ABSORPTION AND SCATTERING OF $\gamma$-RAYS

M. P. Bronstein, Leningrad

§ 1. The question of the absorption and scattering of $\gamma$-rays has recently attracted much attention in connection with the anomalous absorption discovered in 1930 by Tarrant and Chao. This anomalous absorption is of great interest as one of those effects that completely elude explanation within the framework of modern theory and can be explained only after it becomes possible to construct a relativistic quantum theory and, on its basis, a satisfactory theory of the atomic nucleus. A survey of the older work on the absorption and scattering of $\gamma$-rays (up to the discovery of Tarrant and Chao) may be found in the book by Rutherford, Chadwick, and Ellis¹; one paragraph in Gamow’s book² is devoted to the question of anomalous absorption; however, at the present time even this account has already become noticeably outdated in view of the new works that have appeared since then. We therefore consider it useful to give a connected survey of the question in its development and present state.

Already the first investigators of $\gamma$-rays found that, in passing through matter, their intensity decreases according to a law that tends toward an exponential one as the thickness of the absorbing layer is increased. From this it was concluded that the $\gamma$-rays emitted by a given radioactive preparation represent a complex of electromagnetic radiations of different penetrating powers, as a result of which an increase in the thickness of the absorbing layer acts as a filter, making the $\gamma$-rays more homogeneous by transmitting only the most penetrating rays, corresponding to the smallest absorption coefficient. Beginning in 1904 it has been known that the absorption of $\gamma$-rays is accompanied

accompanied by their scattering, i.e., by the emission of secondary γ-rays, the penetrating power of the secondary γ-rays being less than that of the primary ones by an amount that increases noticeably with increasing scattering angle. At the present time it is quite clear that here we are dealing simply with the Compton effect, i.e., with the scattering of γ-rays by electrons. The fact that the Compton effect plays a role in the absorption of γ-rays was confirmed by Skobeltsyn³, who observed in a Wilson chamber tracks of recoil electrons at various angles to the incident beam of γ-rays and measured the velocities of these electrons by curving their paths in a magnetic field; if the velocity of the recoil electron and the angle formed by the direction of this velocity with the direction of the primary γ-rays are known simultaneously, then the frequency of the primary rays can be calculated from Compton’s formula. In this way Skobeltsyn gave a new method for measuring the wavelength of γ-rays, capable of competing with the usual photoelectric method.

Since the study of the absorption of γ-rays is always carried out in such a way that, after passing through an absorbing layer, they enter an ionization chamber, and the decrease of the ionizing current is measured as the thickness of the layer is increased, it is a very essential requirement that the secondary rays, as far as possible, should not enter the chamber and should not superpose their ionization effect on the measured ionization effect of the primary rays. For this it is necessary that the absorber be seen from the chamber under a very small angle. It is therefore understandable that only very powerful sources are suitable for studying the absorption of γ-rays. As such a powerful source, in earlier works radium B + C was used. The first investigators (Soddy and Russell, 1909–1913) found that, after passing through a lead filter of thickness 2 cm, the γ-rays of this source begin to be absorbed according to an exponential law; from this they erroneously concluded that in this way it is possible to obtain γ-rays of a high degree of homogeneity. In reality this is explained only by the fact that any conclusions about the spectral composition of γ-rays based

only on the study of absorption, are extremely unreliable, owing to the low sensitivity of such a method. In reality, however, as is shown by the study carried out by Frille by the method of scattering in crystals, and by Thibaud, Ellis, and Aston by the photoelectric method[^4], the $\gamma$-spectrum of radium B + C consists of a very large number of lines in the region from 0.05 to two and a half million volt-electrons, the hardest rays by no means being distinguished by the greatest intensity. Therefore the production of a homogeneous beam is impossible, and the interpretation of experiments becomes difficult. Far more favorable conditions are found in a preparation of radiothorium in equilibrium with its decay products, among which thorium C″ emits the hardest $\gamma$-line (wavelength $\lambda = 4.66$ X-units, quantum energy $2.649 \cdot 10^6$ volt-electrons). This line is distinguished by a very great intensity; at the same time it is separated by a very wide spectral interval from all the other $\gamma$-lines of the same preparation[^5]. Basting finds, moreover, that about 70 or 80% of the total intensity falls on the hardest line. It is therefore not surprising that a few centimeters of lead are sufficient to isolate practically the line $\lambda = 4.66$ X-units and thus obtain an intense monochromatic beam of $\gamma$-rays. All recent advances in the study of the absorption and scattering of $\gamma$-rays are due to the use of a radiothorium preparation as the source.

With what may the decrease in the energy of $\gamma$-rays be connected when they pass through matter? If we turn to X-rays, it appears that the decrease in the intensity of the primary beam occurs in three ways: 1) an X-ray photon may be absorbed outright with the simultaneous transition of the atom into an excited state or with the ejection of an electron from the atom (photoelectric effect), 2) a photon may be scattered by the whole set of electrons of the atom without a change in wavelength, 3) a photon may be scattered by a single electron, to which it transfers part of its energy and momentum (Compton effect). Since in the case of $\gamma$-rays, as shown—

...as experiment shows (see the addendum to the work cited in note 22), the scattered (secondary) rays contain no noticeable traces of radiation with the original hardness. Therefore, in the region of γ-rays we should expect only the first and third causes of the decrease in intensity, i.e., the photoeffect and the Compton effect. The study of the absorption of X-rays shows that the photoelectric absorption by an atom increases in proportion to the fourth power of the atomic number; however, since it is, in addition, approximately proportional to the third power of the wavelength, we must expect that, in the case of hard γ-rays, even for heavy elements the photoelectric absorption must be many times smaller than the absorption associated with the Compton effect, which, on the contrary, decreases very slowly as the hardness of the rays increases. Thus, we have every reason to expect that the Compton effect will be the chief source of absorption of γ-rays. Therefore, from the very beginning we must recall the results of the theory of the Compton effect.

Since the energy of a γ-ray photon is many times greater than the energy of all, even very deep, electrons surrounding the atomic nucleus, we have the right to regard all atomic electrons in this process as free, possessing no kinetic energy and independent of one another. Therefore the Compton absorption must simply be proportional to the number of all atomic electrons in the volume under consideration. The elementary process consists in the fact that a photon of frequency $\nu$ “collides” with a stationary electron and flies off in a direction forming an angle $\theta$ with the initial direction of its motion. Application of the laws of conservation of energy and momentum leads, as is known, to the result that the frequency of the scattered photon must decrease. If, instead of the frequency $\nu$, one specifies the photon energy $\alpha$, expressed in units of $mc^2$, i.e., the fraction $\dfrac{h\nu}{mc^2}$, then for the energy of the photon scattered through the angle $\theta$ one obtains the well-known expression:

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

A very important question is the probability of the Compton effect, i.e. the intensity of the scattered rays. The theory of this question was developed by Klein and Nishina\(^6\), who applied to it the relativistic Dirac wave equation. They found a solution of this equation for an electron situated in the field of a plane monochromatic electromagnetic wave, and from the electron’s wave function calculated the secondary electromagnetic field produced by it. This makes it possible to calculate the intensity of the scattered rays. If the wave incident on the electron is unpolarized, and through each \(1\ \mathrm{cm}^2\) each second there pass \(n\) photons, then in the directions making angles from \(\theta\) to \(\theta+d\theta\) with the direction of the primary beam the electron emits per second a number of photons equal to:

\[ n\frac{\pi e^4}{m^2c^4}\sin\theta\,d\theta\, \frac{1+\cos^2\theta}{[1+\alpha(1-\cos\theta)]^2}\times \]

\[ \times\left\{1+\alpha^2 \frac{(1-\cos\theta)^2}{(1+\cos^2\theta)[1+\alpha(1-\cos\theta)]} \right\} \tag{2} \]

From formula (1) it is evident that these photons possess energy from \(\alpha'\) to \(\alpha'+d\alpha'\), where \(d\alpha'=-\alpha'^2\sin\theta\,d\theta\). Eliminating the angle \(\theta\) from formulas (1) and (2), we find that the number of photons scattered by the electron per second in the spectral interval from \(\alpha'\) to \(\alpha'+d\alpha'\) is equal to:

\[ f(\alpha')d\alpha'= -n\frac{\pi e^4}{m^2c^4}\frac{d\alpha'}{\alpha^2} \left\{ \frac{2}{\alpha}+\frac{1}{\alpha^2}+\frac{\alpha'}{\alpha} +\frac{1}{\alpha'^2} +\frac{1}{\alpha'}\left(\alpha-\frac{2}{\alpha}-2\right) \right\} \tag{3} \]

Therefore the decrease in the energy of the primary beam, caused by one electron per second, is equal (in ergs) to:

\[ mc^2\alpha \int_{\alpha}^{\frac{\alpha}{1+2\alpha}} f(\alpha')\,d\alpha' . \tag{4} \]

The quantity \(\dfrac{\alpha}{1+2\alpha}\), standing in the upper limit of the integral, is equal to \(\alpha'\) at \(\theta=180^\circ\).

The ratio of the decrease in the energy of the primary beam under the action of one electron to the energy of the primary ...

of a beam that has passed during the same interval of time through \(1\ \mathrm{cm}^{2}\), is called the total Compton absorption coefficient per electron. (It is not difficult to see that the usual volume absorption coefficient, or more precisely that part of it which is associated with the Compton effect, is equal to this ratio multiplied by the number of electrons in unit volume.) Denoting the total absorption coefficient per electron by \(\sigma\), we obtain from (3) and (4):

\[ \sigma=\frac{mc^{2}\alpha}{nh\nu}\int\limits_{\alpha}^{\frac{\alpha}{1+2\alpha}} f(\alpha')\,d\alpha' = \]

\[ =\frac{\pi e^{4}}{m^{2}c^{4}\alpha^{2}} \int\limits_{\frac{\alpha}{1+2\alpha}}^{\alpha} \left[ \frac{2}{\alpha}+\frac{1}{\alpha^{2}}+\frac{\alpha'}{\alpha} +\frac{1}{\alpha'^{2}}+\frac{1}{\alpha'} \left(\alpha-\frac{2}{\alpha}-2\right) \right]d\alpha', \]

or, finally,

\[ \sigma=\frac{2\pi e^{4}}{m^{2}c^{4}} \left[ \frac{2}{\alpha^{2}}+\frac{1+\alpha}{(1+2\alpha)^{2}} +\left(\frac{1}{2\alpha}-\frac{1}{\alpha^{2}}-\frac{1}{\alpha^{3}}\right) \lg(1+2\alpha) \right]. \tag{5} \]

This formula is the celebrated Klein–Nishina formula. Since it is of a very cumbersome character, we illustrate it with a drawing (Fig. 1). For \(\lambda=4.66\) X-units (the hard line \(ThC'\)), calculation by the Klein–Nishina formula gives \(\sigma=1.234\cdot10^{-25}\ \mathrm{cm}^{2}\).

If the absorption of \(\gamma\)-rays were connected only with the Compton effect, then the quantity \(\dfrac{\mu A}{NZ\rho}\), where \(\mu\) is the volume absorption coefficient, \(\rho\) the density, \(Z\) the atomic number, \(N\) Avogadro’s number, and \(A\) the atomic weight, would be equal to \(\sigma\). In fact, it has long been known that for heavy elements \(\dfrac{\mu A}{NZ\rho}\), i.e. the absorption coefficient per electron, is considerably larger than for light ones. This was already established by Soddy and Russell. Kohlrausch and Amaldi\({}^{7}\), whose experiments are described in detail in Rutherford’s book cited above, used radium \(B+C\) as the source. In this case, for light elements the absorption coefficients per electron proved to be

identical. Thus, for example, Kohlrausch found that the absorption coefficient of the $\gamma$-rays of radium $B + C$, passed through a lead filter more than $3.5\ \mathrm{cm}$ thick, is, in carbon, $1.58 \cdot 10^{-25}$, in magnesium, $1.59 \cdot 10^{-25}$, in aluminum, $1.60 \cdot 10^{-25}$, and in sulfur, $1.51 \cdot 10^{-25}\ \mathrm{cm}^{2}$ per electron. We know that the $\gamma$-rays of radium $B + C$, even when filtered in this way, cannot be regarded as entirely homogeneous. However, we may approximately assume that, after passage through the indicated lead filter, the principal role in the absorption begins to be played by

Fig. 1. Dependence of the total absorption coefficient on wavelength according to the Klein–Nishina formula.

the extremely intense line of radium $C$ with wavelength $\lambda = 6.94$ X-units (all the harder lines of radium $B + C$ are considerably less intense). Calculation by the Klein–Nishina formula gives, for this line, $\sigma = 1.57 \cdot 10^{-25}\ \mathrm{cm}^{2}$, which agrees excellently (within the accuracy of the experiment) with Kohlrausch’s numbers. Measurements with the $\gamma$-rays of thorium $C''$ before 1930 were carried out only by Rutherford and Richardson$^{8}$, who found that in aluminum the absorption coefficient of the filtered rays is $1.228 \cdot 10^{-25}\ \mathrm{cm}^{2}$ per electron, which agrees excellently with the number derived above from the Klein–Nishina formula; however, this agreement, as Rutherford notes in his book, should be considered accidental, since in 1913, when these experiments were performed, the measuring technique was very imperfect. Nevertheless, everything indicated that the absorption of $\gamma$-rays by light elements reduces to a single Compton-

effect. The intensity of radiation scattered at an angle \(\theta\) in various directions with respect to the primary beam was studied by Kohlrausch and other authors; however, the inhomogeneity of the \(\gamma\)-rays used does not allow one to assert with certainty that the distribution of the intensity of the secondary rays in the different directions corresponds precisely to the Klein–Nishina formula, and not to any of the previously used formulas for the intensity of Compton scattering. As regards absorption in heavy elements, it already followed from old experiments that the absorption, referred to one electron, is considerably greater than in the case of light elements. A description of these experiments may be found in Rutherford’s book. They all have a very confused character because of the inhomogeneity of the \(\gamma\)-rays used. Ahmad and Kohlrausch found (we shall see below that this is erroneous) that the additional absorption per electron is proportional to the cube of the atomic number; this gave grounds for concluding that the additional absorption in heavy elements is connected with the photoelectric effect. This view is held, for example, by Rutherford, Ellis, and Chadwick in the cited book, published in 1930. However, in that same year, 1930, this opinion was refuted by a series of experimental works in which the decay products of radiothorium were used as the source of \(\gamma\)-rays.

A systematic study of the absorption of monochromatic \(\gamma\)-rays (\(\lambda = 4.66\) X-units) was undertaken by the Chinese physicist Chao \(^{9}\), by Tarrant \(^{10}\), and by Meitner and Hupfeld \(^{11}\), independently of one another. To these also belong the works of Jacobsen \(^{12}\) and Tarrant \(^{13}\). Let us consider the results of these works; moreover, following Tarrant \(^{13}\), who calculated the corresponding corrections for the data obtained in the preceding works, we shall present all results only after first correcting them for the inhomogeneity of the \(\gamma\)-rays remaining after filtration, and for the scattered secondary rays that entered the ionization chamber (“filter correction” and “scattering correction”).

Chao \(^{9}\) measured the absorption coefficients of \(\gamma\)-rays from a radiothorium preparation in equilibrium with the products

decay, after these rays had passed through a lead filter 6.8 cm thick. The results of his measurements are collected in the table.

A more extensive investigation was carried out by Tarrant\(^{10}\). The corresponding table has the form:

(the absorption in hydrogen was calculated from the absorption in paraffin). The circumstance that the additional absorption (over and above the Compton absorption \(1.234 \cdot 10^{-25}\ \mathrm{cm}^2\) per electron) varies very irregularly from element to element convinced Tarrant that the additional absorption is explained not by the photoelectric effect at all, but that here we are dealing with the absorption of \(\gamma\)-rays by nuclei. Subsequently, as we shall see, it turned out that the irregularity in the change of the absorption coefficient with atomic number in fact does not occur and is refuted by more precise measurements; nevertheless, the presence of nuclear absorption is evident at least from the fact that even if all the additional absorption in lead (i.e., the difference between the absorption in lead and the absorption in light elements) were explained by the photoelectric effect, then, varying it in proportion to the cube of the atomic number, we still could explain only an insignificant

part of the additional absorption in tin, antimony, and cadmium. In addition, the fact that the additional absorption of the $\gamma$ rays of thorium $C''$ is greater than in the case of the less hard $\gamma$ rays of radium C also indicates the independence of this phenomenon from the photoelectric effect, which should decrease with increasing frequency. Gray’s experiments[^14] show that the photoeffect produced by $\gamma$ rays of the frequency under consideration in lead is capable of explaining only part of the additional absorption in this element, namely from $0.1\cdot10^{-25}$ to $0.2\cdot10^{-25}$, whereas the additional absorption in lead amounts to about $0.4\cdot10^{-25}$. Thus, in the cited works of Chao and Terrill there was contained the discovery of an entirely new effect—the absorption of $\gamma$ rays by the nuclei of heavy elements. Let us note that this conclusion is not, in general, compulsory, since one may cast doubt on the validity of the Klein–Nishina formula itself: after all, it was derived from the Dirac equation, which is not a true law of relativistic quantum theory but has a certain compromise character; and indeed, Heisenberg[^15] showed that cosmic rays, which at the earth’s surface are electrons with kinetic energy of tens of billions of volts, and at entry into the atmosphere are probably $\gamma$ rays of the corresponding monstrous hardness, subsequently transformed into fast electrons by the Compton effect, are absorbed in this Compton effect many times more strongly than follows from the Klein–Nishina formula. However, in the spectral region under consideration, deviations from the Klein–Nishina formula hardly occur; moreover, various features of the anomalous scattering of $\gamma$ rays, which will be discussed further and which accompanies the anomalous absorption under consideration, indicate that the most probable explanation is a nuclear effect, even if it increases regularly in going from element to element.

Further study of the question was carried out by Meitner and Hupfeld[^11]. The results of their experiments are summarized in the following table:

These data, as can be seen from Fig. 2, where we give the results of all measurements, can hardly be fitted to a smooth curve.

Fig. 2. Dependence of the absorption of the line \(\lambda = 1.66\) X units on atomic number.

A completely different result was reached by Jacobsen[^12]. His method consisted in the following: the absorber was a vessel filled with a solution of the substance under investigation or

of some compound of it in the liquid; the ionization was measured by the fall time of the electrometer leaf. The amount of liquid in the vessel was varied until the same fall time was obtained as in the first substance studied. After this, the number of electrons in the absorber was measured (by means of chemical analysis), which made it possible to find the absorption coefficient per electron. The result was then recalculated for absorption in the pure substance (and not in its solution or its compounds). The absorption of the $\gamma$-rays of radium C and thorium C$''$ was studied. Jacobsen’s results are summarized in the table:

They lie on an entirely smooth curve (the solid curve in Fig. 2), which at $Z = 0$ gives the value corresponding to the Klein–Nishina formula, and with increasing $Z$ shows an additional absorption approximately proportional to the square of the atomic number. This additional absorption consists of the sum of nuclear absorption and the photoelectric effect; regarding the latter we may expect (by analogy with X-rays) that it will proceed (when calculated per electron) proportionally to $Z^3$, and not $Z^2$. In any case, a reliable separation of the two effects is difficult.

Terrent[^13] again undertook careful measurements in order to check his earlier results and to settle definitively the question of whether the dependence of anomalous absorption on atomic number has a regular or irregular character. In doing so he used the same method of direct measurement of absorption as before (in contrast to Jacobsen’s method, in which only the number of electrons was varied, but not the ionization), but a very

significantly increased the accuracy of the experiment by using an ionization chamber filled with nitrogen under very high pressure (100–120 atm): as the pressure is increased, the ionization produced by γ-rays increases, and this made it possible to obtain much larger ionization currents and to measure them more easily than in previous works. Tarrant’s results are collected in the table:

As the figure shows, these results lie very close to a smooth curve (the dashed curve in Fig. 2). This curve, however, differs noticeably from Jacobsen’s curve, leading at \(Z = 0\) to a value noticeably larger than that calculated by the Klein–Nishina formula. This, in Tarrant’s opinion, may be explained by the fact that the γ-rays of thorium \(C''\) have, in addition to the investigated line, also a continuous background; or by the fact that the dependence of intranuclear absorption on the atomic number is not so simple in character, leading to anomalously large values (in comparison with the law \(Z^2\)) for the light elements. In any case, the experiment speaks decisively against the irregular course of anomalous absorption that was obtained in Tarrant’s first work and in the work of Meitner and Hupfeld.

§ 2. The question arises of a theoretical interpretation of nuclear absorption. Beck \(^{16}\) proposed, as an explanation, possible scattering by heavy constituent parts of the nucleus, for example α-particles; Landau \(^{17}\) showed, however, that this scattering must be an enormous number of times smaller

Compton. Therefore, a substantial role must be assigned to intranuclear electrons. Since the behavior of intranuclear electrons lies outside the framework of existing physical theory (i.e., the ordinary, nonrelativistic quantum theory), a theoretical calculation of the absorption produced by them is impossible. In view of the fact that a whole series of circumstances indicates the inapplicability of the law of conservation of energy to electrons inside the nucleus, Gamow18 proposed, as a possible hypothesis, that a photon of $\gamma$-rays absorbed by the nucleus disappears without a trace, leading neither to excitation of the nucleus nor to the corresponding anomalous scattering. This hypothesis, however, does not agree with reality, since experiment indicates the presence of anomalous scattering. We shall now turn to this question.

As early as 1930, Chao19, studying secondary $\gamma$-rays from aluminum and from lead, found that in the case of lead there is added to the rays scattered according to the Compton-effect formulas an anomalously scattered radiation which is absent in the case of aluminum. Chao became convinced of this on the basis of the fact that, when measuring the ionization produced in an ionization chamber by rays scattered at different angles, the ratio of the ionization currents caused by the secondary rays of lead and aluminum is equal to the ratio of the intensities of these rays calculated by the Klein–Nishina formula only at small scattering angles, whereas at large scattering angles the ratio of the ionization effect of the secondary rays of lead to the ionization effect of the secondary rays of aluminum is several times greater than according to the Klein–Nishina formula. Since at small scattering angles the intensity of Compton scattering is much greater than at large ones, this is easily explained as follows: anomalous scattering, approximately the same in all directions, is superposed on the Compton scattering of lead; at small angles it may be neglected in comparison with the Compton scattering, but at large angles it becomes significant and leads to a noticeable increase of the current in the ionization chamber. Wishing to find the wavelength of the ano-

normally scattered rays; Chao studied their absorption in a lead filter. It turned out that, in the case of rays scattered by aluminum, their absorption fully corresponds to the usual change in wavelength occurring in the Compton effect. Passing secondary lead rays, scattered at an angle of \(135^\circ\), through a lead filter of \(1.36\) cm, Chao found that after such filtration there remains approximately monochromatic radiation with an absorption coefficient in lead equal to \(1.5\), which corresponds to a wavelength of \(22.5\) X-units. At the same time the intensity of these anomalously scattered rays was of the same order for different directions; namely, it increased by \(60\%\) when the scattering angle was increased from \(35^\circ\) to \(135^\circ\). Thus Chao succeeded in obtaining a fairly clear indication of the existence of a kind of fluorescence in the region of \(\gamma\)-rays. Chao came to the conclusion that, upon absorption of \(\gamma\)-rays of sufficient hardness, the nucleus may be excited, subsequently emitting a photon of lower frequency in passing to the normal state. It should be noted that such a conception a priori must seem overly conservative, since it assumes the applicability of the usual concepts of wave mechanics to the electrons of the nucleus, by which, as we have seen, anomalous absorption is determined. It would seem more probable (by analogy with continuous \(\beta\)-spectra) that anomalous absorption of \(\gamma\)-rays is accompanied neither by a definite excitation threshold at which absorption begins (as in the absorption of X-rays by atoms), nor by characteristic emission in the form of a line spectrum. Nevertheless Chao thinks, \(^{20}\) that he has found excitation or nuclear-disintegration thresholds connected with anomalous absorption of \(\gamma\)-rays. He proceeded as follows: the \(\gamma\)-rays of thorium C″ were subjected to Compton scattering (apparently in some light element), after which the absorption in lead and in aluminum was studied for rays scattered at different angles and possessing different wavelengths, which can be calculated from the Compton formula. We shall quote from Chao’s work the dependence of anomalous absorption (which is deter-

appeared as the difference between absorption in lead and absorption in aluminum as a function of wavelength:

The measurements are very inaccurate; nevertheless, the existence of an absorption minimum near \(\lambda = 6.6\) X-units may be considered established, although, to be sure, the course of the anomalous absorption with wavelength does not much resemble the analogous picture in the region of X-rays. The existence of absorption bands sharply bounded on the side of large wavelengths in no way follows from these experiments.

Fig. 3. Absorption of anomalously scattered rays (after Gray—Tarrant)

In contrast to Chao, Meitner and Hupfeld ^21^ found that the scattered radiation consists of a mixture of Compton scattering with scattered radiation of the original, unchanged frequency. In order to decide the question of the frequency of the anomalously scattered rays, Gray and Tarrant ^22^ made very careful measurements of the absorption of secondary rays scattered by lead at angles of \(125^\circ\) and \(145^\circ\) to the primary beam, which consisted of \(\gamma\)-rays of thorium \(C''\) or radium \(B + C\). The intensity and hardness of the normal (Compton) scattered rays must be so small at these scattering angles that several millimeters of lead are sufficient for their complete absorption. Therefore the absorption curves obtained in the experiments of Gray and Tarrant, when the thickness of the absorbing layer was varied from several millimeters to \(4 \tfrac{1}{2}\) cm of lead, refer to anomalously scattered rays. In Fig. 3 are shown the curves of absorption by lead of rays scattered by lead itself. Curve \(I\) refers to the absorption of rays scattered at

at an angle of \(125^\circ\), and curve \(II\) at an angle of \(145^\circ\), and in both cases the primary rays belong to thorium \(C'\). The circumstance that the curves refer to anomalously scattered rays is also confirmed by the fact that both curves are parallel to one another (if absorption of Compton secondary rays played a role, this would not be so, since when the scattering angle changes from \(125^\circ\) to \(145^\circ\) their hardness changes very strongly). The absorption curves \(I\) and \(II\) correspond approximately to the absorption of a mixture of two monochromatic radiations, one of which has wavelength \(13.5\) X-units (photon energy \(0.92 \cdot 10^6\) electron-volts), and the other a wavelength of \(27\) X-units (photon energy \(0.47 \cdot 10^6\) electron-volts). Therefore Gray and Tarrant are inclined to think that the anomalous scattering corresponds to the emission of characteristic lines by a nucleus previously excited by the absorption of a photon of sufficiently high frequency. Let us note again that the participation of nuclear electrons in the process of anomalous absorption makes the emission of a line spectrum of \(\gamma\)-rays unlikely and leads one to expect a continuous spectrum by analogy with \(\beta\)-spectra. Of course, analysis of the spectral composition of the radiation on the basis of absorption curves alone is very difficult; therefore the experimental results contradict neither of these points of view.

In order to verify the correctness of the conception of anomalous scattering as characteristic emission of \(\gamma\)-rays by the nucleus, Gray and Tarrant repeated these experiments, replacing the filtered \(\gamma\)-rays of thorium \(C''\) by unfiltered \(\gamma\)-rays of radium \(B + C\). The absorption curve obtained was parallel to the two preceding ones, at least within the limits of the possible experimental errors; unfortunately, the experiment was carried out less carefully than in the case of thorium \(C'\). Therefore the characteristic type of anomalous scattering becomes very probable. The analogy with the absorption of X-rays is manifested also in the fact that there exists a certain minimum frequency at which anomalous absorption can occur, playing the role of an excitation threshold. To verify this assertion, Gray and

Tarrant measured absorption in lead of the rays of radium B + C first in unfiltered form (absorption coefficient \(1.0\ \mathrm{cm}^{-1}\)), and then after filtration through \(1.35\ \mathrm{cm}\) of lead (absorption coefficient \(0.46\ \mathrm{cm}^{-1}\)). With a further increase in the thickness of the filtering layer, the absorption coefficient in lead almost does not change. This can be explained by the fact that the introduction of a \(1.35\ \mathrm{cm}\) filtering layer of lead removes those softer \(\gamma\)-rays which are absorbed by the nucleus without secondary anomalous radiation and therefore have a higher absorption coefficient. From experiments of this kind Gray and Tarrant inferred that anomalous scattering begins when \(\gamma\)-rays with a wavelength of about 7 X-units (photon energy \(1.8 \cdot 10^6\) electron-volts) are absorbed. It is noteworthy that this wavelength is very close to that at which Chao observed the minimum of anomalous absorption.

Another method of determining the position of the same threshold is as follows: since, of the photons emitted by thorium C'', only about one third have energy \(h\nu\) exceeding \(1.65 \cdot 10^6\) electron-volts, and about two thirds have energy less than \(0.79 \cdot 10^6\) electron-volts, it may be asserted that about one third of all the photons emitted by thorium C'' can excite the nucleus. If we take such a radium C preparation that the characteristic emission of \(\gamma\)-rays by lead nuclei caused by it has the same intensity as in the case of the previous thorium C'' preparation, then the total intensity of the \(\gamma\)-spectrum of the radium C preparation will prove, as experiment shows, to be 2.3 times greater than the total intensity of the \(\gamma\)-spectrum of the \(Th\ C''\) preparation. It follows from this that only one seventh of the photons emitted by radium C are capable of exciting lead nuclei. Since the relative numbers of photons in different regions of the \(\gamma\)-spectrum of radium C are known (from the experiments of Ellis and Skobeltsyn), it was therefore possible to conclude that excitation of the lead nucleus is caused only by those \(\gamma\)-photons of radium C whose energy \(h\nu\) is greater than \(1.8 \cdot 10^6\) electron-volts, in agreement with the previously obtained value of the excitation threshold. Therefore Gray and Tarrant conclude that the lead nucleus is capable of absorbing photons with energy from \(1.8 \cdot 10^6\)

volts and higher, after which the nuclei pass into the normal state and emit photons with energies of \(0.92\cdot 10^6\) and \(0.47\cdot 10^6\) electron-volts.

If one assumes (which, it would seem, does not contradict experiment) that the intensity of the anomalously scattered rays is the same in all directions, then from measurements made at angles of \(125^\circ\) and \(145^\circ\) one can calculate the total energy emitted in the form of characteristic radiation and compare it with the energy absorbed by the lead nuclei. It turns out that the emitted energy is equal to \(0.52\) of the absorbed energy. Since the energy of the absorbed photon is \(2.65\cdot 10^6\) electron-volts (the energy \(h\nu\) of the hard line ThC′), it is clear that each excited lead nucleus emits in all \(0.52\cdot 2.65\cdot 10^6 = 1.4\cdot 10^6\) electron-volts, which is precisely equal to the sum of the energies of the two anomalously scattered photons. It is therefore very probable that a nucleus which has absorbed a photon with energy from \(1.8\cdot 10^6\) electron-volts gives up \(1.4\cdot 10^6\) electron-volts of energy in the form of two photons with energies \(0.92\) and \(0.47\cdot 10^6\) electron-volts, or three photons with energy \(0.47\cdot 10^6\) electron-volts, while expending the remaining energy in some unknown way. (The assumption that three photons can be emitted is not only quite probable because the energy of the hard photon is hardly accidentally exactly twice the energy of the soft one, but is even necessary because the number of photons with energy \(0.47\cdot 10^6\) in the anomalously scattered radiation is greater than the number of photons with energy \(0.92\cdot 10^6\) electron-volts.)

Gray and Tarrant also studied the anomalous scattering of the same rays of thorium C′ by tin, iron, and water. It turned out that the frequencies of the characteristic radiation, as well as the threshold frequency, are the same as in the case of lead. At the same intensity of the primary rays, the energy of the anomalously scattered rays, calculated per one nucleus, is approximately proportional to \(Z^2\), which is also confirmed in the case of water if one assumes that only the oxygen atoms in \(H_2O\) take part in the anomalous absorption. In this case the ratio of the number of hard photons (\(0.92\cdot 10^6\) electron-volts) to the number of soft photons (\(0.47\cdot 10^6\) electron-

tons) turns out to be different for different nuclei (for lead 0.332, for tin 0.139, for iron 0.067, for oxygen almost 0, whence it is evident that it decreases as the atomic number decreases). The circumstance that the frequencies of the characteristic radiation and the threshold are the same for all nuclei must have (if only this assertion is correct) enormous significance. It indicates that it is possible to speak of some system of levels common to all nuclei, approximately of the type outlined in Fig. 4. Above the normal state of the nucleus there is an excited level \(A\) with an additional energy of 1.8 (all the data in the figure are given in millions of electron-volts). If the nucleus absorbs a photon with energy greater than 1.8 (for example, 2.65), then it first passes to level \(A\), losing the difference of energies in some unknown manner (but hardly in the form of radiant energy, since the emission of hard photons with energy \(2.6-1.8=0.8\) million electron-volts would have been observed experimentally), after which it further loses energy, passing through the intermediate excited levels \(B, C, D\) in one of the three ways indicated in Fig. 4 by arrows. From state \(D\) it falls to the unexcited state, losing energy in some unknown manner (for example, by emitting a photon with energy \(0.3 \cdot 10^6\) electron-volts, which is rapidly absorbed, being comparatively soft, and therefore escapes observation). The question remains unclear why the nucleus cannot be excited by passing not to level \(A\), but to one of the levels \(B, C, D\). Proceeding from the identity of the levels for all elements, Gray and Tarrant suppose that in anomalous absorption and emission some constituent part of all nuclei takes part, i.e., most likely, an \(\alpha\)-particle. The transition of an \(\alpha\)-particle to state \(A\) could be interpreted as the simultaneous excitation of three protons in it, each of which then

Fig. 4. Level scheme according to Gray and Tarrant (energy differences are given in millions of electron-volts).

passes into the normal state spontaneously (??). At the same time, the question still remains unclear as to why the ratio of the probabilities of emission of two and three photons is different for different elements. Therefore the interpretation proposed by Gray and Tarrant should nevertheless be regarded as improbable. We must not forget that the analysis of the spectrum of anomalously scattered rays on the basis of their absorption curve is of a very doubtful character. Most likely, we are dealing here with effects of the relativistic quantum theory, and the energy relations are probably distinguished by the same uncertainty as in the case of $\beta$-decay. Therefore the development of reliable experimental methods for studying the spectral composition of anomalously scattered radiation is urgently necessary. It is capable of shedding light on many properties of nuclei that have hitherto been unknown, and of providing very important experimental material for the construction of a future relativistic quantum theory.

LITERATURE

1. E. Rutherford, I. Chadwick, C. D. Ellis, Radiations from Radioactive Substances, Cambridge University Press, 1930, chapter XV.

2. G. Gamow, Constitution of Atomic Nuclei and Radioactivity, Oxford University Press, 1931, ch. III, § 6.

3. D. Skobelzyn, Zs. f. Phys. 43, 354, 1927; 58, 595, 1929; Nature 123, 411, 1929.

4. Frilley, Thèse, Paris, 1928; Thibaud, Thèse, Paris, 1925; Ellis and Aston, Proc. Roy. Soc. A 129, 180, 1930. These data are collected in Rutherford’s book (pp. 363 and 365). See also C. D. Ellis, Proc. Camb. Phil. Soc. 22, 369, 1925; C. D. Ellis and W. B. Sinder, Proc. Roy. Soc. A 105, 190, 1924.

5. For the study of the $\gamma$-spectrum of the decay products of radiothorium, see Thibaud, Thèse, Paris 1925; D. H. Black, Proc. Roy. Soc., A 106, 632, 1925; Bastings, Phil. Mag. 5, 785, 1928.

6. O. Klein und Nishina, Zs. f. Phys., 52, 853, 1929.

7. K. W. F. Kohlrausch, Wien. Ber., 126, 441, 683, 1917; Probleme der Strahlung, Braunschweig, Vieweg, 1927; M. Ahmad, Proc. Roy. Soc., A 105, 507, 1924; 109, 207, 1925.

1. Rutherford and Richardson, Phil. Mag. 26, 937, 1913.

2. C. J. Chao, Proc. Nat. Acad. Sci. Amer., 16, 431, 1930.

3. G. T. P. Tarrant, Proc. Roy. Soc., A. 128, 345, 1930.

4. L. Meitner und H. H. Hupfeld, Zs. f. Phys., 67, 147, 1931.

5. J. C. Jacobsen, Naturwiss., 18, 951, 1930; Zs. f. Phys., 70, 145, 1931.

6. G. T. P. Tarrant, Proc. Roy. Soc., A. 135, 223, 1932.

7. L. H. Gray, Proc. Camb. Phil. Soc., 27, 103, 1931.

8. W. Heisenberg, Ann. d. Phys., 13, 430, 1932.

9. G. Beck, Naturwiss., 18, 896, 1930.

10. L. Landau, Naturwiss., 18, 1112, 1930.

11. G. Gamow, l. c. p. 82.

12. C. J. Chao, Phys. Rev. 36, 1519, 1930.

13. C. Chao, Naturwiss., 19, 752, 1932; Proc. Roy. Soc., A. 135, 206, 1932.

14. L. Meitner und H. H. Hupfeld, Naturwiss., 19, 775, 1931.

15. L. H. Gray and G. T. P. Tarrant, Proc. Roy. Soc., A. 136, 662, 1932.