ON THE THEORY OF COSMIC RADIATION¹

W. Heisenberg and H. Euler

The complex phenomena to which cosmic rays owe their origin have, in recent years, been so systematized and clarified that, on the basis of the existing theory, it is possible to outline a coherent picture of these phenomena, although one not yet correct in all its details. The most important success that made this clarification possible was the establishment of the fact that quantum theory describes with great accuracy the behavior of electrons and photons up to the highest encountered values of energy, and the discovery connected with this of new elementary particles responsible for the penetrating component of cosmic radiation and possessing a mass intermediate between the mass of the proton and the mass of the electron. Recently the existence of these particles has also been proved directly from individual photographs in a Wilson chamber. We shall begin the following account with a brief survey of the results that can be obtained from photographs in a Wilson chamber of individual ionizing particles deflected by the action of a magnetic field. This will be followed by an exposition of the existing theory, describing the behavior of light (II) and heavy (III) particles and the secondary effects caused by them. Then a detailed comparison of the theoretical results with experimental data will be given; namely, the spectra of the individual kinds of particles and their transformations in the atmosphere (IV), and the secondary effects produced by them (showers, bursts, nuclear transformations) (V), will be considered.

CHAPTER 1. SURVEY OF THE BEHAVIOR OF INDIVIDUAL PARTICLES

1. Deflection by a Magnetic Field and Measurement of Momentum

Fig. 1 represents a photograph of the path of a particle in a magnetic field in a Wilson chamber. Such a photograph makes it possible, first of all, to measure the curvature of the trajectory. This measurement makes it possible

¹ Ergebnisse der exakten Naturwissenschaften, vol. XVII, 1938. Translated by V. Levich.

find the momentum of the particle \(p\), which is related to the radius of curvature \(\rho\) in the magnetic field by the known relation

\[ pc=eH\rho \tag{1} \]

(\(e\)—charge of the particle, \(c\)—speed of light). Usually the quantity \(p\frac{c}{e}\) is measured in gauss centimeters or in volts. Between the two units there is the relation

\[ \frac{pc}{eH\rho} =1\,\frac{\text{erg}}{\text{charge cgs. gauss. cm}} =300\,\frac{V}{\text{gauss. cm}} . \tag{1'} \]

For a particle whose kinetic energy

\[ E=\sqrt{(mc^{2})^{2}+(pc)^{2}} \]

is large in comparison with the rest energy \(mc^{2}\), the expression \(pc\) is approximately equal to the energy and will therefore most often be expressed in electron-volts (eV). In particular, for electrons in cosmic rays, \(pc\) can practically always be equated to the energy. However, for strongly ionizing particles \(pc\) must be distinguished from the energy.

2. Ionization and velocity measurement¹)

Fig. 1 makes it possible further to calculate the density of droplets in the track of the particle, i.e. makes it possible to measure the energy losses which the observed particle has undergone owing to ionization of the atoms of the air. The ionization losses per 1 cm of path,

\[ -\left(\frac{\partial E}{\partial x}\right)_{j}, \]

are, to a considerable extent, a scale for the velocity \(\beta c\) of the particle. They are expressed by the formula

\[ -\left(\frac{\partial E}{\partial x}\right)_{j}=\frac{a}{\beta^{2}}, \tag{2} \]

where the quantity \(a\), according to Bethe and Bloch \({}^{4,\,15}\), depends only weakly on the mass and increases only logarithmically with the energy.

Fig. 1. Photograph in a Wilson chamber (Anderson). A positron with energy \(6.3\cdot10^{7}\) eV enters from below into a lead plate of thickness 66 mm and leaves it with energy \(2.3\cdot10^{7}\) eV

¹) The literature will be given at the end of the article.

\(E\) particles. The order of magnitude of \(a\) in lead is \(10^7\ \mathrm{eV/cm}\), in water \(2\cdot 10^8\ \mathrm{eV/cm}\). An exact description of the ionization losses is given by the curve in Fig. 2, which represents the energy losses per \(1\ \mathrm{cm}\) of water and lead as a function of the momentum of particles possessing elementary charge and different masses, according to Bloch and Bhabha2.

The cases of different masses considered in the figure (\(m\)—electron mass, \(100m\), \(1840=M\)—proton mass) differ from one another by different rest energies \((mc^2=0.51\cdot 10^6\ \mathrm{eV},\ 100mc^2=0.51\cdot 10^8\ \mathrm{eV},\ Mc^2=0.91\cdot 10^9\ \mathrm{eV})\). From formula (2) and Fig. 2 it is seen that protons and electrons with the same momentum \(pc=2\cdot 10^8\ \mathrm{eV}\) can be clearly distinguished by their ionization: electrons \((mc^2=\frac{1}{2}\cdot 10^6\ \mathrm{eV})\) then have a velocity close to the velocity of light and therefore leave a thin track; protons \((Mc^2=10^9\ \mathrm{eV})\) move at the same momentum with a velocity smaller than the velocity of light and leave a much stronger track. However, the possibility of a clear distinction between protons and electrons by ionization is absent at momenta greater than \(Mc^2=10^9\ \mathrm{eV}\), since then both particles move with a velocity close to the velocity of light and leave a thin track, the density of droplets in which is determined chiefly by the charge. A discussion of finer details will follow in § 9.

Fig. 2. Ionization losses in water and lead according to Bloch’s formula. Abscissa: particle momentum in \(\mathrm{eV}\) or \(\mathrm{gauss\ cm}\). Ordinate: energy losses in \(10^6\ \mathrm{eV}\) per \(1\ \mathrm{cm}\). The three curves belong to particles with different masses: \(m\)—electron mass, \(100m\), \(1840m\)—proton mass.

3. Bremsstrahlung and rest mass[^9]

Finally, from the photograph of Fig. 1 one can find the change in the curvature of the trajectory upon passage through a lead plate. The losses of momentum in the lead plate consist mainly of two parts: ionization losses, which were considered in the preceding paragraph, and losses due to radiation. Whereas a particle moving with the velocity of light always loses, through ionization, the same amount, of the order of \(10^7\ \mathrm{eV/cm}\ \mathrm{Pb}\), through bremsstrahlung such a particle always loses a portion of its energy which is inversely proportional to the square of its mass[^3]. For example, over a path of \(4\ \mathrm{mm}\ \mathrm{Pb}\) an electron with energy greater than \(10^9\ \mathrm{eV}\) always radiates almost half its energy, whereas a proton

on the same path loses to radiation an imperceptible fraction (about 0.01%) of its energy.

Thus, the energy losses to radiation in 1 cm of Pb for an electron far exceed the ionization losses, provided only that its energy exceeds \(10^7\) eV. A more exact treatment of bremsstrahlung will be given in § 7.

4. Heavy and light electrons

We shall now turn to the results obtained from photographs of the paths of individual particles in a Wilson chamber.

1. The statistics of momentum measurements (Kunze \(K^4\), Bleckett \(B^{12}\), Gerth and Scherrer \(H^{10}, H^{11}\), Anderson \(A^2\)), extending up to \(2 \cdot 10^{10}\) eV, gives a continuous spectrum of ionizing particles falling off toward large momenta.

2. The tracks of most particles indicate ionization which differs only very little from that of an electron. Only about 1% \(A^3\) of the particles leave broader tracks. This is an argument in favor of the charge of the observed particles not being capable of being much smaller than the charge of the electron; in what follows it will be assumed that the charge of all particles is equal to the charge of the electron.

From the faintness of the tracks in the region of momenta

\[ pc \leq \frac{1}{2}\cdot 10^9\ \mathrm{eV} \]

it further follows that almost all particles with momenta smaller than

\[ \frac{1}{2}\cdot 10^9\ \mathrm{eV} \]

must be lighter than protons. Indeed, a proton with momentum

\[ pc < \frac{1}{2}\cdot 10^9\ \mathrm{eV} \]

should produce a noticeably thicker track than the large number of light particles found in this region of momenta (Fig. 2).

1. The momentum losses in a lead plate were measured at large momenta by Bleckett and Wilson \(B^{10}, B^{11}\), Neddermeyer and Anderson \(N^1\), and Crussard and Leprince-Ringuet \(C^5\). Bleckett’s measurement results are presented in Fig. 3.

Fig. 3. Energy losses to radiation as a function of momentum. Abscissa: momentum. Ordinate: relative changes of momentum in 1 cm of lead. 1 — Bleckett’s measurements in \(1/3\) cm of lead, 2 — Bleckett’s measurements in 1 cm of lead — course of the theoretical curve for an infinitely thin lead plate (§ 21).

At small momenta \(pc < 2 \cdot 10^8\) eV the losses are as large as those of radiating electrons (§ 3). However, they decrease at larger values of the momentum so that at

\[ \frac{1}{2}\cdot 10^9\ \mathrm{eV} \]

the losses amount to only about \(1/10\) of the initial value.

It follows from No. 1 that, if the radiation theory is considered correct, then most particles with momentum greater than \(pc = 2 \cdot 10^8\) eV must be heavier than the electron. Starting from the fact,\(^{B10,\,B11,\,No.\,1,\,C5}\) that a multitude of particles observed in the momentum interval

\[ 2 \cdot 10^8 \mathrm{eV} < pc < \frac{1}{2} \cdot 10^9 \mathrm{eV}, \]

ionize more weakly than protons and radiate more weakly than electrons, Anderson and Neddermeyer concluded that what is observed here are hitherto unknown “heavy electrons,” whose mass lies between the mass of the electron and the mass of the proton.

In contrast to the interpretation of the measurement results based on the introduction of heavy electrons, Blackett and Wilson, in order to explain these results, at first assumed that all the observed particles are electrons which, at momenta exceeding \(10^8\) eV, lose their radiative capacity.

However, it will be shown below, on the basis of the study of Hoffmann’s showers (§ 24) and measurements in the stratosphere (§ 19), that the formulas of radiation theory remain valid up to very high energies, of the order of \(10^{11}\) eV. Moreover, the unsuitability of the radiation theory in deriving the formula for bremsstrahlung radiation is unclear from the theoretical point of view. Therefore at the present time the first interpretation must be considered correct.

5. Mass of heavy electrons

The mass of heavy electrons can be obtained from a combination of measurements of momentum and velocity. In doing so it is, of course, necessary to observe the heavy electron at the end of its path, where it has a velocity considerably less than the velocity of light, and forms a track which is much stronger than the track of an electron. Despite the rarity of this event, several authors have succeeded in obtaining photographs of tracks of slow heavy electrons. In Fig. 4 is shown a photograph obtained by Williams and Pickup.\(^{W4}\) Here \(d\) denotes the track of the heavy electron, \(e\) the track, shown for comparison, of an electron with energy

\[ \frac{1}{2} \cdot 10^6 \mathrm{eV}. \]

It can clearly be seen that the ionization of particle \(d\) is approximately three times greater than the ionization of electron \(e\). The curvature of track \(d\), on the other hand, indicates that particle \(d\) is not a proton. The curvature and ionization make it possible to calculate the mass of the heavy electron, as given in Table 1 according to Williams and Pickup.

Other authors indicate the following values of the rest mass \(\mu\) (in units of the electron mass):

\[ \frac{\mu}{m} = 250,\quad \text{Corson and Brode}^{C4} \]

\[ = 125,\quad \text{Brode and Starr}^{B19} \]

TABLE 1

Quantities and units	Curvature \(10^5\) \(g\ \mathrm{vss}\) \(\mathrm{cm}\)	Ionization ion. electron per \(10^6\) eV	Momentum \(P\) \(mc\)	Velocity \(c\)	Mass \(m\)	Charge
Track \(a\) . . . .	1.10	5	65	0.30	\(220 \pm 50\)	—
Track \(b\) . . . .	1.83	\(>7\)	107	0.25	\(430\ (<800)\)	\(+^{1}\)
Track \(c\) . . . .	1.47	\(>3\)	85	0.45	\(190 \pm 60\)	\(+\)
Track \(d\) . . . .	1.15	3.3	67	0.41	\(160 \pm 30\)	\(+\)

\[ =160,\quad \text{Street and Stevenson}^{S\,10} \]

\[ \sim 350,\quad \text{Anderson and Neddermeyer}^{A\,3} \]

\[ =120,\quad \text{Ruling and Kren}^{R\,15} \]

\[ =200,\quad \text{Ehrenfest}^{E\,4} \]

\[ =180\text{--}250,\quad \text{Nishina et al.} \]

Assuming that heavy and light electrons occur in cosmic rays, we shall first discuss (Chs. II and III) the theoretical effects that may be expected when such particles enter

Fig. 4. Photograph in a Wilson chamber of the track of a heavy electron (Williams and Pickupp, Nature (Lond.), 141, 684, 1938); \(d\) — track of a heavy electron, \(e\) — track of an electron for comparison

a substance. Then (Chs. IV, V) we shall compare, as a whole, the theoretically predicted effects with experiment, which will allow us, retrospectively, to determine several parameters whose values remained open in the theoretical consideration.

CHAPTER II. THEORY OF LIGHT PARTICLES1

6. Survey of Electromagnetic Processes

If an electron with high energy moves in matter, then, as was indicated above, it experiences above all a twofold action. First, it collides with the electrons of the atomic shells and loses energy to ionization. Second, it is deflected by the Coulomb fields of nuclei, and these deflections lead to the emission of photons when the velocity of the electron approaches the velocity of light.

Electromagnetic forces can, along with the conversion of part of the electron’s energy into radiation, also cause the inverse process, i.e. they can lead to the fact that a quantum encountering an atomic nucleus forms a positron–electron pair. Both processes—bremsstrahlung and pair production—occur sufficiently often if the energy of the quantum or electron appreciably exceeds the electron rest energy \(\frac{1}{2}\cdot 10^6\ \mathrm{eV}\).

TABLE 2

	Pb	Fe	Al	H\(_2\)O	Air
\(X_0\) . . . . . . . . . .	0.4	1.4	7.8	34	27 500	cm
\(E_j\) . . . . . . . . . .	1	3	6	15	15	\(10^7\ \mathrm{eV}\)

To give a clear description of the processes of pair production, bremsstrahlung, and ionization, we shall introduce, following Bhabha and Heitler B 5, C 1, as the unit of length such a segment \(X_0\), in each substance, over which an electron loses to bremsstrahlung, on the average, half of its energy. The first row of Table 2 gives the segments \(X_0\) in various substances. We shall call \(X_0\) the “universal length” and denote by \(l\) the thickness of a layer of matter measured in these units \(\left(l=\frac{x}{X_0}\right)\).

Let us further note B 5, C 1 that for each substance there is a definite energy \(E_j\) at which the electron’s energy losses to radiation will be equal to its losses to ionization. Above the energy \(E_j\) the losses of energy to radiation predominate; below it, the losses of energy to ionization predominate, since the energy losses to radiation grow proportionally to the energy, whereas the energy losses to ionization increase with the energy only logarithmically, i.e. in practice remain constant.

The energy \(E_j\), of the same order of magnitude as the ionization losses over the universal length \(X_0\), is given in the second line of Table 2. In what follows we shall also call \(E_j\) the critical energy.

The universal length \(X_0\) is approximately inversely proportional to the square of the atomic number \(Z\) and to the number of atoms in \(1\ \mathrm{cm}^3\) of matter: \(\frac{\rho}{A}\):

\[ X_0 \approx \operatorname{const}\frac{A}{\rho}\frac{1}{Z^2}. \tag{3} \]

The value of the critical energy is approximately inversely proportional to the atomic number

\[ E_j \approx \frac{\operatorname{const}}{Z}, \tag{4} \]

since the ionization losses, in the first approximation, increase linearly with the number of electrons in the atom \(Z\), whereas the effective cross section of bremsstrahlung increases as the square of the atomic number. A certain deviation from law (4), which becomes noticeable in Table 2, is due to the fact that the ionization potential of the atom depends on the atomic number (§ 9, Fig. 2).

In addition to the processes indicated, positron annihilation and the Compton effect also play a role. However, since these processes become noticeable at low energies (of the order of \(10^6\ \mathrm{eV}\)), they may be neglected when considering phenomena connected with cosmic rays, which occur, generally speaking, in the region of energies greater than \(10^7\ \mathrm{eV}\).

7. Bremsstrahlung Radiation

Let us now suppose that an electron with energy \(E\) falls upon a thin layer of matter \(dl\). We ask: what number of quanta with energy in the interval \(k, k+dk\) will it radiate?

This number, in the energy region most important for cosmic rays, exceeding the rest energy of the electron, will approximately be, according to Bethe and Heitler, \(^{3,5}\)

\[ I(k)\,dl\,dk = dl\,\frac{dk}{k}\ln 2. \tag{5} \]

It follows from this that the total number of quanta emitted over the length \(dl\), with energy exceeding some energy \(E_1\), is equal to

\[ dn = dl \int_{E_1}^{E} I(k)\,dk = dl \cdot \ln \frac{E}{E_1}\cdot \ln 2, \tag{6} \]

and the energy given up on the average along the path \(dl\),

\[ dE = dl \int_{0}^{E} I(k)\, k\, dk = dl E \ln 2. \tag{7} \]

Thus the relative losses to radiation \(\dfrac{dE}{E}\) of an electron with high energy do not depend on the energy, which is precisely in agreement with the assumption we made in § 3.

It follows from this that the mean energy of the radiating electron decreases exponentially with the thickness of the layer traversed,

\[ \overline{E}_l = E_0 2^{-l} \tag{8} \]

(\(E_0\) is the initial energy, \(E_l\) is the energy after passing through a layer \(l = \dfrac{X}{X_0}\)).

In discussing the experiments it is, of course, necessary also to take into account fluctuations in the energy losses. These fluctuations are described by specifying the probability that an electron with initial energy \(E_0\), after traversing a segment \(l\), has an energy lying in the interval \(E_l,\ E_{l+dl}\).

TABLE 3

Mean energy losses of electrons in lead according to Anderson and Neddermeyer (Phys. Rev., 50, 267, 1936)

	7900 gauss, 0.35 cm Pb plate (Pike’s Peak)	7900 gauss, 0.35 cm Pb plate (Pike’s Peak)	7900 gauss, 0.35 cm Pb plate (Pike’s Peak)	7900 gauss, 0.35 cm Pb plate (Pike’s Peak)	7900 gauss, 0.35 cm Pb plate (Pike’s Peak)
Energy interval	\(<50\)	\(50\text{–}100\)	\(100\text{–}150\)	\(150\text{–}200\)	\(10^6\mathrm{eV}\)
Number of tracks	29	65	18	13
Mean initial energy	31	75	123	177	\(10^6\mathrm{eV}\)
Mean energy losses: exp.	42	82	178	191	\(10^6\mathrm{eV}/\mathrm{cm}\ \mathrm{Pb}\)
Mean energy losses: theor.	50	110	175	148

	4500 gauss, 0.33 cm Pb plate (Pasadena)	4500 gauss, 0.33 cm Pb plate (Pasadena)	4500 gauss, 0.33 cm Pb plate (Pasadena)	4500 gauss, 0.33 cm Pb plate (Pasadena)	4500 gauss, 0.33 cm Pb plate (Pasadena)
Energy interval	\(<50\)	\(50\text{–}100\)	\(100\text{–}150\)	\(150\text{–}200\)	\(10^6\mathrm{eV}\)
Number of tracks	22	28	15	16
Mean initial energy	26	71	117	170	\(10^6\mathrm{eV}\)
Mean energy losses: exp.	37	84	124	207	\(10^6\mathrm{eV}/\mathrm{cm}\ \mathrm{Pb}\)
Mean energy losses: theor.	43	105	167	240

According to Bethe and Heitler \(^{33}\), this probability is:

\[ W(E_1)\,dE_1=\frac{dE_1}{E_0}\left(\ln\frac{E_0}{E_1}\right)^{l-1}\cdot\frac{1}{(l-1)!} \tag{9} \]

In particular, in a layer \(l=1\) any loss of energy—from complete loss to a loss equal to zero—is equally probable. The formulas (8) and (9), calculated from the quantum theory of radiation, for the mean energy losses by an electron and the fluctuations of the losses to radiation relative to their mean value, up to energies above \(2\cdot10^8\ \mathrm{eV}\), can be confirmed by direct measurements in a Wilson chamber.

Table 3 presents the mean energy losses in a Pb plate of thickness \(0.35\ \mathrm{cm}\), according to the measurements of Anderson and Neddermeyer \(^{A3}\).

Figure 5 shows the fluctuations of energy losses in the region of energies \(E<2\cdot10^8\ \mathrm{eV}\), according to the measurements of Bleakett \(^{B11}\). In this figure, on the abscissa axis are plotted the relative losses to radiation \(R\) in \(1\ \mathrm{cm}\) of lead. They are obtained from the total losses of the electron energy after subtracting the insignificant losses of energy to ionization \(dX\)

\[ R=\frac{E_0-E_1-dX}{\dfrac{(E_0+E_1)}{2}\cdot X}. \]

Fig. 5. Fluctuations of losses to radiation at \(E<2\cdot10^8\ \mathrm{eV}\) (after Bleakett, Proc. Roy. Soc. Lond., 165, 11, 1938). Abscissa: relative losses \(R\) to radiation in \(1\ \mathrm{cm}\) of lead. Ordinate: frequency of loss to radiation on the path \(dR\) in plates of \(1/3\ \mathrm{cm}\) Pb (left), \(1\ \mathrm{cm}\) Pb (right). \(\varnothing\)—Bleakett’s measurements, — theory according to Bethe–Heitler.

The left part of Fig. 5 shows the frequency of energy losses \(R\) in a thin lead plate of thickness \(1/3\ \mathrm{cm}\), which is somewhat less than the universal unit \(X_0\). Therefore large energy losses prove to be especially rare. In the right part of this figure is plotted the distribution of losses in a plate with thickness of several universal units \(X_0\). Here large energy losses occur especially often. The drawn curves represent the theoretical distribution of losses (9), whose mean value (8) is represented by the dotted line. Those experimental points which correspond to negative losses of momentum arise from errors in measurements of the curvature of tracks and give an approximate measure of the inaccuracy of the momentum measurements. If these paths are regarded as an example of small positive energy losses,

that it turns out that there is a distinct maximum in the region of small energy losses, which cannot be explained by bremsstrahlung and which, evidently, must be ascribed to penetrating particles.

8. Pair Production

The probability that, along a path \(dl\), a quantum will produce a pair, at high energies \(h\nu \gg mc^2\), is independent of the energy of the quantum (cf. \(^{\mathrm{H}9}\))

\[ \omega \cdot dl = 0.6\,dl . \tag{10} \]

At low energies the probability of pair production decreases and reaches a value equal to zero at a quantum energy equal to the rest energy of the pair, \(h\nu = 2mc^2\). This law of pair production, calculated according to Dirac’s theory, was checked in the \(\gamma\)-ray region by direct measurements of Curie-Joliot, Chadwick, Blackett, and Occhialini \(^{\mathrm{H}9}\). In the region of high energies we shall at first assume the validity of its extrapolation (10), which will be verified later by the consequences following from it.

9. Ionization

The ionization of atomic shells can be studied by means of the following phenomena.

A. Energy losses can be obtained from the change in the curvature of the path in a magnetic field, if the electron velocity is so small that bremsstrahlung no longer plays a role. The theory of ionization losses of Bethe \(^{\mathrm{B}4}\) and Bloch \(^{\mathrm{B}15}\), at velocities less than \(1/4\) of the speed of light, was confirmed to a high degree of accuracy by the measurements of Williams \(^{\mathrm{B}4}\). The curve of ionization losses, calculated from Bloch’s formula, is shown in Fig. 2. If the quantity \(a\) entering formula (2) is approximately replaced by a constant (which is quite legitimate up to values \(p \sim 3.5\,\mu c\)), then for the range \(R\) of a particle with mass \(\mu\) and momentum \(p\) we obtain

\[ \left. \begin{array}{c|ccc} & \text{Lead} & \text{Water} & \text{Air} \\ \hline a = & 1.2\cdot 10^{7} & 2\cdot 10^{6} & 2.5\cdot 10^{3}\ \mathrm{eV/cm} \end{array} \right. \]

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

\[ \left. \begin{array}{c|ccccc} \dfrac{p}{\mu c} & 0 & 1 & 2 & 3 & \gg 1 \\ \hline \dfrac{aR}{\mu c^2} & 0 & 0.1 & 0.7 & 1.5 & \dfrac{p}{\mu c}-2 \end{array} \right\} \tag{11} \]

On the Theory of Cosmic Radiation

B. The energy distribution of secondary electrons was measured by Ishino¹² in the energy region \(E < 300\ \mathrm{eV}\). The measurements are in good agreement with the theory, which, for large initial and final energies, leads one to expect that the distribution of secondary electrons with energy \(> E\) has approximately the form \(\dfrac{\mathrm{const}}{E}\) (cf. B⁴, B¹).

C. Secondary ionization, i.e., the number of ions formed directly by the primary electron, can be determined experimentally by counting the number of droplets in a sharp, as yet undiffused, track in a Wilson chamber.

The theory gives, for the number of secondary ions produced by an electron moving with velocity \(\beta c\), per \(1\ \mathrm{cm}\) in a gas, under normal conditions B⁴, B¹,

\[ s=\frac{I}{\beta^{2}}\left(14.24+\ln\frac{\beta^{2}}{1-\beta^{2}}-\beta^{2}\right) \]

\(I=\)	\(\mathrm{N_2}\)	\(\mathrm{O_2}\)	\(\mathrm{H_2}\)
	1	1.15	0.29

ions per \(1\ \mathrm{cm}\) at atmospheric pressure and room temperature. \(\tag{12}\)

For \(p < mc\), secondary ionization, like the energy losses, must decrease inversely proportionally to the square of the velocity. Near the value of the momentum equal to \(3mc\), it has a minimum, after which it increases logarithmically with the momentum

\[ p=mc\,\frac{\beta}{\sqrt{1-\beta^{2}}}. \]

The value of the constant \(I\) can be given reliably only for hydrogen. If one uses the values of this constant according to Barre, given in (12), then we obtain that the secondary ionization in air is minimal for an electron moving with velocity

\[ \beta=0.97,\qquad \frac{p}{mc}=\frac{\beta}{\sqrt{1-\beta^{2}}}=3.5,\qquad H\rho=5.8\cdot 10^{3}\ \text{gauss cm}. \]

The number of secondary ions per \(1\ \mathrm{cm}\) at this minimum is

	Theoretical (12)	Experimental
\(\mathrm{N_2}\)	17	14—18, Corson and Brode C⁴
\(\mathrm{O_2}\)	19.5	20, Williams and Terroux W⁵
\(\mathrm{H_2}\)	4.8	5 » » W⁵

D. Secondary ionization should be distinguished from total ionization \(i\), i.e., from the number of pairs of ions produced by the primary electron and by the secondary, tertiary, etc. elect-

rays. The total ionization has been calculated by Bagge \(^{B1}\) in the nonrelativistic region. The value he obtained for the total ionization is in approximate agreement with the measurements of Gerbes \(^{G6}\).

An electron with energy \(2\cdot 10^4\ \mathrm{eV}\) expends, over its entire path, on the formation of a pair of ions on average

	\(\mathrm{N}_2\)	\(\mathrm{H}_2\)
Theoretical (Bagge \(^{B1}\))	\(28.6\ \mathrm{eV}\)	\(34\ \mathrm{eV}\)
Experimental (Gerbes \(^{G6}\))	\(34\ \mathrm{eV}\)	\(37\ \mathrm{eV}\)

Corson and Brode \(^{C4}\) found in the diffusion trace, after subtracting unseparated droplets, 25 pairs of ions per \(1\ \mathrm{cm}\) of air at minimum ionization \((\beta \simeq 0.96)\). Thus the ratio of total ionization to secondary ionization is greater than \(1.4—1.8\).

Fig. 6. Specific ionization of the electron according to Corson and Brode (Phys. Rev., 53, 215, 1938.) Abscissa: momentum \(H\rho\) of the electron in gauss per \(1\ \mathrm{cm}\). Ordinate: number of ions per \(1\ \mathrm{cm}\) of air under normal conditions. \(\varnothing\) — measurements of Corson and Brode; — theoretical curve (12) with an arbitrary constant multiplier; — — — approximate theoretical curve (7) with \(\alpha=\mathrm{const}\).

In the measurements of Corson and Brode the logarithmic increase of ionization with momentum \(p\) required by the theory was first observed. The experimental points according to Corson and Brode are shown in Fig. 6. The curve in the same figure represents the theoretical course of the ionization. It is obtained by multiplying formula (12) by a suitable factor, which is regarded as constant. The logarithmic increase of the ionization can be followed experimentally only up to \(H\rho=2\cdot 10^5\ \mathrm{gauss\,cm}\), since particles with larger momentum are already in part heavy electrons, whose minimum ionization is superposed on the increase of the ionization of electrons. Now, after the discovery of heavy electrons, we can understand why earlier experiments failed in which attempts were made to establish the logarithmic increase of ionization at \(H\rho\) greater than \(2\cdot 10^8\ \mathrm{gauss\,cm}\).

E. Along with the counting of droplets in an ionization chamber, which makes it possible to find the ionization as a function of momentum, there exists another method of measuring ionization, which, however, gives only the total ionization per \(1\ \mathrm{cm}\), averaged over all momenta of the cosmic radiation \(i\). Stuhlinger \(^{S34}\) compared the number of cos-

mic particles passing through a proportional counter, with ionization, and obtained \(\bar i=30\text{--}35\) ion pairs per \(1\ \mathrm{cm}\) of air for penetrating particles and \(\bar i=50\) ion pairs per \(1\ \mathrm{cm}\) of air for shower particles. Since showers consist chiefly of ordinary electrons (§ 23), while penetrating particles are heavy electrons, the larger number of ions may be regarded as evidence for the logarithmic increase of the number of ions with respect to

\[ \frac{\text{energy}}{\text{rest mass}} \]

\(^{S14}\) (Fig. 2).

The mean total ionization \(\bar i\) can further be judged by comparing the ionization current with the number of coincidences. These measurements give higher values, fluctuating between 70 and 135 ion pairs per \(1\ \mathrm{cm}\) of air \(^{M3,\ G1}\).

However, the latter method of measurement is more difficult to analyze than Schullinger’s method, since it is not excluded that several particles were registered in a single coincidence. This is probably the explanation of the large figure for the number of ion pairs obtained by this method. In a theoretical consideration of ionization losses in thick layers, it is necessary, in addition to the ionization of atomic shells, to take into account ionization by nuclei (§ 16) and diffusion arising as a consequence of elastic scattering. The former will be discussed in one of the following chapters (III); elastic scattering was investigated theoretically by Williams \(^{W6}\) and experimentally by Blackett and Wilson.

It turned out that the effective absorption in passing through thick layers may be considerably increased owing to elastic scattering at small momenta of the incident particles. At higher energies, however, elastic scattering plays an insignificant role.

10. Cascade showers

The processes of bremsstrahlung radiation and pair production are based on the action of electromagnetic forces. According to the quantum theory of the electromagnetic field, they are, generally speaking, single processes, i.e. one quantum produces for the most part only one pair at a nucleus, and an electron produces at a separate nucleus only one photon. Higher-order processes, in which several pairs or quanta arise at one nucleus, are, according to the quantum theory of radiation, much less probable. Their probability is less than the probability of a single process by the fine-structure constant

\[ \frac{e}{\hbar c}=\frac{1}{137} \]

to a high degree and increases with the energy \(E\) of the incident particle only logarithmically \(^{K2}\).

However, despite the fact that the electromagnetic processes are first-order processes, owing to the high frequency of pair production and bremsstrahlung radiation they can lead to the formation, in a layer of matter of finite thickness, of a large number of secondary-

…particles from one primary particle. This was shown almost simultaneously by Carlson—Oppenheimer and Bhabha—Heitler (see also L²).

If an electron with high energy falls on a layer of lead, then in the first 4 mm it emits a photon with energy of the same order of magnitude. This photon forms, in one of the following millimeters, a pair. The electron and positron, in turn, will each emit a photon, and in the end a whole bunch of particles will fly out of the layer of lead.

The processes of emission of bremsstrahlung and of pair production thus lead to a “multiplication” of the particle by the formation of a cascade-like shower. As a precursor of the “cascade” theory of showers one may regard the radiation scheme proposed by Heitler and Fonder in 1935 G³, G¹, which was proposed to explain the order of magnitude of shower phenomena.

Carlson and Oppenheimer, Bhabha and Heitler calculated by a statistical method, from the formulas of §§ 7 and 8, the magnitude and the energy distribution of a cascade shower, i.e. the number \(z\) of electrons and positrons with energy exceeding a certain energy \(E_1\), which are created by an electron with initial energy \(E\) after traversing a layer \(l\)

\[ z=z(l,E,E_1) \]

(in what follows, by “electrons” we shall always mean electrons with positive and negative charge).

Fig. 7. Multiplication of the number of electrons. Abscissae: thickness of the layer \(l\) of matter in the units of Table 2. Ordinate: number \(z(l,y)\) of electrons and positrons that arise from a primary electron with energy \(E_j e^y\) in the energy region \(>E_j\) under a layer \(l\).

The results of their calculations are presented in Fig. 7. Along the abscissa axis is plotted the thickness of the layer \(l\), along the ordinate axis—the number of particles \(z\). The parameter \(y\), referring to the individual curves, denotes the logarithm of the ratio of the initial energy to the final one

\[ y=\ln \frac{E}{E_1}. \]

It is easy to see that the number of particles depends only on this ratio, provided only that \(E_1 \gg E_j\), so that the influence of ionizations may be neglected:

\[ z=z(l,y), \tag{13} \]

since then the relative energy distribution according to (7) and (10) does not depend on the energy (at high energies). The multiplication function \(z=z(l,y)\) shown in Fig. 7 contains, in this case, two pieces of information: first, it indicates the mean magnitude of the shower created by a particle with energy \(E\) after traversing-

thickness of the layer \(l\). This shower magnitude is obtained from the function \(z(l,y)\), if the final energy \(E_1\) is set equal to the ionization limit \(E_j\) (Table 2) of the given substance.

\[ y=\ln\frac{E}{E_j}: \]

\[ \left. \begin{array}{c|cccccc} y= & 3 & 4 & 5 & 7 & 10 & 12\\ \hline \mathrm{Pb}\ldots E= & 2\cdot10^{8} & 5.5\cdot10^{8} & 1.5\cdot10^{9} & 1.1\cdot10^{10} & 2.2\cdot10^{11} & 1.6\cdot10^{12}\,\mathrm{eV}\\[2pt] \mathrm{Fe}\ldots E= & 6\cdot10^{8} & 1.6\cdot10^{9} & 4.5\cdot10^{9} & 3.4\cdot10^{10} & 6.7\cdot10^{11} & 4.8\cdot10^{12}\,\mathrm{eV}\\[2pt] \begin{array}{c} \mathrm{H_2O}\\ \text{air} \end{array} \}E= & 3\cdot10^{9} & 8.2\cdot10^{9} & 2.2\cdot10^{10} & 1.7\cdot10^{11} & 3.3\cdot10^{12} & 2.4\cdot10^{13}\,\mathrm{eV} \end{array} \right\} \tag{14} \]

This means that we approximately take the effect of ionization into account by assuming that the electron is immediately stopped when its energy becomes less than \(E_j\); while at an energy greater than \(E_j\), the electron is not subject to the action of ionization.

For more exact calculations of the shower magnitude, one must add to the number of electrons with energy greater than \(E_j\) also the number of electrons with energy less than \(E_j\). The latter is tabulated in the work of Arley\(^{A4}\) (cf. also \(C^1\)). For \(y=\ln \frac{E}{E_j}=4\), for example\(^{A4}\):

\[ \left. \begin{array}{c|ccccc} \text{For } l= & 1 & 2 & 3 & 5 & 10\\ \hline z(>E_j)=z(l,4)= & 1.84 & 3.35 & 4.66 & 5.17 & 1.11\\ z(<E_j)= & 0.09 & 0.66 & 1.75 & 4.39 & 3.00 \end{array} \right\} \tag{14a} \]

The form of the mean shower magnitude \(z(l,y)\), as a function of the thickness of the layer, which is obtained with the aid of (14) from the curves of Fig. 7, is qualitatively clear.

In thin layers the mean shower magnitude grows with increasing layer thickness, since the initial energy can be fragmented the more strongly, the greater the number of atomic nuclei that enters into the game.

In thick layers, however, energy losses to ionization become appreciable; because of them all particles whose energy decreases through radiation to the ionization limit \(E_j\) are stopped. These energy losses first lead to the establishment of equilibrium between the number of absorbing and arising particles at a certain thickness \(l_m\). With a further increase in the thickness of the layer, the energy losses lead to a fall of the curve \(z(l)\).

Equilibrium occurs, for example, for an electron with energy \(10^{11}\,\mathrm{eV}\) at a layer thickness of \(5\ \mathrm{cm}\) Pb (\(l_m=12\)), after passing through which it creates a shower of 1000 particles. When the layer thickness is increased to \(10\ \mathrm{cm}\), the same absorption predominates, so that at \(10\ \mathrm{cm}\) Pb the shower magnitude at the same initial energy decreases by a factor of 10.

Secondly, the curves in Fig. 7 give information about the spectrum of the various particles in the cascade shower. The form of this secondary spectrum, into which one primary electron is “fragmented,” is obtained from the function \(z(l,y)\), if the layer thickness \(l\) and the initial energy remain constant, while in the expression \(y-\ln \dfrac{E}{E_1}\) the final energy \(E_1\) is varied. In the next paragraph it will be shown that the number of particles in a shower with energy exceeding \(E_1\) decreases approximately as \(\dfrac{\mathrm{const}}{E_1^a}\) \([1<a<2,\ (19),\ (19a),\ (15)]\), when the shower is formed in a thick layer or in the “equilibrium layer” \(l_m\).

If instead of the shower-producing electron there is a photon, then to a considerable extent the same kind of shower arises. Indeed, in the first millimeter of lead the photon creates a pair which, in turn, will cause the process described above. On this basis one may assert that the number of photons in a shower has the same order of magnitude as the number of electrons and positrons.

11. Mathematical supplement to § 10

α) Mean size of a cascade shower.

We must now consider in somewhat greater detail the mathematical properties of the multiplication function

\[ z=z(l,y),\qquad y=\ln\frac{E}{E_j}, \]

shown in Fig. 7, which represents the mean size \(z\) of a shower under a layer \(l\), formed from a particle with initial energy \(E\).

a) The maximum size of the shower \(z_m\) (i.e., the maximum on the curves of Fig. 7) varies proportionally to the initial energy \(E\) of the shower-producing electron\(^{B5}\)

\[ z_m(y)=z(l_m,y)\approx \frac{1}{8}\left(\frac{E}{E_j}\right)^{0.93},\qquad \left(\text{for }\frac{\partial z}{\partial l}=0\right). \tag{15} \]

This formula means that at the maximum the electron energy is fragmented into equal parts, so that the mean energy of each particle is of the order of \(E_j\).

b) The thickness of the equilibrium layer \(l_m\), at which the curves of Fig. 7 have a maximum, grows logarithmically with the initial energy \(E\)

\[ l_m \approx 1.2\,y \approx 3.10\,\lg z_m + 2.7\quad \left(\text{for }\frac{\partial z}{\partial l}=0\right). \tag{16} \]

This circumstance becomes understandable if, following Carlson and Oppenheimer \(^{C1}\), for simplicity we replace the real process of shower formation in the equilibrium layer \(l_m\) by a certain model of this pro—

process. Namely, let us imagine such a model in which, on each segment \(l=1\), a doubling of the number of ionizing particles occurs by division. Then we obtain the values of the shower \(z_m\) under a layer \(l_m\): \(z_m \sim 2^{l_m}\), i.e. the logarithmic course of the thickness of the equilibrium layer \(l_m\) with the number of particles or with the initial energy \(E\).

c) The width of the maximum of the curves in Fig. 7 depends only weakly on the energy. This is most clearly expressed by the relation

\[ \int_0^\infty dl\, z(l,y) \approx \frac{3}{4}\,\frac{E}{E_1}, \tag{17} \]

which we shall often use below.

d) In thin layers and at large initial energies, the influence of energy losses may be neglected in calculating the shower magnitude. Then for \(z(l,y)\) the following expression is obtained:

\[ z(l,y) \approx \sum_{n=0}^{\infty} \frac{x^n}{n!(2n)!}, \quad \left(\text{for } l \leq 2,\ y \gg 1\right) \tag{18} \]

\[ x = 0.83\, y l^2. \]

Each term of this series represents the share contributed by one “generation,” by means of which the acts of increase in the number of particles follow one another.

For large values of \(x\), the sum (18) may be approximated by the expression

\[ z(x) \approx 0.20\, e^{1.89\sqrt[3]{x}} \cdot x^{-1/6}, \quad \left(\text{for } x \gtrsim 1\right). \tag{18a} \]

Therefore the magnitude of showers under thin layers increases with the initial energy, at low energies, only logarithmically. However, at high energies, or in thicker layers, it always grows more strongly with the energy, so as ultimately to reach an almost linear increase with the energy (14) in thick layers.

e) In thick layers, when absorption already begins to predominate, the shower magnitude \(z\) decreases exponentially with the layer thickness \(l\) and increases according to a power law with the initial energy \(E\):

\[ z(l,y) \approx e^{ay - bl - 3.2}. \tag{19} \]

Here \(a\) and \(b\) are quantities varying only slowly with \(y\) and \(l\):

\[ \left\{ \begin{array}{rrrr} l = 9 & 15 & 21 & 30 \\ y = 3 & 5 & 7 & 10 \end{array} \right\} \quad \left\{ \begin{array}{rrrr} a = 2.14 & 2.00 & 1.79 & 1.44 \\ b = 0.48 & 0.44 & 0.37 & 0.25 \end{array} \right\}. \tag{19a} \]

Exact tables of function (13) are given in the works of Carlson—Oppenheimer C¹, Bhabha—Heitler B⁵, and Arley A⁴.

β) Fluctuations of the shower size. The data obtained so far on the mean size of showers, generally speaking, cannot be compared directly with experiment for two reasons: first, the energy of the shower-producing electron is known only in rare cases, and, owing to the continuous spectrum of shower-producing electrons, only its mean value can be observed. Therefore we shall have (in § 11, γ) to average all the expressions so far obtained once more over the electron spectrum. Secondly, by observing the frequency of showers of a definite size, one can find the mean shower size; to calculate it, however, it is necessary to know the fluctuations of the shower size.

We shall therefore assume that the mean size of a shower produced by an electron with energy \(E = E_j t^\nu\) under a layer \(l\)

\[ \overline{N} = z(l,y) \quad (§ 11, \alpha), \]

is known, and we shall be interested in the fluctuation \(\Delta N\) of the shower size relative to its mean value.

\[ \Delta N = \left(\overline{N^2} - \overline{N}^{\,2}\right)^{\frac{1}{2}}, \]

The fluctuation of the shower size was first estimated by Bhabha and Heitler B⁵, who proceeded from the assumption that the formation of individual shower particles may be regarded as independent events. They therefore obtained the Poisson formula

\[ \frac{\Delta N}{\overline{N}} = \frac{1}{\sqrt{\overline{N}}}. \tag{20} \]

Furry F⁷ subsequently connected the phenomenon of shower formation with a model in which each ionizing particle has a definite probability, in a layer of given thickness, of turning into two particles, and obtained much larger fluctuations

\[ \frac{\Delta N}{\overline{N}} = 1. \tag{21} \]

As a more exact investigation E⁶ shows, the true fluctuations can be described by a certain formula intermediate between the two extreme cases (20) and (21). For very thin layers, for very thick layers \((l \gg 2y)\), and at the maximum of the curve in Fig. 7, it passes over into the Poisson formula (20)

\[ \begin{aligned} \frac{\Delta N}{\bar N} &\approx \frac{1}{\sqrt{\bar N}} \quad \text{for } l \ll 1,\\ \frac{\Delta N}{\bar N} &\approx \frac{1}{\sqrt{l y \ln 2}} \quad \text{for } 1 \ll l \leq \frac{1}{2}y \end{aligned} \tag{22a} \]

\[ \begin{aligned} \Delta N &\approx \left|\frac{\partial \bar N}{\partial l}\right| \cdot 0.92 + \sqrt{\bar N} \quad \text{for } \frac{1}{2}y \ll l \ll 2y,\\ \frac{\Delta N}{\bar N} &\approx \frac{1}{\sqrt{\bar N}} \quad \text{for } l \geq 2y \end{aligned} \tag{22b} \]

Formula (22a) is obtained if one takes into account that in the region where absorption is not yet substantial (\(l \ll y\)), the fluctuations of the shower size are due mainly to fluctuations in the number \(n\) of “first-generation” photons

\[ \left[ \frac{\Delta N}{\bar N}\approx \frac{\Delta n}{n}\approx \frac{1}{\sqrt{\bar n}} =\frac{1}{\sqrt{l y \ln 2}} \quad \text{by (6)} \right] \]

In deriving formula (22b), which should be applicable in the region of strong absorption, the essential assumption is that \(y\) photons with high energy may be produced from the primary electron sometimes somewhat earlier, sometimes somewhat later. This leads to the same effect as a fluctuation of the layer thickness \(\Delta l \simeq 0.9\,C^{1}\). The distribution of the shower size for large showers may be represented approximately by a Gaussian distribution: the probability that an electron with energy \(E=E_j e^y\), under a layer of thickness \(l\), produces a shower of \(N\) particles is equal to

\[ W(N,l,y)\,dN= \frac{ e^{-\frac{(N-\bar N)^2}{2(\Delta N)^2}} }{ \sqrt{2\pi}\Delta N }\,dN,\quad [\bar N=z(l,y)]. \tag{23} \]

γ) Frequency of cascade showers. We shall assume that the spectrum of the incident electrons at energy greater than \(E=E_j e^y\) has the form\(^1\)

\[ F(E)=I_0\left(\frac{E_0}{E}\right)^\gamma =I_0\left(\frac{E_0}{E_j}\right)^\gamma e^{-\gamma y}, \tag{24} \]

then we shall find the frequency \(H(N,l)\) of occurrence of showers with a number of particles greater than \(N\) under a layer \(l\).

The function \(H(N,l)\) will also be denoted by \(H(l)\) when the change in the layer thickness \(l\) is considered at constant magni-

\(^1\) The constants \(I_0\) and \(\gamma\), denoting the intensity and the degree of decrease of the spectrum with increasing energy, will later be determined experimentally. \(E_0\) may be chosen arbitrarily, for example, \(E_0: 10^8\ \mathrm{eV}\).

size of showers, and by \(H(N)\)—the “shower distribution curve,” when only the size of the shower varies while the thickness of the layer remains unchanged. The frequency of showers is, in general, given by the formula

\[ \frac{\partial}{\partial N} H(N,l) = \int_{0}^{\infty} dE \, \frac{dF(E)}{dE} \, W\left(N,l,\frac{E}{E_j}\right), \tag{25} \]

where the first factor denotes the number of incident electrons with energies in the interval \(E, E+dE\), and the second factor is the probability, specified by formulas (22) and (23), that an electron with energy \(E\) will produce, in a layer \(l\), a shower of \(N\) particles. As an exact analysis of formula (25) shows, fluctuations are significant only for small showers under thin layers. They may, however, be neglected when considering large showers \(N>100\), or thick layers \(l>3\cdot [[unclear: expression involving]] \lg N E^{6}\). Then (25) becomes the simpler formula

\[ H(N,l) \approx F(E)=I_0\left(\frac{E_0}{E_j}\right)^{\gamma} e^{-\gamma y}; \quad N=z(l,y). \tag{25a} \]

It follows from the last formula that the frequency of occurrence of showers containing more than \(N\) particles under a layer \(l\) is simply equal to the frequency with which electrons fall that possess energies exceeding a certain energy \(E\), where \(E\) denotes the energy at which the electrons would create, on average, under a layer \(l\), a shower of \(N\) particles. The frequency of showers obtained in this way is specified by the curves in Fig. 8 and by the following formulas. Figure 8 shows the curve \(H(l)\) (25a) for \(N=200\) particles, i.e., the frequency of showers with a number of particles greater than two hundred, as a function of the thickness of the layers \(l\) (Table 2) for a spectrum of the form

\[ F(E)=\left(\frac{E_0}{E}\right)^{1.5}. \]

Fig. 8. Abscissa: layer thickness in units of Table 2. Ordinate: theoretical frequency \(H(N,l)\) of occurrence of showers containing more than \(N\) particles. Solid curve: shower size \(N=200\). Dotted curves: small showers \(N=2, 3, 4\) according to Arley. The ordinate scales of the two curves are not comparable.

In Fig. 8 there are also shown the corresponding curves \(H(l)\) (dotted curves) for small showers having more than \(N=2, 3\), and 4 particles, which were calculated by Arley\(^4\), who used, for small showers, a formula (20) suitable for fluctuations and the correction (14a) for electrons with small energies, from a spectrum

\[ F(E)=\left(\frac{E_0}{E}\right)^{1.5} \quad \text{for } E>E_0=2\cdot 10^8 \text{ eV}, \]

\[ F(E)=15\frac{E}{E_0} \quad \text{for } E<E_0. \tag{26} \]

In this case, however, its curve depends only weakly on the form of the spectrum. The general character of the frequency of occurrence of showers, as can be seen in part from the figure, in the case of the spectrum (24) will be as follows2.

a) In thin layers the frequency of large showers increases very strongly with the thickness of the layer. The shower distribution \(H(N)\) then falls off with increasing shower size the more steeply, the thinner the layer (Fig. 25) (18a):

\[ \text{for } 1 \lesssim l \lesssim \text{from } 2 \text{ to } 3,\quad N \lesssim 100 \]

\(H(N,l)\) will have the following form:

\[ H(N,l)\approx I_0\left(\frac{E_0}{E_j}\right)^{\gamma} e^{-\frac{0.18\gamma}{l^2}\left[\ln N+\frac{1}{2}\ln\ln N+1.5\right]^2}, \tag{27} \]

b) The frequency of showers has a maximum at the equilibrium thickness of the layer (16)

\[ l_m\approx 3_{10}\lg N+2.7, \tag{28} \]

at which the shower distribution \(H(N)\) is approximately equal to the distribution in energy \(F(E)\) in the spectrum (15)

\[ H(N,l_m)=H_m(N)\approx I_0\left(\frac{E_0}{E_j}\right)^\gamma \left(\frac{1}{8N}\right)^{\frac{\gamma}{0.93}}. \tag{29} \]

c) In thick layers, according to (19), \(H(l)\) decreases exponentially, so that at a layer thickness doubled in comparison with \(l_m\), \(H(l)\) falls almost to one tenth of the maximum value. The shower distribution \(H(N)\) here is only somewhat less steep than the spectral distribution \(F(E)\), for \(l\gtrsim 5_{10}\lg N\) according to (19)

\[ H(N,l)\approx \left(\frac{1}{N}\right)^{\frac{\gamma}{a}} e^{-\frac{\gamma}{a}bl}\cdot \left(\frac{E_0}{E_j}\right)^\gamma \cdot e^{-3.2\frac{\gamma}{a}}, \tag{30} \]

where the coefficients \(a\) and \(b\) have the following values:

\(a=\)	1.9	1.5
\(b=\)	0.40	0.25
for \(N\approx\)	10	1000
\(l\approx\)	15—20

d) As is seen from formulas (27)—(30), owing to the presence of the multiplier \(\left(\frac{E}{E_j}\right)^\gamma\), the intensity in heavy elements is considerably greater than in light ones.

For example, in Pb the fragmentation of energy may occur down to \(E' = 10^7 eV\), whereas in Al it ceases because of ionization already at \(E' = 6 \cdot 10^7 eV\) (Table 2). In the case \(\gamma = 1\) the intensities in different substances are related approximately as their atomic numbers. For steeper spectra (\(\gamma > 1\)) the ratio of the intensities in different substances is still sharper.

In particular, the function \(H(1,t)\) formed from (25) describes the number of showers consisting of more than one particle, i.e. the number of coincidences caused by electrons in the arrangement of Fig. 9, as a function of the thickness of the layer.

If this absorption curve of coincidences \(H(1,t)\) is calculated for some mean spectral distribution \((24, \gamma \sim 1—2)\) in different substances, then it turns out, as Heitler showed\({}^{6}\), that under layers of equal mass the intensities will be approximately the same.

This approximately mass-proportional decrease in the number of coincidences caused by electrons is produced by the mutual compensation of two processes: although in heavy substances the fragmentation of the initial energy per \(1\ g/cm^2\) is greater than in light ones, nevertheless it can continue down to lower energies.

Thus equal masses have almost the same absorption effect, if one is concerned only with ionization absorption.

12. Bhabha ionization showers

According to the cascade theory of showers, every electron entering a layer thicker than \(1\ cm\) of Pb produces a shower. Heavy particles, however, can practically not cause showers directly, since owing to their large mass they radiate little.

Nevertheless, as Bhabha showed\({}^{6}\), on the basis of the cascade theory of showers one should expect that heavy particles are also sometimes accompanied by small showers.

Indeed, a heavy particle, on passing through matter, loses energy to ionization, i.e. it knocks electrons out of the atoms lying in its path (§ 9). In doing so it will sometimes impart a very large energy to an electron. Such an electron, knocked out with a large energy, will in turn produce a cascade shower. The mean number of electrons and positrons with energy exceeding the critical energy \(E_i\) characteristic for each substance (§ 6), which will thus accompany a heavy electron of energy \(10^{10} eV\), is, according to Bhabha\({}^{6}\), \(10\%\). According to (14a) the total number of electrons and positrons may be regarded as approximately twice as large. More exact figures for the mean number of electrons are given, according to Bhabha’s calculations, for particles with mass \(\mu = 100m\) (\(m\) is the electron mass) at different energies in Table 4\({}^{6}\).

In the following Table 5 is shown the distribution of this number of electrons over various shower sizes. It gives the probabilities \(\frac{d}{dN} Q(N,E)\) that a heavy electron with energy \(E\) is accompanied by a shower of electrons consisting of \(N\) particles. Here \(N\) is understood to mean the number of particles in the shower with energy greater than \(E_j\). According to Bhabha, the true size of the shower is approximately twice as large.

TABLE 4

Average number of electrons produced by a heavy electron with energy \(E\)
(according to Bhabha)

\(E\)	Medium	Number of particles with \(E' > E_i\)	Total number of particles
\(10^8\ \mathrm{eV}\)	Pb	—	—
\(10^8\ \mathrm{eV}\)	\(\mathrm{H_2O}\)	—	—
\(10^{10}\ \mathrm{eV}\)	Pb	0.09	0.19
\(10^{10}\ \mathrm{eV}\)	\(\mathrm{H_2O}\)	0.03	0.07
\(10^{12}\ \mathrm{eV}\)	Pb	0.16	0.34
\(10^{12}\ \mathrm{eV}\)	\(\mathrm{H_2O}\)	0.07	0.15

In a rough approximation the table may be replaced by the following formula for the probability \(Q(N,E)\) that a heavy electron with energy \(E \gg \mu c^2\) is accompanied by an ionization shower containing more than \(N\) particles in water:

\[ Q(N,E) \approx 0.03\,\frac{1}{N} \quad \text{for } E \gg 8NE_j, \]

\[ Q(N,E) \approx 0 \quad \text{for } E \ll 8NE_j. \tag{31} \]

In general terms these formulas are clear: in order for a shower of \(N\) particles to arise according to formula (15), an atomic electron must be given the energy \(8NE_j\). Therefore, in order for a heavy electron to be able to impart such an energy, it must itself have an energy not less than \(8NE_j\). Thus ionization showers consisting of more than \(N\) particles can be formed in matter with criti-

TABLE 5

Probability \(\frac{d}{dN} Q(N,E)\) with which a heavy electron of energy \(E\) is accompanied by a shower of \(N\) particles

\(E\)	Medium	\(N=1\)	2	4	5	10	50
\(10^8\ \mathrm{eV}\)	Pb	—	—	—	—	—	—
\(10^8\ \mathrm{eV}\)	\(\mathrm{H_2O}\)	—	—	—	—	—	—
\(10^{10}\ \mathrm{eV}\)	Pb	0.046	0.014	0.0057	0.0038	0.0012	\(0.34 \cdot 10^{-4}\)
\(10^{10}\ \mathrm{eV}\)	\(\mathrm{H_2O}\)	0.025	0.007	0.0022	0.0011	0.0003	—
\(10^{12}\ \mathrm{eV}\)	Pb	0.047	0.015	0.0060	0.0042	0.0013	\(0.54 \cdot 10^{-4}\)
\(10^{12}\ \mathrm{eV}\)	\(\mathrm{H_2O}\)	0.028	0.009	0.0036	0.0024	0.0008	\(0.29 \cdot 10^{-4}\)

kinetic energy \(E_j\), by heavy particles only when their energy exceeds \(8NE_j\). A distribution of the form \(\frac{1}{N}\) for ionization showers consisting of more than \(N\) particles (31) is obtained, according to (15), because the energy distribution of secondary electrons at energies greater than \(E\) has the form \(\frac{1}{E}\) (§ 9 B).

The mean number of electrons that are in equilibrium with a heavy electron of energy \(E\) will, by (31), be approximately equal to

\[ \int\limits_0^{\frac{E}{8E_j}} \frac{-dQ(NE)}{dN}\,N\,dN = 0.03\ln\frac{E}{8E_j} \quad (\text{for } \mathrm{H_2O}), \tag{32} \]

which is in rough agreement with the data of Table 4. The frequency of ionization showers in other substances may be obtained by multiplying formulas (31) and (32) for water by

\[ \frac{0.8\cdot 10^9\,\mathrm{eV}}{E_j}\,\frac{1}{Z} \tag{33} \]

(\(Z\) is the atomic number and \(E_j\) the critical energy of the substance).

CHAPTER III. THEORY OF HEAVY PARTICLES

13. Heavy electrons and nuclear forces

In the introduction (Chapter 1) the reasons were indicated which make it possible to assert that the penetrating component of cosmic rays consists chiefly of a new kind of elementary particle, whose mass is about 160 electron masses. On the one hand, it cannot be assumed that the penetrating component consists of ordinary electrons, since the latter, according to theory and experimental data, form cascade showers and cannot singly pass through thick layers of matter. Particles of greater mass have no large radiation losses, since the intensity of radiation is inversely proportional to the square of the rest mass of the radiating particle. On the other hand, it follows from ionization measurements that particles, at least with momentum \(pc < 7\cdot 10^8\,\mathrm{eV}\), cannot be regarded as protons, since protons with so small a momentum ionize considerably more strongly than electrons. This argument, together with the already described photographs of heavy elements in a Wilson chamber, makes the hypothesis of the existence of new particles highly reliable. As to the question whether all particles of this kind have the same mass or, on the contrary, whether the penetrating component contains particles of different masses, experiments so far give no answer. Nor has it yet been possible to decide what part of the penetrating particles with very high energies consists of protons. However, all existing

On the Theory of Cosmic Radiation

Experiments carried out up to now, in which the mass of the penetrating particles was determined, can be reconciled with the assumption that the penetrating component consists mainly of a definite kind of particle, whose rest mass is about 160 electron masses, and that, in addition to these particles, it also contains protons and neutrons in comparatively small numbers.

If this assumption is not made, then as yet there exists no theoretical point of view which could give any indications concerning the behavior of the penetrating particles. If, however, one assumes that there exists a definite kind of particle with a rest mass approximately equal to 160 electron masses, then this suggests the idea of connecting these particles with the theory of nuclear forces, which was proposed in 1935 by YukawaY 4, 5, 6, 7, and developed by him and by a number of other authorsF 3, K 1, B 7, B 8, W 3.

Yukawa predicted in this theory the existence of particles of a definite kind and indicated the role of these particles in the general connection of nuclear forces, β-decay, etc. Although it would perhaps be premature to speak of a definite confirmation of Yukawa’s theory by the discovery of heavy particles, it seems natural to analyze the experimental data on the penetrating component of cosmic rays from the point of view of this theory. Therefore, first of all, we shall set forth the basic ideas and most important results of Yukawa’s theory. Yukawa’s theory aims at establishing as close an analogy as possible between electric forces and the forces binding protons and neutrons in the nucleus. Yukawa introduces the field of the forces of the heavy particles forming nuclei, which is described, like the electric field, by a certain wave function satisfying a second-order differential equation. Nuclear forces differ, however, from electric forces in that they have a finite range of action, of the order of the classical electron radius

\[ r_0 = 2.81 \cdot 10^{-13}\ \text{cm}. \]

Therefore, instead of the equation of electrostatics

\[ \Delta \varphi = 0 \]

Yukawa introduces another equation

\[ \Delta \varphi - k^2 \varphi = 0. \]

This equation leads to a potential of the form \(\frac{e^{-kr}}{r}\). The constant \(k\) determines the sphere of action of the nuclear forces and has the order

\[ k \sim \frac{1}{r_0}. \]

Generalizing the equation for the potential into a wave equation, we obtain

\[ -\frac{1}{c^2}\frac{\partial^2 \varphi}{\partial t^2} + \Delta \varphi - k^2 \varphi = 0. \tag{34} \]

The last equation is the wave equation for particles with rest mass \(P_2\), \(\mu=\frac{h k}{c}\). Thus, owing to the finite sphere of action of nuclear forces, in Maxwell’s theory, instead of light quanta there appear particles with rest mass equal to \(\frac{h k}{c}\).

This rest mass is, in order of magnitude, the same as the mass of the heavy electron (about 160 electron masses), as follows from the few experiments available so far. A further distinction between the electric field and the field of nuclear forces must arise because the field of nuclear forces leads to exchange forces between the neutron and the proton. Yukawa achieves this by assuming that the particles corresponding to the nuclear field are charged. Since the charge as a whole must be conserved, the emission of a Yukawa particle is connected with a simultaneous change of the charge of the emitting particle.

From these fundamental conceptions of Yukawa’s theory it follows that, if the heavy electrons observed in cosmic radiation can be identified with Yukawa particles, then, unlike all other charged elementary particles, they must obey Bose–Einstein statistics and possess integral spin. Owing to the close analogy that exists between Yukawa particles and photons, one may expect that Yukawa particles have spin equal to unity. The theory of such particles was developed by Baba in \({}^{5,6}\), by Fröhlich, Heitler, and Kemmer in \({}^{3}\), and by Yukawa in \({}^{7}\).

For reasons of symmetry there must exist positive and negative Yukawa particles, and the absolute value of their charge must amount to one elementary quantum of electric charge.

If the forces between the particles composing nuclei do not depend on charge, as might have been expected on the basis of the experiments of Tuve, Hafstad, and Heydenburg, then it must be assumed that there also exists an uncharged Yukawa particle. However, no firm confirmation of this assumption, whether experimental or theoretical, is as yet possible.

14. Decay of Heavy Electrons

Yukawa further suggested that there also exists a certain interaction of the nuclear field with light particles (electrons and neutrinos). For example, an electron of very high energy, when deflected by the nuclear field, should emit a negatively charged Yukawa particle, while at the same time turning into a neutrino.

A consequence of such an interaction is that the Yukawa particle by itself (without interaction with matter) can decay into an electron and a neutrino. Thus this particle is radioactive with respect to \(\beta\)-decay. Its lifetime is connected with the magnitude of the above-mentioned interaction with the field of light

particles and can be estimated from the calculation of the ordinary β-radioactivity of atomic nuclei. The latter is explained in Yukawa’s theory by means of the assumption that the neutron can transform into a proton with the simultaneous appearance of a virtual particle, which in turn simultaneously decays into an electron and a neutrino. This picture again leads, in general outline, to Fermi’s theory of β-decay and makes it possible to compute the mean lifetime of the heavy electron. According to Yukawa it is

\[ \tau = \frac{1}{2}\cdot 10^{-6}\ \text{sec}. \]

Since the spontaneous decay of heavy electrons is important in considering the behavior of the penetrating component, it should be described somewhat more precisely. The law of conservation of energy and momentum requires that, in the decay of a Yukawa particle at rest into an electron and a neutrino, both light particles fly apart in opposite directions with equal momenta. The sum of the energies of both light particles must be equal to the rest energy of the heavy electron. Owing to the negligible rest mass of the electron and the vanishingly small rest mass of the neutrino, the kinetic energy of the electron and neutrino is

\[ \frac{\mu c^{2}}{2} \]

(where \(\mu\) denotes the mass of the heavy electron). Thus the radioactivity of heavy electrons differs from the β-radioactivity of ordinary nuclei in that, in their decay, an electron with a strictly definite energy of about 40 MeV must appear. In this case the emission of electrons (or neutrinos) will occur with equal probability in all directions.

The decay of a heavy electron will often occur when it possesses a considerable kinetic energy. In this case the momentum of the emitted electron will undergo a Lorentz transformation. Suppose that the heavy electron moves in the direction of the \(X\) axis with velocity \(\beta c\). The angle which the direction of emission of the electron makes with the \(X\) axis, measured in the coordinate system in which the heavy electron is at rest, shall be denoted by \(\varphi\).

Then for the energy of the electron and for the components of its momentum, parallel and perpendicular to the direction \(x\) in the rest system of the heavy electron, we obtain the relations

\[ E'=\frac{\mu c^{2}}{2}; \qquad p'_{\parallel}=\frac{\mu c}{2}\cos\varphi; \qquad p'_{\perp}=\frac{\mu c}{2}\sin\varphi . \]

In the observer’s system the following equalities hold:

\[ E=\frac{\mu c^{2}}{2}\, \frac{(1+\beta\cos\varphi)}{\sqrt{1-\beta^{2}}}; \qquad p_{\parallel}=\frac{\mu c}{2}\, \frac{(\beta+\cos\varphi)}{\sqrt{1-\beta^{2}}}; \qquad p_{\perp}=\frac{\mu c}{2}\sin\varphi . \]

Expressing in these formulas the velocity of the heavy electron in terms of its momentum \(P\), we obtain

\[ E=\frac{c}{2}\left(\sqrt{(\mu c)^2+P^2}+P\cos\varphi\right); \]

\[ p_{\parallel}=\frac{1}{2}\left(P+\sqrt{(\mu c)^2+P^2}\cos\varphi\right);\qquad p_{\perp}=\frac{\mu c}{2}\sin\varphi. \tag{35} \]

If the energy of the heavy electron is much greater than its rest mass, then in most cases in radioactive decay the electron will be emitted almost in the direction of flight of the heavy electron. At the same time its momentum in this direction can, with almost equal probability, take all values between 0 and \(P\).

If we wish to find the probability of decay \(1/\tau\) of the heavy electron after it has traversed a certain path, i.e., if we are interested in the probability of decay per \(1\ \mathrm{cm}\), then, as Bhabha emphasized, we must take into account the change of time that follows from the theory of relativity.

Then for the probability of decay \(w\) per centimeter one can obtain the value

\[ w=\frac{\mu}{\tau P}. \tag{36} \]

15. Secondary processes caused by heavy electrons

The theory of the interaction between protons, neutrons, and heavy electrons according to Yukawa is similar to Fermi’s theory of the interaction between light and heavy particles \(F^{1}\). It leads to certain consequences that are important in considering Hoffmann showers (§§ 23, 24). Namely, the interaction between the three named particles turns out to be the greater, the greater the energy of the colliding particles \(H^{3}\). Hence it follows that, when in a collision there is available an energy considerably exceeding \(10^{8}\ \mathrm{eV}\), this energy will, generally speaking, be used to produce in a single act a number of particles whose energies will lie in the region of several \(10^{8}\ \mathrm{eV}\) \(H^{3}\). Such an explosive shower will, according to Yukawa’s theory (in contrast to Fermi’s theory of \(\beta\)-decay), contain mainly protons, neutrons, and heavy electrons. Photons and ordinary electrons can be formed only with probabilities reduced, respectively, by factors of

\[ \frac{e^{2}}{\hbar c} \quad \text{and} \quad \left(\frac{e^{2}}{\hbar c}\right)^{2}. \]

No more precise information about these explosions can be obtained from Yukawa’s theory, since the appearance of explosions precisely marks the boundary up to which the modern quantum theory is legitimately applicable \(H^{3}, H^{5}, H^{8}\). In outlining the picture of those processes which, according to these theoretical Vorstellungen, must occur when a heavy electron passes through

matter, one can arrive approximately at the following conclusions. First of all, a heavy electron acts on the surrounding matter by virtue of its charge. It ionizes and is therefore slowed down, like an ordinary electron or proton. In this case, for the energy losses per \(1\ \mathrm{cm}\) of path the usual formula (2) will certainly be valid, and for the range as a function of energy—formula (11). Further, in passing through atomic nuclei a heavy electron may enter into interaction with protons and neutrons. This interaction cannot be described by classical forces, but is similar to the interaction between photons and electrons. A heavy electron, therefore, may first be absorbed by a nucleus, which is possible only with the simultaneous transformation of a neutron into a proton (as in the photoelectric effect). Then the heavy particle may be scattered, and its energy, generally speaking, also changes in this process (as in the Compton effect). At a high energy of the heavy electron the latter process will occur more often than the former. In absorption, in order to satisfy the laws of conservation of energy and momentum, the presence of at least two interacting heavy particles is necessary. On the contrary, one heavy particle situated in empty space can already be scattered.

If the energy of the heavy electron becomes of the order of \(10^8\ \mathrm{eV}\), then the effective scattering cross section has the order of magnitude \(10^{-26}\ \mathrm{cm}^2\).

Further, there exist such processes which arise owing to the interaction of nuclear and electric forces, to which Heitler[^8] drew attention. The following may serve as a simple example of such processes: a negative heavy electron collides with a proton. The latter is transformed into a neutron, emitting a quantum of light in the process. The effective cross section of such a process at a particle energy of Yukawa’s of the order of \(10^8\ \mathrm{eV}\), according to Heitler’s estimate, amounts to \(10^{-27}\ \mathrm{cm}^2\).

Finally, if a heavy electron with an energy considerably exceeding \(10^8\ \mathrm{eV}\) collides with a proton or neutron, then, generally speaking, the explosions described above will occur. These explosions may produce, in addition to heavy electrons, as we have already mentioned above, protons and neutrons. Moreover, they may also be produced by photons and electrons, but the probability of the latter processes is reduced, respectively, by a factor of

\[ \frac{e^2}{\hbar c} \quad \text{and} \quad \left(\frac{e^2}{\hbar c}\right)^2 . \]

The effective cross section of explosive showers should be of the order of \(10^{-25}\ \mathrm{cm}^2\). However, it is possible that this cross section again decreases as the energy of the colliding particles increases. Exact information about this cannot be obtained from the existing theory. In all processes known up to now, the losses are to a considerable extent proportional to the masses. This means that the energy losses per \(1\ \mathrm{g}/\mathrm{cm}^2\) must be approximately the same in all substances. Finally, a heavy electron may spontaneously decay into an electron and a neutrino. This process is completely independent of matter,

in which a heavy electron flies, and there can be no question of proportionality between absorption and the mass of the substance. It may, to be sure, be thought that alongside the spontaneous decay there exists still another decay, which is induced by interaction with other particles. However, simple theoretical estimates make it unlikely that this induced decay plays an appreciable role in comparison with spontaneous decay.

16. Secondary effects caused by fast protons and neutrons⁴

In the penetrating component of cosmic rays, protons and neutrons also play a role. To be sure, at the present time it is hardly possible to state what part of the penetrating cosmic rays consists of protons of high energy. In any case, however, experiments always show the presence of heavy particles of small energies R ⁶, A ³. Thus the existence of protons and neutrons in cosmic rays may be regarded as certain. These heavy particles may, according to the theoretical considerations set forth above, first of all cause the following processes: concerning the ionization caused by protons, enough has been said in the introduction. In collisions with atomic nuclei, protons and neutrons may interact with nuclear particles by means of exchange forces. In passing through atomic nuclei, heavy particles behave (in contrast to Yukawa particles) quite analogously to electrons when passing through matter. Owing to the presence of nuclear forces, having a range of action of the order of \(2.8 \cdot 10^{-13}\ \mathrm{cm}\), a proton of high energy, flying through a nucleus, may transfer energy to the nearest protons or neutrons. In this way it forms secondary particles with a definite energy distribution. The energy thereby transferred will for the most part be expended, according to Bohr’s theory B ²¹, in heating the nucleus with which the heavy particle collides. From the heated nucleus protons or neutrons will be emitted (analogously to the process of evaporation), which later may appear as secondary particles.

The energy given up to nuclei by a proton or neutron moving with velocity \(\beta c\) is expressed approximately by the following formula H ⁴:

\[ \frac{\partial E}{\partial x} \approx -\frac{Mc^{2}}{r_{0}\beta^{2}} \cdot 6 \cdot 10^{-3} \tag{37} \]

(\(r_{0}\)—classical radius of the electron,
\(M\)—mass of the proton).

For the range \(R\) of heavy particles as a function of their kinetic energy \(E\), the following expression may be obtained H ⁴ (cf. 2,11):

\[ R = r_{0}\frac{E^{2}}{Mc^{2}(Mc^{2}+E)} \cdot 1.7 \cdot 10^{2}. \tag{37a} \]

It follows from the last formula that protons and neutrons moving with a velocity close to the velocity of light transfer, on average, very little energy (about 20 MV) to nuclei.

On the contrary, at lower velocities heavy particles, for example protons with energy \(3\cdot 10^8\) eV, can lose all their energy in passing through an atomic nucleus, as was just indicated. Intranuclear particles that receive sufficiently large energy from the incident proton or neutron can leave the nucleus, appearing in the phenomenon as secondary particles. The energy distribution of these secondary particles is practically independent of the energy of the incident particle, provided only that the latter is considerably greater than the energy of the secondary particles. It also hardly depends on the size of the nucleus from which the secondary particles are emitted. It turns out that the number of secondary particles \(n(E)\) with energy greater than \(E\) approximately has the form[^4]

\[ n(E) \approx \mathrm{const}\cdot e^{-78\frac{E}{Mc^2}}. \tag{38} \]

Alongside these processes, which are very analogous to the ordinary processes of ionization, there are also other processes that may be compared with the bremsstrahlung of ordinary electrons and which, especially at high energies of the colliding particles, must play the principal role. A proton of very high energy, on colliding with another heavy particle, may emit a Yukawa particle and thereby turn into a neutron. At sufficiently high energy it may create in a single act several heavy electrons and other heavy particles, i.e., produce the above-described showers. The effective cross section of these processes, like the cross section of the corresponding processes with Yukawa particles, must be of the order of \(10^{-26}\ \mathrm{cm}^2\).

(To be continued in the next issue)

1. Cf. H 2. ↩