—

) For this chapter see the reviews by S. Z. Belenky and L. E. Lazareva, “Penetrating Showers in Cosmic Rays,” and by V. L. Ginzburg, “Heavy Particles and Nuclear Disintegrations in Cosmic Rays,” in the collection Meson, Gostekhizdat, 1947. (Ed. note.*)

…of \(10^6\)—\(10^7\) eV. This phenomenon is interpreted as a nuclear explosion requiring the transfer to the nucleus, by an external agent, of energy of the order of \(10^8\) eV. In the majority of photographs there is no visible track of fast ionizing particles. This gives grounds for asserting that most stars are produced by a nonionizing component. According to Hazen, the frequency of photographs on which stars appeared, taken at an altitude of 4300 m, is approximately 5 times greater than the frequency of the appearance of stars at an altitude of 3000 m.

b) Photographs of “penetrating showers” in the Wilson chamber (see, for example, \(F^{2,12,57}\), \(P^4\), \(H^1\), \(R^3\)). In some photographs in the Wilson chamber there are groups of weakly ionizing particles diverging predominantly downward from a common point situated either outside or inside the chamber (however, up to the present not a single photograph has been published of a group of particles produced in the gas). Such groups of particles are called “penetrating showers.” From the specific ionization of the particles of penetrating showers one may conclude that they carry a single charge and have relativistic velocity. Some of these particles may be identified with protons, some with mesons. However, in most cases their nature cannot be determined, and therefore the relative numbers of mesons and protons in penetrating showers are unknown. With the exception, perhaps, of one or two cases \(H^1, R^3\), it is also impossible to establish whether the mesons of penetrating showers are ordinary mesons. Occasionally, from the same center where penetrating particles of high energies are produced, protons of low energies, \(\alpha\)-particles, and heavy fragments of nuclei appear. This shows that, at least in some cases, the formation of penetrating showers is accompanied by nuclear disintegrations. But even if this always occurs, it is possible that the disintegration products of low energies are detected rarely, namely only in the case when the point of origin is separated from the effective volume of the chamber by a sufficiently small thickness of absorber. Penetrating showers are interpreted as reactions in which mesons are produced, while the nucleus breaks up into many fragments of low energy and a small number of nucleons of high energy. For the occurrence of such phenomena it is necessary to transfer to the nucleus energy of the order of \(10^9\) eV or even greater. Penetrating showers, apparently, are produced both by ionizing and by nonionizing particles. Photographs of penetrating showers were obtained both in uncontrolled expansions and in expansions controlled by Geiger–Müller counters. The counters were arranged in such a way that penetrating showers were recorded preferentially over the other phenomena. In uncontrolled expansions, stars appeared considerably more often than penetrating showers.

c) Stars in photographic emulsion. Nuclear explosions of the same type as those causing stars in the Wilson chamber are evidently the cause of the “stars” that are observed…

in a microscopic examination of the photographic emulsion. Since relativistic particles do not leave noticeable tracks in the emulsion, usually only protons of low energies and nuclear fragments are recorded in it. Therefore the photographic method does not make it possible to distinguish nuclear phenomena of high energies from nuclear explosions of low energies. However, recent improvements in photographic technique have made it possible to detect mesons of low energies in nuclear explosions \(^{L1,L2}\). The frequency of star formation increases rapidly with altitude. According to new measurements by Perkins*) \(^{P1}\), the number of stars increases 10-fold from sea level to \(3600\ \mathrm{m}\), and doubles from \(3600\ \mathrm{m}\) to \(4300\ \mathrm{m}\). This corresponds approximately to an exponential dependence on atmospheric depth with a mean free path of \(135\ \mathrm{g\,cm^{-2}}\).

13. Ionization Bursts

It is convenient to study the variation of the frequency of nuclear reactions as a function of altitude and other parameters by means of observations of bursts in an ionization chamber. Bursts in a thin-walled unshielded ionization chamber may be caused either by the passage through the chamber of a large number of electrons of air showers, or by several strongly ionizing particles from stars. The two phenomena can be separated by simultaneous registration of pulses in two or several ionization chambers placed close to one another. Whereas air showers produce pulses of approximately the same magnitude in all chambers, nuclear disintegrations for the most part produce pulses in one of the chambers, more rarely in two, and very rarely in several**). Moreover, the pulses produced by nuclear disintegrations in several chambers usually have different magnitudes. Rossi and Williams \(^{R10}\), using this method, showed that at an altitude of \(3500\ \mathrm{m}\) about 98% of the pulses in a thin-walled ionization chamber of two-liter volume, filled with argon at a pressure of \(5\ \mathrm{atm}\), which corresponded to energy losses exceeding \(6\ \mathrm{MeV}\), were caused by nuclear disintegrations. Air showers were responsible for 2% of the pulses.

The registration threshold and the characteristics of the chamber must be determined accurately, since the ratio of the number of bursts due to showers to the number of bursts produced by nuclear disintegrations increases strongly as the registration threshold and the pressure increase.

Some preliminary results on the frequency of bursts at different altitudes are presented in Fig. 9. These results were obtained by the following investigators: Bridge \(^{B7,B9}\) and Williams \(^{R10}\) at

) See also G. Belovitskii and L. Sukhov, DAN 62, 207 (1948). (Ed. note.)*

) This statement is apparently not entirely correct. This is indicated by the experiments of L. Razorenov and A. Knyazev, DAN 60, 1531 (1948). (Ed. note.)

Fig. 9. Number of counts of unshielded ionization chambers and slow-neutron detectors at various atmospheric depths.

on the ground (both works were carried out at sea level and at various elevations in the mountains); by Bridge\(^{B7}\) in an airplane; by Hulsizer\(^{H6}\) with the aid of radiosondes; by Tatel and Van Allen\(^{T2}\) with cameras mounted in the nose section of a rocket. All experiments, with the exception of Hulsizer’s experiments, were carried out with two-liter cylindrical chambers of the type described above. The chamber used by Hulsizer had the same construction, but smaller dimensions. With Hulsizer’s chamber measurements were carried out at an altitude of \(9000\) m in an airplane, and the results were then used to normalize the data obtained with this chamber to Bridge’s data at this altitude.

For slow neutrons the data obtained by Agnew et al. and by Yuan et al. are normalized at a depth of \(310\ \mathrm{g\,cm^{-2}}\). Fowler’s data on slow neutrons are normalized at \(1030\ \mathrm{g\,cm^{-2}}\).

As was already indicated above, at an altitude of \(3500\) m only a small part of the bursts was caused by showers. On the other hand, the altitude dependence of air showers with a particle density sufficient to create bursts in the lower part of the atmosphere apparently differs little from the altitude dependence of the burst frequency. Above \(5000\) m, air showers increase with increasing altitude more slowly than bursts\(^{K2}\).

Thus, one may confidently suppose that at all altitudes the contribution of air showers to the appearance of bursts is small (see Fig. 9). However, in the rocket experiments, a considerable number of bursts were probably caused by showers produced in the heavy materials situated in the immediate vicinity of the chambers. Since four chambers were used here, the bursts caused by showers can be identified by the approximately equal magnitude of the bursts in two or several chambers. The same selection rule can also be applied to the measurements carried out by Williams at an altitude of \(3500\) m with four ionization chambers of the same type. From a comparison of the two experiments it was found that the frequency of bursts caused by nuclear disintegrations increases from \(3500\) m to the boundary of the atmosphere by a factor of 73. This result is shown in Fig. 9 by a square. The dependence of the burst frequency on depth for altitudes below \(10000\) m is represented, within the limits of experimental errors, by an exponential law \(e^{-x/L_a}\), where \(L_a = 138\ \mathrm{g\,cm^{-2}}\). This mean free path differs little from the mean free path for stars obtained by the photographic method.

The dashed curve in Fig. 9 represents the function

\[ e^{-x/138}+\frac{x}{138}\,Ei\left(-\frac{x}{138}\right). \]

The data shown in Fig. 9 refer to bursts greater than \(8\ \mathrm{MeV}\). It should be noted that, at least for depths greater than \(250\ \mathrm{g\,cm^{-2}}\), approximately the same curve is obtained if the registration threshold of the selected bursts lies in the range from \(5\) to \(10\ \mathrm{MeV}\).

14. Protons, α-particles, and neutrons of low energies

Observations in a Wilson chamber and studies of photographic plates have shown the existence of a large number of protons and α-particles in the atmosphere *). These particles have an approximately isotropic distribution over directions.

The number of single tracks in photographic emulsion, according to Perkins Р¹, increases from sea level up to an altitude of 4300 m, as does the number of stars.

It is natural to suppose that the observed slow protons and α-particles are produced in nuclear disintegrations of the same type as the processes generating stars. Under this assumption one can, from the observed distribution of star particles by ranges and the mean number of particles per star, calculate the ratio of the number of single tracks to the number of stars.

Perkins established that the results of the calculations agree with the experimental data. From the fact that the ratio of the number of single tracks to the number of stars changes little from sea level to 4300 m, it follows that the mean number of particles per star and the distribution of star particles by ranges change little between these two altitudes.

Another effect apparently connected with nuclear disintegrations is the presence in the atmosphere of neutrons of thermal or nearly thermal velocities. These neutrons are usually registered by means of neutron counters. Since the effective cross section of the reaction $(n,\alpha)$ for boron is inversely proportional to the neutron velocity, it may be assumed that the number of counts of a boron detector measures the density of slow neutrons.

Most of the observed neutrons probably have an energy of the same order as the ionizing particles in the stars, i.e. an energy of about $10^7$ eV. They are then slowed down, at first by inelastic collisions and later by elastic collisions with the nuclei of air atoms, until they are absorbed by nitrogen in a reaction of the type $(n,p)$. According to Bethe, Korff, and Placzek В⁴, the mean range of neutrons from the place of their production to the place of absorption is, in order of magnitude, $150\ \mathrm{g\,cm^{-2}}$. Thus, at distances greater than $150\ \mathrm{g\,cm^{-2}}$ from the boundary of the atmosphere, the density of slow neutrons should vary as the number of nuclear disintegrations. From Fig. 9 it is seen that this is confirmed by experiment; in this figure the data on slow neutrons obtained by Fong-Fer F⁴, Agnew, Bright, and Froman А¹, and Yuan and Ladenburg Y² are presented together with data on bursts in thin-walled ionization chambers.

) See also A. Alikhanov, A. Alikhanyan, and S. Nikitin, J. of Phys. 9, 167 (1945), and V. Veksler, N. Dobrotin, and V. Khvoles, J. of Phys.* 9, 277 (1945).

15. Generation of the electron component in nuclear interaction*)

Some of the electrons and photons in the atmosphere are produced in the decay of ordinary mesons or in the electromagnetic interaction of mesons with matter (chiefly in collisions). However, experimental evidence has recently been obtained for the generation of the electron component in other processes, in which the nuclear interaction plays an essential role.

Photographs were obtained in a Wilson chamber in which electrons appeared simultaneously with \(D^1\) stars. In other published photographs one can see showers containing both electrons and penetrating particles \(F^2, B^{10}\).

Fig. 10. Setups for studying the generation of showers by penetrating particles.

In experiments of another type \(B^8\), the ionization chamber was placed under a group of Geiger–Müller counters, with a 15-centimeter layer of lead between the two instruments (Fig. 10, A).

Some of the pulses in the ionization chamber were accompanied by a simultaneous discharge in the Geiger–Müller counters and were interpreted as the result of the passage through the ionization chamber of a group of electrons produced in the lead by ionizing particles. These particles could not have been electrons, since in order to produce under 15 cm of lead a shower of such magnitude (35 particles or more) the latter would have had to possess an energy exceeding \(10^{12}\) eV.

*) For this section see the articles devoted to “special showers.” For example: V. Veksler, L. Kurnosova and A. Lyubimov, ZhETF 17, 1026 (1946); N. Birger, DAN 61, 245 (1948); S. Azimov, N. Birger, A. Gorbunov, DAN (in press). (Editor’s note.)

But electrons of such energy are encountered too rarely for the observed showers to be ascribed to them. The altitude dependence of the effect also excludes the possibility that the observed showers are generated by ordinary mesons in collisions or in radiation processes. Preliminary results\(^ {88}\) show that the frequency of the effect increases by several hundred times from sea level \((1030\ \text{g cm}^{-2})\) to 9000 m \((310\ \text{g cm}^{-2})\). The total intensity of the penetrating component increases over this range of altitudes by only a factor of 6. It may be supposed that the number of mesons with energy sufficiently large to generate showers of the observed magnitude grows still more slowly.

Data on the altitude dependence of showers generated under large thicknesses of lead are presented in Fig. 12.

The generation of showers by penetrating particles was investigated in considerable detail by means of the apparatus shown in Fig. 10, B\(^ {8,10}\). Coincidences of discharges in a group of Geiger–Müller counters and an ionization chamber controlled a Wilson chamber containing eight lead plates, each 6 mm thick. Many photographs were obtained which showed the generation, by penetrating particles, of electron showers in the lead plates. Some of these showers are so large that the lower limit of the energy of the particles generating them must be of the order of \(10^{10}\) eV. Two-thirds of the electron showers in these photographs are accompanied by penetrating particles, or stars, or by both simultaneously. Sometimes some of the penetrating particles can be identified with mesons.

In the remaining part of the photographs the density of electron showers was so great that it proved impossible to detect penetrating particles in them.

To obtain more detailed information on the penetrating component of the showers, the apparatus was modified as shown in Fig. 10, C. To separate the electron component, a block of lead was placed between the ionization chamber and the Wilson chamber. In most of the photographs obtained with such an apparatus, penetrating particles were present. This experimental result agrees with the assumption that, in nuclear events in which electron showers are generated, penetrating particles are always also produced. On the other hand, it is unclear whether penetrating showers are always accompanied by an electron component. From the fact that in most photographs of penetrating showers there are no electron tracks, no conclusion can be drawn about the absence of joint generation of penetrating showers and electrons. Indeed, the electron part of the showers is absorbed much faster than the penetrating part and therefore can be detected only in the case where the point of origin of the shower does not lie too deep in the absorber.

16. Hard Showers

Coincidences of discharges were sometimes observed in Geiger–Müller counters separated from one another by thick layers of lead and not located on a single straight line. Janossy \(J^1\) showed that the cause producing these coincidences is not ordinary cascade showers. Such phenomena were called hard showers. Figure 11 shows two typical arrangements for studying hard showers.

Fig. 11. Two arrangements for studying hard showers. \(A^{J^3}\); coincidences were recorded only when at least one of the counters in each of the seven groups shown in the figure discharged. \(B^{T6}\); coincidences were recorded only in the case when a discharge occurred in at least two counters of each of the three horizontal groups. (The two vertical groups shown in the figure were intended to obtain additional information on the structure of the showers.)

Apparently, the nuclear interaction by which hard showers are generated is of the same type as that which also creates penetrating showers and the electron component. In this connection one must keep in mind the possibility of a subsequent interaction of the particles of hard showers with nuclei. If this actually takes place, then, taking into account the very large amount of material around Geiger–Müller counters, one should conclude that multiple processes may play an essential role in the generation of hard showers. The numerous experimental results obtained in the study of hard showers by Janossy and his collaborators are not discussed here in detail.

We shall only note that the number of hard showers increases rapidly as the depth of the atmosphere decreases. Janossy and Rochester \(J^5\), investigating the barometric effect of hard showers, found a ten-percent increase in the number of hard showers for a decrease of atmospheric pressure by \(1\ \mathrm{cm}\ \mathrm{Hg}\). Figure 12 shows the preliminary results of a recent investigation by Tinlot \(T^6\) of the altitude dependence of hard showers. These data, as well as the barometric effect, correspond to absorption of hard showers in the atmosphere according to the exponential \(e^{-x/L_a}\), where \(L_a = 125\ \mathrm{g}\ \mathrm{cm}^{-2}\). A similar result was obtained by Sala and Wataghin \(S^1\), who used an apparatus that preferentially recorded air penetrating showers rather than showers born in the absorber.

Data obtained by Bridge[^89], who studied the generation of bursts by penetrating particles with the aid of the apparatus shown in

Fig. 12. Hard showers and bursts generated by penetrating ionizing particles, as a function of atmospheric depth. The hard showers were observed with the aid of the apparatus shown in Fig. 11, B; the diameter of the counters was \(2.5\) cm, their length \(25\) cm. The generation of bursts was observed with the aid of the apparatus shown in Fig. 10, A. The counters had a diameter of \(2.5\) cm and a length of \(50\) cm; the chamber had a diameter of \(7.5\) cm and a length of \(52\) cm; the argon pressure in the chamber was 5 atmospheres; bursts that produced ionization in the chamber of more than \(3.2\) MeV were recorded. Along the ordinate axis is plotted the true number of counts per hour.

Fig. 10, A, are included in the graph presented in Fig. 12. It follows from this graph that the number of bursts increases with altitude somewhat faster than the number of hard showers, but the difference does not exceed the limits of experimental error.

17. Origin of Nuclear Phenomena. The \(N\)-Component

It is quite possible that any particles possessing sufficiently high energy can produce the nuclear reactions described in the preceding sections. However, there is no doubt that the effective cross section for nuclear interaction is different for different types of particles. Therefore, from a phenomenological point of view it is convenient to suppose that the particles generating nuclear disintegrations form a separate component, which for brevity we shall call the \(N\)-component. The next problem is to resolve the question of the nature of this component.

a) Mesons with momenta from \(3\cdot 10^8\) to \(10^{10}\) eV/c. These particles constitute the principal part of the hard component at sea level. Since all nuclear effects increase with altitude considerably faster than the hard component, one may conclude that mesons with the above-mentioned momenta play an insignificant role in the generation of nuclear disintegrations at great altitudes.

b) Mesons with momenta less than \(3\cdot 10^8\) eV/c. These mesons cannot produce high-energy nuclear reactions, but their participation in the generation of low-energy disintegrations, for example stars, cannot be excluded. Indeed, it was often assumed that most stars were generated in the process of capture by nuclei of ordinary negative mesons. But, as has already been mentioned, this hypothesis is refuted by the absence of indications of the existence of this effect in the Wilson chamber.

In fact, in the majority of photographs of stars in the Wilson chamber no tracks were noticeable that could be attributed to slow mesons. Moreover, the number of slow mesons increases with altitude less rapidly than the number of stars.

c) Mesons with momenta greater than \(10^{10}\) eV/c. One cannot exclude the possible participation of mesons of very high energies in a considerable part of nuclear disintegrations. In this case it must be assumed that the effective cross section for nuclear interaction of fast mesons is so large that it determines the rapid absorption of these particles in the atmosphere.

d) Electrons and photons. The following experimental facts show that electrons and photons do not play an essential role in the generation of nuclear disintegrations.

1) In a Wilson chamber placed under lead, photographs were obtained of stars and penetrating showers in which electrons were absent.

But it is known that the probability of the emergence from a lead screen of a high-energy photon without an accompanying electron shower is negligibly small.

2) Analysis of photographic plates exposed under thick lead screens shows that the flux of star-like

of the component under lead is insignificantly smaller than over lead[^P].

3) By means of experiments with a group of ionization chambers arranged in such a way that it was possible to distinguish showers from nuclear disintegrations (see § 13), the absorption of the component generating the stars was roughly measured[^B10]. 30 cm of lead reduced the frequency of bursts to 30% of its initial value. The number of electrons and photons of high energies under 30 cm of lead is considerably less than 30% of their number in air.

d) Protons and neutrons. Nucleons of high energies undoubtedly play an important role in the observed nuclear phenomena, although these nucleons cannot cause all disintegrations. It is known from experiments[^H3] that the effective cross section for the nuclear interaction of neutrons with energy close to \(10^8\) eV has the same order of magnitude as the effective interaction cross section of the \(N\)-component obtained from its absorption coefficient. In addition, if the current hypothesis concerning the mesonic character of nuclear forces is correct, one may suppose that mesons of sufficiently high energies are created in the interaction of two nucleons. It is possible that penetrating showers are a manifestation of such a generation process.

e) New particles. It is possible that the \(N\)-component contains particles (charged and uncharged) different from electrons, protons, and ordinary mesons. Recently Occhialini, Powell, and collaborators[^L1] [^L2] experimentally proved the existence of such particles*). It follows from their experiments that these particles interact strongly with nuclei, while ordinary mesons probably do not arise in the process of nuclear disintegrations, but are a product of the decay of heavy particles produced in nuclear interaction.

Recently many theoretical speculations have arisen connected with neutral mesons. It is possible that the hypothesis—which has some theoretical grounds—that the electron showers accompanying nuclear disintegrations arise from photons appearing in the decay of short-lived mesons is correct.

In conclusion it should once more be noted that what we have called the \(N\)-component consists to a considerable extent of high-energy protons and neutrons and also, probably, contains new types of particles. If these particles have a very short lifetime, then the effects produced by them are difficult to distinguish experimentally from the nuclear interaction in which they are generated.

*) Earlier than Occhialini and Powell, A. I. Alikhanov, A. I. Alikhanian, and A. Weissenberg discovered new particles in cosmic rays possessing a mass spectrum. The authors called these particles varitrons. On varitrons see A. Alichanian, A. Alichanov and A. Weissenberg, Journ. of Phys. XI, 97, 199 (1947); A. Alikhanyan, A. Alikhanov and A. Weissenberg, JETP 18, 301 (1948); A. Alikhanyan, A. Alikhanov, V. M. Morozov, G. Muschelischwili and A. Khrimian, JETP 18, 673 (1948). (Ed. note.)

III. GENERAL CONSIDERATION OF PHENOMENA IN COSMIC RAYS

18. Primary component

The study of the structure of cosmic rays, of the interactions between their various components and matter, and of the dependence of various effects on altitude, latitude, and direction provides sufficiently many data to attempt to fill the gaps in our knowledge with well-founded hypotheses and to create a general picture of the phenomena in cosmic rays.

For this purpose we shall first consider the experimental data that help us draw a conclusion about the nature of the primary component of cosmic rays.

a) The latitude effect shows that the primary component contains a large number of charged particles with momenta from \(4.5\) to \(15 \cdot 10^9\ \mathrm{eV}/c\). The shape of the latitude curve indicates that particles with momenta smaller than \(4.5 \cdot 10^9\ \mathrm{eV}/c\) are encountered comparatively rarely, if they exist at all.

b) From the east–west asymmetry it follows that the majority, and possibly all, of the penetrating particles observed at altitudes from sea level up to \(9000\ \mathrm{m}\), originate from the positively charged primary component \(J7;\ S5;\ Y1;\ S6\).

c) Ordinary mesons, owing to their short lifetime, cannot be part of the primary radiation.

d) The altitude dependence of the phenomena considered in §§ 12–16 compels us to suppose that the primary component interacts strongly with nuclei, as a result of which it is rapidly absorbed in the atmosphere. Indeed, no effect produced by cosmic rays can decrease with increasing depth faster than the primary radiation that produced it directly or indirectly.

e) Schein\(^5\) indicated that the primary component apparently does not contain any appreciable number of electrons or photons*). Recently experiments were carried out for the purpose of quantitatively studying this important question \(H6\). A cylindrical ionization chamber 5 cm in diameter and 10 cm long, enclosed in a lead shell 2.5 cm thick, was sent to a great altitude by means of a balloon-sonde. Ionization bursts greater than a certain specified magnitude were recorded. The threshold was chosen equal to the ionization produced by 80 relativistic particles passing through the chamber perpendicular to its axis. This corresponds to the average size of a shower formed in the lead shell by an electron or photon with energy \(4.5 \cdot 10^9\ \mathrm{eV}\).

) A more rigorous proof of the absence of a large number of electrons in the primary component is given in the paper by C. Bricker, S. Vernov, I. Evreinova, G. Sokolov, and T. Charakhchyan, DAN 57, 141 (1947). (Ed. note.*)

At an altitude of 27,000 m (a depth of \(20\ \mathrm{g\,cm^{-2}}\)) 300 pulses per hour were recorded. Comparison with the counting rate in a chamber not shielded by lead shows that, apparently, 50% of these pulses were due to nuclear disintegrations in the walls of the chamber or in the gas filling it (see § 13). If one assumes that all the pulses were due to showers emerging from the lead, and takes into account the geometry of the experimental arrangement, then, as an approximate upper limit for the integral intensity of electrons and photons with energy greater than \(4.5\cdot 10^9\ \mathrm{eV}\), we obtain

\[ J_2 = 3\cdot 10^{-3}\ \mathrm{cm^{-2}\ sec^{-1}} \quad \text{at } x = 20\ \mathrm{g\,cm^{-2}}. \]

Since \(20\ \mathrm{g\,cm^{-2}}\) in air amounts to about \(1/2\) radiation unit of length, the integral intensity of high-energy electrons and photons in the upper part of the atmosphere cannot be much greater than the value given above. On the other hand, the value of \(J_2\) for the total radiation incident on the boundary of the atmosphere is approximately \(0.4\ \mathrm{cm^{-2}\ sec^{-1}}\) (see § 25). Hence it may be concluded that electrons and photons with energy exceeding \(4.5\cdot 10^9\ \mathrm{eV}\) constitute no more than 1% of the primary particles. Indeed, the assumption of the presence in the primary component of high-energy electrons or photons is not necessary, since the showers observed in small numbers may be due to the interaction of fast protons with nuclei (see § 15). In any case, it may be regarded as established that the majority of effects in cosmic rays can be explained without assuming the presence in the primary component of high-energy electrons and photons.

From what has been set forth one may conclude that, in all probability, the primary component consists almost exclusively of high-energy protons § 5. In fact, apart from this, only two other assumptions are possible: a) the primary radiation, at least partly, consists of nuclei heavier than hydrogen nuclei, and b) in addition to electrons and protons, there exist other stable elementary particles. However improbable such assumptions may be, they still cannot be rejected solely on the grounds that such hypothetical components of the primary radiation have never been observed near sea level, since the reason for this could have been their rapid absorption in the atmosphere, which, as is known, is characteristic of primary radiation.

Considering the problems of cosmic radiation phenomenologically, we shall assume that the primary component consists of positively charged particles, different from electrons and ordinary mesons and strongly interacting with atomic nuclei. Despite our conviction that these particles are protons, we shall use for them the less specific name of the primary component of cosmic rays. Proceeding from this, it follows—

giving the general picture of the phenomena in cosmic rays. The primary component, penetrating into the atmosphere, interacts with atomic nuclei. In such interactions the nuclei disintegrate, emitting nucleons of various energies. In these same interactions there arise elementary particles identical with ordinary mesons, or particles which turn into these mesons as a result of decay. In this process electrons or photons are produced, either both together, directly or as a result of the decay of short-lived mesons. The electron component of cosmic rays is due partly to such electrons or photons, and partly arises in decay or in other secondary processes occurring with ordinary mesons. The \(N\)-component consists partly of the primary component that has reached the point of observation, partly of high-energy nucleons arising in collisions with nuclei, and, possibly, of new particles (different from ordinary mesons) formed as a result of these collisions.

Fig. 13. Diagram of the arrangement of Geiger–Müller counters in the head section of rocket \(G^2\).

19. Phenomena in cosmic rays beyond the limits of the atmosphere

Experiments with cosmic rays have recently been carried out at altitudes up to \(160\) km by means of rockets. The results obtained at altitudes greater than that corresponding to a pressure equal to about \(2.5\ \mathrm{g\ cm^{-2}}\) characterize the properties of cosmic rays in free space.

In a number of experiments Geiger–Müller counters were used with absorbers necessary for measuring the penetrating power of cosmic radiation beyond the limits of the atmosphere and the secondary effects produced by it. Here we shall describe in detail the results obtained by Golian and Krause \(^{G2}\), which, at least qualitatively, agree with the results of other authors.

Their setup is shown in Fig. 13. Various combinations of coincidences and anticoincidences between Geiger–Müller counters were recorded. Some of the most important results obtained are given in Table I. One or more numbers in parentheses indicate the numbers of the counters whose coincidences or anticoincidences are recorded. The plus sign denotes coincidences, and the minus sign denotes anticoincidences.

Table I

Frequency of counts recorded by the system of counters shown in Fig. 13, in free space

Type of registration	Number of counts per minute
\((1) + (3) + (6)\)	219
\((1) + (3) + (6) - (2, 4, 5)\)	68
\((1) + (3) + (6) + (10, 11, 12)\)	144
\((1) + (3) + (6) + (13, 14, 15)\)	98
\((1) + (3) + (6) + (13, 14, 15) - (2, 4, 5)\)	40

It was also found that when an event of the type \((1) + (3) + (6) + (7, 8, 9)\), \((1) + (3) + (6) + (10, 11, 12)\), or \((1) + (3) + (6) + (13, 14, 15)\) occurred, a discharge often occurred in more than one counter of the systems \((7, 8, 9)\), \((10, 11, 12)\), or \((13, 14, 15)\).

Analysis of these data leads to the following conclusions:

a) Cosmic particles observed beyond the limits of the atmosphere very intensively produce secondary effects.

b) A coincidence between the counters of the vertical telescope is accompanied in more than 50% of cases by discharges in counters located outside the solid angle defined by the telescope. Thus, in this experimental setup, the majority of coincidences between counters located along a straight line were caused not by one and the same particle passing through the counters, but rather by a group of particles arising in secondary processes. Therefore, from the observed coincidences or anticoincidences it is difficult to determine the number of incident particles.

c) In a very large number of cases, a discharge occurred in the unshielded counters of the telescope \((1, 3, 6)\), while in the shielded counters \((10, 11, 12, 13, 14)\) it was not observed.

Since there were no heavy substances above the setup, in order to explain observation b) it is necessary to assume that the majority of secondary phenomena recorded by the counters occurred in the lead casing. The latter was located under the counters \((1, 2, 3, 4, 5)\). Consequently, one may conclude that many secondary particles are formed at large angles to the direction of the incident particles. However, these angles will not be very large because, outside the atmosphere, the number of incident particles forming an angle from \(\vartheta\) to \(\vartheta + d\vartheta\) with the vertical is proportional to \(\sin \vartheta\, d\vartheta\). Thus,

Thus, a significant fraction of the primary particles entering the setup have an almost horizontal direction. The existence of such interactions of cosmic rays with nuclei, in which the secondary particles produced fly out at a large angle, is confirmed by a number of photographs of penetrating showers in Wilson’s chamber. The considerations given in § 28 (Appendix) show that even for particles of considerable energy one may expect large angles of emission, if one assumes that the nuclear interactions in which secondary particles arise can be described as collisions between two free nucleons.

Result c) may be interpreted as an indication of the existence, in the radiation reaching the boundary of the atmosphere, of particles which are stopped in not very large thicknesses of lead and do not produce any secondary particles capable of causing discharges in the Geiger–Müller counters located under the lead. It is possible that this interpretation is ambiguous. From Table I it is seen that in 80% of the cases, when a discharge occurs in counters 1, 3, 6, and does not occur in counters 13, 14, 15, a pulse is registered in the side counters 2, 4, 5. It is therefore difficult to be certain that those comparatively few cases in which a discharge occurs in 1, 3, 6 and occurs neither in 13, 14, 15 nor in 2, 4, 5, are really due to the passage through counters 1, 3, 6 of particles stopped in the 12 centimeters of lead between 6 and 13, 14, 15. However, if this interpretation is correct, one may conclude that the particles observed at the boundary of the atmosphere are not primary particles. Indeed, it is difficult to imagine that particles possessing a sufficiently large energy to pass through the geomagnetic barrier could be absorbed in 12 cm of lead without producing secondary particles, which would be observed under the lead. The presence of secondary particles at the boundary of the atmosphere can be explained only on the assumption that these particles are formed with a broad angular distribution.

If a charged particle has an upward-directed momentum, but one smaller than that required to overcome the geomagnetic forces, then the particle itself or the charged products of its decay return to the earth under the action of the earth’s magnetic field. Thus, at the boundary of the atmosphere, secondary particles will move in all directions, both upward and downward (with the exception of particles different from ordinary mesons, or particles with a comparable or shorter lifetime, occurring in the downward-directed flux). A theoretical estimate of the energy of the upward-moving particles is given in the Appendix (§ 29).

20. Nuclear interactions of high energies

The \(N\)-component is phenomenologically defined as the component of cosmic rays responsible for nuclear interactions. Therefore any measurement of the frequency of nuclear disintegrations may be considered

be regarded as a measurement of the intensity of the \(N\)-component. It is unnecessary to emphasize that different methods register different parts of the \(N\)-component in different ways and have different sensitivities depending on the direction of the incident radiation. It is therefore not surprising that different nuclear reactions differ from one another in their altitude dependence. Indeed, from the frequency of occurrence of particular nuclear reactions at different altitudes it is possible to determine the altitude dependence of the total intensity, as well as the composition, energy spectrum, and angular dependence of the \(N\)-component.

The experimental data at our disposal at present are so incomplete that for the time being only preliminary conclusions can be drawn.

In the present section we shall consider the results of observations of hard showers and of the production of penetrating-particle showers. Both of these phenomena are due to the high-energy \(N\)-component. In experiments of both types the vertical radiation has the greatest probability of being registered. The altitude dependence of these phenomena will therefore be the same as the altitude dependence of the vertical intensity of the high-energy \(N\)-component. The experimental results shown in Fig. 12 indicate that this intensity is approximately an exponential function of the atmospheric depth, and that the “mean free path” \(L_a\) has a value of the order of \(125\ \mathrm{g\,cm^{-2}}\).

The attenuation of the high-energy part of the \(N\)-component with increasing atmospheric depth cannot be regarded as a simple absorption process. In other words, one cannot assume that the entire \(N\)-component present at some depth consists of primary particles which have not undergone any collisions with nuclei in the upper layers of the atmosphere. Indeed, there is evidence that a large part of the high-energy \(N\)-component observed at a given depth from the boundary of the atmosphere is of secondary origin and probably consists of protons and neutrons produced in nuclear collisions.

Observations in a Wilson chamber show that ionizing and non-ionizing particles are responsible for approximately equal numbers of penetrating showers \(P^4\). The same is true for hard showers registered in experiments with counters \(J^4\)*). On the other hand (see § 18), there is confidence that the primary component does not contain a large number of neutral particles.

There then arises the question of the relation between the penetration depth \(L_a\), which determines the variation of the high-energy \(N\)-component with depth, and the “range” \(L_c\), i.e., the distance traveled by particles between two successive nuclear collisions. Before

) On the existence of a neutral component generating high-energy bursts, see A. Lyubimov, L. Korablyov, and V. Miller, DAN 61, 633 (1948). (Ed. note.)*

In all, it should be noted that \(L_a\) cannot be smaller than \(L_c\). However, \(L_a\) may be considerably larger than \(L_c\). It is possible, for example, that in most nuclear collisions the \(N\)-particle loses only a small part of its energy. It is also possible that the passage of the \(N\)-component through the atmosphere is a cascade process, in which the mean free path depends on the energy spectrum of the radiation, in analogy with what occurs in the case of cascade showers, in which multiplication continues until the energy of the secondary particles becomes less than some “critical energy.”

From the theoretical point of view it is reasonable to assume that the effective cross section for nuclear interactions at high energies is simply equal to the geometrical transverse cross section of the nucleus, whose value may approximately be represented in the form

\[ \sigma=\pi(1.4\cdot10^{-13})^2 A^{2/3}. \tag{9} \]

The mean free path in air corresponding to the geometrical transverse cross section is \(65\ \mathrm{g\ cm^{-2}}\). Hence the following limits can be found for the value of \(L_c\):

\[ 65<L_c<125. \]

High-energy protons may either be part of the \(N\)-component or arise in the interaction of the high-energy \(N\)-component with matter. Therefore one may expect that their number varies with depth as \(e^{-x/25}\). The presently available, very scanty experimental data agree with the assumption that, at least in the interval of depths from 250 to \(1030\ \mathrm{g\ cm^{-2}}\), this is true for protons whose energy is greater than approximately \(1/10\) of their rest energy, and that in this same interval of depths their energy spectrum does not change its form sharply. Table II gives the results of some very rough estimates of intensities, made on the basis of the following experiments.

a) Anderson\({}^{A3}\) recently measured the momentum spectrum of individual positive and negative particles recorded at an altitude of 9000 m by a counter controlling a Wilson chamber. Having found a considerably larger excess of positive particles over negative ones than at sea level, he suggested that this is due to protons. This interpretation is confirmed by the fact that such an excess is observed only for particles whose momentum is greater than approximately \(4\cdot10^8\ \mathrm{eV}/c\), which corresponds to the minimum momentum of a proton capable of passing through all the material lying between the effective volume of the chamber and the lower counter. From Anderson’s measurements it may be estimated that, at an altitude of 9000 m, the number of protons with momenta from \(4\cdot10^8\ \mathrm{eV}/c\) to \(10^9\ \mathrm{eV}/c\) is about 20% of the total number of “hard” particles at this altitude, and that the number of protons with momenta from \(10^9\ \mathrm{eV}/c\) to \(3\cdot10^9\ \mathrm{eV}/c\) is 15%.

b) At an altitude of 1000 m, Leprince-Ringuet[^3] found that from 2.5 to 3% of the particles emerging from 12 cm of lead with momenta from \(3\cdot 10^8\) to \(7\cdot 10^8\ \mathrm{eV}/c\) are protons. On passing from momenta to ranges, the ratio of the number of protons to the number of mesons per unit interval

Table II

Estimate of the number of protons in various energy intervals.
It is assumed that the variation of this number with depth obeys the exponential law \(e^{-x/L_a}\), where \(L_a = 125\ \mathrm{g\,cm^{-2}}\). Momenta are expressed in \(10^8\ \mathrm{eV}/c\), intervals in \(\mathrm{g\,cm^{-2}}\) of air

Quantity	Units	Experimental data	Adopted value at sea level
Number of particles with \(4 < p < 10\) \((6 < R < 100)\)	\(\mathrm{cm^{-2}\ sec^{-1}\ sterad^{-1}}\)	\(\sim 10^{-2}\) at 9000 m (Anderson)	\(3\cdot 10^{-5}\)
Number of particles with \(10 < p < 30\) \((100 < R < 1000)\)	\(\mathrm{cm^{-2}\ sec^{-1}\ sterad^{-1}}\)	\(\sim 7\cdot 10^{-4}\) at 9000 m (Anderson)	\(2\cdot 10^{-5}\)
Differential range spectrum \(i_\nu\), for \(R = 20\)	\(\mathrm{g^{-1}\ sec^{-1}\ sterad^{-1}}\)	\(\sim 5\cdot 10^{-7}\) at sea level (Rochester)	\(5\cdot 10^{-7}\)
Differential range spectrum \(i_\nu\), for \(R = 100\)	\(\mathrm{g^{-1}\ sec^{-1}\ sterad^{-1}}\)	\(\sim 4\cdot 10^{-7}\) at 1000 m (Leprince-Ringuet)	\(1.7\cdot 10^{-7}\)

of ranges at a lead thickness of 12 cm is found to be approximately equal to 6%.

c) At sea level, Rochester and Bound[^2], using a Wilson chamber controlled by anticoincidences rejecting particles stopping in an interval of 2 cm of lead (the lower boundary of this interval was determined by the thickness of the chamber walls and counters, estimated at \(10\ \mathrm{g\,cm^{-2}}\)), found at least 8, and possibly 12, proton tracks in 372 hours of observation. Taking account of the geometry of the experimental setup, the number of protons in the range interval selected by the setup is found to be approximately \(0.5\cdot 10^{-6}\ \mathrm{cm^{-2}\ sec^{-1}\ steradian^{-1}}\).

Very little is known about the behavior of the high-energy \(N\)-component in substances other than air. Bridge made some preliminary measurements at an altitude of 4300 m by means of an experimental setup of the type shown in Fig. 10, A. The coincidence rate between a group of Geiger–Müller counters and an ionization-

was found by means of a cloud chamber as a function of the thickness of lead between the two instruments. The results obtained are shown in Fig. 14.

Fig. 14. Transition curve for the production of showers by ionizing particles, obtained by Bridge\(^9\) on an experimental arrangement similar to that shown in Fig. 10, A.

The maximum, corresponding to approximately 3 cm, is explained by the production of showers by electrons, while the tail of the curve is connected with the production of showers by the high-energy \(N\)-component. From the slope of the curve one obtains the penetration depth in lead, equal to

\[ L_a = 280 \pm 50\ \mathrm{g\,cm^{-2}}{}^{*}). \]

Janossy and Rochester\({}^{13}\), measuring on the apparatus shown in Fig. 11, A, the frequency of occurrence of hard showers as a function of the thickness of lead placed above the upper group of counters, obtained the results presented in Fig. 15. The form of the “transition curve” was explained by the assumption that the observed coincidences are due to showers of penetrating particles produced in the upper layer of lead by radiation whose mean range in lead is of the order of 5 cm. The results of Bridge’s experiment cited above make such an interpretation very doubtful. It seems more probable that, in many cases interpreted as hard showers, the discharges in the counters of the upper group were due to electron showers arising during the passage of penetrating particles. The thickness at which saturation occurs is determined rather by the absorption of these electron showers than by the absorption of the primary radiation.

Fig. 15. Transition curve for hard showers, obtained by Janossy and Rochester\({}^{13}\) on the experimental arrangement shown in Fig. 11, A. Along the abscissa is plotted the thickness of lead above the upper system of counters.

* In reality the range in lead is \(L_a = 430 \pm 90\ \mathrm{g\,cm^{-2}}\). The value given in the article is connected with an error in calculation; see H. Bridge, and B. Rossi, Phys. Rev. 75, 810 (1949). (Translator’s note.)

Rossi and Regener \(^{R5}\), as well as Janossy and Rochester \(^{J4}\), discovered the production of penetrating particles of a nonionizing component. The “range” of the particles of this component, defined in this case as the mean distance between two successive productions of secondary ionizing particles, proved to be \(5—10\) cm of lead. This result is difficult to explain, since the range corresponding to the geometrical cross section of lead nuclei is approximately \(160\ \mathrm{g\,cm^{-2}}\), or \(14\) cm.

21. Nuclear interactions at low energies

Let us now consider observations of stars and single tracks in photographic plates, of bursts in thin-walled shielded ionization chambers, and of effects associated with slow neutrons. These phenomena are connected with the production, in nuclear disintegrations, of particles of comparatively low energy. All these phenomena apparently have one and the same altitude dependence, at least in the lower layers of the atmosphere. As is seen from Fig. 9, for depths greater than \(250\ \mathrm{g\,cm^{-2}}\) this altitude dependence can approximately be represented by the exponential function \(e^{-x/L_a}\), in which the mean path \(L_a\) is equal to \(138\ \mathrm{g\,cm^{-2}}\). The minimum energy necessary for the occurrence of this phenomenon is considerably smaller than the minimum energy at which the phenomena described above arise. In addition, the apparatus used to register the phenomena under consideration is not very sensitive to the direction of the generating radiation. Therefore one may assume that the curve in Fig. 9 represents the altitude dependence of the integral intensity of the \(N\)-component of all energies. From the study of the curve one may conclude that not only the primary component, absorbed exponentially in the atmosphere, is responsible for nuclear disintegrations. Indeed, if the intensity observed at atmospheric depth \(x\) at an angle \(\vartheta\) with the vertical is a function \(J(x/\cos\vartheta)\), then the integral intensity \(J_2\) is related to the intensity in direction \(I\) by the equation

\[ J_2(x)=2\pi x\int_x^\infty I(y)\frac{dy}{y^2}, \tag{10} \]

which, for the case of exponential absorption, when \(I(x)=I_0 e^{-x/L_a}\), becomes

\[ J_2(x)=2\pi I_0 e^{-x/L_a}+\frac{x}{L_a}E_i\left(-\frac{x}{L_a}\right). \tag{11} \]

Since for \(x \gg L_a\) the logarithmic slope of \(J_2(x)\) is equal to \(\frac{1}{L_a}\), then, in order to represent the observations in the lower layers of the atmosphere by means of a function of the type indicated in equation (11), it is necessary to set

$L_a = 138\ \mathrm{g\,cm^{-2}}$. The function $J_a(x)$, calculated for the indicated value of $L_a$, is shown in Fig. 9 by the dotted curve. Despite the fact that the accuracy of measurements at great altitudes is far from sufficient, there is no doubt that the experimental curve at the boundary of the atmosphere falls, as the depth increases, much more slowly than the theoretical dotted curve. This clearly indicates that, when the primary component passes through the atmosphere, many secondary particles are formed (probably mainly protons and neutrons of comparatively low energy), capable of causing nuclear disintegrations. Indeed, it is possible that a sufficient number of these secondary particles is directed upward, as a result of which the frequency of occurrence of nuclear reactions of low energies near the boundary of the atmosphere is appreciably increased.

Other experimental data confirm the picture described above. If it is assumed that protons and neutrons are generated in approximately equal quantities and with equal probability cause nuclear disintegrations, then the number of nuclei disintegrated as a result of the action of each of these particles will be approximately the same. This is true in the case when the energies of the particles are so large that the ionization losses of the protons can be neglected in comparison with their losses in collisions with nuclei. On the contrary, at low energies most protons will be slowed down as a result of ionization losses before they have time to undergo collisions with a nucleus, and most nuclear reactions will be due to neutrons. The transition from one case to the other occurs in the region of energies corresponding to the path between two nuclear collisions. This energy has a value of the order of $5\cdot 10^8\ \mathrm{eV}$. Thus, the circumstance that practically all star formations are apparently caused by the non-ionizing component is in agreement with the supposition that they are caused chiefly by nucleons with energies less than $5\cdot 10^8\ \mathrm{eV}$.

The mean free path for nuclear reactions of low energies ($L_a = 138\ \mathrm{g\,cm^{-2}}$) is apparently somewhat greater than the mean free path for reactions of high energies ($L_a = 125\ \mathrm{g\,cm^{-2}}$). It is possible that $138\ \mathrm{g\,cm^{-2}}$ corresponds to the mean free path in air for neutrons with energies of the order of several units of $10^8\ \mathrm{eV}$. In this connection one may point to the results of recent measurements carried out with the aid of the 184-inch cyclotron in Berkeley [3, 5]. These measurements show that the mean free path for neutrons with energy about $10^8\ \mathrm{eV}$ depends strongly on the geometry of the experimental arrangement. In oxygen the mean free path is equal to $35\ \mathrm{g\,cm^{-2}}$ for “good geometry” and $100\ \mathrm{g\,cm^{-2}}$ for “bad geometry.” Even a still larger mean free path in air for the star-producing component is not in contradiction with the hypothesis that this component consists of neutrons, since the average energy of cosmic neutrons may differ from the energy of the neutrons from the Berkeley cyclotron, and under experi-

ments with cosmic rays the collimation is still worse than in cyclotron experiments with “poor geometry.”

Table III presents an attempt to estimate the probability of occurrence of various nuclear reactions at sea level. The calculation of the probability of formation of stars was made on the basis of the experiments of Perkins \(^{P1}\) and Lattes et al. \(^{L2}\). According to Lattes, in \(1\ \mathrm{cm}^3\) of emulsion

Table III

Estimate of the frequency of various nuclear reactions. It is assumed that the variation of the frequency with depth obeys the exponential law \(e^{-x/L_a}\), where \(L_a = 138\ \mathrm{g\,cm^{-2}}\) at a depth greater than \(250\ \mathrm{g\,cm^{-2}}\)

Type of reaction	Units	Experimental data	Adopted value at sea level
Formation of stars (with 5 tracks or more)	\(\mathrm{g^{-1}\ sec^{-1}}\) (in air)	Perkins’ experiments at sea level, Lattes et al. at an altitude of \(2800\ \mathrm{m}\)	\(10^{-5}\)
Formation of neutrons	\(\mathrm{g^{-1}\ sec^{-1}}\) (in air)	Yoon’s observations at different altitudes	\(2.1 \cdot 10^{-5}\)
Tracks of slow protons \((E < 20\ \mathrm{MeV}?)\)	\(\mathrm{cm^{-2}\ sec^{-1}}\)	Perkins’ measurements at sea level	\(3.5 \cdot 10^{-6}\)

at an altitude of \(2800\ \mathrm{m}\) approximately 10 stars are formed per day. According to Perkins’ data, at sea level 1 star is formed per \(1\ \mathrm{cm}^3\) per day. In Lattes’ work it is noted that only stars with a number of tracks greater than four were included in the calculation. In Perkins’ work there are no indications of the criterion by which he was guided in selecting stars. Both authors used Ilford plates. Perkins states that of 8 stars, 7 represent disintegrations of light nuclei (C, O, N). Since the density of the light elements in the emulsion is close to unity, the number of stars per \(1\ \mathrm{cm}^3\) of emulsion will not differ greatly from the number of stars per \(1\ \mathrm{g}\) of air.

The calculation of the frequency of appearance of tracks of slow protons is based on Perkins’ observations \(^{P1}\), who found at sea level 0.3 track per \(1\ \mathrm{cm}^2\) per day. In the emulsion used, distinguishable tracks could have been left by protons with energies up to 80 MeV. However, apparently, not a single proton track with energy greater than 20 MeV was recorded. Thus, the number of proton tracks per unit area per unit time represents the flux of protons with energy less than 20 MeV.

The data of various authors on the intensity of production in the atmosphere of slow neutrons are not sufficiently definite. The estimate given in Table III is based on the latest measurements made by Yuan and Ladenburg \({}^{Y2}\) in an airplane at different altitudes (see § 14). It should be noted that the estimate given for the intensity of neutron production turns out to be somewhat low in comparison with the intensity of star production, since one must expect that for each star with 5 or more tracks there should be more than 2 neutrons. It is still rather difficult to estimate the value of this discrepancy, since our estimates of the intensity have a very rough character.

22. Analysis of the hard and soft components

According to our definitions, the intensity of the hard component is measured by the frequency of coincidences of two Geiger–Müller counters separated by \(167 \ \mathrm{g \ cm^{-2}}\) of lead. At sea level almost all these coincidences are due to mesons passing through both counters and through the lead located between them. At great altitudes, however, a considerable part of the coincidences is due to high-energy protons or, more generally, to the high-energy \(N\)-component, if one assumes that the \(N\)-component contains ionizing particles other than protons. Part of the \(N\)-component, in passing through lead, will interact with nuclei, so that in many cases the coincidences will be due to the passage of the primary particle through the upper counter and of one of the secondary particles produced in the nuclear interaction through the lower counter. It may also happen that in both counters, both in the upper and in the lower, the discharge will be produced by secondary particles arising in the nuclear interaction of a neutral or charged particle which is part of the \(N\)-component and passes outside both counters.

At the boundary of the atmosphere mesons are absent, and all coincidences are due to primary cosmic radiation. For want of more definite data, we shall assume that the number of coincidences due to the \(N\)-component varies with depth as \(e^{-x/L_a}\), where \(L_a = 125 \ \mathrm{g \ cm^{-2}}\). This assumption is consistent with the experimental data on the altitude variation of the high-energy \(N\)-component considered in § 20.

In order to obtain the vertical intensity of mesons of the hard component, we extrapolate the curve representing the vertical intensity of the whole hard component*) (see Fig. 2) to the point where \(x = 0\), and subtract from it a curve having at \(x = 0\) the same ordinate and decreasing as \(e^{-x/125}\). The difference of the two curves is shown in Fig. 16. It should be noted that, according to our assumptions, the part due to protons (or, more generally, to the \(N\)-compo-

*) The criterion for the possibility of such an extrapolation will be considered in § 23.

In the graph: curves labeled \(\dfrac{p}{\mu c}\) and \(\dfrac{E}{\mu c^{2}}\); horizontal axis labeled

\[ \dfrac{R}{\mu c^{2}}\left(\dfrac{\mathrm{g}\,\mathrm{cm}^{-2}}{10^{8}\,\mathrm{eV}}\right). \]

Fig. 22. \(\dfrac{p}{\mu c}\) and \(\dfrac{E}{\mu c^{2}}\) as functions of \(\dfrac{R}{\mu c^{2}}\) in air; \(p\) is momentum, \(E\) is kinetic energy, \(R\) is range, \(\mu\) is mass. The curves are valid for particles of any mass, provided that radiative losses and losses due to nuclear interactions are negligible in comparison with ionization losses (after Smith\(^{510}\)).

Fig. 23. \(p/\mu c\) as a function of \(R/\mu c^2\) in air, iron, and lead; \(p\) is momentum, \(R\) is range, \(\mu\) is mass. The curves are valid for particles of any mass under the condition that radiative losses and losses due to nuclear interactions are negligible in comparison with ionization losses (after Wick \(^{2}\)).

…tape) the fraction of the total intensity of the hard component at sea level is \(0.4\%\). This value agrees with the estimate of the number of high-energy protons given in Table II.

The soft component, measured with an absorber of thickness \(5\ \mathrm{g\,cm^{-2}}\) of brass between the effective volumes of the counters, contains electrons

Fig. 16. Analysis of the hard component. The curve \(f_m\) shows the vertical intensity of mesons with a range greater than \(167\ \mathrm{g\,cm^{-2}}\) of lead (“fast mesons”). The curve \(p\) shows the contribution of the \(N\)-component (protons of high energy) to the measured intensity of the hard component, but does not give the absolute value of the vertical intensity of the \(N\)-component.

practically of all energies above \(10\ \mathrm{MeV}\), mesons with momenta from \(0.7\cdot 10^8\) to \(3\cdot 10^8\ \mathrm{eV}/c\), and protons with momenta from \(4\cdot 10^8\) to \(10^9\ \mathrm{eV}/c\). The intensities of the mesons and protons that make up the soft component, at atmospheric depths greater than \(250\ \mathrm{g\,cm^{-2}}\), can be estimated, at least approximately, from experimental-

Fig. 17. Analysis of the soft component. The curves show the vertical intensity as a function of atmospheric depth for the following constituent parts: “slow mesons,” or practically all mesons with momenta less than \(3\cdot10^8\ \mathrm{eV}/c\) \((sm)\); protons (or other charged \(N\)-component particles) with momenta from \(4\cdot10^8\) to \(10^9\ \mathrm{eV}/c\) \((P)\); electrons of practically all energies greater than \(10^7\ \mathrm{eV}\) \((e)\).

...data, presented in Fig. 7 and in Table II. The intensity of the electrons can then be obtained by subtracting the intensity of the mesons and protons from the total intensity of the soft component given in Fig. 2. The results of such a calculation are presented in Fig. 17 by the solid curve. Apparently, at all depths greater than \(250\ \mathrm{g\,cm^{-2}}\), the intensity of the protons and mesons constitutes an insignificant part of the total intensity of the soft component. Therefore the still remaining large uncertainty in their values is not substantially reflected in the estimate of the magnitude of the electron intensity.

At the present time it is impossible to estimate with sufficient accuracy the intensity of electrons at depths less than \(250\ \mathrm{g\,cm^{-2}}\). The value of the intensity of the soft component in the upper layers of the atmosphere is unknown partly because the results of measurements of the total intensity (see Fig. 2) do not agree with one another, and partly because in this region it is impossible to determine accurately the intensity of the hard component, which consists mainly of primary particles of high energy. Moreover, there are no measurements of the intensities of mesons and protons of the soft component at depths less than \(250\ \mathrm{g\,cm^{-2}}\). Whereas the intensity of mesons can still be determined with the help of certain general considerations (see § 23), there are as yet insufficient grounds for estimating the intensity of protons.

In considering the energy balance in cosmic radiation, which will be carried out below, it is important to know, for each type of particle, the quantity

\[ \int_{0}^{1030} I_v\,dx, \]

which represents the path length in the atmosphere. The results of calculations of the path length for the various components of cosmic rays are given in Table IV.

The errors indicated in it are estimated from the errors of the various experiments. In determining the path length of the ionizing particles of the \(N\)-component with \(R>100\ \mathrm{g\,cm^{-2}}\), account was taken of the fact that the efficiency of registration by the telescope for the indicated particles is greater than for mesons (see § 19).

23. Energy of the Meson Component

To describe the processes occurring in the atmosphere that are associated with energy exchange, let us introduce, for each group of cosmic-ray particles, a function \(k(x)\), so that \(k(x)d\omega\) represents the loss of energy per unit time in 1 gram of air at depth \(x\) by particles of the given group passing within the solid angle \(d\omega\) about the vertical direction. The quantity \(k(x)\) is measured in \(\mathrm{eV\,g^{-1}\,sec^{-1}\,steradian^{-1}}\).

In air the meson component loses energy as a result of processes of collision and decay (radiative energy losses of mesons at energies less than \(10^{12}\) eV may be neglected). Let us consider separately the mesons constituting the hard and soft components; for brevity we shall henceforth call them, respectively, “fast mesons” and “slow mesons.”

Table IV

Path length of various components of cosmic rays in the atmosphere.

Components	Path length, \(g\,cm^{-4}\,sec^{-1}\,sterad^{-1}\)
All ionizing particles \(R>5\ g\,cm^{-2}\) of lead	\(133 \pm 17\)
Mesons \(5\ g\,cm^{-2}\) of lead \(< R < 100\ g\,cm^{-2}\) of air	\(6 \pm 3\)
Mesons \(R>100\ g\,cm^{-2}\) of air	\(24 \pm 2\)
Protons \(5\ g\,cm^{-2}\) of lead \(< R < 100\ g\,cm^{-2}\) of air	\(6 \pm 3\)
Ionizing \(N\)-component (protons?) \(R>100\ g\,cm^{-2}\) of air	\(8 \pm 4\)
Electrons (as a difference) \(R>5\ g\,cm^{-2}\) of lead	\(89 \pm 18\)

In air, for fast mesons the rate of energy loss by scattering in collisions may be regarded as independent of the energy and equal to \(2\cdot 10^8\ \mathrm{eV}\,g^{-1}cm^2\). Therefore the ionization losses of fast mesons may be represented in the form:

\[ k_c^{(fm)}(x)=2\cdot 10^6 I_\nu^{(fm)}(x)\ \mathrm{eV}\,g^{-1}\,sec^{-1}\,steradian^{-1}, \tag{12} \]

where \(I_\nu^{(fm)}\) is the vertical intensity of fast mesons.

If we restrict ourselves to considering mesons whose energy is large in comparison with their rest energy, then the total energy released in the decay of mesons in a layer of air of \(1\ g\,cm^{-2}\) will not depend on the distribution of the mesons by energy and will be equal to the number of incident mesons multiplied by \(\mu c/\tau\rho\), where \(\rho\) is the density of air\(^5\). Therefore the losses of fast mesons in decay are represented by the expression

\[ k_d^{(fm)}(x)=\frac{\mu c}{\tau\rho}\,I_\nu^{(fm)}(x). \tag{13} \]

In order to find the ionization losses and the losses in decay for slow mesons, let us suppose that at all altitudes from

from 0 to 100 \(g\,cm^{-2}\) the differential range spectrum of mesons will be constant. As was mentioned in § 6, this has indeed proved to be more or less true for all altitudes at which measurements were made. Then the mean ionization losses of slow mesons are obtained by dividing the maximum kinetic energy of slow mesons \((E_m = 2.2\cdot 10^8\ \mathrm{eV})\) by the corresponding range \((R_m = 100\ g\,cm^{-2})\). Thus we obtain:

\[ k_c^{(sm)}(x)=2.2\cdot 10^6 I_y^{(sm)}(x)\ \mathrm{eV}\ g^{-1}\ sec^{-1}\ steradian^{-1}, \tag{14} \]

where \(I_y^{(sm)}\) is the vertical intensity of slow mesons.

Under the same assumption concerning the range distribution, the losses due to the decay of slow mesons are obtained in the following form:

\[ k_d^{(sm)}(x)=1.2\,\frac{\mu c}{\tau \rho}\,I_y^{(sm)}(x). \tag{15} \]

It is necessary, finally, to consider mesons that come to rest in the air and, as a result, decay or are captured by nuclei. The energy lost in these processes by the meson beam is equal to

\[ k^{(mr)}(x)=\frac{I_y^{(sm)}}{R_m}\,\mu c^2, \tag{16} \]

where \(I_y^{(sm)}/R_m\) represents the differential intensity of slow mesons.

The various energy losses experienced by mesons, calculated by means of equations (13), (15), and (16), as well as the results of measurements of the quantities \(I_y^{(fm)}\) and \(I_y^{(sm)}\), are represented in Fig. 18 by solid curves. To extrapolate these curves to the region of smaller depths, let us examine the behavior of the losses due to decay as \(x\to 0\). For this purpose we consider mesons with a given momentum \(p\) and denote by \(K(x)\) the energy per \(1\ g\,sec\,steradian\) required for the production of these mesons at depth \(x\). If \(x\) is sufficiently small, then ionization losses may be neglected, and it may also be assumed that the ratio

\[ \frac{x}{p}=z_0, \]

where \(z_0\) is a constant (Fig. 21). Then the probability that a meson produced at depth \(x\) will decay at depth \(x_1\) in the interval \(dx_1\) will be equal to

\[ \frac{z_0}{\lambda}\left(\frac{x}{x_1}\right)^{z_0/\lambda}\frac{dx_1}{x_1}, \]

where \(\lambda=\dfrac{\tau p}{\mu}\) represents the mean free path before decay. Hence the losses \(k_d\), due to the decay of the meson beam at depth \(x_1\), are represented in the form:

\[ k_d(x_1)=\frac{z_0}{\lambda}\,x_1^{-\frac{z_0}{\lambda}-1}\int_0^{x_1} K(x)\,x^{z_0/\lambda}\,dx. \]

Fig. 18. Losses of the various meson components during decay in the atmosphere, namely fast mesons \((fm)\), slow mesons \((sm)\), and mesons at rest \((mr)\). The curve \(mr\), in addition to the energies of spontaneously decaying mesons, also includes the rest energies of mesons captured by nuclei. The curve \(m\) shows the sum of all three curves.

Expanding \(K(x)\) in a Taylor series near \(x=0\) and integrating, we obtain for \(k_d(x_1)\) the series:

\[ k_d(x_1)=\frac{K(0)}{1+\lambda/z_0}+\frac{K'(0)}{1+2\lambda/z_0}\,x_1+\cdots \tag{17} \]

Consequently, at the boundary of the atmosphere the following relations are valid:

\[ k_d(0)=\frac{K(0)}{1+\lambda/z_0}, \]

\[ \left(\frac{d}{dx}\ln k_d\right)_{x=0} = \frac{1+\lambda/z_0}{1+2\lambda/z_0} \left(\frac{d}{dx}\ln K\right)_{x=0}. \tag{18} \]

It is obvious that at a very small depth of the atmosphere the loss of energy by the meson beam per gram of air is always less than the energy acquired by the meson beam in the same mass of air.*) The values of \(\lambda\) for slow mesons lie between 0 and \(2\cdot 10^5\) cm. On the other hand, at the boundary of the atmosphere \(z_0=6.4\cdot 10^5\) cm. For slow mesons, therefore, the mean value of the fraction \((1+\lambda/z_0)/(1+2\lambda/z_0)\) lies between 1 and 0.8 and, in a first approximation, it may be assumed that the logarithmic derivative of \(k_d(x)\) near the boundary of the atmosphere is equal to the logarithmic derivative of \(K(x)\). According to our assumption, mesons are formed by the high-energy \(N\)-component, whose intensity changes with depth as \(e^{-x/125}\). Hence we conclude that \(\dfrac{d}{dx}\bigl(\ln k_d^{(sm)}\bigr)\) tends to \(1/125\) when \(x\) tends to zero.

On the other hand, Fig. 18 shows that \(\dfrac{d}{dx}\bigl(\ln k_d^{(sm)}\bigr)\) is approximately equal to \(1/125\) already at \(x=250\ \mathrm{g\,cm^{-2}}\). It is therefore natural to assume that \(k_d^{(sm)}\) is represented by the function \(e^{-x/125}\) in the region from 0 to \(250\ \mathrm{g\,cm^{-2}}\), as shown by the dashed curve in Fig. 18.

For fast mesons, a direct determination of \(k_d\) can be made up to considerably greater altitudes than for slow mesons. Since it is clear from the considerations set forth by us that one cannot expect a sharp change in the slope of \(k_d\) near \(x=0\), a linear extrapolation to \(x=0\) of the experimental curve representing \(\ln k_d^{(fm)}\) as a function of \(x\) appears entirely justified. From the extrapolated values of \(k_d^{(fm)}\) and \(k_d^{(sm)}\) one can, by means of equations (13) and (15), calculate the corresponding values of \(f_\nu^{(fm)}\) and \(f_\nu^{(sm)}\). The latter are given in Figs. 16 and 17.

*) This proves the erroneousness of the following, seemingly obvious, proposition: “At the boundary of the atmosphere the density of the air is so small that mesons decay before they pass through an appreciable thickness of the atmosphere; for this reason the energy acquired by the meson component in some layer of the atmosphere is equal to the energy lost by it in that same layer.”

Integrating with respect to \(x\) the functions representing the different energy losses per \(1\ \mathrm{g\,sec\,steradian}\) gives the corresponding energy losses (per \(1\ \mathrm{sec\;steradian}\)) of the meson beam in a vertical column with a cross section of \(1\ \mathrm{cm^2}\). The results are given in Table V.

Table V

Energy losses by mesons in \(\mathrm{MeV\ cm^{-2}\ sec^{-1}\ steradian^{-1}}\)

Depth intervals \((\mathrm{g\ cm^{-2}})\)	\(0 < x < 250\)	\(250 < x < 1030\)	\(x > 1030\)	Total
Ionization losses
Fast mesons	17	31	37	85
Slow mesons	8	6	—	14
Losses in decay
Fast mesons	84	38	—	122
Slow mesons	52	9	—	61
Resting mesons	4	3	—	7
Total energy losses	165	87	37	289

The ionization losses underground \((x > 1030\ \mathrm{g\ cm^{-2}})\) were calculated from experimental data on the meson spectrum at sea level.

For the total energy losses by mesons we obtain:

\[ W^{(m)} = 289 \cdot 10^6\ \mathrm{eV\ cm^{-2}\ sec^{-1}\ steradian^{-1}} . \tag{19} \]

This quantity also represents the total energy of mesons (per \(1\ \mathrm{sec\;steradian}\) in the vertical direction) formed in a vertical column of cross section \(1\ \mathrm{cm^2}\), extending from the boundary of the atmosphere to the maximum depth at which mesons are still formed.

24. Analysis of the electron component

Part of the electrons observed in the atmosphere is due to ionization produced by mesons and to the subsequent multiplication of these electrons. Since the meson intensity changes little with depth, and ionization produces electrons with a small mean energy, the number of electrons arising in this way can be calculated on the assumption that the meson intensity remains approximately constant over a distance equal to the mean range of the shower created by a \(\delta\)-electron. As already mentioned, the minimum energy of an elec-

trons in the soft component is approximately \(10^7\) eV. The number of \(\delta\)-electrons with greater energy can be obtained from the calculations of Rossi and Klapman \(^{R7}\) and Tamm and Belenkii \(^{T1}\). At sea level it amounts to \(6.7\%\) of the number of fast mesons. Slow mesons play no role in this, since the maximum energy that a meson with momentum \(3\cdot 10^8\) eV/\(c\) can transfer to an electron is equal to \(9\) MeV. In analyzing the electron component we shall assume that the ratio of the number of \(\delta\)-electrons to the number of fast mesons remains the same at all altitudes, although in reality this ratio depends somewhat on the meson spectrum, which changes with altitude. The error arising in this way has no noticeable effect when calculating the intensity of electrons of other origin, since the relative contribution of \(\delta\)-electrons to the total electron intensity rapidly decreases with increasing altitude. In addition, we shall neglect \(\delta\)-electrons produced by protons, since the latter can transfer a maximum energy approximately 100 times smaller than mesons with the same momentum.

According to our assumptions, besides ionization processes there are two other, more substantial sources of electrons, namely meson decay and nuclear interactions. It may be assumed with confidence that the mean energy of electrons or photons produced in such processes is large in comparison with the critical energy in air. It can further be shown \(^{R7,T1}\) that the true energy distribution of these electrons or photons has very little effect on the energy spectrum of low-energy electrons arising in their multiplication. Thus, from the number of electrons falling on \(1\ \mathrm{cm}^2\) with energy greater than a given one, it is possible to calculate the energy loss in one gram of air by electrons of all energies. Considering electrons falling vertically and using the results of Rossi and Klapman, we obtain the following relation:

\[ k^{(e)} = 3.26\cdot 10^6 I_\nu^{(e)}\ \mathrm{eV}\ \mathrm{g}^{-1}\ \mathrm{sec}^{-1}\ \mathrm{steradian}^{-1}, \tag{20} \]

where \(I_\nu^{(e)}\) is the vertical intensity of electrons with energy greater than \(10^7\) eV, and \(k^{(e)}\) is the energy loss per \(1\ \mathrm{g}\ \mathrm{sec}\ \mathrm{steradian}\) by electrons of all energies arising as a result of meson decay and nuclear interactions. The quantity \(k^{(e)}(x)\) is shown in Fig. 19 as a function of the atmospheric depth \(x\). The values corresponding to depths less than \(250\ \mathrm{g\ cm}^{-2}\), shown by the dashed curve, as well as the corresponding values of \(I_\nu^{(e)}\), are very uncertain (see § 22).

The total energy loss in the atmosphere by the electron component can be calculated by integrating the curve shown in Fig. 19, which gives:

\[ W^{(e)} = 285\cdot 10^6\ \mathrm{eV}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\ \mathrm{steradian}^{-1}. \tag{21} \]

9*

Fig. 19. The curve \(k^{(e)}\) represents the energy losses in the atmosphere by cosmic-ray electrons (with the exception of \(\delta\)-electrons). The curve \(k_d^{(e)}\) shows the energy losses by electrons arising in meson decay, calculated under the assumption that all the energy released in the decay passes into the electron component.

Since the energy of the electron component at sea level is negligible in comparison with \(W^{(e)}\), the latter quantity represents the total energy (per \(1\ \mathrm{sec}\ \mathrm{steradian}\) in the vertical direction) falling on the electron flux in a vertical atmospheric column of cross section \(1\ \mathrm{cm}^2\).

What relative amount of the energy liberated in meson decay passes into the electron component has not yet been established with certainty\(^*\). Therefore an exact determination is impossible of the relative contributions of decay processes \(\bigl(k_d^{(e)}\bigr)\) and of nuclear interactions \(\bigl(k_n^{(e)}\bigr)\) to the total energy losses by electrons \(k^{(e)}\). The quantity \(k_d^{(e)}\) for the lower layers of the atmosphere was calculated theoretically by the method described by Rossi and Greisen\(^{\mathrm{R6}}\), under the assumption that the above-mentioned ratio is equal to unity. Namely, the energy scattered by electrons in \(1\ \mathrm{g}\) of air at a certain depth \(x\) is assumed equal to the energy received by the electron component in \(1\ \mathrm{g}\) of air at the depth \(x-\bar{x}\), where \(\bar{x}\) is the mean range of the shower produced by electrons or photons arising in the decay. The mean length \(\bar{x}\) is taken to be \(130\ \mathrm{g\ cm^{-2}}\) for decay electrons of fast mesons, \(65\ \mathrm{g\ cm^{-2}}\) for decay electrons of slow mesons, and zero for decay electrons of mesons at rest. The results of the calculations are shown in Fig. 19 by the curve \(k_d^{(e)}\). This curve, as has been noted repeatedly\(^{\mathrm{B1},\mathrm{R6},\mathrm{B3}}\), shows a much less rapid increase with decreasing depth than the observed intensity of electrons. This indicates that the majority of the electrons observed at great heights are not connected with meson decay. At sea level the absolute value of \(k_d^{(e)}\) exceeds the absolute value of \(k^{(e)}\) by more than a factor of two. Taken by itself, this result would seem to show that in meson decay less than one half of all the energy passes into the electron component. However, this conclusion cannot be regarded as final, owing to the still large uncertainty in the estimate of the experimental data. Whereas a reasonable estimate of the inaccuracy gives a possible error of \(20\%\) for the ratio \(k_d^{(e)}/k^{(e)}\), \(k_d^{(e)}/2\) exceeds \(k^{(e)}\) by \(40\%\). In any case, the possibility that all the energy liberated in the decay passes into the electron component must be excluded.

From Table V it is seen that the total energy of the products of meson decay is

\[ 190\cdot 10^6\ \mathrm{eV\ cm^{-2}\ sec^{-1}\ steradian^{-1}} . \]

Since not more than half of this energy passes into the electron component, and since the total energy \(W^{(e)}\) of the electron component is equal

\(^*\) It should be noted that this relative amount is the same both in the laboratory system and in the frame of reference in which the meson is at rest.

\(285 \cdot 10^6\ \mathrm{eV\ cm^{-2}\ sec^{-1}\ steradian^{-1}}\), then the minimum energy of that part of the electronic component which is due to nuclear interactions is

\[ W_n^{(e)} = 190 \cdot 10^6\ \mathrm{eV\ cm^{-2}\ sec^{-1}\ steradian^{-1}} . \]

It is interesting to compare this value with the total energy \(W^{(m)}\) of the meson component, which in § 23 was found to be equal to \(289 \cdot 10^6\ \mathrm{eV\ cm^{-2}\ sec^{-1}\ steradian^{-1}}\). Apparently, \(W_n^{(e)}\) and \(W^{(m)}\) have the same order of magnitude, but their ratio is not precisely determined at present. If one takes into account the inaccuracy in estimating the magnitude of the total energy of the electrons and the inaccuracy in estimating the part of this energy due to meson decay, one may conclude that the value of \(W_n^{(e)}\), in any case, may lie within the limits from \(\dfrac{W^{(m)}}{2}\) to \(W^{(m)}\).

25. Total energy of cosmic rays at latitudes greater than \(45^\circ\)

The total energy of the cosmic radiation incident on the earth (per \(1\ \mathrm{cm^2\ sec\ steradian}\)) can be obtained by summing the energies lost in the various secondary processes caused by this radiation.

The results of such an estimate are given in Table VI *). The ionization losses of particles with ranges greater than \(5\ \mathrm{g\ cm^{-2}}\) of brass, in the atmosphere and underground, were calculated from the data given in the preceding sections. The average ionization losses of the ionizing \(N\)-component (protons?) with \(R > 100\ \mathrm{g\ cm^{-2}}\) of air were taken equal to \(2\ \mathrm{MeV}\) per \(\mathrm{g\ cm^{-2}}\). The average ionization losses of protons with \(R\) from \(5\ \mathrm{g\ cm^{-2}}\) of brass to \(100\ \mathrm{g\ cm^{-2}}\) of air were arbitrarily taken equal to \(6 \cdot 10^6\ \mathrm{eV}\) per \(\mathrm{g\ cm^{-2}}\) (they would be equal to \(4 \cdot 10^6\ \mathrm{eV}\) if the distribution of proton ranges were constant; in reality the differential spectrum of their ranges increases with decreasing \(R\)). The energy of vertically incident primary radiation (per \(1\ \mathrm{cm^2\ sec\ steradian}\)) which is spent on the formation of the particles considered above is not exactly equal to the energy received by these particles, since in the formation of secondary particles there is a change of direction. The difference between these two energies is estimated at approximately \(10\%\) (see the Appendix, § 30).

The losses of energy in nuclear disintegrations include both the energy expended in destroying the nucleus and the energy carried off by neutrons, protons, and other nuclear fragments, formed—

*) A similar estimate of the total energy of cosmic rays was given by H. A. Bethe in July 1947 at the Shelter Island conference. The results of his estimates agree with the results presented by us.

zing in these decays and not registered by the telescope. At present it is impossible to determine exactly the energy losses in nuclear disintegrations. Any estimate of the frequency of nuclear disintegrations at sea level, owing to the discrepancy between the data of experiments with stars and with neutrons (see Table III), is possible only to within a factor of 2. The value adopted by us for air is equal to \(2\cdot 10^{-5}\ \mathrm{g}^{-1}\ \mathrm{sec}^{-1}\). From the curve for the total number of nuclear disintegrations in the atmosphere (Fig. 9) one obtains the value \(3.6\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\). It is also difficult to estimate the mean energy passing, in nuclear disintegrations, into the non-ionizing component. If it is assumed that this energy is equal to \(10^8\) eV, then for the total energy released in the atmosphere we obtain \(3.6\cdot 10^8\ \mathrm{eV}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\). The part of this energy due to the primary component in one steradian is equal to \(3.6\cdot 10^8/\pi = 1.2\cdot 10^8\ \mathrm{eV}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\ \mathrm{steradian}^{-1}\).

Table VI

Estimate of the total energy of cosmic rays in \(\mathrm{MeV}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\ \mathrm{sterad}^{-1}\)

Item	Value
Ionization losses of all mesons in the atmosphere	62
Ionization losses of all mesons under the ground	37
Ionization losses of the ionizing \(N\)-component (protons?) \(R > 100\ \mathrm{g}\ \mathrm{cm}^{-2}\) of air	16
Ionization losses of protons \(5\ \mathrm{g}\ \mathrm{cm}^{-2}\) of brass \(< R < 100\ \mathrm{g}\ \mathrm{cm}^{-2}\) of air	36
Ionization losses of electrons (excluding \(\delta\)-electrons)	285
Sum of all ionization losses	436
With correction for angular distribution	480
Energy losses in nuclear disintegrations	120
Losses for neutrino formation	95
Total incident energy	695

Finally, it remains for us to consider the energy carried away by “neutrino” or by other unregistered neutral particles. We take this energy to be equal to half the total losses in meson decay, or \(95\ \mathrm{MeV}\ \mathrm{g}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\ \mathrm{steradian}^{-1}\). This estimate is a minimum one, since it is possible that less than half of the decay energy passes into the electronic component, or that unregistered particles also arise in other processes.

The error in determining the total ionization losses is estimated at \(60\cdot 10^6\ \mathrm{eV}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\ \mathrm{steradian}^{-1}\). Its cause is chiefly the uncertainty in the path length of all ionizing particles (see Table IV). The uncertainty in the energy losses in nuclear disintegrations is about \(100\cdot 10^6\ \mathrm{eV}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\ \mathrm{stera- <!-- source-page: 061 --> \(\text{dyne}^{-1}\), and the uncertainty in the losses to the “neutrino” in meson decay is about \(50 \cdot 10^6\ \mathrm{eV\ cm^{-2}\ sec^{-1}\ steradian^{-1}}\). Thus

Fig. 20. Depth of the atmosphere as a function of altitude for the standard atmosphere.

The vertical axis is labeled: “Depth of the atmosphere \((\mathrm{g/cm^2})\).”
The horizontal axis is labeled: “Altitude \((\mathrm{km})\).”

Thus, if meson decay is the only process in which unregistered particles are produced, then the total energy of the per-

of cosmic radiation at geomagnetic latitudes greater than \(45^\circ\), is equal to:

\[ W^{(p)}=(695\pm130)\cdot10^6\ \mathrm{eV}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\ \mathrm{steradian}^{-1}. \tag{22} \]

The number of primary cosmic particles can be obtained by dividing the total incident energy by the mean energy of the primary particles. From geomagnetic effects it is known that the minimum energy of primary particles, if they are protons, is about \(4\cdot10^9\ \mathrm{eV}\). From the same effects the mean energy is estimated to be approximately \(10^{10}\ \mathrm{eV}\). If this value is adopted, one may conclude that the intensity in the direction of the primary particles is equal to:

\[ I^{(p)}=(0.07\pm0.013)\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\ \mathrm{steradian}^{-1}. \tag{23} \]

The number given may be compared with the value \(0.12\ \mathrm{cm}^{-2}\times \mathrm{sec}^{-1}\ \mathrm{steradian}^{-1}\) for the “intensity of the hard component” near the boundary of the atmosphere, shown in Fig. 2. It may also be compared with the total intensity at the boundary of the atmosphere, measured recently by Van Allen and Tatel\({}^{V4}\) with the aid of a Geiger–Müller counter placed in the front part of a rocket at a large distance from its body and found likewise to be \(0.12\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\ \mathrm{steradian}^{-1}\). On the basis of the considerations set forth above (see § 19), each of these quantities is probably greater than the true intensity of the primary radiation (if one does not take into account the latitude effect, which should have been taken into consideration in the second measurement, made at latitude \(40^\circ\)). Consequently, the difference between the values of the intensity of the primary component obtained by direct measurements at great altitudes and those estimated from the total energy of cosmic rays does not serve as proof that the amount of energy transferred to neutrinos or to other unregistered radiation is considerably greater than half the energy released in the decay of mesons.

APPENDICES

26. Standard atmosphere

The relations between the depth of the atmosphere, the height above sea level, and the density of the air in the atmosphere are given in Figures 20 and 21.

27. Energy and momentum losses of heavy particles

The dependence of the range for various substances on the energy and momentum of ionizing particles is shown in Figures 22 and 23. They are obtained from the calculations of Wick\({}^{W2}\) and Smith\({}^{S10}\) and are valid for all particles for which the losses of energy to bremsstrahlung and nuclear interactions are negligible compared with ionization losses. Fig. 24 gives \(dR/dE\) and \(dR/dp\) in air as functions of \(R\).

28. Angular distribution of secondary particles formed in nuclear collisions

Consider an inelastic collision between a nucleon with momentum \(p\) and a nucleon at rest, in which two mesons are produced. The equation

\[ \frac{\beta_c}{1-\beta_c^2}=\frac{p}{2Mc} \tag{A1} \]

determines the velocity \(c\beta_c\) of both nucleons in the reference system in which their center of mass is at rest. Let us assume that, in the center-of-mass system,

Fig. 21. \(x/\rho\) as a function of \(x\) in the standard atmosphere; \(x\) is the atmospheric depth, \(\rho\) is the air density.

after the collision both nucleons possess a total energy exceeding their rest energy by a factor \(a\), while the remaining energy is divided equally between the two mesons. The total energy \(E_0\), momentum \(p_0\), and velocity \(\beta_0\) of each particle in the center-of-mass system after the collision are represented by the following equations:

nucleons:

\[ E_0=aMc^2,\qquad p_0=\frac{\beta_0 E_0}{c},\qquad \frac{1}{(1-\beta_0^2)^{1/2}}=a, \]

mesons:

\[ E_0=Mc^2\left(\frac{1}{(1-\beta_c^2)^{1/2}}-a\right),\qquad p_0=\frac{\beta_0 E_0}{c}, \]

\[ \frac{1}{(1-\beta_0^2)^{1/2}} = \frac{M}{\mu} \left[ \frac{1}{(1-\beta_c^2)^{1/2}}-a \right]. \tag{A2} \]

Let us further assume that, in the center-of-mass system, the nucleons and mesons fly out at right angles to the initial direction

motion of the nucleons. Then the energy \(E\) and the emission angle \(\psi\) of each of the particles in the laboratory system will be determined by the following equations:

nucleons:
\[ E=\frac{aMc^2}{(1-\beta_c^2)^{1/2}},\qquad \operatorname{tg}\psi=\frac{\beta_0}{\beta_c}(1-\beta_c^2)^{1/2}, \]

mesons:
\[ E=Mc^2\left[\frac{1}{1-\beta_c^2}-\frac{\alpha}{(1-\beta_c^2)^{1/2}}\right], \]
\[ \operatorname{tg}\psi=\frac{\beta_0}{\beta_c}(1-\beta_c^2)^{1/2}. \tag{A3} \]

The various functions of \(\beta\) appearing in equations (A1), (A2), and (A3) are shown graphically in Fig. 25. As an example,

Fig. 24. \(dr/dE\) and \(dR/dp\) as functions of \(R/\mu c^2\). \(R\) is the range in air, \(E\) is the energy, \(p\) is the momentum.

some values of \(E\) and \(\psi\) corresponding to \(p/Mc=10\) are given in Table VII.

Fig. 25. The quantities \(\dfrac{\beta}{1-\beta^2}\), \(\dfrac{1}{(1-\beta^2)^{1/2}}\), and \(\dfrac{\beta}{(1-\beta^2)^{1/2}}\) as functions of \(1-\beta\).

29. Determination of the upward-directed flux of the energy of cosmic radiation emitted in nuclear interactions

Let us assume that the primary particles are protons. In their collisions with atomic nuclei, mesons and electrons or photons (directly or indirectly) are produced. Mesons with momentum \(p\), moving upward, decay after an average path \(p\tau/\mu\). The mean angle through which they are deflected by the Earth’s magnetic field \(H\) before decay is determined by the mean path divided by the radius of curvature. If the particle trajectories are perpendicular to the field, then \(R=pc/300H\), and the angle of deflection is equal to

\[ 300\,\frac{H\tau}{\mu c}. \]

Since \(H\) is a quantity of the order of \(1/2\) gauss, this angle is of the order of magnitude \(0.1\). In a first approximation we may neglect this deflection and assume, on the basis of the law of conservation of momentum, that the meson decays immediately after its formation.

Table VII

\(\alpha\)	Nucleons \(\dfrac{E}{Mc^{2}}\)	Nucleons \(\psi\)	Mesons \(\dfrac{E}{Mc^{2}}\)	Mesons \(\psi\)
1	2.35	\(0^\circ\)	3.17	\(25^\circ\)
1.2	2.82	\(15^\circ\)	2.70	\(25^\circ\)
1.5	3.52	\(19.5^\circ\)	2.00	\(25^\circ\)
2.0	4.70	\(22^\circ\)	0.82	\(24^\circ\)
2.25	5.29	\(23^\circ\)	0.23	\(0^\circ\)

Let us further suppose that the inelastic collisions in which secondary particles are formed are collisions of two free nucleons, and consider the phenomenon of decay in a frame of reference in which the center of mass is at rest. Let in this frame of reference the total mean energy of the secondary light particles (electrons, photons, neutrinos) be \(E_{0}\), independently of whether they are formed directly or as the result of a decay process. Nothing is known a priori about the angular distribution of these particles, except that, on the average, it must be symmetric with respect to the plane passing through the center of mass perpendicular to the initial direction of motion of the nucleons. We shall therefore consider two limiting cases, in which a) the secondary light particles are emitted in two opposite directions parallel to the direction of motion, and b) the secondary light particles are emitted at right angles to the direction of motion. Let us assume that the secondary light particles have relativistic velocities. In case a) the mean total momentum of the particles moving in each of the two opposite directions is equal to \(E_{0}/2c\). On transforming to the laboratory system we obtain the following values for the mean total energies of both groups of particles:

for the forward direction:

\[ \frac{E_{0}}{2}\,\frac{1+\beta_{c}}{(1-\beta_{c}^{2})^{1/2}} \]

and for the backward direction:

\[ \frac{E_0}{2}\,\frac{1-\beta_c}{(1-\beta_c^2)^{1/2}}, \]

where \(\beta_c\) is the velocity of the center of mass, determined by equation (A1). Therefore the fraction of the total energy corresponding to backward motion will be equal to

\[ \rho=\frac{1-\beta_c}{2}. \tag{A4} \]

Fig. 26. Graph of the function \(f(\psi)\). The quantity \(\operatorname{tg}\psi\, f(\psi)/\pi\) represents the relative number of secondary particles leaving the earth at an angle \(\psi\) to the vertical, on the assumption that these particles are produced by a primary component distributed isotropically in the upper hemisphere.

In case b), the light particles in the laboratory system move at an angle \(\psi\), determined by the equation (see equation (A3)):

\[ \operatorname{tg}\psi=\frac{(1-\beta_c^2)^{1/2}}{\beta_c}. \]

A simple calculation shows that if one assumes that the primary particles fall on the atmosphere with the same intensity in all directions above the horizon, then the relative number of secondary particles emitted upward is given by the expression:

\[ \rho=\frac{\operatorname{tg}\psi}{\pi}\int_0^1 \frac{\arc\cos y\,dy}{(1+y^2\operatorname{tg}^2\psi)^{3/2}}. \tag{A5} \]

Figure 26 presents the graph of the integral entering equation (A5). Table VIII gives the values of \(\rho\), calculated for several values of \(p/Mc\), under each of the assumptions a) and b). The magnitude of the average momentum of the primary protons is probably of the order of \(10^{10}\,\mathrm{eV}/c\). Table VIII shows that the corresponding value of \(\rho\) lies between 5 and 14%.

Table VIII

\(\rho\)—the fraction of the energy radiated upward as a result of nuclear collisions

\(\dfrac{p}{Mc^2}\)	Assumption (a) \(\rho\)	Assumption (b) \(\psi\)	Assumption (b) \(\rho\)
2	0.19	\(52^\circ\)	0.29
5	0.09	\(35^\circ\)	0.19
10	0.046	\(25^\circ\)	0.14
15	0.032	\(21^\circ\)	0.11

In reality, this estimate represents only a lower limit, since in all probability a considerable part of the secondary light particles is formed by secondary nucleons with energies considerably smaller than \(10^{10}\,\mathrm{eV}\). A more exact determination of the fraction of the energy corresponding to upward motion is not ...

is possible as long as we do not possess more detailed data on the secondary processes in which the secondary particles arise. However, the calculations presented show that this part of the energy cannot be neglected.

30. Taking account of the influence of the angular distribution on the total energy of the secondary radiation directed vertically

Let \(W_1\) denote the energy (per \(1\ \mathrm{sec}\ \mathrm{steradian}\)) of all secondary particles produced by the primary component falling vertically on \(1\ \mathrm{cm}^2\) of the atmosphere, and \(W_2\) the energy (per \(1\ \mathrm{sec}\ \mathrm{steradian}\)) of secondary particles moving vertically upward, produced by the entire primary component falling on \(1\ \mathrm{cm}^2\) of the atmosphere.

It is required to calculate the ratio \(W_2/W_1\). For this purpose let us assume that the primary radiation consists of protons and that their interaction with atomic nuclei can be described as collisions between free nucleons. Let us further assume that, in the reference system in which the center of mass of the two colliding nucleons is at rest, all secondary particles produced as a result of any such collision have the same energy \(E_0\), and consequently the same momentum \(p_0\). Since the results turn out not to depend on \(E_0\), this assumption does not restrict the generality of the reasoning.

Let, in the center-of-mass system, \(n_0(\psi_0)\,d\omega_0\) denote the mean number of secondary particles whose directions of motion lie in the element of solid angle \(d\omega_0\), making an angle \(\psi_0\) with the direction of motion of the primary proton. Let, further, \(n(\psi)\,d\omega\), \(E\), \(p\) denote the quantities corresponding to \(n_0(\psi_0)\,d\omega_0\), \(E_0\), \(p_0\) in the laboratory system. If the relative velocity of the two systems is \(\beta_c\), then the following equations hold:

\[ \beta \cos \psi = \frac{p_0 \cos \psi_0 + \dfrac{\beta_c E_0}{c}} {(1-\beta_c^2)^{1/2}}, \]

\[ n(\psi)\sin\psi\,d\psi = n_0(\psi_0)\sin\psi_0\,d\psi_0. \tag{A6} \]

On the other hand, if it is assumed that the intensity of the primary radiation has the constant value \(I^{(p)}\) in all directions above the horizon and is equal to zero in all directions below the horizon, and if by \(p_\nu\,d\omega\) we denote the total momentum of the secondary particles whose directions of motion lie in the element of solid angle \(d\omega\) about the vertical and which are produced by the entire primary component falling on \(1\ \mathrm{cm}^2\) of the atmosphere, then for \(p_\nu\) we obtain the expression

\[ p_\nu = 2\pi I^{(p)} \int_0^{\pi/2} p n(\psi)\cos\psi\sin\psi\,d\psi . \tag{A7} \]

If we use equation (A6) and assume that, by symmetry,

\[ n_0(\psi_0)=n_0(\pi-\psi_0), \]

then equation (A7) takes the form:

\[ p_\nu=\frac{\beta_c E_0/c}{(1-\beta_c^2)^{1/2}}\,I^{(p)}N- \]

\[ -\frac{2\pi I^{(p)}}{(1-\beta_c^2)^{1/2}} \int_{\psi_0'}^{\pi} n_0(\psi_0)\left[p_0\cos\psi_0+\frac{\beta_0E_0}{c}\right]\sin\psi_0\,d\psi_0, \tag{A8} \]

where \(N\) is the total number of secondary particles produced in one collision, and \(\psi_0'\) is the value of \(\psi_0\) corresponding to \(\psi=\pi/2\), and determined from the equation:

\[ \cos\psi_0'=-\frac{\beta_c E_0}{cp_0}. \tag{A9} \]

Since \(\beta_c\) is close to unity, the angle \(\psi_0'\) is close to \(\pi\), and the contribution of the integral in equation (A8) becomes negligible, except in the case when, in the center-of-mass system, most of the secondary particles are formed in directions very close to the direction of the incident proton. Leaving this case aside and further assuming that the secondary particles have relativistic velocities in the laboratory system, we obtain the following expression for the energy \(W_2\), defined above:

\[ W_2=cp_\nu=\frac{\beta_c E_0}{(1-\beta_c^2)^{1/2}}\,NI^{(p)}. \tag{A10} \]

On the other hand, \(W_1\) has the value

\[ W_1=\frac{E_0}{(1-\beta_c^2)^{1/2}}\,NI^{(p)}. \tag{A11} \]

Consequently, the ratio \(W_2/W_1\) is equal to

\[ W_2/W_1=\beta_c. \tag{A12} \]

For protons with momenta of \(10^{10}\,\mathrm{eV}/c\), which corresponds to the mean momentum of the primary component of cosmic rays, the value of \(\beta_c\) is approximately 0.9, so that \(W_2\) is 10% smaller than \(W_1\). In fact, the difference between \(W_2\) and \(W_1\) may be even greater, since it is possible that part of the observed radiation is produced by secondary nucleons whose energy is significantly smaller than the energy of the primary component. It should be noted that in the limiting case, when

\(n_0(\psi_0)\) is nonzero only for \(\psi_0=0\) or \(\psi_0=\pi\); equation (A8) gives the following value for \(W_2\):

\[ W_2=\frac{E_0\beta_c+cp_0}{2(1-\beta_c^2)^{1/2}}\,Nf(p). \]

If, however, the secondary particles have relativistic velocities in the center-of-mass system, then

\[ W_2=\frac{E_0}{(1-\beta_c^2)^{1/2}}\cdot \frac{\beta_c+1}{2}\cdot Nf(p), \]

whence it follows that

\[ \frac{W_2}{W_1}=\frac{\beta_c+1}{2}. \]

In this case, for protons with momenta of \(10^{10}\ \mathrm{eV}/c\), the quantity \(W_2/W_1\) is approximately equal to 0.95.

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