Internal Conversion of $\gamma$-Rays and Determination of the Quantum Characteristics of Nuclear Levels

I. S. Shapiro

Contents

I. Internal conversion on atomic electrons

1. Multipolarity of radiation. Electromagnetic fields of multipoles . . . . . . . . . . . . . . . . . . . . . . 190
2. Physical nature of the phenomenon of internal conversion. Theoretical results . . . . . . . . . . . . . . . . 197
3. Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

II. Internal conversion with pair formation

1. Features of pair conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
2. Principal experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

The study of the internal conversion of $\gamma$-rays is a highly effective method for establishing the characteristics (energy, angular momentum, parity of the wave function) of the levels of radioactive nuclei. Measurement of the energies of conversion electrons by means of magnetic $\beta$-spectrographs is at present one of the most accurate and widely used methods for determining the energy of $\gamma$-radiation from radioactive elements. The application of internal conversion to establish the other quantum characteristics of nuclear levels listed above attracts a large number of investigators for two reasons. First, theoretical calculations of the probability of internal conversion can be carried out without any model assumptions about the structure of the nucleus; thus, by comparing theoretical and experimental results, it becomes possible to obtain information, not based on hypotheses, about the properties of nuclear levels. Second, the experimental means of modern nuclear physics fully permit sufficiently accurate investigations in this field. At the same time, in the results of measurements of the coeffi-

there is still a divergence in the coefficients of internal conversion. The latter circumstance makes it desirable to carry out a comparative analysis of the principal experimental methods employed.

Theoretical calculations of internal-conversion coefficients for the majority of cases of interest were essentially completed in 1948–1949. Since the results of theoretical works, sometimes carried out under different simplifying assumptions and therefore possessing different domains of applicability, are contained mainly in original articles scattered through a large number of journals, a review and discussion of the theoretical data also seems advisable. Consideration of the questions listed above constitutes the subject of the present article.

I. INTERNAL CONVERSION ON ATOMIC ELECTRONS

1. MULTIPOLARITY OF RADIATION. ELECTROMAGNETIC FIELDS OF MULTIPOLES

In this section we shall recall the basic facts connected with the multipolarity of electromagnetic radiation. Let us first consider, from the classical point of view, the electromagnetic field of some system of charges at distances large in comparison with the dimensions of this system. As is well known, the scalar potential of a static system of charges is given by the expression

\[ \varphi=\sum_i \frac{e_i}{|\mathbf r-\mathbf r_i|}. \tag{1} \]

Here \(\mathbf r\) is the radius vector drawn from the origin of coordinates, chosen inside the system, to the point of observation, and \(\mathbf r_i\) is the radius vector drawn from the origin of coordinates to the point occupied by the \(i\)-th charge. If \(|\mathbf r_i|\ll |\mathbf r|\), then, expanding the right-hand side of (1) in a series in powers of \(\frac{r_i}{r}\), we obtain (see, for example, \({}^{1}\)):

\[ \varphi=\frac{\sum e_i}{r}-\mathbf D\,\operatorname{grad}\frac{1}{r} +\frac{1}{6}Q^{\alpha\beta}\frac{\partial^2}{\partial x_\alpha\,\partial x_\beta}\frac{1}{r}+\cdots , \tag{1a} \]

where

\[ \mathbf D=\sum e_i\mathbf r_i;\qquad Q^{\alpha\beta}=\sum_i e_i\left(3x_i^\alpha x_i^\beta-r^2\delta^{\alpha\beta}\right); \]

\[ \delta^{\alpha\beta}= \begin{cases} 0, & \text{for } \alpha\ne\beta,\\ 1, & \text{for } \alpha=\beta. \end{cases} \]

In equation (1a), \(x_\alpha, x_\beta\) are the components of \(\mathbf r\) \((\alpha,\beta=1,2,3)\), \(x_i^\alpha, x_i^\beta\) are the components of \(\mathbf r_i\), and crossed indices denote summation. The vector \(\mathbf D\) is called the dipole moment

systems of charges, the symmetric tensor of rank 2 \(Q^{\alpha\beta}\)—the quadrupole moment. In general, the \((n+1)\)-th term of expansion (1a) can be expressed in an analogous way in terms of a tensor of rank \(n\), composed of the charges and their radius vectors and called the multipole moment of the system. The magnitude of the multipole moment depends on the spatial distribution of the charges. One can imagine systems for which expansion (1a) consists of only a single term, i.e. systems possessing only one multipole moment. Consequently, the static field of any arbitrary system of charges at distances large compared with the dimensions of the system can be represented as a superposition of the fields of various multipoles. An analogous statement holds for the vector potential of the field of steady currents. For example, the magnetic dipole moment is the vector

\[ \boldsymbol{\mu}=\frac{1}{2c}\sum e_i[\mathbf{r}_i\mathbf{v}_i], \tag{1б} \]

where \(\mathbf{v}_i\) is the velocity of the \(i\)-th charge; moreover, the intensity of the steady magnetic field is expressed in terms of the magnetic dipole moment in exactly the same way as the intensity of a static electric field is expressed in terms of the electric dipole moment. Let us now consider the field of arbitrarily moving charges. We shall start from the expression for the retarded potentials:

\[ \varphi=\int \frac{1}{R}\rho_{t'-\frac{R}{c}}\,dV;\quad \mathbf{A}=\frac{1}{c}\int \frac{1}{R}\mathbf{j}_{t'-\frac{R}{c}}\,dV, \tag{2} \]

where \(\mathbf{R}=\mathbf{r}-\mathbf{r}'\) (\(\mathbf{r}'\) is the variable of integration), \(\rho\) and \(\mathbf{j}\) are the charge and current densities, and \(\mathbf{A}\) is the vector potential. If \(r'\ll r\), then, expanding \(R\) in a series in powers of \(\frac{r'}{r}\) and retaining only the first term of the expansion, we obtain:

\[ R=r-\mathbf{r}'\mathbf{n};\quad \mathbf{n}=\frac{\mathbf{r}}{r}. \tag{2a} \]

Let us assume, for simplicity, that we are dealing with monochromatic electromagnetic waves, so that

\[ \rho_{t'-\frac{r}{c}+\frac{\mathbf{r}'\mathbf{n}}{c}} = \rho_0(\mathbf{r}')e^{i\omega t'}e^{-ikr}e^{i\mathbf{k}\mathbf{r}'}, \]

\[ \mathbf{j}_{t'-\frac{r}{c}+\frac{\mathbf{r}'\mathbf{n}}{c}} = \mathbf{j}_0(\mathbf{r}')e^{i\omega t'}e^{-ikr}e^{i\mathbf{k}\mathbf{r}'}, \tag{3} \]

where \(\mathbf{k}\) is the wave vector \((\mathbf{k}=k\mathbf{n})\). Since

\[ e^{i\mathbf{k}\mathbf{r}'}=1+\frac{1}{2}\mathbf{k}\mathbf{r}'+\cdots, \tag{3a} \]

then for

\[ \mathbf{k}\mathbf{r}'\ll 1 \tag{3б} \]

the last factors in the right-hand sides of equation (3) may be set equal to 1. In this case, neglecting in the denominators

of the integrands in (2) by the quantity \(\mathbf{n}\mathbf{r}'\) in comparison with \(r\), and making the substitution

\[ \int \rho_0\, dV = \varepsilon_0,\qquad \mathbf{j}_0=\rho_0\mathbf{V},\qquad t=t'-\frac{r}{c}, \tag{3в} \]

we find:

\[ \left. \begin{aligned} \varphi &= \frac{\varepsilon_0 e^{i\omega t}}{E},\\ \mathbf{A} &= \frac{e^{i\omega t}}{cr}\int \rho_0\mathbf{V}\,dV = \frac{e^{i\omega t}}{cr}\,\frac{d}{dt}\left\{\int \rho_0\mathbf{r}'\,dV\right\} = \frac{e^{i\omega t}}{cr}\,\dot{\mathbf{D}}. \end{aligned} \right\} \tag{3г} \]

Thus, under condition (3б), the potentials of the electromagnetic field are expressed in terms of the total charge and the time derivative of the electric dipole moment of the system. Therefore the radiation field of such a system of charges has received the name dipole radiation. It is easy to show that the expansion (3a) in fact carries out an expansion of the field potentials of an arbitrary system of charges in powers of \(v/c\) (\(v\) is the order of magnitude of the velocity of motion of the charges), with the dipole radiation being given by the term of zeroth order with respect to \(v/c\). The next term of the expansion (3a)—of first order with respect to \(v/c\)—is expressed through the time derivatives of the electric quadrupole and magnetic dipole moments. Taking this term into account, we obtain, for example, for the vector potential:

\[ \mathbf{A}=e^{i\omega t}\left\{\frac{\dot{\mathbf{D}}}{cr}+\frac{1}{6c^2r}\ddot{\mathbf{Q}}+\frac{1}{cr}[\dot{\boldsymbol{\mu}}\mathbf{n}]+\ldots\right\}, \tag{4} \]

where \(\mathbf{Q}\) is a vector whose components are expressed through the quadrupole moment:

\[ Q_\alpha = Q^{\alpha\beta} n_\beta \qquad (\alpha,\beta=1,2,3). \tag{4а} \]

The electromagnetic fields determined by the second and third terms in (4) are called, respectively, electric quadrupole and magnetic dipole radiation. Taking into consideration the remaining terms of the expansion (3a), we obtain radiation fields of higher multipolarities. Thus, the electromagnetic field of an arbitrary system of charges at distances large in comparison with the dimensions of the system can be represented as a superposition of fields of different multipolarities. If in the expansion (4) all quantities depending on the multipole moments, beginning with some one, are comparable with one another, then the most intense radiation will be that of the lowest multipolarity. This follows from the fact that expansion (4) is an expansion in powers of \(v/c\), or, equivalently, in powers of \(r'/\lambda\) (\(r'\) is a quantity characterizing the linear

dimensions of the system, \(\lambda\) is the wavelength of the radiation divided by \(2\pi\)). For example, in the case of emission of \(\gamma\)-rays by a nucleus, the ratio \(\dfrac{r'}{\lambda}\) does not exceed, in order of magnitude, \(10^{-1}\). Thus, the intensity of the radiation determined by the first, next-to-the-lowest nonzero term of the expansion (4) will be 100 times smaller than the intensity of radiation of the lowest multipolarity (since the potentials enter the expressions for the intensities quadratically). Consequently, radiation of higher multipolarities will play a role only when the quantities depending on the moments of the lower multipolarities are anomalously small, or, from the quantum-mechanical point of view, when radiation of the lower multipolarities is forbidden by some selection rules.

Thus, one may expect that in most cases, when \(\gamma\)-rays are emitted by a nucleus, the decisive role will be played by radiation of some single multipolarity or by a mixture of radiations of electric and magnetic multipoles (for example, an electric quadrupole and a magnetic dipole). In this connection, the study of the field structure of a given multipolarity is of particular interest for a whole range of problems and, in particular, for the study of internal conversion. Expressions for the potentials of the fields of electric and magnetic \(2^l\)-poles (\(l=1\) corresponds to a dipole, \(l=2\) to a quadrupole, \(l=3\) to an octupole, etc.) have been obtained in general form by Berestetskii\({}^2\), and for the special case of the gauge \((\varphi = \operatorname{div} \mathbf A = 0)\) by Gaitler\({}^3\). Since we are considering the field at distances large compared with the dimensions of the system, we may regard the radiating multipole as pointlike. Then for an electric \(2^l\)-pole \(\mathbf A\) and \(\varphi\) can be written in the form\(^*\)

\[ \mathbf A = a_l^m i \left[\frac{2}{\pi l(l+1)}\right]^{1/2} f_{l-1}(kr)\,Y_{l,m}^{(-1)}(\theta,\Phi)\,e^{-i\omega t} + \text{complex conjugate}, \]

\[ \varphi = a_l^m \left[\frac{2l}{\pi(l+1)}\right]^{1/2} f_l(kr)\,Y_{l,m}(\theta,\Phi)\,e^{-i\omega t} + \text{complex conjugate}, \tag{5} \]

\[ f_l(kr)= -\frac{H_{l+1/2}^{(1)}(kr)}{(kr)^{1/2}} . \]

Here \(\theta\) and \(\Phi\) are the polar angles of the radius vector of the observation point \(\mathbf r\), \(Y_{l,m}(\theta,\Phi)\) are normalized spherical harmonics of Laplace, \(H_{l+1/2}^{(1)}(kr)\)—

\(^*\) In equation (5), and everywhere below, a convenient system of units is used, in which the unit of mass is the rest mass of the electron \(m\), the unit of length is the Compton wavelength \(\hbar/mc\), and the unit of time is \(\hbar/mc^2\). In this system of units all energies are expressed in \(mc^2\), momenta in \(mc\), while the frequency, energy, and momentum of the photon are given by a single number; the electron charge is \(e=\alpha^{1/2}\) \(\left(\alpha=\dfrac{e^2}{\hbar c}=\dfrac{1}{137}\right.\) is the fine-structure constant). To pass from ordinary units to these, one should put \(\hbar=m=c=1\) in all formulas.

Hankel functions of the first kind. \(\mathbf{Y}^{(-1)}_{l,m}\)—the so-called “spherical vector,” whose “components” \({}_{(i)}Y^{(-1)}_{l,m}\) have the form:

\[ {}_{(i)}Y^{(-1)}_{l,m}=\beta_{mi}Y_{l-1,m+i};\qquad i=0,\ \pm1, \tag{5a} \]

where \(\beta_{mi}\) are coefficients in the expansion

\[ n_iY_{l,m}=\alpha_{mi}Y_{l+1,m+i}+\beta_{mi}Y_{l-1,m+i}, \tag{5б} \]

where the \(n_i\) are expressed in terms of the components of the vector \(\mathbf n=\frac{\mathbf r}{r}\) in the following way:

\[ n_0=n_z,\qquad n_{\pm1}=\mp2^{-1/2}(n_x\pm in_y), \tag{5в} \]

and \(\alpha_{mi}\) and \(\beta_{mi}\) satisfy the relations

\[ \sum_i \alpha_{mi}^2=(l+1)/(2l+1);\qquad \sum_i \beta_{mi}^2=l/(2l+1). \tag{5г} \]

Substituting in (5) the expressions for the “components” of the spherical vector, we obtain linear combinations of the components \(A\):

\[ A_0=A_z,\qquad A_{\pm1}=\mp2^{-1/2}(A_x\pm iA_y). \tag{5д} \]

The use in (5) of Hankel functions that have a singularity at zero is necessary in the case when it is required to obtain a nonzero radiation flux through a sphere of infinitely large radius surrounding a point source of radiation. Precisely such solutions, as will be seen below, are required for calculating the probability of the process of internal conversion. In those problems where consideration of “traveling” waves is not necessary, the Hankel functions in (5) may be replaced by Bessel functions of the same order.

Analogous expressions also hold for the potentials of the field of a magnetic multipole:

\[ \mathbf A=b_l^m\left(\frac{2}{\pi}\right)^{1/2} f_l(kr)\mathbf Y^{(0)}_{l,m}(\theta,\phi)e^{-i\omega t}+\text{complex conjugate},\quad \varphi=0, \tag{6} \]

where

\[ {}_{(i)}Y^{(0)}_{l,m}=\gamma_{m,i}Y_{l,m+i};\qquad \gamma_{m0}=-\frac{m}{[l(l+1)]^{1/2}}, \tag{6a} \]

\[ \gamma_{m,\pm1}=\mp[(l\mp m)(l+1\pm m)/2l(l+1)]^{1/2}. \]

The quantities \(a_l^m\) and \(b_l^m\) determine the moments of the multipoles and are connected with the radiation flux through an infinitely distant sphere surrounding the radiator by the relations:

\[ S_{\text{el}}=\frac{|a_l^m|^2}{\pi^2},\qquad S_{\text{magn}}=\frac{|b_l^m|^2}{\pi^2}; \tag{7} \]

or, if the fluxes are expressed in quanta per second,

\[ S_{\text{el}}=\frac{|a_l^m|^2}{\pi^2 k}\,\frac{\text{quantum}}{\text{sec.}},\qquad S_{\text{magn}}=\frac{|b_l^m|^2}{\pi^2 k}\,\frac{\text{quantum}}{\text{sec.}}, \tag{7a} \]

Let us turn to the consideration of certain questions connected with the quantization of the field. As is known, ordinarily in the quantization the potentials of an arbitrary radiation field are expanded in a series of plane waves, the corresponding amplitudes being regarded as operators satisfying definite commutation relations and having matrix elements that differ from zero only for transitions in which one quantum is emitted or absorbed.

We have seen above that the field potentials of an arbitrary system of charges can be expanded in terms of the potentials of multipole fields given by equations (5) and (6). This circumstance may be used for the quantization of the field, which is carried out in exactly the same way as in the expansion in plane waves. Namely, the amplitudes \(a_l^m\) and \(b_l^m\) must be replaced by operators satisfying the commutation relations:

\[ a_l^{m*} a_l^m - a_l^m a_l^{m*} = b_l^{m*} b_l^m - b_l^m b_l^{m*} = \frac{1}{2k}, \qquad a_l^{m*} b_l^m - b_l^{m*} a_l^m = 0. \tag{8} \]

A photon described by a plane wave possesses a definite momentum. In this sense the expansion of the field potentials in plane waves is entirely analogous to the expansion of the wave function of a particle in the proper wave functions of the momentum operator. As is known, the law of conservation of angular momentum in a system of arbitrarily moving charged particles will be fulfilled only if we take into account the angular momentum of the electromagnetic field. In the classical theory a relation is derived for the angular momentum of the electromagnetic field, which has the form:

\[ \mathbf{M} = \int [\mathbf{r}\mathbf{S}]\, dV, \tag{9} \]

where \(\mathbf{S}\) is the Umov–Poynting vector, and the integration is carried out over the entire volume of the field.

When quantizing the field with the aid of the expansion in “proper waves” (5) and (6), it can be shown\(^3\) that each such “proper wave” possesses a definite angular momentum. For an electric or magnetic \(2^l\)-pole field the eigenvalues of the angular-momentum operator will be:

\[ M^2 = l(l+1);\quad M_z = m;\quad m = -l,\ldots,-1,0,1,\ldots,l. \tag{10} \]

If one uses the commonly employed old terminology, preserved from the time of Bohr quantization, then one may, consequently, say that a quantum of electric or magnetic \(2^l\)-pole radiation carries away angular momentum \(l\). The expansion in multipole fields may now be interpreted as an expansion in states with definite angular momentum,

With the aid of (10) it is easy to obtain the selection rules for angular momentum for transitions of some system (for example, a nucleus) with the emission of a quantum of definite multipolarity. Indeed, it remains only to apply the quantum rules for the addition of angular momenta. Thus, for the emission of an electric or magnetic \(2^l\)-pole quantum we have:

\[ \begin{aligned} \Delta J&=0,\pm 1,\pm 2,\ldots,\pm l,\\ \Delta m&=0,\pm 1,\ldots,\pm l, \end{aligned} \tag{11} \]

where \(\Delta J\) is the possible change of the quantum number \(J\), which determines the total angular momentum of the system \(\bigl(\sqrt{J(J+1)}\bigr)\), and \(\Delta m\) is the change of the magnetic quantum number (the projection of the angular momentum on the \(z\)-axis).

As is known, a very important, specifically quantum characteristic of any system (for example, a nucleus) is the parity of the wave function. Recall that a wave function is called even if, under mirror reflection of the coordinate axes, the sign of the function does not change, and odd if under such a coordinate transformation the sign of the function changes to the opposite one. Parity is an integral of motion, and consequently there are selection rules with respect to parity. We shall characterize even wave functions by the number \((+1)\), and odd ones by \((-1)\). Then from the consideration of (5) and (6) it follows that, upon emission of an electric \(2^l\)-pole quantum, the parity of the wave function of the radiating system changes as \((-1)^l\), while for a magnetic \(2^l\)-pole transition it changes as \((-1)^{l+1}\). For clarity, all the selection rules are collected in Table 1.

Table I

Selection rules for a multipole transition

	\(\Delta J\)	\(\Delta m\)	Change of parity
Electric \(2^l\)-pole	\(0,\pm1,\ldots,\pm l\)	\(0,\pm1,\ldots,\pm l\)	\((-1)^l\)
Magnetic \(2^l\)-pole	\(0,\pm1,\ldots,\pm l\)	\(0,\pm1,\ldots,\pm l\)	\((-1)^{l+1}\)

From the table it is clear that sometimes a transition allowed from the point of view of conservation of angular momentum may be forbidden by the selection rules for parity. For example, if \(\Delta J=2\) and the parities of the initial and final states of the nucleus are different, then an electric quadrupole transition is impossible (although it is allowed by the selection rules (11)), but a magnetic quadrupole transition may occur.

magnetic and electric octupole transitions. Thus, only the combined action of the selection rules for angular momentum and for the parity of the wave function determines the multipolarity of the radiation emitted by a system of charged particles.

Let us now suppose that the nucleus, having emitted a $\gamma$ quantum, has passed from the state $n'$ to the state $n_0$, and that the angular momentum $J_{n_0}$ and the parity of the wave function in the state $n_0$ are known (for example, $n_0$ may be the lower energy state of a stable nucleus). If, in addition, the multipolarity of the $\gamma$ transition is known, then with the aid of the selection rules one can unambiguously establish the parity of the wave function of the excited state $n'$ and obtain information on the possible values of the angular momentum of the nucleus in this state. Indeed, if a $2^l$-pole transition has occurred, this means that radiations of lower multipolarities are forbidden. Suppose that this prohibition is due to the selection rules (11). Then the possible values of the angular momentum of the nucleus in the state $n'$ will be:

\[ J_{n'}=\left|J_{n_0}\pm l\right|. \tag{12} \]

If, however, there is a prohibition by parity (see the example discussed above), then for $J_{n'}$ we obtain:

\[ J_{n'}=\left|J_{n_0}\pm(l-1)\right|,\ \left|J_{n_0}\pm l\right|. \tag{12a} \]

It is obvious that, in order to establish the multipolarity of the $\gamma$ radiation of a nucleus, any effect depending on the multipolarity of the $\gamma$ transition can, at least in principle, be used. One such effect is the internal conversion of $\gamma$ rays.

2. PHYSICAL NATURE OF THE PHENOMENON OF INTERNAL CONVERSION. THEORETICAL RESULTS.

Consider an atom whose nucleus is in an excited state. In the majority of cases, a transition from the excited state to a level with lower energy is possible in two ways:

a) with the emission of a $\gamma$ quantum;
b) with the transfer of the excitation energy to one of the atomic electrons, which will thereby be ejected from the atom.

This latter process is called internal conversion on atomic electrons. Sometimes a radiative one-quantum transition may be absolutely forbidden (for example, if the angular momenta for the levels between which the transition occurs are equal to 0). Then only conversion transitions are possible, or transitions with the emission of several quanta. However, in this article we shall not consider the so-called “$(0-0)$ transitions.”

The task of the theory is to calculate the relative probability of conversion and radiative transitions. To solve this problem, as always in similar problems, one uses quantum perturbation theory. The cause of the conversion transition is the electromagnetic interaction of the atomic electron with the particles of the nucleus. Thus, we must introduce into consideration the radiation field of the nucleus. In the initial state of our system (excited nucleus + electron shell + radiation field) the number of photons is zero. In the final state (de-excited nucleus + electron outside the atom + “remainder” of the electron shell + radiation field) there are likewise no photons. Owing to the properties of the quantum operators of the field (see (8)), whose matrix elements do not vanish only for transitions in which one quantum is created or absorbed, a conversion transition within perturbation theory is a process consisting of two virtual transitions:

I transition: the excited nucleus emits a quantum, passing into a state with lower energy; the state of the electron shell remains unchanged;

II transition: one of the atomic electrons absorbs the quantum, passing into a continuous-spectrum state. Both virtual transitions may take place without conservation of energy, but in the whole process, of course, the energy is conserved. If \(k\) is the energy of the \(\gamma\)-quantum that can be emitted in the radiative de-excitation of the nucleus, then the kinetic energy of the converted electron, just as in the case of the photoelectric effect, will be:

\[ E_{\mathrm{kin}} = k - I, \tag{13} \]

where \(I\) is the ionization potential of the shell in which the electron was located before leaving the atom. It should be borne in mind, however, that photoelectric absorption and internal conversion are two different physical processes. In internal conversion the nucleus transfers its excitation energy to an atomic electron, and, if we were not interested in the computational side of the question, we could avoid introducing \(\gamma\)-quanta altogether. In this respect the name “internal conversion of \(\gamma\)-rays” for the process under consideration does not quite accurately reflect the essence of the matter, since the quantum in the conversion process is emitted and absorbed virtually. The statement frequently encountered in the literature about internal conversion of \(\gamma\)-rays as a process caused, on the one hand, by the “direct” interaction of the particles of the nucleus with an atomic electron, and, on the other hand, by “photoelectric absorption” of the quantum emitted by the nucleus, has only the meaning that any electromagnetic interaction in a system of charged particles can formally be divided into Coulomb and retarded parts.

The authors of all theoretical works on internal conversion, without exception, do not use, in calculating the probability of this process, the rigorous path described above within the framework of quantum perturbation theory. Usually a semiclassical scheme is employed, the equivalence of which to the rigorous path has recently been shown by Berestetskii4. In this scheme the nucleus is replaced by a point classical emitter of monochromatic electromagnetic waves, and the probability is calculated for the transition of an atomic electron into a continuum state as a result of interaction with the field of the multipole radiation. The probability obtained is referred to the flux (in quanta per second) of the multipole radiation emitted through a sphere of large radius surrounding the emitting nucleus. Such a computational scheme, for all its simplicity and, it would seem, clarity, has one very serious shortcoming. Namely, it is not clear what objectively observable quantity corresponds to the artificially introduced radiation flux. In the first calculations5, when internal conversion was regarded entirely as an effect of photoelectric absorption, this radiation flux was identified with the total number of nuclear transitions per unit time (such a conception also gave rise to the semiclassical model described). However, as early as a year after the appearance of the first work, the erroneousness of such a point of view was shown6. It became clear, first, that internal conversion is an additional path of nuclear de-excitation, competing with the radiative transition, and, secondly, that the presence of the electron shell changes the probability of emission of \(\gamma\)-quanta by no means to the extent assumed in the initial interpretation. The question of the influence of the presence of the electron shell on the probability of emission of \(\gamma\)-quanta in the case of small excitation energies has been studied in detail by A. S. Davydov7, whose results are given in Table II.

Table II

Influence of orbital electrons on the probability of emission by the nucleus of a \(\gamma\)-quantum \((N_{\gamma_0} - N_\gamma)\)

\(k\)	\(Z = 20\)	\(Z = 20\)	\(Z = 40\)	\(Z = 40\)
	\(l = 2\)	\(l = 4\)	\(l = 2\)	\(l = 4\)
\(0,1\)	\(1,4 \cdot 10^{-5}\)	\(1,7 \cdot 10^{-7}\)	\(5,8 \cdot 10^{-3}\)	\(1,7 \cdot 10^{-5}\)
\(0,5\)	\(3,5 \cdot 10^{-3}\)	\(1,3 \cdot 10^{-5}\)	\(2,3 \cdot 10^{-4}\)	\(8,1 \cdot 10^{-5}\)

From this table it is seen that the presence of atomic electrons reduces the number of \(\gamma\)-quanta emitted per unit time by an exceedingly small amount, the smaller the greater the multipolarity of the radiation. This means that the presence of the electron shell increases

the total probability of transition of the nucleus, i.e., it decreases the lifetime of the nucleus in the excited state, but almost does not change the probability of emission of \(\gamma\)-quanta. Practically, even in the case of quadrupole radiation at small transition energies (\(\sim 10^5\) keV), one may assume that the probability of emission of a \(\gamma\)-quantum for a nucleus surrounded by an electron shell remains the same as for a “bare” nucleus. This latter circumstance is very essential, for theoretical calculations give precisely the ratio of the number of conversion transitions per unit time \((N_e)\) to the number of radiative transitions of the nucleus in the absence of the electron shell \((N_{\gamma_0})\). Since, however,

\[ N_{\gamma_0}\cong N_\gamma , \tag{14} \]

more precisely

\[ N_{\gamma_0}=N_\gamma+O(a), \tag{14a} \]

where \(N_\gamma\) is the number of \(\gamma\)-quanta leaving the atom per unit time, \(a\) is the fine-structure constant, it may be asserted that theoretical calculations give the value

\[ \frac{N_e}{N_\gamma}, \]

i.e., the ratio of the numbers of conversion electrons and \(\gamma\)-quanta leaving the atom per unit time.

Usually in theoretical papers the ratio

\[ w=\frac{N_e}{N_\gamma} \]

is called the conversion coefficient. As will be seen from what follows (see Section 3), in experimental papers the quantity

\[ W=\frac{N_e}{N_e+N_\gamma}, \]

which is also called the conversion coefficient, is most often measured. Obviously, the two definitions are nearly identical in the case when the conversion probability is small, i.e., if \(N_e\ll N_\gamma\). In the opposite case the quantities \(w\) and \(W\) do not coincide: whereas \(w\) may take the most varied values, both smaller and greater than unity, \(W\) is always \(<1\), and

\[ W=\frac{w}{1+w} \tag{15} \]

\[ (W\approx w\ \text{for } w\ll 1). \]

In what follows we shall call the quantity \(W\) the coefficient of internal conversion, i.e., the ratio of the probability of a conversion transition to the total probability of nuclear decay per unit time.

Let us now turn to the calculation of the quantity \(w\), using the semiclassical model mentioned above. We must calculate the probability of a conversion transition per unit time \(N_e\). According to the rules of perturbation theory this probability is found with the aid of the expression:

\[ dN_e=n_e\,d\Omega_p=2\pi \sum \left|H_{ab}\right|^2\rho_b. \tag{16} \]

\(N_e\) is obtained by integrating (16) over \(\Omega_p\) (\(d\Omega_p\) is an element of solid angle in the momentum space of the electron in the final state):

\[ N_e=\int_{\Omega_p} n_e d\Omega_p . \tag{16a} \]

\(H_{ab}\) is the matrix element of the perturbation-energy operator. In the general relativistic case \(H_{ab}\) has the form:

\[ H_{ab}=\int \psi_a^* H' \psi_b\, d\tau, \tag{17} \]

\[ H'=-\alpha^{1/2}(\boldsymbol{\alpha}\mathbf{A}-\varphi). \tag{17a} \]

\(\psi_a\) denotes the relativistic four-component wave function of the electron in the initial state (a discrete-spectrum state), \(\psi_b\) the wave function of the electron in the final state. \(H'\) is the relativistic operator of the interaction energy of the electron with the radiation field of the multipole (\(\boldsymbol{\alpha}\) is the Dirac matrix velocity vector). The integration in (17) extends over the whole volume and also includes summation over the spin variable. The summation sign in (16) denotes summation over all possible initial and final states of the electron (states with different spin orientations), and the bar denotes averaging over all possible orientations of the multipole moment, determined by the magnetic quantum number \(m\). Such averaging is necessary in order to make it possible to compare theoretical results with experiment, since in experiment radiation from a very large number of nuclei is always observed, whose mechanical moments are oriented chaotically in space (it is easy to show, with the aid of the selection rules, that averaging over the parameter \(m\) in equations (5) and (6) is in fact averaging over all possible orientations of the nuclear spin in the initial and final states). The quantity \(\rho_b\) entering equation (16) gives the “density of states” of the converted electron, i.e. the number of continuous-spectrum states in a unit energy interval:

\[ \rho_b= \frac{(\text{volume element in the electron momentum space})} {(\text{volume of phase space corresponding to one electron state})} \frac{1}{dE} = \frac{pE}{(2\pi)^3}\,d\Omega_p , \tag{17b} \]

where \(p=|\mathbf{p}|\) is the absolute value of the electron momentum in the final state, and \(E\) is the electron energy (including the rest mass). The required quantity \(w\) is now found from relations (16) and (17):

\[ w=\frac{N_e}{N_\gamma}=\frac{\int n_e d\Omega_p}{S}. \tag{18} \]

The quantity unknown to us, \(a_l^m\) (or \(b_l^m\)), which determines the multipole moment and therefore depends on the spatial distribution of particles in the nucleus, enters into \(S\) quadratically. From equations (16) and (17) it is easy to see that the numerator of formula (18) is also proportional to \(|a_l^m|^2\) (or \(|b_l^m|^2\)). Thus, the ratio \(N_e/N_\gamma\) does not depend on \(a_l^m\) (or \(b_l^m\)).

In concrete calculations by formula (18) two kinds of difficulties arise. First, it is necessary to know the wave function of the electron in the atom \((\psi_a)\). Since the exact form of such wave functions is known only for the hydrogen atom, the question arises of choosing satisfactory approximations. This is not difficult to do if one restricts oneself to considering the inner electron shells—the \(K\)- and \(L\)-shells. Incidentally, conversion on just these shells is in the overwhelming majority of cases of practical interest. The second kind of difficulty is connected with the fact that, in calculations by formula (18), generally speaking, one must use relativistic Dirac wave functions and, in particular, relativistic continuum wave functions for an electron in the Coulomb field of the nucleus \((\psi_b)\). But calculations with such wave functions are very complicated even for the simplest case of electric dipole radiation; moreover, the result is obtained by numerical integration, and with increasing multipolarity of the transition the number of expressions subject to numerical integration grows catastrophically. Thus, obtaining in this way a general analytic formula for the coefficient of internal conversion is impossible. Therefore one often resorts to simplifying assumptions in the calculations. Two types of approximations may be indicated.

a) It is assumed that the Coulomb field of the nucleus does not substantially affect the motion of the electron outside the atom, and consequently, in choosing the continuum wave function \((\psi_b)\), one sets \(Z=0\) (\(Z\) is the nuclear charge). Then \(\psi_b\) is represented by a plane wave, and all calculations quite easily lead to simple formulas giving the dependence of the conversion coefficient on the multipolarity and energy of the transition. The assumption \(Z=0\) (the so-called Born approximation) is correct in the case when

\[ \eta=\frac{Z\alpha}{v}\ll 1 \tag{19} \]

(\(v\) is the velocity of the converted electron), i.e. it is applicable for sufficiently light nuclei and comparatively large transition energies (we note that for very heavy nuclei, for example, at \(Z\sim 80\), (19) is not satisfied even when \(v=1\)). Unfortunately, in the case of internal conversion on atomic electrons, the results obtained in the Born approximation are not of great value.

of interest. The point is that, as will be seen from what follows, the coefficient of internal conversion is \(\sim Z^3\) and increases as the transition energy decreases. Thus, precisely in those cases when the conversion probability is large and the effect can be well studied experimentally, condition (19) ceases to be valid. Therefore calculations in the Born approximation are useful chiefly only in the sense that, owing to the exceptional simplicity of the formulas, their results make it very easy to trace qualitatively the behavior of the coefficient of internal conversion.

b) Since the conversion probability on atomic electrons, as has just been noted, increases with decreasing transition energy, it is reasonable to try to solve the problem in the nonrelativistic approximation, i.e. for \(v \ll 1\). In this case one can obviously use wave functions that are solutions of the Schrödinger equation, and the nonrelativistic form of the energy operator for the interaction of the electron with the electromagnetic field. The use of the nonrelativistic approximation simplifies the calculations and makes it possible to obtain results whose range of applicability includes a considerable part of the practically important cases. The formulas derived in the nonrelativistic approximation are valid, first, as indicated above, if the kinetic energy of the electron is considerably less than its rest mass, and, second, for \(Z \lesssim 50\). The latter restriction is connected with the fact that for large \(Z\), when considering the motion of the electron in the atom, allowance for the spin-orbit interaction becomes essential, and as a result it is necessary to use relativistic wave functions of the discrete spectrum \((\psi_a)\).

We shall now present theoretical data obtained by various authors.

A. Electric radiation

a) Conversion on the \(K\)-shell.

In all calculations of the probability of conversion on the \(K\)-shell it is assumed that the motion of the \(K\)-electrons is not distorted by their interaction with one another and with other electrons, i.e. is determined entirely by the Coulomb field of the nucleus. This approximation is quite legitimate; this follows, for example, from the excellent agreement of experimental data on the coefficient of photoelectric absorption with theoretical calculations carried out under the same assumption concerning the motion of the \(K\)-electrons. The negligibly small magnitude of the error introduced by such a simplification is due to the fact that the difference between the binding energies of the \(K\)-electron in the atom, with and without taking into account its interaction with other electrons, is, for all practically interesting cases, many times smaller than the kinetic energy of the converted electron. Exact rela-

tivistic calculations of the conversion coefficient on the \(K\)-shell have been made only for dipole and quadrupole transitions\(^{5,6}\) at \(Z=84\). The results of these calculations are shown in Fig. 1. From Fig. 1 it is clearly seen that the coefficient of internal conversion increases with decreasing transition energy and with increasing multipolarity. For a quadrupole transition the conversion coefficient is approximately 3 times larger than for a dipole transition. For higher multipolarities calculations have been carried out only in the nonrelativistic and Born approximations by Hebb and Uhlenbeck\(^{8}\) (the first five multipoles) and by Dankov and Morrison\(^{9}\) (in general form for any multipolarity). The latter obtained the following formula for \(w\):

\[ w_{K,l}^{\mathrm{el}}= 16a\,\frac{l}{l+1}\,[\Gamma(l+{}^{1}/_{2})]^2 \times \]

\[ \times \left(\frac{2}{k}\right)^{l+1} \frac{ n^4[(l+1)(1+n^2)]^{\,l-2} e^{-2n\operatorname{arc\,ctg} n}-V_l^3 }{ (1+n^2)^{l-2}(l^2+n^2)[(l-1)^2+n^2]\ldots(1+n^2)(1-e^{-2\pi n}) }, \tag{20} \]

where

\[ n=\frac{Za}{[2k-(Za)^2]^{1/2}}; \qquad V_l= \frac{ n^l\displaystyle\prod_{1}^{l}(i^2+n^2) }{ (l+n^2)^3(2l)! }, \]

\[ V_{l+1} = V_l(1+n^2)\frac{l+2}{l+1} + \frac{2^{2l+1}l}{(2l+2)!}\, \frac{1}{(1+n^2)^3} \prod_{1}^{l}(i^2+n^2), \]

\[ V_0=0;\qquad V_1=0,\qquad V_2={}^{1}/_{3},\qquad V_3=\frac{4(3+2n^2)}{15}. \tag{20a} \]

In the case of large \(l\) (\(n^2l\gg 1\)) expression (20) is considerably simplified and can be written in the form

\[ w_{K,l}^{\mathrm{el}} = Z^3a^4\,\frac{l}{l+1} \left(\frac{2}{k}\right)^{l+3/2} + O\left(\frac{1}{l}\right). \tag{20b} \]

Let us now consider the expression for \(w_{K,l}^{\mathrm{el}}\) found in the Born approximation (\(n\ll 1\)):

\[ w_{K,l}^{\mathrm{el}} = \frac{2Z^3a^4}{k^3} \left(\frac{k+2}{k}\right)^{l-1/2} \left[ \frac{(l+1)k^2+4l}{l+1} \right]. \tag{21} \]

As was found above, the results obtained in the Born approximation can be of interest for \(k\gg 1\). Under this assumption (21) becomes

\[ (w_{K,l}^{\mathrm{el}})_{\mathrm{cr.\ rel.}} = \frac{2Z^3a^4}{k}. \tag{21a} \]

Thus, in the extreme relativistic case the ratio \(\dfrac{N_e}{N_\gamma}\) is inversely proportional to the transition energy and does not depend on its multipolarity. The latter can be understood from the fol-

...ing considerations. As is known, the radiation field of any system of charges is usually divided into two parts: the so-called static, or Coulomb, part, which falls off as \(r^{-2}\) at large distances from the radiating system, and the transverse part, proportional to \(r^{-1}\). The region where the second part of the field predominates over the first is called the wave zone. The static region has dimensions of the order of the wavelength of the emitted radiation. Such an artificial division of the field into two parts, justified in many cases, proves, however, to be too crude in solving certain problems. For example, in calculating the angular momentum of the field by means of (9), the field precisely in the intermediate zone plays the chief role,\(^{3,10}\) where the electric-field strength is \(E \sim r^{-2}\), while the magnetic-field strength is \(H \sim r^{-1}\). In other words, the angular momentum of the electromagnetic field is contained in the intermediate zone. Since fields of different multipolarities differ precisely in their angular momenta, then, on the basis of formula (18), one should expect that the coefficients of internal conversion on the \(K\)-shell will in this case depend rather sharply on the multipolarity of the transition, if

Fig. 1a.

Fig. 1b.

the ratio \(\dfrac{\lambda}{a_0}\) does not differ too greatly from 1 (\(a_0\) is the radius of the \(K\)-orbit, i.e., the linear dimensions of the region in which there is an appreciable probability of finding the \(K\)-electron). For \(k \gg 1\) the ratio \(\dfrac{\lambda}{a_0} \ll 1\), and therefore the dependence on the multipolarity disappears.

In Fig. 16, for comparison, the curve of conversion coefficients calculated in the Born approximation for \(l=1\) (dipole transition) is plotted. From a comparison of the exact results with the Born curve it is seen that, in the case of heavy nuclei, the Born approximation gives conversion coefficients that are too large by a factor of 2–3 and therefore cannot be used (a substantially different situation, as we shall see below, occurs for internal conversion with pair production). Likewise, formula (20) cannot be applied to heavy nuclei (\(Z>50\)), as is shown by the curves presented in Fig. 1a (see the note on p. 232).

b) Conversion on the \(L\)-shell. The coefficients of internal conversion of electric \(2^l\)-pole radiation on the \(L\)-shell in the nonrelativistic approximation were calculated by Zavelevich\(^{11}\) and by Hebb and Nelson\(^{12}\) (the results of both works coincide). In choosing the wave functions of the \(L\)-electrons, the screening action of the \(K\)-shell and of the outer electron shells is taken into account. For the ratio

\[ \frac{w^{\mathrm{el}}_{L,l}}{w^{\mathrm{el}}_{K,l}} \]

Zavelevich’s work obtained an expression of the form:

\[ \frac{w^{\mathrm{el}}_{L,l}}{w^{\mathrm{el}}_{K,l}} = 2\,\frac{q^{2}}{d^{2}} \left(\frac{T_{1}}{T_{2}}\right)^{l-2} \prod_{s=1}^{l} \frac{s^{2}+d}{s^{2}+4d} \frac{\left(1-e^{-2\pi\sqrt{d}}\right)} {\left(1-e^{-4\pi\sqrt{d}}\right)} \frac{1}{D_l^{2}} \left\{ A_l^{2} + \right. \]

\[ \left. + \frac{lq\,(l^{2}+4q)}{2l+1}\,B_{l-1}^{2} + \frac{(l+1)(2l+1)} {4\left[(l+1)^{2}+4q\right](1+q)^{2}} \left(B_{l+1}+\frac{l+1}{2l+1}C_{l+1}\right)^{2} \right\}. \tag{22} \]

Here

\[ T_{1}=377.63\cdot 10^{3}k-\frac{(Z-4)^{2}}{4}; \qquad T_{2}=377.63\cdot 10^{3}k-Z^{2}, \]

\[ q=-\frac{(Z-4)^{2}}{4T_{1}}, \qquad d=\frac{Z^{2}}{T_{2}}. \]

The first term in the curly bracket refers to the shell \(L_{\mathrm{I}}\), while the second and third terms refer to the shells \(L_{\mathrm{II}}\) and \(L_{\mathrm{III}}\). They correspond to the two possible transitions of electrons with orbital angular momentum 1:

\[ l\to l-1 \]

(the second term) and

\[ l\to l+1 \]

(the third term)\(^*\).

\[ A_l=a(l+1)[1+(l+3)q](1+q)^{l-3}L_l; \qquad B_l=a(l+2)(1+q)^{l-2}+N_l; \]

\[ C_l=a\left[l+(2l^{2}+7l+4)q\right](1+q)^{l-3}-P_l; \qquad D_l=b(l+1)(1+d)^{l-2}-K_l; \]

\[ a=e^{-4\sqrt{q}\,\operatorname{arc\,tg}\sqrt{q}}; \qquad b=e^{-2\sqrt{d}\,\operatorname{arc\,tg}\sqrt{d}}. \tag{22a} \]

\(^*\) The transition \(l\to l\) in the case of electric radiation is forbidden by the parity selection rules (up to relativistic corrections, which are not taken into account in (22)).

The quantities \(N_l, P_l, K_l\), and \(L_l\) for the first five values of \(l\) are equal to:

\[ \begin{gathered} N_0=0;\quad N_1=1/(1+q);\quad N_2=4/3;\quad N_3=(1/5)(7+3q);\\ N_4=(2/315)(207+122q+11q^2);\\ N_5=(1/2835)(3195+1501q-535q^2-281q^3);\\ P_2=2;\quad P_3=(1/3)(9+5q);\quad P_4=(4/45)(45+49q+16q^2);\\ P_5=(1/315)(1575+2669q+1909q^2+527q^3);\\ K_1=0;\quad K_2=1/3;\quad K_3=(4/15)(3+2d);\\ K_4=(1/105)(141+188d+71d^2);\\ K_5=(2/2835)(2745+5525d+4219d^2+1151d^3);\\ L_1=0,\quad L_l=\frac{1}{2}(P_l-N_l). \end{gathered} \tag{226} \]

In Fig. 2 the dependence of

\[ \frac{w^{\mathrm{el}}_{L,l}}{w^{\mathrm{el}}_{K,l}} \]

on the quantity

\[ \frac{Z^2}{k} \]

and on the multipolarity of the transition is shown graphically. From Fig. 2 it is seen that the conversion coefficients on the \(K\)-shell are, as a rule, larger than on the

Fig. 2.

\(L\)-shell. This is due to the fact that the \(L\)-electrons are farther from the nucleus. The situation, however, changes for small transition energies and for sufficiently large \(Z\) and \(l\). For example, for

\[ l=4,\quad Z\simeq 30,\quad k\simeq 0.1,\quad \frac{w^{\mathrm{el}}_{L,4}}{w^{\mathrm{el}}_{K,4}}\simeq 1.5. \]

The increase of the ratio

\[ \frac{w^{\mathrm{el}}_{L,l}}{w^{\mathrm{el}}_{K,l}} \]

with decreasing transition energy and increasing \(Z\) can be explained qualitatively by the fact that the \(L\)-shell is “drawn into” the static zone of the multipole (since with increasing \(Z\) the size of the \(L\)-shell decreases, and with decreasing \(k\) the wavelength increases), thanks to

as a result of which the difference in the sizes of the \(K\)- and \(L\)-shells has a lesser effect on the conversion coefficient. Therefore the conversion probability calculated per one \(L\)-electron becomes comparable with the probability calculated per one \(K\)-electron. Since, moreover, the number of \(L\)-electrons is four times greater than the number of \(K\)-electrons, it follows that

\[ \frac{w^{\mathrm{el}}_{L,l}}{w^{\mathrm{el}}_{K,l}} \]

can exceed 1. Of course, if the transition energy is less than the ionization potential of the \(K\)-shell, the ratio

\[ \frac{w_{L,l}}{w_{K,l}}=\infty, \]

since conversion on the \(K\)-shell will not occur \((w_{K,l}=0)\).

From the data given above it is clear that measurement of the ratio

\[ \frac{w_{L,l}}{w_{K,l}} \]

can also serve as a good method for establishing the multipolarity of a transition. This is all the more important because in some cases the experimental determination of

\[ \frac{w_{L,l}}{w_{K,l}} \]

turns out to be much less complicated and can be carried out with greater accuracy than the measurement of the quantity \(w_{K,l}\) or \(w_{L,l}\).

B. Magnetic radiation

Exact relativistic calculations of the conversion coefficient on the \(K\)-shell for magnetic multipole radiation have been carried out only for the magnetic dipole transition\(^6\) (see Fig. 1a). In calculating the conversion probability for magnetic multipole radiation, allowance for the spin interaction of the electron with the radiation field becomes essential. Indeed, let us consider, for example, a conversion transition with the ejection of a \(K\)-electron. Since the orbital angular momentum of the \(K\)-electron is zero, while the angular momentum of the radiation of a magnetic \(2^l\)-pole is \(l\), the converted \(l\)-electron can possess only the angular momentum \(l\). The parity of the wave function of an electron with angular momentum \(l\) is \((-1)^l\), whereas the change of parity upon absorption (or emission) of a quantum of a magnetic \(2^l\)-pole is \((-1)^{l+1}\) (see Table 1). Thus, if spin is not taken into account, the transition considered will be forbidden by the parity selection rules. If, however, the electron spin is taken into account, then its total angular momentum in the continuum state may be \(l\pm \tfrac{1}{2}\). Such values of the angular momentum may be due to orbital angular momenta \(l,\, l-1,\, l+1\)*). Since the parity of the coordi-

*) Let us recall that when spin is taken into account, the integral of motion is the total angular momentum, but not the orbital and spin angular momenta separately. Therefore several values of the orbital angular momentum, determined by the rules for addition of orbital and spin quantum numbers, may correspond to a given value of the total angular momentum.

coordinate wave function of the electron, as was just indicated, is determined by the orbital angular momentum, no parity prohibition now arises. In solving the problem in the nonrelativistic approximation it is necessary to apply the nonrelativistic theory of spin; moreover, the correct writing of the nonrelativistic matrix element of the operator of the interaction energy of the electron with the radiation field of the nucleus, taking into account the spin interaction, is very important. In one of the first works[^13] devoted to the conversion

Fig. 3.

of radiation of a magnetic multipole, an error was made in writing the matrix element. Correct calculations of the internal-conversion coefficients on the \(K\)- and \(L\)-shells for magnetic radiation in the nonrelativistic approximation were carried out by Berestetskii[^14] (see also[^15],[^16]).

a) Conversion on the \(K\)-shell. The conversion coefficient on the \(K\)-shell for a magnetic \(2^l\)-pole transition is expressed in terms of the conversion coefficient of \(2^{l+1}\)-pole electric radiation[^14]:

\[ \omega_K^{\mathrm{M}} = \alpha\pi \left\{ \omega_{K,l+1}^{\mathrm{el}}\, \frac{k^5 l(l+2)}{8(l+1)(2l+1)} + \frac{32Z^4\alpha^4}{k^{2l+1}} \left(\frac{l+1}{2l+1}\right)^3 \left[2k-(Z\alpha)^2\right]^l \times \left[ \prod_{i=-l}^{i=+l} \frac{i^2+n^2}{1-e^{-2\pi n}} \right] \right\}. \tag{23} \]

Let us also give the relation obtained in the Born approximation:

\[ w^{M}_{K,l}=\frac{2Z^{3}\alpha^{4}}{k}\left(\frac{k+2}{k}\right)^{l+1/2}. \tag{24} \]

It is easy to see that, just as in the case of electric radiation, for \(k \gg 1\) the internal-conversion coefficient ceases to depend on the multipolarity of the transition, and

\[ \left(w^{M}_{K}\right)_{\mathrm{cr.\ rel}} = \left(w^{\mathrm{el}}_{K}\right)_{\mathrm{cr.\ rel}}. \tag{25} \]

b) Conversion on the \(L\)-shell. The expression for the conversion coefficient on the \(L\)-shell is too cumbersome. Therefore we give here the curves for the conversion coefficients of the first five multipoles (Fig. 3), constructed with the aid of the formulas derived for arbitrary \(l\) in the above-cited work of Berestetskii. In addition, we write down a comparatively simple formula obtained in the Born approximation \(^{12}\):

\[ w^{M}_{L,l} = \frac{Z^{3}\alpha^{4}}{k} \left(\frac{k+2}{k}\right)^{l+1/2} \left\{ 1+ \left(\frac{Z-a}{4}\right)^{2} \left(\frac{k+2}{k}\right) \left[ \frac{l+1}{2l+1} + \frac{l(2l+1)}{4} \left( \frac{2l-1}{2l+1} - \frac{k}{k+2} \right) \right]^{2} \right\}. \tag{26} \]

With this we conclude the survey of the theoretical data and pass on to a consideration of the experimental methods.

3. EXPERIMENTAL METHODS

In this section we shall consider the principal experimental methods for establishing the multipolarity of transitions, using the internal conversion of \(\gamma\)-rays on atomic electrons. We shall omit entirely the description of the method of measuring the conversion coefficients of \(\gamma\)-radiation emitted by nuclei—products of \(\alpha\)-decay—which is based on determining the relative intensities of \(\gamma\)-lines and groups of \(\alpha\)-particles. The reason is that the range of applicability of this method is limited to a narrow group of \(\alpha\)-active isotopes, and, moreover, a detailed discussion of it is contained in the available textbooks (see, for example, \(^{17,18}\)). The objects of our attention will be \(\gamma\)-transitions of nuclei formed as a result of \(\beta\)-decay, and the \(\gamma\)-radiation of isomeric nuclei. In these cases, the use of internal conversion for determining the multipolarity of \(\gamma\)-radiation is carried out by applying the following methods:

a) measurement of the conversion coefficients from the ratio of the areas bounded by the contours of the conversion line and of the continuous \(\beta\)-spectrum (the \(\beta\)-spectrum and the spectrum of conversion electrons are obtained with the aid of a magnetic spectrograph);

b) measurement of the relative probability of conversion in the \(K\)- and \(L\)-shells with respect to the areas of the corresponding conversion lines;

c) measurement of the conversion coefficient by means of the coincidence method;

d) direct measurement of the ratio of the intensities of the radiations of conversion electrons and \(\gamma\)-quanta.

Let us consider each of the listed methods separately.

a) Measurement of the conversion coefficient from the ratio of areas.

This method is very simple in conception and reduces to the following: in a spectrum taken with the aid of a magnetic spectrograph, the areas bounded by the contours of the conversion line and of the continuous \(\beta\)-spectrum are determined graphically. The ratio of these areas gives the total conversion coefficient \(W\), if the lifetime of the nucleus—the product of \(\beta\)-decay—in the excited state is much shorter than the half-life of the initial \(\beta\)-active nucleus (which is always satisfied if the excited state is not metastable), and if there is a “simple” \(\beta\)-spectrum, i.e. decay occurs according to the scheme shown in Fig. 4.

Fig. 4.

For all its simplicity and wide use, the method described has substantial shortcomings, which limit its accuracy to such an extent that at present it is difficult even to cite examples of conversion coefficients reliably measured in this way. Among the shortcomings of the method one must first of all mention the necessity of being certain that a simple \(\beta\)-spectrum is present, which is far from always easy to achieve. In the case of a complex \(\beta\)-spectrum, partial \(\beta\)-spectra have to be separated, and this leads to errors in the determination of the areas. When the intensity of one of the partial spectra is small in comparison with the intensity of another, then an error in the conversion coefficient arising from the complexity of the spectrum, if it does not exceed 15–20%, can, of course, be neglected. But even in this case caution must be exercised. Namely, it is necessary to know that the conversion transition under study follows the intense \(\beta\)-transition. The latter is not always easy to ascertain, especially if

the energy of the γ transition is 50–100 KeV, so that the radiation is strongly converted. In such a situation a rarely occurring γ transition of the nucleus in the conversion spectrum may be represented very intensely, which sometimes misleads the experimenter. A very illustrative example in this respect is the study of the radiation of Xe\(^{131}\). In the spectrum of Xe\(^{131}\), formed as a result of the decay of J\(^{131}\), two γ lines were initially known\(^{19}\)—\(80 \pm 1\) KeV and \(367 \pm 3\) KeV, and it was assumed (mainly on the basis of erroneous experiments on \((\gamma-\gamma)\) coincidences) that the γ quanta are emitted in cascade,

Fig. 5. β-spectrum of Hg\(^{203}\), recorded at different counter-window thicknesses.

and the β spectrum of J\(^{131}\) is simple (upper limit—600 KeV). Under this assumption the conversion coefficient of the γ radiation with energy 80 KeV, measured from the ratio of areas, turned out to be of the same order as the conversion coefficient of the 367 KeV γ transition (\(\sim 1\%\)). Only in the most recent works\(^{20,21,22}\) has it become clear that the Xe\(^{131}\) nucleus remains after the β decay of J\(^{131}\) at the level responsible for the occurrence of the 80 KeV γ line very rarely—only in 6% of the cases of the total number of β decays, and moreover the γ radiation with energy 80 KeV is strongly converted (the latest data on the conversion coefficient—\(80\% \pm 50\%\)). The transition with energy 367 KeV occurs much more often (in 79% of the cases of the total number of decays), but the conversion coefficient of this γ radiation is small (\(1.9\% \pm 0.5\%\)). In a word, application of the method under consideration for determining conversion coefficients is possible only after careful study of the nuclear decay scheme. However, even in the most favorable case, when it is reliably esta-

... a decay scheme of the type shown in Fig. 4 has been established, a number of instrumental effects considerably reduce the accuracy of the results. First, the β-spectrum obtained with the aid of a magnetic spectrograph is always cut off on the low-energy side of the electrons and, consequently, the area bounded by its contour does not give the total number of β-decays. The largest correction associated with this effect occurs in heavy β-active nuclei with a small upper limit of the β-spectrum (of the order of hundreds of KeV). The truncation of the low-energy part of the spectrum is due to the finite thickness of the counter windows used for registering electrons in β-spectrographs. It is very important that the finite thickness of the counter window not only cuts off a certain part of the spectrum, but also leads to a distortion of its shape. As an example one may cite the β-spectrum of \(Hg^{203}\) (upper limit of the spectrum \(205 \pm 10\) KeV, half-life \(43 \pm 0.5\) days).

Figure labels:
\(A\)—\(1\ \text{mg}/\text{cm}^2\)
\(B\)—\(2\ \text{mg}/\text{cm}^2\)
\(C\)—\(5\ \text{mg}/\text{cm}^2\)

Vertical axis: \(\left(\dfrac{N}{f}\right)^{1/2}\)
Horizontal axis: electron energy in KeV

Fig. 6. Fermi plot for the β-spectrum of \(S^{35}\).

In Fig. 5 are shown spectra obtained with different counter-window thicknesses. As is seen from Fig. 5, the transition from a window thickness of \(0.05\ \text{mg}/\text{cm}^2\) to a thickness of \(0.2\ \text{mg}/\text{cm}^2\) strongly affects the shape of the spectrum (the area bounded by the contour of the spectrum changes by 35%). It is characteristic that in both cases the thickness of the windows nominally permits the registration of electrons with energies down to 15 KeV, whereas distortions of the spectrum are still observed at 100 KeV. This testifies to the large role of electron scattering in the counter window, leading to a lengthening of their path in the material of the window (nylon). Distortion of the shape of the “soft” part of the spectrum is also caused by the effect due to the finite thickness of the radioactive source itself and by the scattering of electrons in the backing on which the active substance is deposited. How substantial the distortions associated with these phenomena are...

show, for example, the experiments^24 on the study of the β-spectrum of S^35, carried out with very thin sources (down to \(1\ \mu\mathrm{g}/\mathrm{cm}^2\)), deposited on a collodion film \(3\ \mu\mathrm{g}/\mathrm{cm}^2\) thick. The results of these experiments show (see Fig. 6) that a source only \(5\ \mu\mathrm{g}/\mathrm{cm}^2\) thick already leads to a distortion of the shape of the β-spectrum up to an energy of 70 KeV. It is now easy to understand what distortions of the spectra may occur in most modern works performed with sources of thickness of several \(\mathrm{mg}/\mathrm{cm}^2\). In addition to the reasons listed above, distortion of the shape of the β-spectrum is also caused by scattering of electrons and γ-rays in the walls and on the diaphragms of the spectrograph and, of course, by the finite resolving power of the spectrograph.

A considerable, if not the principal, part of the error in measuring the conversion coefficient by the ratio of areas arises from the inaccuracy in determining the area occupied by the conversion line. In most magnetic spectrographs with semicircular focusing, the conversion line has the form shown in Fig. 7 (in spectrographs with focusing by a longitudinal magnetic field the form of the line is, as a rule, more symmetrical^25). The broad base of the line sometimes makes it difficult to separate unambiguously the areas belonging to the conversion line and to the continuous β-spectrum. The instrumental width of the line is the smaller, the greater the resolving power of the spectrograph. Thus, for as accurate as possible a determination of the conversion coefficient, it is necessary to use β-spectrographs having high resolving power. On the other hand, the thickness of the source is of essential importance. Conversion electrons, slowing down in the substance of the source, may thereby leave the interval \(H\rho\), determined by the resolving power of the spectrograph. In this case the area of the conversion line will not correspond to the total number of conversion transitions. Consequently, the source must be thin for conversion electrons, i.e. its thickness must be much less than the range of the conversion electron, and, moreover, the spread in the electron energies arising from their slowing down in the source must be less than the line width determined by the resolving power of the spectrograph. Just as for electrons of the continuous spectrum, the magnitude

Fig. 7. Shape of a monochromatic conversion line obtained with a magnetic spectrograph with semicircular focusing.

areas cut out by the conversion line, the effect of the finite thickness of the counter window will have an impact. Fig. 8 illustrates the influence of the effect of the electric charging of the backing on which the preparation is deposited\(^{26}\). The solid curve was obtained with a grounded metallic backing, and the dashed curve with a thinner backing made of a nonconducting material (nylon). As can be seen from the figure, the charging effect strongly influences the position and area

Fig. 8. Influence of the source-charging effect on the \(\beta\)-spectrum of \( \mathrm{Lu}^{177} \).

of the conversion line (the area occupied by the \(K\)-conversion line from \(\gamma\)-radiation of 112 KeV changes by approximately a factor of 2).

On the basis of the foregoing, one may conclude that measuring the conversion coefficient by the method of area ratios requires the use of thin sources and, at the same time, spectrographs of high resolving power. But this means that large activities must be used. In short, measuring the conversion coefficient by this method is a difficult experimental problem. It is therefore not surprising that different authors still obtain discordant results. Thus, for example, the presently available data on the conversion coefficient in the \(K\)-shell of the 415 KeV \(\gamma\)-line of \( \mathrm{Hg}^{198} \), obtained by the method of area ratios, are as follows: \(1\%\) \(^{27}\), \(0.7\%\) \(^{28}\), \(2.6\%\) \(^{29}\). Such a scatter of the data does not make it possible to unambiguously establish the mul-

typicality of the transition. Judging from the spectra given in works \(^{27,28,29}\), apparently the last of the listed values is closest to the truth, but the very fact that the results differ by more than a factor of 3 testifies to their sensitivity to the experimental conditions noted above.

When measuring the conversion coefficient relative to areas, we in fact measure, with the aid of one and the same instrument, the relative intensities of the conversion radiation and the \(\beta\)-electron radiation. It would seem, therefore, that a considerable part of the instrumental effects, as in most cases of relative measurements, should be eliminated. However, the relativity of the measurements understood in this sense is, in the present case, to a considerable degree illusory. Distortions in the shapes of the conversion line and of the continuous \(\beta\)-spectrum by no means compensate one another in the calculation of the conversion coefficient, because the causes producing these distortions act differently in different regions of electron energies. Only errors connected with the determination of the luminosity of the instrument are eliminated.

A different situation obtains in determining the relative intensities of two neighboring, but fully resolved, conversion lines.

b) Measurement of the relative probability of conversion on the \(K\)- and \(L\)-shells from the areas of the conversion lines.

The quantity \(\left(\dfrac{w_{L,l}}{w_{K,l}}\right)\) can be measured from the areas of conversion lines much more accurately than the conversion coefficient. Indeed, for a transition energy of the order of one hundred or several hundred KeV and for not very large \(Z\), the \(K\)- and \(L\)-conversion lines will be close to one another in energy. In this case the effects described above, leading to distortions in the line shape, will affect the ratio of the areas of the \(K\)- and \(L\)-conversion lines to a substantially lesser extent than the ratio of the area of the conversion line to the area of the continuous \(\beta\)-spectrum. The use of this method is especially convenient for determining the multipolarity of \(\gamma\)-radiation of nuclei—products of \(K\)-capture and isomeric nuclei. Thus, for example, the multipolarity of the isomeric transition \(\mathrm{Br}^{80}\) \((k = 49\ \mathrm{KeV})\) was determined by Rusinov and Yuzefovich \(^{30}\) from the ratio of the conversion coefficients on the \(K\)- and \(L\)-shells. For \(\dfrac{w_{L,l}}{w_{K,l}}\) the value \(0.35\text{--}0.5\) was obtained experimentally. Theoretical calculations for electric radiation give:

\[ \frac{w_{L,l}}{w_{K,l}} = 0.1;\ 0.2;\ 0.5;\ 1.5;\ 3.1 \]

for \(l = 1,\ 2,\ 3,\ 4,\ 5\), respectively.

Thus, for the transition under study, \(l=3\) is a good fit. Another example of a successful determination of the ratio of the conversion coefficients on the \(K\)- and \(L\)-shells is the study of the \(\gamma\)-radiation of \(\mathrm{Cd}^{111}\) (formed as a result of \(K\)-capture from \(\mathrm{In}^{111}\)), consisting of two cascade-emitted \(\gamma\)-quanta with energies of 173 KeV and 247 KeV \(^{31}\). For the first of the listed \(\gamma\)-lines, \(\dfrac{w_{K,l}}{w_{L,l}}\) turned out to be equal to \(8 \pm 2\), and for the second, \(5 \pm 1\). On the basis of these values, one may conclude (see Table III) that the first

Table III

Conversion coefficients of the \(\gamma\)-radiation of \(\mathrm{Cd}^{111}\)

Radiation energy in KeV	Quantity	Experimental data	Theoretical data: type of radiation	Multipolarity \(l=1\)	Multipolarity \(l=2\)	Multipolarity \(l=3\)
173	\(w_K\)	\((7.1 \pm 0.8)\%\)	Electric Magnetic	2.7% 0.5%	14.3% 2.3%	62% 13.2%
173	\(\dfrac{w_K}{w_L}\)	\((8.0 \pm 2)\%\)	Electric Magnetic	7.9% 7%	5.7% 6.5%	2.7% 5.4%
247	\(w_K\)	\((3.0 \pm 0.5)\%\)	Electric Magnetic	0.9% —	3.6% 1%	11.9% 3%
247	\(\dfrac{w_K}{w_L}\)	\((5.5 \pm 1)\%\)	Electric Magnetic	7.9% 7.3%	6.4% 6.9%	3.9% 6.1%

transition is a mixed electric quadrupole and magnetic dipole transition, while the second is a pure electric quadrupole transition. This conclusion agrees with the values of the conversion coefficients \(W_K\) and \(W_L\), measured by the same authors by the coincidence method.

c) Measurement of the conversion coefficient by the coincidence method.

Let us have a \(\beta\)-active substance decaying according to the scheme shown in Fig. 4, and, for simplicity, let us assume that the nucleus—the product of \(\beta\)-decay—emits only one \(\gamma\)-quantum. Denote by \(W\) the sum of the conversion coefficients over all shells:

\[ W = W_K + W_L + W_M + \ldots \tag{27} \]

Let us further imagine that we register radiation with a β-counter and that the sum of the thicknesses of the specimen and the counter window is much smaller than the range of the least energetic conversion electron. In this case the counter will register all conversion electrons (even from the deepest layers of the source) emitted into the solid angle \(\omega_1\), determined by the mutual arrangement of the counter and the specimen. If \(N_0\) is the activity of the specimen, then the number of counter readings per unit time will be:

\[ N_p = N_0(1+W)\omega_1\varepsilon_1, \tag{28} \]

where \(\varepsilon_1\) is the efficiency of the β-counter. We shall now, with the aid of two β-counters and a selective electronic circuit, register coincidences of pulses caused by β-electrons and conversion electrons. Suppose that the mean lifetime of the excited state of the nucleus—the product of β-decay—is considerably less than the resolving power of the coincidence circuit. Then the number of coincidences is determined by the relation

\[ N_{\beta,e}=N_0W\omega_1\omega_2\varepsilon_1\varepsilon_2, \tag{28a} \]

where \(\omega_2\) is the solid angle subtended by the second counter, and \(\varepsilon_2\) is its efficiency. From (28) and (28a) we find:

\[ W=\frac{N_{\beta,e}}{N_p\omega_2\varepsilon_2-N_{\beta,e}}. \tag{29} \]

Since \(\varepsilon_1\) and \(\varepsilon_2\) are practically equal to unity, \(\omega_2\) can be determined by calculation, or by using standardized preparations. If the investigated γ-radiation arises as a result of a transition from a metastable state whose mean lifetime \((\tau)\) is comparable with the resolving power of the coincidence circuit \((dt)\), then in the right-hand side of (28a) there will appear the factor \((1-e^{-dt/\tau})\). \(\tau\) can be found by means of the delayed-coincidence method\({}^{32}\). Thus, by the method described, it can be determined

Radiation	\(\omega=\Omega/4\pi\): \((e,e')\)-coincidences	\(\omega=\Omega/4\pi\): \((e,\gamma)\)-coincidences	\(\omega=\Omega/4\pi\): \((\gamma\gamma)\)-coincidences
\(e^{113}\)	0.031	0.049	—
\(e^{207}\)	0.031	0.049	—
\(\gamma^{113}\)	—	—	0.074
\(\gamma^{246}\)	—	—	0.074

Fig. 9. Layout of the apparatus for measuring the coefficient of internal conversion of the \( \gamma \)-radiation of \( \mathrm{Cd}^{111} \).

total conversion coefficient. To measure the conversion coefficient on each of the shells, an additional investigation of the composition of the conversion radiation is required. This can be done either by isolating the conversion line with a spectrograph, with simultaneous registration of coincidences between the continuous-spectrum electrons and the conversion electrons forming the isolated line[^19], or by finding \(\dfrac{W_K}{W_L}\) from the areas of the corresponding conversion lines, or, finally, by means of absorption experiments. Let us consider, for example, the determination of the conversion coefficients of the above-mentioned \(\gamma\)-radiation of Cd\(^{111}\). In this case coincidences \((e,e)\), \((\gamma,e)\), \((\gamma,\gamma)\), caused by conversion electrons \((e)\) and \(\gamma\)-quanta emitted in both cascade transitions[^31], were investigated. The relations between the number of observed coincidences, unit counts, and the conversion coefficient are obtained in a manner completely analogous to the derivation of formula (29) given above.

A schematic drawing (to scale) of the apparatus used is shown in Fig. 9. The results obtained in this experiment are summarized in Table III. Directly, by means of the coincidence method, the total conversion coefficient \(W\) was found (see (27)). The conversion coefficients on the \(K\)- and \(L\)-shells were determined from the known \(\dfrac{W_K}{W_L}\), measured from the ratio of the areas of the conversion lines in the spectrum obtained with the aid of a magnetic spectrograph (the terms in (27) due to conversion on the \(M\)-shell and other outer shells may be neglected). For comparison, Table III gives theoretical data. As is seen from the table, the quantities \(\dfrac{w_K}{w_L}\) and \(w_K\) and \(w_L\) separately lead to the same conclusions about the multipolarity of the transitions, which confirms the correctness of the measurements.

c) Direct measurement of the ratio of the intensities of the radiations of conversion electrons and \(\gamma\)-quanta. This method was first used by Ellis and Aston (see[^34]) to measure the conversion coefficients of the \(\gamma\)-radiation of naturally radioactive elements. At the present time the method under consideration is almost not used; however, it may be useful in determining the conversion coefficients of \(\gamma\)-radiation emitted in a transition from a metastable state. The main drawback of the method is that, when it is used, it is necessary to determine the absolute intensity of the \(\gamma\)-radiation. Ellis and Aston measured the intensity of the \(\gamma\)-rays by the number of photoelectrons knocked out by the investigated \(\gamma\)-rays

from lead. The accuracy of the values thus obtained for the conversion coefficients does not exceed 20–30%, but errors of this order still make it possible to compare the experimental results with the theoretical data (see Fig. 10). A modification of the method described, reducing it to a comparison of the relative intensities of γ-lines and conversion lines, was applied by Alikhanov, Latyshev, and others (see\(^ {35}\)). Another acceptable way, in the present case, to measure the absolute intensity of monochromatic γ-rays consists in the use of γ-counters whose efficiency for counting γ-quanta with energy close to that under investigation is known (for example, from the measurement of \((\beta-\gamma)\)-coincidences, if the conversion coefficient of the γ-rays responsible for the coincidences has either been measured or is very small).

In the figure:
\(x\)—experimental points for the γ-radiation of \(Ra(B+C)\)
\(o\)—\(ThC''\)

Fig. 10.

From the material presented in this section it follows that the simplest and at the same time effective experimental methods for determining the multipolarity of γ-radiation, based on the use of internal conversion on atomic electrons, are measurements of the ratios of the conversion coefficients in the \(K\)- and \(L\)-shells and the measurement of conversion coefficients by means of coincidence counting. At the same time it should be noted that examples of accurately measured conversion coefficients are still few, and the existing technique is still far from perfect. Since the combination of the coincidence method with the use of β-spectrographs (for selecting the conversion line producing the coincidence) makes it possible to measure conversion coefficients in complex decay schemes, further progress in the accumulation of data on the multipolarity of the γ-radiation of nuclei is inseparably connected with the improvement of existing types of β-spectrographs, chiefly with an increase in the luminosity

and the development of precise experimental methods for its measurement. Recent work in this field (see, for example, \(^{36*}\)) allows one to hope for serious results in this direction in the near future.

II. INTERNAL CONVERSION WITH PAIR FORMATION

4. FEATURES OF PAIR CONVERSION

By internal conversion with pair formation is meant such a transition of the nucleus from an excited state in which all the energy goes into the formation of an electron–positron pair. Very often, for brevity, one speaks of the formation of a pair by a \(\gamma\)-quantum in the field of the nucleus that emitted this quantum. This expression is just as inaccurate as the interpretation of internal conversion on atomic electrons as photoelectric absorption.

In reality, just as in the process of internal conversion on atomic electrons, the \(\gamma\)-quantum is emitted and absorbed virtually. Everything said in Section 1 about the scheme for calculating the conversion coefficient is also valid for conversion with pair formation. The only difference is that the wave function of the initial state \(\psi_a\) in the present case describes an electron in some state of the continuous spectrum with total negative energy. In addition, the expression for the density of states (176) must also contain a factor depending on the momentum and energy of the positron. A preliminary estimate of the probability of pair conversion for electric dipole and quadrupole transitions was made by Nedelsky and Oppenheimer\(^{37}\). Already from these calculations certain characteristic features of pair conversion became clear, namely, the decrease of the conversion coefficient with increasing multipolarity of the \(\gamma\)-radiation and its increase with increasing transition energy. Thus, the coefficient of pair conversion displays a dependence on the multipolarity and transition energy precisely opposite to that which occurs in the case of conversion on atomic electrons. The calculations of Nedelsky and Oppenheimer were carried out in the Born approximation, i.e., under the assumption \(Z = 0\). Exact relativistic calculations with allowance for the Coulomb field of the nucleus for electric dipole and quadrupole radiation were performed by Hulme and Jaeger\(^{38}\), and for a magnetic dipole transition were very recently carried out by Wang\(^{39}\). The results

*) In the cited work a spectrograph with a magnetic lens is described, whose luminosity is 8% at a resolving power of 1.7%. In experiments using coincidences such resolving power is quite sufficient for a satisfactory measurement of conversion coefficients.

these calculations are shown in Fig. 11. Calculations of this kind are still more cumbersome than those for the coefficient of internal conversion on atomic electrons, since in the present case the initial and final states of the electron belong to a continuous spectrum. Fortunately, the coefficient of internal conversion with pair formation depends extremely weakly on the nuclear charge. In Fig. 11, for comparison, a curve of conversion coefficients calculated in the Born approximation is plotted. As is seen from Fig. 11, the exact results calculated for \(Z = 84\) and the Born curve \((Z = 0)\) differ by only \(15\)—\(20\%\). For smaller \(Z\) this difference will evidently be still smaller, and therefore the conversion coefficients calculated in the Born approximation may be used for comparison with experimental data on the subject of establishing the multipolarity of the transition.

Fig. 11a.

Fig. 11b.

As regards very heavy nuclei \((Z \sim 80)\), the weak dependence of the pair-conversion coefficient on \(Z\) makes it reasonable to try to take into account the influence of the Coulomb field of the nucleus by introducing a certain correction factor into the conversion coefficient calculated in the Born approximation. Such a factor, making it possible to obtain a result close to the exact one, was found by the author \(^{40}\) and has the form:

\[ K = \frac{4\pi^{2} n_{+} n_{-}} {\left(1 - e^{-2\pi n_{-}}\right)\left(e^{2\pi n_{+}} - 1\right)} , \tag{30} \]

where \(n_{+}\) and \(n_{-}\) are determined by equation (19), in which, instead of \(v\), the velocities of the electron and positron must be substituted. The factor \(K\) must be multiplied by the so-called differential conversion coefficient, calculated in the Born

INTERNAL CONVERSION OF γ-RAYS

an approximation giving the probability of formation of a conversion pair with a prescribed distribution of the transition energy between the electron and the positron.

Expressions for the differential coefficient of pair conversion in the general case of a transition of any multipolarity have been obtained by Berestetskii and Shmushkevich^41 and by the author^40 (for individual special cases see also^42, ^43, ^44). These formulas are as follows:

1) Electric \(2^l\)-pole:

\[ w_{\pi,l}^{\mathrm{el}}(k,E_+) \, dE_+ = \]

\[ = \frac{2\alpha}{\pi k^{2l-1}} \left\{(E_+E_-+1)I_{2l-1} +\frac{k}{4}(E_+-E_-)^2 I_{2l-3} -\frac{1}{4}I_{2l+1}+\right. \]

\[ \left. +\frac{e}{2k^3(l+1)} \left[(P_++P_-)^{2l-2} \left(E_+E_-+P_+P_- -1 -\frac{(E_++E_-)^2}{2(l-1)}\right)\right.\right. \]

\[ \left.\left. -(P_+-P_-)^{2l-2} \left(E_+E_- - P_+P_- -1 -\frac{(E_+-E_-)^{2l}}{2(l-1)}\right)\right]\right\}dE_+; \tag{31} \]

\[ I_{2l+1}= \frac{1}{4(l-1)} \left\{(P_+-P_-)^{2l-2}(E_+E_- - P_+P_- -1)-\right. \]

\[ \left. -(P_++P_-)^{2l-2}(E_+E_-+P_+P_- -1)+4lk^2 I_{2l-1}\right\}; \]

\[ I_1=\frac{P_+P_-}{2k^2}, \qquad I_3=\frac{P_+P_-}{2} -\ln \frac{E_+E_-+P_+P_-+1}{k}; \]

\[ l \geqslant 2; \qquad P_+=\sqrt{E_+^2-1}; \qquad P_-=\sqrt{E_-^2-1}. \]

For \(l=1\):

\[ w_{\pi,1}^{\mathrm{el}}(k,E_+) \, dE_+ = \frac{\alpha}{\pi k^3} \left\{(E_+^2+E_-^2) \ln \frac{E_+E_-+P_+P_-+1}{k} +2P_+P_-\right\}dE_+ . \tag{31a} \]

2) Magnetic \(2^l\)-pole:

\[ w_{\pi,l}^{\mathrm{m}}(k,E_+) \, dE_+ = \frac{\alpha}{2\pi k^{2l+1}} \left\{(E_+E_-+1)I_{2l+1} +\frac{k^3(E_+-E_-)^2}{4}I_{2l-1} -\frac{1}{4}I_{2l+3}\right\}dE_+; \tag{32} \]

\[ l \geqslant 1. \]

To obtain the full conversion coefficients, expressions (31) and (32) must be numerically integrated over the energy of the positron \((E_+)\) or of the electron \((E_-)\). In addition to the energy distribution of electrons and positrons of internal conversion, the angular distribution is also of interest, i.e. the probability

formation of a conversion pair with a specified opening angle \(\vartheta\) between the electron and the positron. The angular distribution can be obtained by numerical integration of an expression which, for example in the case of an electric multipole, has the form:

\[ w_{\pi,l}^{\mathrm{el}}(k,E_+,\vartheta)\,d\vartheta\,dE_+ = \]

\[ = \frac{2\alpha}{\pi k^{2l-1}} \left\{ \frac{q^{2l-2}}{k^2-q^2} \left[ E_+E_- + 1 + \frac{k^2(E_+ - E_-)^2}{4q^2} - \frac{q^2}{4} \right] +\right. \]

\[ \left. + \left(\frac{l}{l+1}\right) \frac{q^{2l-2}}{2k^2} \left[ 1-\frac{(E_+ - E_-)^2}{q^2} \right] \right\} P_+P_-\,dE_+\,\sin\vartheta\,d\vartheta, \tag{33} \]

\[ q^2 = P_+^2 + P_-^2 + 2P_+P_-\cos\vartheta . \]

Figure 12 shows, in the form of a polar vector diagram,

Fig. 12. Curve 1—electric dipole \(l=1\), curve 2—electric octupole \(l=3\), curve 3—magnetic quadrupole \(l=2\). \(k=5\). The momentum of the electron is directed along the ray \(\vartheta=0\). The probability of pair formation at \(\vartheta=90^\circ\) for all three curves is taken to be equal to 1.

the angular distribution for the cases \(l=1\) and \(3\) (electric radiation) and \(l=2\) (magnetic radiation). As is seen from Fig. 12, the angular distribution of the electrons and positrons of internal

conversion also depends on the multipolarity of the transition—with increasing \(l\), small values of \(\vartheta\) become predominant. This circumstance can also be used to establish the multipolarity of the transition.

It is easy to see that formulas (31)—(33) are symmetric with respect to the energies and momenta of the positron and the electron. This is quite natural, since they were derived under the assumption \(Z=0\), as a result of which the opposite signs of the charges of the electron and positron have no effect on the result. Taking account of the nuclear charge\({}^{30}\) strongly distorts, for example, the energy distribution of electrons and positrons given by formulas (31) and (32). The energy distribution of the electrons and positrons of internal conversion for \(Z=0\) and for \(Z=84\) is shown in Fig. 13. We see that in the latter case the most probable event is the birth of a positron with energy close to the maximum and of an electron with zero kinetic energy. This form of the energy distribution is due to the fact that, owing to the positive sign of the nuclear charge, the positron is accelerated, while the electron is slowed down by the Coulomb field.

Fig. 13. \(a\)—Born approximation \((l=2)\), \(b\)—\(Z=84\) \((l=2)\).

Up to now we have considered conversion transitions with pair formation, in which the electron born in the conversion process is in a state of the continuous spectrum. However, if there is a vacant level in the electron shell of the atom, then the nucleus may undergo such a conversion transition with pair formation in which the electron born occupies the vacant place in the shell, while the positron carries away all the remaining energy. The energy of the emitted positron will then be equal to:

\[ E_{+}=k-1+P_i \qquad (i=K,\ L,\ M\ \text{and so on}), \tag{34} \]

where \(P_i\) is the ionization potential of the corresponding shell. Thus, depending on which of the shells has a vacant place, the emitted positron will have different energy. Consequently, a discrete spectrum of positrons should be observed, corresponding to the “capture” of electrons into the \(K\)-, \(L\)-, \(M\)-, etc. shells. If it is assumed that there are unoccupied levels in the shell, then the calculation of the probability of formation of a monochromatic group of positrons is carried out in exactly the same way as the calculation of the “ordinary” conversion coefficient with pair formation, with the only difference that, as the wave function

as the final-state electron function \((\psi_b)\), one should choose a wave function of one of the states of the discrete spectrum. Calculations of this kind were carried out by L. A. Sliv\(^{45}\), who was the first to consider this effect theoretically. The results obtained by him reduce to the following (for \(k = 2.74\)):

\[ \eta_{K,l}=\frac{w^{K}_{p,l}}{w_{\pi,l}}= \begin{cases} 1 & \text{for transitions of the type }0\to 0,\\ 1/3 & \text{for an electric dipole,} \end{cases} \tag{35} \]

where \(w^{K}_{p,l}\) is the probability of a conversion transition with the electron set down into a vacant place in the \(K\)-shell, and \(w_{\pi,l}\) is the probability of an “ordinary” conversion transition with pair formation. The values of the ratio (35) for all other multipole transitions will lie between 1 and \(1/3\). As the transition energy decreases, \(\eta_{K,l}\) increases, reaching its greatest value at an energy close to 2. The ratio (35) cannot be measured directly in experiment. This is due to two reasons. The first of them consists in the fact that, generally speaking, there may be no vacant states in the electron shell of an atom. Several processes can be indicated that lead to the formation of vacant places in the shell. Such processes include self-ionization of the atom in \(\alpha\)- and \(\beta\)-decays\(^{46,47}\) and internal conversion on atomic electrons. Thus, if the nuclear transition of interest to us occurs after \(\alpha\)- or \(\beta\)-decay, or if the decay scheme shown in Fig. 4 (cascade \(\gamma\)-transitions) takes place, there will always be some probability \(\gamma_i\) of the formation of vacant places in the electron shell. However, even if, for an excited nucleus, a vacant level has formed in the shell, it may be filled not only as a result of the process under consideration, but also by the transition of an electron from a higher level to this one with emission of an optical or x-ray quantum, or by means of the Auger effect. Thus, other possible paths for filling vacant places in the shells will compete with filling as a result of the conversion transition of interest to us. Therefore the probability that the vacant level will be filled by an electron of the conversion pair will be equal to the ratio of the width of the excited nuclear level \(\Gamma_{\gamma}\) to the total width of the vacant shell level \(\Gamma_i\). Consequently, for the probability of emission of a monochromatic positron we have:

\[ W^{i}_{p,l}=w^{i}_{p,l}\gamma_i\frac{\Gamma_{\gamma}}{\Gamma_i} \tag{36} \]

or

\[ \xi_{i,l}=\frac{W^{i}_{p,l}}{w_{\pi,l}}=\eta_{i,l}\gamma_i\frac{\Gamma_{\gamma}}{\Gamma_i}. \tag{36a} \]

Monochromatic positrons of internal conversion were first observed by Latyshev, Gei, Bashilov, and Barchuk\(^{48}\), whose results are presented below.

5. MAIN EXPERIMENTAL RESULTS

The phenomenon of internal conversion with pair formation was first observed experimentally by Alikhanov and Kozodaev[^49], who discovered the presence of a small number of positrons in the radiation of naturally radioactive elements. Subsequently Alikhanov, Alikhanyan, Kozodaev, Latyshev, Dzhelepov, Spivak, and others used internal conversion with pair formation for the spectroscopy of the $\gamma$-radiation of RaC and Th $(C + C'')$. The methodology of these experiments and their results are discussed in detail in the review article by Latyshev[^35]. Therefore we shall confine ourselves here to only brief remarks.

Fig. 14. Spectrum of positrons of the internal conversion of Th $(C + C'')$. The dotted lines show the theoretical curves.

Table IV gives the results of an investigation of the $\gamma$-spectrum of Th $(C + C'')$, carried out by means of analysis of the spectra of Compton electrons and conversion electrons (see Fig. 14). It has been established—

Table IV

Transition energy (in KeV)	Relative intensities of $\gamma$-lines from the spectrum of Compton electrons	Relative intensities of $\gamma$-lines from the spectrum of positrons of internal conversion	Ratio of the quantities in the third and second columns	Multipolarity of the transition*)
1350	3,6	10,0	2,77	Dipole
1500	3,7	6,5	1,7	”
1600	10,0	11,0	1,1	Quadrupole
1800	6,2	6,5	1,05	”
2200	5,05	10,0	2	”
2620	100	100	1,00	”

*) The transition with energy 2620 KeV is considered quadrupole.

tion of transition multipolarities was carried out by measuring and comparing the relative intensities of the \(\gamma\)-lines from recoil electrons and from the positron spectra of internal conversion. If the multipolarity of any one spectral line is known, then in this way the multipolarities of the remaining transitions can be determined. In the case of Th \((C+C'')\), for example, it follows from Alikhanov’s measurements\(^{50}\) that the transition with energy \(2.62\ \mathrm{MeV}\) is quadrupole (established from measurement of the pair-conversion coefficient, which was found to be \((4—5)\cdot 10^{-2}\%\); the theoretical value according to Hulme and Eger\(^{38}\) is \(4.6\cdot 10^{-2}\%\)*). On the basis of this result, one may, by means of the method described above, determine the conversion coefficients of the remaining \(\gamma\)-lines and compare them with theory (see Fig. 15). In an analogous manner the \(\gamma\)-spectrum of RaC′ was investigated (Table V). In Fig. 16 are shown portions of the positron spectrum of internal conversion of the \(\gamma\)-radiation of RaC′\(^{48}\). The graph clearly shows the peaks of monochromatic conversion positrons formed in transitions with energies \(1527\ \mathrm{KeV}\) (\(K\)- and \(L\)-lines) and \(1620\ \mathrm{KeV}\) (\(M\)-line). The authors of work\(^{48}\) measured the quantities \(\xi_{K,l}\) and \(\xi_{L,l}\) for the transition with energy \(1527\ \mathrm{KeV}\), relative to the area occupied by the monochromatic peak, to the area bounded by the contour of the continuous distribution of positrons produced in this transition. The results of the measurements are as follows:

Fig. 15. Pair-conversion coefficients of the \(\gamma\)-radiation of Th \((C+C'')\). The point corresponding to the \(2620\ \mathrm{KeV}\) \(\gamma\)-line is the reference point.

\[ \xi_{K,l}^{1527}=3\cdot 10^{-3}, \qquad \xi_{L,l}^{1527}=5\cdot 10^{-3}. \tag{37} \]

According to Latyshev’s data\(^{51}\), the \(1527\ \mathrm{KeV}\) transition is electric quadrupole. For this case, according to Sliv’s calculations\(^{45}\),

\[ \eta_{K,l}^{1527}=0.5. \tag{37a} \]

* See also \(^{34}\).

With the large number of γ-transitions that are observed in the RaC′ nucleus, it may be assumed that the probability of ionization of the \(K\)-shell \(\gamma_K\) is close to unity. The width \(\Gamma_K\) can be estimated on the basis of the theoretical and experimental data available for Au \((Z = 79)\). \(\Gamma_K\) turns out to be equal to 60 eV. Similar calculations

Fig. 16. Portions of the spectrum of positrons from internal conversion of RaC′.

can also be carried out for the \(L\)-line. In this way, with the aid of (36a), one obtains:

\[ \Gamma_\gamma^{1527} \simeq (0.4—0.5)\ \text{eV}. \tag{38} \]

From the brief survey given above of the principal experimental results on pair conversion it follows that the use of internal conversion with pair formation, owing to the experimental work of a group of Soviet physicists, has already led to the accumulation of a number of substantial results in the field of γ-radiation spectroscopy of naturally radioactive elements. At the same time

It must be noted, however, that in view of the comparatively weak dependence of the probability of pair conversion on the multipolarity of the transition, measurements of the internal-conversion coefficients with pair formation must be carried out with an accuracy exceeding

Table V

Transition energy (in KeV)	Relative intensities of γ-lines: from the spectrum of Compton electrons	Relative intensities of γ-lines: from the spectrum of positrons of internal conversion	Ratio of the quantities from the third and second columns	Multipolarity of the transition: on the basis of the data of the second and third columns	Multipolarity of the transition: on the basis of the data of the second column and the relative intensities of the lines in the spectrum of K-conversion electrons
2420	0.34	0.36	1.06	Quadrupole
2200 }	1.00	1.00	1.00	Quadrupole	Quadrupole
2090 }					Quadrupole
1820	0.32	0.30	0.94	Quadrupole
1761 }
1690 }	2.51	2.45	0.97	Quadrupole	Quadrupole
1620 }					Quadrupole
1527	0.56	0.52	0.93	Quadrupole
1370 }	1.32	1.19	0.90	Quadrupole	Quadrupole
1290 }
1234	0.41	0.20	0.49	Quadrupole	Quadrupole
1120	1.76				Quadrupole

Curly brackets in the first column denote groups of lines in the spectrum of Compton electrons whose areas cannot be completely separated.

the accuracy of measurements of the conversion coefficient on atomic electrons. For example, some transitions of the nuclei RaC′, ThC, ThC″, which are at present regarded as electric quadrupole transitions, may also be assigned to magnetic dipole transitions because of excessively large errors in the conversion coefficient (20%–25%).

As is seen from Fig. 14, determination of the transition energy from the spectrum of conversion electrons is possible only thanks to the break-like form of the line (see Fig. 13, curve b). In the case of light nuclei such a break will not be observed; the spectrum will have a form close to that shown in Fig. 13, a. This circumstance will make it difficult to separate the lines and to measure the intensities of transitions from the positron spectrum. However, among light artificial-

in radioactive isotopes cases are rarely encountered when the number of \(\gamma\)-transitions with energy sufficient for pair production exceeds 1 or 2. Unfortunately, measurements of the coefficients of pair conversion in light nuclei have so far not been made by anyone, while precisely for small values of \(Z\) the probability of pair conversion may exceed the probability of conversion on atomic electrons, which is proportional to \(Z^3\) and decreases with transition energy. It must be assumed that the phenomenon of pair conversion can be successfully used to determine the multipolarity of \(\gamma\)-transitions of light nuclei, where the coefficient of conversion on atomic electrons is very small and therefore difficult to measure (for example, for \(Z\sim 10\), \(k\sim 5\), \(w_K\sim 10^{-4}\%\), whereas \(w_\pi\sim 10^{-2}\%\)).

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NOTE ADDED IN PROOF

After the present article had gone to press, a communication appeared by Rose, Goertzel, and Spinrad (Phys. Rev. 76, 1883, 1949) on their exact relativistic numerical calculations (with the aid of computing machines) of conversion coefficients in the \(K\)-shell for 23 values of \(Z\) (from \(Z=10\) to \(Z=96\)), for the first five electric and magnetic multipoles, and for 16 values of \(k\) (from 0.3 to 5.0). The results of the calculations are not given in the note.

It follows from the calculations that the range of applicability of the formulas for the conversion coefficient in the \(K\)-shell obtained in the nonrelativistic approximation is much narrower than had been supposed. Even for \(Z \sim 50\) and \(k \sim 0.3\), the values of \(w_K\) given by formulas (20) and (23) may differ from the exact ones by factors of 2 and 3. The reasons for so large a discrepancy are not entirely clear, and this question, as well as the specification of the region of applicability of the nonrelativistic formulas, requires further investigation.