Abstract
In the first part of our article, we analyzed the question of the attenuation of a beam of gamma rays as it passes through matter. Here we shall consider the question of the composition of the scattered radiation arising in matter as gamma rays pass through it; by scattered radiation we shall mean not only true scattered radiation due to the Compton effect, but also the secondary radiation arising from the interaction of electrons and positrons, produced by photons of the primary gamma-ray beam, with the material of the scatterer.
Full Text
ABSORPTION AND SCATTERING OF $\gamma$-RAYS. II
L. V. Groshev, Moscow
In the first part of our article1 we examined the question of the attenuation of a beam of $\gamma$-rays as it passes through matter. Here we shall consider the question of the composition of the scattered radiation arising in matter when $\gamma$-rays pass through it; by scattered radiation we shall understand not only the true radiation scattered as a result of the Compton effect, but also the secondary radiation arising from the interaction of electrons and positrons, produced by photons of the primary beam of $\gamma$-rays, with the material of the scatterer.
§ 1. Introduction
Whereas the attenuation of $\gamma$-radiation as it passes through matter can be subjected to quantitative calculation (see Part I), in the question of scattered radiation we are in a more difficult position: here much is still unclear, there are contradictions in the experimental data obtained in various laboratories, and, moreover, this question has been less developed theoretically. Consequently, at the present time there are only weak indications of a quantitative calculation of scattered radiation.
The experimental study of the question of scattered radiation is complicated by many circumstances. Among these one may name: a) the absence of comparatively strong sources of strictly monochromatic radiation, b) insufficient knowledge of the spectral composition of the $\gamma$-radiation of radioactive elements, as well as its complexity, c) the imperfection of methods of $\gamma$-ray analysis, in particular, of establishing their degree of monochromaticity, d) insufficient knowledge of the operation of instruments used for measuring the intensity of $\gamma$-radiation, for example, the dependence of the sensitivity of the instrument on wavelength, e) the presence of relatively strong parasitic scattered radiation from parts of the apparatus and from the walls of the room in which the measurements are made. To this may also be added the circumstance that, because of the low intensity of the scattered radiation, the scatterer was usually taken in the form of a thick layer (especially in earlier work), for which absorption already affects both the primary and the scattered radiation within the scatterer itself—
the detector. If one takes into account that the composition of scattered radiation is sufficiently complex, especially for the $\gamma$-rays of Ra, it becomes clear how much more complicated the investigation of scattered radiation becomes when thick scattering screens are used. It is true that in studying scattering by thin screens certain complicating circumstances also appear; for example, the positrons produced by $\gamma$-rays leave the scattering screen in which they arose and, upon entering other parts of the apparatus, produce parasitic radiation (through annihilation), which plays a noticeable role because of the small amount of radiation scattered by a thin screen.
In all probability, many contradictions in the experimental results on scattered radiation are due to the fact that the purely methodological shortcomings noted above are sometimes not sufficiently taken into account. Bearing all this in mind, we shall try to clarify the question of scattered radiation, insofar as the work of recent years permits.
§ 2. The Presumed Composition of Scattered Radiation
Six or seven years ago it was believed that scattered radiation is essentially Compton radiation. All the regularities known at that time seemed to support this. However, soon after the discovery of anomalous absorption, the question naturally arose whether this anomalous absorption might not in some way affect the composition of the scattered radiation. Chao², and then others, succeeded in showing that $\gamma$-radiation scattered by bodies in many cases cannot be explained by Compton scattering alone. In addition to Compton scattering, there exists some further scattering, which has been called anomalous. At first it was assumed that anomalously scattered radiation appears as a result of the action of $\gamma$-rays on the nucleus itself or on intranuclear constituents. Further investigations, however, showed that this is not so.
Before proceeding to describe the results of experimental work devoted to the study of scattered radiation, let us try to establish what its composition might be if we proceed from those processes of interaction of $\gamma$-rays with matter that were discussed in the first part of our article.
In accordance with these processes we might expect the following composition of scattered radiation: a) radiation scattered as a result of the Compton effect; b) the so-called annihilation radiation, appearing in the mutual destruction of a positron and an electron; c) bremsstrahlung radiation, arising when electrons and positrons created by photons of the primary beam are slowed down (Compton and photoelectrons, electrons and positrons constituting pairs). We shall consider separately these kinds of radiation, as well as the mechanisms of their origin and the results obtained, and use them in analyzing the experimental data.
§ 3. Compton scattered radiation
As was already noted in the first part of the article, the energy of a photon scattered in the Compton effect depends on the magnitude of the angle (decreases as it increases) at which the scattering occurs. This dependence is given by the following formula:
\[ \alpha'=\frac{\alpha}{1+\alpha(1-\cos\Theta)}, \tag{1} \]
where \(\Theta\) is the scattering angle, and \(\alpha'\) and \(\alpha\) are the energies of the scattered and primary photons, expressed in \(mc^2\). In investigating the question of scattered radiation, however, we shall be interested not only in the magnitude of the energy of the photon scattered at a given angle (and, consequently, in the dependence of the frequency of the scattered radiation on the angle), but also in the probability of this scattering. This probability was calculated on the basis of the relativistic Dirac equation by Klein and Nishina\(^3\) for the case of scattering of \(\gamma\)-radiation by free electrons. They found the following expression for the intensity of radiation scattered at an angle \(\Theta\), calculated per unit solid angle and per one electron:
\[ J=J_0\frac{e^4}{2m^2c^4}\frac{1+\cos^2\Theta}{[1+\alpha(1-\cos\Theta)]^3} \left\{1+\frac{\alpha^2(1-\cos\Theta)^2}{(1+\cos^2\Theta)[1+\alpha(1-\cos\Theta)]}\right\}, \tag{2} \]
where \(J_0\) is the intensity of the primary beam of \(\gamma\)-rays, and \(\alpha\) has the same meaning as in the preceding formula. Formula (2) was later derived by another method by Waller\(^4\), as well as by I. E. Tamm\(^5\) and Vanier\(^6\). In Fig. 1 the dependence of \(J\) on \(\Theta\) is shown graphically for two wavelengths. It is clear from it that the scattered radiation is directed predominantly forward, and the forward directionality is manifested the more strongly, the shorter the wavelength of the radiation being scattered.
Fig. 1.
The validity of formula (2) is confirmed above all by the fact that the formula of Klein–Nishina for the total attenuation coefficient \(\varepsilon^s\), obtained from it (by summation over angles and by some simple additional operations), as we already know (see Part I), agrees excellently with the experimental data. In addition, formula (2) was directly verified by Chao\(^2\) for the case of scattering of \(C''\) \(\gamma\)-rays by aluminum, for which, as we already know, the only—
process of interaction of γ-radiation with matter is the Compton effect. Chao measured, by means of an ionization chamber, the intensity of scattered γ-radiation for various angles. His results are given in Table 1.
TABLE 1
| Scattering angle | 35° | 55° | 90° | 135° |
|---|---|---|---|---|
| $\lambda$ in X-units.* | 9.6 | 15.5 | 29.4 | 47 |
| $N_1$ | 1.0 | 0.493 | 0.249 | 0.180 |
| $N_2$ | 1.0 | 0.504 | 0.254 | 0.188 |
$N_1$ gives the number of photons scattered at the given angle, calculated by formula (2); $N_2$ is the same quantity measured experimentally (in the data of the last line the absorption of the primary and scattered γ-rays in the scatterer is taken into account, as well as the change in the efficiency of the ionization chamber with wavelength). The table presented shows that formula (2) agrees well with the experimental data.
To study radium, formula (2) was checked by Stahel and Ketelaar[^7]. They experimentally determined the ratio of the currents produced in an ionization chamber by radiation coming directly from the source and by radiation scattered by a layer of aluminum; in the second case the source of γ-rays was many times stronger. They then calculated the same ratio on the basis of formula (2). The experimentally found value (1.98) agrees sufficiently well with the calculated one (1.90), thereby confirming the validity of formula (2).
Formula (2) was also checked experimentally by D. V. Skobeltsyn[^8], who used an entirely different method. His method is based on the fact that, in each elementary act of Compton scattering, together with the scattered photon there appears a recoil electron. In this case, between the angles which the scattered photon $(\Theta)$ and the recoil electron $(\vartheta)$ make with the direction of incidence of the primary beam, there is the following relation, following from the laws of conservation of energy and momentum as applied to an elementary act of scattering:
$$ \operatorname{tg}\frac{\Theta}{2}=\frac{\operatorname{ctg}\vartheta}{1+\alpha}. $$
Using this relation, one can associate with each recoil electron formed at an angle $\vartheta$ a photon scattered at an ang—
* The values of $\lambda$ are calculated by formula (1).
\(\Theta\) photon. Therefore, by studying the angular distribution of recoil electrons, one can test formula (2). This was in fact done by D. V. Skobeltsyn. Using a Wilson chamber, he investigated the angular distribution of recoil electrons arising in the gas of the chamber when it was irradiated with \(\gamma\)-rays of Ra\((B + C)\) or ThC″. The angular distribution of the recoil electrons thus established was compared with the distribution obtained from formula (2), after recalculating it [with the aid of relation (3)] for recoil electrons. The spectral distribution of the \(\gamma\)-radiation, which is needed for the recalculation of formula (2), was taken by the author from his earlier works, in which it had been obtained as a result of studying the energy distribution of recoil electrons for the angular interval \(0\)—\(20^\circ\).
The results obtained for the \(\gamma\)-rays of Ra\((B + C)\) are given in Table 2, and for the \(\gamma\)-rays of ThC″ in Table 3.
TABLE 2
| Angular interval in ° | Number of electrons: observed | Number of electrons: calculated |
|---|---|---|
| 0—10 | 117 | 95 |
| 10—20 | 149 | 153 |
| 20—40 | 242 | 264 |
| 40—60 | 215 | 226 |
| 60—80 | 161 | 144 |
| 80—90 | 19 | 21 |
TABLE 3
| Angular interval in ° | Number of electrons: observed | Number of electrons: calculated |
|---|---|---|
| 0—10 | 143 | 96 |
| 10—20 | 110 | 109 |
| 20—40 | 104 | 141 |
| 40—60 | 58 | 78 |
| 60—80 | 42 | 38 |
| 80—90 | 11 | 6 |
As we see, the agreement obtained is fairly good. The author notes that in the details there are discrepancies between the experimental and theoretical data. However, we cannot dwell on them here, all the more since they still have no explanation.
In the material considered, we have sufficient grounds for the validity of formula (2) for the intensity of the scattered radiation.
It should be noted that, besides single Compton scattering, one may also expect multiple scattering, for example double scattering, when the scattered quantum is scattered once more by another electron. In the multiple Compton effect, for one and the same final angle of scattering the energy of the scattered quantum is always greater than in single scattering, as can easily be shown on the basis of equation (1). Therefore such radiation will give a harder component of the scattered radiation. Its intensity can be obtained by successive application of formula (2).
§ 4. Annihilation Radiation
As is known from the first part of the article, when a substance is irradiated with γ-rays of energy greater than 1 MeV, positrons arise. Until now we have been interested only in the probability of their occurrence. We shall now try to trace their fate after they have already arisen. According to Dirac’s theory, a positron can exist in the free state only for a very short time,* since, upon meeting an electron, it can be destroyed together with the latter—annihilate; in this process the kinetic energy of both particles and the energy corresponding to their rest mass are converted into γ-radiation. In Dirac’s interpretation this means the following: a hole in the background of negative levels, representing a positron, cannot remain unoccupied for long. Quite soon after its appearance it is filled by an ordinary electron from one of the positive-energy levels, as a result of which both the positron and the electron disappear.
The energy liberated in annihilation (annihilation radiation) may be emitted in two different ways,** depending on whether the positron annihilates with a free (or weakly bound) electron or with a strongly bound electron. If the annihilation of the positron occurs with a free electron, then the law of conservation of momentum requires that in this case two photons be emitted. If, moreover, both these electrons had very small velocities, then the photons must be emitted in opposite directions and have equal energy, approximately equal to 0.5 MeV \((mc^2)\). In the annihilation of a positron with a strongly bound electron, one photon is emitted, since in this case the conservation laws can be satisfied by the excess momentum being transferred to the nucleus with which the electron was bound. As will be shown below, the first process plays the predominant role. Let us consider both cases.
§ 5. Annihilation of Positrons with the Emission of Two Photons
This case was first investigated by Dirac\(^{10}\) long before the discovery of the positron. Considering the annihilation of a positron*** with
* Thus, for example, for positrons with energy \(10^5\) eV the mean lifetime in water is \(3.6 \cdot 10^{-10}\) sec. For other substances the lifetime (for positrons of the same energy) may be calculated from the value given on the basis of the fact that it is inversely proportional to the electron density.
** We do not consider here the unlikely cases of positron annihilation without the emission of γ-radiation.\(^{9}\)
*** In this work Dirac operated with a “hole” in the background of negative levels, mistakenly identifying it, as was later found, with the proton.
with an electron at rest, he found the following expression for the effective cross section of such a process:
\[ \Phi(\varepsilon)=\frac{\pi e^4}{m^2c^4}\frac{1}{\varepsilon+1} \left[ \frac{\varepsilon^2+4\varepsilon+1}{\varepsilon^2-1} \lg\left(\varepsilon+\sqrt{\varepsilon^2-1}\right) -\frac{\varepsilon+3}{\sqrt{\varepsilon^2-1}} \right], \tag{4} \]
where \(\varepsilon\) is the energy of the positron, expressed in \(mc^2\), and \(e\) and \(m\) are its charge and mass, while \(c\) is the speed of light. From the formula given it is evident that the effective cross section depends on the positron energy, attaining its maximum value for small energies. For a slow positron, where \(\varepsilon^*\) is close to unity,
\[ \Phi\sim \frac{1}{\sqrt{\varepsilon^2-1}}\sim \frac{1}{v}, \tag{5} \]
where \(v\) is the velocity of the positron. For \(v=0\), \(\Phi\) takes an infinite value. This, however, does not lead to any difficulty, since the probability of annihilation per unit time \(w\) is given by the following expression:
\[ w=Nv\Phi, \tag{6} \]
where \(N\) is the number of negative electrons per unit volume. From the formula it is evident that \(w\) remains finite also for the case \(v=0\).
For large energies \(\Phi\) decreases according to the law \(\dfrac{\lg \varepsilon}{\varepsilon}\), as is readily established from formula (4).
On the basis of the fact that \(\Phi\) is large for small \(v\), we arrive at the conclusion that most positrons annihilate only after they have lost all their kinetic energy, expending it on ionization of the atoms of the substance through which they pass. On the other hand, we know that as a result of the annihilation of a slow positron two photons appear, emitted in opposite directions and possessing an energy of about \(0.5\) MeV. Therefore we may expect that the radiation arising in the annihilation of positrons with free electrons will consist mainly of photons with energy \(0.5\) MeV.
A moving positron also has some probability of annihilating with a free electron. In this case, however, two photons with different energies will be emitted, as follows from the law of conservation of momentum. The question of the annihilation of such positrons was treated in considerable detail in Bethe’s work \(^{11}\). In his work Bethe derived a formula for the differential effective cross section for the process of annihilation of a moving positron with
\[ \text{* } \varepsilon \text{ expresses not only the kinetic energy of the positron, but also the energy corresponding to its rest mass. Therefore } \varepsilon=1 \text{ indicates that the positron is at rest.} \]
with the emission of two photons with frequencies \(\nu_r\) and \(\nu_s\), satisfying the following relation:
\[ h\nu_r + h\nu_s = E + mc^2 \quad \text{(law of conservation of energy),} \tag{7} \]
where \(E\) is the total energy of the positron (including the energy corresponding to its rest mass). In his derivation, Bete proceeded from a formula borrowed from Dirac’s work cited above. Analysis of the expression obtained for the differential effective cross section shows that the most probable case is the one in which the photons are emitted in directions making with one another an angle \(\theta\) close to \(180^\circ\); in this case the photon with the greater energy flies in directions close to the direction of motion of the annihilated positron, as follows from the law of conservation of momentum. In the special case when the angle \(\theta\) is exactly \(180^\circ\), the following expressions hold for the frequencies of the emitted photons:
\[ \begin{aligned} \nu_r^{\max} &= \frac{1}{2}\nu + \sqrt{\frac{1}{4}\nu^2 - \frac{1}{2}\nu\nu_0} \approx \nu - \frac{1}{2}\nu_0 = \frac{E + \frac{1}{2}mc^2}{h}, \\[6pt] \nu_s^{\min} &= \frac{1}{2}\nu - \sqrt{\frac{1}{4}\nu^2 - \frac{1}{2}\nu\nu_0} \approx +\frac{1}{2}\nu_0 = \frac{1}{2}\frac{mc^2}{h}, \end{aligned} \tag{8} \]
where
\[ \nu = \nu_r + \nu_s = \frac{E + mc^2}{h} \quad \text{(see (7)),} \]
and
\[ \nu_0 = \frac{mc^2}{h}. \]
\(\nu_r^{\max}\) corresponds to the photon emitted in the direction of motion of the annihilated positron. As can be seen, this photon carries off a large part of the positron’s energy. From all that has been said it may be concluded that, in the annihilation of fast positrons, the hardest component of the radiation is emitted forward, its energy possibly even exceeding the energy of the positrons; in the opposite direction, softer radiation is emitted, with an energy of \(0.25\) MeV \(\left(\frac{mc^2}{2}\right)\) and higher.
In order to judge more clearly the distribution of energy in the annihilation spectrum, we give (Fig. 2) the curve of the intensity distribution of this annihilation radiation by frequencies for the case of annihilation of positrons with kinetic energy \(1.5\) MeV (\(\varepsilon = 4\); the fastest positrons arising from \(\gamma\)-rays of ThC″). The curve is taken from Bete’s work. From it we see that the frequencies closest to \(\nu_r^{\max}\) and \(\nu_s^{\min}\), corresponding to angles \(\theta\) close to \(180^\circ\), are indeed encountered most often.
To clarify the role played in the annihilation radiation by the hard radiation arising in the annihilation of moving positrons, it is necessary to determine what fraction of the total number of positrons undergoes annihilation during their motion. Bethe calculated the probability of annihilation of a positron when its energy decreases
Fig. 2.
from \(\varepsilon\) to \(\varepsilon + d\varepsilon\), and found for it the following expression:
\[ s(\varepsilon)d\varepsilon = \frac{1}{k}\frac{1}{\varepsilon^2(\varepsilon+1)} \left[ (\varepsilon^2+4\varepsilon+1)\lg(\varepsilon+\sqrt{\varepsilon^2-1}) -(\varepsilon+3)\sqrt{\varepsilon^2-1} \right]d\varepsilon = \frac{1}{k}f(\varepsilon)d\varepsilon, \tag{9} \]
where \(\varepsilon\) still gives the energy of the positron in \(mc^2\). The quantity \(k\) is a constant for the given substance in which the positrons are slowed down, but it is not the same for different substances. It is given by the following formula:
\[ k = 4\lg\left(\frac{mc^2}{RZ}\right), \tag{10} \]
where \(Z\) is the atomic number of the substance, and \(R\) is approximately equal to the ionization potential of hydrogen. For hydrogen, air, and lead, \(k\) has respectively the following values: 42; 34.4; and 24.4.
For convenience, the function \(f(\varepsilon)\) is presented graphically in Fig. 3, borrowed from Bethe’s work. Using the curve in Fig. 3 and knowing the constant \(k\), one can easily calculate, for each substance, the function \(s(\varepsilon)=\frac{f(\varepsilon)}{k}\), which, as follows from formula (9), represents the probability of annihilation of a positron with energy \(\varepsilon\) when it loses one unit of energy, i.e. \(mc^2\). From the curve shown it is evident that the greatest probability of annihilation during
of motion, have positrons with a total energy of about \(2mc^2\). However, this probability is not very large. For example, for hydrogen, air, and lead the maximum values of \(s(\varepsilon)\) are as follows: 0.017, 0.02, and 0.03. This means that, for a beam of positrons with kinetic energy of about 500 kV (total energy \(E=2mc^2\)), the number of positrons annihilated when they lose one unit of energy \((mc^2)\) is, for example, for Pb only 3%, i.e. a comparatively small percentage. For elements with smaller atomic number this percentage is still smaller.
Fig. 3.
In what follows we shall also be interested in the probability of annihilation of a positron over the entire time during which it is in motion, i.e. as its energy decreases from the initial value to zero. Denoting this total probability by \(S(\varepsilon_0)\), we have for it:
\[ S(\varepsilon_0)=\int_0^{\varepsilon_0} s(\varepsilon)d\varepsilon =\int_0^{\varepsilon_0} \frac{f(\varepsilon)}{k}\,d\varepsilon =\frac{F(\varepsilon_0)}{k}, \]
where \(\varepsilon_0\) is the initial energy of the positron. The function \(F(\varepsilon_0)\) is also represented graphically in Fig. 3. From the curve shown we find for \(S(\varepsilon_0)\), for example for positrons with kinetic energy \(1.5\) MeV \((\varepsilon=4)\), representing the fastest positrons produced by the \(\gamma\)-rays of ThC′, the following values for hydrogen, air, and lead: 0.044; 0.054 and 0.076, respectively. Thus, for example, in lead \(7.6\%\) of positrons with energy \(\varepsilon=4\) annihilate during their motion. These numbers increase strongly for higher positron energies; for example, for positrons with energy 10 MeV in Pb this quantity is equal to 21%. Therefore, even for positrons produced by the \(\gamma\)-rays of ordinary radioactive-
of elements, we must take into account their annihilation during motion.
Considering all the theoretical data presented on the annihilation of fast and slow positrons with free electrons, we arrive at the following conclusion regarding the composition of the annihilation radiation that arises in this process. The predominant role will be played by radiation with an energy of about 0.5 MeV \((mc^2)\), corresponding to the annihilation of slow positrons. In addition to this band, however, there will be radiation with energy varying from 0.25 MeV \(\left(\frac{1}{2}mc^2\right)\) practically up to energies corresponding to the fastest positrons. This latter broad band is caused by the annihilation of fast positrons and amounts, for example, for Pb (for positrons with energy \(\varepsilon = 4\)), to 7.5% of the total number of photons. If the annihilating positrons are produced by \(\gamma\)-rays, then the energy of these primary rays will differ from the energies of the hardest annihilation radiation by only 0.25 MeV (see 8), which, for example, for the ThC″ \(\gamma\)-rays with energy 2.65 MeV is a comparatively small quantity.
It may also be appropriate here to raise the question of the angular distribution of the entire annihilation radiation. As regards radiation with an energy of 500 keV, it is, of course, distributed isotropically in space. However, with the hard radiation, which is of particular interest to us, the situation is more complicated. We have already seen above that photons of high energy, arising in the annihilation of fast positrons, are emitted predominantly in the direction of motion of the positrons. But, in view of the content of our article, we shall be more interested not in the angular distribution of photons relative to the incident positrons, but in the angular distribution of these photons with respect to the beam of \(\gamma\)-rays that produce the positrons. This question was also treated in Bothe’s work. In investigating the question posed, the principal role is played by the elastic scattering of positrons by atoms, since it is large for the energies encountered in work with \(\gamma\)-rays of radioactive elements. Assuming that the elastic scattering of positrons by atoms occurs in the same way as for electrons, we obtain smaller scattering for substances with small atomic numbers. Therefore, for light elements, the hard annihilation radiation must be more anisotropic (with a preferential forward direction) than for heavy ones. On the basis of statistical calculations of the elastic scattering of positrons, Bothe showed that this indeed takes place. Without making exact calculations of the angular distribution, he found that, for positrons produced by the ThC″ \(\gamma\)-rays and annihilating in air, one half of the hard annihilation radiation is contained in a cone whose generatrix makes an angle of \(60–70^\circ\) with the direction of the primary \(\gamma\)-ray beam. For lead, the hard annihilation radiation is distributed practically isotropically.
§ 6. Annihilation of positrons with the emission of one photon
As has already been noted above, the type of annihilation under consideration occurs only for strongly bound electrons. It follows at once from this that this process is unlikely for slow positrons (in contrast to annihilation with the emission of two photons), since they are unable to reach strongly bound electrons owing to the repulsive action of the atomic electric field. Theoretical calculations confirm this. First Fermi and Uhlenbeck[^12] for the nonrelativistic case, and then Nishina, Tomonaga, and Tamaki[^13] for the relativistic case, showed that the effective cross section for annihilation of comparatively slow positrons with the emission of one photon is small in comparison with the effective cross section for annihilation with the emission of two photons. The question of this type of annihilation was later considered in greater detail in the work of Bethe[^11] and of Bhabha and Hulme[^14]. It was established above all that the effective cross section for annihilation with the emission of one photon is proportional to \(Z^5\). Therefore the radiation appearing as a result of such a process must play a greater role for heavy elements than for light ones.
Fig. 4.
The results obtained by Bethe are most conveniently represented by the curve given in his work (Fig. 4). The figure gives the dependence on the energy of the function \(\chi\), defined by the fact that its product by
\[ \left(\frac{Z}{137}\right)^4 \]
gives the ratio of the probability of annihilation* with the emission of one photon (calculated for \(2k\) electrons) to the probability of annihilation with the emission of 2 photons for one and the same energy. From analysis of the curve presented it follows first of all that for light elements annihilation of positrons with the emission of one photon plays no role, since
\[ \left(\frac{Z}{137}\right)^4 \chi(\varepsilon) \]
is negligibly small for all energies (for example, for air it is equal to \(10^{-5}\)). But even for heavy elements this type of annihilation is improbable.
For example, for lead, for the energy interval corresponding to the maximum values of \(\chi(\varepsilon)\) \((3—20\,mc^2)\), we obtain for
\(\left(\dfrac{Z}{137}\right)^4 \chi(\varepsilon)\) the value 0.16. This means that, for positrons of the given energy, only 16% of the total number of annihilated positrons were destroyed with the emission of one photon. If we also take into account the circumstance that, for these energies, the probability of annihilation with the emission of two photons [see \(s(\varepsilon)\)] is very small in comparison with the total annihilation, then from this one may conclude that, for the complete annihilation radiation, the process considered by us plays practically no role. At the same time it must also be borne in mind that Bethe’s data, obtained by him in the Born approximation, are five times larger than the values given by more exact calculations \(^{15}\). Thus we are entitled to neglect annihilation with the emission of one photon.
On this basis, we may consider that the judgment concerning the composition of the annihilation radiation expressed by us in the preceding section applies not only to the radiation arising in annihilation on free electrons, but also to the complete annihilation radiation.
§ 7. Experimental proof of the existence of annihilation radiation
In the preceding sections we have considered those conclusions to which the theory leads on the question of positron annihilation. Let us now see whether the existence of positron annihilation is confirmed by experimental data.
Soon after the discovery of positrons, Joliot \(^{16}\) and Thibaud \(^{17}\) simultaneously, but independently, showed that positron annihilation in nature does indeed take place. In the experiments of both authors, a beam of positrons was focused by Thibaud’s trochoid method (see, for example, 17 c) onto a layer of substance (Pb and Al in Joliot, and Pt in Thibaud). Joliot in his experiments used layers of substance sufficient for the complete absorption of the positrons. He showed that if positrons are directed onto a substance, an additional \(\gamma\)-radiation arises in it, which is absent in the case when, instead of positrons, negative electrons strike the substance. Measuring, with the aid of a Geiger–Müller counter, the absorption of this additional radiation in lead, Joliot showed that it consists of photons with energy \(485\ \mathrm{keV} \pm 60\ \mathrm{keV}\)*. He was also able to determine, though very roughly, the yield of photons, by measuring for this purpose the number of positrons absorbed in the substance and the number of photons arising in it. The experiments
* In Joliot’s work the positrons were produced in aluminum when it was bombarded by \(\alpha\)-particles; in Thibaud’s, in lead irradiated by \(\gamma\)-rays.
** The energy of the radiation was determined from the magnitude of the absorption coefficient.
showed that for each absorbed positron there are from 1.6 to 3 photons, i.e. on the average about 2. Results almost analogous to those of Thibaud’s experiments were obtained both with respect to the hardness of the additional radiation and with respect to the number of photons per absorbed positron.
The existence of positron annihilation is demonstrated especially vividly by the experiments of Lauritsen and Crane[^18]. In these experiments the source of positrons was an artificially radioactive element produced by bombarding carbon with a stream of fast deuterons. The carbon plate emitting positrons was placed on an ionization chamber with its active side upward. The positrons traveling downward from the active layer were absorbed in the material of the plate and annihilated there, giving rise to \(\gamma\)-radiation, which was measured by the ionization chamber. The positrons emitted upward went into the air to a sufficiently large distance from the source, and therefore the annihilation radiation caused by them was practically not recorded by the chamber. However, they can be made to annihilate near the ionization chamber if a layer of material in which the positrons would be absorbed is placed on the source. For this purpose the authors placed on the source a second, similar carbon plate and found that the current in the ionization chamber, i.e. the intensity of the measured \(\gamma\)-radiation, increased approximately twofold. Under several other conditions Lauritsen and Crane measured the absorption of the resulting \(\gamma\)-radiation in lead. From the value obtained for the absorption coefficient, it was possible to establish that the radiation under investigation was close in hardness to radiation with an energy of 0.5 MeV. Estimating the number of annihilated positrons and the number of photons that appeared, the authors established that for each positron that disappeared approximately two photons arise.
Fig. 5.
Thus all these experiments undoubtedly prove the real existence of annihilation radiation, consisting of photons with an energy of about 0.5 MeV, whose number is approximately twice the number of annihilated positrons. This confirms the theory of positron annihilation. However, the agreement with the theory goes still further.
Klemperer’s experiments[^19] established the simultaneous emission of two photons. The arrangement of his experiments is shown in Fig. 5. Between two semi-cylindrical counters, covered on the inner side with thin foil, there was placed a source of positrons: a graphite plate previously irradiated with fast protons and then covered with a layer of metal sufficient for the complete absorption of positrons. The number of coincidences of pulses in both counters, caused by the simultaneous emission of two photons, was measured. The data obtained show agreement, at least in order of magnitude, between the expected and measured number of double coincidences...
... Klempérer showed, moreover, that under the conditions of his experiments annihilation radiation with energy greater than 0.5 MeV was practically absent.
Recently performed experiments by Alikhanov, Alikhanyan, and Artsimovich ^20, set up to test the law of conservation of momentum in the elementary act, showed that two photons emitted simultaneously fly apart in directions making angles with one another close to 180°. Their experiments are essentially identical with those of Klempérer, with only the difference that the solid angle under which the counters are seen from the source is much smaller in their case than in Klempérer’s. In their experiments only such coincidences could be recorded as are caused by photons with an angle of separation lying within the limits 180–150°. On the basis of this last fact it follows that annihilation takes place for positrons whose energy is in any case less than \(8 \cdot 10^4\) eV.
The experiments listed prove the existence of positron annihilation, but, because of their low accuracy, they do not permit one to draw a rigorous conclusion about the complete spectrum of the annihilation radiation. In particular, concerning its hard component, from these experiments one can conclude only that, if it exists at all, then its intensity is very small.
Here we have described, chiefly, only those works which make it possible to prove the real existence of positron annihilation. A number of works treating the role of annihilation radiation in scattered \(\gamma\)-radiation will be considered later in the analysis of the experimental data on the question of scattering.
§ 8. Bremsstrahlung
Bremsstrahlung arises when electrons and positrons are slowed down by atomic nuclei. Since in our case the electrons and positrons have the most varied directions and velocities, there is no need to calculate a quantitative solution of the question of this type of radiation. For this reason we shall confine ourselves here to certain theoretical indications borrowed from the works of Bethe and Oppenheimer and of Lauritsen ^21; moreover, we shall be interested chiefly in the hard part of the bremsstrahlung radiation, since in the scattered \(\gamma\)-radiation it is superposed only on the weakly intense hard component of the annihilation radiation and therefore can be detected experimentally. The soft part of the bremsstrahlung radiation will play a smaller role because of the presence of the more intense Compton and soft annihilation radiation.
The hard radiation of interest to us arises mainly in the slowing down of the fastest Compton electrons ejected forward. The maximum energy of these electrons in the case of the \(\gamma\)-rays of ThC″ is 2.35 MeV. Consequently, here we can
one should expect radiation with an upper energy limit of about 2.35 MeV*. On moving away from the upper limit toward lower frequencies, the intensity of the radiation increases sharply. Near the upper limit the intensity is proportional to the square of the atomic number. In space it is distributed with a large anisotropy (according to Oppenheimer and Lauritsen, half of the radiation with an energy of about 1.5 MeV lies within a cone with a half-angle of aperture of 15°).
For an approximate quantitative estimate of the intensity of the hard part of the bremsstrahlung, the following data may be cited. The probability of emission of photons with energy greater than \(10^6\) eV is, for electrons with an energy of 1.5 MeV, 0.5%, for electrons with an energy of 2 MeV—2.2%, and for the fastest Compton electrons—4%. These values are somewhat lower than the probability of annihilation of fast positrons. If, however, one takes into account that the \(\gamma\)-rays of ThC″ produce several times more Compton electrons than positrons, then one may come to the conclusion, as Bothe indicated, that hard bremsstrahlung, in its intensity, should be of the same order as hard annihilation radiation. Therefore, in the scattered radiation they should manifest themselves approximately equally.
In concluding this paragraph, let us emphasize once more the complexity of the theoretically expected scattered radiation, which is a continuous spectrum extending from 250 keV practically up to the energies of the primary radiation, with a sharp maximum at about 0.5 MeV. This distribution is further modified somewhat owing to absorption of the radiation in the material of the scatterer itself. Experimental investigation of such complex radiation, carried out moreover under nonidentical conditions, can naturally lead in many cases to different results, especially if one also takes into account the numerous methodological shortcomings noted at the beginning of the article.
§ 9. Experimental data on the scattering of \(\gamma\)-rays
Turning to the consideration of the experimental data known on the question of scattered radiation, it should first of all be noted that, according to the data of all authors, for light elements practically all the scattered radiation is explained by Compton scattering. This is in agreement with the fact that, for them, both the formation of electron pairs by photons and the photoelectric effect have a vanishingly small probability. The situation is different in the case of heavy elements.
* True, besides it there will also be still harder radiation arising in the braking of photoelectrons. However, its intensity will be very small.
In 1930 Chao\(^2\), studying with the aid of a high-pressure ionization chamber the angular distribution of \(\gamma\)-radiation scattered by lead and aluminum from ThC'', filtered through \(2.7\) cm Pb, found that, in contrast to aluminum, for which the angular distribution agrees with formula (2) (see Table 1), in the case of lead a different angular distribution is observed. Chao measured, for various angles, the ratio of the intensities of the radiation scattered by lead and by aluminum, and compared it with the theoretically calculated ratio; in this calculation it was assumed that formula (2) is applicable both to Pb and to Al. Table 4 gives
TABLE 4
| Scattering angle | \(22.5^\circ\) | \(35^\circ\) | \(55^\circ\) | \(90^\circ\) | \(135^\circ\) |
|---|---|---|---|---|---|
| \(\dfrac{E_{\mathrm{Pb}}}{E_{\mathrm{Al}}}\) | 0.70 | 0.72 | 0.70 | 0.57 | 0.38 |
| \(\dfrac{i_{\mathrm{Pb}}}{i_{\mathrm{Al}}}\) | 0.695 | 0.75 | 0.80 | 0.96 | 1.44 |
the data obtained by him. In the second row is given the ratio of intensities calculated by formula (2), with a correction introduced for the absorption of the primary and scattered radiation in the substance of the scatterer. In the last row is given the ratio of the ionization currents caused by the scattered radiation from Pb and Al. As is seen from the table, these ratios are the same only for small scattering angles, whereas for large angles there is a strong discrepancy. From this it was concluded that, in the case of lead, an additional scattering is superposed on the Compton scattering, which received the name anomalous scattering. Measuring the absorption of the scattered radiation in lead, Chao showed that the anomalously scattered radiation has a wavelength different from that of Compton radiation, equal to approximately 22 X-units, which corresponds to a photon energy of about \(0.55\) MeV. The intensity of this radiation is approximately the same in all directions. For this reason it is easiest to observe at large angles, for which the Compton scattered radiation has a long wavelength and very low intensity. According to Chao’s data, for example, for an angle of \(135^\circ\) the anomalously scattered radiation is three times as intense as the Compton radiation. For small angles the intensity of the additional radiation is small in comparison with the intensity of the Compton radiation. This, in all probability, can explain the fact that Gray\(^ {22}\), studying scattered radiation for the angular interval from \(10\) to \(30^\circ\), did not discover anomalous scattering.
Other results were obtained by Meitner and co-workers \(^{23}\). They studied the composition of the scattered radiation with the aid of Geiger–Müller counters, using as the primary radiation the strongly
Fig. 6.
filtered radiation of Ra and MsTh. The scattering angle in their experiments was \(90^\circ\). Iron and lead served as scatterers. Their arrangement is shown in Fig. 6 (two counters were used to increase the efficiency of the setup). To analyze the composition of the scattered
Fig. 7.
radiation, the authors constructed its absorption curves for various absorbing substances. The experiments showed that, for the \(\gamma\)-rays of Ra filtered by \(3\ \mathrm{cm}\) Pb (effective wavelength \(6.7\) X-units), the radiation scattered by iron is wholly Compton in nature.
To study the scattered Pb radiation, the absorption curve in lead is shown in Fig. 7. It is evident from it that in this case, in the scattered radiation, besides the Compton radiation, there is still some harder radiation. By resolving the absorption curve into two components, assuming in doing so that the intensity of the hard component is given by the dotted straight line drawn through the last two experimental points, the authors established that the soft component (□ in the figure) corresponds to Compton scattered radiation, while the hard component corresponds to radiation with the wavelength of the primary radiation. Thus, to the Compton radiation there is added so-called coherent radiation, analogous to the Rayleigh radiation observed for light. Approximately the same results were obtained under the same geometrical conditions for the primary radiation MsTh, filtered by 3 cm Pb. For this radiation, with wavelength 4.7 X-units (in contrast to the Ra radiation), a hard component of the scattered radiation is already present also for an iron scatterer, in accordance with the fact that for Fe, at these wavelengths, anomalous absorption already appears.
According to the authors’ rough estimate, coherent scattered radiation amounts, for \(\lambda = 6.7\) X-units, to about 4%, and for \(\lambda = 4.7\) X-units to about 6–7% of the total intensity of the scattered radiation. This estimate was made on the assumption that the intensity of the Compton radiation is given by formula (2), and the intensity of the coherent radiation by Thomson’s formula, in which the angular distribution is determined by the factor \(1+\cos^{2}\theta\), where \(\theta\) is the scattering angle. In view of the arbitrariness of the latter assumption, the quoted numbers should not be assigned great significance.
The question of scattered radiation was also investigated in detail in several works by Gray and Tarrant \(^{24}\). The authors investigated the scattering of unfiltered \(\gamma\)-rays Ra(B + C) and ThC″ at angles of 125 and 145° for the following substances: C, K, Fe, Cu, Sn, Pb. To measure the scattered radiation they used, as did Chao, an ionization chamber with high gas pressure. Their apparatus is shown schematically in Fig. 8. The scatterer was taken in the form of a cylinder and was placed symmetrically with respect to the ionization chamber. The lead filters serving for the analysis of the scattered radiation were also taken in the form of cylinders and were placed close to the ionization chamber. In order that the intensity of the measured radiation should not be too small, the authors worked with broad beams of scattered rays. Thus, for example, for a mean scattering angle of 125°, \(\gamma\)-rays scattered within the angular interval from 110 to 140° entered the ionization chamber. To obtain the absorption curves of the scattered radiation the authors varied the thickness of the absorbing lead filters from 0 to 4.5 cm. In using such filters, they were dealing practically only with anomalously scattered radiation, since for angles of 125–145° the Compton radiation has considerably lower hardness and lower intensity, and therefore several first millimeters of Pb are sufficient for its absorption.
Carefully performed measurements showed that the absorption curves of the radiation scattered at angles of 125° and 145° are almost parallel to one another. This means that the composition of the anomalously scattered radiation for both directions is approximately the same. An analysis of the absorption curves obtained showed that the anomalously scattered radiation consists of two components. We shall call them the hard and soft components of the anomalously scattered radiation. The results for the case of the primary radiation ThC″ are given in Table 5. It gives the absorption coefficients for both components of the anomalously scattered radiation and the corresponding energies. The last column gives the ratio of the intensities of the hard and soft components.
[In the figure: Source; Absorber; Ionization chamber; Various positions of the scatterer.]
Fig. 8.
It is seen from the table that for all elements in the scattered radiation the soft component has an energy of about 0.5 MeV, while the hard component has an energy of about 1 MeV. At the same time the ratio of the intensities of the hard and soft components increases strongly with increasing atomic number of the scattering element.
For the primary radiation Ra(B + C), the results are somewhat different. They are given in Table 6.
For lead, as we see, the same results are obtained as for the case of ThC″ radiation.
It should be noted that determining the composition of the radiation by decomposing absorption curves is not a sufficiently good
TABLE 5
| Scatterer | Soft component, \(\mu\) in \(\mathrm{cm}^{-1}\) | Soft component, \(h\nu\) in \(10^6\ \mathrm{eV}\) | Hard component, \(\mu\) in \(\mathrm{cm}^{-1}\) | Hard component, \(h\nu\) in \(10^6\ \mathrm{eV}\) | Hard / soft |
|---|---|---|---|---|---|
| C | 2.0 | 0.45 | — | — | 0 |
| K | 2.0 | 0.45 | — | — | — |
| Fe | 2.0 | 0.45 | ? | 0.05 | |
| Cu | 2.0 | 0.45 | ? | 0.07 | |
| Sn | 2.05 | 0.45 | 0.75 | 1.1 | 0.14 |
| Pb | 1.95 | 0.45 | 0.75 | 1.1 | 0.32 |
Absorption and Scattering of γ-Rays
TABLE 6
| Scatterer | Soft component, \(\mu\) in \(\mathrm{cm}^{-1}\) | Soft component, \(h\nu\) in \(10^6\ \mathrm{eV}\) | Hard component, \(\mu\) in \(\mathrm{cm}^{-1}\) | Hard component, \(h\nu\) in \(10^6\ \mathrm{eV}\) | Hard / soft |
|---|---|---|---|---|---|
| C | \((2.6)\) | — | \((0.75)\) | — | 0 |
| Fe | 2.6 | 0.38 | 0.75 | \((1.1)\) | 0.04 |
| Sn | 2.4 | 0.40 | 0.75 | 1.1 | 0.06 |
| Pb | 2.1 | 0.44 | 0.75 | 1.1 | 0.12 |
by the method which, as has already been noted more than once in the first part of the article. Therefore all the values given for the separate components are certain average quantities. Thus, for example, Gray and Tarrant indicate that, with an accuracy of measurement of individual points of the absorption curve of \(2\text{–}3^\circ_0\), as was the case in their experiments, the obtained values \(\mu_1=0.75\ \mathrm{cm}^{-1}\) and \(\mu_2=2\ \mathrm{cm}^{-1}\) (see Table 5) correspond to the following limits of possible errors:
\[ 0.65 < \mu_1 < 0.85\ \mathrm{cm}^{-1}, \]
\[ 1.41 < \mu_2 < 2.45\ \mathrm{cm}^{-1}. \]
As we see, the limits obtained are very broad.
The authors also investigated the intensity of anomalously scattered radiation for various angles in the interval from \(64\) to \(145^\circ\). The experiments led to the conclusion that the intensity is approximately the same for all directions. This made it possible to calculate the total energy of the anomalously scattered radiation. Calculations for the radiation ThC″ showed that, for the same energy of the primary γ-ray beam, the energy of the anomalously scattered radiation, calculated per nucleus, increases in proportion to the square of the atomic number of the scattering element. Table 7 confirms what has been said.
TABLE 7
| Scatterer | Pb | Sn | Cu | Fe |
|---|---|---|---|---|
| \(\dfrac{N K_\gamma}{Z^2}\,10^{27}\) | 0.47 | 0.42 | 0.39 | 0.43 |
In it, \(N K_\gamma\) gives that part of the energy of the primary γ-ray beam which is scattered by one nucleus in the form of the hard and soft components of the anomalous radiation (effective cross section). If one takes
energy only of the soft component, then for it the quadratic dependence is fulfilled even better. For the hard component, instead of \(Z^2\), \(Z^3\) is obtained.
The effective cross section for anomalous scattering \(({}^{N}K_\tau)\) may be compared with those for anomalous absorption \(({}^{N}K=a^\mu-a^\sigma-a^\tau\) (see Part I). For the ratio \(\frac{{}^{N}K_\tau}{{}^{N}K}\) one obtains a value close to unity, which means that almost all the energy (to within \(20\%\)) disappearing as a result of anomalous absorption is emitted in the form of anomalously scattered radiation (in the authors’ first paper, for this ratio a value of about 0.5 was obtained). In the case of Ra(B + C) radiation it is difficult to carry out analogous calculations because of the great complexity of the \(\gamma\)-spectrum.
In order to decide the question of radiation of what energy gives rise to anomalous scattering, the authors carried out the following experiments. They determined the apparent absorption coefficient of the primary radiation by measuring the intensity of the scattered radiation in the presence and absence of an absorbing layer between the source of the primary rays and the scatterer (primary filter). Similar absorption coefficients were determined for various thicknesses of filters placed directly at the ionization chamber (secondary filter). The results obtained are given for the primary radiation ThC″ in Fig. 9.
Fig. 9.
It follows from it that for secondary filters of thickness greater than \(0.5\ \mathrm{cm}\) Pb, the absorption coefficient of the primary radiation for all the elements investigated (Pb, Sn, Fe, C) is the same and does not change with increasing filter thickness. From the equality of the absorption coefficients for different substances it follows that the threshold from which anomalously scattered radiation appears is the same for all the elements investigated. The authors attempted to estimate the energy corresponding to the threshold. For it they obtained a value of about 2 MeV. However, this number cannot be assigned serious significance, since it was obtained under the assumption that the coefficient of anomalous absorption does not depend on the wavelength, which, as we know, is certainly incorrect. The authors themselves, in the notes to their paper, point out that if for the dependence of the anomalous absorption coefficient on frequency one assumes the following form:
\[ K=\alpha(\nu-\nu_e)^4, \]
where \(\nu_e\) is the frequency corresponding to the threshold, then for the threshold an energy of about 1 MeV is obtained.
Comparing the results of the experiments of Meitner, Gray, and Tarrant, we see a sharp contradiction between them. However, the matter stands
where this is not so serious. Under Meitner’s experimental conditions (scattering angle \(90^\circ\)) the Compton scattered radiation had an energy of \(0.4\) MeV, and therefore the soft component of the anomalously scattered radiation (energy \(0.45\) MeV) observed by Gray and Tarrant could hardly have been separated from it. Thus it must be assumed that, with respect to the soft component, there are no contradictions in these experiments. In contrast to Gray and Tarrant, Meitner found coherent scattered radiation with an absorption coefficient in lead of \(0.54\ \mathrm{cm}^{-1}\). However, it must be borne in mind that the hard component was singled out by drawing a straight line through the last two points of the absorption curve (Fig. 7), each of which was measured with a possible error of \(20\%\). For this reason the possible fluctuations in the absorption coefficient must be very large, and therefore its values may fall within the limits indicated by Gray and Tarrant for the absorption coefficient of the hard component observed by them—\(0.65 < \mu < 85\ \mathrm{cm}^{-1}\). Thus, in this respect too, there are no sharp contradictions.
Let us note, incidentally, that Gray and Tarrant carried out special experiments to detect coherent scattering. The experiments showed that radiation without a change of wavelength does not play a noticeable role in the scattered radiation. Thus, for example, for Pb it amounts to no more than \(2\%\) of the total intensity of the scattered radiation.
The conclusions of the works of Gray and Tarrant concerning the composition of the scattered radiation are closely matched by the works of Hueting \(^{25}\), Stahel and Ketelaar \(^{26}\).
Hueting investigated the scattering of ThC\({}^{\prime\prime}\) \(\gamma\)-rays by the following elements: Al, Fe, Cu, Pb. The intensity of the scattered radiation was measured with an ionization chamber containing gas under high pressure. He worked under hermetic conditions close to the working conditions of Gray and Tarrant. However, the angular interval within which the scattered rays entered the ionization chamber was appreciably smaller. For analysis of the scattered radiation the author constructed absorption curves in lead, using filters up to \(5\ \mathrm{cm}\) thick. The results of his work are briefly as follows. For all the elements investigated there is, in the scattered radiation, a component with energy \(0.5\) MeV (\(\mu = 1.75\ \mathrm{cm}^{-1}\)), the intensity of which, calculated per nucleus, increases proportionally to the square of the atomic number. For lead, besides this, there is an even harder component with an energy of about \(1.8\) MeV (\(\mu = 0.59\ \mathrm{cm}^{-1}\)). In its intensity it is noticeably weaker than the soft component.
The conclusions obtained agree fairly well with the data of Gray and Tarrant, except for the value of the energy of the hard component.
In the works of Stahel and Ketelaar, the scattering of Ra radiation was investigated for a number of elements and for different angles. The intensity of the scattered radiation was measured by means of an ionization chamber with gas under high pressure. The authors had at their disposal exceptionally powerful radium preparations (initially
1.6 g, and then—2.6 g and, finally, 7 g). Therefore they were able to measure the absorption curves of the secondary radiation for even greater thicknesses of absorbing screens than Gray and Tarrant. The results they obtained are in many respects interesting. However, the quantitative conclusions of their work are not very reliable because of the complex spectral composition of the radium radiation. Fig. 10 schematically shows their arrangement.
Fig. 10.
Fig. 11.
composition of the radium radiation. In Fig. 10 their apparatus is shown schematically.
Stahl and Ketelaar first of all established that anomalously scattered radiation is produced only by the hardest components
absorbers of the primary radiation. To this end they filtered the primary radiation with lead and measured the intensity of the scattered radiation as a function of the thickness of the filter. The curve thus obtained, II, was compared with the absorption curve of the primary radiation, I. Both curves for a lead scatterer are shown in Fig. 11. In the same figure the curve III is given, showing the dependence of the intensity of the scattered radiation on the thickness of the filter in the primary beam for the case where an additional filter (2.4 cm Pb) was placed between the scatterer and the ionization chamber. It is seen from Fig. 11 that the change in the anomalously scattered radiation (curves II and III) with the thickness of the filter occurs in parallel with the change in the intensity of the hard component of the primary radiation. Similar results are obtained for other scatterers. Hence follows the assertion that anomalous scattering is produced by the hardest component of the incident radiation.
Fig. 12.
To investigate the qualitative composition of the anomalously scattered radiation, the authors used thick scatterers, since in them the scattered radiation is enriched in the hard component owing to its smaller absorption in the scattering body. The experiments showed that, in agreement with the work of Gray and Tarrant, the anomalously scattered radiation of Pb, Au, and W contains hard and soft components. For elements with smaller atomic number (Sn, Zn, Fe, Al), decomposition into two components cannot be carried out with sufficient reliability. For this reason the authors indicate only one component for them.
With respect to the hard component it was shown that the absorption coefficient corresponding to it is much smaller than the absorption coefficient of the primary radiation (Fig. 12). Thus co-
constituent cannot be identified with coherent radiation.
Figure 13 shows the absorption curves of scattered radiation for four different directions. From the parallel course of the rectilinear portions of the curves it follows that the qualitative composition of the hard component for all the directions investigated is approximately one and the same.
Fig. 13.
The same may be said about the composition of the hard component for different scattering elements. This is confirmed by Table 8, obtained from an analysis of the absorption curves for radiation scattered by Pb, Au, W at an angle of \(135^\circ\). From the—
TABLE 8
| Scatterer | Hard component \(\mu\) in \(\mathrm{cm}^{-1}\) | Soft component \(\mu\) in \(\mathrm{cm}^{-1}\) |
|---|---|---|
| Pb \((Z = 82)\) | 0.93 | 2.25 |
| Au \((Z = 79)\) | 0.95 | 2.37 |
| W \((Z = 74)\) | 0.97 | 2.25 |
introduced value of $\mu$ for the energy of the hard component, one obtains a value of about 0.9 MeV.
For the absorption coefficient of the soft component of the anomalously scattered radiation for Pb, Au, and W, a value is obtained that does not differ very strongly from the value given by Gray and Tarrant ($2\ \mathrm{cm}^{-1}$), and in any case lies within the limits they indicate ($1.41—2.45\ \mathrm{cm}^{-1}$). The energy of this component is about 0.4 MeV. For the only components of the radiation scattered by Al, Fe, and Sn, the following energy values are obtained, respectively: 0.4, 0.53, and 0.56 MeV.
In order to determine the quantitative composition of the scattered radiation, the authors carried out experiments with very thin scatterers, for which the intensity of the scattered radiation is proportional to the thickness, i.e., to the number of electrons falling on $1\ \mathrm{cm}^{2}$ of the scatterer. Figure 14 gives curves expressing the dependence of the intensity of the scattered radiation on the thickness of the scattering screen for the case of Pb and Al. Since for Al the scattered radiation is entirely Compton radiation, from such straight lines one can establish what part, for the element under study, is accounted for by the additional radiation (in percent of the Compton radiation). For this it is only necessary to form the difference of the ordinates of the two
Fig. 14.
TABLE 9
| Thickness of the filter between the source and the scatterer | Anomalous scattering in % of Compton |
|---|---|
| $0\ \mathrm{cm}$ Pb | 24 |
| $1.35\ \mathrm{cm}$ Pb | 48 |
| $3\ \mathrm{cm}$ Pb | 53 |
of some corresponding points of the straight lines and divide it by the ordinate for the point of curve A1.
Using a radium source of 7 g, Ketelaar found, for the anomalously scattered radiation of lead (scattering angle \(120^\circ\)), the values given in Table 9.
The same method can be used to establish the dependence of the intensity of anomalously scattered radiation on the angle. For this, it is only necessary to construct a system of similar straight lines for different scattering angles and, from them, to calculate the additional radiation. The results thus obtained for Fe, Sn, and Pb are given in Table 10.
TABLE 10
| Scattering angle in ° | Fe | Sn | Pb |
|---|---|---|---|
| 135 | 0.25 | 0.39 | 1.28 |
| 120 | 0.20 | 0.45 | 1.17 |
| 90 | 0.36 | 0.60 | 1.18 |
| 75 | — | — | 1.00 |
| 60 | — | — | 1.10 |
| Mean | 0.27 | 0.48 | 1.15 |
Anomalously scattered radiation in arbitrary units
It is evident from the table that, for different angles, the intensity of the anomalously scattered radiation is approximately the same.
In order to estimate the ratio of the intensities of the hard and soft components of the anomalously scattered radiation, Ketelaar constructed absorption curves for this radiation, using absorbing screens of up to 6 cm of lead in thickness. These curves were constructed in a somewhat different way than is usually done. Each individual point of a curve was obtained from a separate series of measurements, which consisted in the fact that, for a given thickness of the absorbing screen placed between the scatterer and the ionization chamber, the intensity of the scattered radiation was measured as a function of the thickness of the scatterer. The curves thus obtained initially had a linear course; the deviation from it appeared, at large scatterer thicknesses, the thicker the filter in the path of the scattered radiation was. This latter fact is explained by the circumstance that the presence of absorption of secondary radiation in the scatterer itself (the deviation from linearity) begins to be manifested for hard radiation at larger scatterer thicknesses than for softer radiation. From the system of curves obtained, or more precisely, from their rectilinear portions, it was possible for each
of the thickness of the absorbing screen placed between the scatterer and the recording instrument, to calculate the intensity of the scattered radiation entering the ionization chamber, referring it to one and the same thickness of the thin scatterer. The aggregate of all these values gives the absorption curve of the secondary radiation corresponding to the thin scatterer. Similar absorption curves were constructed for the primary radiation Ra (7 g), filtered by a layer of lead of thicknesses 0, 1.35, and 3 cm. Analysis of the curves obtained in this way made it possible to establish what part of all the radiation scattered by lead is due to the hard component of the anomalously scattered radiation. The results obtained by Ketelaar are given in Table 11.
TABLE 11
| Thickness of filter between the source and the scatterer | Intensity of the hard component relative to the total scattered radiation, in % |
|---|---|
| 0 | 6.6 |
| 1.35 cm Pb | 12.3 |
| 3.0 “ Pb | 14.4 |
Measurements made with very large thicknesses of the absorbing layer (up to 6–8 cm) show, for Pb, the presence of still harder radiation. However, its intensity is very weak. It amounts to about 1% of the total scattered radiation, or 10% of the hard component of the anomalous scattering. For this reason nothing can yet be said about the hardness of this radiation.
If we compare the data of Tables 9 and 11 and carry out some calculations, then for the total radiation scattered by lead at an angle of 120°, we obtain the composition indicated in Table 12.
TABLE 12
| Thickness of filter between the source and the scatterer | Compton scattering, in % | Anomalous scattering, in % | Anomalous scattering, in % |
|---|---|---|---|
| soft component | hard component | ||
| 0 | 81.5 | 12 | 6.5 |
| 1.35 cm Pb | 67.5 | 20 | 12.5 |
| 3.0 “ Pb | 65.5 | 20 | 14.5 |
The results obtained by Stahel and Ketelaar basically coincide with the results of Gray and Tarrant.
In a certain respect, a reconciliation of the contradictory results of all the works considered is the paper by Bothe and Horn[^27]. These authors investigated, with a Geiger–Müller counter, the scattered radiation of ThC″ for scattering angles of 114° and 90°. They worked under such geometrical conditions as required the introduction of the smallest number of corrections. Their arrangement is shown in Fig. 15. In their work Bothe and Horn were interested primarily in the intensity of the scattered radiation for different elements and for different angles; they therefore used chiefly thin scatterers, which themselves absorb the scattered radiation only to an insignificant degree. The results obtained by them may briefly be summarized as follows. The anomalously scattered radiation possesses a sharply expressed spatial
Fig. 15.
Fig. 16.
anisotropy. For example, for lead the ratio of the intensities of this radiation for angles of 114° and 90° is about 10. The intensity of the anomalously scattered radiation for a given scattering angle, within the errors of the experiment, increases proportionally to \(Z^2\).
In the scattered radiation, in addition to the Compton component, there are two others—soft and hard. The first of them, with an absorption coefficient of \(1.67\ \mathrm{cm}^{-1}\), corresponds to an energy of about \(0.5\ \mathrm{MeV}\). The second, in its hardness, is close to the primary radiation. The hard component was separated from the total scattered radiation in the following way. From additional experiments the absorption coefficient of the primary radiation was determined. These experiments consisted in removing the source \(P\), and placing at the position of the scatterer \(S\) a layer of thorium oxide of the same surfaceאַד
sizes. The new source of γ-rays (an oxide layer) gave approximately the same spatial distribution of radiation as the scatterer \(S\) in the preceding experiments. Under normal experimental conditions the absorption coefficient was determined for this radiation. For it a value of \(0.44\ \mathrm{cm}^{-1}\) was obtained. Using this value, the authors constructed the absorption curve of this component, passing it through the last points of the absorption curve of the scattered radiation. In Fig. 16 the curve calculated in this way is shown by a dashed line. Here curve \(I\) refers to a thin lead scatterer (\(\mathrm{Pb}\)) \(2.5\ \mathrm{mm}\) thick. For it, the hard component amounts to about \(5\%\) of the total intensity of the scattered radiation. For a thick scatterer (\(3\ \mathrm{cm}\ \mathrm{Pb}\)) the hard component is approximately \(18\%\) (curve \(II\)). The corresponding figures for the same scatterers, but for a scattering angle of \(90^\circ\), are the following: \(15\%\) for the thin one and \(30\%\) for the thick one. Hence it is evident that the hard component possesses spatial anisotropy.
Fig. 17.
With regard to the soft component of the anomalously scattered radiation, it is of interest to note the following. In Fig. 17 the dependence of the intensity of the scattered radiation on the thickness of the scatterer is presented. The curve refers to Pb and was obtained for a scattering angle of \(114^\circ\). From a comparison of the intensities of the radiation scattered by thin layers of Pb and Al at an angle of \(114^\circ\), the authors found that anomalous scattering under these conditions amounts to \(13\%\) of the Compton scattering. Knowing the absorption coefficients of the Compton and soft components and the ratio of their initial intensities, one can, on the basis of curve \(I\) in Fig. 17, construct for lead the dependence of the intensity of the Compton radiation on the thickness of the scatterer.* This dependence is represented in Fig. 17 by curve \(II\). Taking the difference between \(I\) and \(II\), we obtain a curve (dashed line) giving the change in the intensity of the soft component of the anomalously scattered radiation with the thickness of the scatterer. It shows for
* The hard component is not taken into account in these calculations because of its low intensity.
for small scatterer thicknesses, a quadratic dependence. This means that the radiation corresponding to the soft component is not secondary, but tertiary, i.e., it arises from the primary by means of two successive processes, the probability of each of which is proportional to the thickness. Thus there must exist an intermediate radiation, through the interaction of which with the substance of the scatterer the soft component of the anomalously scattered radiation arises. Such intermediate radiation, evidently, may be positrons, recoil electrons, photoelectrons, and $\gamma$-quanta scattered as a result of the Compton effect. In the following section, in analyzing the experimental data, we shall consider all these possibilities. Here we shall only note that the presence of a multiple Compton effect, i.e., the scattering of photons that have already previously undergone Compton scattering, is confirmed by the following results obtained by Bothe and Horn. In Fig. 16 are shown absorption curves for radiation scattered by a thin ($2.38\ \mathrm{g}/\mathrm{cm}^2$) and a thick layer of graphite. Curve $IV$ refers to the thick scatterer, curve $III$ to the thin scatterer. From them it is seen that for the thick scattering layer of graphite there appears radiation harder than ordinary Compton radiation. It may be explained by the presence of a multiple Compton effect, which is more probable for a thick layer.
§ 10. Analysis of the Experimental Data
In the preceding section the results of the principal works devoted to the analysis of scattered radiation were examined. From them it follows that the $\gamma$-radiation scattered by bodies consists of three components: the Compton component, and the soft and hard components of anomalously scattered radiation.
With regard to the first of the three mentioned components, which appears in a single Compton effect, there are no contradictions. It can be obtained practically in pure form in the case of a thin scatterer with a small atomic number. Its intensity is in good agreement with the theoretical data (formula 2).
As to the question of the soft component of anomalously scattered radiation, there are also more or less concordant results. All authors find for it an energy close to $0.5\ \mathrm{MeV}$. For its intensity a quadratic dependence on the atomic number of the scatterer is obtained. If one recalls that the probability of formation of electron pairs, and consequently also of positrons, increases likewise proportionally to $Z^2$, while the energy of the photons arising in the annihilation of slow positrons is equal to $0.5\ \mathrm{MeV}$, then it at once becomes obvious that the component of the scattered radiation under consideration appears as a result of positron annihilation. Direct evidence that positron annihilation plays a role in the scattering of $\gamma$-rays is provided by the recent experiments of Williams$^{28}$, as well as of Gentner$^{29}$.
Williams, in his experiments, used the arrangement shown schematically in Fig. 18. Here \(S\) is the source of \(\gamma\)-rays, \(J\) is an ionization chamber for measuring the intensity of the scattered radiation, \(CC'\) is a layer of paraffin in which positrons moving from the scattering lead layer \(BB'\) in the backward direction are stopped. The author measured the change in the intensity of the scattered radiation when the lead foil \(BB'\) was inserted, once in the presence of the aluminum layer \(AA'\), and a second time without it. In the first case a change 25% greater than in the second was observed [for the \(\gamma\)-rays of Ra \((B+C)\)]. This excess is caused by the additional radiation that arises upon the annihilation in aluminum of positrons coming forward from the lead foil. In order to make quite certain that annihilation radiation was involved here, Williams measured its intensity for thin foils of various thicknesses. (In these experiments the \(\gamma\)-rays of ThC″ were used, and a lead filter 0.8 cm thick was placed in the path of the scattered radiation; this filter practically completely absorbed the Compton-scattered radiation, allowing only the anomalously scattered radiation to enter the chamber.) If the annihilation hypothesis of the origin of the soft component is correct, then for a foil of thickness comparable with the range of a positron, as the thickness is decreased there should be a more rapid than linear fall in the intensity of the scattered radiation, in view of the fact that for thicknesses smaller than the range, the positrons begin to leave the foil, producing annihilation radiation not recorded by the chamber. Such a sharp decrease in intensity for the soft component was in fact found experimentally. Rough calculations showed that a large part of the energy of the component under consideration must be attributed to annihilation radiation. Similar results were later obtained by Gentner.
Fig. 18.
It may therefore be considered sufficiently well founded that the soft component of anomalously scattered radiation consists mainly of annihilation radiation arising in the destruction of slow positrons. Confirmation of such a conclusion is also provided by the fact that, according to the data of most works, the soft component is distributed in space approximately uniformly.
It should be borne in mind, however, that this component also includes other types of radiation, namely: bremsstrahlung radiation and radiation arising as a result of the multiple Compton effect. Unfortunately, at the present time there are as yet no reliable quantitative calculations taking account of the fraction of these radiations
applicable to the experimental conditions of individual works. With regard to the multiple Compton effect there is as yet only a single work, which appeared after the writing of the present article. The author of the work, Bothe30, calculated the intensity of the radiation scattered as a result of the double Compton effect, for a scattering angle \(\Theta = 114^\circ\), for two lead scatterers of thickness \(1\) and \(2\) mm, arranged symmetrically with respect to the incident and secondary rays (the experimental conditions of Bothe, Horn, and Gentner). His calculations showed that for a scatterer \(1\) mm thick this radiation amounts to \(13\%\) of the singly scattered Compton radiation. The corresponding figure for a scatterer \(2\) mm thick is \(20\%\). Its energy for the scattering angle \(\Theta = 114^\circ\) is close to \(0.5\) MeV, and therefore under these conditions it is indistinguishable from annihilation radiation. As for bremsstrahlung in the energy region close to \(0.5\) MeV, nothing definite can as yet be said.
At the present time, with respect to both of these radiations we can assert only that they play practically no role for very thin scatterers, but may show up noticeably in the case of thick scatterers. In all probability, the superposition of these radiations on the annihilation radiation should explain a certain discrepancy in the results of different works concerning the soft component of anomalously scattered radiation. Possibly this also explains the contradiction with a \(100\%\) yield of anomalously scattered radiation. As was already noted above, Gray and Tarrant established that practically all anomalously absorbed energy is emitted in the form of anomalously scattered radiation. This experimental fact contradicts the annihilation hypothesis of the origin of the soft component, since for ThC″ radiation the yield should not exceed \(40\%\), because the creation of each electron pair consumes \(2.65\) MeV of energy, while upon annihilation of the positron formed in the pair there arise two photons with energy \(0.5\) MeV each. Apparently, for the thick scatterers with which Gray and Tarrant dealt, in addition to annihilation radiation, the soft component of anomalously scattered radiation also included other types of radiation mentioned above, thereby increasing the “apparent” yield of the scattered radiation.
It is quite plausible that the superposition of bremsstrahlung and multiply scattered Compton radiation can also explain the fact that, for the energy of the soft component, Gray and Tarrant obtained values somewhat smaller than the \(0.5\) MeV required by the annihilation hypothesis.
Thus, in conclusion, it may be said that, although some discrepancy in the results of different works on the question of the soft component remains, nevertheless it is not of a fundamental character and is most likely a consequence of the complicated experimental conditions, which it has not yet been possible to take fully into account.
The first quantitative calculations of the scattered radiation, carried out
recently obtained by Gentner for the thinnest scatterers confirm the correctness of the conclusions concerning the annihilation nature of the soft component. Bremsstrahlung and multiply scattered radiation, as well as the hard component of anomalously scattered radiation, played practically no role for those scatterers with which Gentner worked. Therefore here, in the composition of the scattered radiation, one should expect only the singly scattered Compton radiation and annihilation radiation. Gentner carried out his investigations with the thinnest layers using an apparatus shown schematically in Fig. 19. The scattering of radiation from RhC″ was studied. The secondary radiation was observed at an angle of \(114^\circ\), as in the work of Bothe and Horn. Its intensity was measured with a Geiger–Müller counter. In view of the low intensity of the radiation under investigation, a scatterer with a very large surface area \((3740\ \text{cm}^2)\) was used. The results for the thinnest layers of lead and aluminum are given in Fig. 20. It shows the dependence of the intensity of the scattered radiation on the thickness of the scatterer; in doing so, the curve for lead has been corrected in such a way that it includes
Fig. 19.
Fig. 20.
the annihilation radiation of those positrons which leave the scatterer and annihilate at a large distance from the counter. The data presented refer to the case when between the counter
and there was a 2-millimeter lead filter before the scatterer. From the curves it is seen that, up to thicknesses corresponding to \(0.6 \cdot 10^{23}\ \frac{\text{electrons}}{\text{cm}^2}\) (\(\sim 0.02\ \text{cm Pb}\)), there is a linear dependence, i.e. here there is no absorption of the scattered radiation in the scatterer itself.
From a comparison of the rectilinear portions it follows that the ratio of the intensities of the radiation scattered by lead and by aluminum, for the same number of electrons per \(1\ \text{cm}^2\), is
\[ \frac{I_{Pb}}{I_{Al}} = 2.66. \tag{12} \]
If unfiltered scattered radiation enters the \(\gamma\)-counter, then for the very same ratio one obtains a value equal to 2.2. On the other hand, these ratios can be calculated theoretically, starting from the known values for the attenuation coefficients, calculated per atom, for the Compton effect and pair production. Such calculations lead to the following results. In the case of aluminum, for 100 Compton photons there are 18 annihilation photons. For lead these numbers are respectively 100 and 114. If one takes into account the dependence of the sensitivity of the counter on wavelength, then on the basis of the numbers given, for the ratio \(\frac{I_{Pb}}{I_{Al}}\) one obtains the value 2.56 for the case when a 2-millimeter lead filter stands in the path of the scattered rays, and 2.09 when this filter is absent. As we see, the agreement between the theoretical and experimental data is good. However, this is so only for the thinnest scatterers. For thick scatterers such agreement is not obtained, since here it is difficult to take into account radiation of other kinds.
The greatest discrepancies among different investigators occur with respect to the hard component of anomalously scattered radiation. Some authors assert that it is coherent radiation; others ascribe to it a hardness noticeably less than the hardness of the primary radiation. In passing to its consideration, we must bear in mind that it plays a more or less noticeable role only for thick scatterers, for which all the relationships become greatly complicated. Therefore it should not seem strange that precisely on this question the contradictions among different experiments appear so strongly.
There is every reason to assert that these discrepancies are caused chiefly by the fact that the hard component is not a narrow spectral band, but constitutes a very broad region of continuous spectrum, extending from energies greater than \(0.5\ \text{MeV}\) practically up to the energies of the primary radiation. Under different experimental conditions, different parts of this continuous spectrum play the prevailing role, thereby causing disagreement in the results. Moreover, the very character of the entire continuous
spectra are in fact identical for such different arrangements as those indicated in Figs. 6, 8, 10, 15, 18.
In what follows, by the hard component of anomalously scattered radiation we shall mean the entire continuous spectrum with energies greater than 0.5 MeV. It consists of the following kinds of radiation: annihilation radiation, arising in the annihilation of fast positrons, the bremsstrahlung of electrons and positrons, and coherent radiation.
With respect to the last radiation, all authors, with the exception of Meitner and co-workers and Bothe and Horn*, establish that, if it exists at all, then even for thick scatterers it amounts to no more than a few percent of the total radiation (according to Gray and Tarrant, less than 2% for Pb). One of the possible explanations of the contradictions between different authors on this question might be found in the strong anisotropy of this radiation, as was established in the work of Bothe and Horn. According to these authors, the intensity of the coherent radiation decreases from 15% of the total intensity to 5% upon passing from an angle of 90° to 114°. On the other hand, all authors who did not detect coherent scattering worked with scattering angles appreciably exceeding 90°, where this radiation should be strongly weakened as a consequence of the aforementioned anisotropy. A similar anisotropy, as Meitner indicates \(^{31}\), is also obtained theoretically by Delbrück, who considers coherent scattering as the scattering of photons by electrons of negative energies situated near the atomic nucleus \(^{32}\).
The possibility is also not excluded that coherent radiation does not exist at all. The point is that from the decomposition of the absorption curve obtained by Meitner for Ra \((B + C)\) \(\gamma\)-rays, the identity of the hard component with coherent radiation does not follow very convincingly, since the straight line (Fig. 7) is drawn through the last two points, each of which was measured with an accuracy of only up to 20%. It is quite possible that this component has an energy different from that of the primary radiation.
In connection with this it may be mentioned that from Ketteleaar’s recent experiments \(^{26}\), carried out with a very strong source of \(\gamma\)-rays (\(7\ \text{g}\) Ra), it follows that, in addition to the radiation which in Gray and Tarrant and in analogous works was designated as the hard component of anomalously scattered radiation, there exists an even harder radiation. It can be observed only for scatterers with a thickness greater than \(5\ \text{cm}\) Pb. In hardness it is close to the primary radiation, although it cannot yet be said whether it is monochromatic or represents an entire band of continuous spectrum. The intensity of this radiation is less than 8% of the usual hard component, i.e. it is very weak. Apparently, this is the very same component as in Meitner’s experiments
* Let us note that Meitner, as well as Bothe and Horn, used a counter to measure the intensity of the scattered radiation, whereas other authors used an ionization chamber.
and Bothe and Horn. Stahel and Ketelaar \(^{33}\) believe that this component consists mainly of the bremsstrahlung radiation of the fastest photo- and Compton electrons.
On the basis of all that has been said, it should be concluded that the question of coherent scattering has not yet been definitively resolved.
For large scattering angles, at which the majority of investigators have worked, it may be assumed that the hard component of the anomalously scattered radiation consists of bremsstrahlung radiation and annihilation radiation of fast positrons. The presence of hard bremsstrahlung radiation at large angles is confirmed by Gentner’s work \(^{34}\). Studying the scattering of \(\gamma\)-rays by various substances, he found that from the scatterer, even at angles greater than \(90^\circ\), there emerges a rather strong electron radiation, among which there are electrons with energies close to the energy of the primary \(\gamma\)-rays. On entering a filter placed in the path of the secondary radiation, the electrons produce in it bremsstrahlung radiation, the intensity of which depends to a considerable degree on the experimental conditions. In particular, the experiments showed that if between the scatterer and the counter one places a filter consisting of a magnesium plate (thickness \(1\ \mathrm{cm}\)) and a lead plate (\(2\ \mathrm{mm}\)), then the counter detects stronger radiation if the lead plate is turned toward the scatterer. This is explained by the fact that in this case the electrons are slowed down in the lead (with the other position of the filter they are slowed down in magnesium) and therefore produce more intense bremsstrahlung radiation. Gentner points out that, depending on the experimental conditions, the radiation arising in this way may, in its intensity, be of the same order as the hard component of the anomalously scattered radiation.
However, at present we in fact cannot state with sufficient reliability what part of the hard component is due to bremsstrahlung radiation and what part to annihilation radiation. So far there are only the first attempts to calculate these radiations quantitatively. In § 8 we mentioned that, in Bethe’s opinion \(^{11}\), the bremsstrahlung radiation for the case of scattering of the \(\gamma\)-rays of \(\mathrm{ThC''}\) in lead is, in intensity, of the same order as the hard annihilation radiation. On the other hand, this latter radiation, arising in the annihilation of fast positrons, according to the calculations of § 5 amounts to \(7\text{—}8\%\) of the total annihilation radiation. Therefore the hard component of the anomalously scattered radiation should amount to about \(15\%\) of all the annihilation radiation. Williams \(^{28}\), in his experiments on the scattering of the \(\gamma\)-rays of \(\mathrm{ThC''}\) in lead, found that the hard component is approximately equal in intensity to \(15\%\) of the component with energy \(0.5\ \mathrm{MeV}\). From this it may be concluded that, at least in order of magnitude, the theoretical and experimental data agree.
For the \(\gamma\)-rays of radium, quantitative calculations of the hard component of the scattered radiation were carried out quite recently by Stahel, Ketelaar, and Kipfer \(^{35}\). They compared the intensity of the hard component with the sum of the annihilation and bremsstrahlung
radiation. In this, the annihilation radiation was calculated from the data of Bothe11, and the bremsstrahlung—on the basis of results obtained earlier by Stahel and Kipfer36, in studying the bremsstrahlung of UX β-rays.
The results obtained by them for three different filters placed between the source and the scatterer are given in Table 13.
TABLE 13
| Filter | Hard component of the scattered radiation (experiment) | Bremsstrahlung radiation (calculated) | Annihilation radiation (calculated) | Sum of the last two |
|---|---|---|---|---|
| 0.20 | 0.17 | 0.013 | 0.183 | |
| 0.106 | 0.079 | 0.006 | 0.085 | |
| 0.031 | 0.027 | 0.002 | 0.029 |
As the table shows, the agreement with the experimental data is quite good. It is true that no great significance should be attached to the numbers given, since they refer to the radiation of radium, which is very complex in its spectral composition, and many simplifying assumptions were made in the calculations.
Let us note that, in the case of the γ-rays of radium, the annihilation radiation amounts to only 6% of the bremsstrahlung; this is connected with the strong decrease in the yield of electron pairs for this radiation in comparison with the γ-rays of ThC″.
Thus, bearing in mind the data cited, one may regard as sufficiently well founded the assumption that the hard component of the anomalously scattered radiation consists of bremsstrahlung and hard annihilation radiation.
In concluding the exposition, we may say, finally, that, despite numerous contradictions, the qualitative composition of the scattered radiation is fairly well known, although reliable quantitative calculations on this question still cannot yet be carried out. For a complete explanation of all the facts considered here, apparently three mechanisms of interaction of γ-rays with matter are quite sufficient: the Compton effect, the photoelectric effect, and the formation of electron pairs. At the same time all anomalous scattering, in whose explanation the greatest difficulties have been encountered, should be ascribed to secondary effects—the annihilation of slow and fast positrons and the braking of electrons.
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