PASSAGE OF CHARGED PARTICLES THROUGH MATTER1

E. J. Williams, Manchester

Since the discovery of α- and β-particles, questions connected with their penetration into matter, the ionization which they produce, and the radiation which thereby arises have been subjected to serious study. In order to understand these processes it is necessary to know the structure and laws of motion of the particles of matter with which α- or β-particles collide, as well as the laws of these collisions. At first it was assumed that the collisions take place according to the laws of classical mechanics, and the principal aim was to obtain, from the observed effects caused by the collisions, information about the structure of the atom. This period ended with the establishment of the Rutherford model of the atom. Since then the problem of the passage of charged particles through matter has become a problem of atomic mechanics, and from this point of view we shall try to illuminate the course of development and the present state of this problem.

Let us first consider collisions in which the velocities of the incident particles are small compared with the velocity of light. In this case the problem does not require taking account of the principle of relativity, and in this region we may confidently make use of the foundations of quantum mechanics.

1. Nonrelativistic Region

Let us first consider the limits of applicability of classical mechanics and, for this purpose, turn to the uncertainty principle2. This principle establishes the limits within which the motion of a particle can be described with the aid of the concepts of space and time. For example, in order that it should be possible to determine the amount of motion of a particle with an accuracy \(\Delta p\), we must specify the coordinate of the particle with an accuracy no greater than \(\dfrac{\hbar}{\Delta p}\), where \(\hbar\)—constant-

...Planck’s constant divided by \(2\pi\). An infinitely exact definition of all quantities entering into classical calculations proves impossible. We must, therefore, put up with a certain “blurring” of the classical picture, and the applicability of classical mechanics is determined by the magnitude of this blurring. If it is small, the classical treatment is admissible and represents a sufficiently good approximation to the exact quantum theory. On the other hand, the “blurring” may completely destroy the classical picture, and in that case the classical treatment is inapplicable and devoid of any meaning.

Let us apply these considerations to the central problem of collisions—the collision of two free charged particles. Let one of them (\(B\)) initially be at rest, and the other (\(A\)) move with velocity \(v\). Let us denote the corresponding charges by \(Ze\) and \(ze\). In collisions the impact parameter \(\rho\), i.e. the smallest distance between the unperturbed paths of the particles \(A\) and \(B\), may on the average have all values; for a classical calculation we shall first consider a collision with a given definite value of the impact parameter, and then integrate over all values of the parameter. We may, while retaining the classical notions, put up with a small inaccuracy \(\Delta \rho\) in the values of the parameter, provided, of course, that the classical effects do not depend too noticeably on variations of the impact parameter. This obviously requires that \(\Delta \rho \ll \rho\). But this means that the momentum of the particle will be determined only with an accuracy of order \(\Delta p \sim \dfrac{\hbar}{\Delta \rho} \gg \dfrac{\hbar}{\rho}\). Let us now consider the amount by which particle \(A\) is deflected under the influence of particle \(B\) (scattering). This deflection\(^1\) will occur practically only during the time in which particle \(A\) is at a distance of order \(\rho\) from particle \(B\). The time of interaction of the two particles, which we may call the collision time, will therefore be of order \(\rho/v\). The force acting on particle \(A\) during this time will be of order \(\dfrac{Zze^{2}}{\rho^{2}}\), and consequently the momentum acquired, which determines the magnitude of the deflection, is of order \(\dfrac{Zze^{2}}{\rho^{2}}\cdot\dfrac{\rho}{v}=\dfrac{Zze^{2}}{\rho v}\). The preceding considerations show that, for the applicability of classical notions, it is necessary that the expected effect should, in view of the finiteness of the quantum of action, be only insignificantly “blurred,” i.e.

\[ \frac{\hbar}{\rho} \ll \frac{Zze^{2}}{\rho v} \]

or

\[ \frac{Zze^{2}}{\hbar v} = \frac{Zz}{\beta}\frac{e^{2}}{\hbar c} \sim \frac{Zz}{137\,\beta} = \gamma \ll 1, \tag{1} \]

\(^1\) For simplicity, we regard this deflection as small.

where by \(\beta\) is denoted \(\frac{v}{c}\), where \(c\) is the velocity of light, and 137 is the numerical value of the quantity \(\frac{\hbar c}{e^2}\). In the scattering of \(\alpha\)-particles (\(\beta \sim 0.05\), \(z=2\)) by heavy nuclei (\(Z\sim 80\)) we have \(\gamma \sim 30\). The circumstance that this quantity greatly exceeds unity means that consideration of this scattering by means of classical mechanics, as was done by Rutherford, is fully justified, and that the classical formulae satisfy the requirements of quantum mechanics and agree with experiment quite satisfactorily.

Let us now consider the collision of two electrons, or of an electron with a proton. We have \(z=Z=1\). Let the scattered particle be an electron with an energy of \(20\,000\) eV, as occurs in many experiments. The corresponding value of \(\beta\) is of the order of 0.25 and, consequently, \(\gamma \sim 0.03\). This quantity is much less than unity, and the classical calculation does not constitute any approximation to the quantum-mechanical one. It is all the more inapplicable for faster electrons. The notion of classical trajectories in this case is devoid of all meaning.

The scattered particle \(A\), with an arbitrary impact parameter \(\rho\), but a definite velocity, is described in quantum mechanics by a plane de Broglie wave. The field of the particle \(B\) scatters this wave in exactly the same way as a region with a variable refractive index scatters light waves. This method of treating collisions belongs to Born. Classical methods are applicable under the condition \(\gamma \gg 1\), because in this case the scattering field is so strong that the incident wave can be divided into small wave packets, whose refraction along a classical trajectory exceeds many times the diffraction due to their finite dimensions. In the case when \(\gamma \ll 1\), the situation is different, and the wave packets are mixed because of appreciable diffraction. The phase relations then acquire decisive importance, and it therefore becomes necessary to retain the description by means of an infinite wave. In this case the scattering is due chiefly to the interference of secondary waves arriving from various parts of the force field remote from one another. The problem is essentially of a wave character, and the agreement of the final result with Rutherford’s classical formula is merely accidental.

We now turn to a question of exceptional fundamental significance. The point is that the formal agreement of the results of wave and classical mechanics remains valid only in the case where both particles are different and both are free. If, however, both particles are identical, or one of them is bound, as for example an electron in an atom, then classical and quantum mechanics lead to results differing from one another to one degree or another. Thanks to these differences it has been possible to show experimentally that, under the condition \(\gamma \ll 1\), classi-

classical methods are inapplicable, whereas quantum-mechanical calculations lead to correct results.

Let us consider the case of two identical particles, for example two electrons, one of which is at rest before the collision. How can we distinguish after the collision which of them was at rest and which was moving? This is possible only by observing the collision process the whole time, keeping one of the electrons in view. But such observation would certainly produce a disturbance in the motion of the electron. In the case \(\gamma \ll 1\) this disturbance exceeds in magnitude the effect of their interaction, and therefore the only way to determine after the collision which of the electrons was initially at rest proves unsuitable. The fundamental principle of quantum mechanics requires that no more be asserted about a phenomenon than what, if necessary, could in principle be confirmed by experiment. Accordingly, the wave function describing, in the case \(\gamma \ll 1\), the two colliding electrons must not include within itself a distinction between them. A wave function—symmetric or antisymmetric in the coordinates of both electrons—satisfies this condition. Meanwhile the wave function corresponding to Rutherford’s scattering formula does not satisfy this condition, since it is nonsymmetric. The statistics of bound electrons indicate that, in the case of two electrons, we must use an antisymmetric wave function. This was first pointed out by Oppenheimer\(^{1}\), and these considerations were subsequently developed by Mott\(^{2}\). Mott showed that this effect, usually called exchange, for example reduces by a factor of two the probability of scattering through an angle of \(45^\circ\) obtained without taking this effect into account. In the case of \(\alpha\)-particles we must use a symmetric function, and Mott showed that in this case the probability of scattering through an angle of \(45^\circ\), on the contrary, is increased by a factor of two in comparison with what the classical formula gives. The study of the collision of two identical particles thus makes it possible to test experimentally the inapplicability, in the case \(\gamma \ll 1\), of classical methods.

In both cases an experimental test was carried out. For two electrons the necessary material was obtained from observations of the formation of forks in tracks in a Wilson chamber\(^{3}\). Forks are obtained in the case of a collision of an incident electron with an atomic electron, which in this case may be regarded as free. One of the tracks of the fork gives the path of the incident electron, the other that of the atomic electron. Experiment shows that the number of forks (of a given energy) is considerably smaller than classical theory requires, and is in satisfactory agreement with Mott’s quantum-mechanical estimate. Collisions of \(\alpha\)-particles were studied by observing the scattering of \(\alpha\)-particles in helium\(^{4,5}\). In this case the observed scattering at large angles is considerably greater than is given by the classical formula, and also agrees well with the value given by quantum mechanics. These experiments clearly show the inapplicability of classical mechanics in the case \(\gamma \ll 1\),

Let us now consider another effect, which we have already mentioned, in the presence of which, in the case \(\gamma \ll 1\), classical and quantum mechanics give different results: the effect arising in the collision of \(\beta\)- or \(\alpha\)-particles with bound atomic electrons. This is the fundamental problem of the slowing down of particles in matter, since in such collisions the charged particle loses a large part of its energy and, in the end, is slowed down. Before the advent of quantum mechanics, the classical method of treatment, proposed by Bohr\(^6\), was the only method for studying these processes, since the old quantum mechanics gave no indications as to how to take into account the finiteness of the quantum of action in the case of non-periodic motions.

Although Bohr made use of the concepts of classical mechanics, his interpretation of the influence of binding forces is of an entirely general character. As in the theory of the dispersion of light, the electron in the atom is regarded as performing harmonic oscillations with period \(T\) and frequency \(\nu\). In complete analogy with the fact that in the theory of dispersion the electron may be regarded as free if the frequency of the light is considerably greater than \(\nu\), so in the theory of collisions the electron may be regarded as free if the duration of the impact \(\tau = \frac{\rho}{v}\) is small in comparison with the electron period \(T\). If, however, the impact parameter \(\rho\) is so large that the impact time \(\tau\) is considerably greater than the electron’s own period (the adiabatic condition), then the binding forces reduce the loss of energy to a small fraction of that energy which is transferred to a free electron, and such collisions may be neglected in calculating the energy losses. The loss of energy by the electron occurs mainly in “free” collisions, for which

\[ \rho < \rho' \sim \frac{v}{\nu}, \tag{2} \]

where \(\rho'\), in turn, must be large in comparison with the dimensions of the unperturbed orbit of the electron in the atom. Since the size of the latter will be of the order \(\frac{u}{\nu}\), where \(u\) denotes the orbital velocity of the electron, this restriction on \(\rho'\) may be written in the form

\[ v \gg u. \tag{3} \]

This condition is in general satisfied by \(\beta\)-particles, and also by \(\alpha\)-particles if one restricts oneself to collisions with not too heavy atoms. Condition (3) ensures that, in the case \(\rho \gg \rho'\), the excitation energy is the same throughout the space occupied by the atom and is small in comparison with the binding energy of the particles in the atom.

It is important to note that in the case \(\gamma \ll 1\) the quantum calculation gives a smaller value for the energy loss than the corresponding classical one. The classical expression for the energy loss per 1 cm

paths in collisions with impact parameter $\rho < \rho'$ is given by Bohr’s well-known formula

\[ \frac{dT}{dx}=\frac{4\pi z^{2}e^{4}n}{mv^{2}}\,\lg\frac{1.123\,mv^{3}}{4\pi e^{2}\nu}, \tag{4} \]

where the argument under the logarithm is approximately equal to $\dfrac{\rho'}{\rho''}$ ($\rho''$ denotes the distance between particles at which the energy transfer is, in order of magnitude, equal to $mv^{2}$)¹). In this formula $n$ denotes the number of atomic electrons per unit volume and $\nu$ their natural frequency (or, more precisely, the geometric mean of the natural frequencies, since each electron in an atom has a whole set of frequencies). The quantum-mechanical formula for the energy loss over $1\ \mathrm{cm}$ of path has the form:

\[ \frac{dT}{dx}=\frac{4\pi z^{2}e^{4}n}{mv^{2}}\lg\frac{gmv^{2}}{h\nu}, \tag{5} \]

where the argument under the logarithm is approximately equal to $\dfrac{\rho'h}{mv}$.

This formula was first obtained by Bethe⁷, who used Born’s method. The numerical value of the factor $g$ for $\alpha$-particles is equal to 2, and for electrons to $\sqrt{\dfrac{e}{8}}$ ($e$ is the base of natural logarithms).

A comparison of both formulas with experiment is given in Table 1 for $\alpha$- and $\beta$-particles moving in hydrogen⁸, which is the most convenient for comparison, since calculations for hydrogen are free from errors arising from neglects.

TABLE 1

¹) We have substituted for $\dfrac{\rho'}{\rho''}$ its exact value obtained by Bohr. Of course, there is no sharp transition at $\rho'=\rho''$ between free and adiabatic collisions, but $\rho'$ has an exact effective value equal to $\dfrac{1.123\,v}{2\pi\nu}$. The exact value for $\rho''$ is equal to $\left(\dfrac{ze^{2}}{mv^{2}}\right)\left(1+\dfrac{m}{M}\right)$, where $m$ is the mass of the electron, $M$ the mass of the incident particle. For electrons and $\alpha$-particles $\rho''=\dfrac{2e^{2}}{mv^{2}}$, and this value is used in formula (4).

Let us note that although in the case of $\alpha$-particles there is no large discrepancy between the two theoretical values, even in this case the experimental results speak in favor of the quantum-mechanical calculation. As for electrons, for them the observed ranges are 50% greater than those calculated by the classical formula. It was precisely observations of the passage of electrons through hydrogen, carried out with a Wilson chamber 15 years after the derivation of the classical formula, that first clearly showed the limits of applicability of the classical theory to energy loss in collisions. We see again that the quantum-mechanical calculation is in excellent agreement with experiment, since a difference of a few percent can altogether be attributed to observational errors.

In order to consider what happens to an atomic electron in distant collisions, when an electron or $\alpha$-particles pass outside the atom, we may use the concept of the impact parameter even in the case where $\gamma \ll 1$. The mean energy loss in distant collisions, calculated by the methods of quantum mechanics, coincides with the mean energy loss $Q_{\mathrm{cl}}$, calculated classically $^{9,10}$. It is true that, under the condition $\gamma \ll 1$, $Q_{\mathrm{cl}}$ is much smaller than the ionization potential of the atom, and since the losses in distant collisions constitute approximately half of the energy losses given by formula (5), a purely classical treatment gives no ionization at all. On the other hand, according to quantum mechanics all energy losses occur at the expense of excitation and ionization. Therefore the two methods—the classical and the quantum—when calculating the number of ions formed give strongly differing results, although the magnitudes of the mean energy losses coincide. The number of ions formed per 1 cm of path according to quantum theory is considerably greater than follows from the purely classical theory.

For fast electrons the frequency of ion formation is so small that the droplets which they form in a Wilson chamber can be counted directly. In this way one may verify the statements of the theory on losses in different collisions. For electrons moving in hydrogen and having a velocity of $1.5 \cdot 10^{10}\ \mathrm{cm/sec}$, observations give 15 ions/cm $^{11}$. The theoretical value according to the classical theory (Thomson’s formula) is $3.5$ ions/cm. Quantum theory (Bethe’s formula$^{7}$) gives the value 13 ions/cm, in good agreement with the observed number. These observations show that the distribution, as well as the absolute value of the energy losses calculated by quantum theory, is in good agreement with experiment.

It should be noted that, for a quantitative test of the theory to be possible, it is substantially necessary that the velocity of the penetrating particle be greater than the velocity of the atomic electron. The case in which the particle velocity is less than the velocity of the atomic electron has also been subjected to serious study, revealing other “wave” features of the phenomenon considered. However, we shall not touch on this area in our survey, and shall pass on to the consideration of collisio-

...of particles moving with such a high velocity that it becomes necessary to take relativistic effects into account.

2. Relativistic Region

In collision problems it is often convenient to regard one of the particles as the “perturbing” particle and the other as the “perturbed” one. If such a distinction is possible, the inclusion of relativistic effects is often simplified. The essential condition under which such a treatment is possible is that the reaction on the perturbing particle be small. We shall consider two cases of collisions of this kind, encountered in the study of fast electrons in cosmic rays. These are the case of ion formation (primary ionization) and the case of radiation in nuclear collisions. In the case of ionization, the perturbed particle is the electron of the atom being ionized, while the perturbing particle is the fast electron of the cosmic rays. The condition for such a distinction is satisfied, since the energy transfer in the collision is only of the order of the ionization potential of the outer electrons (10–50 V). This quantity may be neglected in comparison with the energy of the ionizing electron, which for fast electrons \((v \sim c)\) reaches the order of \(10^6\) V. Therefore the reaction on it may be neglected, and it may be assumed that it moves throughout with a uniform velocity.

The collision of fast electrons with atoms is accompanied by radiation, since they undergo acceleration in passing through the fields of nuclei. In this case the perturbed particle is the radiating electron, while the perturbing particle is the atomic nucleus, whose motion, owing to its large mass, undergoes practically no change in the collision. Therefore, in that coordinate system in which the electron is initially at rest, it may be assumed that throughout the collision the nucleus continues to move with a constant velocity.

Let us now consider the phenomenon of ionization in somewhat greater detail. In view of the smallness of the energy transfer in ionizing collisions, one may neglect, in comparison with the velocity of light, the velocity of the atomic electron both before and after the collision. The behavior of the atomic electron may in this case be described by the equations of nonrelativistic wave mechanics; the calculation then does not differ from the calculation of ionization by slow electrons. The principle of relativity is taken into account only in calculating the perturbing field of the fast electron. Its motion may be regarded as uniform and rectilinear throughout the collision. The relativistic effect in this case will be the ordinary Lorentz contraction of the moving system. This contraction means that the collision time for a given impact parameter \(\rho\) is shortened by a factor

\[ \xi = (1-\beta^2)^{-\frac{1}{2}} . \]

The impact parameter \(\rho'\) [equation (2), see also equation (4)], for which the collision time is, in order of magnitude, equal to the proper period of the atomic electron, increases...

increases, therefore, by \(\xi\) times. This increases the energy loss and ionization, and the relativistic formula for the magnitude of the primary ionization can be obtained simply by adding the effect of this increase in the parameter \(\rho'\) to the nonrelativistic expression. We obtain

\[ I = G\,\frac{2\pi e^4 n}{mv^2 j} \left[ \lg \frac{gmc^2}{j} + \lg \frac{v^2}{c^2} + \lg \left(1-\frac{v^2}{c^2}\right)^{-1} - \frac{v^2}{c^2} \right] \tag{6} \]

(\(j\) is the ionization potential. For the hydrogen atom, \(G = 0.28\), \(g = 42\)). The relativistic correction is contained in the second part of the expression in brackets. The expression in square brackets increases with increasing velocity \(v\). For

\[ \frac{v}{c}\sim 0.97 \]

this increase begins to be compensated by the term containing \(v^{-2}\), and the whole expression has a minimum (Fig. 1).

Fig. 1.
Initial ionization in relative units. Electron energy (on a logarithmic scale), volts.

There are observations, made with the aid of a Wilson chamber, of the ionizing power of fast \(\beta\)-electrons from various radioactive substances \(^{11}\)

\[ \left(\frac{v}{c}\sim 0.96\right). \]

The results of the observations agree in general with formula (6) and indicate the necessity of taking the relativistic term into account. The energies of the fast \(\beta\)-rays of radioactive elements, however, are insufficient for it to be possible to notice the increase in ionizing power after passing through the minimum. Skobeltsyn’s discovery \(^{12}\), which detected in cosmic rays electrons of much higher energy than in \(\beta\)-rays from radioactive decay, should, however, have made it possible to observe this increase. Investigations by Anderson \(^{14}\), Kunze \(^{15}\), and Blackett and Brode \(^{13}\) showed that in cosmic rays there occur electrons with energies of not less than \(10^9\) V, and possibly even reaching \(10^{10}\) V. The ionizing power of such electrons is above the minimum obtained from the theoretical formula (6). However, the observations that have so far been made do not reveal an increase of ionization beyond the minimum. Before turning to the consequences arising from this disagreement between experiment and theory (we as yet have no precise experimental investigations), let us consider the state of the question of radiation in the relativistic region (bremsstrahlung).

We are interested in the radiation of an electron that initially has velocity \(v\), caused by the acceleration of the electron in a collision with an atomic nucleus. Let us pass to the reference system \(S'\), in which the electron is initially at rest. In this system the atomic nucleus

moves with velocity \(v\), and the perturbing field is of the same type as in the ionization problem. The radiation of the electron under the action of this field may be compared with the radiation of an electron excited by an incident light wave (scattering). Indeed, the radiation in the frame of reference \(S'\) can be calculated if we decompose the field of the nucleus into harmonic components and apply to each component the relativistic scattering formula obtained by Klein and Nishina\(^{16,17}\). A simple Lorentz transformation gives us the expression for the radiation in the original frame of reference, in which the nucleus is at rest. For electrons of very high energy the final formula for the loss of energy by radiation (taking into account the screening of the nucleus by atomic electrons) has the form:

\[ \left(\frac{dT}{dx}\right)_{\mathrm{rad}} = \frac{8\pi e^{6} Z^{2} n z}{hmc^{3}} \lg\frac{g\hbar c}{e^{2} Z^{1/3}}; \qquad g\sim 1. \tag{7} \]

This formula was first obtained by Bethe and Heitler\(^{18}\), who considered the radiation of an electron in passing from one stationary state to another. It turned out that in this energy region the loss of energy by radiation exceeds the energy loss caused by ordinary collisions with atomic electrons, a circumstance to which Heitler\(^{19}\) first drew attention. For electrons with energies not exceeding a million volts, the opposite situation holds, as is indicated by the comparatively small efficiency of X-ray tubes. In them only a very small fraction of the energy of the incident electrons is found in the form of X-rays, even when the anticathode consists of heavy elements.

Anderson\(^{21}\) succeeded, in experiments with cosmic rays, in determining the actual energy losses of fast electrons. These experiments clearly show that, in addition to ionization, there is some other mechanism operating which accounts for energy losses. There is no reason to doubt that this is bremsstrahlung. This is also indicated by the considerable fluctuation of the observed energy losses about the mean losses.

However, a quantitative comparison of the experimental results is not in agreement with the theory. Anderson’s experiments indicate that, for electrons with energies of the order of \(1—2\cdot10^{8}V\), the coefficient in the theoretical formula for radiation is about twice as large as it should be. The disagreement between experiment and theory at higher energies is still more sharply expressed. At energies of the order of \(10^{9}V\), the theory predicts such intense radiation that considerable effects should have been observed. However, the complete absence of such effects indicates that the intensity of the actual radiation amounts to only \(1\%\) of the intensity expected according to the theory.

Let us now consider what consequences follow from these experiments. It is clear from the very beginning that the disagreement between experiment and theory is so considerable that it cannot be due to any ...

be it a neglect or an approximation in the mathematical formulations. In both cases that we have examined, the calculations were based \(^{16a}\):

a) On the relativistic expression for the electromagnetic field of a particle moving uniformly and rectilinearly (at a large distance from the particle in comparison with its classical radius);

b) On the Lorentz expression \(F=(E+[H\cdot v])e\) for the ponderomotive force acting on an electron in a given field;

c) On the quantum mechanics of the electron.

There is no reason to doubt point (a). This proposition follows directly from Coulomb’s law for a stationary charged particle and the Lorentz transformation. The error must, therefore, lie in points (b) and (c). A more detailed investigation shows that, by expanding the electromagnetic field in a Fourier series, we can, on the basis of points (b) and (c), treat all Fourier coefficients independently of one another, which follows directly from the linear expression for the Lorentz force. Before considering the reasons why the Lorentz expression ceases to be applicable, let us note that (contrary to what has often been stated) the inapplicability of theoretical formulas for electrons of high energy is not at all connected with the fact that for them the de Broglie wavelength \(\lambda \sim h/mc\xi\) is equal (or smaller by an order of magnitude) to the dimensions of the classical electron radius \(\sigma \sim e^2/mc^2\). A detailed investigation shows that the ratio \(\lambda/\sigma\) for the cases of interest to us plays no role whatever. This effect would be significant only if the perturbed electron, in the coordinate system in which it was initially at rest, acquired a velocity corresponding to a de Broglie wavelength of order \(\sigma\). In the phenomena of interest to us, however, this does not occur.

Two considerations have been proposed concerning the limitation of the applicability of the expression for the Lorentz force.

a) That it ceases to be applicable when the field acting on the electron (in the coordinate system \(S'\), in which it is initially at rest) varies appreciably over a distance of the order of the classical radius \(\sigma\).

b) When the field strength in the system \(S'\) is greater than \(\dfrac{e^2}{b^3}\), where \(b^2=\sigma\cdot h/mc^{20}\).

If one admits that collisions in which these conditions are fulfilled produce no effect, then formulas are indeed obtained for the radiation and deflection that agree in their general features with experiment. However, neither of these hypotheses can explain the absence of a rise in the ionization curve (Fig. 1), since the above-mentioned conditions are practically never fulfilled in collisions accompanied by ionization. Thus, the behavior of fast electrons agrees poorly with existing theories.

PASSAGE OF CHARGED PARTICLES THROUGH MATTER

Author’s Addendum (September 1937)

Since the time when this article was written (May 1936), the state of affairs has changed substantially, owing to the hypothesis proposed by Neddermeyer and Anderson that not all particles in cosmic rays are electrons. As early as 1932 the author suggested that particles possessing energies above \(10^9\ \mathrm{V}\) are protons. If one calculates the theoretical value of the primary ionization for protons of such energy, it turns out to be approximately equal to the minimum ionization produced by electrons, and therefore agrees better with experiment. However, the observed small value for the energy loss cannot be reconciled with the assumption that the particles are protons. Indeed, although the energy losses by radiation are considerably

Fig. 2.

smaller for protons than for electrons, the energy losses by ionization should, on the other hand, have been much greater for protons. The situation is shown in Fig. 2. Along the ordinate axis is plotted the quantity

\[ R=\frac{(E_1-E_2)}{\frac{1}{2}(E_1+E_2)x}, \]

where \(E_1\) is the initial energy of the particle, and \(E_2\) is the energy of the particle after passing through a layer of \(x\) centimeters of lead.

Curve \(AB\) in Fig. 2 gives the theoretical values of \(R\) for electrons, obtained from the theory of Heitler and Bethe. Curve \(DE\) corresponds to the value of \(R\) for protons (\(x=1\ \mathrm{cm}\) in most experiments). The crosses denote the experimental values for \(R\) obtained by Blackett. We see that for energies greater than \(1\cdot 10^8\ \mathrm{V}\) the experimental points lie below both curves, for protons as well as for electrons. From the experimental data and the theory we can draw only two conclusions: a) that the theory ceases to be applicable at energies of the order of \(1\cdot 10^8\ \mathrm{V}\) and higher, or b) that the particles observed in these experiments are neither protons nor electrons, but particles of an entirely new kind. We have already discussed the first supposition above. The second suppositi-

...position was put forward by Neddermeyer and Anderson (Phys. Rev. Mat., 1937). The number of ions produced per unit path of the particle indicates that their charge is identical with the charge of the electron or proton. To obtain agreement with experiment, we must therefore ascribe to the new particles not a different charge, but a different mass.

Energy losses due to radiation are proportional to

\[ \frac{1}{(\text{mass})^2}. \]

The small loss of energy by the new particles can be explained if they are assigned a mass 5 times greater than the mass of the electron. Then the ratio of the actual mean energy losses to the energy losses calculated from the theoretical formulae is determined by definite relations between the number of new particles and electrons in cosmic rays. On the other hand, the displacement of the curve \(DE\) for protons along the abscissa axis is proportional to the mass, and the corresponding curve for particles with a mass of a smaller order—one fifth of the proton mass—would likewise pass near the experimental points. From these considerations we conclude that the mass \(\mu\) of the new particles lies in the range

\[ 5\,m < \mu < \frac{M}{5}, \]

where \(m\) and \(M\) are the masses of the electron and the proton.

The proof that Neddermeyer and Anderson adduce in favor of the existence of new particles is as follows. If one draws a graph representing the number of particles as a function of their energy loss, two different maxima are found: one in the region of absence of losses (the penetrating group), the other in the region of large losses, when the particle loses almost all of its kinetic energy. In addition, they find that the two groups differ in that the particles of one group are shower particles, whereas the particles of the penetrating group are single particles.

The most recent observations of Bleckett do not confirm these results of Neddermeyer and Anderson. For particles with energies from \(2 \cdot 10^8\) to \(4 \cdot 10^8\) V, Bleckett does not find the existence of two sharp maxima and, on the contrary, finds that both shower particles and individual particles lose energy to the same extent\(^1\). Moreover, Bleckett finds that the ratio of the actual energy losses to the losses calculated for electrons by the theoretical formula depends on the material of the medium, increasing somewhat in the case of light elements. If one accepts the hypothesis of new particles, however, this ratio should depend only on the percentage content of new particles and should not depend on the material of the medium.

In view of these contradictions, we are not at present in a position to decide on the basis of these experiments either in favor of the hypothesis of new particles or in favor of the hypothesis that the theory is inapplicable to fast particles.

There are, however, other arguments that speak rather in favor of the hypothesis of new particles.

\(^1\) In the region of high energies (of the order of \(5 \cdot 10^9\) to \(10 \cdot 10^9\) V), Bleckett likewise does not find any distinction.

We shall consider four such arguments:

1. In addition to measuring the energy losses of cosmic particles in passing through lead, Anderson in his earlier works also measured their scattering.

The theoretical formulae agree well, within the limits of experimental error, with Anderson’s data. In his latest experiments Blackett investigated in greater detail the scattering of cosmic particles with energies in the range from \(10^6\) to \(6 \cdot 10^8\) V. He likewise finds good agreement with theory. The energy losses for particles in this energy range amount to \(95\%\) of the losses calculated on the basis of the theory. From the point of view of general theoretical considerations it seems highly implausible that the theory should be applicable to the calculation of scattering in collisions and inapplicable to calculations of the radiation accompanying these scatterings. Therefore the applicability of the theory to the scattering process argues in favor of the hypothesis of new particles.

1. The cascade theory of showers, developed by Heitler and Baba, Carlson and Oppenheimer, and Landau and Rumer, explains the phenomenon of showers in good agreement with the experimental data. In this theory one has to deal with electrons of energies greater than that at which the theory must be regarded as unsuitable for explaining the results of the experiments of Anderson and Blackett. If the agreement of the cascade theory of showers with experiment is not to be considered accidental, then the modern theory must be regarded as applicable to particles of any energy, which again is an argument in favor of new particles.

2. In some of his experiments of 1936 Anderson finds two or three cases of particles emitted in the process of nuclear disintegration caused by a cosmic particle. These particles ionize strongly, but their range is about three times greater than the range calculated from the curvature in the magnetic field on the assumption that their mass is equal to the proton mass. The range and the curvature of the path indicate that the mass of the particles is of the order of one third of the proton mass. The author’s observations with a Wilson chamber revealed two particle tracks with ionization too great for an electron and too small for a proton. The masses of these particles are of the order of one quarter of the proton mass.

These experiments appear to give direct proof of the existence of new particles. There arises, however, some suspicion that the curvature of the tracks is due not to the magnetic field, but is an accidental consequence of air currents in the chamber.

1. We have already pointed out that, in contradiction with the theory, experiments do not reveal an increase of ionization at energies greater than \(2 \cdot 10^6\) V. For particles with energies in the range \(10^8\)—\(10^9\) V the theory gives, on the assumption that the ionizing particle is an electron, a value \(600\%\) greater than the ionization minimum at \(1.5 \cdot 10^6\) V. On the other hand, a particle with the charge of an electron and with a mass equal to one tenth of the proton mass would have in this region an ionization differing only by \(10\%\) from the minimum for \(1 \cdot 10^6\)—\(2 \cdot 10^6\) V electrons, and, consequently, would give better agreement with experiment.

The arguments in favor of the existence of new particles still do not have the force of proof. Blackett believes, for example, that the absence, according to his experiments, of new particles in the energy region \(1.0 \cdot 10^8\ \mathrm{V}\) (since the observed energy losses in lead agree with the theory if one assumes that all particles are electrons) suggests that the new particles are transformed into ordinary electrons as soon as their energy falls to \(10^8\ \mathrm{V}\). At present, however, we have no possibility of calculating the actual number of new particles in this energy region on the assumption that all particles with energy above \(10^8\ \mathrm{V}\) are new particles.

The state of affairs is rather unclear, but it is of enormous interest. If it turns out that new particles really exist, then a theory will have to be constructed for these particles, explaining their origin, behavior, and annihilation. If, on the contrary, it turns out that we are dealing with electrons of high energy whose behavior is not in agreement with modern theory, then this will require fundamental changes in the modern theory of electrons.

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1. Science Progress No. 121, July 1936. Translation by Yu. B. Rumer. ↩

2. I take the opportunity to express my gratitude to Prof. N. Bohr for discussions on questions of the theory of collisions during my last stay at his institute. ↩