Internal Conversion of $\gamma$-Rays

M. P. Bronshtein, Leningrad

It is known that the $\beta$-rays emitted by radioactive elements exhibit a spectrum of velocities consisting of a number of sharp lines on a continuous background. Such a spectrum can be obtained on a photographic plate on which fall $\beta$-electrons that have undergone deflection in a magnetic field. This method was first applied more than 20 years ago by Baeyer, Hahn, and Meitner,^1 who discovered the existence of sharp lines, without, however, being able to detect the continuous background. Sharp lines, in general, constitute the most conspicuous part of the $\beta$-spectrum, although in the total number of electrons the continuous background considerably exceeds them. This circumstance was first discovered in 1914 by Chadwick,^2 who used, to count the number of electrons at various points of the $\beta$-spectra, an ionization chamber and counter.

Almost simultaneously with this, Rutherford, Robinson, and Rawlinson^3 found that the $\gamma$-rays emitted in the same radioactive decay in which the emission of $\beta$-lines is observed are capable of knocking out from metallic foils groups of secondary electrons with discrete velocities; Rutherford^4 therefore put forward the supposition, fully confirmed by subsequent investigations, that the $\beta$-lines themselves are nothing other than secondary electrons torn from the electron shell of the atom to which the radioactive nucleus belongs by the $\gamma$-rays emitted by this nucleus. Thus the primary electrons (i.e. $\beta$-rays in the proper sense of the word) are not the electrons of the $\beta$-lines, but the electrons of the continuous $\beta$-spectrum. The continuous character of this spectrum is an essential feature of $\beta$-decay and is not connected with any secondary phenomena; this is proved by the experiments of Ellis and Wooster,^5 from which it is seen that the primary $\beta$-rays, already at the very moment of their emission from the radioactive nucleus, possess a continuous distribution of velocities. Therefore the presence of just such a continuous $\beta$-spectrum is a necessary and sufficient sign of $\beta$-decay. The sharp $\beta$-lines, on the contrary, have nothing in common with $\beta$-decay, but represent the result of the partial photoelectric absorption of $\gamma$-rays emitted by nuclei in the electron shells surrounding these nuclei; such photoelectric absorption of $\gamma$-rays has received the name of internal

X-rays. Therefore, a line β-spectrum inevitably accompanies the emission of γ-rays both in the case where the emission of γ-rays is accompanied by β-decay and in the case where it is accompanied by α-decay. A classical example of a radioactive substance emitting such (secondary) β-rays, despite the fact that no β-decay occurs in it, is radium.

The comparative complexity of line β-spectra and their wealth of lines are explained by the fact that one and the same frequency of γ-rays can lead to the appearance of many β-lines, since the absorption of a γ-quantum can take place in different layers of the atom’s electron shell, for example in the \(K\) shell, in the \(L\) shell, \(L_2\), etc. The energy of the electron ejected in the internal conversion of γ-rays will be equal to the energy of the γ-quantum minus the work that must be expended in order to tear the electron out of the corresponding shell. This work is equal to the value \(h\nu\) for the boundary of the corresponding X-ray series. Therefore, in analyzing line spectra of β-rays we must find groups of β-lines possessing the property that the energies of all the lines of a given group are obtained from the energy of the slowest line of this group by adding all possible \(h\nu\)’s of the X-ray \(K\)-series of the corresponding chemical element. Since the X-ray frequencies and the boundaries of the X-ray series are well known, and since the energy of the electrons of β-rays can also be well measured, this makes it possible to determine, with sufficient accuracy, the frequency of the γ-rays emitted by nuclei and partially converted in the electron shells. Such a method of investigating γ-spectra is the most reliable of all existing ones, since it is a matter of measuring the frequency of γ-rays, although, in contrast to methods based on the Compton effect or on the ordinary photoelectric effect, it does not make it possible to determine intensities directly.

However, in order for this method of investigating γ-spectra to be possible, it is necessary to know in precisely which atom the internal conversion occurs—whether in the atom of the original radioactive substance or in the atom of its product. In other words, it is necessary to know whether the emission of γ-rays occurs before the radioactive decay has taken place or after. Meitner \(^{6}\) was the first to express the view that the emission of γ-rays occurs after radioactive decay; in other words, it is the result of the fact that some of the nuclei arising in radioactive decay are in an excited state and only then pass into the normal state, emitting γ-rays, which in the process are partially converted. Thus, for example, if we have some radioactive element with atomic number \(Z\), undergoing α-decay and emitting γ-rays in the process, this means that the γ-rays themselves are emitted already by a nucleus with atomic number \(Z-2\), i.e. by the excited nucleus that has arisen as a result of α-decay; in exactly the same way, if an element with atomic number \(Z\) undergoes β-decay, emitting in the process

...rays, they, in fact, are emitted already in consequence of the $\beta$-decay by a nucleus with atomic number $Z+1$; therefore the $\beta$-lines of which it is usually said that they belong to the element with number $Z$ in reality represent the result of conversion in the electron shell of the atom with atomic number $Z-2$, if $\alpha$-decay occurs, or $Z+1$, if $\beta$-decay occurs. Since the x-ray frequencies increase with increasing atomic number, Meitner’s hypothesis could be tested as soon as it became possible to measure the energies of $\beta$-lines with such great accuracy as to determine with certainty the atomic number to which the corresponding x-ray frequencies are equal to the differences between the energies of the $\beta$-lines. Since the difference between x-ray frequencies is greater when the atomic number changes by 2 than when it changes by 1, it is natural that Meitner’s hypothesis was first tested and confirmed in cases of $\alpha$-decay (we note that at present a further and completely independent confirmation of the same view is provided by the whole complex of phenomena of the fine structure of $\alpha$-rays and its connection with $\gamma$-rays7). Then Ellis and Wooster8 confirmed Meitner’s point of view by carrying out a very accurate measurement of the energies of electrons ejected by the $\gamma$-rays RaB and RaC in platinum (the ordinary photoelectric effect) and in the shells of the atoms RaB and RaC themselves (internal conversion). Let us cite an example from this work: the $\beta$-line, representing the result of internal conversion of one of the $\gamma$-lines of RaC in the $K$-shell, gives $H\rho = 2980$ (we note that the velocities of $\beta$-electrons are usually specified by the quantity $H\rho$—the product of the magnetic field by the radius of curvature of the trajectory—which is determined directly from experiment and is equal to

$$ \frac{c m_0 v}{e \sqrt{1 - v^2/c^2}} $$

(where $m_0$ is the rest mass of the electron, $v$ is its velocity, and $e$ its charge in absolute electrostatic units); the energy of this line is equal to $h\nu - K_{84}$, where $h\nu$ is the energy of the $\gamma$-quantum, if Meitner’s point of view is correct, i.e., if the $\gamma$-quantum is emitted after the RaC nucleus $(Z=83)$ has been transformed into the RaC′ nucleus $(R=84)$, and it is equal to $h\nu - K_{83}$ if the $\gamma$-quantum is emitted before the transformation RaC $\to$ RaC′. The energy of the electron ejected by the same $\gamma$-quantum from the $K$-shell of platinum is equal to $h\nu - K_{78}$. Since all three quantities ($K_{83}$, $K_{84}$, $K_{78}$) are known, one can calculate that, if the $\gamma$-quantum is emitted after the transition RaC $\to$ RaC′, then $H\rho$ for the electron ejected from the $K$-shell of platinum should be 56 greater than for the electron of the $\beta$-line of RaC itself, whereas if this quantum is emitted before the transition, it should be only 46 greater. The measurement gave the number 57, i.e., it fully confirmed Meitner’s point of view. Other methods of verification (the study of x-rays emitted upon restoration of the electron shell from which internal conversion has ejected an electron9, as well as the study of electrons ejected by these x-ray quanta during their conversion in more outer layers of the shell10) also confirmed this point of view. Therefore at present it is accepted, when speaking of the origin of this or that $\beta$- or $\gamma$-line, to indicate at once the two elements from which

the first is the traditional owner of this line, and the second actually emits it: thus, for example, we speak of the γ-lines and β-electrons of RaB·C or ThC·C′, etc.

At the present time the study of line β-spectra constitutes an extensive chapter of nuclear physics. As an example let us point to the work of Ellis¹¹, who measured the energies and intensities (numbers of electrons) of the β-lines of a thorium B preparation in equilibrium with its decay products; in the interval from 0.05 to 3 million V he measured 71 lines of the β-spectrum. Let us note that the β-lines of each available source of radioactive radiations (for example, even of the same thorium preparation, consisting of the elements ThB, ThC, ThC′, ThC″, ThD) are denoted in the following way¹²: the most intense lines are denoted by large Latin letters (for example, the lines \(A\), \(B\), etc., in order of increasing energies); lines of medium intensity lying in the interval between \(A\) and \(B\) are denoted \(Aa\), \(Ab\), \(Ac\), etc., and, finally, quite weak lines lying between \(Ab\) and \(Ac\) are denoted \(Ab1\), \(Ab2\), \(Ab3\), etc. As for γ-lines, they are denoted by adding the letter γ to the name of the most intense β-line representing the result of the conversion of this γ-line. Thus, for example, the intense γ-line with energy 2.62 million V belonging to thorium C″ (or, more precisely, ThC′·D, i.e. emitted by the nucleus of the isotope of lead) and producing (upon conversion in the \(K\), \(L\), \(M\) shells of the Pb atom) the β-lines \(X\), \(Xa\), \(Xa1\), is called γ\(X\).

From all that has been said it is clear that between γ-spectra and the line β-spectra of radioactive substances there exists a close relation, which consists in the fact that to each γ-line there corresponds a series of β-lines representing the result of its conversion in all possible shells of the electronic envelope, and to each β-line there corresponds one γ-line, which can be detected also by some independent method (for example, by the Skobeltzyn method or by the photoelectric method) and as a result of whose conversion the β-line under consideration arises. Of course, in some individual cases exceptions are also possible. Thus, for example, it may happen that the photons of a given γ-line are absorbed so strongly in the electronic shell surrounding the nucleus that only very few of them (and perhaps none) emerge and become accessible to observation by the photoelectric or some other method. An example may be the fact, discovered by Thibaud¹³, that in the line β-spectrum of RaC′ there exist three rather noticeable lines which should be attributed to the conversion, in the \(K\), \(L\), and \(M\) shells of the electronic envelope of RaC′, of a γ-line with energy 1.426 million V, and nevertheless no traces of this γ-line can be detected in the velocity spectrum of electrons ejected by the γ-rays of RaC in various metals. This means that in this case the conversion proves to be complete or, in any case, unusually high. (Let us note that Skobeltzyn¹⁴, applying to the study of the γ-spectrum of Ra(B + C) his method of counting electron tracks in a Wilson chamber, pre-

the ordinary photoelectric method, with its sensitivity, in detecting the presence of γ-radiation; nevertheless, by the accuracy of the wavelength determination corresponding to it, it lies in the region close to 1.4 million V, and more intense radiation in this region would correspond to the weak γ-line \(1.389\cdot 10^6\) V found there in the photoelectric spectrum by Ellis and Aston.^15 It is therefore possible that a weak γ-line with energy \(1.426\cdot 10^6\) V does in fact exist, although, as we shall see below, this does not agree very well with theoretical considerations.) Thus we see that the simplicity of the relation between the β- and γ-spectra may be disturbed by an anomalously high conversion of individual γ-lines. Apparently, inverse cases of anomalously low conversion are also possible, in which no appreciable emission of β-electrons corresponds to the observed γ-line. Thus, for example, in the γ-spectrum of ThC·C′, Skobeltzyn^16 detected by his method the presence of a γ-line with an energy of about \(1.65\cdot 10^6\) V. The existence of a γ-line of approximately this energy is also confirmed by the circumstance that among the excited states in which the majority of ThC′ nuclei find themselves after the β-decay

\(\mathrm{ThC}\to\mathrm{ThC'}\),

there is one with an energy exceeding the energy of the normal state by \(1.78\cdot 10^6\) V (such, according to the measurements of Rosenblum and Valadares,^17 is the difference between the energy of the principal long-range group of α-particles emitted by thorium C′ and the energy of the principal group of α-particles). Nevertheless, Ellis was unable to detect the corresponding conversion lines in the β-spectrum of Th \((\mathrm{B}+\mathrm{C})\). Therefore it may perhaps be concluded that the conversion of this γ-radiation is anomalously small. It is possible, however, that it does exist after all, because Jae Shi-ran^18 finds in the β-spectrum of Th \((\mathrm{C}+\mathrm{C'}+\mathrm{C''})\) a weak line with energy \(1.569\cdot 10^6\) V; if one adds to this \(0.093\cdot 10^6\) V, i.e. the work required to remove an electron from the \(K\)-shell of ThC′ \((Z=84)\), one obtains \(1.662\cdot 10^6\) V, i.e. a number close to that cited by Skobeltzyn. Here, however, as in the case of the γ-line with energy \(1.426\cdot 10^6\) V, the question is not yet fully clarified experimentally (the energy according to Skobeltzyn and Jae Shi-ran is less than according to the fine-structure data), all the more so since the β-line \(1.569\cdot 10^6\) V is, according to Jae Shi-ran, in the vicinity of two other weak β-lines, \(1.441\cdot 10^6\) V and \(1.415\cdot 10^6\) V, whose origin is difficult to understand. Finally, a third possible type of violation of the simplicity of the relation between the β- and γ-spectra may occur when, owing to an accidental numerical coincidence, two different γ-lines, being converted in different shells of the electron cloud, give secondary electrons of one and the same energy. The probability of such a coincidence at the accuracy of present-day measurements is negligible; nevertheless, Ellis^11 found two such curious cases. Thus, for example, the line \(M\) of the Th \((\mathrm{B}+\mathrm{C})\) β-spectrum studied by him consists of electrons ejected by the γ-line \(\gamma L\) \((0.510\cdot 10^6\ \mathrm{V}, \mathrm{ThC''}\cdot D)\) in the \(L_1\) shell of the Pb atom \((=\mathrm{ThD})\), and of electrons ejected by the γ-line \(\gamma M\) \((0.582\cdot 10^6\ \mathrm{V}; \mathrm{ThC''}\cdot D)\) in the \(K\) shell of the same atom. In this case the accidental coincidence proves to be so exact that, at the pre-

the line \(M\) is by no means wider than the neighboring line \(L\) of the same intensity (rather, the opposite is the case). We shall see below how the theory of internal conversion makes it possible to understand such cases.

Summarizing what has been said, we can characterize internal conversion quantitatively in the following way: let there be given some large number of atoms containing an excited nucleus in the state \(i\) (we shall assume that the nuclear states are denoted by \(1, 2, 3\ldots\) in order of increasing energy). Consider that fraction of this number of atoms which makes a transition from the nuclear state \(i\) to the nuclear state \(j\) (\(< i\)). Denote this fraction by \(p_{ij}\) (there is no need that

\[ \sum_{j=1}^{i-1} p_{ij}=1, \]

since it may happen that some part of the nuclei undergoes radioactive decay, emitting, for example, a long-range \(\alpha\)-particle). How can the atom lose, in this process, that excess of energy which it possesses, containing the nucleus in the state \(i\), as compared with its final state, when it contains the nucleus in the state \(j\)? It can do this in several different ways. It can, for example, emit a photon whose \(h\nu\) is equal to the lost excess of energy; it can emit from the \(K\)-shell of its electron envelope an electron with energy \(h\nu-K\), where \(K\) is the work required to tear an electron from this shell, and after this give up the remaining \(K\) ergs either in the form of X-radiation emitted during the restoration of the electron shell, or partly in the form of electrons ejected from higher shells, and partly in the form of X-radiation (such internal conversion of X-rays has been studied by many authors \(^{19}\)); in the same way an electron with energy \(h\nu-L_1\) can be torn from the \(L_1\) shell, while the remaining energy \(L_1\) is given up in the form of X-ray quanta and partly in the form of electrons from higher shells, etc. The final state of the system in all these cases is one and the same, namely—an atom with an undisturbed electron shell and with an unexcited nucleus. (Strictly speaking, in describing all these processes one would also have to take into account the recoil energy of the atom, which, however, here, in contrast to what occurs in the emission of \(\alpha\)-particles, may practically be neglected.)

As Smekal and Rosseland \(^{20}\) first pointed out, in defining internal conversion we must confine ourselves merely to descriptions of the form in which the atom gives up its excess energy, without insisting in this description that the nucleus must first emit a photon with energy \(h\nu\) and that this photon is then absorbed in the electron shell. (To a considerable extent, as we shall see further on, the disputes as to whether internal conversion consists in a preliminary emission and subsequent absorption of a \(\gamma\)-ray photon, or whether there occurs a more direct interaction between the nucleus and the electron shell, are of a philological character.) If we denote by \(p_{ij}(1-\alpha_{ij})\) the probability that the nuclear transition \(i\to j\) will occur and that, in this process,

from the atom a \(\gamma\)-photon will be emitted with complete loss at this \(h\nu_{ij}\), and if we denote by \(p_{ij}\alpha_{ij}\) the probability that a transition \(i\to j\) will occur and that in this case an electron with energy \(h\nu_{ij}-K\) will leave the atom, by \(p_{ij}l_{ij}\alpha_{ij}\) the probability that a transition \(i\to j\) will occur and an electron with energy \(h\nu_{ij}-L\) will leave the atom, etc., then the following relation must hold:

\[ \chi_{ij}=\kappa_{ij}+l_{K}\alpha_{ij}+l_{L}\alpha_{ij}+\cdots \]

The quantity \(\alpha_{ij}\), which may be regarded as the probability of absorption of a \(\gamma\)-photon \(h\nu\) in the electron shell of the atom, is called the total coefficient of internal conversion of this \(\gamma\)-line; the quantity \(\kappa_{ij}\) is its conversion coefficient in the \(K\)-shell, and so on.

The measurement of the internal-conversion coefficients of the \(\gamma\)-lines of radium B + C was carried out by Ellis and Aston1. Their method consisted in the following. A thick-walled platinum tube was filled with radium emanation in equilibrium with its decay products, including RaB and RaC, the quantities of which in this equilibrium state can be calculated from the amount of emanation. The \(\beta+\gamma\) rays emitted by the radium were absorbed in the platinum wall, which therefore emitted photoelectrons. The main group of photoelectrons corresponding to the \(\gamma\)-line \(h\nu\) has energy \(h\nu-K_{\mathrm{Pt}}\); the corresponding line in the \(\beta\)-spectrum of the platinum tube must be somewhat shifted toward lower energies, since the electrons ejected near the inner surface of the wall are retarded in passing through the thickness of the platinum. By photometering the photograph of such a \(\beta\)-line, Ellis and Aston drew conclusions about the number of electrons ejected by \(\gamma\)-quanta from the \(K\)-shell of the atoms of the tube. In order to pass from this to the number of \(\gamma\)-quanta emitted by radium B or C, it is necessary to know the probability of photoelectric absorption. This is the most delicate point of the entire investigation; to determine the photoelectric coefficient, Ellis and Aston used Gray’s empirical formula. As a result of such calculations, the following numbers of \(\gamma\)-quanta emitted per disintegrating nucleus are obtained for the three \(\gamma\)-lines of RaB.

For RaC an analogous table is obtained:

Let us note here that, since photometry gives only relative numbers of electrons of the β-lines, the transition to the absolute numbers of γ-quanta placed in the second column of the table is accomplished by the fact that the total amount of energy emitted in the form of γ-radiation per given number of nuclei can be measured calorimetrically. Ellis and Aston used Gray’s calorimetric measurements.

The number of electrons ejected by a given γ-line from the \(K\)-shell of the electron envelope of the atom itself containing the excited nucleus was determined by Ellis and Aston by photometry of photographs of the β-spectrum emitted directly by a deposit of radium \(B+C\), applied to the outer side of the tube. Dividing the number of β-electrons ejected by the γ-line from the \(K\)-shell, calculated per one disintegration, by the corresponding number of γ-quanta that underwent conversion (i.e., by the number from the second column of the preceding table), we obtain the ratio \(\frac{k\alpha}{1-\alpha}\) for the given γ-line. The results obtained in this way give the following table:

As for the conversion coefficients in other shells, their ratio to the conversion coefficient in the \(K\)-shell can be determined from the relative numbers of electrons of the corresponding β-lines. Measurement of these relative quantities gives the following results:

Such are the experimental results relating to the coefficients of internal conversion. The work of Ellis and Aston, containing these results, is so far the only attempt at a direct determination of the coefficients \(\alpha\). The most difficult part of the problem is, of course, the determination of the intensities of the individual γ-lines. Skobeltsyn’s measurements\(^{11}\) also make it possible

judge the energy distribution in the γ-spectrum of Ra (B + C), and the results obtained by him, although relating only to the general form of the energy distribution and not to individual lines, are in general agreement with the measurements of Ellis and Aston, although, as was already noted above, in the region near \(1.4 \cdot 10^6\) V there is a discrepancy between the results of the measurements which has still not been satisfactorily explained.

The results of the measurements of Ellis and Aston (coefficients of internal conversion in the \(K\)-shell) are plotted as crosses in Figs. 1 and 2. On the abscissa axis are plotted the ratios \(\frac{mc^2}{h\nu}\) for γ-lines, i.e. the wavelengths, measured in units of \(\frac{h}{mc}\), where \(m\) is the electron mass. The corresponding energies in millions of volts are plotted at the top of the drawings. The meaning of the curves drawn in the same drawings is given below.

Fig. 1.

Fig. 2.

We see that in the case of RaB there is a regular (insofar as one can judge from three points) decrease of conversion with increasing hardness, whereas in the case of RaC there is, on the contrary, a very irregular course. At the same time, for RaC all the conversion coefficients are less than \(0.7\%\); the only exception is the RaC line with energy \(1.426 \cdot 10^6\) V, not indicated in the drawing, whose conversion coefficient is very large (more than \(80\%\) for the \(K\)-shell). For RaB the conversion coefficient is, in general, considerably larger than for RaC. (In addition to the lines given in the tables, one may also point to the weak γ-lines \(0.471 \cdot 10^6\) V of radium B.C. and \(0.429 \cdot 10^6\) V of radium C.C′, of which the second is somewhat stronger than the first, whereas the number of converted electrons for the first is five times greater than for the second.)

The theory of internal conversion was first studied by Miss Swirles\(^{23}\) (even before the measurements of Ellis and Aston), Casimir\(^{22}\), and Hulme\(^{24}\). All of them approached the problem as follows: one electron is in the Coulomb field of a nucleus having charge \(Z\); the state of the electron is determined by quantum numbers corresponding to the \(K\)- or \(L\)-shell. The action of the other electrons, i.e. all kinds of screening effects, was completely neglected.

time. A Hertzian vibrator field (a dipole with a variable electric moment) of frequency \(\nu\) is superposed on the Coulomb field as a perturbation. This perturbation leads to a probability of transition of the electron from the initial state (for example, from the \(K\) state) to a state in which it is torn out of the atom and moves with energy \(h\nu-K\). Such a probability is calculated and then multiplied by the number of electrons in the corresponding shell (for example, by 2 in the case of the \(K\)-shell); this gives the theoretical conversion coefficient. The drawback of Miss Swirles’s calculations was that she used the Schrödinger equation, neglecting relativistic effects, which in the present case cannot be neglected. She obtained conversion coefficients about 10 times smaller than those obtained from experiment for most lines (not to mention the anomalous line \(1.426\cdot 10^6\ \mathrm{V}\), where the discrepancy amounts to several hundred times). Casimir was the first to apply the relativistic Dirac equation to the calculation of conversion coefficients, and, in order to simplify the calculation, he assumed that the ratio \(h\nu:mc^2\) is very large, and then extrapolated his result to lower hardness (in other words, he used the asymptotic formula derived for \(h\nu\to\infty\)). He also obtained \(x\)’s that were too small. Therefore the idea arose that internal conversion of \(\gamma\)-rays is a phenomenon of a typically relativistic-quantum character, not fitting within the framework of the existing theory and connected with a special type of interaction between the nucleus and the external electrons, arising from the fact that the external electrons possess a wave function which can take appreciable values also near the nucleus. Such a “mechanical” interaction was first introduced by Fowler \(^{25}\) to explain the conversion of the line \(1.426\cdot 10^6\ \mathrm{V}\); see also the work of Delbrück and Gamow \(^{26}\), where in this connection internal conversion is called infernal conversion instead of internal conversion. However, Hulme (l. c.) obtained more satisfactory results, having managed to carry out very cumbersome calculations without the assumption that \(h\nu \gg mc^2\). Hulme calculated the coefficient \(k^x\) for six lines (\(1.778\cdot 10^6\ \mathrm{Hg}\), \(1.13\cdot 10^6\), \(0.8\cdot 10^6\), \(0.612\cdot 10^6\), \(0.354\cdot 10^6\), \(0.243\cdot 10^6\ \mathrm{V}\)); through the points obtained a curve was drawn (denoted \(H\) in our figures). In doing so it was assumed that \(Z=84\) (the atomic number of the decay product \(\mathrm{RaC}\to\mathrm{RaC}'\); for RaB.C one would have to take \(Z=83\), but the difference is too small and may be neglected). From Fig. 1 it is seen that several experimental points for RaC fit Hulme’s curve rather well, whence it follows that for these lines the proposed explanation of conversion (dipole radiation with subsequent photoeffect) is correct. However, three points of RaC, not to mention the anomalous line, and also all three points of RaB do not fit Hulme’s curve. Hulme also calculated (for the case \(h\nu \gg mc^2\)) the ratio

\[ k^x:l_I^\alpha:l_{II}^\alpha:l_{III}^\alpha \]

and found it equal to

\[ 6.7:10.0086:0.044 \]

(incidentally, according to experimental data \(l_{II}^\alpha\) is always \(> l_{III}^\alpha\)). The ratio \(k^x:l_I^\alpha\) was calculated in a separate

for \(h\nu = 0.72\cdot 10^6\) V and found equal to 7. Therefore the ratio, probably, does not depend on the frequency. This ratio agrees well with the measurements of Ellis and Aston.

Taylor and Mott\(^{27}\) carried out an extensive investigation in which the physical meaning of the assumption that the radiation has a dipole character was clarified to a considerable extent. If we assume that the process of excitation of a nucleus consists in the fact that one charged particle (for example, an \(\alpha\)-particle), moving in the unchanging field of the remaining parts of the nucleus, passes into a state of higher energy, then the radiation emitted by it in the reverse transition has, as quantum mechanics shows, a dipole character in the case where the azimuthal quantum number (angular momentum, measured in units of \(h/2\pi\)) changes by 1; if, however, it does not change or changes by 2, then the field of the emitted radiation will have not a dipole, but a quadrupole character. In dealing with atoms, we are accustomed to the fact that the probability of quadrupole transitions is very small (the so-called forbidden lines); however, in the case of a nucleus it may be that the probability of quadrupole transitions is quite comparable with the probability of dipole transitions (only a future theory of the nucleus, in which the properties of nuclear electrons will be explained, can give a satisfactory explanation of this; from the point of view of ordinary wave mechanics this means only that the dipole moment of the nucleus is very small). Considering transitions of an \(\alpha\)-particle from one state to another, Taylor and Mott introduce the interaction of this \(\alpha\)-particle with an external electron. In doing so, generally speaking, one may neglect the influence of the electric field inside the nucleus, in which the \(\alpha\)-particle moves, on the wave function of the electron, i.e. the “mechanical” interaction of the electron with the nucleus. The only exception is the transition of an \(\alpha\)-particle in which the azimuthal quantum number is equal to zero before and after the transition, i.e. a transition of the type \(S \to S\). Such a transition, accompanied by radiation, is absolutely forbidden by the selection rules of wave mechanics; it can take place only under the influence of the “mechanical” interaction of the electron with the nucleus, the electron then taking up all the energy released in the transition. Fowler (l. c.) had already explained in this way the anomalous line \(1.426\cdot 10^6\) V. In all other cases the interaction of an \(\alpha\)-particle moving inside the barrier with an electron outside the barrier can be calculated by Meller’s method,\(^{28}\) which makes it possible to take account of the retardation effect as follows: the charge and current density corresponding to the given transition of the \(\alpha\)-particle are calculated, and the retarded potentials arising from this charge and current are substituted as a perturbation in the relativistic wave equation for an electron situated outside. This method is at the same time a description of the mechanical interaction between the \(\alpha\)-particle and the electron and a description of the photoelectric absorption of the \(\gamma\)-rays emitted by the \(\alpha\)-particle; therefore, as Taylor and

Mott, the difference between the two points of view in this case has only a philological character.

The curve of quadrupole radiation, calculated by Taylor and Mott in addition to Helm’s dipole curve, is denoted in our figures by the letters TM. We see that all the points fall close either to one or to the other curve. Thus, for example, of the two points lying at \(\alpha \sim 0.006\) and denoted by straight crosses (this is how we have marked those measurements which Ellis considers especially reliable), one lies on Helm’s curve, and the other on the Taylor–Mott curve. In the case of RaB the agreement is somewhat worse: the conversion coefficient calculated by Taylor and Mott is one third smaller than the measured one, but the variation with frequency \(v\) is reproduced correctly here. Since the measurements are very difficult, one cannot be certain that these discrepancies are real. Thus, in general it may be said that the measured conversion coefficients are satisfactorily explained by the photoelectric absorption in the electron shell of \(\gamma\)-photons emitted in nuclear transitions with a change of the azimuthal quantum number by 1, 2, and 0.

As an example of the services rendered by the theory of internal conversion in the analysis of \(\beta\)-spectra, let us give an analysis of the origin of the above-mentioned “double” \(\beta\)-line of thorium preparation M (Ellis)\(^{11}\). Line \(L\) in the \(\beta\)-spectrum of this preparation, belonging to ThC′,D, has energy \(0.42198 \cdot 10^6\) V, while line \(M\), belonging to the same nucleus, has energy \(0.49417 \cdot 10^6\) V. If to the energy of line \(L\) one adds the work of removing an electron from the \(K\)-shell of lead, equal to \(0.08750 \cdot 10^6\) V, one obtains \(0.50948 \cdot 10^6\), and if to the energy of line \(M\) one adds the work of removal from the \(L_I\)-shell, equal to \(0.01582 \cdot 10^6\) V, one obtains \(0.50999 \cdot 10^6\)—an agreement sufficient for speaking of the existence of a \(\gamma\)-line with energy \(0.51 \cdot 10^6\) V. But (in the arbitrary units adopted in Ellis’s work) the intensity of line \(L\) is equal to 1.70. Since the ratio \(\kappa^2 : L_1^2\), on the basis of experiment and theory, must be equal to seven, one might expect that the intensity of line \(M\) should be equal to 0.24. However, it is equal to 1.70. Therefore one must assume that the remaining 1.46 has some other origin. If we add to the energy of line \(M\) the work of removal from the \(K\)-shell, we obtain \(0.58167 \cdot 10^6\) V. There is, in the same \(\beta\)-spectrum, a line \(N\), which gives, if to it one adds the work of removal from the \(L_I\)-shell, the energy \(0.58240 \cdot 10^6\) V. It is therefore possible to speak of the existence of a \(\gamma\)-line with energy \(0.58 \cdot 10^6\) V, the conversion of which in the \(K\)-shell explains part 1.46 of line \(M\), and the conversion of which in the \(L_I\)-shell explains line \(N\). If this explanation is correct, then the intensity of line \(N\) should be equal to 0.2. Experiment gives 0.3, which for such a comparatively weak line may be regarded as good agreement. Further confirmation is the existence of the lines \(M\alpha\) and \(M\alpha_1\), which can be interpreted as the result of conversion of those

of the two γ-lines, \(0.51 \cdot 10^6\) and \(0.58 \cdot 10^6\) V in the \(M_1\)-layer, their intensities are equal to 0.05 and 0.07, which is approximately the required number of times smaller than the intensities 1.70 and 1.46. Thus, not only energy considerations, but also a calculation of intensities based on theoretical conceptions of internal conversion helps to disentangle the regularities of γ-spectra.

According to Ellis and Mott,^29 the theory of internal conversion is in a position to help in interpreting a system of nuclear levels when the question is to find the azimuthal numbers corresponding to these levels. As an example they consider the system of levels of ThC′ found by Rosenblum and Valadares^17 in their investigation of the fine structure of the γ-rays emitted in the transition ThC→ThC′. According to the measurements of Rosenblum and Valadares, 77% of the ThC nuclei transform into ThC′ with an excitation energy of \(0.041 \cdot 10^6\) V (the energy is reckoned from the normal level), 2.2% transform into ThC′ with an excitation energy of \(0.332 \cdot 10^6\) V, then 0.4% give ThC′ with an excitation energy of \(0.477 \cdot 10^6\) V, and, finally, 1.5% give ThC′ with an energy \(0.498 \cdot 10^6\) V exceeding the energy of the normal level. The levels of ThC′ have been assigned the names \(\alpha_1\) (normal level), \(\alpha_0\) (\(0.041 \cdot 10^6\) V), \(\alpha_2\) (\(0.332 \cdot 10^6\) V), \(\alpha_4\) (\(0.477 \cdot 10^6\) V), \(\alpha_3\) (\(0.498 \cdot 10^6\) V), \(\alpha_5\) (\(0.626 \cdot 10^6\) V).

Since the number of ThC nuclei passing into a definite state, for example \(\alpha_2\) of the ThC′ nucleus, must be equal to the number of ThC′ nuclei passing from the state \(\alpha_2\) into deeper states \(\alpha_0\) and \(\alpha_1\), minus the number of ThC′ nuclei passing into the state \(\alpha_2\) from higher states \(\alpha_4, \alpha_3, \alpha_5\), we therefore have certain equalities which the numbers of ThC′ nuclei passing from one state to another must satisfy. Let us denote these numbers, calculated per one decay ThC → ThC′, by the letters \(p\). We must, for example, have
\(p(\alpha_2 \to \alpha_0) + p(\alpha_2 \to \alpha_1) - p(\alpha_4 \to \alpha_2) - p(\alpha_3 \to \alpha_2) = 0.022\).
At the same time \(p(\alpha_2 \to \alpha_0)\) is equal to the number of γ-quanta with energy
\(\alpha_2 - \alpha_0 = 0.291 \cdot 10^6\) V, emitted per one decay ThC→ThC′, and so on. The numbers \(p\) are not measured, but the corresponding numbers of electrons emitted in the conversion of these lines in a definite shell (for example the \(K\)-shell) of the ThC″ atom are measured, i.e. the products of the numbers \(p\) by the corresponding conversion coefficients. In order to determine the numbers \(p\), it is necessary to make the hypothesis as to whether the corresponding γ-line is dipole or quadrupole, which makes it possible to determine the conversion coefficient theoretically from the Heitler or Taylor–Mott curve. We give a table of those transitions of the ThC″ nucleus in which γ-rays of appreciable intensity are actually emitted, and give in it the numbers of γ-quanta per one decay, according to the quadrupole and dipole hypotheses:

How is one to choose between the two hypotheses? Let us consider, for example, the level \(a_2\). We must have \(p(a_2 \to z_1)+p(a_2 \to z_0)=0.022\). It is clear that, of all the possibilities afforded us by our table, only the possibility \(p(a_2 \to a_0)=0.025,\ p(a_2 \to a_1)=0.0081\) makes it possible to satisfy this requirement approximately. Thus it is clear that the \(\gamma\)-lines \(a_2 \to a_0\) and \(a_2 \to a_1\) also have a quadrupole character.

If one assumes—though quite arbitrarily—that the azimuthal quantum number of the ground level \(a_1\) is equal to zero, then one must set the azimuthal quantum numbers of the levels \(a_0\) and \(a_2\) equal to 2, since the transition \(0 \to 0\) is forbidden. In an analogous way Ellis and Mott derive the azimuthal quantum numbers of the other levels. Further experimental investigations must verify the correctness of these conclusions.

Let us note that Gamow has recently also made partial use of the theory of internal conversion in constructing the level scheme of the nucleus RaC\(^1\).\(^{30}\)

One may draw the general conclusion that, despite the existence of some discrepancies between theory and measurements which are as yet difficult to explain, the theory of internal conversion can serve as a valuable auxiliary tool in deciphering \(\gamma\)-spectra and systems of nuclear levels.

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1. Ellis and Aston. ↩