TOWARD A THEORY OF COSMIC RADIATION¹

W. Heisenberg and H. Euler

IV. SPECTRUM AND ABSORPTION OF COSMIC RAYS

17. Basic experimental data (cf. M3, §8)

In a phenomenological description of cosmic rays it is convenient, following Auger, to distinguish hard and soft components. By the “hard component” is meant that part of the ionizing rays which passes through more than a 10-centimeter layer of Pb; by the “soft component,” that part which is absorbed in 10 cm of Pb^A5, A6.

This distinction is suggested by experiments with coincidences, the arrangement of which is shown in Fig. 9 (above). In Fig. 9 the number of coincidences of counters placed on one straight line is presented as a function of the thickness of the layer of lead placed between the counters. From Fig. 9 it is seen that the coincidence rate decreases sharply in the first 10 cm of Pb, after which a slow fall of the absorption curve occurs. The two branches of the absorption curve may be interpreted as the result of the action of two components of the radiation, one of which is absorbed in a thin layer, whereas lead has little effect on the other.

Fig. 9. Absorption of coincidences in lead according to Woodward and Street. Abscissa: absorber thickness. Ordinate: coincidence rate according to measurements by Woodward and Street at two different altitudes, corresponding to 52 and 76 cm Hg.

From the analysis of photographs of particle tracks in the chamber, carried out in the introductory chapter, it follows that the soft component consists of electrons and positrons, which already in the first centimeter of lead undergo transformations and are absorbed; the hard component, however,

¹ See Uspekhi Fizicheskikh Nauk, XXI, 130, 1939.

consists of heavy electrons, i.e. particles with a mass of the order of 100—200 electron masses.

The question of whether protons and other heavy particles with momenta greater than \(10^9\) eV enter in appreciable quantity into the hard component has not yet been resolved.

Fig. 10. Penetrating and soft components in the lower layers of the atmosphere according to Bowen, Millikan, and Neher (Phys. Rev., 46, 645, 1934). Abscissa: thickness of the atmosphere in meters of water. Ordinate: ionization without shield 1 and with a shield of 10 cm Pb 2 according to Millikan’s measurements (in California). The shaded region is the soft component.

For the time being we shall assume that the hard component consists chiefly of heavy electrons, and we shall test the consequences that follow from this assumption. The fraction of the soft component in the total number of ionizing particles at sea level is about 30%, as is evident from the curves in Fig. 9. According to the measurements of Auger and his collaborators, the fraction of the soft component falls to 15% under a one-and-a-half-meter layer of water and to 7% at a depth of 8 m underground, and rises to 50% at an altitude corresponding to 50 cm Hg\(^\text{A5}\). From the ionization measurements carried out by Bowen, Millikan, and Neher (Fig. 10), it follows that at great heights above the earth’s surface the ratio of the soft component to the hard one continues at first to increase slowly, and then more rapidly. The change of the total

intensity with height is shown in Figs. 11 and 12.

Figure 11 represents the coincidence curve obtained by Pfotzer with counters raised on balloons\(^\text{P1}\). The vertical intensity at first increases, at an altitude corresponding to 50 cm Hg reaches a value twice as large as at the earth’s surface, and finally reaches in the stratosphere

Fig. 11. Vertical intensity of cosmic rays in the atmosphere according to Pfotzer (Z. Physik, 102, 23, 1936). Abscissa: thickness of the atmosphere in centimeters Hg. Ordinate: number of coincidences.

maximum, exceeding the intensity at sea level by approximately 40 times. In still higher layers of the atmosphere, corresponding to pressures less than 8 mm Hg, the number of coincidences again decreases.

Fig. 12 presents the number of coincidences in Lake Boden according to Ehmert1. It turns out that here the intensity decreases much more slowly, by 1 g/cm², than in the atmosphere. The law of decrease of intensity with depth \(T\) can be represented by a power function \(T^{-\gamma}\), where the exponent \(\gamma\), according to Ehmert, at a depth greater than 50 m of water, is equal to \(1.87\)1.

Fig. 13. Ionization in the atmosphere at three different latitudes, according to the measurements of Bowen, Millikan, and Neher (Physic. Rev., 52, 80, 1937). \(1A\)—Sith Forney \(50^\circ N\), \(2A\)—Kepner Stevens Anderson, \(3B\)—Fort Sam Houston \(33.5^\circ N\), \(C\)—Madras \(3^\circ\)

Fig. 12. Vertical intensity in Lake Boden according to Ehmert (Z. Physik, 106, 751, 1937). Abscissa: depth in meters of water from the boundary of the atmosphere (on a logarithmic scale). Ordinate: number of coincidences (on a logarithmic scale). The lowest point of the B F measurements was taken from Forro’s measurements in a mine. 1—in iron, 2—in water, 3—in air was taken by Ehmert with the aid of oblique coincidences, 4—showers

From measurements of ionization, i.e. measurements of the intensity of rays coming from all directions, a somewhat larger ratio of the intensity in the stratosphere to the intensity in Lake Boden is obtained than for vertical coincidences, as is evident from the upper curve in Fig. 13. Vertical intensity and ionization

may be converted one into another, according to Gross \(G^7\), with the aid of several simplifying assumptions concerning the absorption of oblique rays in the atmosphere (cf. \(P^1\)).

From the integral of ionization over the atmosphere one obtains the total energy coming from outer space \(B^{18}\). This energy will ultimately be expended on the formation of ions when it penetrates sufficiently deeply into the atmosphere. Regener \(R^1\) found, for complete ionization, the figure \(6.1 \cdot 10^9\) ion pairs in 1 min per \(1 \text{ cm}^2\), which corresponds (§ 9) to an energy flux of \(1.9 \cdot 10^{11}\) eV/\(\text{min}\cdot\text{cm}^2\). Bowen, Millikan, and Neher \(B^{18}\) give another value for the energy flux—\(1.7 \cdot 10^{11}\) eV/\(\text{min}\cdot\text{cm}^2\). We shall compare the distribution of intensity with depth with the spectral distribution of intensity over different energies.

In Fig. 16 the spectrum of penetrating particles at sea level according to Blackett \(B^{12}\) is shown (cf. also \(K^4, B^{12}, H^{10}, A^2\)). Along the abscissa are plotted the values of the momentum \(H\rho\), and along the ordinate—the frequency of occurrence of particles in the given interval of momenta. From this curve the total energy of penetrating particles at sea level may be estimated as \(6 \cdot 10^9\) eV/\(\text{min}\cdot\text{cm}^2\).

18. Qualitative theoretical discussion

The course of the vertical intensity in the upper layers of the atmosphere can be understood on the basis of the cascade theory of showers, if it is assumed that electrons or photons \(B^5, C^1\) fall upon the surface of the earth’s atmosphere from outer space. These electrons multiply in the very upper layers of the atmosphere. In the deeper layers of the atmosphere there occurs absorption of the particles because of fragmentation of the initial energy. In this way an intensity curve is formed in the atmosphere (Fig. 11), which has a form similar to that of the curves of the cascade theory of showers (Fig. 7).

Since the universal length \((l = 1)\) (Table 2) in air (275 m of air under normal conditions) corresponds to a pressure difference of \(2.6\) cm Hg, or \(0.34\) m of water, a change of intensity by an order of magnitude must always occur over sections corresponding to several centimeters of mercury.

From the position of the maximum at 8 cm Hg \((l \simeq 3)\) it follows, on the basis of Fig. 7 and Table 14, that the average initial energy is about \(10^9\) eV. A more exact comparison of the radiation in the upper layers of the atmosphere with the cascade theory of showers will be carried out in § 19. It will then appear that, although in the upper layers of the atmosphere the formation of cascade showers may be attributed to electrons, the presence of electrons in the lower layers of the atmosphere is due to another process connected with the penetrating component.

In considering the penetrating component we shall proceed from Yukawa’s theory (ch. III). According to this theory, heavy electrons are unstable, i.e., they can decay in empty space. We must therefore assume that they cannot exist in outer space for a long time and that heavy

toward a theory of cosmic radiation

the electrons that we find at the surface of the earth are first formed in the atmosphere by the soft component W³, B¹⁸.

Although the occurrence of penetrating particles has not yet been proved by balloon experiments (in that case, in the very upper layers of the atmosphere a decrease in the intensity of penetrating particles should be observed), nevertheless the formation of penetrating particles should be regarded as possible B¹⁸ on the basis of the fact that the energy contained in the penetrating component on the earth, \(6 \cdot 10^9\) eV/min·cm², is small in comparison with the total energy \(2 \cdot 10^{11}\) eV/min·cm², which falls on the boundary of the atmosphere¹).

The theoretical possibility of the creation of heavy electrons in the atmosphere is due, within Yukawa’s theory, to a process inverse to the process considered in § 15: a photon with large energy, upon collision with a heavy particle, can form one or several heavy electrons W³, H⁸. However, the course of this process in the region of high energies cannot be calculated because of the limitations of the present theory. Therefore we shall consider here the occurrence of heavy particles semi-empirically.

We shall expect that heavy electrons are most often created in the stratosphere, since here the soft component has the greatest intensity. Therefore we shall make the somewhat schematized assumption that the spectrum of heavy particles, which depends on the energy in approximately the same way as the spectrum of electrons, arises near the ionization maximum (\(\sim 8\) cm Hg). Whether this simple assumption is correct we shall check later by means of the Blekett spectrum at the surface of the earth. This spectrum of heavy electrons will undergo the following changes while passing through the atmosphere and the layer of the earth.

First, the heavy electrons will be slowed down by ionization. Therefore, from the theory of ionization there can be obtained, in a known approximation, the law of absorption of penetrating particles (§ 20).

Second, heavy electrons will spontaneously decay into electrons and neutrinos. It turns out that this process makes a significant contribution to the absorption of the penetrating component and that the electrons thereby created constitute an essential part of the soft radiation in the lower layers of the atmosphere. The intensity of the latter makes it possible to estimate the decay constant in Yukawa’s theory (§ 21).

Third, upon collision with nuclei the heavy particles may cause the processes considered in § 15. As analysis of the experimental material shows, the soft radiation arising in these processes will, in the first instance, be observed together with ionization showers under water, whereas in the lower layers of the atmosphere it recedes into the background in comparison with the electrons formed in spontaneous decay.

¹) Blekett’s remark.

19. Cascade spectrum of electrons in the upper layers of the atmosphere. The latitude effect

In a more precise discussion of cascade radiation we proceed from measurements of ionization at various latitudes, which were carried out by Bowen, Millikan, and Neher \(^{B17}\). The results of these measurements are given in Fig. 13. The large difference in intensity in the stratosphere at different latitudes can be explained, according to Störmer \(^{S13,S14}\), Lemaitre and Vallarta \(^{L1}\), if it is assumed that the radiation coming from outer space contains a considerable number of charged particles.

Charged particles can overcome the earth’s magnetic field at geomagnetic latitude \(\varphi\) only when their momentum is greater than a certain value characteristic of each latitude \(^{S13}\):

\[ pc = E_{\varphi}=1.9\cdot 10^{10}\ \mathrm{eV}\cdot \cos^{4}\varphi \]

\(\varphi\)	0	20	40	50	80	90°
\(E_{\varphi}\)	1.9	1.5	0.65	0.12	0.0017	\(0\cdot 10^{10}\ \mathrm{eV}\)

\[ \tag{39} \]

Here it is assumed that the charge of the particle is equal to the elementary charge.

Thus, at a given height at the equator fewer electrons can arrive than at high latitudes. Therefore the ionization maximum at the equator is lower than in the polar regions. Further, the average energy of the incoming particles at the equator is greater than in the polar regions. Therefore the flat ionization maximum at the equator occurs at a greater thickness of the equivalent layer of water than at high latitudes \(^{H7,N8}\). According to (39), we must understand the latitude-independent part of the ionization, i.e. the difference of the curves for the various latitudes \(53—38^\circ\), \(38—3^\circ\), as the result of the action of electrons in a narrow energy interval

\[ 2.5\cdot 10^{9}\ \mathrm{eV} < E < 6.7\cdot 10^{9}\ \mathrm{eV}, \]

\[ 6.7\cdot 10^{9}\ \mathrm{eV} < E < 17\cdot 10^{9}\ \mathrm{eV}, \]

in other words, as the result of the action of electrons with an almost definite energy \(^{B18}\):

\[ E \approx 4\cdot 10^{9}\ \mathrm{eV}, \qquad E \approx 10^{10}\ \mathrm{eV}. \tag{39a} \]

On the other hand, the cascade theory of showers gives (Fig. 7) the course of the intensity produced by electrons of these energies, \(y=3.3\); \(y=4.2\) in the atmosphere (Fig. 7).

A comparison of the experimental and theoretical course of the latitude effect is given in Fig. 14, which is borrowed from the work of Bowen, Millikan, and Neher \(^{B18}\). We see that in the upper layers of the atmosphere the course of the measured curve agrees with the curve of the cascade theory of showers, if one disregards a small discrepancy

positions of maximum N⁷, caused by the choice of the mean value (39a). On the contrary, in the lower layers of the atmosphere the intensity, according to the cascade theory of showers, is considerably less than the true intensity of the soft component, which, according to Auger and Leprince-Ringuet A⁷, is the same at all latitudes and amounts at sea level to about \(1/3\) of the total intensity.

It is therefore necessary to conclude that electrons in the lower layers of the atmosphere constitute a secondary formation of the penetrating component B¹⁸ ¹). We shall arrive at the same result later, proceeding from the altitude dependence of Hoffmann’s coincidences (§ 23).

Let us now consider the effects produced by electrons with energy exceeding the magnetic cutoff energy \(E_{\varphi}\). This question was considered by Nordheim N⁶ and Heitler H⁷. For this purpose we shall assume that electrons arrive from outer space whose energy spectrum has the form

Fig. 14. Comparison of the ionization produced by a fraction of cosmic rays depending on latitude with the cascade theory according to Bhabha, Millikan, and Neher (Phys. Rev., 52, 80, 1937). Difference of the curves of Fig. 13 on a logarithmic scale. 1—according to the cascade theory. 2—measurements. \(B-C\)—Fort Sam Houston—Madras, \(E_{0}=10^{10}\ \mathrm{eV}\). \(A-B\)—Fort Churchill—Fort Sam Houston: \(E_{0}=4\cdot10^{9}\ \mathrm{eV}\)

\[ F_{1}(E',0)=I_{0}\left(\frac{10^{8}\mathrm{eV}}{E'}\right)^{\gamma} \tag{40} \]

for energies greater than \(E'\). The constants \(\gamma\) and \(I_{0}\) must at first remain undetermined. We are then interested in the spectrum \(F_{1}(E,l)\), which is established at a depth \(l\) below the boundary of the atmosphere, in particular at sea level \((l=29)\).

As Nordheim N⁶ showed, in deep layers \((l \gg 15)\) the form of the spectrum \(F_{1}(E,l)\) is again represented by the power law (40). The inten-

¹) This follows even more clearly from Fig. 14, if one takes into account the Gross transformation (§ 17).

sity of the spectrum \(F_1(E,l)\) will decrease exponentially in the atmosphere, the exponent being the larger, the larger the value of \(\gamma\). Then, according to § 10:

\[ F_1(E,l)=\int^\infty z\left(l,\ln\frac{E'}{E}\right) \frac{-\partial F_1(E',0)}{\partial E'}\,dE', \quad (E \ge E_j) \]

\[ \begin{aligned} E'&=E_\varphi \quad &&\text{for } E<E_\varphi,\\ E'&=E \quad &&\text{for } E>E_\varphi, \end{aligned} \]

or, since \(\dfrac{E'}{E}=e^{y'}\), with the aid of (40) we obtain

\[ F_1(E,l)=I_0\left(\frac{10^8\mathrm{eV}}{E}\right)^\gamma \int^\infty z(l,y')e^{-\gamma y'}\gamma\,dy'. \]

\[ \begin{aligned} y'&=\ln\frac{E_\varphi}{E} \quad &&\text{for } E<E_\varphi,\\ y'&=0 \quad &&\text{for } E>E_\varphi . \end{aligned} \]

For large \(l\) the value of the integral depends almost not at all on \(E\) and, consequently, on the latitude \(\varphi\), and, according to (19), it decreases exponentially with \(l\); numerical calculation gives:

\[ F_1(E,l)\approx 4I_0\left(\frac{10^8\mathrm{eV}}{E}\right)^\gamma e^{-f(\gamma)(l-10)}, \tag{41} \]

where

\[ f(\gamma)\approx 0.36\sqrt{\gamma-1};\quad (\gamma\lessgtr 5). \]

According to Nordheim’s more exact calculations, for \(l\to\infty\):

\[ f(\gamma)=\frac{4}{3}-\frac{1}{\gamma+1} -\sqrt{\left(\frac{2}{3}-\frac{1}{1+\gamma}\right)^2 +\frac{4}{3(\gamma+1)\gamma}}\, . \tag{42} \]

Calculation of the intensity for thin layers leads to a maximum lying at our latitudes (for \(\gamma=1.5\text{--}2.5\)) at \(l=3\text{--}4\). The intensity at the maximum is \(I_1\sim 4I_0^{H7,H6}\). The constant \(I_1\) may be taken from Pfotzer’s measurements in the stratosphere:

\[ 4I_0\approx I_1\approx 40/\mathrm{min}\cdot\mathrm{cm}^2. \]

It is more difficult to determine the constant \(\gamma\). From the condition that the intensity of the cascade spectrum at the surface of the earth (\(l=29\)) must be no greater than the true intensity of the soft component (electrons) \(\left(\sim \frac{1}{2}/\mathrm{min}\cdot\mathrm{cm}^2\right)\), one obtains first of all the lower limit of the exponent: \(\gamma>1.3\).

Further data for estimating \(\gamma\) may be obtained from the latitude effect. If one assumes that the spectrum of particles arriving from outer space consists, chiefly, of

On the Theory of Cosmic Radiation

electrons and positrons and only a relatively small number of photons, then for the energy brought by them we obtain

\[ \int_{E_\varphi}^{\infty} -\frac{\partial F_1}{\partial E}\, E\, dE, \qquad \text{or, by (40), } \left(\frac{1}{E_\varphi}\right)^{\gamma-1}; \]

where \(E_\varphi\) denotes the limiting energy (39).

From the table compiled by Johnson\(^5\) on the basis of the measurements of Bowen, Millikan, and Neher, it follows that \(\gamma \approx 1.8\).

Latitude	\(3^\circ\)	\(39^\circ\)	\(52^\circ\)
Ions/\(\mathrm{cm^2/min}\)	\(1.8\cdot 10^9\)	\(3.2\cdot 10^9\)	\(6\cdot 10^9\)
Energy eV/\(\mathrm{cm^2/min}\)	\(0.6\cdot 10^{11}\)	\(1.0\cdot 10^{11}\)	\(1.9\cdot 10^{11}\)
\(E_\varphi\)	\(15\cdot 10^9\)	\(8\cdot 10^9\)	\(2\cdot 10^9\)

Up to now it has been assumed that the form of the primary spectrum arriving from world space can be represented by a purely power-law expression (40). A spectrum of a more general form is best considered by resolving it into a sum of expressions of the form (40)\(^6\).

It is also possible to construct a spectrum of such a form that it represents the intensity of the soft component in the whole atmosphere\(^7\), \(^\text{H}\). However, as Heitler\(^\text{H7}\) showed, such a spectrum gives, for the soft component at sea level, a latitude effect of less than 1%, whereas in reality, according to the measurements of Auger and Leprince-Ringuet\(^\text{A7}\), it amounts, for the soft component, to about 10%.

Hence one may conclude, as before, that the electrons present at sea level cannot be obtained from the cascade spectrum, but arise from the penetrating component. In Fig. 15 the course of the various constituent parts of cosmic radiation in the atmosphere is presented.

Fig. 15. Decomposition of cosmic radiation in the atmosphere. 1 — measurements of the vertical intensity according to Pfotzer; 2 — extrapolation of the penetrating part. \(A\) (shaded region) “cascade electrons,” formed from electrons arriving from world space by multiplication (§ 19, \(\gamma = 1.9\)). \(B\) — heavy electrons formed from \(A\) in the atmosphere (§ 20). \(C\) — decay electrons arising from \(B\) (§ 21).

The solid curve is a repetition of the curve of vertical intensity as a func-

height by Pfotzer (Fig. 11). The lower dotted curve represents the extrapolation of that part of it which we can attribute to the penetrating component (§ 20). The shaded region \(A\) gives the theoretical number of cascade electrons which are obtained from the primary spectrum (40), falling from the earth’s atmosphere, where \(\gamma = 1.9\).

It is evident that measurements of the intensity in the upper layers of the atmosphere are very well represented by this cascade spectrum. However, in the lower layers of the atmosphere there still remains a certain part \(C\), which for the reasons indicated above must have another origin. This part of the cosmic radiation was first noticed in the experiments of Regener and his pupils \(^{P1}\). It will be considered in § 21.

§ 20. Penetrating component

In what follows it will be assumed that heavy electrons are produced in the stratosphere (more precisely, at \(8\ \mathrm{cm}\ \mathrm{Hg}\)). The distribution of heavy electrons by energy must be similar to the distribution by energy of the soft component. Therefore the distribution of heavy electrons in the stratosphere having momentum greater than \(p\) must have the form:

\[ F_s(p)=I_0\left(\frac{2\cdot 10^9\ \mathrm{eV}}{pc}\right)^\gamma . \tag{43} \]

We shall calculate the absorption of such a spectrum of heavy electrons in the atmosphere, due to ionization and spontaneous decay. In calculating the ionization we shall proceed from formula (11), according to which a heavy electron with momentum \(p\) has a range \(R\), approximately equal to:

\[ R(p)\approx \frac{\mu c^2}{a} \left( \frac{2+\left(\frac{p}{\mu c}\right)^2} {\sqrt{1+\left(\frac{p}{\mu c}\right)^2}} -2 \right) \]

\[ (\mu = 160\,m \text{ — the mass of the heavy electron; } \mu c^2 \sim 8\cdot 10^7\ \mathrm{eV}). \]

TABLE 6

	Lead	Water	Air
\(a\)	\(1.2\cdot 10^7\)	\(2\cdot 10^6\)	\(2.5\cdot 10^3\ \mathrm{eV/cm}\)

A particle which initially had momentum \(p_0\), after passing through a layer \(T\), has a smaller momentum \(p\), which can be found from the equation:

\[ T = R(p_0)-R(p), \]

i.e., for \(p_0 \gg 3\mu c\),

\[ T \simeq \frac{p_0c-2\mu c^2}{a}-R(p). \]

In this case the spectrum (43), after passing through a layer \(T\), owing to absorption, is transformed into a spectrum poorer in energy of particles with momentum exceeding \(p\):

\[ F_s(T,p)=I_0\left(\frac{\alpha^{-1}\cdot 2\cdot 10^9\ \mathrm{eV}}{T+R(p)+\frac{2\mu c^2}{a}}\right)^\gamma \tag{44} \]

(because of the condition \(p_0>3\mu c\), at small momenta this formula is suitable only for \(T>\mu c^2/a \sim 40\ \mathrm{cm}\ \mathrm{H_2O}\)).

Fig. 16. Spectrum at sea level. Abscissa: momentum \(p\) of the particles. Ordinate: number \(\partial F/\partial p\) of particles in a unit interval of momentum. The curve \(\partial F_s/\partial p\): theoretical spectrum of heavy electrons. The curve \(\partial F_2/\partial p\): theoretical spectrum of light electrons (§ 21). The curve \(\partial(F_s+F_2)/\partial p\): complete theoretical spectrum. Points: results of measurements of the complete spectrum according to Bleckett. The upper part of the figure gives spectra at small values of the momentum. The lower part gives the spectrum of penetrating particles at large momenta.

This integral spectrum of penetrating particles at large momenta \((p>3\mu c)\) decreases as

\[ F_s(T,p)\approx I_0\left(\frac{2\cdot 10^9\mathrm{eV}}{aT+pc}\right)^\gamma \tag{45} \]

(\(aT\approx 2\cdot 10^9\ \mathrm{eV}\) at sea level). Thus the number of heavy electrons with momentum greater than \(p\), at depth \(T\), is equal to the number of primary particles with momentum greater than \(p\) plus the momentum lost along the path \(T\).

At small momenta one may expand formula (43) in a series in \(R(p)\), and then for the number of particles with momentum less than the momentum \(p\) we obtain:

\[ I_0\gamma\frac{(2\cdot 10^9\mathrm{eV})}{(aT+2\mu c^2)^{\gamma+1}}\cdot aR(p). \tag{46} \]

Since the mean free path rapidly decreases with decreasing momentum, formula (46) directly explains why heavy electrons with small momentum are so rarely observed. In Fig. 16 the course of the theoretical curve is presented for the differential spectrum at a momentum less than \(pc=7\cdot 10^8\ \mathrm{eV}\).

Up to now we have considered the absorption of penetrating particles caused by ionization, and have neglected absorption due to their spontaneous decay. As an estimate of the decay period of a heavy electron according to Yukawa shows \(\left(\tau\sim \frac{1}{2}\cdot 10^{-5}\ \mathrm{sec};\ \text{see § 14}\right)\), this neglect is permissible when the absorption of the penetrating component in water or in solids is considered. By contrast, in the atmosphere spontaneous decay plays an essential role.

When considering spontaneous decay, we may confine ourselves to the spectrum at large momenta \((pc>2\cdot 10^8\ \mathrm{eV})\), since at small momenta \((pc<2\cdot 10^8\ \mathrm{eV})\) spontaneous decay only slightly changes the qualitative course of (46). Taking into account simultaneously the absorption due to ionization and spontaneous decay, for the change of the differential spectrum

\[ f(T,p)=-\frac{\partial F(T,p)}{\partial p} \]

with depth \(T\) we obtain the differential equation

\[ \frac{\partial f(T,p)}{\partial T} = \frac{a}{c}\cdot \frac{\partial f(T,p)}{\partial p} - \frac{b}{pT}f(T,p). \tag{47} \]

Here \(T\) denotes the depth in centimeters of water, measured from the beginning of the atmosphere, and \(a\) is the energy loss per \(1\ \mathrm{cm}\) of water (see Table 6 on p. 270).

The probability of decay per \(1\ \mathrm{cm}\) of water in the atmosphere will be the greater, the smaller the density of the air. Therefore the second term of (47) contains \(T\) in the denominator, on the assumption of the validity of the barometric formula. The quantity \(b\) is related to the decay period \(\tau\) of the heavy electron by the formula

\[ b=\frac{\mu}{\tau}\, \frac{\text{density of water}\cdot (T_{\text{sea level}})} {\text{density of air at sea level}}. \]

The solution of the differential equation (47) has the form

\[ f(T,p)=g(pc+aT)\left(\frac{pc}{aT}\right)^{\frac{bc}{pc+aT}}, \tag{48} \]

where \(g(pc+aT)\) represents an arbitrary function of \(pc+aT\). It is determined from the condition that in the stratosphere, more precisely at \(T\simeq 100\), \(f(T,p)\) must pass into the differential spectrum obtained by differentiating (43). Then for the dependence of the differential spectrum on depth we obtain

\[ f(T,p)=\gamma I_0 \frac{(2\cdot 10^9\ \mathrm{eV})^\gamma \cdot c} {[pc+a(T-100)]^{\gamma+1}} \left[ \frac{100\,pc}{T[pc+a(T-100)]} \right]^{\frac{bc}{pc}}. \tag{49} \]

In particular, at sea level \((T \sim 1000)\) the spectrum will have the form

\[ f(1000,p)=\gamma I_0 \frac{(2\cdot 10^9\mathrm{eV})^\gamma c}{(pc+900a)^{\gamma+1}} \left[\frac{pc}{10(pc+900a)}\right]^{\frac{bc}{pc+1\,000a}} . \tag{50} \]

The second factor on the right-hand side of the last formula contributes to the fact that the spectrum of the penetrating particles falls off more slowly than the spectrum of the soft component. This expresses the fact that, on the given segment, spontaneous decay occurs more often at small values of the momentum than at large ones. At very high energies \((pc \gg 900a \sim 1.8\cdot 10^9\ \mathrm{eV})\) the falloff of the spectrum is again determined only by the exponent \(\gamma\). These conclusions are confirmed by Blekett’s measurements \(^{B12}\). In Fig. 16 (p. 271) (lower part) the theoretical spectrum, computed for the values \(\gamma=1.87,\ \dfrac{bc}{1000a}=0.37\), obtained from Ehmert’s measurements \(^{E1}\), which will be discussed below, is compared with Blekett’s measurement results. The theory satisfactorily conveys the general course of the spectrum, although, of course, not in all details.

Further, formulas (49) and (50), as Kuilenkampf \(^{K5}\) observed, give a simple explanation of the surprising results of the observations of Ehmert \(^{E1,E2}\) and Kuilenkampf \(^{K5}\), who found that absorption in water proceeds more slowly than in air (Fig. 12 on p. 263). The thickness of the traversed layer of air in Ehmert’s experiments varied because, with the observation point unchanged (approximately at sea level), the angle \(\varphi\) formed by the incident rays with the vertical was changed. This was achieved by changing the inclination with respect to the vertical of the apparatus operating on coincidences.

Let us call the quantity \(T'=\dfrac{1000}{\cos\varphi}\) the apparent depth and compute the intensity of the penetrating component registered by such an apparatus. For this it is necessary to replace, in equation (47), \(a\) by \(\dfrac{a}{\cos\varphi}\) and \(b\) by \(\dfrac{b}{\cos\varphi}\). Then for the distribution of the intensity at sea level as a function of \(T'\) we obtain the following expression:

\[ f_{\mathrm{air}}(T',p)=\gamma I_0 \frac{(2\cdot 10^9\mathrm{eV})^\gamma c} {\left(pc+\frac{9}{10}aT'\right)^{\gamma+1}} \left[ \frac{pc}{10\left(pc+\frac{9}{10}aT'\right)} \right]^{\frac{bc}{pc\frac{1000}{T'}+1000a}} . \tag{51} \]

For measuring the intensity in water, where spontaneous decay no longer plays a role, on the basis of (48) \((b=0)\) and (50) it follows that

\[ f_{\mathrm{water}}(T,p)=\gamma I_0 \frac{(2\cdot 10^9\mathrm{eV})^\gamma c} {[pc+a(T-100)]^{\gamma+1}} \left[ \frac{pc+a(T-1000)}{10[pc+a(T-1000)]} \right]^{\frac{bc}{pc+aT}} . \tag{52} \]

Integrating (51) and (52) over the entire spectrum, we obtain the dependence of the intensity on the depth \(\left(\delta=\dfrac{hc}{1000a}\right)\)

\[ F_s(T',0)_{\text{air}} = \gamma I_0 \left( \frac{2\cdot 10^9 eV}{\frac{9}{10}aT'} \right)^\gamma \int_0^\infty \frac{du}{(1+u)^{\gamma+1}} \left[ \frac{u}{10(1+u)} \right]^{ \frac{\delta}{1+\frac{\gamma}{10}u} }, \tag{53} \]

\[ F_s(T,0)_{\text{water}} = \gamma I_0 \left( \frac{2\cdot 10^9 eV}{a(T-100)} \right)^\gamma \times \]

\[ \times \int_0^\infty \frac{du}{(1+u)^{\gamma+1}} \left[ \frac{1}{10(1+u)} \left( u+\frac{T-1000}{T-100} \right) \right]^{ \frac{\delta\cdot 1000}{u(T-100)+T} }. \tag{54} \]

Under very thick layers, the expressions for the intensity are in a constant ratio to one another, which, according to Ehmert (Fig. 12), is approximately \(2:1\). In this case the decrease of the intensities with depth follows the law \(T^{-\gamma}\). The value of the constant \(\gamma\) according to Ehmert, \(\gamma\sim 1.87\), is in good agreement with that value of it which was determined in § 19 from other considerations.

From the constant value of the ratio of the intensities one can determine the quantity \(\delta\) and, consequently, the mean lifetime of the heavy electron. It turns out that

\[ \delta\sim 0.37;\quad \tau\sim 2.7\cdot 10^{-6}\ \text{sec.} \tag{55} \]

This value is about 5 times larger than that calculated from Yukawa’s theory (§ 14). Taking into account the inaccuracy of many details in Yukawa’s theory, in particular the inaccurate value of the mass of the heavy electron, this agreement is satisfactory. It should also be noted that the experiments give directly the ratio of the lifetime to the mass of the heavy electron. Thus a change in the value of the mass of the heavy electron entails a corresponding change in the experimental value of the lifetime.

Finally, let us also compare the theoretical formulas for the decrease of the intensity with depth, (49), (54), with the measurements of Auger\(^5\), who obtained a somewhat slower decrease of the intensity than Ehmert. The theoretical values of the intensity were calculated with the constants \(\gamma\) and \(\delta\) taken from Ehmert’s measurements. It turns out that the theory correctly conveys the corresponding slower decrease of the intensity; however, for a more accurate representation of Auger’s experiments it is necessary to choose a somewhat smaller value of \(\gamma\).

In the preceding calculations of the spectrum of penetrating particles and their absorption, the processes considered in § 15 were not taken into account, the existence of which follows from the presence of forces acting between heavy

particles and nuclei. However, Figs. 16 and 14 give the impression that these nuclear processes apparently affect only the details in the course of the spectrum of penetrating particles. A more exact theoretical investigation of these effects, considered in § 15, is still impossible.

TABLE 7

	$I$ in meters of water	Intensity by Oke	Theoretical value of intensity
Jungfraujoch	6.6	108	120
Paris . . .	10	75	75
Basement . . .	30	28	17
Catacombs . . .	75	5	3

The penetrating component at sea level has a latitude effect of the order of $10^0/_{0}$,2 The fact that this latitude effect is small may be regarded as an independent argument in favor of the assumption that the penetrating component is produced by the soft component: penetrating particles lose in the atmosphere about $2 \cdot 10^9 \ \mathrm{eV}$ to ionization. If we regard the primary particles as having an energy of $2 \cdot 10^9 \ \mathrm{eV}$, then one should expect the presence of a latitude effect of the order of 100%, since, whereas in our latitudes all particles with energy greater than $2 \cdot 10^9 \ \mathrm{eV}$ pass through the earth’s magnetic field, at the equator only particles with energy greater than $10^{10} \ \mathrm{eV}$ pass through it. However, if the penetrating component is regarded as the secondary formation of the soft component, then it must be assumed that the penetrating particles observed at the earth’s surface are created by photons at the maximum of intensity in the stratosphere $(l \sim 4)$. These photons, which must have energy exceeding $2 \cdot 10^9 \ \mathrm{eV}$, are in turn created by electrons arriving from cosmic space, with energy exceeding on the average $2 \cdot 10^{10} \ \mathrm{eV}$ (owing to the fragmentation of energy in the stratosphere). Electrons flying from cosmic space with such energy will not, however, be disturbed by the earth’s magnetic field (§ 12). It is natural to expect that the penetrating component depends only weakly on the magnetic field, since the soft radiation is responsible for the creation of the penetrating component not only at the maximum of intensity in the stratosphere, although, because of the great intensity of the soft radiation, a large part of the penetrating particles is created precisely there.

21 Electrons produced in the decay of heavy electrons in the lower layers of the atmosphere

According to Yukawa’s theory, the penetrating component creates secondary soft radiation in two ways.

First, penetrating particles can spontaneously decay into a neutrino and electrons, which then multiply by means of a cascade mechanism. Second, heavy electrons, upon colliding with atomic nuclei, can create photons which

They then in turn give rise to cascade showers (§ 15). Electrons produced from nuclear processes will be encountered almost equally often in air and under large layers of solid bodies, since the formation and absorption of secondary particles occur approximately in proportion to the mass of the substance (§ 11 γ).

On the contrary, electrons formed as a result of spontaneous decay are encountered in air in a considerably larger quantity than under water or under layers of other dense substances, since the region from which secondary particles formed in air can emerge is geometrically larger than the corresponding region in water. It is evident that in this larger region, owing to spontaneous decay, a greater number of electrons will be produced.

The experimental value of the ratio of the soft component to the hard one in air at sea level is larger than under a layer of solid substance (§ 17). Therefore we shall assume that the electrons present in the lower layers of the atmosphere, which according to § 19 constitute a secondary formation of the hard component, are created mainly owing to spontaneous decay. The electrons arising in nuclear processes, together with ionization showers, become noticeable under water or under thick layers of dense substances.

Let us calculate the spectrum of electrons formed as a result of spontaneous decay:

\[ F_2(T,p), \]

which is in equilibrium with the spectrum of the penetrating particles

\[ F_s(T,p) \]

\[ [F(T,p)=\text{ the number of particles at depth }T\text{ from the boundary of the atmosphere with momentum greater than }p]. \]
We shall divide the electrons formed from spontaneous decay into two categories: first, electrons formed as a result of the decay of heavy electrons previously stopped owing to ionization; second, electrons arising in the decay of moving heavy particles.

The former have a very small energy, less than \(4\cdot10^7\ \mathrm{eV}\), since the rest energy of the heavy particle going into the formation of an electron and a neutrino is \(\mu c^2=8\cdot10^7\ \mathrm{eV}\). According to the conclusions of §§ 9 and 11, these electrons have a range of the order

\[ \frac{\mu c^2}{2a}\approx 20\ \mathrm{g/cm^2}. \]
The total number of such electrons \(\bar A\) is therefore approximately equal, according to (46), to

\[ \bar A\sim \frac{1}{2}\frac{\mu c^2}{aT}\cdot F_s'(T,0), \tag{56} \]

which at sea level amounts to about \(2\%\) of the penetrating particles,

A large number of electrons can be formed as a result of the decay of moving heavy electrons. According to § 14, a heavy electron with momentum \(p\) has a decay probability

\[ \frac{\mu}{p\tau} \quad \text{per } 1\ \mathrm{cm}, \]

where \(\tau\) denotes the lifetime of the heavy particle in that frame of reference in which the heavy particle is at rest.

In the decay of a penetrating particle with momentum \(p\), an electron will be formed with a component of momentum in the direction of motion of the penetrating particle equal, on the average, to \(\frac{p}{2}\) (§ 14). This electron, in turn, will create a shower containing \(z\left(l,\frac{p/2}{p'}\right)\) electrons with momentum \(p' > E_j\) after traversing a layer \(l\). Here \(z\) denotes the multiplication function (§§ 10, 11), \(l\) is the thickness of the layer of matter in the system of units given in Table 2, which is characterized by the units of length \(X_0\) and of energy \(E_j\) in each substance.

Thus there arises a certain energy spectrum of electrons \((p'c > E_j)\) in equilibrium with the spectrum of penetrating particles \(F_s(T,p)\):

\[ F_2(T,p')=\int_{p=2p'}^{\infty} dp\,\frac{-\partial F_s(T,p)}{\partial p} \int_{l=0}^{\infty} dl\,z\left(l,\frac{p/2}{p'}\right)\frac{X_0}{\tau c}\frac{\mu c}{p}+A, \]

or, from (17),

\[ F_2(T,p')=\frac{\mu c}{2p'}\frac{3}{4}\frac{X_0}{\tau c}F_s(T,2p')+A. \tag{57} \]

We shall first consider the total intensity, and then the spectral distribution of these decay electrons \(F_2(T,p')\).

The total number of electrons will be, by (14a), approximately equal to \(3F_2\left(T,\frac{E_j}{c}\right)\). For the total relative intensity, i.e. for the ratio of the number of electrons to the number of penetrating particles, from (57) and (56) we obtain

\[ \varkappa \approx \frac{3F_2\left(T,\frac{E_j}{c}\right)}{F_s(T,0)} \approx \frac{9}{4}\frac{X_0}{\tau c}\frac{\mu c^2}{2E_j} + \frac{1}{2}\frac{\mu c^2}{aT}. \tag{58} \]

Formula (58) gives, after adding the ionization electrons (§ 12), the ratio of the intensity of the soft component to the intensity of the hard component at sea level \((X_0=275\ \mathrm{m})\)

\[ \varkappa=\frac{1}{3}, \]

that agrees with the experimental value of this ratio, if for the decay time \(\tau\) one takes:

\[ \tau = 2 \cdot 10^{-6}\ \mathrm{sec}. \tag{59} \]

This determination of the decay period of heavy electrons from the intensity of the soft component in the lower layers of the atmosphere is in approximate agreement with the determination of the decay period from measurements of absorption according to Ehmert (55).

The universal length \(X_0\), entering into formula (58), accounts for the above-mentioned dependence of the intensity of decay electrons on the specific volume of the substance in which equilibrium with the penetrating component is established. According to (58), the ratio of the intensity of the soft component (decay electrons) to the hard one increases with height inversely proportional to the pressure, which, it would seem, is in agreement with the results of measurements presented in Figs. 10 and 15. This increase continues up to the maximum of the intensity of vertical coincidences (Figs. 11, 15), observed by Regener and his students at a pressure of \(30\ \mathrm{mm}\ \mathrm{Hg}\). This maximum should signify, according to formula (48), the entry into play of decay electrons.

Under water (\(X_0 = 34\ \mathrm{cm}\)) or under a layer of another dense substance, the number of decay electrons is reduced to a small addition to the second term in (58). Thus, here the Babha cascade showers predominate, caused by ionization electrons (§ 12) and by random electrons arising as a result of nuclear transformations.

The form of the differential spectrum of decay electrons

\[ \frac{\partial}{\partial p} F_s(T, p), \]

i.e. the number of decay electrons with energy lying in the interval \(dp\) at sea level, is shown in Fig. 16. Here for \(\tau\) the value \(\tau = 2 \cdot 10^{-6}\ \mathrm{sec.}^{E7}\) was adopted. This spectrum at high energies falls approximately one power faster than the spectrum of penetrating particles. The ratio of the two spectra as a function of momentum can be compared with experiment, since it is determined by the average energy losses by particles of different masses possessing the same momentum \(p\). Namely, heavy particles suffer in a thin layer, generally speaking, only small ionization losses.

By contrast, the energy losses of radiation electrons are very large, and according to (7) the relative losses do not depend on energy and amount to \(1.72\) per \(1\ \mathrm{cm}\ \mathrm{Pb}\). Therefore the relative losses in thin layers, averaged over the particles, are

\[ \frac{ 1.72\, \dfrac{\partial F_2(p)}{\partial p} }{ \dfrac{\partial F_2(p)}{\partial p} + \dfrac{\partial F_s(p)}{\partial p} } \tag{60} \]

per \(1\ \mathrm{cm}\ \mathrm{Pb}\). This theoretical value of the relative losses per \(1\ \mathrm{cm}\ \mathrm{Pb}\) as a function of momentum is plotted in Fig. 3. It is easy to see,

that it is in agreement with Blekett’s measurements[^B11]. Whereas in a thin layer the relative energy losses of heavy electrons to ionization are very small, they may be appreciable in thick layers. For example, in a 2 cm layer of gold not only electrons, but also all heavy electrons with momentum less than a certain value, amounting according to (11) to \(pc = 1.3 \cdot 10^8\) eV, must be stopped. This value may accidentally increase somewhat further owing to scattering or nuclear processes.

According to Blekett’s measurements[^B11], all particles with momentum greater than \(pc = 2.4 \cdot 10^8\) eV can pass through a gold plate 2 cm thick, and all particles with momentum less than \(pc = 1.65 \cdot 10^8\) eV are stopped in it. At first it seems that these results indicate the presence of induced decay. However, they can also be qualitatively reconciled with the theory of spontaneous decay.

Chapter V. Secondary Effects

22. Survey

On the basis of what has been set forth in the preceding sections, we must expect cosmic rays to produce the following secondary effects.

First, the soft component will create cascade showers (§§ 10, 11). These cascade showers develop in layers of 1–5 cm Pb (Fig. 8), depending on whether they contain several particles or several thousand particles. In heavy substances they are formed more often than in light ones (§ 11). The energies \(E_j\) and lengths \(l = 1\), characteristic for the formation of cascade showers in various substances, are given in Table 2.

Second, under thick layers of matter the penetrating component will be accompanied by Bhabha ionization showers (§ 12). The number of small ionization showers (\(N < 20\) in Pb, \(N < 5\) in Al) should be almost independent of the substance; on the contrary, the frequency of large ionization showers should be greater in heavier substances[^B6]. If one integrates the frequency of ionization showers (31), calculated by Bhabha, over the spectrum (45) of penetrating particles, then we obtain the relative number

\[ q(N)=\frac{1}{F(0)}\int_0^\infty Q(N,E)\frac{-dF(E)}{dE}\,dE \]

of showers containing more than \(N\) particles per one penetrating particle:

\[ q \sim \frac{0.03}{N}\quad \text{for } N \ll \frac{aT}{8E_j}, \]

\[ q \sim \frac{0.03}{N}\left(\frac{aT}{8NE_j}\right)^\gamma \quad \text{for } N \gg \frac{aT}{8E_j}, \tag{61} \]

\(aT \simeq 2 \cdot 10^9\) eV at sea level; \(\gamma\) from 1 to 2 according to (50).

Thirdly, we shall expect that cosmic rays form explosive showers (§ 15). Since the effective cross section for the formation of explosions does not differ very much for different particles, at sea level there will be observed, predominantly, those explosions which are produced by the penetrating component. Indeed, the formation of rare explosions by the soft component will for the most part be impeded by strong absorption due to the much more frequent radiation processes.

Since an explosion occurs already on a single nucleus, the frequency of explosive showers in thin layers of matter, independently of the magnitude of the shower, increases with the thickness of the layer. On the contrary, in the case of cascade showers, whose formation is connected with the presence of a large number of nuclei, the complete development of the shower requires a thickness of the layer of matter that is the greater, the larger the dimensions of the shower itself [cf. formula (27)]. At large thicknesses the intensity of explosive showers will only slowly decrease with increasing layer thickness, which is connected with the weak absorption of the penetrating particles that cause the explosions. Conversely, the intensity of cascade showers will rapidly fall to zero (Fig. 8). This “saturation” of explosive processes occurs at that layer thickness which corresponds to the range of the particles formed in explosions (§ 15). Finally, nuclear processes will often be added to the showers that are produced on a single nucleus.

23. Small Showers (see G¹, M³, G⁵)

A. A direct solution of the question whether the above-described processes of cascade-shower formation actually exist has recently become possible with the aid of photographs in a Wilson chamber. Namely, photographs were obtained of rays which had to pass through several absorbing plates placed inside the chamber^F8.

Fig. 17, kindly placed at our disposal by Fussell, represents a typical picture of a cascade shower: a particle arriving from above quadruples in the first plate, 6 mm thick, and then again quadruples in the second plate of the same thickness. Of the 16 particles emerging from the second plate, only a few are then scattered in the thin third plate (0.7 mm thick). This successive increase in the number of particles corresponds to the cascade theory, provided only that the initial particle is assigned an energy of about \(2 \cdot 10^9\) eV (Fig. 7).

The following photograph (Fig. 18), likewise obtained by Fussell, represents a typical explosive shower. The explosive character of this shower follows, first, from the fact that the particle forming the shower passed through the first lead plate (6 mm thick) without producing any effects, and then created the entire shower in the second plate (also 6 mm thick). Secondly, the explosive character of the shower follows from the circumstance that the shower contains several strongly ionizing particles. These strongly ionizing фразырываются?

To the article by W. Heisenberg and H. Euler

Fig. 17. Cascade shower according to Fussell. Thickness of the plates from top to bottom: 6.3, 6.3, and 0.7 mm Pb

Fig. 18. Explosive shower according to Fussell. Absorbers the same as in Fig. 17

Fig. 18a. Explosive shower according to Fussell. Thickness of the plates: 0.7, 6.3, and 6.3 mm Pb (particle tracks retouched for clarity)

to the theory of cosmic radiation

The particles may be either protons, which are ejected from the surface of the plate as a result of “evaporation” from nuclei in secondary nuclear reactions caused by the shower, or they may be slow heavy electrons that are formed directly in the burst shower. The question of whether the burst shower consists entirely of heavy electrons cannot be resolved with the aid of the photograph shown in Fig. 18.

To resolve this question it is necessary to have such photographs in which all shower particles pass also through one lead plate several millimeters thick. Then it will be seen whether the shower particles produce further cascades, i.e., whether they are electrons, or, on the contrary, whether all of them pass through the absorber without producing new showers, as is to be expected for heavy electrons.

In favor of the latter supposition is the photograph shown in Fig. 18a, likewise obtained by Fussell. The shower presented in this photograph must also be a burst shower, since it contains several strongly ionizing particles and arises in a lead plate only 0.7 mm thick, in which the formation of a cascade shower is very improbable. At the same time, however, it seems that the majority of the shower particles are heavy electrons, since the greater part of them pass through the next plate, 6 mm thick, and only one particle produces a cascade shower.

Of 900 photographs representing cascade showers, Fussell F⁸ was able to detect burst showers in only three. Trumpy T³ established, in any case, the predominance of cascade showers. This ratio indicates that the greater part of small showers from lead are cascade showers. This becomes understandable if one recalls that each electron or photon in a layer of lead several millimeters thick produces a shower, whereas penetrating particles have a very small effective cross section for the production of burst showers.

B. A statistical study of small showers is possible on the basis of the study of coincidences by the Rossi method (R³⁴; cf. G¹,M³), i.e., by observations of simultaneous discharges of several counters forming a triangle, caused by shower particles passing through them. The frequency of such coincidences as a function of the thickness of the layer in which the shower develops, according to the measurements of Morgan and Nielsen M⁶ in lead and iron, is shown in Fig. 19. The frequency of coincidences first rises to a maximum, which occurs at 1.5 cm Pb \((l = 4)\) and, correspondingly, 4.5 cm Fe \((l = 3)\). After the maximum, the coincidence curve falls to a certain value, which then changes only little with increasing layer thickness. It will be shown below that coincidences in Rossi’s arrangement at the maximum are caused by cascade showers, and that at large thicknesses of matter they arise chiefly because of Bhabha ionization showers and, probably, only to an insignificant extent because of processes related to burst showers.

Experimental separation of purely cascade showers from showers that contain more penetrating secondary particles was carried out by Schwegler[^52]. Schwegler compared the number of ordinary coincidences in the Rossi arrangement with the number of coincidences in the same arrangement, but with a lead block 10 cm thick placed between the counters (lower curve in Fig. 20). It then turns out that all coincidences occurring under thick layers of material above the counters are also retained when a 10-centimeter lead block is placed between the counters. Since the behavior of the number of coincidences upon

Fig. 19. Coincidences in the Rossi arrangement according to Montgomery and Nielsen (Phys. Rev., 52, 564, 1937). Abscissa: thickness of the shower-producing layer in grams per square centimeter. Ordinate: number of coincidences per 1 sec. 1 — lead, 2 — iron

Fig. 20. Experimental separation of cascade showers from the remaining showers according to Schwegler (Z. Physik, 96, 62, 1935). Abscissa: thickness of the lead layer above the counters. Ordinate: coincidence frequency. Upper curve without a lead absorber \(Q\) between the counters. Lower curve—with a lead absorber \(Q\) 10 cm thick. Shaded—the part corresponding to cascade showers

placing a 10-centimeter Pb block between the counters as a function of the thickness of the lead layer above the counters in which the showers are formed has the character of the curve for Bhabha’s ionization showers, we shall ascribe the predominant part of the coincidences in the Rossi arrangement under large thicknesses of lead to ionization showers produced by penetrating particles. To be sure, a certain part of the coincidences under thick layers must be attributed to other processes, as follows from the new measurements of Schmeiser and Bothe[^51], which will be considered in § 24.

The difference between Schwegler’s measurements without and with the lead block (shaded in Fig. 20), i.e. the predominant part of the coincidences in layers less than 5 cm Pb, has the form indicating a cascade ...

toward the theory of cosmic radiation

origin (Fig. 8), which is obtained on the basis of the following, highly convincing considerations:

a) The dependence on the material deviates from the course proportional to the mass in the sense of Table 2. This circumstance was cited by Geiger and Fünfer, even before the creation of cascade theory, as an argument in favor of the fact that these showers arise owing to the braking of radiation and the formation of pairs \(G^3, G^1\).

b) From Geiger’s experiments \(G^5\) with a large number of counters it follows that the showers from lead are larger than the showers from aluminum. This can be understood on the basis of cascade theory, since the multiplication in lead continues up to energies of \(10^7\ \mathrm{eV}\), whereas in aluminum it ceases owing to ionization at an energy of \(6\cdot 10^7\ \mathrm{eV}\) (Table 2).

c) If one proceeds from cascade theory, then the results obtained by Chu Chin Shan \(H^8\) in measuring absorption become understandable. Namely, he found that the shower particles from aluminum possess greater energy than the shower particles from lead.

The experimental determination of the numerical value of the frequency \(H(N,l)\) of showers with a number of particles greater than \(N=2,3,4\ldots\), as a function of the thickness of the layer \(l\), which is easily calculated theoretically, is still impossible for small showers (\(N<10\)). At present it is possible to find numerical values of the following quantities: the frequency of \(n\)-fold coincidences in various arrangements. From this one can find the counting probability (Ansprechwahrscheinlichkeit \(G^5, G^2\)) of a counter in a definite arrangement, i.e. the relative number of cases in which this counter counts simultaneously with a group of other counters. From this probability one can draw a conclusion about the average magnitude of a shower, and from the frequency of coincidences one can draw a conclusion about the frequency of the “average shower,” whose magnitude varies with the thickness of the layer. Arley \(A^8\) considered experiments with coincidences, proceeding from the assumption that the frequency of double coincidences is equal to the frequency of showers containing more than two particles. It turned out that in general there is quantitative agreement with cascade theory in the sense of Fig. 20. A more exact consideration is possible on the basis of Geiger’s idea of the “counting probability.”

Stulinger \(S^{14}\), with the aid of proportional counters, measured the distribution of showers, i.e. the relative number \(H(N,l)\) of showers containing more than \(1,2\ldots N\) particles under a lead layer of thickness \(1.5\ \mathrm{cm}\) (\(l=4\)). His results can be approximately represented by the law

\[ H(N,4)=\frac{\mathrm{const}}{N}\quad \text{for } 1<N<100. \]

Of course, these results still cannot be directly compared with the theory set forth in § 11γ and § 19, since we do not know the geometrical conditions for the angles of divergence of the rays.

The theoretical curve of the shower distribution \(H_m(N)\), which we expect on average at the equilibrium thickness of the layer \(l_m\sim 3\lg_{4.0} N+2.7\) (§ 11γ), is shown in Fig. 24. In the following paragraphs it will be considered more precisely.

The number of photons capable of producing showers, i.e., photons with energy greater than the critical energy $E_f$ (Table 2), contained in the soft component of cosmic rays according to cascade theory, is approximately equal to the number of electrons. This is in agreement with the measurements of Auger, Leprince-Ringuet, and Ehrenfest (Fig. 21)$^{A6}$, according to which the absorption curve of the soft component in direct coincidences increases several-fold if the absorber is moved from the interval between the counters into the space above them. In the latter position of the absorber, all photons with energy greater than $E_f$ are converted into electrons.

Fig. 21. Measurements of the number of coincidences on the Jungfraujoch (Auger, Leprince-Ringuet—Ehrenfest, J. Physique et Radium, 2, 58, 1936). Abscissa: absorber thickness. Ordinate: number of coincidences. The lower curve represents coincidences for which the absorber was placed between the counters in such a way that coincidences could be caused only by charged particles. The upper curve corresponds to measurements with the absorber above the counters, so that photons with energy $> E_f \sim 10^7$ eV could be converted into electrons and cause coincidences (cf. Fig. 9). The dashed line is the extrapolated part of the hard component.

On the contrary, according to radiation theory, the number of photons with small energy, lying below the critical energy, should be much greater than the number of electrons of equal energy. Indeed, electrons with energy $E < E_f$ will be slowed down by ionization. By contrast, photons begin to undergo energy losses through Compton and photoelectric effects only at quite small energies, of the order of $10^6$ eV. Therefore, whereas the differential spectrum

$$ \frac{\partial F(E)}{\partial E} $$

of electrons in the soft component of cosmic rays (Fig. 16 on p. 271) (or in secondary shower rays) below the critical energy $E_f$ will remain approximately constant, the corresponding photon spectrum at small energies $k < E_f$ will increase as

$$ \frac{dk}{k} $$

(§ 7).

Thus radiation in a shower will be accompanied by a large number of photons of small energy, whose number may exceed the number of electrons many times over. According to the coincidence measurements of Geiger and Zeiler $G^8$, the number of photons falling on one electron is 50. The measured by Gei-

... by Terom and his collaborators the effect of back radiation (Rückstrahleffekte) was ascribed by them precisely to photons of small energies.

24. Large showersS6, M3

Information on large showers can be obtained from the so-called Hoffmann bursts, i.e., from the sudden appearance of a large number of ions, which is observed in an ionization chamber. Hoffmann bursts can sometimes be observed in a Wilson chamber. Photographs show that they consist of a shower of 10–1000 weakly ionizing particles (Fig. 22).

Fig. 22. Photograph of a burst-like shower containing more than 300 particles, with a total energy greater than \(1.5\cdot 10^9\ \mathrm{eV}\), according to Anderson and Neddermeyer (Phys. Rev., 50, 263, 1936)

For the present, it is true, one can only rather inexactly infer, from the magnitude of the ionization, the size of the shower \(N\), and, from the frequency of bursts in the chamber, the frequency of showers per 1 min per \(1\ \mathrm{cm}^2\). However, such estimates, within certain limits of error—with an accuracy to a factor of 2–3 for the shower size and the shower frequency—can provide a number of data on the statistics of large showers. The estimates used here for converting the experimentally observed quantities into theoretical ones are presented in Table 8 (p. 286). In all cases the effective area of the absorber was set equal to the cross-section of the chamber. The ionization was assumed to be equal to 70 ion pairs per \(1\ \mathrm{cm}\).

In Fig. 23 the frequency of bursts containing more than \(N=200\) particles is shown as a function of the absorber thickness according to HeitlerN2, N3 in various substances. The course of the curve is similar to that of the corresponding curve for small showers (Figs. 19, 20). The form of the curve changes little with the size of the bursts \(N\)M2, C2, B16, Y1,2. Below we shall show that the shaded part at the maximum of the curve of ionization bursts must be ascribed to cascade showers, and that the remaining ionization bursts, and above all bursts under thick ...

...with thick and thin layers arise on account of bursts, if the results of the available measurements are confirmed under cleaner conditions \(E^6\). In Fig. 23 the separation of both kinds of ionization bursts has been carried out. From each experimental point the theoretical value of the frequency of large cascade showers has been subtracted (Fig. 8, § 11 γ), with a suitable value being chosen for the undetermined common factor in the intensity.

TABLE 8

Author	Effective surface in \(\mathrm{cm^2}\)	Number of particles per \(10^6\) pairs of ions
Bergild	700	40
Ni	1 600	50
Messerschmidt	500	20
Jung and Street	50	—
Carmichael	2 000	—

In favor of the cascade nature of the ionization bursts at the maximum of the curve (shaded in Fig. 23) the following arguments may be adduced:

a) The position \(l_m\) of the maximum is transmitted approximately correctly by cascade theory, as Table 9 shows. A discrepancy between theory and experiment larger than could be expected as a result of inaccuracy in the conversion factor (Table 8) was observed only in Bergild’s measurements.

b) The width of the maximum, which can be checked first of all from the measurements of Jung and Street \(E^6\), is also represented correctly. Especially correctly is it predicted in the case of Fe, if one ascribes to cascades the theoretically expected part of the ionization bursts, smaller than in Pb (Fig. 23).

Fig. 23. Frequency of Hoffmann bursts according to Ni. Abscissa: thickness of the layer in which bursts are formed, in \(1\ \mathrm{g/cm^2}\). Ordinate: frequency of bursts with number of particles \(N = 200\). \(1\)—measurements by Ni in Pb, Fe, Al. Shading: theoretical value of the cascade part according to Fig. 8, § 11; \(2\)—result of subtracting the cascade part.

c) Apparently, the ratio of the intensities at the maxima of the ionization-burst frequency curve is correctly transmitted by that electron spectrum which can be theoretically expected on the basis of §§ 21 and 19. The latter follows, in particular, from the curve in Fig. 24, where the experimentally observed frequency \(H_m(N)\) of cascade bursts at the maximum in lead is presented, i.e. the shaded part of the curve in Fig. 23 in Pb (11 γ), as a function of the magnitude of the bursts.

According to formula (15), this function should represent a direct measure of the spectrum of electrons producing ionization ...

TABLE 9

Position of the maximum of the burst-frequency curve

Burst magnitude	\(N=10\)	20	30	40	80	200	300	400
Young and Street \(^{Y1,2}\) in Pb, \(l_m=\)	\(4\pm2\)	\(5\pm2\)	\(5.5\pm2\)	—	—	—	—	—
Beggild \(^{B16}\) in Fe, \(l_m=\)	—	\(3\pm1.5\)	—	\(4\pm1.5\)	\(5\pm1.5\)	—	—	—
Ni \(^{2}\) in Pb, \(l_m=\)	—	—	—	—	—	\(7.5\pm4\)	\(9\pm4\)	\(10\pm4\)
Ni \(^{2}\) in Fe, \(l_m=\)	—	—	—	—	—	\(8\pm2\)	\(9\pm2\)	\(10\pm2\)
Theoretically (§ 11 γ), \(2.7+3.0\lg N\)	\(N=5.7\uparrow\)	6.6	7.2	7.5	8.5	9.6	10.2	10.5

showers, since in the case under consideration the frequency of bursts with a number of particles greater than \(N\) is equal to the frequency of those cases in which the energy of an electron falling on a lead plate exceeds \(E=(8N)_{1.07}E_s\ \mathrm{eV}\). Therefore, in Fig. 24, along with the particle scale, an energy scale is also indicated.

Fig. 24. Spectrum and frequency of cascade showers. Abscissa: momentum \(pc\) in eV (logarithmic scale). Ordinate: number \(\dfrac{F}{2}\) of particles with momentum greater than \(p\), per minute per \(1\ \mathrm{cm}^2\) at sea level (logarithmic scale).
\(1—F_1\): number of cascade electrons (§ 19, \(\gamma=1.85\)).
\(2—F_s\): number of penetrating particles (§ 20).
\(3—F_2\): number of decay electrons (§ 21).
Since each electron with energy \(E\) produces in lead a cascade shower whose magnitude \(N\) at the maximum \(l_m\) is given by the expression

\[ N=\frac{1}{8}\left(\frac{E}{10^7\mathrm{eV}}\right)^{0.93} \]

(§ 11, γ), the figure may also be read in the following way: abscissa: the magnitude \(N\) of cascade showers; ordinate: the frequency \(H_m\) of cascade showers from \(N\) particles at the maximum, per 1 min per \(1\ \mathrm{cm}^2\). Individual points on the curve represent the results of measurements by the following authors: \(YS\)—Young and Street; \(B\)—Beggild; \(M\)—Messerschmidt; \(N\)—Ni; \(C\)—Carmichael. \(4\)—experimental inaccuracy in determining the frequency and magnitude of bursts.

As is seen from Fig. 24, the different segments of curves, which represent the results of measurements of the magnitude of showers in various ionization chambers \(Y^2, M^2, G^2, B^2, N^{2,3}\), form a single curve. If in at least one case the absolute frequency of showers were known exactly, this curve would give the spectrum of electrons incident on the earth’s surface. This empirical electron spectrum apparently does not differ very much from the theoretical electron spectrum that was considered in §§ 21 and 19. The latter consisted of two parts: the spectrum of decay electrons (§ 21), indicated in Fig. 24 by crosses, which predominates at small energies, and the spectrum of cascade electrons (§ 19), indicated in Fig. 24 by circles. The spectrum of cascade electrons at sea level is found only at high energies.

However, for a final decision on the character of these spectra one must still await the obtaining of more accurate curves for the frequency of cascade showers.

d) The observed strong dependence of the Hoffmann showers on altitude at the maximum of the frequency curve in lead \(Y^3, M^4, S^{16}\) can be understood from the form of the spectrum of the electrons producing the cascades. At the same time, the weak altitude dependence of showers in thick layers of matter \(Y^1, M^3\) indicates that the latter owe their origin to the penetrating component. The increase in the frequency of showers at the maximum (\(\sim 2\)—\(4\) cm Pb) for a difference in altitude from 76 to 45 cm Hg is (Table 10):

TABLE 10

	Woodward \(W^8\)	Young and Street \(Y^{2,3}\)	Young and Street \(Y^{2,3}\)	Young and Street \(Y^{2,3}\)	Montgomery \(M^4\)
Magnitude of showers . . . .	\(N < 10\)	10	20	30	40
Increase factor . . .	8.5	10	17	22	26

This increase, the larger the greater the magnitude of the showers, is understandable if one starts from the superposition of the electron spectra considered in §§ 21 and 19. Then at small energies and, consequently, for small cascade showers at sea level, the spectrum of decay electrons predominates (indicated by crosses in Fig. 24), slowly increasing with altitude (cf. § 21). At large energies and large cascade showers, the spectrum of cascade electrons predominates (indicated by circles in Fig. 24), which increases very rapidly with altitude [(41) § 19] (Table 11).

TABLE 11

Increase factor for a specified height difference \((e=12)\) . .		20	40	63
For constant § (19) . . . . .	\(\gamma\)	1.5	1.7	1.9

At great heights above the earth’s surface one may expect the cascade spectrum to predominate at all energies. Further

measurements of the altitude dependence of large Hoffmann bursts may more precisely indicate the form of the electron spectrum, which in §§ 19 and 20 was determined only inaccurately and indirectly \((\gamma = 1.8—1.9)\).

The altitude dependence of Hoffmann cascade bursts (Table 10) gives yet another independent argument in favor of the production, at sea level, of electrons from the penetrating component. Indeed, if we wish to construct the spectrum of electrons that arrive from outer space, then undergo transformations in the atmosphere as a result of shower formation and, in the lower layers of the atmosphere, create Hoffmann cascade bursts, then it is necessary to ascribe to this spectrum such a form that the degree of falloff of the spectrum at large energies is greater than at small energies (since large cascade bursts increase with altitude more strongly than small ones) (§ 19). If, however, in doing this we want to construct the spectrum so that it represents the number of electrons in the upper and lower layers of the atmosphere, then it is necessary to assume a weaker falloff of the spectrum at large energies than at small ones (§ 19) Н7,Н6.

Both of these conditions obviously cannot be satisfied simultaneously, and this means that the soft radiation in the lower parts of the atmosphere must arise from penetrating rays. The course of the burst intensity with altitude corresponds to the course of its change with depth. According to the measurements of Bergild В10 in a mine, the decrease of the sharp cascade maximum is greater than the decrease of intensity at large thicknesses. Weizsädel’s measurements W1 showed that in Lake Boden the burst frequency, which may be regarded as the burst frequency under thick layers, decreases similarly to the decrease of the penetrating component. Corresponding experiments were also carried out with coincidences in the Rossi arrangement (cf. A5), so that in the case of bursts, as in the case of coincidences, one may assume that the maximum of the frequency is produced by the soft component. At the same time, the correctness of the theory of radiation (§ 7) is confirmed up to energies of the order of \(10^{11}\ \mathrm{eV}\).

В. Whereas part of the bursts at the maximum of the frequency curve possesses all the properties characteristic of a cascade, the bursts under large thicknesses of matter must be ascribed to other causes.

e) This is indicated, first of all, by the circumstance that, according to cascade theory, the burst frequency in large layers rapidly falls to zero (Fig. 8), whereas in reality the bursts under large layers of matter have a finite frequency. In the case of bursts in lead, their frequency under thick layers amounts to half the frequency at the maximum (Fig. 23).

f) The strongest argument in favor of the non-cascade nature of the greater part of the bursts is their dependence on the material of the layer. Whereas, according to cascade theory, the frequencies of the corresponding bursts in \(\mathrm{Pb}:\mathrm{Fe}:\mathrm{Al}\) should be approximately as \(1:3^{-2}:6^{-2}\) (§ 11 \(\gamma, \gamma \approx 2\)), in reality, on the contrary, the frequency of bursts under thick layers of light substances is greater than under layers of heavy substances (Fig. 23). If, thus, we count 50% of all bursts as cascades, then at the maxima of the curve for Fe and Al we may count as cascades, respectively, only 6 and

1.5% of the bursts. Bursts under thick and thin layers and, above all, the predominant number of bursts in light substances must, therefore, be of a different origin. The above-mentioned dependence of Hoffmann bursts on the substance also excludes the possibility that bursts under thick layers of matter arise as a result of Baba’s ionization avalanches. Indeed, large ionization avalanches should arise in heavy elements more often than in light ones, approximately in the ratio of the atomic numbers. In the bursts, however, rather the opposite ratio of intensities is observed (Fig. 23).

g) The starting point for elucidating the nature of non-cascade bursts may be their behavior in thin layers.

Fig. 25. Distribution of bursts in thin layers. Abscissa: magnitude of bursts (logarithmic scale). Ordinate: frequency of bursts with a number of particles greater than \(N\) in 1 min. per \(1\ \mathrm{kg}\). \(1\)—theoretical: according to the cascade theory (§ 11 g; for \(\gamma = 1.5\), the overall multiplier arbitrary), \(2\)—measurements: \(M\)—Messerschmidt, \(N\)—Ney, \(B\)—Bertilida. \(3\)—inaccuracy in comparing the measurements \(M\), \(N\), and \(B\).

In Fig. 25 is shown the experimental frequency of bursts consisting of more than \(N\) particles in 1 min. per \(1\ \mathrm{kg}\), as a function of the magnitude of the burst \(N\) in thin layers. The fact that the experimental curves referring to different thicknesses almost coincide means that in this case there is an almost linear increase of the burst frequency with the thickness of the layer. If bursts in thin layers of light substances were cascades, then quite the opposite should be expected—a much stronger dependence of the burst frequency on the thickness of the layer. This is shown by the dashed lines in Fig. 25. For example, according to cascade theory, bursts in 1 min. per \(1\ \mathrm{cm}^{2}\) in a 30-centimeter layer of Al should occur more than \(10^{5}\) times more often than in a 10-centimeter layer of Al. In reality, however, bursts (in 1 min. per \(1\ \mathrm{cm}^{2}\)) of 200 particles in 30 cm Al, according to Ney’s observations, are only 2–3 times more frequent than the same bursts under 9 cm Al. If these results of measuring the burst frequency under thin layers are confirmed also under clean conditions excluding the influence of the walls of the room, then it follows from this

conclude that non-cascade bursts arise in an explosive manner, since only when a burst arises on a single nucleus can one understand the linear increase of the frequency with the thickness of the layer.

The question of whether the secondary particles arising in explosions are light or heavy electrons can be resolved in experiments with bursts. Although at present no clear answer to this question has yet been given, it seems that the available experiments point rather to the occurrence of heavy electrons than to the predominant formation of light electrons. Experiments with coincidences of Hoffmann bursts may provide a solution to the question. According to Ni’s measurements3, the number of showers with a number of particles exceeding 100 from 10 cm Fe, which simultaneously enter two ionization chambers, decreases to one fifth of the initial number if a layer of lead 9 cm thick is placed between the chambers.

This can be understood rather in the case when heavy electrons, and not light electrons, are produced in the explosion. Indeed, in the latter case, owing to strong absorption, one would expect a decrease in the number of showers to 1% of the initial value5. However, a reliable solution of the question is still impossible.

The nature of the secondary particles can further be clarified from “transition effects”: a layer of lead several centimeters thick should multiply the bursts obtained in a 10–20-centimeter layer of Al if they consist of electrons, and should change them only weakly if they consist of heavy electrons.

Such measurements of transition effects have so far been carried out only by Bethegeld6, and moreover under such layers under which the bursts undoubtedly represent cascades (namely, at thicknesses corresponding to the maximum in lead).

Bethegeld’s measurements confirmed that the bursts at the maximum of the shower frequency in lead consist of electrons and, consequently, are cascades. Corresponding measurements under such layers as correspond to explosions (for example, 10–20 cm Al) will be able to provide information on the nature of the secondary particles in explosive bursts. In doing so, it is of course necessary to take into account the circumstance that, even if at first only heavy particles arose in the shower, explosive bursts under thick layers will be accompanied by a considerable number (about 20–40%) of electrons, since heavy electrons stopping in the last centimeter will admix their decay products to the shower. This admixture of electrons can be avoided only when the bursts are observed under thin layers of 10–20 cm Al or 2–4 cm Fe, since in this case few heavy electrons can stop and decay in the material.

If we assume for the time being that in explosions heavy electrons are produced predominantly, then from the thickness of that layer up to which an increase in the number of explosive bursts occurs (10 cm in Fig. 23), we can judge the range and, consequently, the energy of the heavy shower-producing electrons in the shower4. From the length of the sec[[unclear: word cut off at page bottom]]

of the rise of the burst-frequency curve (10 cm Al, Fig. 23 on p. 286), we then obtain (Table 6) that the mean energy of the secondary particles in the burst is about \(E_k \approx 10^8\ \mathrm{eV}\), in agreement, in order of magnitude, with the critical energy indicated in § 15. For the formation of a burst of \(N\) particles, consequently, a heavy electron with an average energy

\[ E \sim N \cdot 10^8\ \mathrm{eV}. \]

is required.

A heavy electron producing a shower of \(N\) particles must therefore have approximately the same energy as a light electron forming a cascade with the same number of particles at the maximum of the frequency curve in lead,

\[ E = 8 \cdot N \cdot E_j = 8 \cdot 10^7 \cdot N \cdot \mathrm{eV} \sim 10^8 N\ \mathrm{eV}. \]

The effective cross section for the formation of burst showers by heavy electrons can now be obtained by comparing the experimental intensity of bursts in thick layers of lead with the experimental intensity of cascades at the maximum of the frequency curve in lead. These intensities, according to the curves in Fig. 23, are related approximately as \(1 : 1\), and this ratio changes remarkably little from small bursts of 30 particles \(Y^{3,B16}\) to very large ones consisting of 1,000 particles \(C_2^2\). Therefore, according to (57) and (59), the number of electrons \(F_2(E)\) with energy greater than \(E \lesssim 10^{11}\ \mathrm{eV}\) is related to the number \(F_s(2E)\) of penetrating particles with energy greater than \(2E\) at sea level as

\[ F_2(E) : F_s(2E) = \frac{10^7\ \mathrm{eV}}{E}. \]

Since, moreover, each electron produces a shower, it follows that the probability that a heavy electron of energy \(E\), on a path of \(10\ \mathrm{cm}\) Fe, will produce a burst is, approximately,

\[ \frac{10^7\ \mathrm{eV}}{E} \quad (\text{in } 10\ \mathrm{cm}\ \mathrm{Fe}). \]

Since \(1\ \mathrm{cm}^3\) of Fe contains \(8.5 \cdot 10^{22}\) atoms, each proton or neutron of the nucleus has an effective cross section for the formation of a burst by a heavy electron with \(E > 10^9\ \mathrm{eV}\)

\[ Q \approx \frac{1}{2}\cdot 10^{-27}\ \mathrm{cm}^2 \cdot \left(\frac{10^9\ \mathrm{eV}}{E}\right). \tag{62} \]

The order of magnitude of this effective cross section is in precise agreement with that predicted theoretically (§ 15). The empirical law according to which the effective cross section decreases at high energies as though proportional to the first power of the energy cannot, of course, yet be theoretically justified.

If the law (62) is correct, then one should expect that for very large showers (\(N > 1000\)) the ratio of the maximum intensity to the saturation intensity (Fig. 23) will again increase, since the cascade spectrum then begins to play a role (Fig. 24).

Thus the Hoffmann bursts under thick layers of matter are also understandable, at least in such an approximation in which one may assume that they occur equally often in all substances, and that their saturation (10 cm Fe; \(>30\) cm Al) sets in under layers of equal mass. In fact, the formation and absorption of explosive showers should proceed approximately in proportion to the mass of the layer. In reality, however, showers under large layers of light elements have a greater intensity than under thick layers of heavy elements, and, accordingly, saturation in light elements sets in at somewhat greater masses than in heavy ones (Figs. 19, 23, pp. 282, 286). It is possible that a way toward understanding this still unclear dependence of showers on the nature of the substance may be provided by taking into account the possibility that the appearance of shower particles in the explosion is “retarded” because of nuclear processes caused by particles still in the nucleus \(E^6\) (see §§ 16, 24).

This “retardation,” which is essential primarily for slow heavy electrons in the shower, could play a larger role in the heavy nuclei of lead than in the light nuclei of aluminum, and could cause the observed advantage of light nuclei. However, a theoretical treatment of this phenomenon is still impossible. The just-mentioned, still unclear dependence of Hoffmann bursts on the nature of the substance has much in common with the dependence on the nature of the substance of a special kind of coincidence in Rossi’s arrangement, investigated recently by Schmeiser and Bothe\(^{S1}\). Various authors\(^{K3, M1, D1, H15}\), when counting coincidences in Rossi’s arrangement, found indications of the presence of a second maximum under 15–20 cm Pb. Schmeiser and Bothe\(^{S1}\), in observations at sharp angles, succeeded in obtaining this second maximum with particular intensity. It turned out that under 17 cm Pb or 30 cm Fe there is a quite distinct second maximum on Rossi’s coincidence curve (Fig. 26). The following circumstances make probable the assumption that the second maximum is produced by small explosive showers, the larger variety of which is observed in the form of non-cascade Hoffmann bursts.

a) The second maximum on the coincidence curve is produced by the penetrating component. This was proved by the measurements of Schmeiser and Bothe in a mine, where the first maximum was weakened owing to the absorption of the soft component, whereas the second maximum changed only very slightly.

b) The second maximum occurs at thicknesses (17 cm Pb, 30 cm Al) comparable with the thicknesses of the saturation layers (3–8 cm Pb, 10 cm Fe) of the non-cascade Hoffmann bursts. Namely: the second maximum occurs at thicknesses roughly three times greater than the saturation thickness in Hoffmann bursts. If the second maximum of the coincidence curve owes its origin to heavy electrons, then this permits one to conclude that the energy of the secondary particles is about \(3 \cdot 10^8\). It is also probable that in the showers observed by Schmeiser and Bothe, heavy electrons are formed first, since it would be difficult to understand how electrons can

have such a large range (17 cm) in lead. However, because of the decay of these heavy electrons, ordinary electrons would again be admixed with them in the observed showers.

c) On the basis of the conservation laws, for explosive showers with a total energy considerably greater than the rest energy \(10^9\) eV of the constituent particle of the nucleus, one should expect small angles of divergence. Moreover, subsequently the shower particles can undergo only weak deflections due to elastic scattering (since they, in contrast to the particles in cascade showers, have large momenta). With this circumstance must be connected the small angle of aperture of the showers in the second maximum observed by Schmeiser and Bothe. It also makes it possible to expect that, in the case of Hoffmann’s bursts, explosive showers differ from cascade showers by small angles of aperture of the shower cone.

Fig. 26. Second maximum on the curve of triple coincidences in Rossi’s arrangement at an acute angle of \(7^\circ\), according to Schmeiser and Bothe (Naturwiss., 25, 833, 1937) (cf. Fig. 19). The abscissa is the thickness of the lead; the ordinate is the number of coincidences per hour.

d) Finally, the second maximum on the coincidence curve reveals the same incomprehensible dependence on the character of the substance as do explosive bursts.

The dependence on the substance of the secondary effects considered here has a certain similarity to the still unexplained “transition effects” in thick layers\(^{59}\). By a “transition effect” is meant a rapid change in the course of the absorption curve of the intensity, measured in an ionization chamber, which occurs upon passage from one absorbing substance to another\(^{54}\). Transition effects in thin layers (less than 100 g/cm\(^2\)) consist in an increase of ionization if lead is placed under aluminium, and in a decrease of it if aluminium is placed under lead. In thick layers, however, transition effects occur in precisely the reverse order. According to the theory of radiation\(^{G1}\) and, in particular, the cascade theory\(^{C1,B3}\), transition effects in thin layers are understandable: since lead can still multiply particles emerging from aluminium, whereas particles from lead can no longer be further multiplied in aluminium, one should expect the observed course of the change in intensity in thin layers. However, other secondary effects, possessing the same dependence on the character of the substance as Hoffmann’s bursts under thick layers and the showers in the second maximum, must be responsible for the transition effects under thick layers of various substances.

25. Nuclear Processes

In cosmic rays, heavy particles have often been observed which ionized considerably more strongly than ordinary electrons.

Their presence was proved, on the one hand, with the aid of the Wilson chamber, in particular in the well-known work of Anderson and Neddermeyer^A3, and also in the work of Brode and Starr^B19; on the other hand, from the tracks left by strongly ionizing particles in the light-sensitive layer of a photographic plate. Various investigators (Herzog and Scherrer^H10, Rumbaugh and Locher^R6, von Fifer^F5, Schopper^S15, Blau and Wambacher^B14, Taylor^T3), partly in laboratories and partly at high altitude, placed photographic plates which for a long time were exposed to the action of cosmic rays. Later, tracks were found in the plates which

Fig. 27. Evaporation from a nucleus according to Blau and Wambacher
(Nature, 140, 585, 1937)

could most simply be interpreted as the paths of protons. Blau and Wambacher^B14 further observed “stars” in such tracks, i.e., points on the photographic plate from which several protons emerged simultaneously (Fig. 27). Finally, Taylor^T3 observed, in several places on his plate, “swarms” (Haufen) of such tracks.

For comparatively slow heavy particles, what is characteristic above all is the strong increase in their yield with altitude. Anderson and Neddermeyer, on Pikes Peak, found almost 12 times more tracks of slow heavy particles than at sea level. Similar values are also obtained for proton tracks in photographic plates. For the “stars,” however, Taylor observed an even stronger increase with altitude. It follows from this that tracks with strong ionization are in any case produced by strongly absorbed radiation.

Until now most investigators have thought that what is involved here must be, chiefly, slow protons. It is not excluded, however, that slow heavy electrons are also partly responsible for tracks with strong ionization. It is possible, furthermore, that in individual cases α-particles may also be involved.

particles. The tracks in the “stars” in the photographs of Blau and Wambacher, however, are probably caused for the most part by protons.

As for the radiation which indirectly produces these tracks, it is known, first of all, that it is very strongly absorbed in the atmosphere. It follows from this that heavy electrons and protons of very high energies, as well as neutrons, can participate in the formation of only an insignificant fraction of the slow protons. Therefore, as the rays creating the slow protons, only electrons and photons can appear. In what follows one need not take account of fast protons and neutrons.

Electrons or photons can create heavy particles in the nucleus with a sufficient degree of probability only in the case when processes arising as a result of the interaction of electrical and nuclear forces play a role (§ 15). For example, one should take account of the inverse of the process considered in § 15. A photon with sufficient energy, in collision with a neutron, creates a negatively charged heavy electron, while the neutron is simultaneously transformed into a proton, taking away part of the energy of the light quantum. If the energy of the photon is sufficiently large, then instead of this process there will figure a multiple process of the “explosion” type. The effective cross section of this process was estimated above and is about \(10^{-27}\ \mathrm{cm}^2\) (§ 15).

The formation of an analogous process by an electron apparently occurs more rarely by \(\sim \dfrac{e^2}{\hbar c}\) times. The photographs of Anderson and Neddermeyer^A3 show that slow protons are observed most often simultaneously with cascade showers. This circumstance indicates that slow protons are in fact produced by light quanta. At large energies of the latter, the formation of slow protons apparently can occur only with the aid of the process indicated above.

At small photon energies one may also think of the direct nuclear photoeffect observed by Bothe and Gentner^B20. However, in the nuclear photoeffect there must fly out of the nucleus protons with very small energy, which will be so rapidly absorbed in the substance in which they were created that in most cases they will not reach the Wilson chamber. Thus it remains still very disputable whether the nuclear photoeffect plays a noticeable role in the formation of the observed slow protons. Likewise, if we take as a basis the other processes mentioned above, it seems difficult to explain the relatively high frequency of the appearance of slow protons in observations. For the present the theoretical estimates of their effective cross section are so unreliable that the significance of this small discrepancy with experiment cannot yet be discussed.

Slow protons may also be produced in part by neutrons or by protons of intermediate energies. In this case one should think that, in the processes leading to the formation of the penetrating component of cosmic radiation, which also belong

belong to the above-mentioned type, protons and neutrons of medium energies are always produced. In particular, in showers that are produced by electrons or photons of very high energies, heavy particles with an energy of \(10^8—10^9\ \mathrm{eV}\) are often produced. These particles, on their way from the stratosphere, where they predominantly arise, are absorbed very strongly and, possibly, are responsible in part for the formation of slow protons.

Indeed, Anderson and Neddermeyer believe that it can be proved that one of the nuclear transformations in their photographs was caused by a neutron, and that this neutron, in turn, came with a large shower. It is possible that this is a large explosive shower, but this cannot be concluded from the photograph with complete clarity.

The “stars” observed by Blau and Wambacher may perhaps naturally be explained as secondary effects caused by protons or neutrons of medium energies (\(10^8—6\cdot 10^8\ \mathrm{eV}\)), i.e., as “nuclear ionization.”

Here we are dealing either with a process in which a heavy particle with such an energy enters a nucleus from outside and forms secondary particles in it, or with a process in which a heavy particle is produced in the nucleus by a photon arriving from outside and, on its way out of the nucleus, forms secondary particles. In the latter case, simultaneously with the proton one proton flies out of the nucleus, and in the case where the incident photon has sufficient energy—several heavy electrons. Thus we again arrive at explosive processes which, naturally, take place chiefly in a single nucleus and are associated with the emission of secondary particles from this nucleus. The emission of secondary particles from the nucleus is also accompanied, in all known cases, by “evaporation from the nucleus” of the same type as in ordinary nuclear transformations. If one assumes that explosions are rarer processes and that, in the case of the Blau–Wambacher stars, we are dealing mainly with secondary particles produced by heavy particles of medium energy, then the energy distribution of these secondary particles can be compared with the theoretical distribution (38). The measurements made so far agree well with the theory. As for the “roars” observed by Taylor, any reasonably firm theoretical judgments are apparently still impossible.

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1. Ehmert. ↩↩

2. As in the original text, the notation appears as \(10^0/_{0}\), i.e. ten percent. ↩