Recent Research in the Field of Cosmic Rays

V. G. Levich, Moscow

§ 1. Introduction

Cosmic rays and the phenomena associated with them constitute one of those areas of physics to which, over the last 10–15 years, the attention of researchers has been especially drawn. The past year has been marked by a number of significant successes, to a brief survey of which the present article is devoted.

In order to assess the significance of these successes, it is necessary to characterize the state of the question at the beginning of the current year. The work of numerous investigators had provided a whole series of valuable data on cosmic rays. These include, first of all: the determination of the composition of the primary rays and the study of showers. The existence of the latitude effect, i.e. the variation of intensity with geographic latitude, very convincingly indicates that the primary cosmic rays include charged particles deflected by the earth’s magnetic field. Direct investigations with Geiger–Müller counters and the work of various authors with the Wilson chamber led to the same conclusion. These, as well as certain other facts—for example, the approximate independence of the transition effect from latitude, and the presence of east–west asymmetry—finally established that at least the main mass of primary rays consists of charged particles.

If there are neutral particles in the primary rays, then the ionization caused by them at sea level is insignificant in comparison with the ionization produced by charged particles. It was further established that the primary rays include particles of both signs, the number of positive ones being somewhat greater than the number of negative ones.

Finally, it was assumed that the primary rays consist of positrons, electrons, and protons, and also, in small quantity, of photons.

The energies of cosmic particles were estimated from their deflection by the earth’s magnetic field. It can be shown that electrons reaching

striking the surface of the earth in the vertical direction, at the equator must have an energy of no less than \(20\cdot 10^9\) eV. The energies of the particles were also determined from their deflection by the magnetic field in Wilson’s chamber. The results of these measurements coincided with the results obtained by the first method, and it was considered established that cosmic particles have energies of the order of \(10^9\) eV, but that individual particles with greater energies, up to \(10^{10}\)—\(10^{11}\) eV, are encountered.

A very important question is that of the energy losses of electrons of such high energies in passing through matter. The measurements of Anderson and Neddermeyer led to the following conclusion: up to energies of the order of \(70\cdot 10^6\) eV, the energy losses show good agreement with the theoretical values obtained by Bethe and Heitler. However, for high energies this agreement is sharply violated, and the energy losses become considerably smaller than follows from the theory.

From this it was concluded that in the region of energies greater than \(150\)—\(300\,mc^2\), quantum mechanics is no longer applicable; to explain the phenomena occurring in this energy region, a new theory must be created.

For a long time, another remarkable phenomenon—showers caused by cosmic rays—appeared equally inexplicable from the point of view of quantum theory.

Blackett and Occhialini, Skobeltsyn, Anderson, and others, in studying cosmic rays, observed the appearance of entire groups of particles, usually issuing from one point of the substance surrounding the chamber. This phenomenon was given the name of showers.

Photographs show the existence of peculiar “foci” of showers—separate centers giving bundles of tracks diverging approximately from one point. The showers consist, for the most part, of light particles—electrons and positrons of high energies; heavy, strongly ionizing particles are rarely encountered. The number of particles in a shower varies from three to five particles up to several hundreds.

The occurrence of showers was ascribed to some special “explosions” of nuclei, leading to their disintegration into a large number of particles.

The principal difficulty in explaining the origin of showers consisted in the fact that, from the point of view of quantum mechanics, the occurrence of a large number of secondary particles in a single act was exceedingly improbable.

According to ordinary quantum mechanics, the energy of interaction between a particle (an electron, a positron) and the field is proportional to the charge \(e\) of this particle.

This interaction may lead to the formation of new particles (pair production). The probability of the occurrence of a new particle is a quantity having the dimension \(\mathrm{sec}^{-1}\) and proportional to the square of the matrix element of the interaction energy, i.e. proportional to \(e^2\).

The probability of the simultaneous formation of two particles is proportional already to \(e^4\) and likewise has the dimension \(\mathrm{sec}^{-1}\). Finally, the probability of the simultaneous formation of \(n\)-particles is proportional to \((e^2)^n\).

We see that the probability of particle formation is some function of \(e^2\). Since, however, an expression for a probability, for dimensional reasons, cannot contain \(e^2\) by itself, the charge must enter only together with other quantities forming with it a dimensionless combination. The only dimensionless quantity containing only \(e^2\) and universal constants is the combination \(\dfrac{e^2}{\hbar c}\). Therefore the probabilities of particle formation must be proportional to various powers of \(\dfrac{e^2}{\hbar c}\).

In particular, the probability of the simultaneous formation of \(n\)-particles must be proportional to \(\left(\dfrac{e^2}{\hbar c}\right)^n\). Substituting the numerical values of the universal constants, we find that the probability of the simultaneous formation of \(n\)-particles is proportional to \(\left(\dfrac{1}{137}\right)^n\). Thus, indeed, the probability of emitting a large number of particles turns out to be very small.

Heisenberg\(^{1}\) attempted to overcome this difficulty by assuming that the interaction leading to the formation of a shower is not electromagnetic in character, but is similar to the interaction between heavy and light particles that leads to \(\beta\)-decay.

According to Heisenberg, the process of shower formation itself is a kind of higher-order \(\beta\)-decay with the simultaneous emission of several electrons and positrons. According to Fermi, \(\beta\)-decay may be represented by the scheme

\[ \text{neutron} \to \text{proton} + \text{electron} + \text{neutrino}. \]

Then the shower may be represented in the form of a series of transformations:

\[ \begin{aligned} &\text{neutron} \to \text{proton} + \text{electron} + \text{neutrino} \to \\ &\to \text{neutron} + \text{positron} + \text{antineutrino} + \\ &+ \text{electron} + \text{neutrino} \to \text{proton} + \\ &+ 2\,\text{electrons} + \text{positron} + 2\,\text{neutrinos} + \text{antineutrino} \to \\ &\to \ldots \to \text{neutron} + n\,\text{positrons} + n\,\text{electrons} + \\ &+ n\,\text{neutrinos} + n\,\text{antineutrinos}. \end{aligned} \]

Thus the shower differs from \(\beta\)-decay only in the number of emitted particles or, in other words, represents the same \(\beta\)-decay of higher order.

The expression for the energy of interaction between heavy and light particles in Fermi’s theory of \(\beta\)-decay contains a certain

a new universal constant \(f\), having the dimension \(\mathrm{cm}^2\). Accordingly, the probability of emitting one particle must be proportional to \(f^2\), two—to \(f^4\), and, finally, \(n\) particles—to \((f^2)^n\).

Since the probability must have the dimension \(\mathrm{sec}^{-1}\), \(f\) must occur only in the form of a dimensionless combination with other physical quantities. Such a dimensionless combination is \(fk^2\), where \(k=\dfrac{2\pi}{\lambda}\) is the wave vector of the emitted particles. Thus the probability of simultaneous emission of particles must be proportional to \((fk^2)^{2n}\).

If particles of very high energies are emitted, for which \(k\) is also very large, the probability of emitting a large number of particles may become no smaller than the probability of emitting a small number of particles.

With the aid of certain additional artificial assumptions about the character of the interaction between particles, Heisenberg succeeded in obtaining qualitative agreement with experiment. Recently, however, Pauli \(^2\) showed that the use of higher approximations in Fermi’s theory, as carried out by Heisenberg, is illegitimate. The higher approximations turn out to be divergent and give infinities for transition probabilities and other quantities. Therefore Heisenberg’s theory of showers must be recognized as unsatisfactory. As we shall see below, at present a new, so-called avalanche theory of showers has been developed, in which the difficulties connected with the simultaneous emission of several particles have been eliminated.

§ 2. Bremsstrahlung and Pair Production

Before proceeding to the exposition of the new theory of showers, one should become acquainted with two closely related processes: bremsstrahlung and the production of pairs by \(\gamma\)-rays.

Let us consider an electron flying with a very high velocity in the field of a nucleus with charge \(Ze\). Owing to the interaction between the nucleus and the electron, there exists a certain probability that the electron will pass from a state with momentum \(p_0\) and energy \(E_0\) into a state with momentum \(p\) and energy \(E\), emitting a photon in the process. This phenomenon is called bremsstrahlung (Bremsstrahlung). It is necessary to find the probability of such a process.

An exact expression for the probability of bremsstrahlung was obtained by Bethe and Heitler and by other authors with the aid of quantum perturbation theory. Their calculations are of a rather complicated character. It is possible, however, from quite intuitive considerations, to obtain the correct order of magnitude by using the method proposed by Williams \(^4\) and Weizsäcker \(^5\).

We shall first carry out an approximate calculation in the case when the classical approximation is applicable. Here by the word “classical” is meant relativistic, not quantum-mechanical.

Let the electron fly past in the field of the nucleus at a distance \(r\) from it. The quantity \(r\) is called the impact parameter. We shall assume the velocity of the electron \(v_0\) to be almost equal to the speed of light \(c\), so that

\[ \gamma=\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}\gg 1 . \tag{1} \]

We shall take the direction of flight of the electron as the \(z\)-axis. Owing to the interaction of the passing electron with the field of the nucleus, it will be deflected from the initial direction of motion, i.e., will experience an acceleration. This acceleration must be accompanied by radiation. Since the formula for the radiation of a fast electron is very cumbersome, it is more convenient to introduce a new coordinate system in which the electron is at rest, while the nucleus moves with velocity \(v_0\). The distance between the nucleus and the electron remains the same in both coordinate systems, since it is perpendicular to the \(z\)-axis and remains invariant. The system in which the nucleus moves we shall call the system of the nucleus, and all quantities measured in this system will be provided with primes. The old coordinate system, in which the nucleus was at rest, will be called the system of the electron.

The interaction between the electron and the nucleus is very brief, owing to the large velocities. One may approximately assume that the time of interaction in the electron system \(\tau\) will be

\[ \tau \simeq \frac{r}{c}. \tag{2} \]

In the system of the moving nucleus its field undergoes Lorentz contraction. Therefore the time of interaction in the nucleus system will be shortened and will be equal to

\[ \tau'=\frac{\tau}{\gamma}=\frac{r}{\gamma c}. \tag{2′} \]

The field acting on the electron will have, roughly speaking, an appreciable value in a region whose dimensions are proportional to \(r\). In the nucleus system, accordingly, the dimensions of this region will decrease in the ratio \(\frac{r}{\gamma}\).

The component of the electric field in the direction \(r\) (perpendicular to the \(z\)-axis), upon transformation to the nucleus system, will be transformed from \(E_r\) into

\[ E'_r=E_r\gamma=\frac{Ze}{r^2}\gamma \tag{3} \]

The acceleration acquired by the electron in the nucleus system is equal to

\[ W=\frac{eE'_r}{m}=\frac{Ze^2}{mr^2}\gamma . \tag{4} \]

Since in the nuclear system the electron was initially at rest, the energy radiated by it per unit time will be

\[ \mathcal{E}'=\frac{e^2 W'^2}{c^3} =\frac{Z^2 e^6}{m^2 c^3 r^4}\gamma^2 . \tag{5} \]

The total energy radiated by the electron, for a given \(r\), is

\[ \Delta E'=\mathcal{E}'\tau' =\frac{Z^2 e^6}{m^2 c^4 r^3}\gamma =\frac{Z^2 e^2 r_0^2}{r^3}\gamma , \tag{6} \]

where \(r_0=\dfrac{e^2}{mc^2}=2.8\cdot 10^{-13}\) is the classical radius of the electron.

In the electron system the total energy radiated for a given \(r\) will be

\[ \Delta E=\Delta E'\gamma=\mathcal{E}'\tau'\gamma =\frac{Z^2 e^2 r_0^2}{r^3}\gamma^2 . \tag{6'} \]

The probability that the electron will pass at a distance between \(r\) and \(r+dr\) is proportional to \(r\,dr\) (the area of the corresponding ring). Therefore, for the total radiated energy over all values of the collision parameter \(r\), we have

\[ E=\int \Delta E\,r\,dr =Z^2 r_0^2 e^2 \gamma^2\int \frac{dr}{r^2}. \]

Dividing the total radiated energy by the initial energy of the flying electron \(E_0\), we obtain the effective cross section for bremsstrahlung radiation

\[ \psi_{\mathrm{rad}.}=\frac{E}{E_0} =\frac{Z^2 r_0^2 e^2 \gamma^2}{E_0} \int \frac{dr}{r^2}. \tag{7} \]

The integration should be carried out over all values of the collision parameter, i.e. from \(0\) to \(\infty\). Up to now, however, we have treated the electron classically. Now its quantum properties must be taken into account. Namely, the electron constitutes a certain wave packet. The initial energy of this wave packet is determined only with a certain degree of accuracy, say with accuracy up to \(\Delta E_0\). The uncertainty in the energy \(\Delta E_0\) cannot be greater than \(mc^2\), since in that case the number of electrons becomes uncertain because of pair creation. Therefore let us assume that the energy is determined with an uncertainty \(\Delta E_0\sim mc^2\). Then the dimensions of the wave packet must be no smaller than

\[ \Delta x=\frac{\hbar}{\Delta p_0}\sim \frac{\hbar c}{\Delta E_0}=\frac{\hbar}{mc}. \]

If the uncertainty in the value of the initial energy were still smaller, the minimum dimensions of the wave packet, obviously, would have to be still larger.

The classical treatment is admissible when the external field does not change appreciably over the extent of the wave packet. Thus

as the field has an appreciable magnitude over a distance \(\dfrac{r}{\gamma}\), it is necessary that

\[ \frac{r}{\gamma} > \Delta x = \frac{\hbar}{mc}. \tag{8} \]

Therefore formula (7) is valid only up to \(r \gg \dfrac{\hbar}{mc^{2}}\gamma\). Substituting this smallest value of \(r\) as the lower limit of the integral in (7), we have

\[ \psi_{\mathrm{rad}} = \frac{Z^{2} r_{0}^{2} e^{2} \gamma^{2}}{E_{0}} \int_{\frac{\hbar}{mc}\gamma}^{\infty}\frac{dr}{r^{2}} = Z^{2} r_{0}^{2}\frac{e^{2}}{\hbar c} = \bar{\psi}, \tag{9} \]

where \(E_{0}=mc^{2}\gamma\) has been substituted.

We see that the effective cross section for radiation is obtained as constant and proportional to \(Z^{2}\). A direct transfer of the preceding classical calculation to the quantum-mechanical case is difficult. This difficulty can, however, be avoided by regarding the field of the nucleus as a certain radiation field, represented by a superposition of plane waves propagating with the velocity of light. Of course, such a treatment is possible only approximately; but what is important for us are the spatial and temporal variations of the field only in the small region in which the electron is located, and it can be shown that the error thereby committed is of order \(\dfrac{1}{\gamma}\) and is permissible for \(\gamma \gg 1\). Then the radiation of the electron caused by its interaction with the nucleus should be treated as the scattering of the primary waves according to the Klein–Nishina formula.

The effective cross section for bremsstrahlung radiation can be obtained as follows. For each value of the collision parameter, the field of the nucleus passing by the stationary electron is expanded into a Fourier integral. Each component of the expansion, representing a plane wave of definite frequency \(\nu\), is scattered and passes to a frequency \(\nu'\), related to \(\nu\) by the known formula

\[ \nu'=\frac{mc^{2}\nu}{mc^{2}+\nu(1-\cos\theta)}. \tag{10} \]

Then a Lorentz transformation is made to the electron system, and the scattered rays of the same frequency are added.

The resulting formula must be integrated over all scattering angles and all values of the collision parameter. It turns out that an essential role is played not by all primary frequencies (components of the expansion), but only those for which

\[ \nu \ll \frac{mc^{2}}{\hbar}. \]

Large frequencies, however, may be neglected, since the scattering coefficient of hard \(\gamma\)-rays is small in comparison with the scattering coefficient of soft rays.

Calculations give for the effective cross section

\[ \psi_{\mathrm{rad}}=2Z^2r_0^2\left(\frac{e^2}{\hbar c}\right)\lg\gamma=\psi\lg\gamma . \tag{11} \]

The exact calculations of Bethe and Heitler\(^6\) for high energies give

\[ \psi_{\mathrm{rad}}=4\psi\left(\lg 2\gamma-\frac{1}{3}\right). \tag{11'} \]

We see, therefore, that even the classical calculation [formula (9)] gives the correct order of magnitude for \(\psi_{\mathrm{rad}}\).

Fig. 1. Effective cross section \(\psi_{\mathrm{rad}}\) for losses of energy to radiation, in units
\[ \bar{\psi}=\frac{r_0^2Z^2}{137}. \]
The dashed lines show energy losses to ionization in the same units. The left upper curve represents \(\psi_{\mathrm{rad}}\) without taking screening into account.

The expressions for \(\psi_{\mathrm{rad}}\) (11) and (11′) were obtained under the assumption of a Coulomb interaction between the nucleus and the electron. It turns out, however, that the screening role of the atomic electrons becomes significant at high energies of the incident electrons \((E_0\gg mc^2)\) and for heavy atoms. Taking screening into account with the aid of the Thomas–Fermi model, one can obtain for the effective cross section of bremsstrahlung radiation

\[ \psi_{\mathrm{rad}}=\psi\left\{4\lg\left(\frac{183}{Z^{1/3}}\right)+\frac{2}{9}\right\}. \tag{12} \]

where \(\psi\) is, as before, equal to \(Z^{2}r_{0}^{2}\dfrac{e^{2}}{\hbar c}\), or \(\psi = \dfrac{10^{-25}}{137}\). The cross section for radiation \(\psi_{\text{rad}}\) is plotted on a logarithmic scale as a function of the initial energy in Fig. 1 for various substances. In the same figure, dashed curves, in the same units, show the curves of energy loss in inelastic collisions (ionization). The solid curve on the left represents \(\psi_{\text{rad}}\) without allowance for screening; in our units it is the same for all elements.

We see that at small energies the dominant role is played by ionization losses. But at a certain critical energy \(\varepsilon\), different for each substance, the curves intersect, and the losses to radiation and to ionization become of the same order.

As is seen from Fig. 1, for lead \(\varepsilon \sim 20mc^{2}\), and for water \(\varepsilon \sim 300mc^{2}\).

Fig. 2. Energy losses per unit path in lead (in units of \(mc^{2}\)) as a function of the initial energy \(E_{0} — mc^{2}\), on a logarithmic scale for an electron and a proton. The dashed curves represent the component parts of the total energy loss of the electron.

At energies exceeding the critical one, energy losses to ionization become insignificant, and bremsstrahlung begins to play the main role.

The role of screening, as is seen from formula (12), consists in the fact that, owing to its existence, the cross section does not increase without bound with increasing electron energy, but tends to a quite definite limiting value for each substance. In the region of very large energies \((E_{0} \gg mc^{2})\) it may be regarded as constant [see formula (12)].

In Fig. 2, again on a logarithmic scale, the energy losses of an electron and a proton in lead are plotted. The dashed curves represent the component parts of the electron losses—to radiation and

ionization. Here we have indicated only that the relation between the role of ionization and that of radiation at different energies appears especially clearly. A heavy particle (a proton) loses energy only in inelastic collisions and does not radiate at all. It can be shown that the probability of radiation for a proton, owing to its large mass, is utterly negligible. Radiation by a proton becomes appreciable only at energies of the order of \(Mc^2\), where \(M\) is the mass of the proton.

The second process which, as we shall see below, plays a large role in the theory of showers is the formation of pairs by \(\gamma\)-rays.

As is known, according to Dirac’s theory, the formation of a pair may be regarded as the transition of an electron, under the action of some perturbation, from a state with negative energy to a state with positive energy. The “hole” which remains in the completely filled states of negative energy is represented to the observer as a positron.

The perturbation causing such a transition may in particular be a sufficiently hard \(\gamma\)-quantum. The energy of this quantum must exceed \(2mc^2\)—the minimum energy necessary for the transition of an electron from the state with \(E=-mc^2\) to the state \(E=+mc^2\).

It is not difficult to see that, for the simultaneous fulfillment of the laws of conservation of energy and momentum, the existence of a third body is necessary. Indeed, suppose, for example, that the pair is created by a photon with energy \(2mc^2\) and the corresponding momentum, equal to \(2mc\). The energy of the pair formed will be equal to \(2mc^2\), but its momentum will be zero, since both particles are at rest.* The third body may receive or give up the excess momentum. Such a third body may be, for example, a nucleus. Therefore the passage of hard \(\gamma\)-rays \((h\nu>2mc^2)\) through matter must be accompanied by the formation of pairs.

Let us find the effective cross-section for pair formation. For this purpose we shall consider the process inverse to pair formation—the annihilation of the pair in the presence of a nucleus.

In annihilation, an electron moving in the field of a nucleus passes from a state with positive energy into an unoccupied state with negative energy, emitting one \(\gamma\)-quantum. We see that such annihilation differs from bremsstrahlung radiation only in the sign of the energy of the final state. This corresponds to replacing all quantities in the calculations of bremsstrahlung radiation by their complex conjugates. Therefore the course of the calculations for the inverse process—pair formation—does not differ in any way from their calculation in the case of bremsstrahlung radiation.

Neglecting screening, for very hard \(\gamma\)-quanta \((h\nu \gg mc^2)\) one obtains the following expression for the effective cross-section for pair formation\(^6\)

\[ \psi_{\text{pair}} = Z^2 r_0^2 \frac{e^2}{\hbar c} \left( \frac{28}{9}\lg\frac{2h\nu}{mc^2} - \frac{218}{27} \right) = \psi \left( \frac{28}{9}\lg\frac{2h\nu}{mc^2} - \frac{218}{27} \right), \tag{13} \]

where \(\psi\), as before, is equal to

\[ \frac{Z^2 r_0^2}{137}. \]

If screening is taken into account, then in the same case of hard \(\gamma\)-quanta one obtains

\[ \psi_{\mathrm{pair}}=\psi\left(\frac{2^3}{9}\lg 183 Z^{-1/3}-\frac{2}{27}\right). \tag{14} \]

In Fig. 3 the effective cross section for pair production is presented on a logarithmic scale as a function of the energy of the incident \(\gamma\)-quanta. We see that the effective cross section increases rapidly with \(h\nu\), but then tends to a definite limiting value characteristic of the given substance. The solid left-hand curve represents \(\psi_{\mathrm{pair}}\) without taking screening into account.

Fig. 3. Effective cross section for pair production \(\psi_{\mathrm{pair}}\) as a function of the frequency of the incident \(\gamma\)-quanta on a logarithmic scale. Units

\[ \psi=\frac{Z^2 r_0^2}{137}. \]

The dashed curves represent the effective cross section for the Compton effect in the same units. The upper left-hand curve shows the behavior of \(\psi_{\mathrm{pair}}\) without taking screening into account.

For comparison, the cross section of the Compton effect is drawn with a dashed line in the same units. For small \(h\nu\) the probability of pair production proves to be much smaller than the probability of Compton scattering. At a certain critical energy \(\varepsilon\), the two processes become equally probable. For lead the critical energy \(\varepsilon\) is of the order of \(10mc^2\). Finally, at high energy the Compton effect plays practically no role, and \(\gamma\)-quanta only produce pairs.

§ 3. New Particles

The first important success obtained this year is the discovery by Anderson and Neddermeyer\(^3\), among cosmic rays, of new particles which have received the name of semi-heavy particles.

As was mentioned above, the early measurements of energy losses in the passage of fast electrons through matter, carried out by these authors, led to a discrepancy between theory and experiment.

In 1936–1937 Anderson and Neddermeyer[^3] made new control measurements of the losses. They measured the energy losses in the passage of fast particles, forming part of the shower, through a platinum plate 1 cm thick. The plate was placed inside a Wilson chamber controlled by counters.

Fig. 4. Energy loss in 1 cm of platinum

The results of these measurements are presented in Fig. 4. On the abscissa axis are plotted the initial energies of the particles, and on the ordinate axis—the relative losses. The observed losses are indicated by points for both negative and positive particles. We see that all the particles fall quite distinctly into two groups: a group with a large (mean) value of the losses and a group with a smaller value of the losses. The first group should be identified with electrons and positrons. It should, however, be noted that in connection with the new shower theory the meaning attached to the term “energy loss” has changed (see § 4). We shall call this group the soft component (in the sense of large absorptivity, but not in the sense of small energies).

It is natural to try to identify the second, penetrating component with heavy particles, for example, protons. But protons of the corresponding energies should have ionized much more strongly than was observed experimentally. Therefore this supposition falls away. Then only two possibilities remain:

1. The penetrating component also consists of light particles, but quantum mechanics proves inapplicable to them.

2. There exist some new, semi-heavy particles with a charge equal to the charge of the electron (in absolute value), but with a mass greater than that of the electron and smaller than that of the proton. Anderson and Neddermeyer estimate the mass of the semi-heavy particles at \(50 m_e\), where \(m_e\) is the mass of the electron.

With the aid of semi-heavy particles it is possible to explain the observed facts—the great penetrating power and small ionization of the penetrating component.

Of course, the hypothesis of the existence of semi-heavy particles has a number of serious shortcomings. These include, first of all, the fact that the origin of these particles and their relationship to their...

with light particles. Further, it is unclear why they are not observed under ordinary conditions.

However, only with the aid of this hypothesis can a number of observed facts be explained while remaining within the framework of quantum mechanics. Moreover, as we shall see below, the theory of showers has put forward serious arguments in favor of the existence of semi-heavy particles. Therefore, without forgetting, of course, due caution, one may say that at the present time the existence of semi-heavy particles appears very probable.

§ 4. The Avalanche Theory of Showers

The second major success is the construction of the theory of showers. Recently, Heitler and Bhabha^7 and, independently, Carlson and Oppenheimer^8 succeeded in constructing a new theory of showers, excellently explaining almost all the observed phenomena. Moreover, they showed that the existence of showers is a direct consequence of long-known facts—the formation of pairs and bremsstrahlung—and can be explained by ordinary quantum mechanics without any additional hypotheses. Thanks to this, the phenomenon of showers loses the character of an exceptional and mysterious phenomenon, and in a certain sense it is strange that until very recently the origin of showers had not been clarified.

The essence of the new ideas about showers is as follows. Let us consider a very fast electron with an energy much greater than the critical one, which has entered a layer of some substance.

According to the conclusions of the preceding paragraph, owing to the interaction between the electron and the nuclei of the substance, there is a probability that the electron will emit a γ-quantum.

It follows from the theory that the energy of the emitted quantum will be of the same order as the energy of the electron, and the direction of flight must approximately coincide with the direction of motion of the electron. The emitted γ-quantum will therefore have an energy considerably exceeding the critical one. Consequently it will form pairs. Both components of the pair that has formed will still have very large energies. Roughly speaking, the energy of the quantum will be divided in half between the positron and the electron, i.e., each of them will again have an energy greater than the critical one.

The direction of flight of the positron and the electron will be approximately the same as the direction of flight of the photon, as follows from the theory. Since the energy of the electron and positron that have arisen is very great, they in turn must emit γ-quanta. These γ-quanta will create new pairs, and so on.

We see, therefore, that there is produced, as it were, an avalanche of particles flying approximately in the direction of flight of the primary electron. The number of particles in the avalanche must increase rapidly, while the energy falling to the share of each particle must decrease.

Particles which have received an energy less than the critical one will cease radiating and will lose energy through ionization and, in the end

eventually be absorbed. In exactly the same way, photons with energy less than the critical energy will lose energy through the Compton effect and be absorbed. Thus the formation of new particles will cease with time, and the shower will stop growing.

The number of particles in the shower depends on the initial energy of the incident particle and on the thickness of the layer traversed, and may be very large—up to 1000 and more.

Since the ranges of particles and $\gamma$-quanta in matter are very small, the greater part of the shower is produced over a small section of the path; for lead, for example, over a path of about $1\ \mathrm{cm}$ at an initial energy of the order of $10^{10}\ \mathrm{eV}$.

All particles of the shower that has formed will have almost the same direction of flight. Owing to this, a quite negligible error in the measurement of angles is sufficient for all the tracks in a Wilson chamber, obtained from the moving shower, to appear as if they originate from a single point.

Thus the main difficulty connected with the simultaneous formation of many particles is completely removed in the new theory. Showers are formed not in a single act, but in a series of successive acts, and their existence in fact follows directly from previously known facts. The formation of showers turns out to be a kind of interaction of particles with matter characteristic of very high energies, just as ionization is characteristic of comparatively low energies.

The new theory of showers not only explains the origin of showers and the phenomena connected with them, but also makes it possible to obtain a number of interesting consequences. In particular, it may be applied to the study of the composition of cosmic rays.

The quantitative calculations of Heitler and Bhabha, and of Oppenheimer and Carlson, are of a rough, approximate character. More exact calculations, which made it possible to obtain a number of interesting consequences, were recently carried out by Landau and Rumer.^2 We shall now turn to the exposition of the results obtained by them.

Let us consider a fast electron entering a sufficiently thick layer of matter. If its energy $E_0$ exceeds the critical energy $\varepsilon$ characteristic of each substance, then along its path it will create a shower. The shower will contain particles (electrons, positrons) and $\gamma$-quanta of various energies.

Let $\Pi(E)\,dE$ be the number of particles with energy in the interval $E$, $E+dE$ at a distance $x$ from the surface of the substance. Then the number of particles $\Pi(E)$ will change, on the one hand, as a result of the emission of $\gamma$-quanta by particles, and, on the other hand, because of the formation of pairs. In the first process the particles lose energy, and this leads to a decrease of $\Pi(E)$. The second process, on the contrary, increases $\Pi(E)$.

Let also $\Gamma(E)$ be the number of photons with energy in the interval $E$, $E+dE$ at a distance $x$ from the boundary of the substance. $\Gamma(E)$, just like $\Pi(E)$, will experience a twofold tendency: on the one hand, the formation of pairs will decrease $\Gamma(E)$; on the other hand, it will increase at the expense of photons emitted by the particles.

As a result, a certain equilibrium distribution of particles and photons in the shower must be established. We therefore need to find the equilibrium distributions of particles and photons at depth \(x\).

For the rates of change \(\Pi(E)\) and \(\Gamma(E)\) with distance from the surface, equations are obtained that contain unknown functions under the integral sign. These so-called integro-differential equations cannot be solved directly. It proves convenient to introduce, instead of the former energy variable, a special variable \(S\), in which the integral equations reduce to differential ones. This variable \(S\) is related to the former variable by the integral equation

\[ fs=\int_0^\infty f(E)E^S\,dE, \tag{15} \]

where \(f(E)\) is the sign of an arbitrary function of the energy.

In addition, it proves convenient to measure length in dimensionless units \(t\), related to the ordinary length by

\[ t=Ax, \]

where \(A\) is a certain constant of dimension \(\mathrm{cm}^{-1}\), entering into the formulas for the probabilities of bremsstrahlung emission and pair production, whose numerical value is known for each element. In the new variable \(S\), the distribution functions \(\Pi_S(t)\) and \(\Gamma_S(t)\) will have the following meaning: \(\Pi_S(t)\) is the number of particles with \(S\), lying in the interval \(S,\ S+dS\) at depth \(t\); \(\Gamma_S(t)\) is the corresponding quantity for photons. Then, in the new variable, the equations take the form:

\[ \left. \begin{aligned} \frac{d\Pi_S}{dt} &= -A(S)\Pi_S + B(S)\Gamma_S,\\ \frac{d\Gamma_S}{dt} &= C(S)\Pi_S - D(S)\Gamma_S \end{aligned} \right\} \tag{16} \]

Here \(A(S)\) is the operator corresponding to the decrease in the number of particles due to radiation, \(B(S)\) is the operator corresponding to the appearance of new particles through pair production, and \(C(S)\) and \(D(S)\) have an analogous meaning for photons.

The solutions of these equations have the form:

\[ \left\{ \begin{aligned} \Pi_S(t)&=a_S e^{-\lambda_S t}\left(1+b_S e^{-(\mu_S+\lambda_S)t}\right),\\ \Gamma_S(t)&=c_S e^{-\lambda_S t}\left(1+d_S e^{-(\mu_S+\lambda_S)t}\right), \end{aligned} \right. \tag{17} \]

\[ \tag{17'} \]

where \(\lambda_S\) and \(\mu_S\) are the roots of the secular equation

\[ \left| \begin{array}{cc} -A(S)+x & B(S)\\ C(S) & -D(S)+x \end{array} \right|=0. \tag{18} \]

For a sufficiently large penetration depth of the shower one may write, approximately,

\[ \left\{ \begin{aligned} \Pi_S(t)&=a_S e^{-\lambda_S t},\\ \Gamma_S(t)&=c_S e^{-\lambda_S t}. \end{aligned} \right. \tag{19} \]

\[ \tag{19'} \]

It can be shown that \(\lambda_S\) is the coefficient of absorption of particles in matter. Where \(\lambda_S\) is less than zero, the number of particles increases. This corresponds to the formation of new particles in the shower. As soon as \(\lambda_S\) becomes greater than zero, the number of particles begins to decrease. The constants \(a_S\) and \(c_S\) may be found from the initial conditions. If at the initial moment there was one electron with energy \(E_0\), then for \(t=0\)

\[ \Pi(E)=\delta(E-E_0)\quad \text{and}\quad \Gamma(E)=0. \tag{20} \]

It remains now to pass to the former energy variable \(E\). In principle this is always possible by using relation (15). But it turns out to be more convenient to measure the energy \(E\) in the new variable

\[ y=\lg \frac{E_0}{E}. \]

Then to the critical energy \(\varepsilon\) there corresponds the quantity

\[ \eta=\lg \frac{E_0}{\varepsilon}. \]

Let us introduce a new distribution function, common to particles and photons—\(e^{\varphi(y,t)}\). \(N(y,t)=e^{\varphi(y,t)}\) denotes the number of photons and particles in the shower with energy in the interval \(y, y+dy\), at depth \(t\) from the surface of the substance. Accordingly, henceforth by the word “particles” we shall understand both material particles and photons. \(e^{\varphi(y,t)}\) is connected with \(\Pi(E)\) and \(\Gamma(E)\) by the relations

\[ [\Gamma(E)+\Pi(E)]\,dE=e^{\varphi(y,t)}\,dy. \tag{21} \]

Taking into account the initial conditions (20), one can find \(\varphi(y,t)\) as a function of both variables. Knowing \(\varphi(y,t)\), we obtain the number of particles in the shower \(N(y,t)\) as a function of the energy \(y\) and the penetration depth \(t\).

From physical considerations one may expect that \(N(y,t)\), as a function of \(y\) for fixed \(t\), must have a sharp maximum.

Namely, it can be shown that the probability of the formation of new particles is the greater, the smaller their energy. Therefore a large part of the particles formed will have a small energy in comparison with the initial energy \(E_0\), i.e. the number of particles in the shower with small energy will be much greater than the number of particles with energy comparable to the initial one.

On the other hand, at energies smaller than the critical energy, the probability of ionization (or of the Compton effect for quanta) becomes much greater than the probability of emission of bremsstrahlung (or, correspondingly, of pair production). As a result of ionization or of the Compton effect the particles lose energy and are absorbed.

Thus one may expect that the number of particles in the shower will increase as the energy decreases, down to an energy equal to \(\varepsilon\). At an energy equal to the critical one, the number of particles in the shower will begin to decrease sharply because of energy losses to ionization and the Compton effect and the subsequent absorption of the particles.

In Fig. 5 the curve \(N(y,t)\) is shown for \(t\) fixed, on a logarithmic scale.

Fig. 5. Distribution of particles in a shower as a function of energy on a logarithmic scale for \(t=\mathrm{const}\).

A decrease of the energy \(E\) corresponds to an increase of \(y\); the critical energy \(\varepsilon\) corresponds to the value \(\eta\). It can be shown that, starting from the point \(y=\eta\), where the curve has a maximum, it passes into a straight line.

Starting from this point, the number of particles decreases as the energy decreases. Physically this means that formation of the shower has ceased and only its absorption is taking place.

Of great interest is the distribution of particles in the shower along its path in the substance, i.e. \(N(y,t)\) at a fixed value of \(y\). Since we are naturally interested in the distribution of the maximum number of particles along the path, we shall consider the behavior of \(N(y,t)\) for \(y=\eta\).

From the mechanism of the shower-formation process one can predict such a behavior of the curve \(N(y,t)\): where \(\lambda_s\) is negative, the avalanche of particles grows rapidly. At a certain depth \(t_m\) the largest number of particles is formed. Farther on, \(\lambda_s\) becomes positive, the formation of new particles ceases, and rapid absorption of the shower begins. Consequently, at some point \(t_{\max}\) the curve \(N(\eta,t)\) must have a maximum. At another value of \(t\), equal for example to \(t_0\), the shower must be completely absorbed. Thus \(t_0\) gives the maximum range of the shower in the substance. The behavior of \(N(\eta,t)\) is shown in Fig. 6.

For the quantity \(t_{\max}\), the calculations of Landau and Rumer give

\[ t_{\max}=3{,}8\,\lg \frac{E_0}{\varepsilon}. \tag{22} \]

One can calculate what fraction of the particles with a given initial energy \(E_0\) penetrates through a specified layer of matter \(t\). At a depth \(t\) the number of particles with energy \(E_0\) will be equal to \(e^{\varphi(E_0,t)}\).

For example, let us calculate what fraction of the particles entering the atmosphere with an initial energy \(10^{10}\ \mathrm{eV}\) can reach sea level.

Fig. 6. Distribution of particles in a shower as a function of distance on a logarithmic scale for \(\eta=\lg 100\).

The thickness of the atmosphere corresponds to \(t=26\), and for air \(\varepsilon=10^8\ \mathrm{eV}\); hence

\[ \eta=\lg \frac{E_0}{\varepsilon}=4.6 . \]

From the table given in the cited work one can find the corresponding value \(\varphi=-4.7\). Thus \(e^{-4.7}\) particles reach sea level, i.e. approximately \(1\%\) of the particles entering the atmosphere.

For an initial energy \(3\cdot 10^9\), the number of particles reaching sea level is still smaller, approximately \(0.1\%\).

From all the numerical calculations presented, one can immediately draw a very important conclusion: if showers are produced only by electrons and positrons entering the atmosphere from outer space, then at sea level we should not observe showers at all.

All showers produced by these particles with the most probable energies \(10^9\)—\(10^{10}\ \mathrm{eV}\) must be absorbed high in the atmosphere. The assumption that the primary cosmic rays contain a significant number of electrons with high energies appears doubtful. In any case, it is entirely unclear why the frequency of occurrence of showers in lead does not dimin—

decreases to zero with increasing thickness of the plate, while \(a\) tends to a definite limit.

Thus, in order to preserve the mechanism of pair formation proposed by Heitler and Bhabha, it is necessary to suppose that the showers observed at sea level or in great thicknesses of matter are produced not by electrons, but by some particles with greater penetrating power.

One may suppose that such particles are protons. These protons can, for example, emit photons or knock electrons out of atoms, which in turn create showers. However, the probability of radiation for a proton, as was indicated above, is very small. Likewise, at high energies the probability of ionization becomes small. Therefore, assuming that protons are the agent causing showers (at sea level), we obtain too small a probability of shower formation. Hence the conclusion suggests itself that the penetrating particles causing showers are the semi-heavy particles of Anderson and Neddermeyer.

The semi-heavy particles themselves cannot create showers, but must produce along their path light particles which, in turn, would create showers.

In this case there are, generally speaking, two possibilities:

1) The semi-heavy particles emit photons, which create showers.

2) The semi-heavy particles produce electrons along their path, which then cause showers. There are some grounds for thinking that the second case occurs more often, namely: the equilibrium number of showers (at sea level) is approximately independent of the atomic number \(Z\) of the substance in which they arise. This same circumstance is characteristic of showers produced by the second route. The avalanche theory of showers, correctly describing the interaction of very fast particles with matter, puts forward a very strong argument in favor of the reality of semi-heavy particles.

Indeed, light particles with energies of the order of \(100\text{--}500\,me^2\), observed by Anderson and Neddermeyer in the experiments described above, should have lost energy in the formation of showers (which was in fact observed).

The particles of the hard component, however, which passed through the plate without noticeable losses and did not produce showers along their path, cannot be identified with electrons or positrons. But, on the other hand, they could not have been protons either, since the latter would have had to ionize considerably more strongly.

Thus there remains only one way out—to admit the existence of the semi-heavy particles of Anderson and Neddermeyer.

The existence of showers at the surface of the earth is yet another confirmation of this hypothesis.

At the same time, our views on the composition of cosmic rays and on the nature of showers are radically changed. Summarizing, one may characterize the resulting picture as follows.

Primary cosmic rays consist mainly of light pa

These rays interact with matter by means of showers, which arise along their path and are absorbed high in the atmosphere. The cosmic rays that reach the Earth consist of semi-heavy particles. These semi-heavy particles indirectly produce the showers observed here.

The origin of the semi-heavy particles is at present entirely unknown. In particular, it is unknown whether they enter into the composition of the primary rays that come into the atmosphere from cosmic space, or whether they are produced already within the atmosphere. There is no doubt that the resolution of this question is of very great importance for physics.

LITERATURE

1. Heisenberg, Z. Physik, 101, 533, 1936.
2. Pauli (in press).
3. Anderson and Neddermeyer, Phys. Rev., 51, 884, 1936.
4. Williams, Phys. Rev., 45, 729, 1934.
5. Weizsäcker, Z. Physik, 88, 612, 1934.
6. Bethe and Heitler, Proc. Roy. Soc., 146, 83, 1934; Heitler, The Quantum Theory of Radiation, Oxford, 1936.
7. Heller and Bhabha, Proc. Roy. Soc., 159, 432, 1937.
8. Carlson and Oppenheimer, Phys. Rev., 51, 220, 1937.
9. Landau and Rumer, ZhETF (in press).