Absorption and Scattering of $\gamma$-Rays

L. Groshev, Moscow

The $\gamma$-rays emitted by certain radioactive substances are, as is known, electromagnetic radiation of very short wavelength. Passing through matter, $\gamma$-rays are partly absorbed in it and partly scattered. The purpose of our article is to consider the question of the absorption and scattering of $\gamma$-rays as they pass through matter. In the exposition, the main emphasis is placed on the work of recent years, which has made it possible to explain the so-called anomalous absorption and anomalous scattering of $\gamma$-rays. The results of earlier work are mentioned only very briefly. A more detailed account of them may be found in the book by Rutherford, Chadwick, and Ellis[^1], and also in Kohlrausch[^2]. The question of the discovery of the anomalous absorption of $\gamma$-rays is set forth in considerable detail in the article by M. P. Bronstein[^3]. The question of the absorption and scattering of hard $\gamma$-rays is also considered in Chapter III of D. V. Skobeltsyn’s book Cosmic Rays[^4], which will soon appear in print.

In the first part of our article the question of the weakening of a beam of $\gamma$-rays as it passes through matter is analyzed; in the second, the question of the scattered $\gamma$-radiation arising in this process.

I

§ 1. When passing through matter, a beam of $\gamma$-rays diminishes its intensity. This weakening of the beam is connected, on the one hand, with the absorption of photons by the matter, and on the other hand with their scattering. The weakening of a parallel beam of a given $\gamma$-radiation in some substance is characterized by the linear attenuation coefficients $\mu$, defined by the following formula:

$$ \frac{\Delta I}{I}=-\mu\,\Delta x, \tag{1} $$

where $\frac{\Delta I}{I}$ indicates the degree of decrease of the beam intensity in passing through a layer of the given substance of thickness $\Delta x$ (with $\mu$ measured in $\text{cm}^{-1}$). The attenuation coefficient $\mu$ may be divided into two parts, one of which ($\tau$) determines the weakening of the beam of $\gamma$-rays due to their absorption, and the other ($\sigma$)—due to their

scattering (removal of photons from the beam without their complete absorption).

Thus we have

\[ \mu=\tau+\sigma . \tag{2} \]

\(\tau\) is the coefficient of true absorption, \(\sigma\) is the scattering coefficient.*

For a parallel monochromatic beam of \(\gamma\)-rays, whose qualitative composition does not change in passing through matter, formula (1) can be written in the usual integral form:

\[ I=I_0 e^{-\mu x}, \tag{3} \]

where \(I_0\) is the initial intensity of the incident beam, \(I\) is the intensity of the beam that has passed through a layer of matter of thickness \(x\).

If the absorption of \(\gamma\)-rays of a monochromatic beam occurs by means of several different mechanisms, then formula (3) remains valid. However, the attenuation coefficient in this case is

\[ \mu=\tau_1+\tau_2+\tau_3+\ldots+\sigma, \tag{4} \]

where \(\tau_1, \tau_2, \tau_3\) are absorption coefficients corresponding to the different absorption mechanisms. For a parallel monochromatic beam, \(\mu\) is also a constant in this case.

For nonmonochromatic \(\gamma\)-rays the attenuation of the beam may be described by the same law (1) as for monochromatic rays; however, in this case \(\mu\) will not be a constant. Its value will depend on the thickness of the layer through which the \(\gamma\)-rays have passed, since with a change in the thickness of the layer the spectral composition of the \(\gamma\)-radiation under consideration will also change. For radiation of known composition, the attenuation coefficient for a thin layer can be calculated by the following formula**:

\[ \mu=\frac{\sum_i \mu_i I_i}{\sum I_i}, \tag{5} \]

* More precisely, part of the total scattering coefficient \(\sigma\) should be included in the coefficient of true absorption (\(\sigma_a\)—see below). However, in what follows we shall adhere to the notation introduced above.

** If in the formula \(I_i\) is understood to mean the ionization current corresponding to the \(i\)-th component of the radiation under investigation, then each term of the numerator and denominator must be multiplied by the “sensitivity function” of the instrument with which the current is measured, taken for the wavelength corresponding to each component.

where \(I_i\) and \(\mu_i\) are the intensities and attenuation coefficients of the individual components of the \(\gamma\)-radiation.

In what follows we shall have to deal chiefly with attenuation, absorption, and scattering coefficients calculated per one atom or one electron. We shall therefore dwell here on their definition. In most cases, for calculations one uses the so-called mass attenuation coefficient \(\frac{\mu}{\rho}\), where \(\rho\) is the density of the substance. \(\frac{\mu}{\rho}\) indicates the relative attenuation of a beam on passing through a layer for each \(1\ \mathrm{g}\) of the given substance per \(\mathrm{cm}^2\). The mass attenuation coefficient is, for a given substance, a characteristic quantity independent of the physical state of the substance. Dividing the mass coefficient by the number of atoms contained in \(1\ \mathrm{g}\) of the substance, we obtain the attenuation coefficient calculated per atom, \({}_a\mu\):

\[ {}_a\mu=\frac{\mu}{\rho}\frac{A}{N}. \tag{6} \]

Here \(A\) is the atomic weight, \(N\) is Avogadro’s number.

For the coefficient calculated per electron, we analogously obtain

\[ {}_e\mu=\frac{{}_a\mu}{Z}=\frac{\mu}{\rho}\frac{A}{NZ}, \tag{7} \]

where \(Z\) is the atomic number; \({}_a\mu\) and \({}_e\mu\) are measured in \(\mathrm{cm}^2\), as follows from formulas (6) and (7), if one takes into account that \(\mu\) is measured in \(\mathrm{cm}^{-1}\).

In an entirely analogous way one can find the true absorption coefficients and the total scattering coefficients calculated per atom or electron. In this case

\[ \begin{aligned} {}_a\mu &= {}_a\tau+{}_a\sigma \\ {}_e\mu &= {}_e\tau+{}_e\sigma \end{aligned} \Bigg\}. \tag{8} \]

\({}_a\tau\), \({}_e\tau\), \({}_a\sigma\), and \({}_e\sigma\) may be regarded as effective cross sections for the corresponding processes (absorption of a photon or its removal from the beam as a result of a scattering event).

§ 2. Measurement of the attenuation coefficient is usually carried out in the following way. Between the source of \(\gamma\)-rays (in older work predominantly Ra, in newer work—ThC″) and the indicator—an ionization chamber or counter—a layer of absorbing substance is placed, and the intensity of the \(\gamma\)-radiation that has passed through the substance is measured for various thicknesses of the layer. In such measurements, in most cases an approximately parallel beam of \(\gamma\)-rays is isolated by means of lead blocks with channels arranged in the proper manner. If the investigated

the γ-radiation under study is monochromatic and if certain additional conditions are fulfilled, which will be discussed below, then for the dependence of the logarithm of the intensity on the thickness of the absorbing layer one obtains a straight line, whose slope gives the value of the attenuation coefficient \(\mu\) for the given substance [see formula (3)].

Fig. 1. Mass defects of nuclei as a function of the number of particles.

In some cases, when it is known in advance that the γ-radiation under study is close to monochromatic (for example, strongly filtered radiation from radioactive preparations), the attenuation coefficient can be determined more rapidly in a somewhat different way, by measuring the intensity of the γ-radiation in the presence and in the absence of an absorbing layer of one definite thickness. Then \(\mu\) can be calculated directly from relation (3).

For nonmonochromatic radiation, which was what was usually dealt with in almost all works before 1930 (Ra with its decay products), the dependence of the logarithm of the intensity on the thickness of the absorbing layer is expressed by a curve concave with respect to the coordinate axes. An example of such a curve is given in Fig. 1 (according to Kohlrausch), where the dependence of the logarithm of the intensity of the γ-radiation of Ra and its decay products on the thickness of a layer of lead is shown. Such curves are usually used in order to separate the γ-radiation under investigation into several more or less monochromatic components. This separation is based on the following fact. Even the first measurements of the attenuation of the γ-radiation of Ra (with

its decay products) in lead showed that for large thicknesses of the absorbing layer the dependence of the logarithm of the intensity on the thickness of the layer is expressed by a straight line. This is explained by the fact that the soft components are absorbed in lead more rapidly than the hard ones, and therefore in the filtered radiation the hard components play the predominant role. By extrapolating to the origin the straight line obtained at large thicknesses of the absorbing layer, one can separate from the total intensity of the beam the part corresponding to the hard component. Then, subtracting it from the total intensity, one can construct the curve of the dependence of the logarithm of the intensity of the remaining part of the radiation on the thickness of the layer. The resulting curve, in turn, can be decomposed by the same method, and so on. Thus, for example, in Fig. 1 shown here the hard component with attenuation coefficient $\mu = 0.543\ \mathrm{cm}^{-1}$ (I), corresponding to RaC, and the softer component of the same element $\mu = 1.43\ \mathrm{cm}^{-1}$ (II), have been separated. In addition to them, there is also a very soft component belonging to RaB and manifested by the presence of a drop at the beginning of curve II. We have dwelt on the decomposition of the absorption curve in somewhat greater detail, since this method is very often used even at the present time. It should be noted, however, that such an analysis of $\gamma$-radiation is very imperfect because of the low sensitivity of the decomposition to changes in the spectral composition of the radiation (owing to the logarithmic dependence); therefore, in some cases it may lead to erroneous conclusions. Indeed, D. V. Skobeltsyn’s experiments$^{4b}$ on determining the spectral composition of $\gamma$-radiation by means of recoil electrons (see below) showed that the $\gamma$-radiation of RaC filtered through $3\ \mathrm{cm}$ of Pb is not monochromatic, despite the fact that the absorption curve after $3\ \mathrm{cm}$ of Pb is practically a straight line.

For nonmonochromatic radiation, for which the relative intensities and attenuation coefficients of all lines of the spectrum are known, the total attenuation coefficient can be calculated from formula (5).

We noted above that, in determining $\mu$, certain additional conditions must be fulfilled. In the old works of various authors, discordant values were often obtained for $\mu$. This is explained, on the one hand, by the fact that the radiation used was not monochromatic, and, on the other hand, by insufficiently clean geometrical conditions under which the experiments on studying the attenuation of $\gamma$-radiation were carried out. Geometrical conditions also play an essential role in experiments with practically monochromatic ThC$''$ radiation (filtered through several $\mathrm{cm}$ of Pb). The point is that the simple formula $I = I_0 e^{-\mu d}$ is applicable only to a parallel beam of $\gamma$-rays, which can be obtained only at large distances of the source from the absorbing material (which requires strong sources of $\gamma$-rays). In old works this was not always the case. Another, more essential circumstance is that the indicator—an ionization chamber or counter—in addition to the radiation that has passed through the substance

without changing direction, registers scattered and secondary radiation incident on it from other parts of the absorbing substance. This additional radiation is the greater, the larger the solid angle under which the indicator is seen from the place where the absorbing layer is located. All this is further complicated by the fact that the scattered and secondary radiation have a wavelength different from that of the incident radiation, and therefore the indicator reacts to them differently. In work on the precise determination of the attenuation coefficient it is necessary to pay attention to maintaining pure geometrical conditions that exclude the action of this additional radiation. Failure to observe these precautions may lead to incorrect values of $\mu$.

§ 3. By 1930, the numerous studies of the attenuation of $\gamma$-rays in their passage through matter had established the following concerning the magnitude of the attenuation coefficient. For all light elements, up to $Z = 50$, the attenuation coefficient, calculated per electron, remains the same for a given composition of $\gamma$-radiation, whereas for heavier elements it increases monotonically with increasing atomic number of the element. To understand this fact, let us turn to an elucidation of the mechanism of attenuation of $\gamma$-rays in their passage through matter.

By 1930 two processes were known that cause attenuation of a beam of $\gamma$-rays. These are the Compton effect and the photoelectric effect. In the Compton effect we observe scattering of photons by electrons. In this process one part of the energy of the incident photon is converted into the kinetic energy of the recoil electron, and the other into a photon of lower energy, scattered at some angle $\theta$ to the direction of motion of the primary photon. The application of the laws of conservation of energy and momentum to the elastic collision of a photon with an electron makes it possible to calculate the energy of the scattered photon and the energy of the recoil electron.* For the energy of a photon scattered at an angle $\theta$, one obtains, as is known, the following expression:

\[ \alpha' = \frac{\alpha}{1 + \alpha(1 - \cos \theta)}, \tag{9} \]

where $\alpha'$ and $\alpha$ are the energies of the scattered and primary photons, expressed in units of $mc^2$ $\left(\alpha'=\frac{h\nu'}{mc^2}\ \text{and}\ \alpha=\frac{h\nu}{mc^2}\right)$. The formula given shows that, for a given frequency of the incident radiation, the frequency of the scattered radiation is the smaller, the larger the angle at which the scattering of the photon takes place. This dependence, characteristic of the Compton effect, of the hardness of the scattered radiation on the angle for $\gamma$-rays was known long before the discovery of the Compton effect itself.

As a result of Compton scattering, a beam of $\gamma$-rays passing through matter will be attenuated. We shall characterize this attenuation of the beam by the total scattering coefficient $_e\sigma$, cal-

* It is assumed here that the scattering occurs on a free electron which, before the act of scattering, was at rest.

calculated per electron (the effective cross-section for removing an incident photon from the beam). \(e^\sigma\) shows how much the intensity of the incident beam of \(\gamma\)-rays is weakened by one electron. The total scattering coefficient \(e^\sigma\), sometimes called the absorption coefficient by scattering, can be divided into two components. One of them, which we shall denote by \(e^\sigma_s\), shows what part of the energy removed from the beam of primary photons is converted into the energy of scattered photons (\(e^\sigma_s\) is also sometimes called the coefficient of true scattering). The other component shows what part of the energy of these photons removed from the beam is converted into the kinetic energy of recoil electrons. Denoting this component by \(e^\sigma_a\), we have:

\[ e^\sigma = e^\sigma_s + e^\sigma_a . \tag{10} \]

\(e^\sigma_s\) and \(e^\sigma_a\) are also calculated per electron.*

In 1928 Klein and Nishina\(^5\), applying the relativistic Dirac equation, theoretically investigated the question of the scattering of \(\gamma\)-radiation in the Compton effect and calculated both the coefficient of true scattering \(e^\sigma_s\) and the total scattering coefficient \(e^\sigma\). In their study all atomic electrons were regarded as free, and therefore the magnitude of the energy loss in scattering was the same for each electron. The assumption that, with respect to the Compton effect, atomic electrons behave as free ones is fulfilled rather well for the region of \(\gamma\)-radiation, since the energy of the photons here is large in comparison with the binding energy of the electrons in the atom.**

For unpolarized \(\gamma\)-radiation Klein and Nishina found the following expression for the total scattering coefficient:

\[ e^\sigma = \frac{2\pi e^4}{m^2 c^4} \left[ \frac{1+\alpha}{\alpha^2} \left\{ \frac{2(1+\alpha)}{1+2\alpha} - \frac{1}{\alpha}\ln(1+2\alpha) \right\} + \frac{1}{2\alpha}\ln(1+2\alpha) - \frac{1+3\alpha}{(1+2\alpha)^2} \right], \tag{11} \]

where \(m\) and \(e\) are the mass and charge of the electron, \(c\) is the speed of light, and \(\alpha\) is the energy of the incident photon expressed in units of \(mc^2\). This is the famous Klein—Nishina formula. It shows that the coefficient of total scattering \(e^\sigma\) depends exclusively on the wavelength of the scattered radiation. In Fig. 2 the dependence of \(e^\sigma\) on wavelength is shown. From the curve it is evident that the total scattering coefficient decreases as the wavelength decreases.

* \(e^\sigma_a\) may be regarded as the coefficient of true absorption (see the note on p. 788), since the corresponding energy of the \(\gamma\)-radiation is wholly converted into another form of energy (absorbed).

** In a number of later works the question of the scattering of \(\gamma\)-radiation by bound atomic electrons was considered. However, only a small correction to the Klein—Nishina formula is obtained, lying within the limits of experimental error.

For hard monochromatic X-rays the Klein—Nishina formula was experimentally verified with great accuracy in the work of Read and Lauritsen,^6 who measured the absorption of these rays in light elements (C and Al). Confirmation of the validity of formula (11) for γ-rays is provided first of all by the fact that, for all light elements, the attenuation coefficient calculated per electron \((\varepsilon^\mu)\) is a constant quantity, independent of \(Z\), and its numerical value coincides with the value of \(\varepsilon^\sigma\) calculated from the Klein—Nishina formula. Hence, incidentally, it follows that for light elements the attenuation of a beam of γ-rays is explained by Compton scattering alone.

Fig. 2.

The conclusions of Klein—Nishina were also confirmed by investigations of the angular distribution of the scattered radiation. However, we shall examine this question in greater detail in the second part of the article.

§ 4. In the photoelectric effect, as is well known, the photon is absorbed completely, transferring its energy (less the binding energy of the electron in the atom) to the electron ejected from the atom. By superposing photoelectric absorption on Compton scattering, at first one explained the fact that for heavy elements, in contrast to light ones, the attenuation coefficient calculated per electron becomes greater than \(\varepsilon^\sigma\), as given by the Klein—Nishina formula. However, a systematic investigation of the attenuation of filtered γ-radiation of ThC″* in passing through matter, carried out in the works of Chao,^7 Meitner and Hupfeld,^8 Tarrant^9 and Jacobsen,^10 led to the conclusion that the additional absorption \((\varepsilon^\mu-\varepsilon^\sigma)\), observed for heavy elements and increasing with increasing \(Z\) (according to Jacobsen and Tarrant^9b as \(Z^2\)), cannot be explained by the presence of the photoelectric effect alone. This is confirmed by at least the following fact. Photoelectric absorption, as is known from experiments with X-rays, decreases with decreasing wavelength. In actuality, however, Meitner and Hupfeld showed that the more

* The radiation of ThC″ consists of an intense line with energy \(h\nu=2.65\ \mathrm{MeV}\) \((\lambda=4.66\ \mathrm{X})\) and a number of less intense and softer lines located far from the main line. By filtering such radiation with several centimeters of lead, one can remove the soft components entirely, thereby obtaining practically monochromatic radiation. It is true that there are indications,^54 that in the radiation of RdTh, which is usually used in work with the γ-radiation of ThC″, besides the intense line with energy \(h\nu=2.65\ \mathrm{MeV}\), there are also lines with energies \(h\nu \simeq 2\ \mathrm{MeV}\) and \(h\nu=1.65\ \mathrm{MeV}\). If this is so, then these components are rather difficult to separate from the main line by filtration.

ABSORPTION AND SCATTERING OF γ-RAYS

the shorter the wavelength of the radiation, the lower the \(Z\) at which the additional absorption begins to appear; i.e., for a given \(Z\) it increases as the wavelength decreases. This fact was also confirmed by Jacobsen.

Thus, the work of the authors mentioned establishes the existence of a new kind of absorption of \(\gamma\)-rays, which has been given the name anomalous. We shall not describe in detail the experiments of Chao, Tarrant, Meitner and Hupfeld, and Jacobsen (see the article by M. P. Bronstein), and shall confine ourselves only to presenting the curve expressing the dependence of the attenuation coefficient, calculated per electron, on \(Z\), constructed from the data obtained in these works (Fig. 3). The solid curve is drawn from Jacobsen’s data, the dashed curve from Tarrant’s data. The arrow on the ordinate axis indicates the value of \(e\sigma\) given by the Klein—Nishina formula. The large scatter of the points is due to the low accuracy of the measurements.

§ 5. At first the nature of anomalous absorption was quite unclear. It was explained by the presence of nuclear absorption of \(\gamma\)-rays; at the same time, various assumptions were made about the character of the absorption. The discovery of the positron brought an unexpected solution to the nature of anomalous absorption.

Blekett and Occhialini\(^{11}\), studying cosmic rays with the aid of a Wilson chamber placed in the path of the cosmic rays themselves, established the presence of a large number of positrons in so-called showers—aggregates of a large number of particles passing simultaneously through the chamber or arising in it. In an article devoted to the exposition of these results, Blekett and Occhialini expressed the supposition that the anomalous absorption of \(\gamma\)-rays by heavy nuclei might be connected with the formation of positive electrons. Soon after these experiments, the same authors, together with Chadwick\(^{12}\), established that radiation from beryllium, consisting of neutrons and \(\gamma\)-quanta and irradiated by \(\alpha\)-particles, when incident on a lead plate placed next to the Wilson chamber, produces in it, along with negative electrons, also positrons. The same fact was

Fig. 3. \(I\)—Chao\(^{7a}\), \(II\)—Tarrant\(^{8a}\), \(III\)—Meitner and Hupfeld\(^{8c}\), \(IV\)—Jacobsen\(^{10}\), \(V\)—Tarrant\(^{9}\)

It was discovered by Curie and Joliot ^13, and also by Meitner ^14. Curie and Joliot ^13, filtering beryllium radiation with a layer of lead, showed that the cause of the appearance of positrons is $\gamma$-radiation, and not neutrons. Soon thereafter Curie and Joliot ^15, Meitner and Philipp ^16, and also Anderson and Neddermeyer ^17, almost simultaneously showed that $\gamma$-radiation from $\mathrm{ThC}^{\prime\prime}$ does indeed create positrons. At the same time a very essential circumstance was noted: that in many cases the positron and the electron arise simultaneously at one and the same point in space, thus forming a so-called pair. The mechanism of formation of electron pairs may be represented on the basis of Dirac’s theory, created even before the discovery of the positron. In it, as is known, the positron is identified with the so-called “hole,” i.e. an unoccupied level in the continuous background of electron levels corresponding to negative kinetic energies of electrons. The formation of a pair is interpreted in Dirac’s theory in the following way. A photon with sufficiently large energy can tear an electron out of one of the occupied levels corresponding to negative kinetic energies of electrons. As a result of such a process (which, by analogy with the tearing of electrons by light from ordinary levels of positive energies, may be regarded as a photoelectric effect from levels of negative kinetic energy), an electron from a level of negative kinetic energy is transferred to one of the levels of positive energy. Thus in this process a “hole”—a positron—is formed, and a new electron appears, which up to that time had not manifested itself in any way. Since, according to Dirac’s theory, the minimum distance between the levels of positive and negative energies is equal to $2mc^2$, where $m$ is the mass of the electron and $c$ the speed of light, pair production can occur only from photons whose energy is not less than $2mc^2$ ($\sim 1\ \mathrm{MeV}$). This consequence may also be obtained without Dirac’s theory, directly from the law of conservation of energy-mass. According to the well-known relation between mass and energy, the rest energy of an electron is equal to $mc^2$, and therefore the minimum energy necessary for creating an electron and a positron is equal to $2mc^2$. A quantum with such an energy corresponds to the production of a pair, both components of which possess zero kinetic energy. For quanta with greater energy $h\nu$, the energy $2mc^2$ is expended on creating the electron and the positron, while the remaining part $(h\nu - 2mc^2)$ appears in the form of the kinetic energy of the two particles forming the pair.

The conclusions obtained from Dirac’s theory are confirmed by experimental data. Indeed, the experiments of Meitner and Philipp ^16 showed that in the case of polonium $\gamma$-radiation, which possesses an energy less than $1\ \mathrm{MeV}$, positrons do not appear, although for the radiation of $\mathrm{ThC}^{\prime\prime}$, under the same conditions, positrons do arise. Moreover, in studying the energy distribution of positrons knocked out from lead by $\gamma$-rays of $\mathrm{ThC}^{\prime\prime}$ with an energy of $2.65\ \mathrm{MeV}$, it was found that the maximum energy of these positrons is about $1.6\ \mathrm{MeV}$, which is also in agreement with Dirac’s theory,

since these positrons correspond to the case when all the kinetic energy \((2.6—1)\) is transferred entirely to the positron. An estimate of the energy of the pairs formed in the gas by the \(\gamma\)-radiation of \(\mathrm{ThC}''\) shows that their energy is also approximately \(1.6\ \mathrm{MeV}\), as was to be expected if the above arguments are valid. (The pairs observed by some authors—in a small number—with energies greater than \(1.6\ \mathrm{MeV}\) are explained, in all probability, by the presence of a small number of high-energy photons.)

From all that has been said one may conclude that the production of pairs by photons plays a definite role in the absorption of \(\gamma\)-radiation. The question now arises whether the anomalous absorption of \(\gamma\)-radiation can be explained completely by the production of pairs by photons. To resolve this question it is necessary to try somehow to separate photoelectric and anomalous absorption and then compare the latter with the absorption due to pair production. For this purpose let us turn to a quantitative calculation of the photoelectric effect and of the production of electron pairs by photons.

§ 6. Although at the present time there are no final formulae for calculating photoelectric absorption for arbitrary photon energies and arbitrary atomic numbers of the absorbing substance, nevertheless there are some data that make it possible to carry out this calculation. First of all, let us note that the formula usually used for calculations of the photoelectric absorption of X-rays,

\[ a^{\tau}=cZ^{4}\lambda^{n}, \tag{12} \]

(where \(a^{\tau}\) is the coefficient of photoelectric absorption, calculated per atom, and \(n\) is approximately equal to 3) gives incorrect results for \(\gamma\)-rays. For different photon energies and for different elements the exponents of \(Z\) and \(\lambda\) are not the same. In all probability, for \(\gamma\)-rays \(\tau\) cannot be expressed by a single term with a power factor of \(\lambda\). Therefore here one has to seek a more complicated dependence of \(a^{\tau}\) on \(\lambda\).

Let us first consider Gray’s empirical formula\(^{18}\) for the dependence of the coefficient of photoelectric absorption on wavelength in the case of lead. This formula may be written as follows:

\[ \tau_{\mathrm{Pb}}=4.472\cdot 10^{-3}\lambda^{(1+0.48\lg \lambda)} \tag{13} \]

or, in a more convenient form,

\[ \lg \tau_{\mathrm{Pb}}=\overline{3}.6505+1.0\lg \lambda+0.480\lg^{2}(\lambda), \tag{14} \]

where \(\lambda\) is measured in X-units.

The numerical coefficients in this formula have been chosen so that it corresponds in the best way to the experimental data. From this formula one can find the coefficient of photoelectric—

of absorption, calculated per atom. For it the following expression is obtained:

\[ a_\tau^{\mathrm{Pb}}=1.349\cdot 10^{-25}\lambda^{(1+0.480\lg\lambda)}. \tag{15} \]

The validity of Gray’s formula for hard X-rays with wavelengths between 80 and 25 X-units was confirmed by the work of Kosman and Alikhanian\({}^{19}\). It is true that Read\({}^{20}\), in a carefully carried out study on the absorption of hard X-rays (\(\lambda\) from 20 to 52 X-units), asserts that Gray’s formula gives somewhat smaller values in comparison with the data obtained by him for the photoelectric absorption coefficient for lead. This discrepancy reaches 20% for the shorter wavelengths.

The shortcoming of Gray’s formula is that it is applicable only to the case of lead and therefore does not make it possible to calculate \(a_\tau\) for other elements.

There have also been attempts to calculate theoretically the coefficient of photoelectric absorption for \(\gamma\)-rays. Such a calculation on the basis of the relativistic Dirac equation was first carried out by Sauter\({}^{21}\) and Hulme\({}^{22}\) for light elements, for which

\[ \frac{Z}{137}\ll 1. \]

According to these authors’ data one can calculate the absorption coefficient for various \(h\nu\), but only for small \(Z\). For example, for \(h\nu \gg mc^2\) Sauter found the following expression for the coefficient of photoelectric absorption, calculated for 2 atomic electrons of the \(K\) shell:

\[ \tau_K \approx 1.16\cdot 10^{-23}\lambda Z^5\ \mathrm{cm}^2. \tag{16} \]

Both Sauter and Hulme note that if one uses, for calculations of photoelectric absorption, a formula of the form (12), then for different regions of hard X-rays and \(\gamma\)-rays the exponent \(n\) must be assigned different values, decreasing with decreasing wavelength and reaching 1 for \(h\nu \gg mc^2\) (see formula [16]).

The results obtained by Sauter and Hulme give the correct order of magnitude for \(\tau\)*. However, they cannot be verified experimentally with sufficient accuracy, since they are valid only for light elements, where photoelectric absorption plays a negligible role in comparison with Compton absorption.

* Photoelectric absorption for photons of high energies was also calculated by Hall and Oppenheimer\({}^{23}\). However, their results differ greatly from the experimental data.

Later Hall[^24], again proceeding from the relativistic Dirac equation, found for \(\tau_K\) a somewhat modified expression:

\[ \tau_K=\frac{1{,}16\cdot 10^{-23}R}{a^{2\alpha}e^{\varepsilon(\pi-2\alpha)}} , \tag{17} \]

where

\[ \alpha=\frac{2\pi e^2}{hc}Z;\qquad R=\frac{k_0' r_e^2}{k'^4}\left[\frac{4}{3\varepsilon} +\frac{\varepsilon\cdot 2}{\varepsilon+1} \left(1+\frac{1}{2\varepsilon k_0'}\lg\frac{\varepsilon-k_0'}{\varepsilon+k_0'}\right)\right]; \]

\[ \varepsilon=k'+(1+\alpha^2)^{\frac12},\qquad k'=\frac{h\nu}{mc^2},\qquad k_0'=(\varepsilon^2-1)^{\frac12}. \]

The formula he obtained is applicable for any values of \(Z\), but only for \(h\nu\gg mc^2\). Therefore, in the region of \(\gamma\)-rays from ordinary radioactive sources it must be used with great caution.

In his next work[^25], together with Rarita, Hall also calculated photoelectric absorption for the atomic electrons of the \(L_1\) shell. Relative to absorption in the \(K\) shell it amounts to \(20\%\). Assuming that photoelectric absorption in the other atomic shells does not play a noticeable role, we may write for the photoelectric absorption coefficient, calculated per atom, the following expression:

\[ a\tau=\tau_K+\tau_L+\ldots=\frac54\tau_K, \tag{18} \]

where \(\tau_K\) is given by the preceding formula. The factor \(\frac54\) apparently also follows from experimental data.[^26]

Recently, Hulme, McDougall, Buckingham, and Fowler[^27] carried out a rigorous theoretical calculation of the photoelectric absorption coefficient for the \(K\) shell. Owing to the large amount of numerical work, the authors were able finally to calculate \(\tau_K\) only for two wavelengths and for three \(Z\), although in principle this can be done for any \(Z\) and \(h\nu\). The results of their calculation are given in Table 1; in parentheses are the values of \(\tau_K\) calculated by Hall’s formula.

TABLE 1

Here \(\theta=\dfrac{mc^2}{h\nu}\). If one takes into account the circumstance that Hall’s formula is strictly applicable only for \(\theta \ll 1\), then the agreement between the values given must be regarded as good.

Using the data given in the table, Hulme et al. constructed the dependence of \(\dfrac{\tau_K}{Z^5}\) on \(Z\) for two wavelengths \((\theta=0.452\) and \(\theta=1.443)\), making use of additional values of \(\tau_K\) obtained from Sauter’s calculations for \(Z=0\), where his solutions are exact. From the curves constructed it was possible to find \(\tau_K\) for two wavelengths for any values of \(Z\). Having supplemented the data obtained in this way with the value of \(\tau_K\) for \(\theta=0\) \((h\nu=\infty)\) and \(\theta=0.194\) \((h\nu=2.65\,\mathrm{MeV})\), taken from Hall’s formula, which in this case is applicable with sufficient accuracy, the authors constructed a system of curves expressing the dependence of \(\dfrac{a\tau}{Z^5}\) on wavelength for various \(Z\); in this passage from \(\tau_K\) to \(a\tau\) was carried out by multiplying \(\tau_K\) by \(\dfrac{5}{4}\) (see above). This system of curves is shown in Fig. 4.

Fig. 4. The numbers by the curves indicate the atomic number of the element to which the given curve applies.

On the curve corresponding to \(Z=82\), crosses mark three values of the atomic coefficient of photoelectric absorption obtained from Gray’s empirical formula. As we see, the agreement

theoretical and experimental data is very good.

Thus we come to the conclusion that, with sufficient accuracy, the photoelectric absorption for any \(h\nu\) and \(Z\) can be determined from the system of curves of Helm et al.

§ 7. Let us now turn to the question of how to calculate the absorption due to the production of pairs by photons. First of all it must be remembered that this kind of absorption will manifest itself only for energies \(h\nu > 2mc^2\), as was already noted above.

Soon after the discovery of the positron, Oppenheimer and Plesset \(^{28}\), on the basis of Dirac’s theory, calculated the probability of the appearance of pairs under the action of photons. In their theory it was assumed that the production of pairs is a consequence of the photoelectric effect on electrons of negative levels, occurring in the Coulomb field of the nucleus. In the case of the appearance of a pair from a single photon, the atomic nucleus plays an auxiliary role,* being a kind of catalyst. It takes up part of the momentum of the primary photon, without appreciably affecting the energy balance of the process. Neglecting relativistic effects, i.e. considering the case when the emerging negative and positive electrons constituting the pair have small velocities in comparison with the velocity of light—which corresponds to pair production by photons with energy only slightly exceeding \(2mc^2\)—Oppenheimer and Plesset were able to calculate rigorously the effective cross section (absorption coefficient, calculated per atom) for such a process. They also carried out an approximate calculation for photons of high energies, obtaining for the effective cross section a formula which was subsequently somewhat modified by calculations of other authors. Later the question of pair production was analyzed in the theoretical works of Heitler and Sauter \(^{29}\), Bethe and Heitler \(^{30}\), Nishina, Tomonaga, and Sakata \(^{31}\), and also in the work of Racah \(^{32}\); the calculations were made in various approximations.

For photons with energy only slightly exceeding \(2mc^2\), the effective cross section for pair production \((\alpha^\chi)\), calculated for one atom (nucleus), was computed for two approximations, in both of which \(\frac{v_1}{c} \ll 1\) and \(\frac{v_2}{c} \ll 1\), where \(v_1\) and \(v_2\) are the velocities of the positive and negative electrons forming the pair. In the first of the mentioned approximations, satisfying the conditions that \(\frac{v_1}{c} \ll \alpha Z\), \(\frac{v_2}{c} \ll \alpha Z\), and \(\alpha Z \ll 1\), the following expression was obtained for the effective cross section:

\[ \alpha^\chi = k\,\frac{Z^5}{(137)^4} \left(\frac{e^2}{mc^2}\right)^2 g^{\frac{3}{2}} e^{-\frac{2\pi\,\frac{Z}{137}}{\sqrt{2g}}}, \tag{19} \]

* Pair production can also occur in the absence of an atomic nucleus. In this case there must be a collision of two photons, and therefore such an effect has a negligibly small probability.

where \(g=\dfrac{h\nu-2mc}{mc^2}\), i.e., the kinetic energy of the constituents of the pair, expressed in units of \(mc^2\); \(k\) is a numerical factor, for which Oppenheimer and Plesset, Nishina et al. give somewhat different values.

In another approximation, the so-called Born approximation, satisfying the conditions \(\dfrac{v_1}{c}\gg\dfrac{Z}{137}\) and \(\dfrac{v_2}{c}\gg\dfrac{Z}{137}\), Nishina et al. and Racah found the following expression for the effective cross section:

\[ a^\chi=\frac{\pi}{12}\frac{Z^2}{137}\left(\frac{e^2}{mc^2}\right)^2 g^3, \tag{20} \]

where \(g\) has the same value as in the preceding formula. The formulas given for \(\chi\), however, are not of great significance for us, since they are applicable in those energy regions where the probability of pair production by photons is negligibly small.

The question of the production of pairs by photons whose energy appreciably exceeds \(2mc^2\) has been considered most thoroughly in the work of Bethe and Heitler. Starting from Dirac’s theory and applying the Born approximation \(\left(\dfrac{v}{c}\gg \alpha Z\right)\), the authors calculated the effective cross section for pair production. An analytic expression for \(a^\chi\) could be obtained only for energy regions for which \(h\nu\gg mc^2\). Thus, for example, for the region where \(mc^2\ll h\nu\ll 137mc^2 Z^{-1/3}\), they found for the total effective cross section:

\[ a^\chi=\frac{Z^2}{137}\left(\frac{e^2}{mc^2}\right)^2 \left[\frac{28}{9}\lg\frac{2h\nu}{mc^2}-\frac{218}{27}\right]. \tag{21} \]

The same formula was later obtained by Nishina\(^{31}\) et al. and Racah\(^{32}\). (The formula proposed by Oppenheimer and Plesset differs in that it lacks the logarithmic factor.)

The effective cross section for energies from \(h\nu=2mc^2\) to \(h\nu\gg mc^2\) was obtained by Bethe and Heitler by numerical integration. The results they obtained are shown in Fig. 5, in which the ordinates give the effective cross sections in units of \(\dfrac{Z^2}{137}\left(\dfrac{e^2}{mc^2}\right)^2\), and the abscissas give the photon energies, expressed in \(mc^2\). From the Bethe and Heitler curves one can calculate the effective cross section for pair production for photons of any energies ordinarily encountered under experimental conditions.

In deriving formula (21), the screening action of the external atomic electrons on the nuclear field was not taken into account. However, as Bethe\(^{33}\) showed, it affects the magnitude of the effective cross section only for energies \(h\nu\) much greater than \(mc^2\). Therefore, when screening is taken into account, the curves of Fig. 5 will not noticeably change their course in the energy region of interest to us.

Using the results of Bethe and Heitler, one must always bear in mind that they were obtained in the Born approximation \(\left(\dfrac{v}{c} \gg \dfrac{Z}{137}\right)\), i.e., they are applicable with sufficient accuracy only for not very heavy elements. Thus, for example, for lead \((Z=82)\) one may expect that the approximation will already prove insufficiently good.

Helm and Jaeger1, using Dirac wave functions, carried out a rigorous calculation of \(a^\chi\). In view of the great complexity and duration of the computations, they calculated \(a^\chi\) only for lead at two values of the energy. The data they obtained are given in Table 2, in which the last column gives the values yielded by the Born approximation.

Indeed, for lead, especially for small energies \(h\nu\), we have a discrepancy between the values given by the rigorous theory and by the Born approximation.

Fig. 5. The scale to the left of the ordinate axis refers to curves \(I\), the scale to the right—to curves \(II\); curves \(I\) and \(II\) labeled “Compton effect” give, for comparison, the effective cross section \(a^\sigma\) for the Compton effect.

Subsequently Jaeger2 carried out analogous calculations of \(a^\chi\) for \(Z=50\) and \(Z=65\) at \(h\nu=3mc^2\). The results, together with those obtained earlier, are given in Table 3. The lower line gives the values obtained in the Born approximation.

TABLE 2

From consideration of the data presented one may conclude that in the Born approximation smaller values are obtained for \(a^\chi\) than those given by the rigorous theory. However, these dif-

TABLE 3

deviations decrease strongly with decreasing atomic number and with increasing \(\gamma\)-ray energy.

The values obtained by Jaeger for \(a^x\) lie on the following curve:

\[ a^x \cdot 10^{24} = 0.95 \left( \frac{Z}{137} \right)^2 + 2.54 \left( \frac{Z}{137} \right)^4 , \tag{22} \]

where the first term corresponds to the values obtained in the Born approximation, while the second gives the correction introduced into them by the exact calculation. From the last term one could estimate the correction to the value of Bethe and Heitler. Unfortunately, this relation was obtained only for \(h\nu = 3mc^2\) and cannot be applied for other energies. Therefore, in what follows, in calculating \(a^x\) we shall use the data of Bethe and Heitler, bearing in mind, however, that they are somewhat lower than the true values of \(a^x\).

§ 8. In order to see how the conclusions obtained by Bethe and Heitler agree with the experimental data, let us first dwell on those regularities to which \(a^x\) is subject. Both from the curves in Fig. 5 and from formula (21) it follows that the effective cross section for pair production increases proportionally to the square of the atomic number of the absorbing substance and, moreover, increases strongly with increasing energy of the absorbed photons. The following simple Table 4, borrowed from Blackett’s review article\(^{36}\) on the positron and constructed from the data of Curie and Joliot\(^{15}\), Greenberg\(^{37}\), and Chadwick, Blackett, and Occhialini, already shows that these regularities do in fact exist. Table 4 gives the ratio of the number of positrons (in percent) to the total number of observed negative electrons. It is assumed that each positron that appears corresponds to the formation of an electron pair, and therefore the relative numbers given in the table may serve for a rough estimate of the probability of pair production.

There is also more direct evidence that the probability of pair production by photons increases proportionally to \(Z^2\). Benedetti\(^{38}\) investigated how the number of produ-

TABLE 4

…of positrons arising upon irradiation of different substances by one and the same γ-radiation (RdTh in equilibrium with its decay products). In doing so it was assumed that the positrons are constituents of electron pairs and that therefore, from their number, one may judge how the number of pairs produced (i.e., $\chi$) changes as a function of $Z$. In Benedetti’s experiments the positrons formed upon irradiation of a plate by γ-rays were focused by the Tibó trajectory method and entered a Geiger–Müller counter with thin walls, placed at the opposite end of a diameter passing through the source of γ-rays and the center of the magnet bending the positrons. The results obtained by Benedetti are given in Table 5. In the first row is given the number of pulses in the counter (referred to one and the same interval of time) caused by positrons,

TABLE 5

arising in plates of different substances with the same mass per $1\ \mathrm{cm}^2$. In the second row are given the same numbers divided by

$$ \frac{Z^2}{A}. $$

$A$ is introduced into the denominator in order to refer the number of pulses in the counter to one and the same number of nuclei in the irradiated plates.

It is seen from the table that the number of positrons, and consequently also of pairs, calculated per atom, is proportional to \(Z^2\).

Benecke \(^{39}\), using a Wilson chamber, determined the yield of positrons for various elements when they were irradiated with ThC″ \(\gamma\)-rays. Taking into account the absorption of positrons in the irradiated substance and making the corresponding recalculations, he computed the ratio of the effective cross sections for pair production in the cases of lead and aluminum and found for it a value equal to 43. The theory of Bethe and Heitler gives for this ratio the value \(40\left[\left(\frac{82}{13}\right)^2\right]\). As we see, here too the quadratic dependence of \(a^x\) on \(Z\) is well satisfied.

In addition, there are also indirect proofs of the validity of the quadratic dependence of \(a^x\) on \(Z\). For example, they follow from the study of annihilation radiation for various substances. This will be discussed in the second part of our article.

The dependence of \(a^x\) on the photon energy is seen from the fourth column of Table 4. This dependence can also be illustrated by Table 6, given by Bethe and Heitler in their work. It refers to the case of lead.

TABLE 6

In the last line is given the ratio of the number of positrons to the number of Compton electrons and photoelectrons. In the penultimate line is given the ratio of the corresponding effective cross sections, calculated theoretically. In parentheses are given the ratios obtained for thick layers of the irradiated substance, where the relation between the number of positrons and the number of electrons is more complicated than in the case of thin layers. The ratio for ThC″ has been recalculated for a thin layer. As is seen from the table, the agreement between the theoretical and experimental data is quite good, especially if one takes into account that the data of the last line are very rough.

In order to compare the numerical values of \(a^x\) obtained experimentally and theoretically, one may use the following indication of Chadwick, Blackett, and Occhialini \(^{40}\). They state,

that for the case of lead and photons with energy \(2.65\) MeV the number of positrons amounts to from 20 to \(30\%\) of the number of Compton electrons and photoelectrons. These numbers were obtained as the result of a whole series of calculations and therefore can give only a rough estimate. If this ratio is taken equal to \(25\%\), then for the effective cross section \(a^{\kappa}\) one obtains for lead the value \(2.9\cdot 10^{-24}\,\mathrm{cm}^{2}\), whereas according to the data of Bethe and Heitler \(a^{\kappa}=2.4\cdot 10^{-24}\,\mathrm{cm}^{2}\). Since the theory of Bethe and Heitler gives somewhat underestimated values for \(a^{\kappa}\), the agreement between the theoretical and experimental data may be considered quite good.

§ 9. On the basis of all the data presented above we can calculate the probability of the process of pair production. We now return to the question of whether pair production is sufficient for a complete explanation of anomalous absorption. To answer this question we must first of all single out from the total attenuation coefficient \(a^{\mu}\) the part corresponding to anomalous absorption. For this purpose it is sufficient to subtract from \(a^{\mu}\) \((a^{\sigma}+a^{\tau})\), i.e. the part of the attenuation caused by the Compton effect and the photoelectric effect. Before proceeding to such calculations, one should note the circumstance that the attenuation coefficient \(a^{\mu}-(a^{\sigma}+a^{\tau})\) will take into account, besides the formation of electron pairs by photons, all the other absorption mechanisms about which we have not spoken so far. It is known, for example, that when a beam of \(\gamma\)-rays passes through matter, so-called coherent scattering (without change of wavelength) occurs. As a consequence of this scattering the primary beam will undergo attenuation. However, as will be shown later, this attenuation is very small in comparison with the attenuation caused by the principal mechanisms—the Compton effect, the photoelectric effect, and the process of formation of electron pairs. On this basis, in quantitative calculations we shall not take into account this fourth mechanism of attenuation of a beam of \(\gamma\)-rays.

To the mechanisms of absorption of \(\gamma\)-rays considered there has recently been added one more mechanism, which, however, plays, like the preceding one, a very insignificant role. This is the nuclear photoelectric effect—the disintegration of a nucleus by a photon. Chadwick and Goldhaber \(^{41}\) first established that, upon irradiation of heavy water by \(\gamma\)-rays of ThC\(''\), the nucleus of heavy hydrogen is disintegrated with emission of a proton and a neutron. Some time later, Szilard and Chalmers \(^{42}\) discovered that beryllium nuclei are also disintegrated by photons with emission of neutrons. The question of the nuclear photoelectric effect was then investigated in more detail in a number of other works \(^{43}\). In what follows, when calculating the attenuation of \(\gamma\)-rays in their passage through matter, we shall not dwell on this absorption mechanism, since in comparison with the others it plays a quite small role (certainly less than \(1\%\) of the total attenuation, which, at the accuracy of present measurements, lies within the limits of error) and, moreover, appears only in two cases (\(\mathrm{D}_{2}\mathrm{O}\), Be).

§ 10. Passing to the analysis of anomalous absorption, one should

first of all to note that it begins to manifest itself only for energies greater than 1 MeV ($\lambda \sim 12$ X-units), as was to be expected if it is assumed that it is due to the production of pairs by photons. In Fig. 6 a curve is given for the dependence of the anomalous absorption $[\mu-(a\sigma+a\tau)]$ on the wavelength for the case of lead. This curve was obtained by Gentner $^{44}$, who investigated the dependence of $a^\mu$ on $\lambda$. In Gentner’s experiments, $\gamma$-radiation of different wavelengths was obtained by Compton scattering of practically monochromatic ThC″ radiation on aluminum. Aluminum was taken as the scatterer because, for light elements, the scattered radiation practically consists of a single Compton component and therefore

Fig. 6.          Fig. 7.

for a given scattering angle it is more homogeneous than it would be with a heavy scatterer. Measurements of the attenuation coefficient for lead were made at scattering angles of $0;\ 18;\ 23;\ 30$ and $36^\circ$, which corresponds to wavelengths of $4.7,\ 5.9,\ 6.6,\ 7.9$ and $9.3$ X-units. From the obtained values of $a^\mu$, $\sigma^s$ and $\tau$ were subtracted, calculated respectively from the Klein—Nishina formula and from Gray’s empirical formula. The values obtained in this way are plotted on the curve in Fig. 6. The comparatively large scatter of the points is explained by the fact that the scattered radiation used for the measurement was very weak, which leads to large statistical errors.

To judge the dependence of the anomalous absorption on the atomic number of the absorbing substance, we shall present the following two tables, borrowed in somewhat modified form from another work by Gentner $^{45}$. Table 7 is constructed from the data of Ketelaar, Piccard, and Stahel $^{46}$ for the case of strongly filtered $\gamma$-radiation of very

TABLE 7

\[ \lambda = 6.4\ \text{X-units}\quad \left(\varepsilon^{\tau}=1.48\cdot 10^{-25}\right) \]

TABLE 8.

\[ \lambda = 7\ \text{X-units}\quad \left(\varepsilon^{\tau}=1.56\cdot 10^{-25}\right) \]

of an intense radium preparation (1.7 g); Table 8—according to Rogers’ data[^47] for less filtered radium \(\gamma\)-rays. In both tables the effective wavelength of the \(\gamma\)-radiation is determined by the Klein—Nishina formula from the attenuation coefficient for light elements. In the second rows are given the experimentally determined values \(a^{\mu}-a^{\sigma}\); \(a^{\tau}\) was calculated from the graph of Hulme et al. (Fig. 4). The penultimate rows give the anomalous absorption. In the last rows are given \(a^{\chi}\), calculated from Bethe and Heitler’s data. From the table one can see that \(a^{\chi}\) varies with \(Z\) approximately in the same way as \(a^{\mu}-(a^{\sigma}+a^{\tau})\). Better agreement is difficult to expect, since the \(\gamma\)-radiation of Ra, even after strong filtration, is not monochromatic.

Matters are more favorable for the practically monochromatic radiation ThC″. Considering this case, we

we shall proceed somewhat differently, namely: we shall compute \(a^\sigma\) by the Klein–Nishina formula, \(a^\tau\) from Helm’s curves, and \(a^\chi\) from the curves of Bothe and Geitler, and then compare their sum with the experimentally found values for \(a^\mu\). In Fig. 7, borrowed from Gentner’s article \(^{48}\), the results of such a calculation are plotted, with the sole difference that \(\sigma\), \(\tau\), and \(\chi\) are calculated not per atom but per electron, i.e. quantities smaller by a factor \(Z\) have been taken. In accordance with this, for example, \(\chi_e\) is expressed by a straight line, since \(a^\chi \sim Z^2\). The points indicate the values of \(e^\mu\), measured by Gentner and Starkiewicz \(^{49}\) under very good geometrical conditions (source–counter distance \(3\ \mathrm{m}\), solid angle of the counter \(0.5^\circ\)). The vertical strokes indicate the measurement errors. For Al, Mg, Ag, and Pb, moreover, measurements of \(e^\mu\) were carried out by the comparison method, which made it possible to reduce the measurement errors to \(1\%\). From the curve presented it is seen that the agreement between the experimental and theoretical data within the limits of error is complete, i.e. the anomalous

Fig. 8. I — Meitner and Hupfeld, II — Rogers,
III — Ketelaar, Pickard and Stahel, IV — Gentner

absorption can be explained entirely by the formation of electron pairs. Let us note, incidentally, that the linear course of \(e^\mu - e^\sigma\) for light elements is explained entirely by pair production, since the photoeffect for them begins to play any noticeable role only for \(Z\) not less than 30.

In order to establish how the anomalous absorption changes with change in wavelength, we shall proceed in the same way as in the preceding case. We shall compute \(\sigma\), \(\tau\), and \(\chi\) for each wavelength, form their sum, and compare it with the experimentally found value \(\mu\). The corresponding results for lead are given in Table 9.

In Fig. 8 the curve \(e^\sigma + e^\tau + e^\chi = e^\mu\) is constructed by summing the ordinates of three curves \(e^\sigma\), \(e^\tau\), and \(e^\chi\), calculated respectively by the Klein–Nishina formula, from the graphs of Helm et al., and from the curves

...Bothe and Gentner. The experimental values of \(\alpha^h\) were obtained on scattering by crystals (Meitner, Hupfeld), triangles (Rodgers), squares (Ketelar et al.), and circles (Gentner). As is seen from the data presented, agreement between theoretical and experimental...

ABSORPTION AND SCATTERING OF \(\gamma\)-RAYS

TABLE 9

experimental values excellently. Let us note the following feature in the curve expressing the dependence of the attenuation coefficient \(\mu\) on the wavelength. In view of the fact that \(\chi\) increases strongly with decreasing wavelength, while \(\sigma\) and \(\tau\), on the contrary, decrease, the curve \(\mu=f(\lambda)\) has a minimum, after which, toward short wavelengths, \(\mu\) begins to increase sharply because of the high probability of pair formation. As a rule, two values of \(\lambda\) correspond to a given attenuation coefficient, and therefore in many cases one cannot judge unambiguously from the magnitude of the attenuation coefficient the energy of the \(\gamma\)-radiation under investigation[^53]. In such cases it is necessary to measure the attenuation coefficients for two different elements, a heavy one and a light one.

From consideration of all the material presented, we arrive at the conclusion that the anomalous absorption of \(\gamma\)-rays is entirely explained by the formation of electron pairs, and therefore three processes—the Compton effect, the photoelectric effect, and pair formation by photons—are sufficient for a complete quantitative calculation of the total attenuation of a beam of \(\gamma\)-rays as it passes through matter. The small excesses of the experimental values \(e^\mu\) over the theoretical ones, observed in all cases, can in all probability be explained by the fact that, in calculating the probability of pair formation, we used the Born approximation, which gives values of \(a^\chi\) that are certainly somewhat underestimated. However, to verify this supposition, the theoretical data on \(\chi\) now available are insufficient, and the experimental values of the attenuation coefficients are still obtained with insufficiently high accuracy.

LITERATURE

1. E. Rutherford, J. Chadwick, C. D. Ellis, Radiations from Radioactive Substances, Cambridge University Press, 1930. XV.
2. F. Kohlrausch, Handbuch der Experimentalphysik, B. 15 K. II, F. Kohlrausch, Probleme der \(\gamma\) Strahlung, Braunschweig, 1927.
3. M. P. Bronstein, Uspekhi fizich. nauk, XII, 649, 1932.
   4a. D. V. Skobeltsyn, “Cosmic Rays,” ONTI, 1936.
   4b. D. Skobelzyn, C. R. 194, 1568, 1932.
4. O. Klein and Nishina, Z. Physik, 52, 853, 1929.
5. J. Read a. Lauritsen, Phys. Rev., 45, 433, 1934.
   7a. C. J. Chao, Proc. Nat. Acad. Sci. Amer., 16, 431, 1930.
   7b. C. J. Chao, Proc. Roy. Soc., A 135, 206, 1932.
   8a. L. Meitner u. H. Hupfeld, Naturwiss., 18, 534, 1930.
   8b. L. Meitner u. H. Hupfeld, Physik. Z., 31, 947, 1930.
   8c. L. Meitner und H. Hupfeld, Z. Physik, 67, 147, 1930.
   9a. G. T. B. Tarrant, Proc. Roy. Soc., A 128, 345, 1930.
   9b. G. T. B. Tarrant, Proc. Roy. Soc., A 135, 223, 1932.
   10a. J. C. Jacobsen, Naturwiss., 18, 951, 1930.
   10b. J. C. Jacobsen, Z. Physik, 70, 145, 1931.
6. P. M. S. Blackett a. G. P. S. Occhialini, Proc. Roy. Soc., A 139, 639, 1933.
7. J. Chadwick, Blackett a. Occhialini, Nature, 131, 473, 1933.
8. J. Curie et F. Joliot, C. R. 196, 1105, 1933.
9. L. Meitner und Philipp, Naturwiss., 21, 286, 1933.
   15a. J. Curie et F. Joliot, C. R., 196, 1581, 1933.

15b. J. Curie et F. Joliot, J. d. Physique, 4, 494, 1933.
16. L. Meitner u. Phylipp, Naturwiss, 21, 468, 1933.
17. C. D. Anderson a. S. H. Neddermeyer, Phys. Rev., 43, 1034, 1933.
18. L. H. Gray, Proc. Camb. Phil. Soc., 27, 103, 1931.
19. A. Alichanjan u. M. Kosman, Z. Physik, 90, 779, 1934.
20. J. Read, Proc. Roy. Soc., A 152, 402, 1935.
21. F. Sauter, Ann. d. Phys., 9, 217, 1931; 11, 454, 1931.
22. H. K. Hulme, Proc. Roy. Soc., A 133, 381, 1931.
23. J. K. Oppenheimer a. H. Hall, Phys. Rev., 38, 57, 1931.
24. H. Hall, Phys. Rev., 45, 620, 1934.
25. H. Hall a. W. Rarita, Phys. Rev., 46, 143, 1934.
26. E. Rutherford, J. Chadwick, C. D. Ellis, Radiations from Radioactive Substances, p. 464.
27a. J. McDougall a. H. R. Hulme, Nature, 132, 352, 1933.
27b. H. R. Hulme, J. McDougall, R. A. Buckingham a. R. H. Fowler, Proc. Roy. Soc., A 149, 131, 1935.
28. J. K. Oppenheimer a. M. S. Plesset, Phys. Rev., 44, 53, 1933.
29. W. Heitler a. F. Sauter, Nature, 132, 892, 1933.
30. H. Bethe a. W. Heitler, Proc. Roy. Soc., A 146, 83, 1934.
31. J. Nishina, S. Tomonaga a. S. Sakate, Supp. Sci. Pap. Inst. Phys. Chem. Res., 24, 1, 1934.
32. G. Racah, Nuovo Cimento, XI, 477, 1934.
33. H. Bethe, Proc. Camb. Phil. Soc., 30, 524, 1934.
34. J. C. Jaeger a. H. R. Hulme, Proc. Roy. Soc., A 153, 443, 1936.
35. J. C. Jaeger, Nature, 137, 781, 1936.
36. P. M. S. Blackett, Nature, 132, 917, 1933.
37. Grinberg, C. R., 197, 318, 1933.
38. S. Benedetti, C. R., 200, 1339, 1935.
39. T. Benecke, Z. Physik, 96, 571, 1935.
40. J. Chadwick, P. M. S. Blackett a. G. P. S. Occhialini, Proc. Roy. Soc., A 144, 235, 1934.
41. J. Chadwick, M. Goldhaber, Nature, 134, 237, 1934. See also Uspekhi fiz. nauk, XIV, issue 8, 1934.
42. L. Szilard a. T. A. Chalmers, Nature, 134, 494, 1934.
43. A. Brasch, F. Lange, L. Szillard a. others, Nature, 134, 880, 1934; W. Gentner, C. R., 199, 1211, 1934; 200, 311, 1935; L. N. Ridenour, K. Skjanhaara, M. Jost, Phys. Rev., 47, 718, 1935; L. Arzimowitsch u. Palibin, Sow. Phys. 7, 245, 1935; J. Chadwick a. M. Goldhaber, Proc. Roy. Soc. A 151, 479, 1935.
44. W. Gentner, J. d. Phys. V, 49, 1934.
45. W. Gentner, J. d. Phys. VI, 274, 1935.
46. H. Ketelaar, A. Picard et F. Stael, J. d. Phys. V, 385, 1934.
47. J. S. Rogers, Proc. Phys. Soc., 44, 349, 1932.
48. W. Gentner, Z. techn. Phys., 16, 416, 1935.
49. W. Gentner et I. Starkewicz, J. d. Phys., VI, 340, 1935.
50. C. Y. Chao, Phys. Rev., 36, 1519, 1930.
51. L. Meitner u. H. Hupfeld, Z. Physik, 75, 705, 1932.
52. K. F. W-Kohlrausch, Wiener Ber., 126, 705, 887, 1917.
53. H. R. Crane, L. A. Delasasso, W. A. Fowler, C. C. Lauritsen, Phys. Rev. 46, 531, 1934.
54. D. Skobelzyn, C. R., 194, 1486, 1932.
55. H. Casimir, Helv. Phys. Acta, VI, 287, 1933.
W. Franz, Z. Physik, 90, 623, 1934; Z. Physik, 98, 314, 1935.
Z. Physik, 95, 652, 1935.