ISOMERISM OF ATOMIC NUCLEI1

L. I. Rusinov, Leningrad

INTRODUCTION

In recent years, as a result of a number of investigations, it has been shown that, alongside isotopes and isobars, there exist nuclear isomers, i.e., nuclei having identical atomic numbers and mass numbers, but possessing certain differences in physical properties.

It is interesting to note that Soddy[^2], as early as 1917, while developing the question of the isotopic composition of the elements, pointed out the possibility of the existence of nuclei of this kind. In subsequent years Meitner[^3] attempted experimentally to detect representatives of “isotopy of a higher order” in the decay products of naturally radioactive elements, but reached no definite conclusions.

Only in 1921 did Hahn’s investigations[^4] first show the presence of higher-order isotopes, or, as they subsequently came to be called, isomers, in a series of radioactive transformations of uranium ($UX_2$ and $UZ$). For many years the so-called “uranium fork” was the only known example of the phenomenon of nuclear isomerism.

In 1935 Kurchatov, Rusinov, and others[^5] discovered a new period of artificially radioactive bromine. Subsequently, the result of this investigation served as the basis for establishing nuclear isomerism in one of the isotopes of this element. The discovery of bromine isomerism gave impetus to the search for other examples of nuclear isomerism among artificially radioactive nuclei.

A considerable increase in the number of cases of nuclear isomerism became possible thanks to the remarkable experiments of Goldhaber, Hill, and Szilard[^6], who in 1938 showed for the first time (using indium as an example) that nuclear isomerism can also exist in stable nuclei.

The presence of nuclear isomerism in a comparatively large number of artificially radioactive elements gave experimenters new objects of investigation and opened broad possibilities for studying this new phenomenon.

By the present time a large body of experimental material on nuclear isomerism has been accumulated, and certain theoretical—

theoretical notions about its physical nature. It is therefore of interest to give a review of the current state of the question of the isomerism of atomic nuclei.

1. METHODS OF ESTABLISHING NUCLEAR ISOMERISM

Usually the phenomenon of nuclear isomerism is established by the difference in the radioactive properties of isomeric nuclei. As is known, the decay period of a given radioactive nucleus is its characteristic individual constant. However, among nuclear reactions there are cases in which two different periods have to be assigned to one and the same unstable isotope, and in this case the phenomenon of nuclear isomerism is ascertained.

Isomerism in stable nuclei is established from the fact that radioactive properties have to be assigned to a nucleus having the same composition as one of the stable isotopes of the given element.

The first indication of the presence of nuclear isomerism is most often the circumstance that the number of radioactive periods for a given element exceeds the number of its possible radioactive isotopes. This method of ascertaining nuclear isomerism is especially suitable for elements with an odd atomic number, which possess a small number of stable isotopes. Obviously, in this way one can establish only the fact of the existence of nuclear isomerism in a given element, but there is no possibility of determining in precisely which isotope the isomeric state occurs.

To assign a definite mass number to a given isomeric pair, the so-called method of crossed nuclear reactions is used. It consists in attempting to obtain one and the same radioactive period as the result of different nuclear reactions, by irradiating the element under investigation and the elements neighboring it with activating particles of various kinds. (Examples are given in the last column of Table 1.) It should be noted that even when this method is used, owing to the presence of a large number of neighboring isotopes and for other reasons, great difficulties often arise both in establishing the very fact of nuclear isomerism and in assigning mass numbers to the isomers. To this day there remain a number of cases of isomerism in which the mass number of the nuclear pair has not yet been definitively established, for example, \( \mathrm{Ag}_{47}^{107} \) or \( \mathrm{Ag}_{47}^{109} \), \( \mathrm{Ma}_{43}^{99} \) or \( \mathrm{Ma}_{43}^{101} \), etc.

In recent times great successes have been achieved in the field of isotope separation. Work with separated isotopes should provide great conveniences for the study of many nuclear transformations and, in particular, for the investigation under pure conditions of questions of nuclear isomerism. Experiments performed by Reddemann\(^6\) with a separated isotope of strontium under natural conditions have already made it possible to obtain a number of new data on the isomer \( \mathrm{Sr}_{38}^{87} \).

The method of establishing the isomerism of atomic nuclei from radioactive properties is very sensitive, but it is not the only possible one.

According to existing ideas about the nature of nuclear isomerism (which will be considered below), one of two isomeric nuclei is in an excited state. Therefore two isomeric nuclei, with equal atomic numbers and mass numbers, nevertheless differ somewhat in their atomic weights; they must also differ in their mechanical and magnetic moments, etc. At present, however, we do not yet have experimental possibilities that would allow the fact of nuclear isomerism to be established from differences of this kind.

2. THE DISTRIBUTION OF NUCLEAR ISOMERISM AMONG THE ELEMENTS OF THE PERIODIC SYSTEM

Table 1 presents all the cases of nuclear isomerism known up to the present time and established with greater or lesser certainty.

The isomeric state of nuclei is encountered throughout the entire periodic system of the elements—from nuclei with atomic number 20 to nuclei with atomic number 91. Among the known cases of isomerism, the decay periods of the excited nucleus vary from several tens of seconds (40 sec. for \({}_{47}\mathrm{Ag}^{107,109}\)) to several months (90 days for \({}_{52}\mathrm{Te}^{127}\)). The excitation energies of isomeric nuclei have values from \(\sim 50\ \mathrm{KeV}\) (in the cases of bromine and krypton) to \(\sim 500\ \mathrm{KeV}\) (in the cases of strontium and zinc).

In this connection it should be noted that the number of nuclear isomers discovered at the present time is, to a certain extent, accidental in comparison with the actual prevalence of this phenomenon. Naturally, it was primarily those cases of nuclear isomerism that were discovered in which the magnitudes of the decay periods and the magnitudes of the excitation energies proved convenient for experimental observation. The table of isomers presented here, in the course of further investigation of nuclear isomerism, is continually being expanded and supplemented with new examples.

One may suppose that there also exist in nature such isomeric nuclei as are practically stable. If the excitation energy of such a nucleus amounts to several kilovolts (processes of internal conversion on the \(K\)- and \(L\)-shells cannot occur in this case), then, in the presence of even a not very large difference in angular momenta between the excited and ground states, this nucleus will have a very long lifetime, for example of the order of \(10^{10}\) years. To detect such isomers experimentally is very difficult and is possible only by weakly intensive \(\gamma\)-radiation or by the fact that, for a given stable nucleus, two different mechanical moments will be established, or several slightly different atomic weights will be found.

The multitude of cases of nuclear isomerism already known at present definitely shows that nuclear isomerism is not a rare exception but, on the contrary, is a phenomenon highly characteristic of atomic nuclei.

3. THEORETICAL CONCEPTIONS OF THE NATURE OF NUCLEAR ISOMERISM

The first attempt at a theoretical explanation of nuclear isomerism belongs to Gamow. In 1934 Gamow⁷ put forward an idea explaining the then-known cases of protactinium isomerism and the presumed isomerism of \(\mathrm{Pb}_{82}^{210}\) by the existence of a metastable excited state of the nuclei of these elements. The cause of the metastability, according to Gamow, lay in the existence in certain nuclei of hypothetical particles—antiprotons.

In 1936 Weizsäcker⁸, developing Bohr’s ideas, proposed a new theory of nuclear isomerism, which proved to be very fruitful. According to this theory, isomerism is likewise explained by the existence of a metastable excited state of nuclei. The cause of the metastability, however, consists in the large difference between the angular momenta corresponding to the excited and ground states of the given nucleus. Therefore, by virtue of the quantum-mechanical selection rules, the radiative transition of one isomer into the other becomes unlikely.

The calculation carried out by Weizsäcker, which considered the radiating system—the nucleus—according to the “single-particle model,” showed that the probability of transition of a slightly excited state of the nucleus to the ground state by \(\gamma\)-emission will be very small when there is a large difference between the angular momenta corresponding to these states.

Subsequently Bethe⁹ performed an analogous calculation, considering nuclear radiation in a more general form, and obtained results qualitatively coinciding with Weizsäcker’s.

The lifetimes of nuclei obtained according to these calculations may have the same order of magnitude as the periods of decay of isomers observed experimentally.

Recently, in 1940, Zaks¹⁰ proposed a new hypothesis which, possibly, in a number of cases could explain the phenomenon of nuclear isomerism. According to Zaks, one of the isomeric nuclei is also in a metastable state, but the cause of the metastability is that the two isomeric nuclei have zero angular momentum and opposite parity. In this case a \(\gamma\)-transition, as well as an ordinary conversion transition, will be forbidden. De-excitation of the metastable nucleus is then possible only by the simultaneous emission of two photons or two conversion electrons. This process will have a low probability and, as Zaks’s estimates show, can likewise explain the lifetimes of isomeric nuclei observed experimentally.

The difference between the hypotheses of Weizsäcker and Zaks will show up experimentally in the energy distribution of the \(\gamma\)-quanta or conversion electrons emitted in the isomeric transition. According to the first theory, the spectrum of photons or conversion electrons should be line-like, whereas according to the second it should be continuous.

The mechanism of nuclear isomerism proposed by Zaks does not exclude Weizsäcker’s mechanism, and it may be assumed that in some cases it occurs.

4. NUCLEAR ISOMERISM AND INTERNAL CONVERSION

A consequence of the Bohr–Weizsäcker theory of the nature of nuclear isomerism set forth above should be the existence of a new type of radioactive decay—independent γ-radioactivity with a relatively long mean lifetime, associated with the transition of a metastable nucleus to the ground state.

In the study of nuclear isomerism, for a long time it was not possible to detect γ-radiation of this kind from isomeric nuclei.

In 1938, Rusinov and Yuzefovich¹¹, in the laboratory of Prof. I. V. Kurchatov, using isomeric bromine as an example, and also, independently, Pontecorvo¹² using radium as an example, discovered conversion radiation of isomers. These experiments explained the reason for the difficulties with which the detection of γ-radioactivity in nuclear isomerism had been connected. It turned out that they were due to the nature of the phenomenon, namely, to the fact that the discharge of the metastable level proceeds mainly by processes of internal conversion.

The results of these works for the first time provided, to a certain extent, experimental proof of the validity of the Bohr–Weizsäcker ideas on the nature of nuclear isomerism.

It was very significant that these data were in qualitative agreement with the theoretical calculations by Hebb and Uhlenbeck¹³ that appeared at that time, according to which the discharge of a nucleus at low excitation energies and a large difference of angular momenta proceeds predominantly by internal conversion.

The conversion radiation of isomers proved to be a very valuable instrument for elucidating many details of isomeric transformations.

Study of the spectrum of conversion electrons made it possible, for most isomers, to determine the magnitude of the energy difference between the metastable and ground levels of isomeric nuclei. We note that in all cases of isomerism known up to now, conversion electrons are observed that have a line spectrum corresponding to conversion on the \(K\)-, \(L\)-, and also \(M\)-shells of the atom. This fact indicates that so far there are no examples of nuclear isomerism agreeing with Sachs’s theory.

Further, study of the decrease in intensity of conversion radiation makes it possible to determine which of the two radioactive periods belongs to the excited isomer.

Moreover, as a result of investigating the characteristic X-radiation¹⁴ accompanying the processes of internal conversion, it is possible to determine the atomic number of the metastable nucleus and to clarify the scheme of isomeric transitions. For this purpose, the ingenious method of chemical separation proposed by Segrè et al.¹⁵, in which the recoil of the atom produced upon emission of a conversion electron is used, also proved very valuable. In a number of cases this method makes it possible to prove that a scheme of “successive decay” of isomers takes place (see Section 6).

But the most essential result of the investigation of the conversion radiation of isomers is the possibility of quantitative co-

comparison of experimental data on internal-conversion processes with theoretical calculations. This comparison, as will be shown below, makes it possible to determine the magnitude of the difference of angular momenta between the metastable and the ground states of an isomeric nucleus and thereby fully to verify the validity of existing theoretical notions concerning the nature of nuclear isomerism.

5. MECHANISM OF FORMATION OF ISOMERIC NUCLEI

All known nuclear isomers are obtained as the result of one or another nuclear reaction, i.e. they belong to artificially radioactive nuclei. An exception is the isomerism of protactinium, which is formed as the result of the decay of naturally radioactive uranium.

Let us consider several cases of the formation of isomeric nuclei. A compound nucleus obtained, for example, as a result of a reaction with slow neutrons can “de-excite” by cascade emission of γ-quanta not only to the ground state, but also to some metastable state. In this case two isomeric nuclei are formed. The relative probability of the transition of the compound nucleus to the metastable or the ground level is determined by the disposition of the intermediate energy levels, the type of radiative transitions, and the selection rules for these transitions.

One may note an interesting example of the manifestation of these selection rules. The metastable nucleus \({}^{114}_{49}\mathrm{In}\) (\(T = 50\) days) is readily formed in the reaction \((n,\gamma)\); however, it is not possible to obtain the same nucleus \({}^{114}_{49}\mathrm{In}\) in the ground state (\(T = 72\) sec.) by the same reaction, despite special experiments carried out for this purpose\(^{16}\). In this case the selection rules lead to the fact that the compound nucleus practically always de-excites to the metastable level.

In nuclear reactions with various kinds of fast particles, the compound nucleus obtained may have different excitation energies and an angular momentum very different from that of the initial nucleus. In these reactions the relative probability of the transition of the compound nucleus into the metastable and ground states changes rather strongly depending on the particle energy and the type of reaction. For example, the relative probability \(\xi\) of formation of a metastable bromine nucleus upon capture of thermal neutrons is \(0.5^{17}\); in a reaction with photons having an energy of 12 MeV, \(\xi = 0.4\); with 17 MeV photons \(\xi = 1.0^{18}\); and under the action of fast neutrons with energies up to 14 MeV \(\xi = 2.0^{19}\). In the case of rhodium isomerism, for a reaction with thermal neutrons \(\xi = 0.1\), while for a reaction with fast neutrons from an \((\mathrm{Rn}+\mathrm{Be})\) source \(\xi = 0.2^{20}\).

The facts set forth are in complete agreement with the existing ideas of Bohr’s statistical nuclear theory.

In the formation of an isomeric state of stable nuclei, a feature is revealed that is connected with the circumstance that the probability of excitation of a stable nucleus directly to a metastable level is very small. The isomeric state of a stable nucleus is practically always obtained as a result of a reaction with neighboring elemen-

by beta rays, or by exciting the original stable nucleus (with the aid of hard X-rays or processes of inelastic scattering of heavy particles) to a higher activation level \(^{5,21}\). This level is then discharged by \(\gamma\)-emission, and there is some probability that part of the transitions will lead to the formation of metastable nuclei.

By exciting the nucleus \(\mathrm{In}_{49}^{115}\) with hard X-ray quanta, Waldman et al. \(^{21}\), as well as Korsunsky et al. \(^{22}\), determined the energies of the activation levels for this element.

It is possible that in a number of cases the isomeric state of stable nuclei under bombardment by charged particles is formed as a result of the action of the electric field of these particles flying near the nucleus \(^{23}\).

On the basis of an analysis of all cases of formation of nuclear isomers known so far (the data given in Table 1), the following general conclusion may be drawn. All types of nuclear reactions, insofar as they lead to the occurrence of excited states of the nucleus, prove in one case or another to be suitable also for the formation of nuclear isomers. To these reactions should be added those cases in which excitation of nuclei is obtained as a result of \(\beta\)-decay, \(K\)-capture, and fission of heavy nuclei.

All the experimental material on the mechanism of isomer formation is in qualitatively good agreement with the picture that regards isomeric nuclei as two nuclei situated in different energy states, one of which is metastable.

In principle, it is possible, by investigating the mechanism of isomer formation, to determine the difference of angular momenta for nuclei in the metastable and ground states. For example, the angular momentum of a compound nucleus formed as a result of an \((n,\gamma)\) reaction will differ by \(1/2\) from the moment of the initial nucleus. If, further, it is possible to ascertain the types of cascade \(\gamma\)-transitions of the compound nucleus, then it may be assumed that in a number of cases it will be possible to determine the difference \(\Delta l\) of the angular momenta corresponding to the two isomeric nuclei. Similarly, by studying the processes of excitation of stable nuclei by \(\gamma\)-quanta to the activation level and the subsequent radiative transitions of the nucleus to the metastable level, it will apparently also be possible to determine the value of \(\Delta l\) for particular isomers. However, from the experimental material presently available on the mechanism of isomer formation, it is not yet possible to draw any quantitative conclusions on the question of the magnitude of \(\Delta l\).

6. DECAY SCHEMES OF THE METASTABLE STATE

In the preceding section the question was considered of how metastable states of nuclei are formed. We shall now indicate the transformations undergone by a nucleus that is in a metastable state.

Two schemes of discharge of a nucleus from a metastable level may be proposed: “successive” and “independent” decay. Both of these schemes are encountered experimentally.

a) Scheme of successive decay

In a number of cases a metastable nucleus, by means of $\gamma$-emission or conversion radiation, passes into the ground state, and then $\beta$-decay occurs only from the ground level. One of the two $\beta$-decay periods observed experimentally is due to the fact that the $\beta$-radiation follows the isomeric transition (Fig. 1, a): as is known, if the period of this transition is much longer than the period of the $\beta$-radiation, then two periods of $\beta$-decay will be observed. Naturally, the limits and shapes of the $\beta$-spectra corresponding to these periods will be identical.

Fig. 1

It should be noted that in those cases where the period of the isomeric transition is shorter than the period of $\beta$-decay of the nucleus from the ground state, establishing the fact of nuclear isomerism from the difference in the $\beta$-decay constants is practically impossible. In this case the fact of isomerism can be established by detecting the $\gamma$- or conversion radiation corresponding to the isomeric transition.

In the majority of known examples of nuclear isomerism, the discharge of the metastable state occurs according to the scheme of successive decay (see the isomers in Table 1 for which one of the $\beta$-decay periods is indicated in parentheses). In these cases, by using the spectrum of conversion electrons, it is possible to determine the energy difference for the two isomeric nuclei. The decay period of the excited isomer into the ground one can readily be determined from data on the decrease in the intensity of the conversion radiation.

The group of isomers under consideration also includes the known cases of isomerism of stable nuclei, in which the discharge of the metastable level likewise takes place by means of $\gamma$-emission or a conversion transition (see in Table 1 the nuclei marked with an asterisk).

b) Scheme of independent decay

In this type of transition, the discharge of the metastable nucleus occurs by $\beta$-decay to an excited level of the neighboring nucleus. Only a few cases of nuclear isomerism are known in which decay proceeds according to this scheme: $\mathrm{Mn}^{52}_{25}$, $\mathrm{Ag}^{106}_{47}$, $\mathrm{In}^{116}_{49}$ and, possibly, $\mathrm{Pa}^{234}_{91}$.

It is obvious that, in the scheme of independent decay, the shapes and limits of the $\beta$-spectra corresponding to the two isomeric periods will, generally speaking, be different (Fig. 1, b). This type of decay has been little studied. In the case of such a scheme it is not possible to establish which of the decay periods belongs to the metastable nucleus, and it is also not yet possible to determine the magnitude of the excitation energy of this nucleus.

We note that the statement by Pool et al. $^{24}$ on the existence of positron–electron isomerism of $\mathrm{Ag}^{106}_{47}$ is erroneous. In the present case,

apparently, a decay scheme with the emission of positrons by both isomeric nuclei takes place.

The occurrence, in one case or another, of a scheme of successive or independent decay is evidently determined by which of the discharge processes of the metastable nucleus has the greater probability—whether the transition of the metastable nucleus by means of γ-radiation or conversion radiation to the ground level of the same nucleus, or the transition of the metastable nucleus by means of β-decay to a suitable level of the neighboring nucleus.

Among the known isomers, not a single case has been found in which β-decay of each of the two isomeric nuclei directly to the ground level of the neighboring nucleus would be observed (Fig. 1, c). This fact is in full agreement with the views connecting nuclear isomerism with a large difference in angular momenta between the metastable and the ground state. Indeed, if such a difference exists, then decay according to the scheme of Fig. 1, c would be associated with the transfer of a large moment to the β-electron of one of the periods. The probability of β-decay with a large change of angular momentum is, however, very small.

7. EXPERIMENTAL TEST OF THE THEORY OF THE METASTABLE STATE OF NUCLEI

All the experimental material considered above on the question of the isomerism of atomic nuclei shows that two isomeric nuclei differ from one another in their energy states, of which one is metastable.

For a complete proof of the Bohr–Weizsäcker ideas on the nature of nuclear isomerism, it is necessary to confirm their hypothesis on the cause of the metastability of nuclei, i.e. to show that the small probability of transition of nuclei from the metastable state to the ground state is due to the large difference of the angular momenta corresponding to these states. This point of the theory has remained open until recently. It can be checked by direct experiments on the study of the hyperfine structure of the optical spectra of isomers, by direct measurement of mechanical moments in Stern–Gerlach experiments, etc. But at present we still do not possess sufficiently intense radioactive sources for experiments of this kind.

It is possible, however, to try to test this aspect of the theory by studying the processes of internal conversion in nuclear isomerism.

The change \(\Delta l\) of angular momentum in the transition of a nucleus from the metastable state to the ground state can be determined from the order of the multipolarity of the radiation associated with this transition. The quantities of the internal-conversion coefficients depend, to a large extent, on the order of multipolarity of the radiation, as follows from the theory of internal conversion.

It is essential to note here that the theory gives values for the probabilities of internal-conversion processes that do not depend on a particular model of the nucleus. Therefore, the determination of the change of angular momentum

in an isomeric transition, when using this method, preference should undoubtedly be given to the one in which the theoretically derived relation between the mean lifetime of the metastable nucleus and the value \(\Delta l\) is used.

Theoretical calculations carried out by Hebb and Uhlenbeck \(^{13}\) and by Dancoff and Morrison \(^{25}\) make it possible in a number of cases to calculate the coefficients of internal conversion on the \(K\)-shells of atoms. Zavelevich’s calculations \(^{26}\) make it possible also to compute the relative conversion coefficients (the ratio of conversion on the \(L\)- and \(K\)-levels of the atom). The results of the calculations show that the total and relative internal-conversion coefficients, especially at small excitation energies, increase very rapidly with increasing order of the multipolarity of the radiation. Since there is such a sharp dependence, it should be assumed that in a number of cases, by comparing the internal-conversion coefficients obtained from calculation with experimental data, one can uniquely determine the order of the multipolarity of the radiation and the corresponding change in the angular momentum of the nucleus.

We note that at small excitation energies (\(\sim 100\) KeV) it is more convenient to determine the order of the multipolarity of the radiation from the relative conversion coefficient, since the total coefficient already at the first orders of the multipolarity of the radiation practically reaches unity. For higher excitation energies of isomers, determination of the order of the multipolarity of the radiation is possible both from the total and from the relative conversion coefficients. For isomers with still higher excitation energy, establishing the order of the multipolarity of the radiation is possible only from the total conversion coefficient, since the relative coefficient in this case, apparently, depends only to a small degree on the order of the multipolarity of the radiation.

Everything said above pertains to radiation of electric multipoles. However, it may be supposed that in a number of cases the radiation of the nucleus has a magnetic character. From the theory of internal conversion it follows that the dependence of the total and relative conversion coefficients on the order of the multipolarity of the radiation will be different for magnetic and electric transitions. Therefore, by simultaneously comparing the total and relative conversion coefficients obtained experimentally with theory, one can determine whether the nuclear radiation in the given case has a magnetic or an electric character.

At present we do not yet have calculations of internal-conversion coefficients for the case of magnetic multipoles and small excitation energy of isomers, which are needed for comparison with experiment. (The relativistic calculation for magnetic transitions performed by Dancoff and Morrison \(^{25}\), neglecting the binding of the orbital electrons in the atom, is applicable only in cases of very high excitation energies, not encountered in practice.) This gap in the theory of conversion makes it difficult to establish the character of the nuclear radiation from the experimental values of the internal-conversion coefficients. There exists, however, a number of general theoretical considerations \(^{25}\), according to which in the case of magnetic radiation the conversion coefficients on the \(K\)-shell will be small, whereas the relative

the conversion coefficients will be anomalously high. This circumstance may make it possible in a number of cases to indicate the character of the radiation and to determine the order of its multipolarity.

Let us turn to the question of the theoretical calculation of the dependence between the lifetime of an isomeric nucleus, its excitation energy, and the magnitude of the difference of the angular momenta of the nucleus in the excited and ground states. It should be noted that this calculation is only qualitative, in view of the fact that at present it is impossible to estimate accurately the values of the matrix elements for nuclear transitions.

The formula proposed by Bethe^9 for the mean lifetime of a nucleus with respect to \(\gamma\)-emission has the form:

\[ \tau_{\gamma}=5\cdot 10^{-21}(\Delta l)^2\left(\frac{20}{\Delta E}\right)^{2\Delta l+1}\ \text{sec.}, \tag{1} \]

where \(\Delta l\) is the magnitude of the difference of the angular momenta for the corresponding states of the nucleus, expressed in units \(h/2\pi\); \(\Delta E\) is the magnitude of the excitation energy of the nucleus in millions of electron-volts.

When comparing the lifetime given by Bethe’s formula with experiment, it is necessary to take into account the additional probability of decay of the metastable state produced by processes of internal conversion.

The Weizsäcker^8 formula for the lifetime is based on a very approximate calculation and, in comparison with Bethe’s formula, gives values for the lifetime smaller by a factor \(Z^2(\Delta l)^2\); thus, for nuclei with medium atomic number and at \(\Delta l=4\), the calculated values of \(\tau\) differ from one another by approximately \(10^8\) times.

Prof. Ya. I. Frenkel, in his recent work^27, pointed out the inconsistency of Bethe’s calculation and noted that the interpretation of metastable levels given by Bethe on the basis of the “drop” model of the nucleus must be rejected.

In any case, it may be stated that at the present time theory has not yet provided a justified and more or less exact calculation for the lifetimes of metastable nuclei; in other words, a quantitative theory of nuclear isomerism still does not exist.

Below we give a comparison of the value \(\tau\), calculated by Bethe’s formula with allowance for conversion, with the lifetime of the metastable nucleus measured experimentally. This comparison was made in the following way. As is seen from formula (1), in order to calculate the mean lifetime of some isomer it is necessary to know for it the quantities \(\Delta E\) and \(\Delta l\). The value of \(\Delta E\) is known for many isomers from experimental data on the spectrum of conversion electrons, and we determine the quantity \(\Delta l\) by comparing the experimentally measured conversion coefficients with the results of theoretical calculations.

Let us turn to consideration of those few cases of isomerism for which such a comparison can be made, since for them there are more or less detailed data on the coefficients of internal conversion.

Bromine 80. This element was investigated by Rusinov, Yuzefovich, and Grinberg^28. According to the data of these authors, in the nucleus \(\mathrm{Br}^{80}_{35}\) there occur two successive transitions. The upper transition

has a decay period of 4.5 hours. The energy difference for this transition is 49 KeV; the total conversion coefficient is equal to \(\sim 1\), and the relative conversion coefficient is \(0.35—0.5\).

In the present case a large value of the total coefficient of internal conversion and a small value of the relative conversion coefficient are observed. Therefore it may be assumed that here radiation of an electric character takes place.

It follows from the magnitude of the total coefficient of internal conversion that in the case of bromine the order of multipolarity of the radiation is greater than two units, since for \(\Delta E = 49\) KeV and \(Z = 35\) the total conversion coefficient according to theory reaches \(\sim 1\) already at \(\Delta l = 2\). From a comparison of Zavelevich’s calculations concerning the relative conversion coefficients with the experimental data it follows that, in the case of bromine, the change of angular momentum in the isomeric transition is equal to three units. If, using Bethe’s formula (with allowance for conversion), one calculates the lifetime of the metastable nucleus, substituting the values \(\Delta E = 49\) KeV and \(\Delta l = 3\), one obtains a value of \(\tau\) smaller by \(10^7\) than that known from experiment.

Mercury 99 or 101. This element was studied by Segrè, Ziborg, and others.\(^{29}\) The excitation energy of the isomer under consideration is 136 KeV, and the decay period is 6.6 hours. The total conversion coefficient is \(\sim 0.5\). The relative coefficient was not measured, but the authors note a large conversion in the \(K\)-shell in comparison with the \(L\)-shell. Therefore in this case also it should be assumed that the radiation has an electric character. The change of angular momentum, determined from the magnitude of the total conversion coefficient, is equal to three units. The lifetime calculated by Bethe’s formula (with allowance for the indicated value of the conversion coefficient) turns out to be \(10^8\) times smaller than that observed experimentally.

Silver 107 or 109. This isomer was studied in detail by Alvarez, Nelson, and others.\(^{30}\)

In the present case \(\Delta E = 93.5\) KeV, \(T = 40\) sec., the total conversion coefficient is 0.98, and the relative one \(1.0 \pm 0.1\). The magnitude of the total conversion coefficient indicates that in the present case also an electric transition occurs and that the change of angular momentum is greater than three units. From the data on the relative conversion coefficient it follows that the change of angular momentum is equal to four. (The authors of the article give this value of \(\Delta l\), referring to unpublished theoretical calculations of relative conversion coefficients.)

A calculation of the lifetime carried out analogously to the preceding cases gives a value for \(\tau\) which agrees with that observed experimentally.

Indium 114. This isomer was studied in detail by Lawson and Cork.\(^{16}\) Its excitation energy is 192 KeV. The total conversion coefficient is \(1.0 \pm 0.15\); the relative coefficient is 1.0; the decay period of the metastable level is 50 days. From the magnitude of the total conversion coefficient it follows that here too electric radiation takes place, and in such a case the change of the angular momentum of the nucleus is equal to or greater than five. The value of the relative conversion coefficient gives \(\Delta l = 5\).

The theoretically calculated lifetime coincides with the experimental value.

Indium 113 and 1152. The excitation energy of these isomers is too high (393 and 338 KeV) for nonrelativistic formulas to be applicable, but it is much lower than the limit of applicability of the relativistic formulas.

Using the method of graphical interpolation, it is nevertheless possible to estimate the values of the internal conversion coefficients for electric multipoles. Since we do not have the necessary formulas for calculating conversion coefficients in the nonrelativistic region for magnetic multipoles, we cannot carry out an analogous interpolation for them, although such a calculation would be of interest, since in the case under consideration we have no grounds for choosing the character of the radiation (magnetic or electric).

Let us note, however, that if the theoretical calculation of the lifetime, even under the assumption of electric radiation, gives large discrepancies with experiment (see Table 2), then for magnetic radiation this discrepancy would be still greater, since the observed value of the conversion coefficient would correspond to an even greater difference in angular momenta.

All the examples considered are collected in Table 2.

Table 2

As can be seen, only for six cases out of the total number of known isomeric nuclei have the values of the internal conversion coefficients been measured, making it possible to carry out this kind of comparison of experimental data with theoretical calculations.

In a number of cases—bromine, silver, indium—it was possible, in order to determine the order of the multipolarity of the radiation and the change in angular momentum associated with it, to use both the total and the relative conversion coefficients. In the remaining cases this determination had to be made only from the values of the total conversion coefficients.

All the material presented here concerning the determination of the magnitude of the difference in angular momenta for a number of isomeric nuclei from data on internal conversion is of great significance. It shows that the Bohr–Weizsäcker theory of the nature of nuclear isomerism is confirmed also on the point that has until now remained open, namely, that the metastability of isomeric nuclei is indeed due to the difference in the angular momenta of the corresponding states. This result constitutes a major success achieved recently in the study of nuclear isomerism.

The comparison, carried out by us in the manner indicated, of theoretical calculations with experimental data on the mean lifetimes of isomers shows that in a number of cases there is a very considerable discrepancy. The greatest discrepancy is given by Weizsäcker’s formula (up to \(10^{16}\) times), which, however, is also erroneous from the theoretical point of view. A noticeable disagreement with experiment is also given in a number of cases by Bethe’s formula.

The discrepancy obtained may be connected with the circumstance that in some of the examples considered radiation of magnetic character nevertheless takes place, or with the fact that in a number of cases the authors have exceeded the limit of applicability of their calculation of the conversion coefficients. There is reason to suppose that the discrepancy in the calculation of \(\tau\) is due to the great inaccuracy of the existing theoretical formulae for the lifetime.

In connection with the existing disagreement between experimental and theoretical data, further experimental study of the total and relative conversion coefficients for other nuclear isomers, and the performance of a number of theoretical calculations of internal-conversion coefficients—primarily for magnetic transitions at low excitation energies—are highly urgent.

These data, and possibly also other methods mentioned above, will make it possible to determine the magnitude of the difference in angular momenta for a considerably larger number of isomers.

As a result of further investigations, more complete material will be accumulated on angular momenta, as well as on mean lifetimes, the character of transitions, and the excitation energies of metastable nuclei; this will serve as the basis for constructing a quantitative theory of nuclear isomerism and will give us a number of new and valuable data on the system of energy levels in atomic nuclei.

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1. Report at the 1940 Conference on the Atomic Nucleus; see p. 241 of this issue. ↩

2. Visible reference marker in the source text. ↩