PASSAGE OF COSMIC RAYS THROUGH MATTER

L. V. Mysovskii, Leningrad

Introduction

The question of the nature of cosmic rays is closely connected with the question of the passage through a material medium of fast protons, fast electrons, and quanta of high energy. By studying the $\alpha$-, $\beta$-, and $\gamma$-rays of radioactive elements, we can, partly by extrapolation and partly by theoretical calculations, obtain an idea of how corpuscles and photons with energies of tens and hundreds of millions of volts should behave. Before proceeding to a detailed theoretical analysis of the question that concerns us—the passage of cosmic rays through matter—let us recall that the principal quantity characterizing the passage of a beam through a material medium, the absorption coefficient $\mu$, can be represented with sufficient completeness by the formula

\[ \mu = - \frac{1}{I}\frac{dI}{dx} \tag{1} \]

only in the simplest cases. From this formula, as is known, one can, for example, calculate the coefficient $\mu$ for a parallel homogeneous beam absorbed uniformly over the whole extent of its path. We shall obtain a much more general expression for the absorption coefficient if we set the intensity of a parallel beam $I = Z\varepsilon$, where $Z$ is the number of corpuscles or photons in the ray, and $\varepsilon$ is the energy possessed by one photon or one corpuscle. The absorption coefficient $\mu$ for the beam $I = Z\varepsilon$ can now be written in the following form:

\[ \mu = - \frac{1}{I}\frac{dI}{dx} = - \frac{1}{Z\varepsilon}\frac{d}{dx}(Z\varepsilon) = - \frac{1}{Z\varepsilon}\left(\varepsilon dZ + Z d\varepsilon\right) \]

or

\[ \mu = - \left(\frac{1}{Z}\frac{dZ}{dx} + \frac{1}{\varepsilon}\frac{d\varepsilon}{dx}\right). \]

We see that in this case absorption can proceed in two ways. The first way is a decrease in the number of elements of the beam, $Z$. The second way is a decrease in the energy of each element, $\varepsilon$.

In accordance with this, we obtain two absorption coefficients:

\[ \mu_1=-\frac{1}{Z}\frac{dZ}{dx} \]

\[ \mu_2=-\frac{1}{\varepsilon}\frac{d\varepsilon}{dx}. \]

A beam of \(x\)-rays from a radioactive substance is absorbed chiefly through the gradual loss of energy of each individual \(x\)-particle. Only an insignificant number of \(\alpha\)-particles is scattered through a large angle and leaves the beam. Hence the fundamental property of \(x\)-particles—their definite range for a given initial velocity. Conversely, \(\gamma\)-rays are absorbed mainly according to an exponential law (photoelectric effect or Compton effect). In the case of \(\beta\)-rays we encounter both kinds of absorption simultaneously. Part of the energy of \(\beta\)-rays is lost gradually to ionization along the path, while another part goes into the formation of secondary electrons which, in turn, move with high velocity. Thus the study of the absorption of fast electrons turns out to be the most complicated, and the absorption of photons somewhat less complicated. A special article by Heisenberg was devoted to the theoretical consideration of the absorption of electrons and photons with the energy of cosmic rays. The principal points of this article we shall consider in the present review.

I. Passage through Matter of Fast Electrons

Heisenberg considers only electrons with energy \(E\), which is much greater than \(mc^2\):

\[ E \gg mc^2 . \tag{1} \]

First of all he dwells on the absorption of electrons by loss of energy, or, as is sometimes said in this case, on the braking of the electron.

a) Braking of Fast Electrons

Let an electron whose velocity is very close to the velocity of light and equal to \(v\) move along the \(x\)-axis (Fig. 1). From the theory of the passage of \(\beta\)-rays through matter it is known that a free electron at rest, situated at a distance \(y\) from the path of the fast electron, receives an impulse

\[ p_y=\frac{2e^2}{vy} \tag{2} \]

and an energy \(\varepsilon\) equal to

\[ \varepsilon=\frac{2e^4}{mv^2y^2}. \tag{3} \]

Let us now find an expression for the energy lost by the primary electron per unit path length. Let us first express this energy in terms of \(y\). To obtain it, it is evidently necessary to multiply the number of collisions (7) by \(\varepsilon\):

\[ \frac{\Delta E}{\Delta x}=-\int 2\pi y\,dy\,N\cdot \varepsilon . \]

Replacing \(\varepsilon\) by its expression as a function of \(y\) on the basis of formula (3), we have:

\[ \frac{\Delta E}{\Delta x} = \frac{4\pi e^{4}N}{mc^{2}}\int \frac{dy}{y} = -\frac{4\pi e^{4}N}{mc^{2}}\ln \frac{y_{\max}}{y_{\min}} . \]

We may, however, express \(\dfrac{\Delta E}{\Delta x}\) not in terms of \(y\), but in terms of \(\varepsilon\), if instead of (7) we use formula (8):

\[ \frac{\Delta E_e}{\Delta x} = \frac{2\pi Ne^{4}}{mc^{2}}\frac{d\varepsilon}{\varepsilon^{2}}\cdot \varepsilon = \frac{2\pi e^{4}N}{mc^{2}}\int \frac{d\varepsilon}{\varepsilon} = \frac{2\pi e^{4}N}{mc^{2}}\ln \frac{\varepsilon_{\min}}{\varepsilon_{\max}} . \tag{10} \]

It is obvious that the energy \(\varepsilon_{\max}\) which can be given up is equal to the entire energy reserve of the primary electron, i.e. \(E\). Consequently:

\[ \varepsilon_{\max}=E. \]

Hence, on the basis of (3):

\[ y_{\min}=\frac{e^{2}}{mc^{2}}\sqrt{\frac{2mc^{2}}{E}} . \]

The minimal energy \(\varepsilon_{\min}\) may be determined from the condition, given by Bohr, which consists in the fact that the duration of the collision

\[ \tau=\frac{y}{c}\frac{mc^{2}}{E} \]

must be small in comparison with the proper period of the electron,

\[ T=\frac{h}{E_a}. \]

\(E_a\) may be regarded as the ionizing potential for the given secondary electron.

We shall find the maximum value of \(y\) from the equation obtained by equating \(\tau\) and \(T\):

\[ \frac{y_{\max}}{c}\cdot \frac{mc^{2}}{E}=\frac{h}{E_a}. \]

Hence for \(y_{\max}\) we find:

\[ y_{\max}=\frac{E}{E_a}\cdot \frac{h}{mc}. \]

Knowing \(y_{\max}\), with the aid of (3) we find \(\varepsilon_{\min}\):

\[ \varepsilon_{\min} = \frac{2e^{4}}{mc^{2}y_{\max}^{2}} = 2\left(\frac{e^{2}}{hc}\frac{E_a}{E}\right)^{2}\cdot mc^{2}. \]

Substituting \(\varepsilon_{\min}\) and \(\varepsilon_{\max}\) into formula (10), we obtain the final expression for the slowing down of the primary electron:

\[ \frac{dE}{dx} = \frac{2\pi e^4 N}{m v^2} \ln \frac{\varepsilon_{\max}}{\varepsilon_{\min}} = \frac{4\pi e^4 N}{m c^2} \ln \frac{E}{E_a} \sqrt{\frac{E}{2mc^2}}\cdot \frac{hc}{e^2}. \tag{11} \]

Since different electrons in an atom possess different ionization potentials, for formula (11) it is necessary to find the corresponding mean value. Before doing this, Heisenberg, on the basis of the calculations of Bethe and Bloch, introduces into this formula a correction in the form of an additional coefficient \(2e^2/hc\) under the logarithm sign. It is not difficult to see that the logarithm will then have the following form:

\[ \ln \frac{2E}{E_a}\sqrt{\frac{E}{2mc^2}}. \tag{11a} \]

In order to obtain the total slowing down of an electron, the quantity (11a) must be summed over all the electrons of the atom. Let us note that in doing so one must also take into account the electrons inside the nucleus. Heisenberg divides the electrons in one atom with ordinal number \(Z_a\) into three groups, assigning to the electron of each group a certain average energy. Table 1 may serve as an illustration of his assumptions.

TABLE 1

In this table \(Z_a\) denotes the ordinal number of the element, \(Z_k\) the number of nuclear electrons, \(E_a\) the mean value of the energy, and \(Rh\) the energy of the hydrogen atom. As is seen from the table, Heisenberg assigns to each outer electron the energy \(Rh\), to each “free” nuclear electron the energy \(mc^2\), and to each electron in an \(\alpha\)-particle the energy \(30mc^2\). Omitting the intermediate calculations, we give the final result obtained by Heisenberg:

\[ \frac{dE}{dx} = -\frac{4\pi e^4 sL}{mc^2}\,B, \tag{12} \]

where \(s\) is the specific gravity of the medium, \(L\) is Loschmidt’s number, and

\[ B = 0.35 + 2.303\cdot \left\{ \frac{Z_a}{Z_a+Z_k} \left(4.876-\lg_{10} Z_a\right) + \lg_{10}\frac{E}{2mc^2} + \frac{1}{2}\lg_{10}\frac{E}{15mc^2} \right\}. \]

On the basis of (11) and (12), the range \(R\) of the primary electron can also be calculated. It turns out, with some approximation, to be equal to

\[ R=\frac{E}{mc^{2}}\left(4\pi sL\left(\frac{e^{2}}{mc^{2}}\right)^{2}B\right)^{-1} =\frac{E}{mc^{2}}\cdot\frac{1.67}{s\cdot B}\ \text{cm}. \tag{13} \]

Table 2 gives the ranges in water and lead calculated on the basis of formula (13) for various values of \(E/mc^{2}\).

TABLE 2

In Heisenberg’s opinion, even for the largest values of \(E/mc^{2}\), the values given by formula (13) cannot differ from the true ones by more than a factor of 2, provided that no other, as yet unknown phenomena have to be taken into account.

b) Scattering of Fast Electrons

Besides the gradual braking of electrons as they pass through matter, their scattering is also observed. If deviations from the path occur fairly often, the range of an electron may prove to be much smaller than that calculated by formula (13). In the case of multiple scattering, as Bothe has shown, one may speak of an exponential absorption law. To calculate the mean scattering range \(R_s\), Heisenberg uses the formula:

\[ \frac{1}{\alpha}=R_s=\left(\frac{E}{mc^{2}}\right)^{2}\cdot \frac{Z_a+Z_k}{sZn^{2}}\cdot 0.81\ \text{cm}. \tag{14} \]

Here \(\alpha\) is the absorption coefficient. Numerical values of \(R_s\) for lead and water are given in Table 3.

TABLE 3

In reality, when a primary electron passes through a material medium, both phenomena occur—both brak-

tion and scattering. Taking this circumstance into account, Heisenberg introduces the quantity of effective scattering \(R_{eff}\), connected with \(R_s\) and \(R\) by the approximate formula:

\[ R_{eff}=R_s\left(1-\frac{R_{eff}}{R}\right)^2 . \tag{15} \]

Table 4 gives the values of \(R_{eff}\) calculated by this formula for water and lead.

TABLE 4

From a comparison of the data in Tables 2, 3, and 4, one can see that the value \(R_{eff}\) for fast electrons coincides with \(R\); on the contrary, for slow electrons moving through a medium with small atomic weight, \(R_{eff}\) is determined chiefly by scattering, and not by braking. Directly connected with this is also the circumstance that, after passing through an absorbing screen, slow electrons are strongly deflected from their path, while fast ones are deflected very little.

c) Distribution of secondary electrons

Since the angular distribution of secondary electrons is extremely important for the study of cosmic rays, Heisenberg considers this question separately. Let \(z\) primary electrons fall in 1 sec on \(1\ \mathrm{cm}^2\) of absorbing plane in the perpendicular direction. We shall denote the initial energy of the incident electron by \(\varepsilon_0\). The question is: what will be the number of secondary electrons with energy between \(\varepsilon\) and \(\varepsilon+\Delta\varepsilon\)? Since fast electrons are of greatest interest to us, we may assume that they will be deflected from the primary ones only by a small angle.

Let us suppose that the energy \(\varepsilon\) is proportional to \(R_{eff}\). Then, if a secondary electron possessing energy \(\varepsilon\) has traversed a path \(x\) after its appearance, its energy at the moment of appearance will be equal to

\[ \varepsilon\left(1+\frac{x}{R_{eff}(\varepsilon)}\right). \]

The number of electrons with energy between \(\varepsilon\) and \(\varepsilon+\Delta\varepsilon\) and arising on the path between \(x\) and \(x+\Delta x\) will be:

\[ \frac{2\pi e^4 N\,\Delta x}{mc^2}\cdot \frac{\Delta\varepsilon}{\varepsilon^2\left(1+\frac{x}{R_{eff}(\varepsilon)}\right)^4}. \]

The maximum value of \(x\) is found from the condition

\[ \varepsilon_0\left(1+\frac{x_{\max}}{R_{eff}(\varepsilon_0)}\right)=E, \]

whence

\[ x_{\max}=R_{eff}(\varepsilon_0)\frac{E-\varepsilon_0}{\varepsilon_0}. \]

The total number \(\Delta z_1\) of secondary electrons with energies between \(\varepsilon\) and \(\varepsilon+\Delta\varepsilon\) is found by integrating with respect to \(x\) from \(0\) to \(x_{\max}\). Thus we have:

\[ \Delta z_1 = z\frac{2\pi Ne^4}{mc^2}\cdot \frac{\Delta\varepsilon}{\varepsilon^2} \left( R_{eff}(\varepsilon) - \frac{R_{eff}(\varepsilon)} {1+\dfrac{x_{\max}}{R_{eff}(\varepsilon)}} \right) \tag{16} \]

or, since for \(\varepsilon_0 \ll E\) we always have \(x_{\max}\gg R_{eff}(\varepsilon)\):

\[ \Delta z_1 = z\frac{2\pi Ne^4}{mc^2}\cdot \frac{\Delta\varepsilon}{\varepsilon^2} R_{eff}(\varepsilon). \tag{17} \]

For cosmic-ray electrons, instead of \(N\) one may put the sum of the outer and nuclear electrons contained in \(1\ \mathrm{cm}^3\). Then, bearing in mind the relation \(N=Ls\), we obtain:

\[ \Delta z_1 = z\cdot 0.30\cdot s\cdot R_{eff}(\varepsilon) \frac{mc^2\,\Delta\varepsilon}{\varepsilon^2}. \tag{18} \]

From formula (18) it can be calculated that, for small \(\varepsilon\), the number of secondary electrons in lead is smaller than in water. For \(\varepsilon>100\,mc^2\), on the contrary, the number of secondary electrons is greater in lead than in water. The general course of the number of secondary electrons with increasing energy \(\varepsilon\) is shown by the curve in Fig. 2. This curve shows that the number of secondary electrons rapidly decreases with increasing \(\varepsilon\).

Fig. 2.

If we assume that the quantity \(\dfrac{R_{eff}s}{\varepsilon}\) does not depend on \(\varepsilon\) (\(\varepsilon\) is approximately proportional to \(R_{eff}\)), then the number of secondary electrons can be obtained by integrating (18) between the limits \(\varepsilon_1\) and \(\varepsilon_0\):

\[ z_1 = z\cdot 0.30 \frac{R_{eff}(\varepsilon)s}{\varepsilon_1} mc^2\ln\frac{\varepsilon_0}{\varepsilon_1}. \tag{18a} \]

Putting, for example, \(\varepsilon_0=5000\,mc^2\), \(\varepsilon_1=5\,mc^2\), we obtain that for water \(z_1=0.35\,z\),

II. Absorption and Scattering of Hard γ-Rays

a) Klein–Nishina Formula

The number \(\Delta z_1\) of secondary scattered quanta with energy \(h\nu'\), according to Klein–Nishina, is expressed by the formula:

\[ \Delta z_1 = zN\Delta x \frac{e^4\pi}{mc^2 h\nu^2} \left( \frac{\nu'}{\nu}+\frac{\nu}{\nu'} \right) \Delta \nu' . \tag{19} \]

Here \(\nu\) is the frequency of the primary quanta and \(z\) is their number.

The angle between the direction of motion of the primary and secondary quantum can be determined from the relation:

\[ \nu'= \frac{\nu}{ 1+\frac{h\nu}{mc^2}(1-\cos\theta) }. \tag{20} \]

Let us assume that \(\nu' \ll \nu\); indeed, freeing ourselves from the denominator and dividing both sides by \(\nu\), we find:

\[ \frac{\nu'}{\nu} + \frac{2h\nu'}{mc^2}(1-\cos\theta) = 1. \tag{20a} \]

Putting \(\frac{\nu'}{\nu}=0\), we shall have:

\[ \sin \frac{\theta}{2} = \sqrt{\frac{mc^2}{2h\nu'}} . \tag{21} \]

The total number of quanta \(z_1\) will be found by integration. The limits are found from (20a). Indeed, on the basis of (20a) and putting \(h\nu \gg mc^2\), we have the right to write:

\[ \nu \gg \nu' \gg \frac{mc^2}{2h}. \tag{21a} \]

TABLE 5

Consequently, \(z_1\) will be expressed by the formula:

\[ z_1 = zN\Delta x \frac{e^4\pi}{mc^2h\nu} \int_{\frac{mc^2}{2h}}^{\nu} d\nu' \left( \frac{\nu'}{\nu} + \frac{\nu}{\nu'} \right) = zN\Delta x \frac{e^4\pi}{mc^2h\nu} \left( \frac{1}{2} + \ln\frac{2h\nu}{mc^2} \right). \tag{22} \]

Since the formation of secondary quanta is coupled with the scattering of primary ones, the absorption coefficient of the primary quanta is found from the formula:

\[ \mu = \frac{1}{z}\frac{dz}{dx} = - N \frac{e^4\pi}{mc^2h\nu} \left( \frac{1}{2} + \ln\frac{2h\nu}{mc^2} \right). \tag{23} \]

Denoting by \(f\) the ratio of the number of electrons participating in the scattering to the total number of electrons in the atom \((z_a+z_k)\), we may write formula (23) as follows:

\[ \mu = s\cdot f\, \frac{e^4\pi}{(mc^2)^2}\cdot \frac{mc^2}{h\nu} \left(\frac{1}{2}+\ln\frac{2h\nu}{mc^2}\right) = s\cdot f\,\frac{mc^2}{h\nu}\cdot 0.15 \left(\frac{1}{2}+\ln\frac{2h\nu}{mc^2}\right). \]

As is known, the mean free path of a photon is usually taken to be the quantity reciprocal to the absorption coefficient. Consequently,

\[ L_{h\nu} = \frac{1}{\mu} = \frac{h\nu}{mc^2}\cdot \frac{0.67}{s\cdot f\left(\frac{1}{2}+\ln\frac{2h\nu}{mc^2}\right)} . \tag{24} \]

It is interesting to note that, in Bohr’s opinion, the Klein–Nishina formula is valid up to \(h\nu=(800)^2 mc^2\). Heisenberg indicates, however, that if the radiation of the electron in the Compton effect is taken into account, then the limit of applicability of this formula must be very considerably lowered.

b) Scattering by the atomic nucleus

If all extranuclear electrons, with respect to quanta of high energy, may be regarded as free, this is by no means the case for electrons located inside the nucleus. Heisenberg supposes that for wavelengths greater than the diameter of the nucleus, the nuclear electrons will scatter coherently, and therefore the scattering will be proportional to the square of the number of free electrons. For example, for \(h\nu=100\,mc^2\), or \(\lambda=2.4\cdot 10^{-12}\) cm:

\[ f= \frac{ Z_a+\left(\frac{Z_k-Z_a}{2}\right)^2 }{ Z_a+Z_k }. \tag{25} \]

In this case, for oxygen \(f=1/2\), for lead \(f=6.6\). If \(h\nu>1500\,mc^2\), then \(f\) must be set equal to 1. Therefore one should expect that the scattering of quanta of medium hardness will be especially intense. For very hard rays the Klein–Nishina formula must be applied in its entirety. However, Heisenberg immediately makes a reservation here as well and points to the circumstance that almost all our assumptions about the behavior of nuclear electrons are hypothetical in character. In reality, electrons inside the nucleus may behave quite differently from how we presently imagine. There is all the more reason for such doubt since we still cannot, with full justification, apply to nuclear electrons the conclusions of modern quantum mechanics.

c) Distribution of secondary electrons

Denote by \(\Delta z_\varepsilon\) the number of secondary electrons with energy \(\varepsilon\). Since \(h\nu'=h\nu-\varepsilon\), then, dividing this equality by \(h\nu\), we find:

\[ \frac{\nu'}{\nu}=1-\frac{\varepsilon}{h\nu}. \]

Substituting \(\gamma/\nu\) and \(\gamma/\nu'\) in (19), we obtain:

\[ \Delta' z_2 = zN\Delta x\,\frac{e^4\pi}{mc^2h^2\nu^3} \left( 1-\frac{\varepsilon}{h\nu} + \frac{1}{1-\frac{\varepsilon}{h\nu}} \right)\Delta\varepsilon . \tag{26} \]

Here \(z\) denotes no longer primary electrons, but primary quanta. The angle \(\vartheta\), made by the direction of motion of the secondary electron with the direction of the primary radiation, is given by the equality:

\[ \sin\vartheta = \sqrt{ \frac{ 1-\frac{\varepsilon}{h\nu} - \varepsilon\,\frac{mc^2}{2(h\nu)^2} }{ 1+\frac{\varepsilon}{2mc^2} } }. \tag{27} \]

From this relation we see that for \(\varepsilon \gg mc^2\) the angle \(\vartheta\) will be very small.

Analogously to the preceding case (see secondary electrons produced by primary electrons), we may assume that the energy of a secondary electron at the moment of its formation at a distance \(x\) from the given point will be

\[ \varepsilon\left(1+\frac{x}{R_{\mathrm{eff}}(\varepsilon)}\right). \]

Since the quanta are absorbed along the path \(x\), their number at the moment of formation of the secondaries must be greater by a factor \(e^{\mu x}\). Taking all that has been said into account, we obtain for the number of secondary electrons at distances \(x\) and \(x+\Delta x\) the expression:

\[ ze^{\mu x}N\Delta x\,\frac{e^4\pi\Delta\varepsilon}{mc^2h^2\nu^3} \times \]

\[ \times \left( 1-\frac{\varepsilon}{h\nu}\left(1+\frac{x}{R(\varepsilon)}\right) + \frac{1}{ 1-\frac{\varepsilon}{h\nu}\left(1+\frac{x}{R(\varepsilon)}\right) } \right). \tag{28} \]

In order to obtain the total number of secondary electrons, this expression must be integrated with respect to \(x\). The upper limit of integration \(x_{\max}\) is found from the equality:

\[ \varepsilon\left(1+\frac{x_{\max}}{R_{\mathrm{eff}}(\varepsilon)}\right) = h\nu-\frac{mc^2}{2}. \tag{29} \]

Thus we can find the number of secondary electrons in “equilibrium” with the primary radiation:

\[ \Delta z_2 = \int_0^{x_{\max}} dx\cdot ze^{\mu x}\cdot N\frac{e^4\pi\Delta\varepsilon}{mc^2h^2\nu^3} \cdot \left( 1-\frac{\varepsilon}{h\nu}\left(1+\frac{x}{R}\right) + \frac{1}{ 1-\frac{\varepsilon}{h\nu}\left(1+\frac{x}{R}\right) } \right). \tag{30} \]

Introducing the new variable

\[ \xi = 1-\frac{\varepsilon}{h\nu}\left(1+\frac{x}{R}\right) \]

and reversing the limits of integration, we find:

\[ \Delta z_2 \int_{\frac{mc^2}{2h\nu}}^{\,1-\frac{\varepsilon}{h\nu}} d\xi = \frac{Rh\nu}{\varepsilon}\, zN\frac{e^4\pi\Delta\varepsilon}{mc^2h^2\nu^3} \left(\xi+\frac{1}{\xi}\right) \cdot e^{\mu R\left(\frac{h\nu}{\varepsilon}-1-\frac{h\nu}{\varepsilon}\xi\right)}, \]

\[ \text{* See 21a.} \]

If we assume that \(\mu R_{\varepsilon}^{h\nu}\) is very small, then, taking the integral, we obtain:

\[ \Delta z_2=\frac{R}{\varepsilon}z = \frac{e^4 z N \,\Delta \varepsilon}{mc^2\cdot h\nu} \left\{ \frac{1}{2}\left(1-\frac{\varepsilon}{h\nu}\right)^2 -\frac{1}{2}\left(\frac{mc^2}{2h\nu}\right)^2 +\ln \frac{2h\nu}{mc^2}\left(1-\frac{\varepsilon}{h\nu}\right) \right\}. \tag{31} \]

In Fig. 3 a curve is shown which gives the dependence of \(\dfrac{\Delta z_2}{\Delta \varepsilon}\) as a function of \(\varepsilon\). The curve is taken for water, and \(h\nu=5000\cdot mc^2\).

To calculate the total number of secondary electrons one may use the approximate formula:

\[ z_2 \sim \left(\frac{R}{\varepsilon}\right)_{\frac{h\nu}{2}} z = \frac{e^4 z L s}{h\nu\, mc^2}\ln \frac{2h\nu}{mc^2} = \left(\frac{R}{\varepsilon}\right)_{\frac{h\nu}{2}}\cdot mc^2\cdot 0.15\cdot s\cdot \ln \frac{2h\nu}{mc^2}. \tag{32} \]

For water and \(h\nu=5000\,mc^2\) this formula gives:

\[ z_2=0.19z. \]

It is interesting to note the difference between secondary electrons produced by primary quanta and primary electrons. In the first case, as we have seen, the intensity of the secondary electrons rapidly decreases with increasing energies; in the second case the secondary electrons are distributed almost uniformly over the whole spectrum (Fig. 3).

Fig. 3.

III. Comparison of the Theory with Experiment

In his paper Heisenberg indicates that the aim of his work consists only in analyzing the most important experimental data on cosmic rays from the point of view of modern physical theories. He does not attempt in any way to go beyond the limits of contemporary quantum mechanics, believing that at present this is scarcely possible. Nevertheless, the brilliant analysis of the experimental facts made by him is extremely interesting and will undoubtedly help to clarify the new material on cosmic rays that is being continuously accumulated by experiment. In what follows we shall somewhat change the order of exposition adopted by Heisenberg and shall begin with the question of the nature of the primary rays.

a) The Absorption Curve of Cosmic Rays

The question of the nature of cosmic rays has already been discussed more than once in the pages of Uspekhi fizicheskikh nauk, and therefore there is hardly any need to recall that this question in its...

lead one to ask whether the primary rays consist of electrons or of quanta of radiant energy. The first observations of cosmic rays, made with ionization chambers, led to an absorption curve which was explained as the absorption curve of quantum radiation. Observations with Geiger–Müller counters, and especially observations with Wilson’s chamber, showed that in cosmic rays we encounter electrons possessing an enormous store of energy. The most important experiments with Geiger counters and the first observations of cosmic rays in Wilson’s chamber have already been described in previous surveys. Therefore here we shall say only a few words about those works which confirm Heisenberg’s conclusions but appeared after his article had been printed. Let us first dwell on the work of the American physicist Anderson^13. In order to obtain, in Wilson’s chamber, long tracks of electrons from cosmic rays, the chamber was placed vertically. To determine the energy of the electrons, the chamber was placed between the poles of a strong magnet, producing a field with intensity up to 17 thousand gauss. The action of this field on an $\alpha$-particle with a range of 10 cm is shown in Fig. 4 (see the plate). We see that even the path of an $\alpha$-particle is appreciably curved. Electrons from $\alpha$-rays in such a field, however, give paths in the form of circles of small diameter. Some electron paths have so small a radius that, after several revolutions, their path merges into a small spot. The situation is otherwise with electrons from cosmic rays. Some of them possess such energy that they are almost not deflected at all by the magnetic field over the diameter of the chamber. For others the curvature of the path is still noticeable.

As an example, let us first give a photograph of the track of an electron, made by Anderson at a field intensity of 12 thousand gauss (Fig. 5). In this photograph it is evident that the electron has managed to make one and a half revolutions in Wilson’s chamber. The displacement of the ring is explained by the fact that the magnetic field at the center of the chamber is 10% stronger than at the edges. The energy of this electron, judging from the curvature, is equal to 8 million V. In the following Fig. 6 is shown the track of an electron which passed through a lead plate 6 mm thick, placed inside the chamber. On passing through the lead plate the track was displaced by 0.5°. Anderson estimates its energy at 600 million V. The photograph was taken at the same field intensity, 12 thousand gauss. A similar photograph is also given in Fig. 7. Here the displacement of the track after penetration through the plate proved somewhat larger, namely 0.8°. Anderson assigns it an energy of 450 million V. It must be said that Anderson does not venture to assert that the straight tracks observed by him belong to negative electrons. In some cases Anderson ascribes the track to a positive proton. However, the presence of protons in cosmic rays would have to lead to the appearance of comparatively thick tracks. The protons themselves must have a definite range, and at the end of their path

To L. V. Mysovskii’s article

Fig. 4.

Fig. 5.

Fig. 6.

Fig. 7.

Fig. 9.

in no way differ from the \(H\)-particles well known to us. Meanwhile, in Wilson’s chamber, traces resembling the \(H\)-particles from cosmic rays were not observed. True, in some cases Anderson had occasion to observe that the curvature of the path was directed to the side opposite to the curvature of the path of a negative electron, but, as we shall see below, another explanation must be given to this phenomenon. Soon after Anderson, a similar investigation was carried out by Kunze. Kunze’s apparatus differed somewhat from Anderson’s in that in his case the magnetic field was produced by a solenoid through which a current with a power of 500 kW was passed. Kunze in general confirmed the results obtained by Anderson. We shall not dwell here in any greater detail on the description of these very interesting investigations, but shall confine ourselves only to stating the fact that electrons of very great energy are present in cosmic rays. Whether these electrons originate from primary \(\gamma\)-quanta, or whether they themselves constitute the primary cosmic rays—this is what the question of the nature of cosmic rays at the present time comes down to. Heisenberg indicates what the absorption curves should look like in the first and in the second cases. If the primary cosmic rays are hard \(\gamma\)-quanta, then at very great altitude the ionization observed in an ionization chamber must be negligible and practically equal to zero. As the altitude decreases, the absorbing layer of the atmosphere will increase, and along with this the number of secondary electrons will increase until equilibrium is established between the primary \(\gamma\)-radiation and the secondary electrons (Fig. 8; the curve in the figure has a maximum). Subsequently the curve will show the absorption of the primary \(\gamma\)-radiation. If, however, the primary cosmic rays consist of electrons with energy greater than \(100\,mc^2\), then the course of the intensity curve will already be somewhat different. This curve should begin with some finite value and, at the moment of equilibrium with the secondary radiation, reach a maximum, and then fall as the number of primary electrons decreases (Fig. 10). Since first the electrons with lower energy will be absorbed, and then those with ever greater energy, the nature of the decline of the curve, evidently, will depend on how the energy is distributed among the primary electrons. Measurements of the intensity of cosmic rays at very great altitudes during flights of observers themselves into the stratosphere (Piccard) or of sounding balloons (Regener) could give us some information about the nature of cosmic radiation. Unfortunately, the data currently available are still far from sufficient for the solution of this question.

b) Deflection of cosmic rays in the earth’s magnetic field

As we have seen, it has recently proved possible to apply a magnetic field to the study of various electrons in cosmic ...

rays. Observations were also made indicating a deflection of cosmic rays in the Earth’s magnetic field. Geisenberg was not yet aware of these new data, and he considered this question only from a purely theoretical point of view, proceeding from the theory of polar auroras given by Störmer. According to this theory, the angle \(\Theta\) (measured from the magnetic pole), within which electrons can be encountered, is given by the equation:

\[ \sin \Theta=\sqrt{\frac{2R}{a}+\frac{R^{4}}{4a^{4}}+\frac{R^{2}}{2a^{2}}}. \]

Here \(R\) is the radius of the Earth, and the characteristic length associated with the magnetic moment of the Earth \(M=8.5\cdot 10^{25}\) CGS and the energy of the fast electrons \(E\) is expressed by the equation:

\[ a=\sqrt{\frac{Me}{E}}. \]

TABLE 6

Table 6 gives the values of the angle \(\Theta\) corresponding to different values of \(E\).

From this table we see that only electrons whose energy is greater than \(20\,400\,mc^{2}\) (or more than 10 billion V per electron) can reach the Earth’s surface in all regions. However, even for these electrons the dependence of the intensity on the latitude of the place should be quite noticeable. In fact, Compton, and also Clay, found that near the equator the intensity of cosmic radiation is 16% less than at other latitudes. Millikan, however, insists on his earlier conclusion that cosmic radiation is independent of the latitude of the place. In the very latest issue of Physical Review there is printed a report to the American Physical Society by Millikan and Neher, in which, on the basis of their observations with improved apparatus, the authors categorically insist that cosmic rays consist of photons. At the same time they believe that in the upper layers the photons of cosmic rays are not accompanied by secondary electrons, at least not in such quantities as could affect the uniform distribution of the intensity of cosmic radiation. It must be supposed that further experiments on the distribution of intensity with the latitude of the place and with altitude will give us sufficient material for a final clarification of the question of the nature of cosmic rays. At present, in Geisenberg’s opinion, one can only assert that if cosmic rays consist of electrons with energy \(>20\,000\,mc^{2}\) (only then will their distribution be uniform), then the formulas derived for the braking of electrons give values that are too small to explain the experimental absorption curve. If cosmic rays are considered to consist of photons capable of producing secondary electrons with an energy of \(10^{9}\) V, then the Klein–Nishina formula gives,

in turn, 25 times smaller in value than would follow from agreement with experiment. Leaving aside the coincidences in the Geiger–Müller counters, we shall dwell in greater detail on the second assumption and its consequences.

Counting coincidences in cylindrical counters. If the question of the nature of the primary cosmic rays still presents us with difficulties not yet overcome, then the nature and properties of the secondary rays fit more easily within the framework of existing theories. Let us consider first of all the experiments with Geiger–Müller counters. From the schematic drawing (11), given by Heisenberg, one can form an idea of the conditions for coincident readings of two Geiger–Müller counters. Coincidences may be observed in those cases when the same electron passes through both counters, or when the electron passes through only one counter but, by means of collisions, gives rise to another electron which, in turn, enters the second counter. In both cases a coincidence will be registered.

Fig. 8.

Fig. 10.

The most interesting, in Heisenberg’s opinion, is the long-known fact that, by placing screens between the counters and thus studying the absorption of cosmic rays, we obtain an absorption coefficient which, within the limits of experimental accuracy, agrees with the ordinary absorption coefficient measured by means of an ionization chamber. This circumstance indicates that, at sea level, in all our measurements we are dealing chiefly with electrons. If the primary cosmic rays are quanta, then they must be absorbed in the upper layers of the atmosphere, and consequently only the secondary electrons produced by them reach us. Such an assumption inevitably leads to the conclusion that very hard γ-quanta, capable of producing the fast electrons observed by us, must be absorbed approximately 25 times more strongly than follows from the Klein–Nishina formula. A possible way out of these difficulties with absorption will be indicated by us in the discussion of the new work of Blackett and Occhialini*, now [[unclear: text continues off page]].

* See the previous article in the present issue, p. [[unclear: page number]].

relating to the explanation of phenomena depending on secondary rays.

In Fig. 12, the absorption curve is given for the passage of cosmic rays from water into lead, and in Fig. 13, conversely, from lead into water. This effect of the transition layer is readily explained by the disturbance of the equilibrium between secondary and primary electrons. From formulas (18) and (18a) it follows that the number of secondary electrons in lead must be much smaller than in water. Experiments with transition layers show that the intensities in the transition layer—lead–water—change in the ratio \(8:1\), whereas the theoretical value obtained from the formulas proves to be equal to \(10:1\). Taking into account the assumptions made in deriving the formulas, it must be considered that the agreement between the experimental data and the theoretical data explains the transition-layer effect quite satisfactorily.

Fig. 11.

Fig. 12.

Fig. 13.

Conclusion

Bursts Caused by Cosmic Rays

The transition-layer effect just discussed once again confirms that the theoretical formulas known to us are fully applicable to secondary electrons observed at sea level. However, these same formulas give values that are too small both in the case of the hypothesis of primary quanta and in the case of the hypothesis of primary electrons. Observations made

... Blackett and Occhialini, apparently, provides the key to explaining the reasons for this discrepancy.

The advantage of Blackett and Occhialini’s apparatus consisted in the fact that it combined the action of counters and a Wilson chamber. Above and below the Wilson chamber, placed vertically, there was a Geiger cylindrical counter. The apparatus was arranged so that “coincidences” in the counters caused the pistons in the Wilson chamber to descend and actuated the shutter of the photographic camera. Thus photographs of tracks in the Wilson chamber were taken only in the case when a cosmic ray passed through the counters and the chamber. With the aid of such an apparatus it proved possible to show that, when a cosmic ray passes through, whole streams of electrons emerging from a single point are observed. The number of electrons in one bundle reached 20. Blackett and Occhialini called such streams of electrons “showers.” The application of a magnetic field showed that among the electrons making up the “showers” and emerging from a single center there are also positive electrons, whose existence followed from Dirac’s theory. As is known, such electrons had not yet been observed until now. Without being able, in this survey, to dwell in detail on the various very interesting particulars of this newly discovered phenomenon of “showers” and “explosions” of atoms under the action of cosmic rays, let us indicate only the significance which this phenomenon must have for the theory of absorption. In deriving all the formulae given here the phenomenon of the “explosion” was not taken into account. If the formation of an “explosion” must expend a much greater energy than the acceleration of only one electron, then it becomes quite clear why all the theoretical formulae derived earlier gave values that were too small for the absorption coefficients. The phenomenon of “showers” and “explosions” of atoms has only just been discovered; this phenomenon has been confirmed also in Anderson’s letter, printed in the latest issue of Physical Review; but no quantitative relations between the ray and the bundle of electrons it produces have yet been established. However, one can hardly doubt that the establishment of such a relation will shed entirely new light not only on the laws of absorption of cosmic rays, but also on their nature and origin.

At the same time, we shall evidently obtain still new and very interesting data on the structure of the nuclei of atoms of various elements.

References

1. Heisenberg W., Ann. d. Physik 5 F., Bd. 13, 1933.
2. Bothe H., Zs. Physik 76, 293, 1932.
3. Bothe W., Zs. Physik 54, S. 161, 1929.
4. Klein O. u. Nishina J., Zs. Physik 52, 853, 1929.
5. Anderson Carl. D. Phys. Rev. 41, 405, 1932.
6. Kunze P., Zs. Physik 79, 203, 1932; 80, 559, 1933.
7. Piccard, Naturwiss. 20, 1932.
8. Regener, Naturwiss. 20, 1932.
9. Compton, Phys. Rev. 41, 111, 1932.
10. Clay, Naturwiss. 20, 657, 1932; 21, H. 3, 1933.
11. Millikan R. A. a. Neher H. Vict., Phys. Rev. 43, 381, 1933.
12. Blackett P. M. S. a. Occhialini G. P. S., Pr. Roy. Soc., A. v. 139, 699, 1933 (a complete translation of this paper is to be found in the preceding article in the present issue of UFN).
13. Anderson Carl. D., Phys. Rev. 43, 368, 1933.