COSMIC RAYS AND NEW “SEMI-HEAVY” PARTICLES

D. D. Ivanenko, Tomsk

1. Investigation of Cosmic Rays

The question of the nature of cosmic rays has in recent years been coming ever more strongly to the fore and is beginning to exert an ever deeper influence both on the central problem of our day—the physics of the atomic nucleus—and on the development of the general principles of quantum theory. Let us recall that, in exactly the same way, the successful construction of the picture of the atom and of Bohr’s theory became possible only after Rutherford’s discovery of the atomic nucleus in 1911. However, the further assistance given to atomists by nuclear physics, which at that time was a problem of the future, was perhaps weaker than the conditioning of the modern conception of matter by the numerous discoveries in the field of cosmic rays.

The purpose of the present article is to characterize the present state of the question of the nature of cosmic rays and, chiefly, the change in the point of view on cosmic rays which occurred during the past year under the influence of the new shower theory, put forward on the basis of all the remarkable experimental data of recent years: the work of Blackett and Anderson with Wilson-chamber photographs of cosmic rays, the discovery of the latitude effect by Compton, and also the results obtained by Regener in ascents of self-recording instruments. At the same time, both the mathematical side of the theory and a detailed exposition of the experimental material are left aside in this review, which may, if desired, be regarded as an introduction.

Apart from the exceptional importance of the study of cosmic rays in itself, they provide us with the only possibility of using particles possessing energies of billions and more electron-volts, i.e. thousands and millions of times greater than all nuclear projectiles available in laboratories (in Lawrence’s cyclotron protons and $\alpha$-particles are accelerated up to $10^6\ \mathrm{eV}$; the hardest radiation quanta in nuclear reactions have energies of only about $20 \cdot 10^6\ \mathrm{eV}$). The fact that the energy of cosmic rays reaches the order of the proper energy of a heavy

COSMIC RAYS AND NEW “SEMI-HEAVY” PARTICLES

particles—protons or neutrons \((Mc^2\)—approximately \(10^9\) eV), and even exceeds it, clearly shows that we are dealing here with non-nuclear and, still more, non-atomic processes. Atomic nuclei, consisting of heavy particles—protons and neutrons—obviously cannot, in any transitions or reactions whatsoever, emit energy of the order of the rest energy of the proton itself (neutron); similarly, atomic electrons—light particles—do not emit \(\gamma\)-rays with energy of the order of the rest energy of the electron (i.e. \(mc^2\)—approximately \(0.5 \cdot 10^6\) eV), capable of producing those same electrons and positrons, i.e. of changing the number of particles in atoms. Radiation with quantum energy \(Mc^2\) could change the number of particles in the nucleus, which, of course, is impossible.

Let us note at once that the question of the origin of cosmic rays is still by no means clarified. Since no atomic or nuclear processes can lead to the appearance of particles of such great energy, in searching for their sources physicists have to turn to astrophysical phenomena of the type of an explosion with the formation of new or supernova stars, or to other mechanisms of stellar or cosmological order, which might give rise to such particles or at least accelerate them to the observed velocity. Doubts concerning the extraterrestrial origin of cosmic radiation, expressed as recently as 10–15 years ago, have been so clearly and definitively refuted by all subsequent research that there is now no need to dwell on them. In particular, Wilson’s recent hypothesis concerning the possibility of the formation of such fast particles during thunderstorms has had no success.

What, then, can be said about the nature of cosmic rays? Undoubtedly, the entire flux, both primary—at the upper boundary of the atmosphere—and enriched, in passing through the earth’s atmosphere, by secondary, tertiary, etc., particles and electromagnetic radiation, is not something homogeneous either in composition or in magnitude of energy. The energies of cosmic rays are distributed over a large interval up to \(10^8\) and \(10^{10}\) eV, as can be directly verified from photographs in the Wilson chamber obtained after Skobeltsyn, Anderson, Blackett, Kunze, Leprince-Ringuet, and others. Indirect considerations indicate the presence of cosmic particles with energies of \(10^{11}\)—\(10^{12}\) eV, and for the time being there is no question of any upper limit of the spectrum. The enormous energy of cosmic rays corresponds to their quite extraordinary penetrating power, allowing them to pass through layers of matter of more than a meter of lead, or 500–700 m of water, whereas all natural radioactive radiations: \(\alpha\)-, \(\beta\)- and the most penetrating \(\gamma\)-rays, are absorbed already in a layer of approximately 10 cm of lead. As is known, the presence of residual ionization in instruments shielded from radioactive rays, and also its increase upon ascending upward after the minimum caused by attenuation of terrestrial sources, initiated the work that then led Hess (shortly before the war) to the hypothesis of the existence of extraterrestrial cosmic radiation.

of great rigidity. At first glance the explanation of penetrating power presents no difficulties, since it seems quite obvious that charged particles of greater energy will be less retarded by the fields of the surrounding atoms, and the reduced interaction will allow them, while losing an insignificant fraction of their energy, to penetrate to a great depth. Moreover, uncharged particles of the electromagnetic field, i.e. photons, of very high frequency or energy would seem to be even more penetrating radiation, losing energy only in rare collisions with electrons (the Compton effect) or with atoms (the photoelectric effect), while the fraction of the latter collisions will decrease as the photon energy increases. Indeed, at high frequency, i.e. small wavelength, the effective “dimensions” of the photon are small, and the atom will appear to it as “empty,” so that the photon will collide “directly” with electrons. In this case the probability of collision with atoms and electrons will in fact decrease with increasing photon energy.

Let us briefly recall the corresponding formulas, which are especially well analyzed in Heitler’s excellent book.

The effective cross section for the photoelectric effect with ejection of an electron from the \(K\)-shell will be inversely proportional to the photon frequency \(\nu\) to the power \(3/2\); at very high photon energies it decreases more slowly with increasing energy, namely inversely as the first power of the frequency

\[ \sigma_{\mathrm{ph}} \sim r_0^2 \frac{Z^5}{137^4}\frac{mc^2}{h\nu} \sim 10^{-23} Z^5\lambda\ \mathrm{cm}^2 \quad (h\nu \gg mc^2) \tag{1} \]

\[ \left( r_0=\frac{e^2}{mc^2}\sim 10^{-13}\ \mathrm{cm} \text{ is the classical radius of the electron, } Z\text{—} \right. \]

the atomic number, \(m\)—the electron mass,

\[ \frac{1}{137}=\frac{2\pi e^2}{hc} \]

—the fine-structure constant). Here and below we discard numerical coefficients of order unity\(^1\).

\(^1\) As is known, classical electrodynamics, quite independently of all disputes about the structure of the electron (point, little sphere, etc.), automatically assigns to a free electron, in the scattering of light, an effective cross section \(\sigma\) of the order of the square of the electron “radius” \(r_0\). Under the action of the wave field the electron will move according to the equation

\[ \ddot{x}=\frac{e}{m}E \]

and will emit a secondary scattered wave, whose intensity at a distance \(r\), according to the well-known formula, is equal to

\[ I \sim \frac{e^2}{R^2}\frac{|\ddot{x}|^2}{c^3} \sim \frac{e^4E^2}{m^2c^3R^2} \sim \left(\frac{e^2}{mc^2}\right)^2\frac{1}{R^2}I_0 \sim \frac{r_0^2}{R^2}I_0 \]

The effective cross-section for the scattering of light by an electron (the Compton effect) is given by quantum theory by the Klein–Nishina formula, which at low frequencies coincides with Thomson’s classical formula

\[ \sigma_{\mathrm{sc}} \sim r_0^2 \sim 7\cdot 10^{-25}\ \mathrm{cm}^2 \quad (h\nu < mc^2), \tag{2,1} \]

i.e., the electron behaves like a sphere of radius \(r_0\).

Fig. 1. Course of the energy loss in lead on a logarithmic scale as a function of the value of the primary energy for an electron and a proton. The sharp increase in the energy loss by the electron from \(20\,mc^2\) is due to the intensifying bremsstrahlung; if only inelastic collisions were present, in this region we would obtain the dotted curve, i.e., almost no increase of the losses.

At high frequencies, however, the effective cross-section, or the number of scattered photons, decreases inversely proportionally to the photon energy,

\[ \sigma_{\mathrm{sc}} \sim r_0^2 \frac{mc^2}{h\nu} \quad (h\nu > mc^2). \tag{2,2} \]

To obtain the absorption coefficient \(\tau\), the effective cross-section must be multiplied by the number of atoms, or correspondingly electrons, in \(1\ \mathrm{cm}^3\):

\[ \tau_{\mathrm{ph}} = N\sigma_{\mathrm{ph}},\qquad \tau_{\mathrm{sc}} = NZ\sigma_{\mathrm{sc}}. \tag{3} \]

(Thomson formula), where

\[ I_0=\frac{c|E|^2}{8\pi} \]

is the intensity of the primary wave. In this way we arrive directly at the electron radius

\[ r_0=\frac{e^2}{mc^2}. \]

The fact that the probability of the photoelectric effect and the Compton effect decreases with increasing energy led Millikan 10 years ago to put forward the hypothesis that all primary cosmic rays are a stream of high-frequency \(\gamma\)-rays, as the most penetrating radiation. This hypothesis, allegedly confirmed by the coincidence of the actually measured absorption coefficient of cosmic rays with that theoretically derived for \(\gamma\)-rays of the given energy, turned out, however, to be refuted on all points. First, subsequently the experimentally obtained absorption coefficient of cosmic rays was much smaller than that admitted by Millikan, namely of the order of \(0.03\) per \(1\ m\) of water (a penetrating power one hundred times greater than that of the hardest \(\gamma\)-rays). Second, the theoretical expression for scattering which Millikan used proved unsuitable in the region of energies greater than \(10^6\ \mathrm{eV}\), and was replaced by another expression obtained on the basis of Dirac’s equation (the Klein–Nishina formula).

We leave aside here that aspect of Millikan’s hypothesis which speaks of the source of cosmic rays, allegedly arising in the synthesis of the atomic nuclei of elements, chiefly helium, from primary elementary particles (as was then supposed, the proton and the electron) upon collisions of the latter in interstellar space. Quite apart from the complete fictitiousness of the “coincidence” of the energies of \(\gamma\)-rays, calculated from an incorrect formula, with those obtained from—subsequently found to be preliminary—experiments on the absorptivity of cosmic rays, and apart from the fantastic arbitrariness in assigning to particular components of cosmic radiation the values of nuclear mass defects (obtained from an incorrect proton–electron model), in addition to all this, the frequency of possible collisions in interstellar space is insufficient to provide the observed total energy of the cosmic flux, which is of the same order as the total radiation energy of all stars.

Leaving aside the question of their origin, let us nevertheless ask whether the primary cosmic rays entering the upper boundary of the atmosphere can consist entirely of \(\gamma\)-photons.

Let us return to Dirac’s theory, which predicts the possibility of the formation of an electron–positron pair in the interaction (“collision”) of a \(\gamma\)-ray with an atomic nucleus. The probability of this process increases strongly with increasing photon energy. The differential effective cross-section for the formation of a pair with particle energies \(E'\) and \(h\nu - E'\) in the collision of a hard photon \(h\nu \gg 2mc^2\) with a nucleus of charge \(Z\) is given essentially by the formula

\[ \sigma_{\mathrm{pair}}\, dE' \sim \frac{8}{3}\frac{Z^2}{137} r_0^2 \lg \frac{2h\nu}{mc^2}\frac{dE'}{h\nu}. \]

The total cross-section is

\[ \int_{0}^{h\nu}\sigma_{\mathrm{pair}}\, dE' \sim \frac{8}{3}\frac{1}{137} Z^2 r_0^2 \lg \frac{2h\nu}{mc^2}, \tag{4} \]

that is, it increases with the photon energy. If, however, the screening of the nucleus by electrons is taken into account, then calculations give, for extremely high energies,

\[ \sigma'_{\mathrm{pair}}\sim \frac{8}{3}\frac{Z^2}{137}\, r_0^2 \lg \frac{183}{Z^{1/3}};\quad \tau_{\mathrm{pair}}=N\sigma'_{\mathrm{pair}}. \tag{4,1} \]

Hence, with the aid of the absorption coefficients \(\tau'_{\mathrm{pair}}\), we obtain an expression, important for what follows, for the probability that a photon will traverse a distance \(l\),

\[ W=e^{-\tau' l} \]

(taking into account only the absorption due to pair production). Thus the value of the reciprocal absorption coefficient \(l_0=\dfrac{1}{\tau'_{\mathrm{pair}}}\) plays the role of the mean free path of a photon, independent of energy at very high energies and when screening is taken into account. Thus, finally, we see that as the energy is increased the penetrating power of photons first rapidly increases owing to the decrease of the absorption coefficient for the photoeffect; then it increases more slowly owing to the decrease of the probability of the Compton effect, reaches a maximum in the region \(5\text{--}20\,mc^2\), after which the penetrating power begins to decrease because of the growth of photon absorption in pair production. At extremely high energies the penetrating power reaches an approximately constant value. Thereby the grounds for ascribing a photon nature to cosmic rays disappear.

The final blow to the hypothesis of a purely photon composition of cosmic rays was dealt by the discovery of the geomagnetic latitude effect by the Dutch physicist Clay, and also by Compton. Compton’s numerous expeditions (many dozens of them), contrary to previous contradictory results (including Millikan’s erroneous observations), proved with final convincement (1932) that the intensity of cosmic radiation at sea level at the equator is approximately \(10\%\) less than at high latitudes (\(50^\circ\) and higher). Moreover, what is involved is a magnetic equator and a regular course of the intensity of cosmic rays with magnetic latitude, quite similar to the regions of equal intensity of the aurora borealis. The explanation of this fundamental discovery is obvious in essence: the Earth’s magnetic field, acting as a barrier, deflects part of the electrically charged cosmic rays toward the poles, creating a more intense “northern cosmic aurora.” This scheme is entirely analogous to the well-known interpretation of ordinary visible auroras, which are associated with the action of much slower electrical particles, owing to which their region is also limited by direct proximity to the pole.

In order to cause a noticeable deflection of fast particles, the very weak magnetic field of the Earth (fractions of a gauss at the Earth’s surface—in comparison with fields of 17,000 gauss, used by Anderson and Kunze) must begin to act on the particle from afar, long before it enters the atmosphere, at distances comparable with the radius of the Earth; that is, the geomagnetic latitude effect must necessarily be connected with primary and, evidently, charged particles, and not with photons and the secondary particles produced by them. At high altitude the latitude effect becomes still more noticeable, that is, an even larger percentage of particles is deflected toward the poles; this was observed, in particular, by Cosyns during the flight in the Piccard stratospheric balloon. Consequently, charged particles play an even greater role in the primary radiation. In addition to the latitude effect, a number of other geomagnetic effects are observed; for example, Johnson observed an excess of several percent of cosmic rays from the west, which indicates a large fraction of positively charged primary particles.

A detailed analysis carried out by a number of authors, especially by Compton, who sharply objected to all attempts by Millikan to revive the hypothesis of a photonic composition of a substantial part of the primary cosmic radiation, shows that at present there are no grounds for admitting an appreciable number of primary photons (and, at the same time, of other neutral particles as well: neutrons and neutrinos).

What part of the cosmic flux, then, falls to each of the known kinds of elementary particles of modern physics: protons, electrons, and positrons? Wilson photographs give an equal number of particles of both signs (Blackett), perhaps with some predominance of positive ones at the highest energies (Leprince-Ringuet).

First of all we must dwell on electrons and positrons. Despite the indisputable presence of electrons and positrons in Wilson photographs—on which, as is known, positrons were first discovered—the assumption of these light particles as primary cosmic rays long seemed completely excluded on the basis of the following theoretical considerations, developed by Heitler, Bethe, and Sauter. Electrons (or positrons) of high energy—of many millions of eV—when passing through a layer of matter lose only an insignificant fraction of their energy in inelastic ionizing collisions with atoms, but instead emit very intense γ-rays of the so-called bremsstrahlung radiation in interaction (“collisions”) with atomic nuclei (just as less fast electrons, striking an anticathode, emit X-rays). As the energy of the flying electron increases, the probability of emission of a bremsstrahlung γ-photon rapidly grows. The situation here is quite analogous to the question of the loss of energy by hard γ-rays, which, as we indicated above, likewise most “readily” dissipate their energy

not on the photoelectric effect (or ionization) or the Compton effect, but on collision with nuclei with the production of pairs.

Let us dwell briefly on the basic formulas. The loss of energy by an electron in inelastic collisions over an interval \(dl\) will be equal to the number of collisions, determined by the probability or effective cross section, multiplied by the energy loss in each collision. With \(N\) atoms of atomic number \(Z\) per cubic centimeter, recalling the value of the effective cross section \(\sigma\) (compare the Rutherford scattering formula) and the fact that in an impact the electron transfers to the medium an energy of the order \(\Delta E \sim mv^2\), we have

\[ -\frac{dE}{dl}=NZ\sigma\cdot \Delta E, \tag{5,1} \]

i.e.

\[ -\frac{dE}{dl}\sim NZ\left(\frac{e^2}{mv^2}\right)^2\cdot mv^2 \sim NZr_0^2 mc^2\frac{c^2}{v^2}. \tag{5,2} \]

Thus, as the energy of the particle increases, the energy loss will decrease. In the relativistic case, for \(v\sim c\), our crude formula will give a constant energy loss independent of the energy of the particle. In fact, the exact formulas contain a logarithmic

Fig. 2. Absorption coefficient of \(\gamma\)-rays in lead as a function of photon energy on a logarithmic scale. The dotted lines indicate the fractions contributed by the photoelectric effect, the Compton effect, and the formation of pairs.

dependence on the energy, so that the energy loss to ionization will, at high velocities, increase slowly with the energy.

The differential effective cross section or probability of emission by a fast electron of a bremsstrahlung photon with energy between \(E\) and \(E+dE\) in collision with a nucleus of charge \(Z\) is given, in the main, by the expression

\[ \sigma_{\mathrm{rad}}\,dE \sim 4\,\frac{Z^{2}r_{0}^{2}}{137}\lg \frac{183}{Z^{1/3}}\,\frac{dE}{E} \tag{6} \]

(where screening has already been taken into account by the characteristic factor \(\lg \frac{183}{Z^{1/3}}\)), i.e., it does not depend on the energy of the electron.

The total energy loss by an electron to bremsstrahlung radiation over a segment \(dl\), at nuclear density \(N\), will be equal to

\[ -\frac{dE}{dl}\sim N\int_{0}^{E_{0}}\sigma_{\mathrm{rad}}\,dE\cdot E \sim \frac{NZr_{0}^{2}}{137}\,E_{0}\lg \frac{183}{Z^{1/3}} \tag{7} \]

(for the electron can emit energy from zero up to the value of its initial energy \(E_{0}\)).

The quantity \(4\frac{Z^{2}r_{0}^{2}}{137}\lg \frac{183}{Z^{1/3}}\), independent of the energy, when screening and ultrarelativistic velocities are taken into account, may be called the total effective cross section for bremsstrahlung radiation. Formula (7) gives us the fundamental result that the radiative energy loss grows with increasing particle energy and is practically the sole cause of losses at high velocities, in view of the insignificance of ionization losses.

It is interesting to dwell on the question why the effective cross sections or probabilities of two completely different processes—the production of a pair by a photon and the emission of a bremsstrahlung photon by an electron—are given, in the main, by identical formulas (4) and (7) (somewhat more precisely: \(\frac{\sigma_{\mathrm{pair}}}{\sigma_{\mathrm{rad}}}\sim 0.6\); as Heitler and Bhabha note, the use of approximate formulas of the indicated type leads to errors of the order of 30%).

The order of an effect in quantum electrodynamics is determined by the number of photons participating in the process. For example, emission or absorption of light, as well as the photoeffect, are first-order processes. The Coulomb interaction of two particles arises as the result of a second-order process, being realized by the emission of a (longitudinal) photon by one charge and its absorption by another. The scattering of light by an atom (Raman effect) or by an electron (Compton effect) is a second-order effect, visibly corresponding to the disappearance or absorption of one primary and the emission

of another secondary photon. For the production of a pair by a photon incident on a nucleus, the following two stages of the process are necessary: 1) absorption of the photon, 2) transition of an electron from a negative level to a positive one under the influence of the Coulomb interaction with the nucleus, which at the same time means the appearance of a positron, i.e. here we have an effect of the third order, since the second part of the process is in itself an effect of the second order. According to the general rules, the probability of a third-order process is obtained, essentially, by multiplying the probability of a second-order effect by the dimensionless quantity of the fine-structure constant, i.e. by

\[ \frac{2\pi e^2}{hc}=\frac{1}{137}. \]

Such is the origin of the factor \(\frac{1}{137}\) in formulas (4) and (7), distinguishing them from expressions (2) and (5).

The emission of a bremsstrahlung photon is also a third-order effect, which may be resolved in the following way: 1) the Coulomb interaction of the passing electron with the nucleus, causing the deceleration of the electron (a second-order effect), 2) the emission of a bremsstrahlung photon; from this it is clear that the formulas for pair production and bremsstrahlung emission in the collision with a nucleus of a photon or, respectively, an electron must be very similar.

Repeating the argument concerning the penetrating power of photons, we may say that with increasing electron energy the loss of energy at first decreases because of the decrease in inelastic collisions, reaches a minimum for some value of the energy \(E_i\), characteristic for each substance, and then begins to increase on account of the increase in losses due to bremsstrahlung emission of \(\gamma\)-rays.

Starting with energies of the order of \(10^6\) eV and higher in passing through lead, and of the order of \(1.5\cdot10^8\) eV in passing through water or air, practically the entire loss of energy of the electron will be due to bremsstrahlung in collisions with nuclei, and not to ionizing collisions with atoms, since for so rapid a particle, “slipping” through the atom with its weak fields almost as through empty space and coming under the action of the nucleus at a short distance, the outer shell can manifest itself only in the form of screening of the nuclear charge.

The same considerations are, of course, also valid for protons, but only beginning with energies large in comparison with the proton’s rest energy (i.e. tens of billions of electron-volts). In the region up to approximately \(50\cdot10^6\) eV for lead, a proton of the same energy as an electron will obviously have (in view of its large mass) a smaller velocity and will, passing comparatively slowly through atoms, interact strongly with them, losing energy to ionization. This means that the tearing out of individual electrons, i.e. its loss of energy according to formula (5), will decrease with increasing energy (Fig. 2). With a further increase of energy and velocity the protons will interact ever less intensely with atoms,

but they are still far from becoming capable of intensive bremsstrahlung radiation or pair production. As a result, theoretical analysis, overturning all intuitive notions, predicts for protons in the energy region above \(50\cdot 10^6\ \mathrm{eV}\) (but below \(10^{10}\ \mathrm{eV}\)) a smaller loss of energy in passing through matter than for electrons. Heavy protons prove to be more penetrating than electrons and positrons, or even \(\gamma\)-rays.

Thus the process of formation of electron and positron pairs, which might yield these light particles in large numbers, is, for protons in the velocity region of interest to us (hundreds—thousands of electron-volts), very improbable; likewise the emission by the proton of hard photons, which in turn might form pairs, is quite insignificant. Therefore, although in view of the absence of noticeable energy losses by radiation and by pair formation the most penetrating protons would seem, a priori, to be the most suitable particles as primary cosmic rays (the old 1934 hypothesis of Williams, Compton, and Bethe), this very circumstance does not give them the possibility of being accompanied by the required significant number of secondary electrons and positrons, i.e. it is a counter-argument.

Moreover, protons, like other heavy particles, have not so far been directly detected as components of the cosmic flux in Wilson-chamber photographs, although in a number of photographs (for example, those of Brode and Starr) the tracks of heavy particles are distinctly visible, arising in connection with the passage of cosmic rays as the result of some secondary nuclear disintegrations. It should be emphasized, however, that the entire world collection of Wilson cosmic photographs (probably about 25 thousand photographs) is still not so large as to permit any categorical conclusions. Cosmic rays still conceal many surprises. Our present task is merely to attempt to outline, perhaps even in an exaggerated way, the characteristic scientific situation in this field, trying not to become lost in the abundance of often contradictory experimental material and by no means seeking to set forth all the contending points of view.

All the more, at the present time there is certainly no proof of the presence in cosmic rays of antiprotons, hypothetical particles with the mass of the proton but with negative charge. We emphasize that the very existence of such particles can be inferred only from a visual analogy with the electron-positron pair, without any theoretical proofs analogous to Dirac’s famous reasoning, which led him to the idea of the existence of antielectrons, i.e. positrons. Dirac’s argument cannot be transferred directly from the electron to the proton, for the latter is certainly governed not by the quantum Dirac equation, but by some more complicated equation which must describe, in particular, the decay of a proton into a neutron possible in the field of a nucleus.

positron and neutrino. A brief but telling argument against the direct application of the Dirac equation to the proton is the value of the latter’s magnetic moment, approximately three times greater than the value that the Dirac equation would give. However, the theory of elementary particles is in such a mysterious state that we cannot deny the possibility of the existence of antiprotons.

Thus, the indicated considerations apparently make it impossible to admit electrons or positrons as primary particles, owing to their small penetrating power. Even for an electron with an energy of \(10^{12}\,\mathrm{eV}\), the range would be only \(2\) km of air, whereas the entire thickness of the atmosphere is equivalent to \(8\) km of normal air. The probability for such an electron, with an energy exceeding all measured values for cosmic-ray particles, to reach sea level and retain an energy of \(100\cdot 10^6\,\mathrm{eV}\) is exceedingly small—only of the order of one hundred-thousandth—whereas the quite sufficient penetrating power of heavy particles, i.e. protons, would allow them to pass calmly through the atmosphere. Heavy particles are excluded, however, for other reasons noted above.

2. The Question of the Limits of Quantum Mechanics

After the exclusion of all known particles as primary cosmic rays, a doubt naturally arises as to the correctness of the theoretical predictions in the region of interest to us, namely the necessity of large energy losses to radiation, increasing with energy. Are we entitled to apply quantum mechanics (the special relativistic Dirac wave equation and quantum electrodynamics, i.e. the theory of the electromagnetic field or photons), constructed, strictly speaking, with application to atomic processes, to phenomena with such great energy?

Taught by the experience of the theory of relativity and quantum mechanics, physicists of our day are not inclined unconditionally to guarantee the applicability of all the propositions of a theory when it is extended into a new region. We know well that, for example, classical mechanics is not correct or, more precisely, is unsuitable in the region of large velocities comparable with the speed of light, and must be replaced by relativistic mechanics. In exactly the same way, classical mechanics is inapplicable in regions of small dimensions, of atomic order, where quantum mechanics comes into play. In both cases the classical theory proves to be a legitimate special case. The old primitive point of view of nineteenth-century physics, on the contrary, consisted in the more or less clearly expressed conviction of the unlimited applicability of all the laws of macroscopic ordinary mechanics, electrodynamics, and so on.

Where, then, do the limits of the modern quantum theory lie? It is clear that a precise answer to this question can be given only after the construction

of a more general theory applicable to an even wider range of phenomena; but it is very difficult to settle the question while remaining within the framework of the old scheme. Even now, however, it is already clear that the future “superquantum” theory will have, in particular, to give a proof of the existence of these or other elementary particles, to find theoretically the value of the charge, the masses of the electron, proton, etc. All these quantities, as well as the very fact of the presence of elementary particles, are used in the present theory as empirical material. Into the Dirac equation (i.e., the quantum equation of motion), for example, one may substitute at will the mass of the electron, proton, or some other particle, and we have no a priori arguments against the inapplicability of the Dirac kinematic law in this or that case. In fact, however, the Dirac equation proves unsuitable for a quite exact description of the proton (it gives an incorrect value of the magnetic moment), and for each kind of particle there must probably be its own special equation. Thus the formal apparatus of quantum theory proves helpless in the domain in which the differences between elementary particles become manifest, or the structure of the particles themselves begins to play a role. This situation raises no particular doubts; the dispute begins when one attempts to indicate the boundary of energies, lengths, etc., from which the laws of the future “superquantum” theory come into force. Whether this boundary will be characterized by the smallest length, below which there is “quantization of space,” or by a maximal field, as in Born’s theory, or by a maximal energy, etc., we do not know.

Elementary particles—the electron (positron), proton, and neutron—must, as is known, be assigned certain dimensions, characterized by a radius of about \(r_0 \sim 10^{-13}\) cm, and moreover, apparently, all of the same order. Of course it would be absurd to picture the electron, in a crudely visual way, as a little sphere, but in collisions with photons and other particles the electron, undoubtedly, appears as it were as a disk with dimensions of the order of the square of the radius just mentioned. Thus we are inclined to expect that the present theory will not prove irreproachable in its application to dimensions smaller than \(10^{-13}\) cm. Here it is not only a question of small distances, but also of small wavelengths. To every wavelength there corresponds a certain frequency and energy, and to the critical length of the electron radius there will correspond an energy 137 times greater than the electron’s own energy, i.e. \(137mc^2\), or about \(70 \cdot 10^6\) eV. Indeed, to the Compton wavelength

\[ \lambda_0=\frac{h}{mc}\quad(\sim 10^{-11}\ \text{cm}) \]

there corresponds the energy

\[ E=\frac{hc}{\lambda}=mc^2, \]

whereas the electron radius

\[ \frac{hc}{e^2}\approx 137 \]

is times smaller than \(\lambda_0\).*

* The critical value of the energy, when various details were taken into account, turned out to be sometimes greater, sometimes smaller than \(137mc^2\), but it was always connected with the number 137.

It gives the impression that at energies of \(100 \cdot 10^6\) eV and higher, i.e. precisely in the region of cosmic-ray energies, one should expect one or another deviation from the theory in the sense, for example, that the energy loss by fast electrons may turn out to be less than that predicted by the theory, and consequently their penetrating power greater. It may be that electrons, contrary to the theory, will be able to penetrate even through the entire atmosphere. In this way, cosmic-ray physics proves to be connected with the problem of the limits of quantum theory.

The most critically minded physicists emphasized in this connection that Dirac’s theory of positrons and quantum electrodynamics cannot be regarded not only as fully completed, but even as deserving great confidence, since they lead to a number of absurd conclusions, such as the existence of an infinite unobservable density of electricity, an infinitely large zero-point energy of the electromagnetic field, and a number of others. To this one may object that such absurd conclusions, in essence, concern rather the principled formal considerations of the theory, fortunately not reflecting themselves in all concrete predictions concerning the probability of pair production by one method or another, the probability of radiation in the most complicated processes, etc., which are wholly confirmed by experiment, at least at energies up to \(10^6\) eV. To a theorist making use of quantum electrodynamics, it seems that he is constantly walking along the edge of an (“infinite”) abyss, but each time he manages safely to pass the dangerous place and even to open new horizons.

In connection with this, one cannot fail to note that Bohr’s insistent suggestions to analyze more precisely the limits of quantum mechanics in effect amounted to a tendency to restrict the domain of application of the theory and in recent years gave rise to a very noticeable fear of going beyond certain “permitted” bounds. These tendencies of the Copenhagen school manifested themselves not only in the problem of high energies. It is known that Bohr at first rose up against Dirac’s entire brilliant hypothesis of identifying the positron with holes in filled states of negative energy (essentially on the basis of arguments about the observability of the Dirac vacuum)—after the discovery of the positron and the quantitative confirmation of Dirac’s formulae, these objections of Bohr, of course, fell away. In exactly the same way, proceeding from empirical data on radioactive \(\beta\)-decay, Bohr not only pointed to the principled possibility of nonfulfillment of the law of conservation of energy in the region of the atomic nucleus, but directly stimulated the development of nonconservation theories of electron emission, stellar structure, etc. As is known, modern physics rejected these absolutely unfruitful tendencies and successfully proceeded both along the path of refining the positron model and along the path of developing the hypothesis of a new particle, the neutrino, eliminating the apparent visible nonconservation of energy in \(\beta\)-decay.

Shankland’s experiments of 1936, which supposedly indicated nonconservation

the change in energy in the scattering of hard \(\gamma\)-rays were likewise refuted by a number of experimenters, including Jacobsen at Bohr’s Institute, and were then acknowledged as erroneous by the author himself, who had previously referred to Bohr’s ideas. Niels Bohr was forced, in a special article, to come out with an admission of the necessity of following the path of the conservation law and the neutrino theory. The presence of such an unfruitful tendency in a physicist of such stature as Bohr, who at the same time continued to work very successfully on problems of contemporary theory, seems truly astonishing.

Be that as it may, in recent times the not very convincing and rigorous argumentation of Williams and Weizsäcker\(^1\) in favor of the applicability of the formulae for pair production and bremsstrahlung at energies above \(137\,mc^2\), i.e. approximately \(10^8\ \mathrm{eV}\), has been opposed by the conviction that there is a limit to the modern theory somewhere in this same region.

Moreover, experiment also seemed to speak unequivocally against the formulae for bremsstrahlung. In 1934 Anderson and Neddermeyer (and then Blackett) succeeded in directly measuring the loss of energy by cosmic rays in the energy region up to 400 million eV, at first only for 9 electrons, whose tracks were observed in the same remarkable Anderson Wilson chamber with a lead plate about a centimeter thick inside. In full agreement with the quantum theory of bremsstrahlung, the energy loss proved to increase rapidly with the energy of the particles, up to approximately energies of 100–150 million eV; the last three points, however, for energies of 150–400 million eV, gave a sharp decrease of the energy loss, indicating, it would seem, with unquestionable obviousness, the inapplicability of the theory to such high energies.

Here it is appropriate to mention a typical variant of the theory with a restricted application of quantum electrodynamics, specially “tailored” to explain the results of Anderson—Blackett. Under the influence of Born’s theory, which introduces a certain maximum electromagnetic field

\[ F_{\max}\sim \frac{e}{r_0^2}, \]

Nordheim (who, however, has now abandoned his theory and has adopted the viewpoint of Heit-

\(^1\) Instead of the radiation of an electron in the field of a nucleus at rest, one considers the equivalent problem of the scattering by an electron at rest of electromagnetic waves, to which the field of a nucleus moving in the opposite direction reduces; according to the main theory, the scattering takes place with energies of the order of \(mc^2\); at larger energies \((h\nu \gg mc^2)\) the effective cross section, or the intensity of scattering, i.e. the Compton effect, rapidly decreases according to formula (2.1), so that even when energies greater than \(mc^2\) are discarded we nevertheless obtain practically the whole required amount of scattering, and together with it the bremsstrahlung for our moving electron. From this one could suppose that higher energies in general play no essential role in the formula for bremsstrahlung, and that this formula holds also at energies greater than \(137\,mc^2\).

supposed that for fields greater than the critical field

\[ F_{\mathrm{cr}}=\frac{e}{r_0 \frac{h}{mc}}=\frac{e}{r_0^2}\frac{1}{137} \]

(where \(r_0\) is the electron radius) quantum electrodynamics, and with it the formula for bremsstrahlung, are inapplicable. An electron with energy exceeding the critical value

\[ E_{\mathrm{cr}}=(137)^3 Z^{-\frac{5}{3}} mc^2, \]

will not be able to approach the nucleus to distances smaller than the critical distance and come under the action of a field greater than the critical one; for energies above \(E_{\mathrm{cr}}\) the electron will no longer undergo energy losses to bremsstrahlung; at energies close to \(E_{\mathrm{cr}}\) these losses will be small. Another version of a similar theory with a limiting critical value of the energy (depending, as in Nordheim, on the atomic number), developed by Blackett and Wilson, gives a slower decline of the theoretical curve for energy losses with increasing energy, whereas in Nordheim’s theory an electron, for example, with energy \(8 \cdot 10^9\) eV and higher should not emit bremsstrahlung photons at all when passing through lead. By disposing of an arbitrarily chosen value of the critical energy, Blackett manages to draw the theoretical curve through his experimental points.

Marx’s recent attempt to introduce the smallest possible wavelength, following the (generally speaking, reasonable) idea of the quantization of space, and to forbid energy loss into excessively hard bremsstrahlung photons with wavelengths smaller than the permitted one (as being nonexistent in general), and thereby to explain the apparent decrease in energy loss observed by Anderson and Blackett, is based to a considerable extent on an incorrect interpretation of empirical data and does not take into account Anderson’s latest measurements or Heitler’s theory.

The hasty interpretation, as we shall see, of the apparent decrease in energy losses as indicating the inapplicability of the theory of bremsstrahlung, and hence of all quantum electrodynamics, clearly delayed the sound development of cosmic-ray physics, without at the same time assisting reasonable attempts to construct a “super-quantum theory.”

All the above-mentioned and many other attempts to introduce limiting lengths, fields, energies, etc., proved completely unsuccessful and brought no benefit to the study of cosmic rays. It became clear, as we shall now see, that modern quantum theory had by no means yet exhausted its possibilities, and that all restrictions on it were quite premature.

3. Shower Theory

The problem of cosmic rays is extremely interesting—and exceptionally difficult—for the reason that here theoretical physicists must mobilize all the most refined methods of relativistic quantum theory, while experimentalists must work with the strongest magnetic fields, the most advanced automatic Wilson chambers, and so forth. In addition, the abundance, confusion, and not infrequently contradictory nature of the empirical data impede progress and demand, above all, the introduction of simple working hypotheses.

One of the most important results of the study of cosmic rays was their division into two groups: hard and soft, the necessity of which was especially emphasized by Auger. Cosmic rays in the stratosphere consist almost entirely of rays of the soft group; at an altitude of 3500 m the intensities of soft and hard rays are approximately the same; at sea level the soft rays are still more retarded by absorption in the atmosphere and make up only a third of the hard rays. The division into two groups is made by measuring the absorption coefficient; at sea level, after passing through 10–15 cm of lead, the intensity of cosmic rays decreases by 25%, but then the absorption diminishes, and after passing through 1 m of lead there still remain 30% of the particles, owing to the greater penetrating power of the remaining hard rays, since the soft rays have already had time to be absorbed. In rough figures, the absorption coefficient of soft rays is 0.6 per 1 m of water (one half is absorbed in 1 m of water), and for hard rays 0.02 (one half is absorbed in 30 m of water).

Many attempts have been made to decompose the entire intensity curve of cosmic radiation at different altitudes into separate components absorbed according to an exponential law, but up to now only the general division into soft and hard rays has proved essential. Ascribing fundamental significance to the latter, we must reformulate our main question as follows: what is the nature of the hard and soft rays, and are the predictions of the theory fulfilled for each of the two components separately?

A. Cascade showers. The decisive step in the understanding of the whole problem was the work of Heitler and Bhabha (and also the paper of Oppenheimer and Carlson) in April 1937, where the bold hypothesis was clearly put forward and justified that soft cosmic rays are electrons and positrons, to which quantum mechanics is applicable without restriction. Needless to say, this work was the culmination of the efforts of many investigators. The successes of Heitler’s theory¹ thus lie both in the direction of the rehabilitation of quantum mechanics in the region of high energies and in the physical ex—

¹ In what follows, for the sake of brevity, we shall limit ourselves to mentioning the name of only one author, especially since it is precisely Heitler who deserves the credit for clarifying the whole problem of bremsstrahlung.

deciphering the exceptionally tangled empiricism of cosmic rays. Heitler’s theory is, so to speak, the “normal” quantum theory of the passage of an electron through matter, based on the previously derived formulas of Dirac’s theory and quantum electrodynamics (chiefly with the participation of Heitler himself), without assuming any new processes whatever in the interaction of the electron with matter.

In the region of interest to us, losses of energy to ionization may be neglected, and a fast electron, passing through a substance—for example, the atmosphere—emits only very hard bremsstrahlung \(\gamma\)-photons, and practically only in the direction of its motion. A hard photon, interacting with the surrounding atomic nuclei (at high photon energies we neglect the photo- and Compton effects), has a large probability of forming an electron–positron pair, again moving approximately in the initial direction. The electron and the positron, in their turn, emit bremsstrahlung photons, and so on. As a result of such a cascade process we obtain an entire shower of electrons and positrons, which can be directly observed in a Wilson chamber in the form of “showers” discovered by Blackett (as is known, even Skobeltsyn’s classic first photographs had already revealed the tendency of cosmic rays not to appear singly).

It is important to note that what is involved here is the cascade formation of a shower, a kind of multiplication of the number of particles as the result of repeated successive pair formation, and not the simultaneous emission of all the shower particles. The probability of the latter process as a high-order effect according to the laws of quantum electrodynamics is very small, of the order of \(\left(\frac{1}{137}\right)^n\) times the probability of forming one pair (here \(n\) is the number of particles in the shower). The latter circumstance is connected with the absence in quantum electrodynamics of any constants having the dimension of length, so that the expansion in perturbation theory proceeds, as has already been noted, in powers of the dimensionless fine-structure constant

\[ \frac{2\pi e^2}{hc} = \frac{1}{137}. \]

The failure of the attempt to interpret a shower as a simultaneous higher-order process caused by electromagnetic forces further strengthened the conviction noted above that quantum theory was unsuitable, but at the same time led to new fundamental possibilities for particle emission, of which we shall speak below.

Heitler’s theory describes, in general terms, the cascade process of the gradual splitting of the primary energy of an electron (or positron) \(E_0\) in the following way. After traversing a certain distance \(l_0\), characteristic for each substance (\(0.4\) cm for lead, \(34\) cm for water, \(275\) m for air, etc.), the electron collides with a nucleus and loses on the average approximately one half (more exactly, 70%) of its energy in emitting a bremsstrahlung photon. The bremsstrahlung photon, having traversed on the average the same layer of substance \(l_0\), in turn collides with its own

in turn with the nucleus and produces, on the average (more precisely, with probability 0.6), one electron–positron pair. Such subdivision of the energy and multiplication of the number of particles will continue until the energy of each of the numerous components of the shower that has formed becomes so insignificant that the processes of production of new components, i.e. bremsstrahlung emission and pair formation (by electrons and photons, respectively), become less probable than inelastic-collision processes for the electron. We have already indicated above that for each substance there exists a characteristic energy boundary \(E_i\), at which the losses of the electron to ionization are comparable with the radiative losses; it is equal to \(10^7\ \mathrm{eV}\) for lead, \(1.5 \cdot 10^8\ \mathrm{eV}\) for air or water, \(3 \cdot 10^8\ \mathrm{eV}\) for iron, \(6 \cdot 10^7\ \mathrm{eV}\) for aluminum. Similarly, there exists a characteristic boundary at which the Compton effect and then the photo-effect become more probable for the photon than the pair-production effect; for lead it is approximately \(5 \cdot 10^6\ \mathrm{eV}\).

When the energy, as a result of the subdivision, reaches the indicated boundary, the electron will lose this “remainder” of energy by ionizing through inelastic collisions over a very small path, and in any case will become trapped somewhere within a segment \(l_0\) characteristic for each substance.

It is clear that the total number of secondary, tertiary, etc. particles in such an electron–positron avalanche or shower will be determined by the ratio \(\dfrac{E_0}{E_i}\) and will at first increase rapidly with the layer of matter \(l\) into which the primary electron penetrates, owing to multiplication of the number of particles; upon further penetration the total number of particles will begin to decrease comparatively slowly owing to the trapping of electrons because of ionization losses. Thus, for each initial energy \(E_0\), at some depth characteristic of the given substance, we shall obtain a maximum of the total number of secondary, tertiary, etc. particles. The visual intuition suggesting that the total number of all secondary particles \(N\) should be proportional to the ratio of the values \(E_0\), is confirmed by the Heitler–Bhabha theory, which gives the following formula for the maximum number of particles with energy greater than a certain \(E\):

\[ N_{\max}=0.06\left(\frac{E_0}{E}\right)^{0.93}. \tag{8} \]

With an increase of the initial energy \(E_0\), the depth \(l_{\max}\) at which the maximum number of particles will be reached will obviously increase somewhat, for at a larger primary value of \(E_0\) the multiplication process must proceed farther before the ionization energy boundary \(E_i\) is reached (or some other fixed small value of the energy \(E\), if we are interested in observing particles with energy above a certain \(E\)).

Fig. 3a. (after Heitler—Bhabha). The total number of secondary, tertiary, etc. particles \(N\) with energy not less than \(E\) as a function of the thickness of the layer of matter traversed, \(l\), for a given primary energy \(E_0\). The variation in the number of primary particles themselves is drawn with a dotted line. The two curves for \(n=3\) (3a) and \(n=5\) (3b)

\[ \left(y=\lg \frac{E_0}{E}\right) \]

characterize the rapid growth in the number of secondary particles and the shift of the maximum as the primary energy \(E_0\) is increased. The dimensionless unit of length \(l\) corresponds to \(0.4\ \mathrm{cm}\) of lead, \(34\ \mathrm{cm}\) of water, \(275\ \mathrm{m}\) of air; the entire atmosphere (8 km of normal air) is equivalent to a thickness \(l=29\).

Fig. 3b.

It turns out that even the simplest analysis correctly conveys the order of magnitude. Since each process of energy loss by an electron or photon leads, on average, to a doubling of the number of particles,

on the segment \(l_0\), then after traversing a layer of matter \(l=l_0 n\) we shall have \(2n\) particles. Since the multiplication of the number of particles will, on the other hand, proceed until the ionization energy limit \(E_i\) is reached, we obtain the following equality:

\[ 2^n \sim \frac{E_0}{E_i}. \tag{9} \]

Thus the depth at which the maximum is reached will increase approximately logarithmically with the initial energy, which is also confirmed by the exact theory.

\[ l_{\max} \sim l_0 \lg \frac{E_0}{E_i}. \tag{10} \]

The total number of all particles rapidly increases with the initial energy of the electron, and, for example, at \(E_0=2\cdot 10^{11}\ \mathrm{eV}\), after passage through a lead plate of \(5\ \mathrm{cm}\), up to 600 pairs of electrons and positrons with energies above \(10^7\ \mathrm{eV}\) arise. In this way the theory gives a qualitative and quantitative explanation of the cosmic showers discovered by Blackett in 1933 (see, for example, the remarkable photographs of showers of several hundred particles by Auger and Ehrenfest1).

\[ \frac{dN_1}{dt}=-\lambda_1N_1;\ldots\quad \frac{dN_k}{dt}=-\lambda_kN_k+\lambda_{k-1}N_{k-1};\ldots . \]

The solutions are sums of exponential terms

\[ N_1=N_1^0 e^{-\lambda_1 t},\quad N_2=A(\lambda_1\lambda_2)e^{-\lambda_1 t} +B(\lambda_1\lambda_2)e^{-\lambda_2 t},\quad N_3=\ldots . \]

In our case of the transformation of photons into pairs and the emission of photons by electrons, the role of the decay probabilities \(\lambda_k\) will be played by the differential effective cross sections \(\sigma_{\mathrm{rad}}\,dE/E\) and \(\sigma_{\mathrm{pair}}\,dE/E\). The main complication consists in the fact that, instead of a discrete series of decaying radioactive elements, we now have a continuous series of mutually transforming particles. Actually, a given sort of particle should evidently be understood as electrons or photons with a given energy in the interval between \(E\) and \(E+dE\). In bremsstrahlung emission, electrons of a given energy will transform into electrons of another sort and photons, and so on. Thus the basic equations of the multiplicative theory of showers will be written in the form

\[ \frac{dN_{\mathrm{el}}(E)}{dx} = 2\int_E^\infty N_{\mathrm{ph}}(u)\frac{\sigma_{\mathrm{pair}}}{u}\,du + \int_E^\infty N_{\mathrm{el}}(u)\frac{\sigma_{\mathrm{rad}}}{u}\,du - \int_0^E N_{\mathrm{el}}(u)\frac{\sigma_{\mathrm{rad}}}{u}\,du, \]

\[ \frac{dN_{\mathrm{ph}}(E)}{dx} = \int_E^\infty N_{\mathrm{el}}(u)\frac{\sigma_{\mathrm{rad}}}{u}\,du - \int_0^E N_{\mathrm{ph}}(u)\frac{\sigma_{\mathrm{pair}}}{u}\,du. \]

Geitler’s theory also directly gives an explanation of one of the most basic empirical facts: the distribution of the intensity of cosmic rays in the atmosphere with height, investigated in especially great detail by Regener and his colleague Pfotzer, who succeeded in launching sounding balloons with self-recording instruments to a height of 29 km (one cannot help expressing legitimate admiration for this experiment, carried out with the aid of the most delicate technique and leading to such a direct interpretation). Recently, a number of American investigators, repeating Regener’s experiment to approximately the same height, succeeded in transmitting the counter readings to the ground by radio.

Fig. 4. Total intensity of cosmic rays traveling in the vertical direction as a function of height in the atmosphere according to Regener—Pfotzer. The dotted curve gives the theoretical curve for the equator.

In fact, electrons and positrons in the energy interval \(E, E+dE\) will be produced by photons with all energies from \(E\) to \(\infty\) with probability

\[ \frac{\sigma_{\mathrm{pair}}}{E'}\,dE \]

(the first term); their number will also be replenished at the expense of electrons losing energy by radiation (the second term), and will decrease because of bremsstrahlung emission of photons of any energy from \(0\) to \(E\) (the third term of the first equation). The number of photons \(N_{\mathrm{ph}}\) of the given energy interval between \(E\) and \(E+dE\) will increase through the emission of radiation by electrons with energies from \(E\) to \(\infty\) (the first term of the second equation) and will decrease because of the production of pairs with component energies from \(0\) to \(E\) (the second term of the second equation). Although instead of a system of linear integro-differential equations we here have aggregate integro-differential equations (let us also recall that \(\sigma_{\mathrm{pair}}\) and \(\sigma_{\mathrm{rad}}\), under more exact consideration, are functions of the energies of the producing and produced particles), the character of the very complicated solution remains essentially quite similar to the familiar picture of the gradual growth of a secondary radioactive element and then its decay with time. The maximum at the atoms of a radioactive element at a certain moment of time \(t\) will correspond in our case to the maximum of the number of particles at some depth \(l_{\max}\). By analogy with the problem of radioactive decay, for the number of electrons or photons we shall have a solution of the form \(N = N_0 e^{-\lambda(E)l}\), but, as has already been emphasized above, our \(\lambda(E)\) will now be a function of energy and, it turns out, can take both positive and negative values. For \(\lambda(E)<0\) the number of particles increases with depth \(l\); for \(\lambda(E)>0\) the number of particles decreases.

In Regener’s curve, the existence of a maximum in the number of cosmic rays (measured by Pfotzer every 4 min by the number of triple coincidences of counters) is extremely significant. Contrary to doubts expressed more than once and to earlier uncertain observations, Regener and Pfotzer indisputably found, at a mercury-column pressure of \(8\ \mathrm{cm}\), i.e. approximately at an altitude of \(16\ \mathrm{km}\), a maximum in the number of particles and a considerable decrease in the number of particles upon further ascent, i.e. on approaching the boundary of the atmosphere. Extrapolation to the boundary of the atmosphere from the altitude corresponding to a pressure of \(1\ \mathrm{cm}\) of mercury (\(29\ \mathrm{km}\)) proves to be quite unambiguous and, with complete obviousness, shows a considerable number of charged particles in the cosmic flux outside the atmosphere. At sea level, at a latitude of about \(50^\circ\), the number of triple coincidences in 4 min proved to be 7; at the maximum, 250; at the boundary of the atmosphere, approximately 100 coincidences.

The initial rise of Regener’s curve from the boundary of the atmosphere is explained, evidently, according to Heitler, by the rapid increase in the number of secondary, tertiary, etc. particles, which then, having penetrated into the atmosphere and reached a lower energy, are no longer capable of further multiplication and begin to be strongly absorbed owing to energy losses by ionization. Heitler and Nordheim succeeded in selecting, for the primary soft radiation, a theoretical energy spectrum beginning at \(3\cdot 10^9\ \mathrm{eV}\) and decreasing toward higher energies inversely proportional to the second or third power of the energy

\[ \left(\text{the number of particles is proportional to } \frac{dE}{E^{2.5}}\right). \]

Thus the boundary of applicability of the formulas of bremsstrahlung, and therefore also of quantum electrodynamics, is pushed back at least to \(3\cdot 10^9\ \mathrm{eV}\), i.e. much farther than \(137\,mc^2\) (\(\sim 7\cdot 10^7\ \mathrm{eV}\))*).

A completely analogous, and moreover also quantitative, explanation is obtained by Rossi’s curves, who measured the number of triple coincidences in counters placed under a lead plate as a function of the thickness of the lead. As the thickness is increased, the number of coincidences again first increases, reaching a maximum at a certain plate thickness, and then begins to decrease. One may say that Rossi’s curve on a small scale—thanks

*) If, however, one adopts for a moment the point of view of Blackett, who insists on the absence of significant losses to radiation at high energies beginning with \(5\cdot 10^9\)–\(10\cdot 10^{10}\ \mathrm{eV}\) for air or \(2\cdot 10^8\)–\(3\cdot 10^8\ \mathrm{eV}\) for lead, and does not attribute these points, together with Anderson, to hard particles, then even then the theoretical explanation of Rossi’s curve and Regener’s curve, etc., according to Heitler will in the main remain valid; but the phenomena at sea level will, of course, receive a different interpretation, since, at sea level according to Blackett, primary electrons of such high energy can arrive that they no longer obey the quantum theory and do not emit bremsstrahlung photons; according to Heitler—Anderson, either secondary, tertiary, etc. particles, or particles of the hard component, can reach sea level.

toward greater absorption in lead—repeats Regener’s curve for absorption in the atmosphere.

The fact that the characteristic ionization boundary \(E_i\) has, for aluminum, for example, a larger value than for lead [in view of the fact that ionization losses increase proportionally to the atomic number \(Z\), while radiative losses are proportional to \(Z_2\), see formulas (5) and (6)], directly leads to an explanation of a number of typical phenomena in cosmic rays. According to the Heitler–Bhabha cascade theory, the number of multiplication stages in lead will obviously be greater than in aluminum, i.e. showers with a larger number of particles will emerge from a lead plate. The mean energy of the shower components will, however, in the case of aluminum be higher, which is also fully confirmed by the experiments of Chien-Shan and his collaborators.

The so-called transition processes observed when cosmic rays pass from one substance into another receive an analogous interpretation. For example, a lead plate placed behind an aluminum one will, according to the foregoing, promote further fragmentation of the energy (from \(6\cdot 10^7\) eV to \(10^7\) eV) and an increase in the number of particles. Rays, however, that have passed through lead will no longer be able to undergo multiplication in an aluminum plate, but, because of the higher ionization boundary, will undergo absorption in it—again in agreement with the experiments of Steinke, Schindler, and others on comparatively thin plates.

Thus, we may assume that the soft rays, i.e. the most considerable part of the primary cosmic radiation, consist of positrons and electrons mainly with an energy of about \(3\cdot 10^9\) eV (for precisely at such an initial energy Heitler’s theory gives a maximum of the proper magnitude and at the required altitude). These primary particles themselves do not reach sea level, but secondary particles generated by the shower process do reach the latter.

In view of the fact that \(\gamma\)-rays participate in the multiplication process on equal terms and in equal number, admitting them in some quantity as part of the primary radiation is possible, but is not required by any facts.

The primary electron must have the same energy, \(3\cdot 10^9\) eV, in order to reach the top of the atmosphere at latitude \(50^\circ\). At the equator the minimum initial energy must be, in view of the deflecting magnetic field of the Earth, considerably greater, namely about \(3\cdot 10^{10}\) eV. Thus the majority of the primary rays do not reach the equator, which agrees well, for example, with Clay’s observations, which found a considerable decrease in ionization at great altitudes near the equator in comparison with observations at latitude \(50^\circ\). In addition, because of the large initial energy, equatorial electrons must give a maximum of secondary and so forth particles at a greater depth of the atmosphere, which also agrees with Millikan’s observations. By connecting the formation of showers with the rays of the soft component, the theory obviously predicts a more rapid increase in the number of showers with altitude than the increase of the total radiation (i.e. the sum of the soft and

rigid rays). The observations of Woodward in the mountains and of Braddick and Gilbert in an airplane up to an altitude of 10 km in fact confirm the rapid increase in the number of showers in parallel with the increase in the intensity of the soft rays.

B. Explosive showers. Since, according to the Heitler–Bhabha theory, showers of electrons–positrons can be attributed only to the soft component, i.e. to the very primary electrons–positrons, the question remains unexplained of showers at great depth under water, undoubtedly connected with hard rays. Further, according to the observations of Johnson and Read, the intensity of showers gives a geomagnetic latitude effect, approximately half the usual one, i.e. 6–10%. Such showers must also be connected with the penetrating component, for soft rays in general cannot give a latitude effect at sea level (which, incidentally, was directly shown by Leprince-Ringuet, who found for hard rays, after filtering out the soft ones, a latitude effect of the same magnitude as for the entire radiation).

Although the whole problem of the hard component is only beginning to be outlined within its bounds, it is of interest to indicate here other fundamental possibilities for the mechanism of showers. In contrast to the cascade showers of Heitler–Bhabha, one may try to consider showers as a simultaneous act, if one introduces a connection of a new, non-electromagnetic type between the particle producing the shower (say, for definiteness, a proton) and the emitted particles. It is known that a proton can, interacting with an atomic nucleus, emit a pair: a positron and a neutrino, and turn into a neutron. Fermi’s theory of β-decay, although still far from complete, makes it possible, at least for the present rather qualitatively than quantitatively, to describe similar processes and predicts that the subsequent transformation of the neutron back into a proton with emission of an electron and a neutrino may with high probability immediately follow the first. Thus, as Heisenberg pointed out, a very fast proton (or neutron), flying through a layer of matter, may, interacting with some atomic nucleus (β-forces, and not electromagnetic interaction), be transformed many times into a neutron, back into a proton, and so on, emitting an entire shower of electrons, positrons, and neutrinos. It is remarkable that, owing to a different, Fermi (and not the usual quantum-electrodynamical), mechanism, the probability of such an \(n\)-fold simultaneous process (“redressing” of the proton) will, at high energies, beginning from some critical one, be of the same order as the probability of a single formation of an electron–neutrino pair; for, unlike quantum electrodynamics, in the theory of nuclear forces there is a new constant of the dimension of length, connected with the constant \(g\) of Fermi’s theory of β-decay, according to which the decomposition occurs in calculating higher-order effects. Multiple processes with a wavelength \(\lambda\) of the order of the critical

\[ \lambda_0=\sqrt{\frac{g}{hc}} \]

(in the simplest va-

variant of Fermi’s theory) turn out to be just as probable as single acts, owing to the fact that the decay parameter \(\frac{\lambda}{\lambda_0}\) will no longer be small, unlike the parameter \(\frac{1}{137}\) in the case of electromagnetic forces. Such simultaneous emission of many particles will correspond to an “explosive,” and not a cascade or stepwise, formation of the shower. Thus, experiments should in particular reveal the presence of hard neutrinos in cosmic rays, if Heisenberg’s theory is to be believed.

Apparently, the assumption of an analogous mechanism for the emission by a proton of an electron–positron pair (and not an electron–neutrino pair) has just as reasonable a meaning. Such Fermi emission of a pair may be called “direct,” in contrast to the Dirac, quantum-electrodynamical one, where in essence the proton first emits a \(\gamma\)-quantum, which then forms the pair. For illustration, let us write one of the forms of the interaction energy of a proton with a pair, which leads to the direct emission of an electron–positron: \(U=\alpha \psi_{\mathrm{el}}\psi_{\mathrm{pos}}\); in Fermi’s case the energy is \(u=g\psi_{\mathrm{el}}\psi_{\mathrm{neutr}}\).

Moreover, there is no reason not to apply the same mechanism of “direct” emission to the radiation of a pair by the electron itself. A very fast electron, interacting with an atomic nucleus, may either emit a hard braking quantum of light, or emit the same energy in the form of a pair: electron–positron. Heisenberg’s argumentation is wholly applicable here, and we obtain a new, purely “nonlinear” (the electron produces an electron) mechanism of shower formation in the form of a simultaneous “explosion.” The multiplication of particles will here be a nonlinear effect, somewhat analogous to the multiplication and demultiplication of frequencies in radio engineering. Perhaps there is also a share of truth in the rather fantastic picture of showers drawn by Born, where the proton “broke up” into parts, turning into a positron (precisely because here, too, a nonlinear effect is involved), if one excludes Born’s unacceptable hypothesis of the electromagnetic character of the proton mass. The latter, according to modern views, must be due to specific nuclear \(\beta\)-forces, which determine the connection between the neutron and the proton and lead to \(\beta\)-decay.

It is possible that a small part of the showers even of the soft component, namely showers with a very large number of particles, arises not by the cascade multiplication process, but by the nonlinear explosive path described; the investigation of this, independently of cosmic rays, is of great interest as a possible extension of quantum mechanics. A number of ambiguities in experiments on the scattering of fast electrons, for example Skobeltsyn’s observations, which do not fit within the framework of the theory, compel us to take into account all new possibilities of interaction of the electron with matter.

In contrast to the numerous quantitative confirmations

according to Heitler–Bhabha’s cascade theory of showers; for the direct or explosive production of showers (nonlinear, with the formation of electron–positron pairs, as we propose, or Heisenberg’s—from heavy particles with the participation of neutrinos) we still have only unclear evidence. Thus, Fussell, experimenting with several lead plates inserted into a Wilson chamber, along with typical multiplication processes, observed in 3 cases out of 900 showers as if of an “explosive” type, moreover containing tracks of heavy particles. The latter possibly indicates the participation of Fermi $\beta$-forces.

Nishina’s experiments further show that thin plates of $10$–$20\ \text{cm}$ of aluminum or $2$–$4\ \text{cm}$ of iron give large showers (or Hoffmann bursts) with hundreds of particles much more often than follows from cascade theory, or from comparison with showers from lead if the latter are explained wholly by Heitler multiplication; it is thus possible that here we are dealing with explosive showers. According to Wataghin, the question can be resolved by investigating showers in thin layers as thoroughly as possible: the number of explosive showers should increase linearly with the thickness, while that of cascade showers—more rapidly.

4. Discovery of the Heavy Electron

Without dwelling on other aspects of Heitler’s shower theory, the entire turning-point significance of which at the present day in the field of understanding cosmic rays is hard to overestimate, let us return to the problem of the hard component and to the paradoxical results of Anderson and Blackett, which seemed to refute the theory. It is clear that the explanation of the nature of hard rays now becomes still more difficult, since electrons and positrons have been assigned definitively to the soft component, while penetrating protons had not yet been detected. In this state of affairs a natural step was taken by Anderson and Neddermeyer, and also by Street and Stevenson, at the end of 1936 and the beginning of 1937, who, on the basis of repeated measurements of absorption, advanced the hypothesis of the existence of a new “semi-heavy” particle.

Continuing his investigations in a Wilson chamber controlled, after Blackett, by two counters placed before and after the chamber (“the particle itself gives notice of its passage through the chamber and the counters”), Anderson uses the data of a series of 6000 photographs taken with a platinum plate of $1\ \text{cm}$ in the chamber (equivalent to $1.96\ \text{cm}$ of lead), and plots a curve of the dependence of the energy loss on the magnitude of the energy in the region up to $500\cdot 10^6\ \text{eV}$ for 55 points (compare the earlier 9 points up to $300\cdot 10^6\ \text{eV}$). The decisive step consists in dividing all the values into two groups: the first is associated with particles producing showers or entering into the composition of showers. As was to be expected from the Heitler–Bhabha theory, the energy loss continually increases with increasing energy of these

shower particles, evidently soft particles. The other group of points, belonging to single particles not connected with showers, i.e., evidently to hard penetrating cosmic rays, shows a considerably smaller loss of energy. For them all the energy losses are apparently due only to inelastic collisions, i.e., ionization. The misunderstanding with Anderson’s earlier experiments is now explained very “simply”: the last three points out of 9 belonged to the hard group and, naturally, showed a smaller energy loss in comparison with the preceding points of the soft group1.

Fig. 5. Results of measurements of energy loss as a function of energy according to Anderson and Neddermeyer. The values for shower points (crosses) fall on the theoretical straight line for electrons and positrons (the soft component of cosmic rays). Single particles (points) turn out to be considerably more penetrating.

By removing the contradiction with theory and quantitatively confirming the formulas for the ever-increasing magnitude of the energy loss for electrons, Anderson and Neddermeyer further note that the tracks of penetrating, or single, particles do not differ noticeably from the tracks of soft rays, i.e., cannot be attributed to protons, which would have to ionize much more strongly.

A concrete example is furnished by one of the single tracks of the penetrating group, for which, according to Street and Stevenson, the product of the magnetic field \(H\) and the radius of curvature \(\rho\) was \(4.5 \cdot 10^5\) eV, and the total range only \(1\ \mathrm{cm}\), whereas the track in the chamber has a length of \(7\ \mathrm{cm}\). The same conclusion about the impossibility of attributing the observed tracks to protons can be drawn on the basis of a calculation of the ionization density, to facilitate which Street and

Stevenson photographed tracks with a delay of one second, which allowed the ions to disperse considerably.

The way out of the difficulty, as was mentioned, is to admit a new particle with a mass greater than that of the electron by about a factor of 100. Having a mass greater than that of the electron, the new “semi-heavy” particle (“heavy electron”) will not emit braking photons so intensely, i.e., it will be more penetrating, approaching in its properties the heavy protons, since the radiation is inversely proportional to the square of the mass; on the other hand, owing to the value of the mass of the “semi-heavy” particle being smaller than that of the proton, in this range of velocities it will not ionize as intensely as protons. Nishina, together with Takeuchi and Ichimiya, also arrives at a similar conclusion about the necessity of introducing “heavy” electrons in a recent paper; on the other hand, Bhabha still appears inclined to ascribe the hard rays to electrons, for which the theory at ultrahigh energies of the order of \(10^{12}\) eV is “after all” not applicable (although Bhabha has not yet spoken in print about Anderson’s hypothesis).

It must be emphasized very strongly that the solution of the question of a new particle at the present moment depends entirely on experiment, since theorists can say nothing either about the necessity of a new (possibly unstable) particle, or “forbid” it on the basis of any reasonable considerations. The theoretical situation in this respect differs sharply from the situation at the discovery of the positron, the proof of whose existence, as is known, was given on the basis of a completely analogous analysis of Wilson tracks. It is not excluded that the future field theory, along with solving the question of the masses of the electron and proton, will have to be able to describe hypothetical intermediate “semi-heavy” particles as well, possibly even with different masses. The small number of measured tracks of the new particle and the complexity of their interpretation compel us to treat the question of its existence with the greatest caution. It is all the more premature to decide the questions of where semi-heavy particles go when they enter terrestrial conditions, and, still more, where they come from. Since the tracks of individual particles have both signs of curvature, they probably arise in pairs, like electrons–positrons. The reduction of ionizing ability in the energy region around a million volts in comparison with electrons could be achieved not only by increasing the particle mass, but also by reducing the charge. The latter, however, appears considerably less probable.

We see that the past year has brought a substantial turning point in the understanding of cosmic rays. It is for the future to determine whether showers are formed by hard rays, or whether part of the showers are formed by soft rays through an explosive nonlinear or other mechanism; in any case, in its essentials the multiplicative picture of the enormous majority of showers from particles of the soft component, i.e., from electro-

... of electrons and positrons. The success of the theory of bremsstrahlung and of the shower theory broadens the limits of applicability of quantum theory and, most importantly, eliminates the general, unproductive tendency to seek the boundaries of the theory.

Further, especially for experimentalists, there remains a difficult but most rewarding task: the final clarification of the cause of the apparent decrease in energy losses at ultrahigh energies—whether it is caused by the inapplicability of quantum theory with its formulas for bremsstrahlung (Blackett), or whether quantum mechanics must be applied here to a new object: the “heavy electron” (Anderson’s new point of view). Just as in resolving the question of the apparent nonconservation of energy in the case of \(\beta\)-decay, modern physics will apparently decidedly prefer to leave quantum theory in force (as in \(\beta\)-decay—the conservation law), introducing the hypothesis of a new “semihard” particle (as in \(\beta\)-decay—the neutrino hypothesis), rather than proceed along the path of renouncing the applicability of our basic theoretical conceptions. It is needless to emphasize that the final admission of the “heavy” electron, on equal rights, into the family of elementary particles will extraordinarily enrich our conception of the structure of matter.

Addendum at proof correction

The system of fundamental integro-differential equations of the multiplicative theory of showers is so complicated (even when the Compton effect and ionizing collisions are neglected, and with the choice of simple effective cross sections admissible in the region of high energies) that for a long time it was not possible to obtain an explicit solution. Bhabha and Heitler confined themselves to a numerical solution, presenting the results in the form of curves; Oppenheimer and Carlson replace the system of basic equations by a certain auxiliary one, the meaning of which is not especially clear.

Without citing all the calculations (recently carried out by Sokolov and the author), let us note that the simplest path to the required formula for the distribution of the number of particles as a function of energy and depth can be obtained if one uses two mathematical devices. First, it is convenient to apply the so-called Laplace–Mellin transformation, which is a kind of “Fourier expansion” in different powers of the energy \(E\)

\[ f(E)=\frac{1}{2\pi i E}\int_{\delta-i\infty}^{\delta+i\infty} f(s)E^{-s}\,ds;\qquad f(s)=\int_{0}^{\infty} f(E)E^s\,dE \]

of any function \(f(E)\), for example, the number of particles \(N(E)\). Then our integro-differential equations for \(N_{el}(E)\) and \(N_{ph}(E)\) pass into ordinary differential equations for the coefficients of the expansion \(N_{el}(s)\). These latter equations also have the above-mentioned known solutions in the form of exponential functions of the type \(N(s)=Ae^{-\lambda(s)l}\), etc.

Applying then, in evaluating the integral over \(s\), the saddle-point method, we finally obtain for the number of particles the formula:

\[ N(E)=\frac{1}{E\sqrt{2\pi l\lambda''(s_0)}}\, e^{-\lambda(s_0)l+\gamma s_0} \left(1+\text{small terms of order } \frac{1}{e}\right). \]

The saddle point \(s_0\) is determined from the condition \(\lambda'(s)=0\), where \(\lambda(s)\) is a known function of \(s\), constructed from the given expressions for the effective cross sections. The formula obtained makes it possible simply to estimate the errors of the various derivations and shows that the whole theory has a reasonable meaning at depths \(l\) that are not too small. This circumstance is in full agreement with the physical meaning of the problem, since at small depths (i.e., in the first stages of the cascade process) the fluctuations of the multiplication process will play an essential role, so that the use of the basic equations, which have a statistical character, will be unreasonable.

LITERATURE

Quantum electrodynamics:

W. Heitler, The Quantum Theory of Radiation, Oxford, 1936.
D. V. Skobeltsyn, Cosmic Rays, L., 1936.

Experimental surveys:

A. H. Compton, Rev. Sci. Instruments, 7, 71, 1936; Phys. Rev., 50, 1119, 1936.
P. M. S. Blackett, Cosmic Radiation, Kharkov, 1935.
E. Regener, Naturwiss., 25, 1, 1937.

Theory of showers:

H. J. Bhabha and W. Heitler, Proc. Roy. Soc., 159, 432, 1937.
W. Heitler, Proc. Roy. Soc., 161, 261, 1937.
J. F. Carlson and J. R. Oppenheimer, Phys. Rev. 51, 220, 1937.
W. Heisenberg, Z. Physik, 101, 553, 1936.
H. Euler, Physik. Z., 38, 943, 1927.
D. Iwanenko, Sow. Phys. (in press).

New particle:

J. C. Street and E. C. Stevenson, Phys. Rev., 52, 1003, 1937.
S. H. Neddermeyer and C. D. Anderson, Phys. Rev., 51, 884, 1937.
I. Nishina, M. Takeuchi and T. Ichimiya, Phys. Rev., 52, 1198, 1937.

1. On the other hand, Blackett and Wilson, in their latest work, do not make a distinction between shower and single particles and therefore still insist on a decrease of the energy loss at high energies, contrary to the bremsstrahlung formula, and thereby support the viewpoint that the quantum theory of bremsstrahlung is unacceptable for high energies. Since Blackett offers no objections to Anderson’s latest work and does not take into account the later theory of Heitler, we are not inclined to regard Blackett’s results as any new arguments. In its tendency this work still stands wholly on the old, fruitless point of view. ↩↩