NUCLEAR ISOMERISM*)

E. Segrè and A. Helmholz

CONTENTS

1. History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
2. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
3. Radiation of isomeric nuclei. a) Purely isomeric transitions. b) Beta radioactivity and isomerism . . . . . . . . . . . . . . . . . . . 362
4. Weizsäcker’s hypothesis. a) The qualitative aspect of the question. b) Multipole radiation. c) Selection rules. d) Half-life of excited states. e) Internal conversion. f) 0—0 transitions. g) Applications of the theory . . . . . . . . . . . . . . . . . . . . . 369
5. Production of isomeric nuclei . . . . . . . . . . . . . . . . . . . . . 389
6. Separation of isomers . . . . . . . . . . . . . . . . . . . . . . . . . . 397
7. Some cases of isomerism . . . . . . . . . . . . . . . . . . . . . . . . 400
   Cited literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

1. HISTORY

In 1917, Soddy$^{81}$, developing his general considerations on isotopy, suggested the possibility of the existence of nuclei having the same charge and mass, but nevertheless different. Such nuclei may be isotopic (same charge) and isobaric (equal masses) and differ only in other features, in particular in their radioactivity. They must have the same chemical properties, and even with the aid of a mass spectrograph they could be distinguished only if the difference in their masses reached a value of the order of $10^{-5}$ mass units. Such nuclei were then called $P^{1}$ isotopes of the second kind, in distinction to ordinary isotopes, which have different masses. We now call these nuclei isomers.

These suggestions were not confirmed by experiment until 1921, when Hahn$^{11}$ discovered the radioactive substance UZ ($\mathrm{Pa}_{91}^{234}$), which was isotopic and isobaric with the well-known substance $\mathrm{UX}_2$ and differed from it in half-life (6.7 hours instead of 1.15 min.) and in the type of radiation emitted. The evidence

*) E. Segrè and A. C. Helmholz, Reviews of Modern Physics 21, 271 (1949).

the isomers \(UZ\) and \(UX_2\), introduced by Hahn, was based on the fact that both isomers originated from \(UX_1(^{90}\mathrm{Th}^{234})\) by \(\beta\)-decay and, consequently, were isobars with it and had atomic number \(90 + 1 = 91\).

This example of isomerism remained the only one for many years. In view of the serious difficulties in the theoretical explanation of this phenomenon, it was highly desirable to find other similar examples. Subsequent work \(^{H2, G1, W1}\) confirmed Hahn’s results, and recently the case of \(UZ\) was studied in detail \(^{P2, B7}\).

The discovery of artificial radioactivity in 1934, and especially the expansion of the field of investigation as a result of obtaining many new radioactive substances by neutron bombardment, contributed to the finding of new examples of isomerism. The first indications of the existence of isomerism were found in India by Szilard and Chalmers \(^{S3}\). Isomerism was first established for the case of bromine by Kurchatov \(^{K1}\) and his collaborators *), and also by Fermi and his colleagues \(^{A1}\).

They found that radioactive isotopes of bromine obtained by neutron bombardment gave three different radioactive periods: 18 min., 4.4 hours, and 34 hours. All of them were “water-sensitive,” i.e. their production was achieved more effectively by the action of slow neutrons than of fast ones, and they therefore were formed as a result of neutron capture. On the other hand, bromine has only two stable isotopes, one with mass 79 and the other with mass 81. The three radioactivities must therefore be assigned to bromine isotopes with masses 80 and 82, and consequently at least one of these isotopes must occur in two isomeric states. By means of a special investigation, Blewett \(^{B1}\) eliminated the alternative explanation associated with the possible existence of a third rare stable isotope (with an abundance of less than \(1/3000\)), and consequently bromine isomerism could be considered proven. Subsequent work \(^{B2, S2}\) confirmed the previous results and showed that the periods of 18 min. and 4.4 hours must both be assigned to \({}_{35}\mathrm{Br}^{80}\).

Thus evidence for isomerism began to accumulate, and the next important step was taken by Weizsäcker \(^{W2}\), who was the first to give a theoretical explanation of isomerism in agreement with the experimental data and with generally accepted theoretical ideas. It had often been assumed that nuclear isomers might be nuclei in an excited metastable state, but these hypotheses seemed unacceptable, since it appeared that

) For more detail on the discovery of nuclear isomerism and further investigations in this field carried out by Soviet physicists, see the article by I. V. Kurchatov and L. I. Rusinov, “Isomerism of Atomic Nuclei” (Jubilee Collection of the Academy of Sciences of the USSR Dedicated to the 30th Anniversary of the Great October Revolution, part I, p. 285). (Translator’s note.)*

there is no mechanism ensuring a sufficiently long half-life for the metastable state. Weizsäcker showed that if the spins in the excited and ground states differ by several units (up to 5) at an excitation energy of up to several hundred kilovolts, then the half-life (return to the normal state occurs by way of $\gamma$-radiation) becomes sufficiently long for the existence of isomers to be detected. Before Weizsäcker’s work such an explanation was considered untenable, since the half-life for $\gamma$-radiation was estimated to be of the order of $10^{-13}$ sec., so that all nuclei should practically always be found in the normal state.

Weizsäcker’s theory gave a new impetus to the study of isomerism, since it pointed to certain consequences concerning internal conversion, etc. $^{2,\,5}$, which were confirmed experimentally.

Finally, direct proof of the formation of one isomeric state from another was given by Segrè and his collaborators by means of chemical separation $^{55}$.

At the present time more than 75 pairs of isomeric nuclei are known, and their number is steadily increasing.

2. INTRODUCTION

As was said above, we call two nuclei “isomeric” if they have the same charge and mass but differ in other nuclear properties, for example in half-life. The word “isomer” is borrowed from organic chemistry, where it denotes substances with the same formula but with a different grouping of atoms, which leads to different properties of these substances. An explanation of isomerism on the basis of structural differences is not applicable to atomic nuclei. In organic molecules the nuclei of the atoms composing them are practically at rest or, more precisely, oscillate about definite equilibrium positions with amplitudes small in comparison with the distances between the nuclei. The nuclei thus form a quasi-rigid permanent framework within which the electrons move. An entirely different situation obtains in nuclei. Here it is assumed that neutrons and protons interact very strongly and move throughout the whole nucleus in such a way that no definite configurations of them can exist for any appreciable length of time. Whereas an organic molecule is similar to a small crystal, the nucleus may be compared with a drop of liquid. This assumption is confirmed by considerations using the uncertainty relation and the known data on the sizes of nuclei and the magnitude of nuclear forces $^{3}$.

The possibility that a nucleus, considered as a drop of liquid, can exist in a stable spherical, as well as

in a stable ellipsoidal state was discussed by FlüggeF1. Calculations by FinbergF13 and WeizsäckerW19 showed that both these stable forms can exist only for

\[ 44 \lessgtr \frac{Z^2}{A} \lessgtr 50 \]

and for \(U^{238}\) \(\left(\frac{Z^2}{A}=36\right)\). Further calculationsW20 showed that even these states may prove to be unstable with respect to transition into other geometric forms. Consequently, although this idea may have some significance in considering very heavy nuclei, it cannot be relevant to the problem of isomerism of the known elements. Some other theories were proposed, discussed by Flügge, but none of these theories seems to us satisfactory in the light of the existing data (see the discussion of \(0—0\) transitions, Section 4e).

According to Weizsäcker’s hypothesis, one of the pair of isomers is in an excited metastable state. Further, the half-life for the return of the nucleus from the excited state by \(\gamma\)-emission, for the objects studied up to the present time, ranges from \(10^{-13}\) sec. to several months. Although no one would call a state with a half-life equal to \(10^{-13}\) sec. metastable, it is quite clear that there is no qualitative difference between excited states with very short and long half-lives. However, for practical purposes it is convenient to speak of isomers only in the case where the half-life is sufficiently long for direct observations to be made; in other cases we shall speak of ordinary excited states. We might conventionally choose as the boundary a half-life of about one second, but the use for measurements of the newest electronic technique with delayed coincidences pushes this boundary down to \(10^{-7}\) sec. (seeD22). Up to now only a few such cases have been found, but the discovery of a considerably larger number of them is possible*).

The usual method for determining whether two radioactive nuclei belong to isomeric nuclei, or, in general, the method for finding isomeric nuclei, is their production by means of different types of nuclear bombardment. Bombardment with slow neutrons

*) De Benedetti and McGowanD22 investigated 60 activities in the range from \(10^{-3}\) to \(10^{-6}\) sec. and obtained a positive result only in 4 cases. A larger number of cases can normally be expected. Holmes, May, and TurgelH41 unsuccessfully attempted to observe isomers with half-lives from \(10^{-3}\) sec. to 1 sec. in 16 elements bombarded by slow neutrons. TressiT8 did not find isomers with a short half-life in 6 elements bombarded with X-rays of energy 1 MeV.

gives, except for very few cases among light nuclei, only reactions of the type \((n,\gamma)^*)\), i.e., neutron-capture reactions. Gamma rays of the order of \(8\) MeV \(^{**)}\) give reactions of the type \((\gamma,n)\), which are very easy to interpret. In the case of bromine (see above), the stable isotopes have masses 79 and 81; by bombardment with slow neutrons one can obtain only masses 80 and 82, whereas by \(\gamma\)-bombardment one can obtain masses 80 and 78. Activities with periods of 18 min. and 4.4 hours were obtained by both types of bombardment and, consequently, may be assigned to \(\mathrm{Br}^{80}\).

As another example let us consider two periods, equal to 21 min. and 6.5 days, obtained as a result of bombarding chromium with protons and iron with deuterons. Both periods belong to manganese isotopes. Further, chromium has stable isotopes with masses 50, 52, 53, 54 and, after bombardment with protons, as a result of a reaction of the type \((p,n)\), should give manganese with the same masses. On the other hand, iron has stable isotopes with masses 54, 56, 57, 58 and, after bombardment with deuterons of the energy used, can practically give manganese only by reactions of the type \((d,\alpha)\). This should lead to masses 52, 54, 55, 56. Consequently, the two above-mentioned periods must be assigned to \(\mathrm{Mn}^{54}\) or \(\mathrm{Mn}^{52}\). On the other hand, \(\mathrm{Mn}^{54}\) is known from other reactions and has a period equal to 310 days. Hence it is clear that, of the three manganese isotopes with periods of 21 min., 6.5 days, and 310 days, two must be isomers. The periods equal to 21 min. and 6.5 days must both be assigned to \(\mathrm{Mn}^{52}\), since they cannot be obtained from \(\mathrm{V}^{51}\) by means of a reaction of the type \((\alpha,n)\), or from \(\mathrm{Cr}^{53}\) by a reaction of the type \((d,n)\) (at the same time it is known that practically all energetically possible reactions are observed in actuality).

Gamma rays with energies of \(1\)—\(3\) MeV can bring nuclei into an excited state and, consequently, can be used to obtain isomeric states of stable nuclei, but in this case no nuclear particles are emitted. The same excited states may, moreover, arise in inelastic collisions with the nucleus of particles \(e^{-}\), \(p\), \(\alpha\), and \(d\).

*) We denote, following Bothe, the bombarding particle first, and the emitted particle (or quantum) second; \((n,\gamma)\), thus, denotes

\[ {}_{Z}A^{M}+{}_{n}0^{1}={}_{Z}A^{M+1}+\gamma, \]

where \(M\) is the mass of the nucleus.

**) This applies to isotopes of elements of medium weight.

3. RADIATIONS OF ISOMERIC NUCLEI

a) Pure isomeric transitions

Let us dwell briefly on the radiations emitted in an isomeric transition from the higher state 2 to the lower state 1. There are two types of such radiations: gamma rays and conversion electrons*). In the case of emission of gamma radiation with frequency \(\dfrac{\omega}{2\pi}\), we have

\[ \hbar \omega = E_2 - E_1 \tag{1} \]

(\(E_2, E_1\) are respectively the energies of levels 2, 1). When electrons are emitted from the \(K\), \(L\), etc. shells of the atom, their energy will be

\[ \hbar \omega - K = E_K, \tag{2} \]

\[ \hbar \omega - L = E_L \quad \text{etc.}, \tag{3} \]

where \(E_K, E_L\) are the energies of individual monokinetic groups of emitted electrons; \(K, L\) are the binding energies of atomic electrons in the \(K, L\) shells of the atom undergoing the isomeric transition. From the standpoint of the law of conservation of energy, one may consider that gamma radiation with frequency \(\dfrac{\omega}{2\pi}\) produces a photoelectric effect in the emitting atom. Therefore this effect is sometimes called the internal photoelectric effect or internal conversion. However, such a picture does not give a complete description of the physical process actually taking place.

The ratio of the number of ejected \(K\)-electrons to the number of emitted gamma quanta is called the coefficient of incomplete internal conversion for the \(K\)-shell\({}^{1}\):

\[ \alpha_K = \frac{N_K}{N_\gamma}. \tag{4} \]

In an analogous way \(\alpha_L\), etc. are defined. The ratio of the total number of emitted electrons to the number of emitted quanta is called simply the coefficient of internal conversion. Thus the coefficient of internal conversion is equal to the sum of the partial coefficients of incomplete internal conversion:

\[ \alpha = \alpha_K + \alpha_L + \alpha_M = \frac{N_e}{N_\gamma}, \tag{5} \]

where \(N_e\) is the total number of emitted electrons.

*) In the transition from an excited state, an electron–positron pair may be emitted (an electron and a positron). For example, \(O^2\) can serve as \(Ne^{20}\). This type of transition is very rapid. Another possibility, in which two quanta are emitted, will be discussed below (Section 4e).

The value of the internal-conversion coefficient may vary between zero and infinity. Sometimes in the older and even in the modern literature the internal-conversion coefficient has been defined as

\[ \frac{N_e}{(N_e+N_\gamma)}, \]

i.e., as the ratio of the number of emitted electrons to the number of nuclear transitions. The first definition (4) is more convenient, and we shall follow it.

If an atom loses a \(K\)- or \(L\)-electron, then it must emit the \(K\) and \(L\) X-ray series, as well as Auger electrons. The emitted X-rays belong to the spectrum corresponding to the atomic number of the substance undergoing the isomeric transition. For example, in the case of \(\mathrm{Br}^{80}\) one can observe the characteristic radiation of bromine. It should be noted that in other nuclear processes leading to the secondary emission of X-rays after radioactive decay, the X-rays are characteristic of the decay products. Thus, in the capture of a \(K\)-electron the X-rays are characteristic of the preceding element in the periodic table, while in beta decay they are characteristic of the element following the decaying one. The study of the emission of characteristic X-rays is therefore a good method for identifying isomeric transitions.

Fig. 1. Absorption curves of the X-rays and \(\gamma\)-rays of \(\mathrm{Tc}^{99}\). \(a\)—absorption of X-rays in Al; \(b\)—absorption of X-rays in Zr; \(c\)—absorption of X-rays in Cb; \(d\)—absorption of X-rays in Mo. The figure is taken from Phys. Rev. 55, 808 (1939).

When characteristic absorbers are used for the identification of X-rays, comparatively small radiation intensities are sufficient for an unambiguous determination of the atomic number. In Fig. 1 is given the absorption curve of the X-rays emitted in the isomeric transition by element 43 (technetium). The same X-rays were photographed by Abelson with the aid of a crystal spectrograph\(^{47}\); a microphotograph of the spectrum is shown in Fig. 2. A quantitative

measurement of the number of emitted conversion $K$-electrons can be carried out by determining the yield of fluorescence Cl. Internal-conversion electrons can usually be detected with the aid of thin-walled ionization chambers or counters.

Fig. 2. Microphotograph of the $K_{\alpha}$ lines of Tc. The three peaks on the upper and lower calibration curves correspond to the $K_{\alpha}$ doublet of Cb, Mo, and Ru. The main peak on the curve in the center is the $K_{\alpha}$ line of Tc. The peak on the left is the $K_{\alpha}$ line of molybdenum, arising as a result of the decay of Tc with a half-life equal to 2 days. The figure is taken from Phys. Rev. 56, 753 (1939).

The absorption curve of soft monoenergetic electrons is often sufficiently characteristic for a given instrument, although it depends strongly on the geometry of the apparatus, etc. Fig. 3 gives an example of an absorption curve for electrons with an energy of 115 kev and for very soft beta rays, for a standard ionization chamber. It is also useful to note that, in the case of ordinary beta decay, low-energy electrons are associated with a long half-life; therefore the presence of electrons of only low energy with a short half-life (for example, 200 kev and 1.3 sec.) is evidence for isomerism.

The best results were obtained with the aid of beta spectrographs of various types. The main difficulty here is obtaining sufficiently thin and intense sources. Owing to this difficulty, electron spectra obtained up to now with artificial radioactive substances are inferior to electron spectra obtained with natural radioactive substances, which are ideal for obtaining extremely thin sources with considerable activity. Photographic plates, counters, and ionization chambers were also used for detecting electrons. After proper microphotometric calibration or counter calibration, reliable quantitative results were obtained (see L1, V1, H8, D9). In the case of electrons of very low energy, below $\sim 20$ kev, all difficulties increase.

Finally, a Wilson chamber was also used, but not very widely, since scattering in the chamber leads to the appearance of “lines” similar to the continuous spectrum of beta rays. Modern proportional counters will also, of course, be used for such measurements (see, for example, C19 and H38).

Knowledge of the internal-conversion coefficient is very important from the theoretical point of view, but, unfortunately, measurement of this quantity is very difficult. The problem consists in determining the number of emitted quanta and the number of emitted electrons. It must be added that often both of these types of radiation have energies below 100 keV, which increases the difficulties associated with absorption in the sources, absorption in various windows, etc. However, the most difficult matter is the calibration of the measuring instrument. In the case of ionization chambers or counters, γ-rays with an energy of 500 keV can be calibrated relative to electrons by using a positron emitter and directly comparing the effect of positrons with the effect of annihilation radiation G². Counters operating in coincidence F², D¹, R⁵, D⁹ are also very convenient.

In cases where the transition being measured follows β-decay or precedes it (see the following section), a β-spectrograph makes it possible to count both the conversion electrons and the decay electrons, whose number is equal to the number of conversion electrons plus the number of γ-quanta (see, for example, L¹). To determine how many times the β-radiation follows (or precedes) the conversion electron, β—β coincidences may be used. To find how often the γ-radiation follows (or precedes) the β-radiation, β—γ coincidences are used. Formulas for the treatment of such measurements were given by Wiedenbeck and Chu W²¹, who, however, used the coefficient \(\alpha\) in the old sense of this concept (see equation (5) and below). They also give experimental results for certain γ-rays.

Fig. 3. \(a\) and \(v\)—absorption of electrons from Tc⁹⁹ in aluminum; \(b\)—absorption in aluminum of beta rays of Co⁶⁰. The figure is taken from Phys. Rev. 55, 808 (1939).

From the theoretical point of view it is also important to measure the ratio between the various partial coefficients of internal conversion. Usually it is necessary to consider only the ratio \(\dfrac{\alpha_K}{\alpha_L}\).

This ratio is accurately measured with the aid of a beta spectrograph, and the difficulties of calibrating the results are comparatively small, since the energy difference between the two groups of conversion electrons is in most cases small in comparison with their energy and, consequently, the detection efficiency of these groups is essentially the same.

b) Beta radioactivity and isomerism

In cases of isomers of stable nuclei, such as \(\mathrm{In}^{115m}\), \(\mathrm{Kr}^{83m}\), \(\mathrm{Sr}^{87m}\)*), the radiations associated with the isomeric transition can be observed in the absence of any other radiations. However, in many cases nuclei possessing isomeric states also turn out to be beta-radioactive. Let us describe, by means of a few imaginary simple examples, some interesting features that arise with such a coincidence. In Fig. 4 a typical level diagram is given for two nuclei \(A\) and \(B\). \(A\) transforms into \(B\) by means of \(\beta\)-emission and has an excited state 2 approximately 100 keV above its ground state 1. \(B\) also has an excited state. The probability of transition per unit time from state 2 with respect to beta decay is \(\lambda_{2\beta}\), and the probability of transition from state 2 to state 1 is \(\lambda_{2\gamma}\). The half-life of state 2 is then given by the equation

Fig. 4. Examples of energy levels of isomeric nuclei. \(a\) — general case; \(b\) — type II (\(p \gg 1\)); states 2 and 1 decay independently as a result of beta emission; \(c\) — type I (\(p \ll 1\)); state 2 decays into state 1, from which \(\beta\)-decay then proceeds.

\[ \tau_2=\frac{0.69}{\lambda_{2\beta}+\lambda_{2\gamma}} . \tag{6} \]

The ratio between the numbers of beta and gamma transitions is equal to

\[ p=\frac{\lambda_{2\beta}}{\lambda_{2\gamma}} . \tag{7} \]

*) We denote by \(m\) the excited states of stable nuclei (see \(^{20}\)).

Now let us consider two limiting cases encountered in practice.

First suppose \(p \gg 1\) (type II). Then state 2 decays practically only by \(\beta\)-emission, and nuclei in state 2 or in state 1 behave as different independent radioactive nuclei, each of which emits its own characteristic beta spectrum and gamma rays following the \(\beta\)-decay.

In the other limiting case, for which \(p \ll 1\) (type I), state 2 practically goes over only into state 1, which in turn undergoes \(\beta\)-decay. States 2 and 1 behave in this case as two radioactive substances, with 1 being the daughter product of 2. In such a case we speak of a pair of genetically related isomers. The beta spectrum of a pair of genetically related isomers corresponds to the transition beginning from state 1 with the decay constant \(\lambda_{1\beta}\). Now suppose that by means of some process we have obtained equal amounts of nuclei in state 2 and in state 1 and that, as very often happens, the isomeric transition from state 2 to state 1 is accompanied by radiation that can be easily separated, for example by absorption. When only the beta spectrum is investigated, two different periods with an identical beta spectrum were sometimes observed in the case \(\lambda_{2\gamma} < \lambda_{1\beta}\). At first the \(\beta\)-rays are emitted chiefly directly from state 1, and the activity decreases approximately with the decay constant \(\lambda_{1\beta}\). Then radioactive equilibrium is established between states 2 and 1, and the beta decay gives the constant \(\lambda_{2\gamma}\), corresponding to the transition from state 2 to state 1, while the beta spectrum remains that corresponding to beta decay from state 1.

This situation arises in the majority of presently known genetically related pairs of isotopes, for example \(Br^{80}\), \(Rh^{104}\). In Fig. 5 the absorption curve is given for the radiation of \(Zn^{69}\) with periods equal to 13.8 hours and 57 min. The beta-absorption curves are clearly identical for both states, whereas the period of 13.8 hours also gives gamma radiation corresponding to the transition \(2 \to 1\) (the transition between isomeric states) in Fig. 4.

In this particular case no \(\gamma\)-rays following the \(\beta\)-decay were observed. In other cases such radiation may occur, and the observation of coincident beta and gamma rays is especially convenient for distinguishing gamma rays accompanying beta decay from gamma rays participating in the isomeric transition.

From all that has been said above one may conclude that the existence of an identical beta spectrum associated with different periods is very valuable from the point of view of identifying and determining isomeric pairs.

In the case when $\lambda_{2\gamma}>\lambda_{1\beta}$, no such characteristic features arise, but one can immediately notice an anomaly in the law of beta decay. An example of such a case may be the isomerism of $\mathrm{Ba}^{133}$, where the half-life of the $\gamma$ transition is 38 hours, while for the $\beta$ transition the half-life is approximately 20 years $^{\mathrm{K9}}$.

Quantitatively, the ionization per unit time $I$, arising as a result of the radiation emitted in the decay of a pair of genetically related isomers, is equal to

\[ I=k\lambda_{1\beta}\left[ N_2\frac{\lambda_{2\gamma}}{\lambda_{1\beta}-\lambda_{2\gamma}}e^{-\lambda_{2\gamma}t} + \left(N_1-\frac{\lambda_{2\gamma}}{\lambda_{1\beta}-\lambda_{2\gamma}}\right)e^{-\lambda_{1\beta}t} \right], \tag{8} \]

where $N_1$ and $N_2$ are the initial populations of states 1 and 2. The coefficient $k$ takes into account the ionizing capacity of the emitted radiation, the geometrical conditions, etc. The ionization caused by the radiation corresponding to the isomeric transition $2—1$ is not included here, since it can easily be separated. Formula (8) is obtained by direct application of the laws of radioactive decay. The initial populations $N_1$, $N_2$ depend on the method of formation of the isomeric nuclei; we shall discuss this question below.

Fig. 5. Absorption curves (in aluminum) for $\mathrm{Zn}^{69}$. $a$ — 13.8-hour activity; $b$ — 57-minute activity. The figure is taken from Phys. Rev. 56, 1095 (1939).

In practice one may also expect cases intermediate between the limiting ones (“a” and “b”) discussed above. They occur when, by chance, $\dfrac{\lambda_{2\gamma}}{\lambda_{2\beta}}$ does not differ too much from 1. Since $\lambda_{2\gamma}$ and $\lambda_{2\beta}$ vary independently of one another over wide limits, the possibility that they will have the same order of magnitude is rather small. Further complications in the analysis of radiations are connected with the fact that the gamma rays accompanying $\beta$ emission sometimes have a complex spectrum. This phenomenon is not accidental; it should be expected in accordance with Weizsäcker’s hypothesis and with the selection rules of the theory of $\beta$ decay. We shall return to this question in Section 5.

4. WEIZSÄCKER’S HYPOTHESIS

a) Qualitative aspect of the question

Let us consider the fundamental problem connected with nuclear isomerism, i.e. the question of the mechanism that prevents the emission of the energy of the excited state in a very short time. The mean lifetime of a state under dipole emission of gamma rays is expressed by the well-known formula of electromagnetic theory:

\[ T_\gamma=\frac{3}{4}\,\frac{\hbar c^3}{\omega^3 M_{nm}^2}, \tag{9} \]

where \(M_{nm}\) is the matrix element of the electric moment of the nucleus in the \(mn\)-transition. In this formula the only indeterminate quantity is \(M_{nm}\). If \(M_{nm}\) is taken equal to the electronic charge multiplied by \(10^{-13}\) cm, which is quite consistent with nuclear charges and nuclear dimensions, then, for \(\omega\) corresponding to an energy of 100 kev, we obtain \(T_\gamma=4\cdot 10^{-12}\) sec. This mean lifetime is \(10^{18}\) times shorter than some of those observed experimentally. Hence it is clear that very powerful selection rules must be operating, preventing radiative transitions. Bohr considered the possibility that the electric dipole moment of the nucleus may be much smaller than the value estimated above, owing to the fact that in the nucleus protons are closely bound to neutrons, so that when some of the elementary particles move inside the nucleus, the electric center of gravity approximately coincides with the mechanical one. This should considerably reduce the dipole moment of the nucleus. However, it is difficult to believe that this effect can be substantial enough for a sufficient reduction in the magnitude of the matrix elements.

As mentioned in Section 1, Weizsäcker \({}^{w2}\) pointed out that if the angular momenta of two nuclear levels differ by more than one unit \(\dfrac{h}{2\pi}\), then the transition between them by means of dipole radiation is strictly forbidden, and usually, if the spin difference is \(l\), the first allowed transition will be due to an electric or magnetic multipole of order \(2^l\).

Now the ratio of the intensities of quadrupole and dipole radiation at a given frequency will be of the order

\[ \frac{x^2}{\lambda^2}, \]

where \(x\) is the length corresponding to the dimensions of the nucleus, and \(\bar{\lambda}\) is the wavelength of the emitted radiation divided by \(2\pi\). In general, the radiation intensity of an electric pole \(2^l\) or a magnetic pole \(2^{l-1}\) will be of the order

\[ \left(\frac{x}{\bar{\lambda}}\right)^{2(l-1)} \]

of the dipole radiation. Since the ratio \(x/\bar{\lambda}\) is of the order \(1/300\) for a quantum with energy 100 kev, it follows that,

the intensity of quadrupole radiation is approximately \(10^5\) times less than the intensity of dipole radiation; an analogous relation is obtained also for higher multipoles. Assuming that the change in angular momentum taking place in the transition is sufficiently large, one can explain the existence of an arbitrarily long half-life period. The qualitative considerations presented can be developed quantitatively.

b) Multipole radiation

The starting point of the theory is provided by the classical expressions for the scalar \((\varphi)\) and vector \((\mathbf A)\) potentials associated with charges moving in the nucleus

\[ \mathbf A(\mathbf R,t)=\int \frac{\mathbf j_1(\mathbf r',t^*)}{cr}\,d\tau', \tag{10} \]

\[ \varphi(\mathbf R,t)=\int \frac{\rho_1(\mathbf r',t^*)}{r}\,d\tau', \tag{11} \]

where \(r\) is the distance between the point \(P\) with coordinates \(x,y,z\) (Fig. 6) and the volume element under consideration, \(t^*=t-\dfrac{r}{c}\), \(c\) is the velocity of light. The integrals are taken over all space; \(\rho_1\) is the charge density at the point with coordinates \(x',y',z'\) at the instant \(t^*\), and \(\mathbf j_1\) is the current density at the same point and at the same time; \(d\tau'\) corresponds to \(dx'\,dy'\,dz'\).

Fig. 6. Arrangement of the vectors \(\mathbf R\), \(\mathbf r\), \(\mathbf r'\), and \(\mathbf n\).

Now suppose that \(\rho_1\) and \(\mathbf j_1\) are harmonic functions of time with frequency \(\nu=\dfrac{\omega}{2\pi}\). Then they can be represented as the real parts of the expressions:

\[ \rho(\mathbf r')e^{i\omega t}=\rho_1 \tag{12} \]

and

\[ \mathbf j(\mathbf r')e^{i\omega t}=\mathbf j_1. \tag{13} \]

These quantities are related by the continuity equation

\[ \frac{\partial \rho_1}{\partial t}=-\operatorname{div}\mathbf j_1, \tag{14} \]

which is equivalent to the equation

\[ i\omega \rho=-\operatorname{div}\mathbf{j}. \tag{15} \]

Now let us calculate the vector and scalar potentials at points of space located at a distance from our system of charges large in comparison with the dimensions of the system itself. From Fig. 6 we have

\[ \mathbf{r}=\mathbf{R}-\mathbf{r}', \tag{16} \]

\[ r^2=R^2+r'^2-2r' \cos\vartheta, \tag{17} \]

or, approximately for \(R \gg r'\),

\[ r=R-r'\cos\vartheta. \tag{18} \]

Substituting (12), (13), and (18) into (10) and (11), we obtain

\[ \mathbf{A}(R,t)= \frac{e^{i\omega\left(t-\frac{R}{c}\right)}}{c} \int \frac{\mathbf{j}(\mathbf{r}')\,e^{i\mathbf{k}\mathbf{n}\cdot \mathbf{r}'}}{R-r'\cos\vartheta}\,d\tau', \tag{19} \]

\[ \varphi(R,t)= e^{i\omega\left(t-\frac{R}{c}\right)} \int \frac{\rho(\mathbf{r}')\,e^{i\mathbf{k}\mathbf{n}\cdot \mathbf{r}'}}{R-r'\cos\vartheta}\,d\tau'. \tag{20} \]

Here, as also in the subsequent complex expressions for \(\mathbf{E}\), \(\mathbf{H}\), etc., we mean that only the real part is to be considered; \(\mathbf{n}\) is the unit vector in the direction of \(R\), and

\[ k=\frac{\omega}{c}=\frac{2\pi}{\lambda}=\frac{1}{\lambda}, \]

where \(\lambda\) is the wavelength of the emitted radiation.

From (19) and (20) one can obtain the electric and magnetic vectors with the aid of the well-known relations:

\[ \mathbf{E}=-\operatorname{grad}\varphi-\frac{1}{c}\frac{\partial \mathbf{A}}{\partial t}, \tag{21} \]

\[ \mathbf{H}=\operatorname{rot}\mathbf{A}. \]

Differentiating with respect to the coordinates of the point \(P\), we may regard \(\mathbf{n}\) as constant, and also replace \(R-r'\cos\vartheta\) in the denominators by \(R\). We obtain

\[ \mathbf{E}=+ \frac{i\omega}{cR} e^{i\omega\left(t-\frac{R}{c}\right)} \int \left(\rho\mathbf{n}-\frac{\mathbf{j}}{c}\right) e^{i\mathbf{k}\mathbf{n}\cdot\mathbf{r}'}\,d\tau', \tag{22} \]

\[ \mathbf{H}=+ \frac{i\omega}{cR} e^{i\omega\left(t-\frac{R}{c}\right)} \int \left[\frac{\mathbf{j}}{c}\,\mathbf{n}\right] e^{i\mathbf{k}\mathbf{n}\cdot\mathbf{r}'}\,d\tau'. \tag{23} \]

Using the relation following from the continuity equation (15),

\[ i\omega \int \rho e^{ik\mathbf n\cdot \mathbf r'}\,d\tau' = -\int \operatorname{div}\mathbf j\, e^{ik\mathbf n\cdot \mathbf r'}\,d\tau' = ik\int \mathbf j\cdot \mathbf n\, e^{ik\mathbf n\cdot \mathbf r'}\,d\tau', \]

the expression for \(\mathbf E\) can be transformed into

\[ \mathbf E = -\frac{i\omega}{c^2}\, \frac{e^{i\omega\left(t-\frac{R}{c}\right)}}{R} \int \mathbf j_{\perp} e^{ik\mathbf n\cdot \mathbf r'}\,d\tau'. \tag{24} \]

Here \(\mathbf j_{\perp}\) is the projection of the vector \(\mathbf j\) onto the plane perpendicular to the vector \(\mathbf n\), or

\[ \mathbf j_{\perp}=\mathbf j-(\mathbf j\cdot \mathbf n)\mathbf n . \tag{25} \]

Formulas (22), (23), (24) show that \(\mathbf E\) and \(\mathbf H\) are perpendicular to each other and to the vector \(\mathbf n\), and that they have equal absolute values, inversely proportional to \(R\). This is the well-known property of the radiation field, satisfying the equations

\[ \mathbf E\cdot \mathbf n=0;\qquad -\mathbf H=[\mathbf E\mathbf n];\qquad \mathbf E\cdot \mathbf H=0 . \tag{26} \]

Dropping the prime on \(r\), we expand the exponential in (24) in spherical harmonics\(^6\):

\[ e^{ik\mathbf n\cdot \mathbf r} = \sum_{l=0}^{\infty} i^l(2l+1)P_l(\cos\theta) \left(\frac{\pi}{2kr}\right)^{\frac12} J_{l+\frac12}(kr), \tag{27} \]

where \(\theta\) is the angle between \(\mathbf n\) and \(\mathbf r\), and \(J_{l+\frac12}(kr)\) is a Bessel function of order \(l+\frac12\). If \(kr\ll 1\), i.e. if the wavelength of the emitted radiation is large in comparison with the dimensions of the region in which the system of charges is located, then expression (27) can be approximated by replacing \(J_{l+\frac12}\) by the lowest term in the expansion of this function in a power series. Thus we obtain (see\(^7\))

\[ \left(\frac{\pi}{2kr}\right)^{\frac12} J_{l+\frac12}(kr) = \left(\frac{\pi}{2kr}\right)^{\frac12} \frac{(kr)^{l+\frac12}} {2^{l+\frac12}\Gamma\left(l+\frac32\right)}, \]

and consequently, since

\[ 2^{l+1}\Gamma\left(l+\frac32\right) = 1\cdot3\cdot5\cdots(2l-1)\times(2l+1)(\pi)^{\frac12}, \]

we have

\[ e^{ik\mathbf n\cdot \mathbf r} = \sum_{l=0}^{\infty} \frac{i^l(kr)^l}{1\cdot3\cdot5\cdots(2l-1)} P_l(\cos\theta). \tag{27a} \]

When (27a) is substituted into (24), we find \(\mathbf E\) as a sum of integrals containing different powers of \(kr\):

\[ \mathbf E=\mathbf E_1+\mathbf E_2+\mathbf E_3+\ldots, \tag{28} \]

where the integral \(\mathbf E_i\) contains under its sign \((kr)^{i-1}\). The first terms in (28) are equal to

\[ \mathbf E_1=-\frac{i\omega}{c^2R}\,e^{i\omega\left(t-\frac{R}{c}\right)}\int \mathbf j_{\perp}\,d\tau, \tag{29a} \]

\[ \mathbf E_2=\frac{\omega}{c^2R}\,e^{i\omega\left(t-\frac{R}{c}\right)}\int \mathbf j_{\perp}\,kr\cos\theta\,d\tau, \tag{29b} \]

\[ \mathbf E_3=\frac{i\omega}{3c^2R}\,e^{i\omega\left(t-\frac{R}{c}\right)}\int \mathbf j_{\perp}(kr)^2\left(\frac{3\cos^2\theta-1}{2}\right)d\tau, \tag{29c} \]

where the prime on \(d\tau\) has been omitted.

\(\mathbf E_1\) corresponds to the field of electric dipole radiation; \(\mathbf E_2\), to the field of “magnetic dipole and electric quadrupole” radiation; \(\mathbf E_3\), to the field of “magnetic quadrupole and electric octupole” radiation, etc.

The reason for the adopted names of the radiations becomes clear after integrating by parts the integrals in \(\mathbf E_1, \mathbf E_2\), etc., using the continuity equation. Multiplying (15) by \(x\) and integrating by parts, we obtain

\[ \int j_x\,d\tau=i\omega\int \rho x\,d\tau=M_x, \tag{30} \]

where \(M_x\) denotes the \(x\)-component of the electric dipole moment of the system of charges. Analogous equations are obtained for \(M_y\) and \(M_z\).

Substitution of (30) into (29) shows that \(\mathbf E_1\) is determined simply by the vector \(\mathbf M_{\perp}\), i.e., by the projection of \(\mathbf M\) perpendicular to \(\mathbf n\). \(\mathbf E_1\) corresponds to the radiation of an electric dipole situated at the origin. Substitution of (30) into (29b) gives a more complicated expression. Under the integral we obtain a second-rank tensor with components

\[ a_{ik}=j_i x_k=b_{ik}+c_{ik},\qquad i,k=x,y,z. \tag{31} \]

It is convenient to express this tensor as the sum of two parts: one, \(b_{ik}\), symmetric with respect to interchange of \(i\) and \(k\), and the other, \(c_{ik}\), antisymmetric with respect to the same interchange. The antisymmetric part gives the electric field corresponding to the magnetic dipole

\[ \mathbf N=\frac{1}{2}\int [\mathbf r\mathbf j]\,d\tau, \tag{32} \]

whence the name “magnetic dipole radiation” is clear. The electric field corresponding to the symmetric part of the tensor is the field of an electric quadrupole.

Analogous considerations may also be applied to \(\mathbf{E}_3\), which is the sum of magnetic quadrupole and electric octupole fields, etc. The names quadrupole, octupole, etc., are connected with the fact that fields similar to the symmetric part of \(\mathbf{E}_2\) can be formed by two equal dipoles with opposite phases, displaced with respect to one another; the resulting system of 4 charges, having no dipole moment and containing 4 poles, is called a quadrupole. Similarly, from 2 quadrupoles one can compose an octupole, etc.

The expansion of the field produced by our system of charges into \(\mathbf{E}_1\), \(\mathbf{E}_2\), etc., is especially applicable if \(kx \ll 1\), since each term is, in order of magnitude, \(kx = \dfrac{x}{\lambda}\) times smaller than the preceding one, and the series converges rapidly. Here \(x\) denotes a length corresponding to the size of our system of charges; for a nucleus, the nuclear radius may be chosen approximately as \(x\). This confirms the statement concerning the order of magnitude of the various types of radiation made at the beginning of the present section.

The preceding consideration is purely classical\({}^{H40,S36}\), but it can be generalized by replacing the amplitudes, dipole moments, quadrupole tensors, etc., by the corresponding matrix elements. Such a calculation was carried out completely by Duncombe and Morrison\({}^{D2}\).

c) Selection Rules

A detailed discussion of the form of the electromagnetic field generated by an arbitrary multipole was carried out by Heitler\({}^{H3}\). He gives expressions for the fields not only in the wave zone, where they vary as \(R^{-1}\), but also near the multipole itself, where the decrease of the field occurs more rapidly; and, by quantizing the field, he also shows that electromagnetic fields arising from electric or magnetic \(2^l\)-poles have angular momenta \(l\hbar\) with respect to the center of the multipole. The corresponding calculations are rather complicated, and in carrying them out the consideration of the field in the intermediate zone is essential.

The fact that the radiation carries away angular momentum (moment of momentum), in combination with the application of the law of conservation of angular momenta to the radiating system (the nucleus) and to the radiation, leads to the establishment of a selection rule. If \(I\) and \(I'\) are the angular-momentum vectors (in units of \(\hbar\)) of two nuclear states between which an electric or magnetic \(2^l\)-pole transition takes place, then

\[ |I-I'|\leq l. \tag{33} \]

This selection rule can also be obtained directly by expanding the electric field in a series of spherical harmonics, as was done above (see (29)) and in greater detail in Heitler’s article.

Further, if the nucleus has an excited state with angular momentum \(I'\) and a ground state with angular momentum \(I\), the radiation of the lowest order associated with this transition, i.e., the most probable radiation, will be electric or magnetic multipole radiation of order \(2^l\) (see the parity considerations given below), for which

\[ l = |I - I'|. \tag{34} \]

This follows directly from (33). Indeed, one can show that

\[ |I + I'| \geq l \geq |I - I'|, \tag{35} \]

whence the minimum value of \(l\) is \(|I - I'|\).

Another selection rule can be obtained by considering the parity of the eigenfunctions of the nucleus. The parity of a given eigenfunction indicates what happens to it if the sign of all particle coordinates is changed. We have

\[ \psi(q_1, q_2, \ldots, q_n)=\pm \psi(-q_1,-q_2,\ldots,-q_n), \tag{36} \]

where \(q_i\) are the coordinates of the \(i\)-th particle (excluding its spin coordinates). If equation (36) holds with the plus sign, the eigenfunction is called even; in the opposite case, odd.

Each matrix element of the type

\[ M_{mn}=\int \psi_m^*(q_1,q_2,\ldots)X^rY^sZ^t\psi_n(q_1,q_2,\ldots)\,dq_1\,dq_2\ldots \tag{37} \]

is equal to zero if the functions \(\psi_m\) and \(\psi_n\) have the same parity and the sum \(r+s+t\) is odd, or if they have different parity and the sum \(r+s+t\) is even. In this integral \(X=\sum x_i\), etc.; all matrix elements determining the multipole radiation are of the type (37).

These results are a generalization of the well-known spectroscopic Laporte rule. Hence it follows that electric dipole radiation occurs only between states of different parity, and, in general, electric \(2^l\)-pole radiation and magnetic \(2^{\,l-1}\)-pole radiation arise only in transitions between states of the same parity if \(l\) is even, and only between states of different parity if \(l\) is odd.

It is useful to consider the lowest power of the quantity \(\left(\dfrac{x}{\lambda}\right)^2\) entering into the formulas for the intensity. We shall call this power the order of the transition and denote it by the symbol \(\Delta\).

The order of the transition is equal to 1 for an electric dipole, 2 for an electric quadrupole or magnetic dipole, etc.

Both selection rules, one connected with angular momentum and the other with the parity of the eigenfunctions, are given in Table I, which indicates the multipoles of minimum order between states of given parity, for which

\[ |l-l'|\leqslant L \leqslant |l+l'|. \]

To Table I there must be added the rule following from (35), according to which a transition between two states with \(l=0\) is forbidden for radiation of any type (see Section 4e).

Table I

Selection rules for multipole radiation

Degree of the quantity \(x/\lambda\) in the expression for the intensity	2	4	6	8	10
Electric radiation	Dipole	Quadrupole	Octupole	16-pole	32-pole
\(\left.\begin{array}{l}\|l+l'\|\geqslant \\ \|l-l'\|\leqslant\end{array}\right\}\)	1	2	3	4	5
Change in parity	Yes	No	Yes	No	Yes
Magnetic radiation	—	Dipole	Quadrupole	Octupole	16-pole
\(\left.\begin{array}{l}\|l+l'\|\geqslant \\ \|l-l'\|\leqslant\end{array}\right\}\)		1	2	3	4
Change in parity		No	Yes	No	Yes

g) Half-life of excited states

Bethe B3 roughly estimated the probability of radiation by excited nuclei by replacing the integrals in equations (29a), (29b), etc., by the expression

\[ \eta e x\omega\left(\frac{\omega x}{c}\right)^{\Lambda-1}, \]

NUCLEAR ISOMERISM

where $\eta$ is a quantity of order unity, $e$ is the electron charge, and $x$ is the nuclear radius.

Calculating the energy flux, integrating and dividing by $\hbar\omega$, we obtain the averaged number of quanta emitted per unit time by one nucleus, i.e. $\lambda_\gamma$:

\[ \lambda_\gamma = \left(\frac{\omega}{c}\right)^{2\Lambda+1} \frac{e^2}{\hbar}\eta^2 \frac{x^{2\Lambda}}{[1\cdot 3\cdot 5\cdot \ldots \cdot (2\Lambda-1)]^2}. \tag{38} \]

If we put $\eta^2=1$ and $x=1.4\cdot 10^{-13} A^{1/3}$, where $A$ is the mass of the nucleus ($O^{16}=16$), which is in agreement with the available information on nuclear radii, formula (38) gives

\[ \lg \lambda_\gamma = 20.30 - 2\lg(1\cdot 3\cdot \ldots \cdot 2\Lambda-1) - \]

\[ - (2\Lambda+1)\,1.30\lg E - 2\Lambda\left(0.84-\frac{1}{3}\lg A\right), \tag{38a} \]

where $E$ is the energy of the $\gamma$-radiation in MeV and $\lambda_\gamma$ is expressed in sec.$^{-1}$. Here $\Lambda$ is the multipole order for electric radiation. Since magnetic $2^l$-pole radiation has, from the standpoint of the calculation just carried out, the same probability as electric $2^{l+1}$-pole radiation, the decay constant of magnetic $2^l$-pole radiation is obtained from the formula given above with $\Lambda=l+1$.

This formula can be regarded only as a very rough approximation, since the replacement of the matrix elements of the corresponding degree of the product of the nuclear radius by the electron charge is justified only by dimensional considerations. However, a better approximation to reality can be obtained only through a detailed consideration of a quantitatively correct nuclear model, which at present is still impossible.

Nevertheless, some authors, proceeding from various assumptions, have undertaken more detailed calculations of nuclear half-lives. Thus, Hebb and UhlenbeckH5 assumed that the emitting object is a single alpha particle moving in the field of the remaining part of the nucleus, while KoenumaK8 took the moving emitting charge to be a single proton. LoewenL12, FierzF14 and BergelB11 used the liquid-drop model, assuming that the radiation is caused by vibrations of a charged nucleus. FlüggeF1 gave a formula for the half-life based on the assumption that the radiation is connected with the rotation of a charged drop. Some of the available experimental data were compared with these theoretical expressionsF1, B11, W16. From our point of view, existing knowledge in the field of nuclear structure is insufficient to justify the use of any particular nuclear model, and we have therefore used formula (38a). However, one point appears beyond doubt*, namely,

* This was pointed out by Dankov (see D21).

that agreement between the theoretical expressions and the experimental material is good, or acceptable, only when the correction associated with internal conversion is taken into account.

If the atom were completely deprived of all its electrons, then the decay constant of the nuclear isomeric state would be given by formula (38). The presence of atomic electrons, however, makes possible nuclear transitions in the nucleus that are accompanied by the emission of atomic electrons (internal conversion). This effect naturally increases the decay constant, and in a very good approximation we may suppose that the decay constant is

\[ \lambda=\lambda_{\gamma}+\lambda_{e}, \tag{39} \]

where \(\lambda_{\gamma}\) is given by formula (38), and \(\lambda_{e}\) is the decay constant corresponding to the probability of a transition caused by electrons.

The half-life of the excited state \(\tau\) is determined by the relation

\[ \tau=\frac{0.69}{\lambda_{e}+\lambda_{\gamma}} =\frac{0.69}{\lambda_{\gamma}}(1+\alpha), \tag{39a} \]

and since the coefficient of internal conversion \(\alpha\) can have values very large in comparison with unity, it is clear that internal conversion may be an essential factor in determining the half-life of an excited state.

Fig. 7. \(\lg \tau\) as a function of \(\lg E\) for \(Z=35,\ A=80\). — \(\tau_{\gamma}\) according to (38a); — — — \(\tau\) according to (39a). Logarithms to base 10 are used; the values of \(\Lambda\) are indicated in the figure.

In Fig. 7 the values \(\tau_{\gamma}=\dfrac{0.69}{\lambda_{\gamma}}\) and \(\tau\) according to (38) and (39a) are plotted as functions of the gamma-radiation energy for \(A=80,\ Z=35\). The values of \(\alpha\) in (39a) were taken from theoretical calculations (see the following sections). From Fig. 7 the importance of internal conversion is clear from the point of view of lowering the half-life of states with low energy. Despite the extremely rough approximations, it appears probable that, for a given energy, transitions with different \(\Lambda\) have half-lives sufficiently different from one another to make it possible to find \(\Lambda\) for

of this transition, with the known half-life and energy.

Apparently it is expedient to make a few remarks concerning the various formulas for the decay constant given above. Bethe’s formula (38) is based only on the most general considerations. In any theory the probability of emission of an electric \(2^l\)-pole quantum is proportional to \(\omega^{2l+2} x^2\), where \(\omega\) and \(x\) have the meanings indicated above. If we simply divide the corresponding expression by \(\omega\), then for the decay constant we obtain equation (38). However, in the formulas of Fierz, Lowen, Flugge, and Berthelot a particular model of the nucleus was used—the liquid drop; in this case the energy of oscillations in the drop is proportional to \(\omega^2\)*). Therefore the decay constant becomes proportional to \(\omega^{-l}\) instead of \(\omega^{2l+1}\). Furthermore, the expression for the energy of oscillations contains the square of the nuclear radius, and therefore the resulting decay constant is proportional to the nuclear radius to the power \(2l-2\). Finally, since it is assumed that the radiation is determined by the entire nuclear charge \(Z\), \(\lambda\) depends on \(Z^2\), whereas in Bethe’s formula there is no dependence on \(Z\).

An attempt was made to compare the experimental results with the formulas of Bethe and Fierz \(^{F14}\). In the table of isomers given below, the theoretical values of the half-lives are obtained from Bethe’s formula, which, generally speaking, gives somewhat overestimated values of the half-lives. Fierz’s formula usually gives too small values of the half-life. Of 25 cases for which the data are considered reliable, the two formulas give different \(\Lambda\)’s only in 4 cases. The experimental data are not good enough to establish by what law \(\Lambda\) depends on \(\omega\) and on the nuclear radius. A more detailed comparison of theory with experiment will become possible after more accurate data have been obtained on the conversion coefficient, which remains in the calculations a somewhat uncertain quantity, since \(\alpha\) for magnetic \(2^l\)-pole radiation is not equal to \(\alpha\) for electric \(2^{l+1}\)-pole radiation**).

d) Internal conversion

Fortunately, the calculations of \(\alpha\), although complicated, are based on a considerably more reliable foundation than the estimates of \(\lambda\). The point is that the estimate of \(\lambda\) is connected with a very inaccurate determination of the nuclear matrix elements, which depend on the details of the structure of the nucleus.

* Just as the energy of a harmonic oscillator \(x = a \cos \omega t\) is equal to
\[ \frac{1}{2} m a^2 \omega^2 . \]

** Alex and Dankov \(^{A11}\) found good agreement between 50 experimental results and a theoretical formula having the same dependence on the energy and radius of the nucleus as equation (38).

The calculation of \(\alpha\), on the contrary, is based on electrodynamics and on knowledge of the structure of the atom and can claim an accuracy comparable with the accuracy of most ordinary spectroscopic calculations.

The corresponding calculations were first carried out by Hulme, Mott, Taylor, and F. Oppenheimer \(^{\mathrm{H4,T1,T2}}\) for heavy elements in the case of dipole and quadrupole radiation. Calculations for light elements and for multipoles of higher order were performed by Hebb and Uhlenbeck \(^{\mathrm{H5}}\), Dankov and Morrison \(^{\mathrm{D2}}\), and Hebb and Nelson \(^{\mathrm{H6*}}\).

The basic idea of such calculations is to determine the probability of a transition of \(K\)-, \(L\)-, and other atomic electrons into continuum states under the influence of the perturbation produced by the electromagnetic field of a multipole located at the origin of coordinates (at the center of the nucleus) and emitting one quantum per second. A characteristic feature of this probability is its substantial dependence on the electric and magnetic field in the vicinity of the nucleus, where the \(K\)- and \(L\)-electrons are situated. In this region the field decreases according to the law \(r^n\), where \(n\) is related to the order of the multipole \((n = l + 1\) for an electric \(2^l\)-pole), and, consequently, the study of internal conversion makes it possible to determine the order of the multipole. The usual nonstationary perturbation theory is applied, and the main task is the calculation of matrix elements for the perturbing potential that connects the \(K\)- and \(L\)-states with continuum states.

Calculations covering all possible cases must be too complicated, especially when relativistic effects are taken into account, which are significant for fast \(K\)-electrons (heavy elements) or fast emitted electrons. However, for the majority of known measured transitions the energy of the emitted electrons and the kinetic energy of the \(K\)-electrons (for example, \(18\ \text{keV}\) for \(Z = 40\), \(43\ \text{keV}\) for \(Z = 60\)) are small in comparison with \(mc^2\), so that relativistic effects are not very significant \(^{**}\).

Conversion on the \(K\)-shell. Dankov and Morrison give formulas for the conversion of electric \(2^l\)-pole radiation

* The most recent calculations of magnetic conversion coefficients were published in \(^{\mathrm{S37,D25,L20}}\), and for electric dipole conversion coefficients in \(^{\mathrm{G23}}\). (See also I. S. Shapiro, “Internal conversion of \(\gamma\)-rays and the determination of quantum characteristics of nuclear levels,” UFN 40, 189 (1950). In this review, in particular, a number of works by Soviet authors not reflected in the article by Segrè and Helmholz are covered. Translator’s note.)

** For \(Z > 60\), \(E > 250\ \text{keV}\), and for magnetic transitions, relativistic effects must be taken into account.

on two \(K\)-electrons in the nonrelativistic approximation. The formula is valid for \(\dfrac{v}{c}\ll 1\) (\(v\) is the velocity of the emitted electrons):

\[ \frac{16\alpha l}{l+1}\left[\Gamma\left(l+\frac{1}{2}\right)\right]\left[\frac{2}{\nu}\right]^{l+1} \frac{n^{4}}{(1+n^{2})^{l-2}}\times \]
\[ \times \frac{\left[(l+1)(1+n^{2})^{l-2}e^{-\frac{2n}{\operatorname{ctg}^{-1} n}}-V_l\right]^2} {[l^{2}+n^{2}][(l-1)^{2}+n^{2}]\ldots[1+n^{2}]\,(1-e^{-2\pi n})}, \tag{40} \]

where \(V_l\) satisfies the equation

\[ V_{l+1}=V_l(1+n^2)\frac{l+2}{l+1} +\frac{2^{l+1}l}{(2l+2)!}\left(\frac{1}{1+n^2}\right)^l \prod_{i=1}^{l}(i^2+n^2) \tag{41} \]

and

\[ V_0=0. \tag{42} \]

In these formulas \(\alpha\) is the fine-structure constant, \(\nu\) is the energy of gamma radiation in units of \(mc^2\), and

\[ n=\frac{Z\alpha}{(2\nu-Z^2\alpha^2)^{\frac{1}{2}}} \]
\[ =\left(\frac{\text{binding energy of the }K\text{-electrons}} {\text{kinetic energy of the conversion electrons}}\right)^{\frac{1}{2}} =\frac{Ze^2}{h v}. \tag{43} \]

Formula (40) is considerably simplified if the condition \(n\ll 1\) is satisfied, i.e., if the binding energy of the \(K\)-shell is small in comparison with the kinetic energy of the conversion electrons or, better, if the velocity of the \(K\)-electrons is small in comparison with the velocity of the conversion electrons. In this case

\[ \alpha_{Kl}=Z^3\alpha^4\frac{l}{l+1}\left(\frac{2}{\nu}\right)^{l+\frac{5}{2}}. \tag{44} \]

If the influence of spin is neglected, magnetic multipole radiation is not converted on the \(K\)-shell.

In the relativistic approximation, i.e., when Dirac wave functions are used and the binding energies of the \(K\)-electrons are neglected (\(n\ll 1\)), Dankov and Morrison found

\[ \alpha_{Kl}=\frac{2Z^3\alpha^4}{\nu^3} \left(\frac{\nu+2}{\nu}\right)^{l-\frac{1}{2}} \left[\frac{(l+1)\nu^2+4l}{l+1}\right], \tag{45} \]

\[ \beta_{Kl}=\frac{2Z^3\alpha^4}{\nu} \left(\frac{\nu+2}{\nu}\right)^{l+\frac{1}{2}}, \tag{46} \]

where \(\alpha_{Kl}\) and \(\beta_{Kl}\) are the conversion coefficients on the \(K\)-shell for electric \(2^l\)-pole and magnetic \(2^l\)-pole radiation.

Quantitative results for gamma rays with energies below \(0.2\,\mathrm{MeV}\) and \(Z<40\) are given by formula (40). For \(Z<30\), high energies, and not too high multipole orders, formulas (45) and (46) give acceptable estimates. For \(Z>50\), numerical calculations are necessary in order to obtain exact results. These calculations, important for nuclear physics, can be carried out comparatively easily by using the Fermi–Thomas potential for the atomic field and by numerically integrating the Schrödinger equation with the aid of the latest computing machines. In the formulas given above, hydrogen wave functions were used.

Conversion on the \(L\)-shell and numerical results. Analogous calculations connected with conversion on the \(L\)-shell were performed by Hebb and Nelson \({}^{6}\). They also give numerical results for the conversion coefficient on the \(K\)- and \(L\)-shells in the case of an electric multipole.

The final formulas, based on the same approximations as in the case of the \(K\)-shell, are more complicated, and we shall not present them here. Hebb and Nelson give a very convenient table by means of which \(\alpha_K\) and \(\alpha_L\) can be calculated. A graph of \(\alpha_K\) as a function of the energy for \(Z=35\) is given in Fig. 8. Figure 9 presents, borrowed from the paper of Hebb and Nelson, a graph of \(\dfrac{\alpha_K}{\alpha_L}\) as a function of \(\dfrac{Z^2}{E}\).

Fig. 8. Internal conversion coefficient on the \(K\)-shell as a function of the energy of \(\gamma\)-rays. Electric radiation, \(L=35\).

It is necessary to note that the mentioned table \({}^{6}\) gives \(\lg[\gamma^{2l+2}\alpha_K]\) as a function of \(\dfrac{W}{\gamma^2}\), where \(W\) has the same meaning as the quantity \(\nu\) introduced above, i.e. \(W\) is the energy of \(\gamma\)-radiation in units of \(mc^2\), and \(\gamma=\dfrac{Z}{137}\).

Consequently,

\[ \frac{W}{\gamma^2}=\frac{(137)^2E}{mc^2Z^2}, \]

or

\[ \frac{36.7 E}{Z^2}, \]

if \(E\) is expressed in keV. To obtain the best results in computations of

\[ \frac{\alpha_K}{\alpha_L}, \]

a table should be used, not a graph; in all the preceding formulas the approximation can be improved if the screening of the nuclear charge for the various orbits is taken into account, as is done in the theory of x-rays.

Fig. 9. Curves for \(N_K/N_L\) as a function of \(Z^2/E\) for electric multipole radiation. The curve is taken from Phys. Rev. 58, 489 (1940).

This is achieved by replacing \(Z\) by \(Z_{\mathrm{eff}} = Z - \sigma\), where \(\sigma = 0.30\) for the \(K\)-shell and \(\sigma = 4.15\) for the \(L\)-shell. The results given by Hebb and Nelson are exact for \(Z = 35\), and the deviations for \(25 < Z < 50\) should lie within the limits of 10–20%.

For magnetic multipole radiation and the \(L\)-shell, the conversion coefficient is given by the expression

\[ \beta_L = \frac{Z^3 \alpha^4}{4\nu} \left( \frac{\nu + 2}{\nu} \right)^{l+\frac{1}{2}} \left\{ 1 + \frac{Z^2 \alpha^2}{4}\, \frac{\nu + 2}{\nu} \left[ \frac{l + 1}{2l + 1} + \frac{l(2l + 1)}{4} \left( \frac{2l - 1}{2l + 1} - \frac{\nu}{\nu + 2} \right)^2 \right] \right\}. \tag{47} \]

As can be seen, this expression contains \(\beta_K\) as a factor (see formula (46)). In this case as well, the approximation can be improved by taking screening into account.

e) 0—0 transitions

A special case occurs in transitions between two states with \(I=0\). Such a transition is strictly forbidden for all types of electromagnetic radiation.

If the two states have the same parity, atomic electrons can be emitted directly, and the probability of the transition depends on the penetration of the atomic electrons into the nucleus. R. H. Fowler\(^{F3}\) calculated this probability. The RaC gamma radiation with energy \(1.426\) MeV, according to Ellis and Aston\(^{E2}\), is completely converted and is usually regarded as a \(0—0\) transition, completely forbidden in radiation\(^{F3}\). The same penetration effect also acts in cases where radiative transitions are possible, but in this case it is insignificant\(^{T2}\). Another possible case was reported by Bowe et al.\(^{B13}\), who found in Ge\(^{72}\) (formed in the decay of Ga\(^{72}\)) strongly converted \(\gamma\)-radiation \((\alpha>1)\) with an energy of \(0.7\) MeV and a half-life of \(5\cdot 10^{-7}\) sec., and no agreement with the Weizsäcker formula was obtained.

If, however, both states have different parity, then even this type of transition is absent; the transition can then occur only as a result of a second-order effect, for example as a result of the simultaneous emission of two quanta or two electrons. Sachs\(^{S6}\) calculated the probability of such effects, which may be observed in certain cases. In the case of the emission of two electrons, it would be very characteristic that these electrons would be emitted simultaneously and would have a broad spread of energies instead of the fine line that occurs in ordinary transitions, in which conversion electrons are emitted from monochromatic gamma radiation. Up to now, a possible experimental indication of the existence of such transitions exists only in the case of the isomer Ir\(^{192}\) with a half-life of \(1.5\) min., which apparently emits both a continuous spectrum of \(\gamma\)-rays and conversion electrons. Goldberger\(^{G11}\) carried out calculations for the case of the emission of one electron and one \(\gamma\)-quantum for various values of the energies of the intermediate states and various combinations of parities.

g) Application of the theory

Summarizing the results of the preceding section, we see that the half-life of a nucleus in an excited state depends on the nuclear charge, on \(E\) and \(\Lambda\), i.e. on the atomic number, the energy, and the order of the transition to the ground state. However, the theoretical determination of this dependence is very inaccurate, since it requires more complete knowledge of the structure of the nucleus than that which we possess.

The conversion coefficients \(a_K, a_L, \beta_K, \beta_L\) are also functions of \(Z, E, l\), but their values can be calculated more accurately than \(\tau\), since the corresponding calculations are based on the use of electrodynamics and atomic theory.

Experimentally one can measure \(Z, E, \tau\), the numbers of \(K\)- and \(L\)-conversion electrons \((N_K\) and \(N_L)\), and the number of gamma quanta \(N_\gamma\) emitted in the decay. Usually the first three of these quantities and the ratio \(\dfrac{N_K}{N_L}\) can be measured accurately. Measurements of \(N_K\) and \(N_\gamma\), or of their ratio, are considerably more difficult and, consequently, are known only approximately, except in the case when the conversion electrons are accompanied by decay electrons (see, for example, the cases of \(\operatorname{In}^{114}\) and \(\operatorname{In}^{115}\), studied by Cork and Lawson \(^{L1}\)).

Let us apply the theoretical results obtained above first to the case in which magnetic multipole radiation may be neglected in comparison with electric multipole radiation. This, generally speaking, occurs for even \(l\) and a transition between states of the same parity, or for odd \(l\) and a transition between states of different parity (see Table I). In such cases the number of electrons ejected in the conversion of magnetic multipole radiation may also be neglected.

The theory gives the following functional relations:

\[ \tau = \tau(Z, E, \Lambda) \quad \text{according to formulas (39a), (38);} \tag{48} \]

\[ a_K = a_K(Z, E, l) \quad \text{according to the table in } {}^{H6}; \tag{49} \]

\[ a_L = a_L(Z, E, l) \quad \text{according to the table in } {}^{H6}. \tag{50} \]

Therefore, having measured \(Z, E, \tau, a_K, \dfrac{a_K}{a_L}\), we have three independent estimates of \(\Lambda\) (for this case \(\Lambda = l\)), which must be in agreement with one another.

Another method for determining \(l\) for successive \(\gamma\)-quanta is the angular correlation between these successively emitted \(\gamma\)-quanta. The theory in this case was developed by Hamilton \(^{H15}\) and Goertzel \(^{G12}\). Brady and Deutsch \(^{B14}\) showed that this effect exists in \(\operatorname{Co}^{60}\) and \(\operatorname{Sc}^{46}\) (see the discussion in Section 7). Angular correlation is sensitive to the values of \(I\) for the three states participating in the two transitions. The possibility of obtaining further information on the values of \(I\) is connected with observation of the correlation between the polarizations of successively emitted \(\gamma\)-quanta; the corresponding calculations were carried out by Falkoff \(^{F8}\). This effect, in contrast to angular correlation, depends on the electric or magnetic character of the radiation. Polarization correlation of annihilation quanta was

experimentally confirmed by Bleuler and Bradt \(^{B12}\). Deutsch and Metzger \(^{D24}\) successfully measured this effect for the \(\gamma\)-rays of Rh \(^{106}\). When both of the above-mentioned effects are used, rapid emission of the second \(\gamma\)-quantum is assumed, and consequently in isomeric transitions nothing can be obtained in this way.

In the case when the intensity of magnetic multipole radiation is comparable with the intensity of electric multipole radiation, i.e., for odd \(l\) and transitions between states of the same parity or for even \(l\) and transitions between states of different parity, the quantity \(\Lambda\) can be determined from the value of \(\tau\) and from the energy according to formula (38), neglecting the influence of \(\alpha\) or using a roughly approximate value of this quantity. In this case, however, the ratio \(\frac{\alpha_K}{\alpha_L}\) does not correspond to its value given in Fig. 9. This discrepancy will indicate the presence of magnetic multipole radiation, and we therefore assume that any discrepancy between the experimental value of \(\frac{\alpha_K}{\alpha_L}\) and the value given in Fig. 9 is associated with magnetic multipole radiation. The ratio between the number of conversion \(K\)-electrons \((N_K)\) and the number of conversion \(L\)-electrons \((N_L)\) is equal to

\[ \frac{N_K}{N_L} = \frac{N_{\gamma e}\alpha_K + N_{\gamma m}\beta_K} {N_{\gamma e}\alpha_L + N_{\gamma m}\beta_L}, \tag{51} \]

where \(N_{\gamma e}\) and \(N_{\gamma m}\) are the numbers of quanta emitted in electric and magnetic multipole radiation*). The quantities \(N_{\gamma e}\) and \(N_{\gamma m}\) have the following meaning: if we can decompose the electric field corresponding to our radiation into the sum of two fields \(E_e\) and \(E_m\), produced respectively by electric and magnetic multipoles, then

\[ \frac{N_{\gamma e}}{N_{\gamma m}}=\frac{E_e^2}{E_m^2}. \]

In formula (51) the quantities \(\alpha_K, \alpha_L, \beta_K, \beta_L\) are taken from Table 6 and from formulas (46) and (47), while the ratio \(\frac{N_K}{N_L}\) is measured directly. Thus we find the ratio \(\frac{N_{\gamma e}}{N_{\gamma m}}\), i.e., the ratio of the intensities of electric and magnetic multipole radiations.

*) According to M. Nelson, there are no interference effects that could make formula (51) unacceptable.

Even more interesting than the ratio \(\dfrac{N_{\gamma e}}{N_{\gamma m}}\) is the ratio \(\dfrac{\lambda_e}{\lambda_m}\) between the numbers of nuclear transitions caused, respectively, by the electric and magnetic multipoles of the nucleus.

This ratio is related to the ratio \(\dfrac{N_K}{N_L}\) by the formula

\[ \frac{\lambda_e}{\lambda_e+\lambda_m} = \]

\[ = \frac{1+a_K+a_L}{1+a_K+a_L+(a_K N_K-a_L N_L)(1+\beta_K+\beta_L)(\beta_L N_K-\beta_K N_L)} . \tag{52a} \]

For the practically important limiting case in which all the internal-conversion coefficients \(a_K, a_L, \beta_K, \beta_L\) are large in comparison with unity, this formula gives

\[ \frac{\lambda_e}{\lambda_e+\lambda_m} = \frac{(a_K+a_L)\left(\dfrac{N_K}{N_L}-\dfrac{\beta_K}{\beta_L}\right)N_L} {(N_K+N_L)\left(\dfrac{a_K}{a_L}-\dfrac{\beta_K}{\beta_L}\right)a_L}. \tag{52b} \]

An example of such mixed radiation may be the radiation of \(\mathrm{Te}^{131}\) with a half-life of 30 hours \(^{\mathrm{H}6}\)*); such radiation will probably also be found in the complex spectra of naturally radioactive substances (see, for example, \(^{\mathrm{M}1}\)). At present, however, for such heavy elements we do not have sufficiently detailed calculations of the conversion coefficients that are necessary for a systematic comparison of the theory with the experimental material. Details concerning the application of these ideas to certain special cases are given in Section 7.

A more direct test of the theory would be the measurement of nuclear spins before and after the isomeric transition, either by a spectroscopic method through the study of the hyperfine structure, or by the molecular-beam method. For the corresponding measurements the nucleus \(\mathrm{Kr}^{83}\) and some others could be used, although an attempt undertaken in this direction

*) In \(^{\mathrm{H}6}\) the quantity \(\dfrac{\lambda_e}{(\lambda_e+\lambda_m)}\) was calculated, and in the case of \(\mathrm{Te}^{131}\) there is a misprint. After correction of this misprint, the corresponding passage in \(^{\mathrm{H}6}\) reads: “The experimental data on the radiation can be explained if the \(\gamma\)-radiation is assumed to be \(\dfrac{2}{3}\) magnetic multipole radiation with \(l=4\) and \(\dfrac{1}{3}\) electric multipole radiation with \(l=5\).”

(Mrozovskii and Segre, 1940) the measurement of nuclear spins by the spectroscopic method was not crowned with success because of the insufficient intensity of the radiation. There is no doubt that the large quantities of radioactive substances obtained by modern methods will make it possible to carry out such an experiment successfully.

Another consequence of Weizsäcker’s theory, deserving mention, is the small probability of the existence of two metastable states of one and the same nucleus. Taking the minimum difference between the spins of the excited and ground states of the nucleus to be equal to four units, we see that in the most favorable case, when the spin in the ground state is equal to 0, we must expect, for the excited states, spins equal to 4 and 8. A spin equal to 8 must be regarded as exceptionally large, even after Schuler and Gollnow’s discovery in S¹⁰ of a spin \(I = 7\) in Lu. Of the seventy known cases with odd mass number, only in one case, Sb¹²⁴ (not counting Cd¹¹¹ and Te¹²¹, for which the half-life is \(\sim 10^{-8}\) sec.), were indications obtained of the existence of two states M¹⁶. This question is discussed in Section 7.

A very important question concerning excited states is their connection with the nuclear model. The most successful in this respect is the liquid-drop model, and although such a model is still a very crude approximation, it makes it possible to understand, at least qualitatively, some spectroscopic features that allow the measurements to be explained. It is necessary to consider two basic types of motion of the drop: rotations and surface oscillations. Rotation of the nucleus as a whole should lead to the appearance of rotational levels*):

\[ E = \frac{\hbar}{2J} I(I+1), \tag{53} \]

where \(J\) is the moment of inertia of the nucleus. If we represent the nucleus in the form of a solid sphere with mass \(M\), then \(J\) is equal to \(\frac{2}{5}MR^2\), where \(R\) is the radius of the nucleus.

Using the relation of mass to radius, we have

\[ E = \frac{2.43 \cdot 10^6}{A^{5/3}} I(I+1)\ \text{electron-volts}. \tag{54} \]

The probabilities of transitions between rotational states must be extremely small, since the rotating sphere is entirely

*) The rotational levels determined by formula (53) will occur only for a solid (rigid) nucleus. In the case of a liquid drop there are no grounds for using formula (53). (Author’s note.)

does not radiate, and the radiation is caused only by the deviation of the drop’s shape from the spherical one. However, the energy levels associated with vibrations also lie close to one another, so that the rotational levels cannot be regarded as “fine structure” in comparison with the vibrational levels, as is the case in molecular spectra. Therefore the interaction between rotation and vibrations is very substantial, and Frenkel\(^{F6}\) attempted to relate the lower nuclear levels precisely to the rotational-vibrational levels of the droplet model.

For a nucleus possessing certain symmetry elements, the Pauli principle, even in the case of pure rotation, excludes many energy levels for reasons similar to those which exclude half of the rotational levels in diatomic molecules with identical nuclei of zero spin. This question was discussed by Teller and Wheeler\(^{T3}\).

Mattauch\(^{M12}\) drew attention to the fact that there are no isomeric pairs with an even number of protons and with an even number of neutrons. As a possible interpretation of this fact, he suggested that for nuclei of this type (for even-even nuclei) the distance between the lower levels is large, and therefore the probability of transition between them never becomes sufficiently small for isomerism to be observed. A violation of this rule is the case of Pb\(^{204}\), which, however, has not yet been thoroughly studied; if the limits within which we consider nuclear states isomeric are extended, the nucleus Ge\(^{72}\) is also an exception. Thus the “Mattauch rule” cannot be regarded as strict, and, probably, its chief significance in the study of isomerism lies in the possibility of indicating, in a case where the mass of an isomeric nucleus is unknown, the more probable isotope.

5. PRODUCTION OF ISOMERIC NUCLEI

Isomeric nuclei are formed in various ways. We may divide the methods of producing them into a) electromagnetic excitation and b) nuclear reactions. Electromagnetic excitation has been studied only recently; excitation was achieved directly by the action of high-energy x-rays\(^{P5}\), or by the action of the electric field produced by rapidly moving charged particles. For this purpose \(\alpha\)-particles\(^{L2}\), protons\(^{B4}\), and electrons\(^{C2,M22}\) are used. The second method of producing isomers as a result of nuclear reactions is older; as we have already said, the first pair of isomers was discovered in the products of beta decay\(^{H1}\). At present, to obtain isomers by means of nuclear transformations, slow-neutron capture, inelastic collisions of fast neutrons, the photoelectric effect, and also bombardment of nuclei by charged particles are used.

In some cases isomeric nuclei are formed directly (direct formation); however, in the vast majority of cases the formation of isomers proceeds in two stages: first, the formation of a highly excited state of the nucleus, and second, the decay from this highly excited state into a metastable state as the result of cascade processes (indirect formation).

In cases of direct formation the isomeric state is usually obtained as the result of beta decay. Direct formation by the method of electromagnetic excitation is impossible, since the same selection rules that ensure the metastability of the isomeric state prevent a nucleus in the ground state from absorbing radiation with a transition to the metastable state. This assumption is confirmed by the experiments described at the end of the present section. In the case of transformations in which particle capture occurs, direct formation is also practically impossible, since after capture the nucleus remains in a state with an excitation energy of several MeV, which is considerably higher than the excitation expected for any isomeric state.

Fig. 10. Nuclear levels. Starting from level \(A\), the metastable level \(B\) can be reached by a cascade process. The ground state \(C\) can be obtained from \(A\) by another cascade process or as the result of a single jump. Only electric dipole radiation is considered.

In cases of indirect formation the primary process brings the nucleus into a highly excited state, from which, as a result of cascade processes, it passes into a metastable state. In order, in a short time, to reach states with a noticeable difference of spins by means of dipole or quadrupole radiation, many stages with \(l = 1\) or \(2\) are required. A very schematic diagram qualitatively explaining this cascade process is given in Fig. 10. In this diagram the levels with a plus sign are even levels, and the levels with a minus sign are odd levels. Suppose that the nucleus formed by means of a transformation is initially in state \(A\), which has \(l = 1\) and is even. Then the nucleus begins to lose excitation, which can occur in many,

competing paths. In Fig. 10 are shown the jumps corresponding to three competing processes. These processes are as follows: a single jump bringing the nucleus into the ground state, a series of jumps bringing it to the same state, and, finally, a series of jumps bringing it to the metastable state \(B\). In the last case the nucleus remains in an excited state, and the half-life corresponding to the jump from \(B\) to \(C\), owing to the large difference in spins, must be very long.

Although this cascade process has not yet been studied in detail, it is detected experimentally by the appearance of several quanta emitted upon the capture of one neutron, the energy of these gamma quanta associated with the capture of a slow neutron being very low. The existing energy measurements are in some contradiction with one another \(^{R1,F5,M11}\). The latest measurements \(^{M11}\) show that the most probable energy of the \(\gamma\)-rays emitted in the reaction \(\mathrm{Cd}(n,\gamma)\) may reach \(5\) MeV.

The cascade process is substantially broadened in connection with the broadening and overlapping of strongly excited levels; there are still very few corresponding experimental data.

Among the few experiments devoted to this question we may mention those connected with the study of the behavior of slow and resonance neutrons in the formation of isomers. Most of the experimental material relates to genetically related isomers, and the cross sections must be calculated from the experimental data with this circumstance taken into account.

Formula (8) expresses the ionization caused by a pair of genetically related isomers. It is still necessary to know the relation between the initial populations of the levels \(N_1\) and \(N_2\) and the cross section for their formation. This relation is as follows:

\[ N_2 = \Phi \frac{\sigma_2}{\lambda_{2\gamma}}\left(1 - e^{-\lambda_{2\gamma} t}\right), \tag{55} \]

\[ N_1 = \Phi \left\{\frac{\sigma_1+\sigma_2}{\lambda_{1\beta}}\left(1 - e^{-\lambda_{1\beta} t}\right) + \frac{\sigma_2}{\lambda_{2\gamma} - \lambda_{1\beta}} \left(e^{-\lambda_{2\gamma} t} - e^{-\lambda_{1\beta} t}\right)\right\}, \tag{56} \]

where \(\sigma_1\) and \(\sigma_2\) are the cross sections for the formation of states 1 and 2, \(\Phi\) is proportional to the number of bombarding particles incident per unit surface area of the sample under investigation per unit time, and \(t\) is the irradiation time. The activity measured at any subsequent moment is determined by formula (8). Formulas (55) and (56) follow directly from the law of radioactive decay.

If \(t \gg \dfrac{1}{\lambda_1\beta}\) and \(t \gg \dfrac{1}{\lambda_2\gamma}\), we have

\[ N_2=\Phi\left(\frac{\sigma_2}{\lambda_2\gamma}\right), \tag{57} \]

\[ N_1=\Phi\frac{(\sigma_1+\sigma_2)}{(\lambda_1\beta)}. \tag{58} \]

Unfortunately, many experimental data were processed without taking into account the genetic relationship between isomers, and at the same time the bombardment time is not indicated in the corresponding articles, so that the cross sections cannot be recomputed on their basis.

Measurements of the relative cross sections for the formation of the isomers \(Rh^{104}\) and \(Br^{80}\) showed that the resonance energy is the same for both levels \(S^{28,A2,F10,P6}\). This may be regarded as evidence that slow neutrons in both isomers are captured by a single level and that the process leading to the formation of the isomers follows this capture. Pontecorvo also determined the influence, on the relative cross section, of a change in the energy of the neutrons producing radioactive rhodium. On the basis of his data one can obtain the ratio of the cross sections for the formation, by thermal neutrons, of the excited and ground states of rhodium

\[ \frac{\sigma_2}{\sigma_1}=\frac{1}{13.5}. \]

For neutrons with mean energy less than \(0.1\) MeV, but considerably exceeding thermal energy, he found \(\dfrac{\sigma_2}{\sigma_1}=\dfrac{1}{7.4}\). This indicates that capture by many levels leads, on the average, to cascade processes different from absorption by a single level or by a few levels. For still faster neutrons (\(2.4\) MeV) Redeman \(R^2\) found \(\dfrac{\sigma_2}{\sigma_1}=\dfrac{1}{2.6\pm}\). Similar experiments were carried out for \(Br^{80}\) by Zoltan and Wertenshtein \(S^{21}\), and for \(Br^{80}\), \(Zn^{69}\), and the isomeric pair Pt by Nag \(N^1\). Their results agree qualitatively with those given above for Rh.

If the energy of the captured neutrons is increased, so that capture occurs on one of many levels with all possible angular momenta, then one may expect that the influence of the level on which capture occurs will be smeared out, and in the limiting case the cross section for the formation of isomers should be determined only by the statistical weights \((2I+1)\) of the isomeric states themselves.

In the case of slow neutrons the absolute magnitude of the cross sections for the formation of isomers fluctuates strongly, as

this generally occurs in reactions of capture of slow neutrons. Cross sections for the formation of isomers in a number of cases of slow-neutron capture have been measured. Table II gives the results obtained in S^14 and constituting the most complete data in this field. In all cases for which the upper and lower states are known, the upper state is indicated first.

The following considerations are relevant to the formation of isomers as a result of capture of a slow neutron. State \(A\) in Fig. 10 must have a spin differing by \(\dfrac{1}{2}\) (by the angular momentum of the neutron) from the spin of the capturing nucleus. On the basis of the isomerism and \(\beta\)-decay of the isomers obtained, one can determine which of the two isomeric states \(B\) or \(C\) has the larger spin and which the smaller. In Table II the state having a spin closer to the spin of state \(A\) should have the larger cross section. In Section 7 these arguments are applied to Co^60 and Br^80.

The data of Table II, with the exception of one or two cases, agree with the considerations set forth.

Returning to the primary process, i.e., to the formation of a highly excited state that initiates cascade processes, let us consider electrical excitation.

Pontecorvo and Lazard P^5 discovered the simplest and very typical case of the formation of In^115m as a result of irradiating ordinary indium with a continuous spectrum of x-rays with a maximum energy of 1.85 MeV. Miller and Waldman M^22 showed that this reaction has a threshold equal to \(1.04 \pm 0.02\) MeV. The known isomeric state has an energy 0.34 MeV above the ground state. This result can be interpreted by assuming that In^115 has, at 1.04 MeV, an excited level from which transitions to both states—the ground and the isomeric—are possible. This level is excited directly by the x-rays and is the starting point for the cascade process by which the metastable level is reached. The excitation corresponds to a narrow absorption line, as was shown by Guth G^5, and measurement of the excitation function under such conditions characterizes only the x-ray spectrum but gives no indication of the nuclear process. However, if the energy of the x-rays is greatly increased, other nuclear levels may become effective, becoming starting points of cascade processes; then one may expect a jump in the cross section for isomer formation. This effect was in fact observed by Miller and Waldman M^22.

Excitation by electron impact is directly connected with excitation by x-rays. Collins and Waldman C^2

Table II

Neutron cross sections

Radioactive isomer	Half-life	Cross section for thermal neutrons (in units of $10^{-24}\ \mathrm{cm}^2$)	Cross-section ratio	Radioactive isomer	Half-life	Cross section for thermal neutrons (in units of $10^{-24}\ \mathrm{cm}^2$)	Cross-section ratio
$_{20}\mathrm{Ca}^{49}$	30 min.	0.55	2.68	$_{49}\mathrm{In}^{116}$	54 min.	144.06	2.79
$_{20}\mathrm{Ca}^{49}$	150 min.	0.205	2.68	$_{49}\mathrm{In}^{116}$	13 sec.	51.8	2.79
$_{22}\mathrm{Ti}^{51}$	6 min.	0.141	3.6	$_{53}\mathrm{Te}^{127}$	90 days	0.073	0.094
$_{22}\mathrm{Ti}^{51}$	72 days	0.039	3.6	$_{53}\mathrm{Te}^{127}$	9.3 hours	0.78	0.094
$_{27}\mathrm{Co}^{60}$	10.7 min.	0.66	0.030	$_{53}\mathrm{Te}^{129}$	32 days	0.0154	0.116
$_{27}\mathrm{Co}^{60}$	5.3 years	21.7	0.030	$_{53}\mathrm{Te}^{129}$	72 min.	0.113	0.116
$_{30}\mathrm{Zn}^{69}$	13.8 hours	0.31	0.288	$_{53}\mathrm{Te}^{131}$	30 hours	$<0.008$	0.036
$_{30}\mathrm{Zn}^{69}$	57 min.	1.09	0.288	$_{53}\mathrm{Te}^{131}$	25 min.	0.222	0.036
$_{32}\mathrm{Ge}^{71}$	40 hours	0.073	0.0162	$_{55}\mathrm{Cs}^{134}$	3 hours	0.016	0.000625
$_{32}\mathrm{Ge}^{71}$	11 days	$\sim 0.45$	0.0162	$_{55}\mathrm{Cs}^{134}$	1.7 years	25.6	0.000625
$_{34}\mathrm{Se}^{81}$	57 min.	0.033	0.072	$_{63}\mathrm{Eu}^{152}$	9.2 hours	1380	1.73
$_{34}\mathrm{Se}^{81}$	19 min.	0.46	0.072	$_{63}\mathrm{Eu}^{152}$	5–8 years	796	1.73
$_{35}\mathrm{Br}^{80}$	4.4 hours	2.76	0.34	$_{66}\mathrm{Dy}^{165}$	1.25 min.	120 (possibly less)	0.0458
$_{35}\mathrm{Br}^{80}$	18 min.	8.1	0.34	$_{66}\mathrm{Dy}^{165}$	140 min.	2620	0.0458
$_{45}\mathrm{Rh}^{104}$	4.2 min.	11.6	0.085	$_{73}\mathrm{Ta}^{182}$	16.2 min.	0.034	0.00165
$_{45}\mathrm{Rh}^{104}$	44 sec.	137	0.085	$_{73}\mathrm{Ta}^{182}$	117 days	20.6	0.00165
$_{47}\mathrm{Ag}^{110}$	22 sec.	97	47	$_{77}\mathrm{Ir}^{192}$	1.5 min.	260	0.26
$_{47}\mathrm{Ag}^{110}$	225 days	2.3	47	$_{77}\mathrm{Ir}^{192}$	70 days	1000	0.26
$_{48}\mathrm{Cd}^{115}$	43 days	0.14	0.127	$_{78}\mathrm{Pt}^{197}$	18 hours	1.1	0.244
$_{48}\mathrm{Cd}^{115}$	2.5 days	1.1	0.127	$_{78}\mathrm{Pt}^{197}$	3.3 days	4.5	0.244

demonstrated that excitation of this type occurs in the case of In\(^{115m}\) (indium was bombarded with electrons of energy \(1.3\,Mev\)); the cross section was found to be of the order of \(10^{-32}\,cm^2\). The theory of electrical excitation by electrons was developed by Wick\(^{24}\). Similar calculations were recently published by Sneddon and Touschek\(^{30}\). In these calculations an essential role is played by the density of nuclear levels as a function of energy and by the matrix elements of the nuclear electric dipole and quadrupole moments. The uncertainty of these quantities leads to the fact that the final results for the absolute values of the cross sections must be regarded as estimates of order of magnitude. However, the ratio between the cross sections for electrical excitation by x-rays and by electrons can be calculated much more accurately, since it is to a considerable extent independent of the structure of the nucleus. Roughly speaking, a quantum is more effective than an electron of the same energy by a factor of the order of 137.

Fig. 11. Examples of type-II isomerism.
a — level \(D\) has a spin close to the spin of level \(B\); b — level \(D\) has a spin close to the spin of level \(A\).

The case of excitation by the electric field of heavy charged particles was analyzed by Weizsäcker\(^{W4}\). The known uncertainty of his calculations, as well as of Wick’s calculations, is connected with ignorance of the nuclear matrix elements.

Experiments on the electrical excitation of isomers by particles and quanta will undoubtedly contribute to improving our knowledge of the essential nuclear quantities; Goote gave an example of how experimental results can be used for this purpose.

In the case of the formation of isomers by \(\beta\)-decay, the Weizsäcker theory in combination with the theory of beta decay makes it probable that, with the exception of a few special cases, beta decay should directly lead only to the formation of a single isomeric state. This becomes clear from Fig. 11, a, in which level \(C\) of a beta-radioactive nucleus is shown, transforming by decay into another nucleus situated at level \(A\). Since the selection rules for beta decay allow transitions with a spin difference of 0 or 1, then, according to the various modifications of Fermi’s theory\(^{K6}\), we must suppose that level \(A\) has the same spin as level \(C\), or a spin close to that of the nucleus at level \(C\). On the other hand,

the isomeric level \(D\) must have a spin very different from the spin of level \(C\), and consequently also from that of level \(A\). This makes a direct beta transition from \(D\) to \(A\) improbable and makes the transition from \(D\) to \(B\), indicated in the figure, considerably more probable.

From Fig. 11,b an alternative possibility is clear, in which the spins of levels \(C\) and \(D\) are interchanged in comparison with the case of Fig. 11,a. The case of Fig. 11,b occurs in \(\mathrm{Cd}^{115}\mathrm{H}^{16}\), where the transition \(D—A\) corresponds to an isomer with a half-life of 43 days, the transition \(C—B\) corresponds to an isomer with a half-life of 2.3 days, and the transition \(B—A\) corresponds to the isomer \(\mathrm{In}^{115}\) with a half-life of 4.5 hours.

Possible exceptions might be observed for cases in which the spin of level \(C\) is intermediate between the spins

Fig. 12. Nuclear levels of Rb and Sr.

of levels \(A\) and \(B\), or for cases in which the influence of the energy difference of levels \(A\) and \(B\) compensates the influence on the beta-decay probability of the spin difference, or even leads to the beta transition being impossible. An example of such a situation may be the case of \(\mathrm{Sr}^{87m}\), which was carefully investigated by Du Bridge and Marshall\({}^{D3}\). These investigators found an excited state of \(\mathrm{Sr}^{87}\) lying \(0.36\ \mathrm{MeV}\) above the ground state. The half-life of this state is 2.75 hours. On the basis of these data one may expect that \(\Lambda = 5\). The spin of \(\mathrm{Sr}^{87}\) is equal to \(\frac{9}{2}\), and therefore it is probable that the spin of \(\mathrm{Sr}^{87*}\) is equal to \(\frac{1}{2}\), and the parities of the states are opposite, unless we wish to admit that the spin difference is \(\frac{17}{2}\). The spin of Rb is equal to \(\frac{3}{2}\) \({}^{K2,M2}\), and the beta transition from \(\mathrm{Rb}^{87}\) to \(\mathrm{Sr}^{87m}\) is energetically impossible, since the levels are arranged as shown in Fig. 12, and the beta transition from \(\mathrm{Rb}^{87}\) to \(\mathrm{Sr}^{87m}\) would have to proceed with absorption of energy.

Various examples of beta decay of antimony leading to tellurium isotopes that have isomeric states are observed among the fission products of uranium; the study of the corresponding branching processes is of interest.

6. SEPARATION OF ISOMERS *)

A chemical method for separating nuclear isomers, as applied to bromine, was proposed by Segrè, Halford, and Seaborg S5, and also by Vault and Libby D4. Later this method was applied to tellurium by Seaborg, Livingood, and Kennedy S7, and by LeGoff and Segrè L3 to selenium. The physicochemical aspect of the question was also intensively studied, especially by Willard, Seaborg, Friedlander, and Kennedy W5, S8.

When, in a molecule, one of the atoms composing it undergoes an isomeric transition, various effects may arise that lead to the rupture of chemical bonds. First of all, one should note here the simple recoil from the emission of a gamma quantum, as a result of which the atom undergoing the transformation acquires a kinetic energy \(E\), equal (in eV) to

\[ E=\frac{\hbar^{2}\omega^{2}}{2Mc^{2}}=0.54\cdot 10^{-3}\frac{\varepsilon^{2}}{M}, \tag{59} \]

where \(\varepsilon\) is the energy of the gamma quantum in keV and \(M\) is the atomic weight of the nucleus \((O=16)\). If the gamma quantum undergoes internal conversion, then the recoil energy is, of course, greater, since it corresponds to the recoil associated with the emission of a particle having a nonzero rest mass. In this case the recoil energy is equal to

\[ E=\left(\frac{4.80}{M}\right)\cdot 10^{-5}(H\rho)^{2}. \tag{60} \]

Here \(H\rho\) is the electron momentum in gauss·cm, and \(E\) in equations (59) and (60) is expressed in eV.

These recoil energies can sometimes be sufficient to destroy a molecule; however, for example, in the case of bromine 80, the recoil energy is only \(0.0155\) eV in the case of gamma radiation, or \(0.034\) eV in the case of recoil associated with the ejection of a conversion electron. These energies are certainly less than the energy of a chemical bond and are insufficient to destroy the molecule.

However, in the case when internal conversion takes place, another effect accompanies the chemical changes: a free electron orbit appears in the inner shell

) A review devoted to this question was written earlier by K. Starke (Phys. Zeits. 42*, 184 (1941)).

of the atom, and a fall of the outer electrons occurs, down to the valence electrons. In this process the chemical bond is destroyed, and the atom undergoing the isomeric transition separates from the molecule, becoming an ion or changing its valence. After this, the atom can be isolated by chemical methods, as is done in the well-known Szilard–Chalmers method for separating the products of neutron bombardment. For example, if we prepare bromobenzene or some other organic bromine compound, using for this purpose \( \mathrm{Br}^{80} \) in the higher isomeric state (\(\tau = 4.5\) hours), we can then remove the organic compound with the aid of water and a small amount of sodium sulfite. In the aqueous fraction, after precipitation with silver nitrate, we find pure \( \mathrm{Br}^{80} \) in the lower isomeric state (\(\tau = 18\) min.). The molecules in which the isomeric transition \( \mathrm{Br}^{-} \) has occurred have been destroyed and have liberated bromine. In water this bromine passes into \( \mathrm{Br}^{-} \) and is precipitated by silver nitrate, whereas bromine that has not undergone isomeric transition remains in the organic molecule. Finally, one can also observe an increase of \( \mathrm{Br}^{80} \) (\(\tau = 18\) min.) in the organic compound after separation, which directly confirms the genetic connection between the two bromine isomers.

As indicated above, it seems probable that what is essential in this method is internal conversion. This was pointed out by Szilard \(^{75}\) and by Feifeisaber \(^{77}\) and was confirmed experimentally by Seaborg, Friedlander, and Kennedy \(^{88}\). These authors prepared diethyl zinc with \( \mathrm{Zn}^{69} \) in its higher state and diethyl tellurium with \( \mathrm{Te}^{127} \) or \( \mathrm{Te}^{129} \), likewise in their higher isomeric states. They then evaporated the indicated compounds and introduced them into a vessel containing two electrodes with a potential difference of several hundred volts. In the case of tellurium, the atoms in the lower isomeric state were thereby separated from the vapor. In the case of zinc, Kennedy, Seaborg, and Segrè observed no separation. Since both initial compounds are similar and the chemical bonds of zinc and tellurium are approximately the same, it is natural to ascribe the difference in their behavior to the fact that tellurium has a very large coefficient of internal conversion, whereas zinc has a very small conversion coefficient.

This result is still more surprising if one takes into account the fact that the energy difference between the isomeric states in the case of zinc is 439 kev, while in the case of tellurium it is only 100 kev. The recoil energy of the gamma quantum in the first case is 1.5 ev, while in the second case the recoil energy of the conversion electron is only about 0.7 ev. Despite this substantial difference, chemical separation in tellurium is considerably more effective than in zinc; this also indicates that internal conversion, and not recoil, plays an essential role in the separation process. This circumstance can be explained by various mecha-

mechanisms. One of them has already been indicated—it is associated with the successive fall of electrons into the places vacated as a result of the emission of a conversion electron; moreover, the falling continues until even the valence electrons fall onto the inner shell, as a result of which the chemical bond is broken.

In addition, the emission of a conversion electron is a process that is very rapid in comparison with the period of molecular vibrations. Therefore, after the emission of a conversion electron the molecule may find itself in a predissociative state. This becomes clearer if one takes into account that the emission of \(K\)- or \(L\)-electrons suddenly increases by one unit the effective charge of the inner part of the atom, which causes large changes in the wave functions of the outer electrons*). Figure 13 shows potential-

Fig. 13. Potential energy as a function of the distance between nuclei in a molecule. \(a\)—normal state; \(b\)—state in which one of the atoms has undergone an isomeric transition with internal conversion.

-energy curves for a molecule as a function of the distance between the nuclei (curve \(a\)); applying the Franck–Condon principle, we see that a molecule in the fundamental vibrational state, as a result of an isomeric transition, makes a jump from curve \(a\) to curve \(b\) and thus passes into a state of predissociation. This phenomenon was studied theoretically by Cooper\({}^{13}\). We do not yet have detailed experimental investigations of the separate mechanisms of destruction of the molecule during an isomeric transition, and on the basis of the existing data it is still impossible to draw any definite conclusions on this question. However, from

*) This effect may be enhanced in connection with the emission of Auger electrons, which arises after the appearance of a vacancy in the \(K\)-shell.

experiments of Willard W6, Seaborg, Friedlander, and Kennedy S8, and Kennedy, Seaborg, and Segrè K3, it appears probable that there is a close interrelation between the yield for the separation of isomers and the internal-conversion coefficients. In the ideal case, when only molecules undergoing an isomeric transition with emission of a conversion electron are decomposed and removed, we should have

\[ \alpha=\frac{p}{1-p}, \tag{61} \]

where \(\alpha\) is the internal-conversion coefficient and \(p\) is the fraction of the total activity isolated by the use of chemical methods.

De Vault and Libby D5 measured \(p\) for several reactions involving \(\mathrm{Br}^{80}\), and found values of \(p\) varying between 0.026 and 0.095. It is natural to suppose that the highest observed values of \(p\), according to (61), give a lower limit for \(\alpha\). Consequently, in the case of \(\mathrm{Br}^{80}\), \(\alpha > 20\). This is in agreement with the experiments of Greenberg and Rusanov G4 (see Section 7), who in this case found no unconverted 47-kilovolt \(\gamma\)-rays.

Thus the chemical method makes it possible to measure the internal-conversion coefficient indirectly. In the case of three tellurium isomers and \(\mathrm{Se}^{81}\), the application of the chemical method also indicates that the internal-conversion coefficient is large*). The lower values of \(p\), observed in some cases L4, D5, must almost certainly be attributed to the chemical peculiarities of the reactions used.

Chemical separation of isomers can also be achieved in some cases by an exchange reaction; this was shown by Imre I1, who concentrated \(\mathrm{Br}^{80}\) with a half-life of 18 min. on the surface of a precipitate of silver bromide suspended in an aqueous solution containing \(\mathrm{Br}^{80}\) ions with a half-life of 4.5 hours. In an analogous way Segrè S9 concentrated \(\mathrm{Br}^{80}\) with a half-life of 18 min. on the surface of metallic silver immersed in bromobenzene containing \(\mathrm{Br}^{80}\) with a half-life of 4.5 hours. Electric fields in nonconducting solutions can also be used to separate isomers that are in lower isomeric states C15, G13.

7. SOME CASES OF ISOMERISM

In the present section the information at our disposal on certain isomers is considered in detail. Exam—

*) In the work of R. R. Williams W23 the maximum yield in the case of Te isomers proved to be considerably less than 100%. However, this question still remains open.

were chosen somewhat at random, but in part the choice was determined by the availability of a sufficient amount of information. Three cases of isomerism of type I are given, although a considerably larger number of them have been subjected to careful study. Two cases of type II are included, which, apart from the case \(UX_2—UZ\), are the only representatives of isomers of this type that have been investigated in detail.

The case of \(Br^{80}\)

The case of \(Br^{80}\) is a classic one in the history of artificial radioactivity. The discovery of the isomerism of \(Br^{80}\) has already been discussed in section 1. Attempts to determine which state is the higher one by measuring the reaction threshold were inconclusive, because, as will be seen, the isomeric levels are very close to one another. In addition, it was unknown whether this case belonged to type I or to type II. However, the chemical separation method proposed by Segrè, Seaborg, and Halford \(S^5\) showed definitely that the isomer with a half-life of 4.4 hours passes by decay into the isomer with a half-life of 18 min., thereby establishing its belonging to type I (see the discussion of this method in section 6). Soon afterward Valley and McCreary \(V^1\) photographed the conversion-electron lines of two \(\gamma\)-radiations. They observed three electron lines, which could be assigned to the \(K\)- and \(L\)-lines of \(\gamma\)-radiation with energy 49 kev and to the \(L\)-line of \(\gamma\)-radiation with energy 25 kev, or to the \(K\)- and \(L\)-lines of \(\gamma\)-radiation with energies 37 kev and 49 kev. This question was clarified by the experiments of Greenberg and Rusinov \(G^4\), who showed, by means of absorption measurements, that a \(\gamma\)-quantum with an energy between 35.9 and 37.4 kev is emitted. They observed no quanta with energy 49 kev and measured that, for \(\gamma\)-radiation with energy 37 kev, the conversion coefficient is less than unity

\[ \left( \frac{N_e}{(N_e + N_q)} = 0.5 \right). \]

On the basis of these data it becomes quite obvious that \(Br^{80}\) has a state lying 86 kev above the ground state, and that the direct transition from this state to the ground state is very strongly forbidden. However, the transition to the state with energy 37 kev has a half-life of 4.4 hours, and the transition from the state with energy 37 kev is apparently fast.

The question was studied further in detail by Berthelot \(B^{11}\), who again measured the conversion coefficient of the \(\gamma\)-radiation with energy 37 kev and determined the \(K\)- and \(L\)-conversion coefficients of each type of \(\gamma\)-radiation. To separate the electrons he used absorption, which gives considerably lower accuracy than

the use of a beta spectrograph with a counter having a “window” (the sensitivity of photographic film to electrons in this energy region changes rapidly). Assuming again that the conversion of the $\gamma$-radiation with energy 49 kev is complete ($a=\infty$), he found $a_K/a_L=7.3$ for the $\gamma$-radiation with energy 49 kev and $a_K/a_L=7$ for the $\gamma$-radiation with energy 37 kev at $a_K+a_L=0.64$. Comparing these results with the theory of Hebb and Nelson$^{\mathrm{H}6}$, Berthelot concluded that the $\gamma$-radiation with energy 49 kev is magnetic octupole, and the $\gamma$-radiation with energy 37 kev is magnetic dipole. The calculated half-life has the correct order of magnitude, but a strong objection arises against the scheme he adopted.

Fig. 14. Decay scheme.

According to the selection rules (Table I), the parity of the level with energy 86 kev and of the ground state must be the same, and therefore the transition between these two states must occur as a $2^4$-pole transition with a half-life shorter than 4.4 hours, since in this case a higher energy is emitted. It therefore seems more probable that, for example, the $\gamma$-radiation with energy 37 kev is electric dipole, as was assumed by Greenberg and Roussinov$^{\mathrm{G}4}$; the parities of the states and the spins correspond to those indicated in Fig. 14. According to Konopinski$^{\mathrm{K}6}$, the $\beta^-$ transition with a period of 18 min. and the corresponding $\beta^+$ transition, discovered by Barber$^{\mathrm{B}15}$, are once-forbidden. Since Kr$^{80}$ has spin equal to 0, it is most logical that the spins are distributed as indicated in Fig. 14.

Careful measurements of $a_K/a_L$ with a $\beta$-spectrograph may clarify this question.

The spin of Br$^{79}$ is equal to $\dfrac{3}{2}$, so that the Br$^{80}$ nucleus formed as a result of the capture of a slow neutron will have spin equal to 1 or 2.

On the basis of the considerations given above, in the subsequent cascade process one should rather expect the formation of the state with a period of 18 min. than of the state with a period of 4.4 hours. In fact, as is seen from Table II, this is true, and the corresponding ratio is equal to 2.9.

The Case of Ag$^{107}$

Isomerism in Ag$^{107}$ was discovered by Alvarez, Helmholz, and Nelson A$^{3}$. They drew attention to the fact that the small value of the ratio $\alpha_K/\alpha_L$, observed for $\gamma$-radiation with energy 93 kev emitted by Cd with a half-life of 6.7 hours, indicates a large change of spin in the corresponding transition of the Ag nucleus formed as a result of $K$-capture. They further found an Ag activity corresponding to this transition with a period of 40 sec. This case was completely investigated by Bradt et al. B$^{10}$, who found that the period is 44.3 sec. The same isomeric state was obtained by Weidenbeck W$^{13}$ by means of excitation with X-rays, and it was assigned by Helmholz to Ag$^{107}$ on the basis of experiments with bombardment of separated Cd isotopes H$^{27}$. Figure 15 gives the level scheme proposed by Bradt et al., who found positrons with an upper energy limit of 0.32 Mev and $\gamma$-rays with energy 0.846 Mev. They compared the number of unconverted $\gamma$-quanta with the number of annihilation quanta obtained from positrons, and also compared the number of positrons with the number of internal-conversion electrons. From these two ratios, knowing the relative efficiency of the counters for positrons and annihilation radiation, they calculated $\alpha = 16 \pm 3$. The value of $\alpha$ for electric $2^4$-pole radiation is equal to 165, and for magnetic $2^3$-pole radiation $\alpha = 24$. Therefore it is most probable to assume that magnetic $2^3$-pole radiation takes place. However, the ratio $\alpha_K/\alpha_L$ is equal to 0.92, whereas the theoretical ratio for a magnetic transition is 4.2, and for an electric transition 0.58. From this, the radiation must be 80% electric and 20% magnetic. This contradiction possibly arises because, in this region, calculation of the conversion coefficients in the nonrelativistic approximation is no longer accurate. The half-life calculated on the basis of equation (39a) is 1500 times longer than required, whereas for $\Lambda = 3$ it proves to be

Fig. 15. Decay scheme.

4000 times shorter than that observed. Taking all the data into account, it seems probable that the change of spin is equal to 3 in the presence of a prohibition for electric octupole radiation, which gives, for the state with energy 93 kev, a spin equal to \(\frac{7}{2}\), whereas it is known that for \(\mathrm{Ag}^{107}\) in the ground state the spin must be equal to \(\frac{1}{2}\). Gamma radiation with energy \(0.846\) Mev is electric dipole radiation. The ground state and the state with energy \(0.093\) Mev must have different parity (since electric \(2^3\)-pole radiation is forbidden); the \(\beta^+\)-transition and \(K\)-capture are allowed transitions.

The case of \(\mathrm{Mn}^{52}\)

The isomerism of \(\mathrm{Mn}^{52}\) belongs to type II. The isomers of \(\mathrm{Mn}^{52}\) were discovered by Livingood and Seaborg\(^{L5}\), who obtained them as a result of the reaction \(\mathrm{Fe}^{54}(d,\alpha)\mathrm{Mn}^{52}\). Gemmendinger\(^{H7}\) obtained these isomers as a result of the reaction \(\mathrm{Cr}(p,n)\mathrm{Mn}\) and studied their radiation. These activities were studied fully by Peacock and Deutsch\(^{P13}\) and by Osborn and Deutsch\(^{O4}\), using a \(\beta\)-spectrograph and the coincidence method; the results given below are taken from their papers. The isomers have periods of 6.5 days and 21 min. The period equal to 6.5 days is associated with the emission of positrons with an upper limit of \(0.582\) Mev, followed by three cascade \(\gamma\)-radiations with energies \(0.734\), \(0.940\), and \(1.46\) Mev. The independence of the \(\beta-\gamma\) coincidences from the energy of the \(\beta\)-particles shows that the spectrum of \(\beta\)-rays is simple. This isomer also undergoes \(K\)-capture, accompanied by the same three \(\gamma\)-radiations. The ratio \(\lambda_K/\lambda_\beta\) is equal to 0.54, which apparently indicates that the transition is allowed with \(\Delta I=0\) or 1 and without change of parity\(^{G14}\).

The period equal to 21 min., studied by Osborn and Deutsch\(^{O4}\), has a positron spectrum with an upper limit equal to \(2.66\) Mev, and is accompanied by a single \(\gamma\)-radiation with energy \(1.46\) Mev. This radiation is almost certainly the same as that observed for the longer period; this indicates that the first excited state participating in these transitions has energy \(1.46\) Mev. Comparing the decay energies, we see that the level responsible for the 21-min. period lies \(2.66-0.528-0.734-0.940=0.40\) Mev above the level responsible for the activity with a period of 6.5 days. Osborn and Deutsch confirmed this assumption by finding, in \(2\cdot 10^3\) decays, one conversion electron from \(\gamma\)-rays with energy \(0.392\) Mev. The conversion coefficient for \(\gamma\)-radiation with such a high energy must be low, and thus there must be considerably more \(\gamma\)-quanta than electrons. In the case of magnetic

\(2^4\)-pole radiation, the conversion coefficient must be equal to 0.039, and correspondingly the half-life must be equal to \(21 \times 2 \cdot 10^3 \times 0.039\), which gives about 1600 min. A transition with \(\Lambda = 5\) gives a half-life of \(1.6 \cdot 10^5\) min., which, although not in good agreement with experiment, is considerably better than a transition with \(\Lambda = 4\), giving a period equal to 20 sec. Therefore we must assume that the two isomeric states differ by 4 spin units and that they have different parity, so that electric \(2^4\)-pole radiation is forbidden. In order not to take spin values greater than 4, it seems logical to adopt \(I = 0\) for the level with a period of 21 min. and \(I = 4\) for the level with a period of 6.5 days. This leads to the level scheme shown in Fig. 16. The \(\beta^+\)-transition with a period of 21 min. is allowed to the state with energy 1.46 Mev, but is forbidden to the ground state, since the \(0—0\) transition is, at least, doubly forbidden by Teller’s selection rule. The three excited states of \(\mathrm{Cr}^{52}\) must have increasing spins, so that direct transitions to the ground state, which has spin 0, are sufficiently forbidden. To clarify the arrangement and character of the levels it is necessary to have accurate information on the state with energy 1.46 Mev.

Fig. 16. Decay scheme.

Case of \(\mathrm{Co}^{60}\)

Cobalt has only one stable isotope, \(\mathrm{Co}^{59}\), which, as a result of bombardment by slow neutrons, shows two radioactive periods, equal to 10.7 min. and 5.3 years. These periods must be attributed to \(\mathrm{Co}^{60}\). Deutsch, Elliott, and Roberts\({}^{13}\) investigated these activities in detail. The activity with period 5.3 years is, of course, the easier one to study. Careful measurements with the aid of a \(\beta\)-spectrograph and \(\beta\)—\(\gamma\) and \(\gamma\)—\(\gamma\) coincidences showed that the \(\beta\)-radiation with maximum energy 0.308 Mev is accompanied by two cascade \(\gamma\)-radiations with energies 1.1 and 1.3 Mev. Study of the angular correlation of the two \(\gamma\)-radiations (Brady and Deutsch\({}^{14}\)) showed that each of them is quadrupole and that the states participating in the transition probably have spins 0; 2 and 4, since

the ground state of \( \mathrm{Ni}^{60} \), which is the product of the decay of \( \mathrm{Co}^{60} \), undoubtedly has spin equal to 0. These assumptions were confirmed by later observations by D[^20], according to which in \(10^5\) decays fewer than one \(\gamma\)-quantum with energy \(2.4\) MeV is observed; in other words, the direct transition from the state with energy \(2.4\) MeV to the ground state is strongly forbidden. On the basis of these data, the level scheme shown in Fig. 17 appears possible. Since the correlation between the directions of the \(\gamma\)-rays does not make it possible to distinguish electric radiation from magnetic radiation, the parity of the states is undetermined and has been chosen so as to ensure the minimum value of \(I\) for the state with period 10.7 min. The transition from the \( \mathrm{Co}^{60} \) state with period 5.3 yr to the ground state of \( \mathrm{Ni}^{60} \) is, of course, strongly forbidden, and the transition with energy 0.308 MeV is forbidden once or twice.

Fig. 17. Decay scheme.

Deutsch, Elliott, and Roberts D13 found, for the period 10.7 min, conversion electrons of \(\gamma\)-radiation with energy 0.056 MeV, which must be attributed to an isomeric transition of type I. This transition occurs in almost 90% of cases. In the remaining 10% of cases a \(\beta\)-particle is emitted with maximum energy 1.28 MeV (from the following data this energy should be equal to 1.45 MeV), accompanied by a \(\gamma\)-quantum which, as was discovered by Peacock D20, is identical to the \(\gamma\)-quantum with energy 1.3 MeV emitted in the case of activity with a long period. If, as mentioned above, one assumes that the level with energy 1.3 MeV (see Fig. 17) has spin 2, then the level whose decay period is 10.7 min must have spin 1 with the same parity*). Then the \(\beta\)-transition will be allowed, the \(\gamma\)-transition will then be electric 23-pole radiation, and the parities of the two isomeric levels will be opposite. The theoretical half-life for this transition is about 0.2 sec, which is in good agreement with the “partial half-life,” equal to 12 min. However, electric 24-pole radiation gives \(3 \cdot 10^5\) sec, which gives an even stronger discrepancy.

*) Another possible scheme is as follows: for the level with period 10.7 min, the spin is 2, parity \(\pm\); for the level with period 5.3 yr, \(5 \mp\); for the ground state, \(0 \pm\); the remaining levels are unchanged.

Measurement in this case of the ratio of the numbers of \(K\)- and \(L\)-conversion electrons may give the order of the \(\gamma\)-transition, but for such a short half-life this measurement is difficult to carry out. The level scheme in Fig. 17 appears to be in agreement with the experimental data.

One more question should also be considered. Both isomers, with periods of 10.7 min and 5.3 yr, are observed upon capture of slow neutrons by \(\mathrm{Co}^{59}\), for which \(I=\dfrac{7}{2}\). The compound nucleus will therefore have \(I=3\) or 4, and, as Deutsch, Elliott, and Roberts indicated, if the transition to the isomeric state occurs as a result of the emission of a small number of quanta, the state with a period of 5 yr (\(I=4\)) should be represented more strongly than the state with a period of 10.7 min (\(I=1\)). In fact, experimental measurements (see Table II) show that the cross section for the formation of the isomer with a period of 5 yr is 33 times greater than the cross section for the formation of the isomer with a period of 10.7 min.

The case of \(\mathrm{Sb}^{124}\)

The last complex and interesting case is the triple isomerism of \(\mathrm{Sb}^{124}\), discovered by Mateosian et al. \(^{\mathrm{M16}}\). They found three radioactive periods associated with \(\mathrm{Sb}^{124}\), formed upon the capture of slow neutrons in samples enriched in \(\mathrm{Sb}^{123}\). One of the periods is the well-known 60-day period, whose decay scheme, proposed by Scharff-Goldhaber and Meyerhof \(^{\mathrm{M17}}\), is given in Fig. 18. Of the two remaining periods, one, equal to 21 min, is associated with the emission of \(\beta\)-rays and conversion electrons of very low energy, corresponding to \(\gamma\)-radiation with an energy of about 20 keV; with the second period, equal to 1.3 min, there is associated the emission of energetic \(\beta\)-rays with an upper limit of 3.0 MeV and of a small number of low-energy conversion electrons, corresponding to \(\gamma\)-radiation with an energy

Fig. 18. Decay scheme.

15 keV. No \(K\)-X-rays of Sb were observed, which indicates that the available energy is insufficient for \(K\)-conversion.

Both states with periods of 1.3 min and 21 min must be assigned to higher states of \(\mathrm{Sb}^{124}\), since they both emit strongly converted \(\gamma\)-rays. Let us suppose that both states decay into the state with a period of 60 days, which has a total transformation energy in \(\mathrm{Te}^{124}\) equal to \(0.6+1.7+0.7=3.0\) MeV. Since the transition with a period of 21 min emits converted \(\gamma\)-radiation with an energy of 20 keV, we may place it 3.020 MeV above \(\mathrm{Te}^{124}\) and assume that the observed \(\beta\)-rays are daughter rays of the 60-day period. The 21-minute half-life can be explained if the spin difference is 2 and quadrupole radiation is forbidden. Theoretically, in this case the period is equal to \(\frac{1}{2}\) min for magnetic quadrupole radiation and only 0.04 sec for electric \(2^3\)-pole radiation with a conversion coefficient significantly greater in the latter case. In any event, the \(\beta\)-transition to the ground state of \(\mathrm{Te}^{124}\) must be strongly forbidden. The state with a period of 1.3 min must have a partial half-life for decay into the state with a period of 60 days that is very long compared with 1.3 min, since the predominant type of decay is \(\beta\)-emission, leading directly to the ground state of \(\mathrm{Te}^{124}\). Again, a spin difference of 2 with forbidden electric radiation can provide the necessary half-life. For a purely magnetic quadrupole transition the calculated half-life is 13 min. The beta transition to the ground state of \(\mathrm{Te}^{124}\) is probably once forbidden. It can be seen that these data fit into several decay schemes. One of these schemes is that presented in Fig. 18. The transition with energy 0.7 MeV from the level with a period of 60 days, according to the formulas given by Konopinski, is once forbidden, while the transition with energy 2.4 MeV is forbidden at least twice. Correspondingly, for the state with a period of 60 days the spin is 3, so that the transition to the ground state of \(\mathrm{Te}^{124}\) is thrice forbidden. Moreover, if the level with a period of 1.3 min has spin equal to 0, with parity opposite to that of \(\mathrm{Te}^{124}\), the \(\beta\)-transition is twice forbidden, and the \(\gamma\)-transition occurs with \(\Lambda=4\) instead of \(\Lambda=3\), as was assumed above. To refine the scheme of the transitions under consideration, several experiments were proposed. For example, from the correlation of the directions of \(\gamma\)-radiations with energies of 1.7 and 0.6 MeV in the case of the 60-day period, one can judge the spins of the corresponding states. Additional data on the selection rules in \(\beta\)-decay should also be very useful. The conversion coefficients for isomeric \(\gamma\)-rays are so large that measuring them is very difficult.

EXPLANATIONS TO TABLE III

Table III gives data on all known isomers as of January 1, 1949. The construction of the table is the same as that used by Bette and recently by Seaborg^S20^. The first two columns require no explanation. In the third column, “Classes,” the designations are the same as in Seaborg:

A — isotope known (mass number and element known),
B — isotope probable, element known,
C — one of several isotopes, element known.

Isotopes falling under the classification D, E, F, and G (see S20) were rejected, with one or two exceptions. In these few cases the evidence for isomerism appears very convincing, but the isotopic composition is unknown. Such cases arise, for example, when the isomers are produced by excitation with x-rays. As a result of this selection, some genuine cases of isomerism were certainly rejected, but we suppose that there are few of them.

In the fourth column the type of radiation is indicated by the following designations:

$\beta^-$ — negative beta particles,
$\beta^+$ — positive beta particles (positrons),
$\gamma$ — gamma rays,
$e$ — internal-conversion electrons,
$\alpha$ — alpha particles,
$K$ — $K$-capture (or, in general, capture of an orbital electron).

Annihilation radiation and x-rays accompanying internal conversion are not indicated.

As for half-lives, we did not intend to include all measured values, although in some cases several values were included. In several cases limits are given within which the measured values lie. The references to the literature are by no means exhaustive (but most of the recent papers are cited).

In the column “Type of isomer,” type I means the decay of a metastable state to the ground state by means of $\gamma$-radiation or internal conversion; type II means the decay of both isomeric states to a neighboring element by means of $\beta^-$-, $\beta^+$-, or $K$-processes (see Section 3b). In addition, the letter $U$ indicates the upper of the isomeric states, $L$ the lower. In the case of three states, $U''$ denotes the highest state, $U'$ the middle state. In doubtful cases a question mark is put. In the absence of definite evidence these letters are not entered. In cases of isomers of stable elements, the stable isotopes are also given. For further explanations to Table III, see p. 429.

Table of Isomers

Table III

Z	A	Class	Type of radiation	Half-life	Type of isomer	Radiation energy, particles	Radiation energy, γ-rays	Notes
20	Ca\(^{49}\) Ca\(^{49}\)	A C	\(\beta^{-}, \lambda(e^{-}?)\) \(\beta^{-}\)	2.5 hours (W9) 30 min. (W9)	II (?) II (?)	2.3 (W9) abs.	0.8 (W9)	May be Ca\(^{41}\); in (H30) its formation by the \((\gamma,n)\)-reaction is reported.
21	Sc\(^{44}\)	A	\(e^{-}, \gamma\)	2.2 days (W8) (S19) 2.44 days (H17)	I \(U\)		0.269 (S19) spect.	\(\alpha = 0.08;\ \alpha_K/\alpha_L = 8,\) but in this region it is insensitive to \(l\). Theory gives \(\alpha = 0.15,\ \tau \sim 10\) min. for \(E_\gamma = 924\). The value \(\Lambda = 4\) is preferable.
21	Sc\(^{44}\)	A	\(\beta^{+}\)	4.1 hours (W8)	I \(L\)	1.47 (S19) spect.	1.33 (H17) abs., spect.
21	Sc\(^{46}\)	A	\(e^{-}, \gamma\)	20 sec. (G22)	I \(U\)		0.18 abs. (G22)	Conversion electrons and γ-rays were observed. Probably \(\Lambda = 4\). Theory predicts that \(\tau \sim 1\) hour.
21	Sc\(^{46}\)	A	\(\beta^{-}, \gamma\) \(K\)(W10)	85 days (W10)	I \(L\)	0.36, 1.49 (P12) spect.	0.88, 1.12 (P12)	Probably a \(0,2,0\) cascade in the Ti\(^{46}\) nucleus (B23).
22	Ti\(^{51}\)	A	\(\beta^{-}, \gamma\)(W8)	6 min. (S9)	II \(U\) (?)	1.6 (S9) abs.		The state is higher from the energetic point of view.
22	Ti\(^{51}\)	A	\(\beta^{-}, \gamma\)	72 days (W10)	II \(L\) (?)	0.36 (W10) abs.	1.0 (W13) abs.; lower energy (M8)	The \(\beta\)-spectrum is probably simple. Sometimes cascade γ-rays (M8).

$Z^\*$
	25	Mn$^{53}$	A	$\beta^+,\gamma$	21 min. (L5)	I, II $U$	2.66 (O4) spect.	0.392 1.46 (O4) spect.	See discussion in Section 7.
		Mn$^{52}$	A	$\beta^+,K,\gamma$(G14)	6.5 days (L5)	I, II $L$	0.58 (P13) spect.	0.73, 0.94, 1.46 (P13) spect.
	27	Co$^{58}$	B	$e^-$	9.3 hours (S35)	I $U$		0.023 (S35) spect.	Probably, $\Lambda=3$.
		Co$^{58}$	A	$\beta^+,\gamma$, $K$(G14)	72 days (L14)	I $L$	0.47 (D23) spect.	0.805 (D23) spect.
		Co$^{60}$	A	$\beta^-,\gamma,e^-$	10.7 min. (L14)	I, II $U$	1.56 (P7) spect.	0.056, 1.30 (D13) (P12) spect.	See discussion in Section 7.
		Co$^{60}$	A	$\beta^-,\gamma$	5.3 years (L14)	I, II $L$	0.31 (D12)	1.16, 1.30 (P7) (J1) spect.
	30	Zn$^{69}$	A	$e^-,\gamma$	13.8 hours (L7)	I $U$		0.439 (H8) spect.	$0.1>a>0.01$, which indicates $\Lambda=5$. Theory gives $\tau \simeq 270$ hours, $Э^{25}$ or $M^{24}$. Spin, probably, $\dfrac{9}{2}\pm$.
		Zn$^{69}$	A	$\beta^-$	57 min. (L7) (K3)	I $L$	1.0 (K3)	no $\gamma$ (K3)	Spin, probably, $\dfrac{1}{2}\mp$.
	32	Ge$^{71}$	A	$\beta^+$	40 hours (S11); 36 hours (H17)	II $U$ (?)	1.2 (S11) abs.		The state is higher from the energetic point of view.
		Ge$^{71}$	A	$K,e^-$(S11) $\beta^+$(?) (M13)	11 days (S11)	II $L$ (?)	0.6 (M13) spect.	0.5 (M13) spect. 0.6 (S23) abs.	As a result of the $(\gamma,n)$ reaction this isomer is not observed (H37).
		Ge$^{72}$	A	$e^-$	$5\cdot 10^{-7}$ sec. (B13)	I $U$		0.70 (B13)	May be a $0—0$ transition. See Section IVe.
		Ge$^{72}$	A		stable	I $L$
		Ge$^{77}$	A	$\beta^-$(A8)	59 sec. (A8)	II	2.8 (A8) abs.		To elucidate the $U$ and $L$ states, it is necessary to study $\gamma$-rays.
		Ge$^{77}$	A	$\beta^-$(S17)	12 hours (S11)	II	1.9 cam. (S17) (S22)

Continuation

Z	A	Class	Type of radiation	Half-life	Type of isomer	Energy of radiation: particles	Energy of radiation: γ-rays	Notes
33	As71 As71	A B	K β+	60 hr. (H32) 52 min. (H31)	II II
34	Se77	A	e−	17.5 sec. (G15)	I U		0.15 (A8) abs.	The theory gives τ ∼ 2 hours for Λ = 4.
34	Se77 Se81 (L6)	A B	e−(L3)	stable 57 min. (L3)	I L I U		0.99 (H8) spect.	From the half-life it is probable that Λ = 4; \(a_K/a_L = 4\), which indicates 50% E2, 50% M2. Transition to the ground state Br83, probably, is allowed.
34	Se81	B	β−	19 min. (L3)	I L	1.5 (L3) abs.
34	Se83	B	β−, γ(A8)	67 sec. (A8)	II	3.4 (A8) abs.
34	Se83	A	β−, γ	30 min.	II	1.5 (G21) abs.	0.17, 0.37, 1.1 (G21) abs.	Transition to the ground state Br83 is strongly forbidden.
35	Br80	A	e+, γ	4.4 hr. (B11) (S5)	I U		0.049, 0.037 (V1) spect. (G4) abs. 0.5 (S2) (B16) abs.	See the discussion in Section 7.
35	Br80	A	β−, β+ (B15)	18 min. (S2) (S5)	I L	β−2.0 (A10) spect. β+0.7 (B15) abs.
36	Kr79,81	C	e−, γ, no β+	13 sec. (C5)	I U		0.187 (C5) spect.	The isomeric transition is undoubted; assignment to Kr is doubtful. Probably Λ = 4. 13 sec.
36	Kr79,81 Kr79	C A	e−, γ, no β+ e+(2%), K (98%) (W25)	55 sec. (C5) 3± hr. (B17) (C6)	I U I L	∼0.9 (30%) ∼0.6 (70%) H(31) abs.; 1.0 (W25) abs.	0.127 (C5) spect. 0.2 (H34) abs.	Lower state for transitions with periods of 13 sec. or 55 sec.

$Z$	Nuclide	Type	Radiation	Half-life	Level	$\beta$ energy	$\gamma$ energy	Remarks
	$\mathrm{Kr}^{83}$	A	$e^-$	113 min. (L3)	I $U$		0.029 or 0.046 (H8) spect.	Both $\gamma$ rays are present. $\alpha_K/\alpha_L \simeq 1$ for the $\gamma$ ray with energy 0.046; in this case $\Lambda=4$, 60% E2 and 40% M2. The calculated value is $\tau \sim 50$ h. Spin $\dfrac{3}{2}\pm$.
	$\mathrm{Kr}^{83}$	A		stable	I $L$			Spin $\dfrac{9}{2}\pm$.
	$\mathrm{Kr}^{85}$	A	$\beta^-, \gamma$ (H34)	4.5 hours (H34) (W25)	II $U$ (?)	1.0 (H31) abs.	0.17, 0.37 (H34) abs.	The higher state from the standpoint of the energy of the $\beta$ particles.
	$\mathrm{Kr}^{85}$	B	$\beta^-$	$\sim 10$ years (H33)	II $L$ (?)	0.74 (H33) abs.	No $\gamma$ (H33)
38	$\mathrm{Sr}^{85}$	A	$e^-, \gamma$ (D3)	70 min. (D3)	I $U$		0.170 (D3) spect.	For $\Lambda=4$, $\tau \sim 150$ min. Knowledge of $\alpha$ and $\alpha_K/\alpha_L$ makes it possible to determine $l$.
	$\mathrm{Sr}^{85}$	A	$K, \gamma$ (D7)	65 days (D7)	I $L$		0.8 (D3) abs.	$\alpha_K/\alpha_L = 6$, $\alpha \sim 0.15$. These values, like the value of $\tau$, indicate $\Lambda=5$, E2. Spin $\dfrac{1}{2}\pm$.
	$\mathrm{Sr}^{87}$	A	$e^-, \gamma$ (D7) (R3)	2.7 hours	I $U$		0.386 (H8) spect.	Spin $\dfrac{9}{2}\mp$. The evidence is unreliable.
	$\mathrm{Sr}^{87}$	A		stable	I $L$
39	$\mathrm{Y}^{87}$	B	$e^-, \gamma$ (D3)	14 h. (D3) (S12)	I $U$		0.5 (D3) abs.	Decays into $\mathrm{Sr}^{87}$ with a period of 2.7 hours.
	$\mathrm{Y}^{87}$	A	$K$ (D7)	80 h. (D7)	I $L$		no $\gamma$ (?) (D3)
	$\mathrm{Y}^{88}$	A	$K, \gamma$ (D3) (H19); $\beta^+$, 0.19% (P14)	105 days (D3)	II $U$ (?)	0.83 (P14) spect.	0.908, 1.89 (D14) spect.; 2.8 (1%) (G16) D $(\gamma,n)$	0.908 and 1.89 are cascaded. The $\beta^+$ transition to $\mathrm{Sr}^{88}$ is strongly forbidden.
	$\mathrm{Y}^{88}$	A	$\beta^+$	2 h. (S12)	II $L$ (?)	1.65 abs. (O1)		Lower limit of the $\gamma$-ray energy 0.15 MeV.

Continuation

Z	A	Class	Type of radiation	Half-life	Type of isomer	Radiation energy: particles	Radiation energy: γ-rays	Notes
40	Y⁹¹	A	e⁻, γ	50 min. (G9) (S18)	I U		0.61 (F19) abs.	Probably, Λ = 5. Theoretical τ ∼ 2 hours.
40	Y⁹¹ (H20)	A	β⁻ (B8)	57 days (G9) (S18) (H10) (G17)	I L	1.6 (B8) abs.; 1.53 (L4) spectr.	γ and e⁻
40	Zr⁸⁹	A	e⁻, γ or K	4.5 min. (D3) (D7)	I (?)		0.555 (H38) spectr.	Apparently there is no β⁺. Selective absorption of X-rays may make it possible to distinguish types I and II.
40	Zr⁸⁹	A	β⁺ (S13)	78 hours (D3)	L (?)	1.0 (S13) cham. (D3) abs.	no γ
41	Cb⁹¹	A	e⁻, γ (B24)	62 days	I U		0.13 (?) (M20) abs.	Values Λ = 4 and Λ = 5 disagree by a factor of 10³ with the experimental data. Accurate values of the γ-energy and \( \frac{a_K}{a_L} \) are necessary.
41	Cb⁹¹			long				Radioactivity not detected.
41	Cb⁹²	A	β⁻, γ	10.1 days (K11); 11 days (S31)	II	1.38 (S31) cham. 1.38 (K11) abs.	1.0 (K11) abs.	Accurate measurements of β- and γ-rays are necessary.
41	Cb⁹²	A	β⁻, γ (W26)	21.6 hours (W26)	II	1.2 (W26) abs.	0.6 (W26) abs.
41	Cb⁹³	A	e⁻	42 days (W15)	I U			Obtained by excitation with X-rays. Other reports of this activity may refer to Cb⁹⁵.

	Cb\(^{93}\)	A		stable	I	\(L\)			Spin \(\dfrac{9}{2}\).
	Cb\(^{94}\)	A	\(e^{-}; \beta^{-}(0.01\%)\) (G18)	6.6 min. (G18)	I, II	\(U\)	1.3 abs.	0.058 abs. (G18), 1.0 (C11) abs.	Theory gives \(\tau \sim 900\) min. \(\dfrac{\sigma_K}{\sigma_L}=\dfrac{1}{6}\) for E24. Probably, \(\Lambda=4\).
	Cb\(^{94}\)			\(>100\) years	I, II	\(L\)			Radioactivity has not yet been detected.
	Cb\(^{95}\)	A	\(e^{-}\)	90 h. (L11)	I	\(U\)		0.216 (H35) spect.	The values \(\Lambda=4\) and \(\Lambda=5\) both disagree with experiment. The question may be resolved by measuring \(\dfrac{\sigma_K}{\sigma_l}\).
	Cb\(^{95}\)	A	\(e^{-}, \gamma, \beta\)	35 days	I	\(L\)	0.146 (H35) spect.	0.758 (H35) spect.	Formed in 98.6% of cases in the decay of Zr\(^{95}\), \(\alpha=2.4\cdot10^{-3}\) for a \(\gamma\)-ray with energy 0.758 MeV.
43	Tc\(^{94}\)	B	\(e^{-}\)	53 min. (H28)	I	\(U\)		0.0334 (H28) spect.	\(\Lambda=4\). Theory gives \(\tau \sim 20\) h.; \(\dfrac{\sigma_K}{\sigma_L}\) should be very small.
	Tc\(^{94}\)	B	\(\beta^{+}, k, \gamma\)	53 min. (H28) (M4)	I	\(L\)	2.45 (H28) spect.	0.38, 0.87, 1.48, 1.85, 2.74 (H28) spect.
	Tc\(^{95}\)	B	\(K, \gamma, e^{-}\) (E3) (H29); \(\beta^{+}\) (1%) (H29)	62 days (H29); 52 days (E3)	II		0.4 (H29) cham.	0.201, 0.570, 0.810, 1.017 (H29) spect.	Must be 1.4 MeV above the ground state of Mo\(^{95}\).

Continuation

\(Z\)	\(A\)	Class	Type of radiation	Half-life	Isomer type	Radiation energy: particles	Radiation energy: \(\gamma\)-rays	Notes
43	\(\mathrm{Tc}^{95}\)	A	\(K, e^{-}, \gamma\)	20 h (E5) (M4)	II		0.762, 0.932, 1.07 (M21) spectr.	Isomerism is doubtful, since none of these states has been observed in \(\mathrm{Mo}^{95}\).
43	\(\mathrm{Tc}^{97}\)	A	\(e^{-}\)	93 days (C7)	I (M15) \(U\) (E3)		0.097 (H8) spectr.	\(\dfrac{\alpha_K}{\alpha_L}\sim 2\). The theoretical values of \(\tau\) do not agree with the experiment within a factor of 1000 for \(\Lambda=4\) or 5. Probably mixed \(E\)- and \(M\)-radiation (H8).
43	\(\mathrm{Tc}^{97}\)	A	\(\beta^{+}\) or \(K\)		I \(L\)			The half-life is too long for detection at the present time.
43	\(\mathrm{Tc}^{99}\)	A	\(e^{-}, \gamma\) (S4)	6.6 h (S4)	I \(U\)		0.136 (S4) spectr.	The experimental evidence is unreliable. The value \(\Lambda=4\) indicates \(\tau\sim 9\) h. Measurements of \(\alpha\) and \(\dfrac{\alpha_K}{\alpha_L}\) are necessary.
43	\(\mathrm{Tc}^{99}\)	A	\(\beta^{-}\)	\(9.4\cdot 10^{5}\) years (M14)	I \(L\)	0.32 (M14) abs.
45	\(\mathrm{Rh}^{103}\)	A	\(e^{-}\)	45—48 min. (F12) (W14)	I \(U\)		0.0942 (F15) abs.; 0.059 or 0.037 (H21) spectr.; 0.0659 (G8) spectr.	According to (G8), \(K\)-X-rays and electrons with energy only 42.7 kev and greater are observed. Absence

	\(\mathrm{Rh}^{103}\) \(\mathrm{Rh}^{104}\)	A A	\(e^-\)	stable 4.3 min. (A4) (P2) (F12)	I I	\(L\) \(U\)		The \(L\)-electron value looks strange. With the exception of the ratio \(\dfrac{\alpha_K}{\alpha_L}\), there is good agreement with the value \(\Lambda=4\), which gives \(\tau\sim 20\) h. Apparently either \(K\)- or \(L\)-conversion is absent. The value measured in (A6) indicates \(M2\). \(\Lambda=4\) gives \(\tau\sim 20\) min., but \(\dfrac{\alpha_K}{\alpha_L}\) should be \(\sim 1\).
47	\(\mathrm{Rh}^{104}\) \(\mathrm{Ag}^{106}\)	A A	\(\beta^-\) \(\beta^+\)	44 sec. (A4) (P2) 24.5 min. (D6) (P8)	I II	\(L\)	2.3 (C8) cam.; 2.6 (H21) spect. 2.04 (F2) abs.	no \(\gamma\) (F2) More precise measurements of \(\beta^+\)- and \(\gamma\)-radiations are needed. If all \(\gamma\)-rays are cascading, then the 8.2-day period is the upper (\(U\)) one.
	\(\mathrm{Ag}^{106}\)	A	\(K, e^-, \gamma\)	8.2 days (F2) (P8) (H22)	II			0.72, 1.06, 1.63 (D11) (E1) spect. 0.093 (H8) (B10) spect.
	\(\mathrm{Ag}^{107}\)	A	\(e^-, \gamma\)	40 sec. (A3); 44.3 sec. (B10)	I	\(U\)		See discussion in Section 7. \(I=\dfrac{7}{2}\pm\).
	\(\mathrm{Ag}^{107}\)	A		stable	I	\(L\)		\(I=\dfrac{1}{2}\pm\).
	\(\mathrm{Ag}^{109}\)	A	\(e^-, \gamma\)	39 sec. (B10)	I	\(U\)		0.088 (H9) (B10) spect. \(\alpha_K+\alpha_L=19\), which indicates \(M2\), but \(\dfrac{\alpha_K}{\alpha_L}=1\)

Continuation

\(Z\)	\(A\)	Class	Type of radiation	Half-life	Type of isomer	Radiation energy: particles	Radiation energy: \(\gamma\)-rays	Notes
47	\(\mathrm{Ag}^{109}\)	A		stable	I \(L\)			gives 65% \(E2^4\), 35% \(M2^3\). Theory gives \(\tau \sim 1\) hour for \(\Lambda=4\), \(I=\dfrac{7}{2}\pm\).
47	\(\mathrm{Ag}^{110}\)	A	\(\beta^-,\ \gamma\)	22 sec. (A4); 24 sec. (H23); 28 sec. (F12) (P8)	II	2.6 (H23) pos. 2.8 (G6) cam.		\(I=\dfrac{1}{2}\pm\). On the basis of \(\sigma\) (Table II) the spin is probably small. There is not enough information to determine \(U\) or \(L\).
47	\(\mathrm{Ag}^{110}\)	A	\(K,\ e^-\)	225 days (L8) (R4)	II		0.65, 0.925, 1.51 (D11) spect.	On the basis of \(\sigma\) (Table II) the spin is probably large.
48	\(\mathrm{Cd}^{111}\)	A	\(e^-,\ \gamma\)	48.7 min. (G15) (W17) (D8)	I \(U''\)		0.149, 0.247 (H24) (H18) spect.	For the 0.149 \(\gamma\)-ray \(\Lambda=4\). The 0.247 \(\gamma\)-ray is cascade.
48	\(\mathrm{Cd}^{111}\)	A	\(e^-,\ \gamma\)	\(8\cdot10^{-8}\) sec. (D20)	I \(U'\)		0.247	Probably \(E2^2\).
48	\(\mathrm{Cd}^{111}\)	A		stable	I \(L\)
48	\(\mathrm{Cd}^{113}\)	A		2.3 min. (H25) (T5)	I \(U\)
48	\(\mathrm{Cd}^{113}\)	A		stable
48	\(\mathrm{Cd}^{115}\)	A	\(\beta^-,\ \gamma\)	43 days (S23)	II \(U\)	1.67 (H16) spect.	0.5 (S23) pos.	\(\beta^-\)-transition to the ground state of \(\mathrm{In}^{115}\). The transition to the isomer \(\mathrm{In}^{115}\) is very weak. No transition to the ground state of \(\mathrm{Cd}^{115}\) was observed. Probably, \(\Lambda=6\).

No.	Nucleus	Type	Radiation	Half-life					Remarks
	Cd¹¹⁵	A	β⁻, γ	2.33 days (L1); 2.5 days (G3)	II	L	1.10 (H16) spect.; 1.13, 0.6 (L1) spect.	0.52 (H16) spect.; 0.65 (M9) spect.	β⁻ transition to the isomeric state In¹¹⁵.
49	In¹¹²	B	e⁻	23 min. (B18) (S29)	I	U		0.16 (B18)	Λ = 4 gives τ ∼ 15 min., $\dfrac{\alpha_K}{\alpha_L}=2$ and $\alpha=6$ for E2⁴. Measurements of $\dfrac{\alpha_K}{\alpha_L}$ will make it possible to choose between E- and M-radiations. Spin 4 or 5.
	In¹¹³	B	β⁺, β⁻, K	9 min. (B18) (S29)	I	L	β⁺ 1.7 (L1) capt. β⁻ 1.		Spin, probably 1.
	In¹¹³	A	e⁻, γ	105 min. (L1)	I	U		0.393 (L1) spect.	$\dfrac{\alpha_K}{\alpha_L}=5.4,\quad \alpha=0.7.$ Most probably M2⁴. Theory gives $\dfrac{\alpha_K}{\alpha_L}=6,\ \alpha=0.3;\ \tau \sim 140$ hr. Spin $\dfrac{1}{2}\pm$.
	In¹¹³	A		stable	I	L			Spin $\dfrac{9}{2}+$ .
	In¹¹⁴	A	e⁻	48 days (L1)	I	U		0.192 (L1) spect.	$\dfrac{\alpha_K}{\alpha_L}=1,\ \alpha \sim 100.$ Theory gives for $\lambda=5$ — $\dfrac{5}{6}$ E2⁵, $\dfrac{1}{6}$ M2⁴, $\alpha=12$ and $\tau \sim 1600$ days. Spin, probably, $5\pm$.

Continuation

\(Z\)	\(A\)	Class	Type of radiation	Half-life	Type of isomer	Energy of radiation: particles	Energy of radiation: \(\gamma\)-rays	Notes
49	\(\mathrm{In}^{114}\)	A	\(\beta^{-}\)	72 sec. (L1) (B6) (L9) (L10)	I \(L\)	1.98 (L1) spectr.		Spin, probably \(1\pm\).
49	\(\mathrm{In}^{115}\)	A	\(e^{-},\ \gamma\)	4.5 hours (L1), 4.1 hours (B4) (G3)	I \(U\)		0.338 (L1) spectr.	\(\dfrac{\sigma_K}{\sigma_L}=4.8-5.3,\ \alpha \simeq 1.\) \(M24\) gives \(\tau \sim 500\) hours, \(\dfrac{\sigma_K}{\sigma_L}=5.75,\ \alpha=0.6.\) Spin, probably \(\dfrac{1}{2}\pm\).
49	\(\mathrm{In}^{115}\)	A		stable				Spin \(\dfrac{9}{2}\mp\).
49	\(\mathrm{In}^{116}\)	A	\(\beta^{-}\)	13 sec. (A4) (C9)	II	2.8 (C9) cham.	no \(\gamma\) (M5)	Both transitions are resolved; further investigations are necessary.
49	\(\mathrm{In}^{116}\)	A	\(\beta^{-},\ \gamma\)	54 min. (A4) (C9) (L9)	II	0.85 (C10) spectr.	0.428, 1.12, 1.31, 2.32 (D11) spectr.	The 54-min. period is probably the upper one (\(U\)).
50	\(\mathrm{Sn}^{119}\)		\(e^{-},\ \gamma\)	13 days (L19)	I \(U\)		\(\sim 0.250\) (L19) abs.; \(e^{-},\ \gamma\)	The mass number 119 is doubtful, but it is definitely an isomer, since it emits tin X-rays; probably \(I=5\).
50	\(\mathrm{Sn}^{119}\)	C A		stable	I \(L\)
50	\(\mathrm{Sn}^{121-}\)	B	\(\beta^{-}\)	36 min. (N2)	II	2.5–3.0 (N2) abs.	no \(\gamma\) (?) (N2)
50	\(\mathrm{Sn}^{121}\)	A	\(\beta^{-}\)	28 hours (L18)	II	0.4 (L18) abs.	no \(\gamma\) (L18)
51	\(\mathrm{Sb}^{120}\)	B	\(K,\ e^{-},\ \gamma\)	6.0 days (L18)	II		1.1 (L18) abs.
51	\(\mathrm{Sb}^{120}\)	B	\(\beta^{+}\)	17 min. (L15)	II	1.53 (A9) cham.

	Nuclide		Radiation	Half-life					Notes
	Sb\(^{123}\)	A	\(e^{-}\)	3.5 min. (M16)	I	\(U\)		0.140 (M16) abs.	The theory gives for \(\Lambda=4\) \(\tau \sim 12\) sec. \(\alpha\) should be \(\sim 12\).
	Sb\(^{122}\)	A	\(\beta^{-}, \gamma, e^{-}\)	2.8 days	I	\(L\)	1.91, 1.36 (M13) spect.	0.58 (R6) spect.; 0.80 (M9) spect.
	Sb\(^{124}\)	A	\(e^{-}\)	21 min. (M16)	I	\(U''\)		0.02 (M16) abs.	See discussion in Section 7.
	Sb\(^{124}\)	A	\(\beta^{-}, e^{-}\)	1.3 min. (M16)	I, II, \(U'\)	3.2 (M16) abs.	0.015 abs.
	Sb\(^{134}\)	A	\(\beta^{-}, \gamma\)	60 days (L15)	I, II, \(L\)	2.4, 0.7 (M17)	1.7 (M17)	Possibly the decay scheme is considerably more complex. See, for example, (K5).
52	Te\(^{121}\)	A	\(e^{-}, \gamma\)	143 days (P10); 125 days (S7) (B19)	I	\(U''\)		0.082 (H36) spect.	\(\dfrac{\alpha_K}{\alpha_L}\) for 0.082 indicates \(\Lambda=3\). The half-life requires, at least, \(\Lambda=4\).
52	Te\(^{121}\)	A	\(e^{-}, \gamma\)	\(5\cdot 10^{-8}\) sec. (B20)	I	\(U'\)		0.213 (H36) spect.	\(\dfrac{\alpha_K}{\alpha_L}\) indicates \(\Lambda=3\). \(\Lambda=2\) gives \(\tau \sim 10^{-9}\) sec.
52	Te\(^{121}\)	A	\(K, \gamma\)	17 days (P10);	I	\(L\)		0.61 (P10) abs.
52	Te\(^{125}\)	A	\(e^{-}\)	\(\sim 60\) days (F20)	I	\(U\)		0.125 abs. (F20)	\(\Lambda=5\) gives \(\tau \sim 4000\) days, \(\dfrac{\alpha_K}{\alpha_L}\sim 0.2\). The \(\gamma\)-ray energy was obtained under the assumption that the absorption edge corresponds to \(L\)-electrons.

Z	A	Class	Type of radiation	Half-life	Isomer type	Radiation energy: particles	Radiation energy: γ-rays	Notes
52	Te¹²⁵	A		stable	I L
52	Te¹²⁷	A	e⁻	90 days (S7)	I U		0.086 (H8) spect.	$\dfrac{a_K}{a_L} \sim 0.75$; $\Lambda = 5$ gives slightly better agreement. $\dfrac{a_K}{a_L}$ gives 50% E, 50% M.
52	Te¹²⁷	A	β⁻	9.3 hours (S7)	I L	0.76 (S7) abs.	no γ
52	Te¹³⁹	A	e⁻	32 days (S7)	I U		0.102 (H8) spect.	$\dfrac{a_K}{a_L} \sim 1$. Probably E and M. $\Lambda = 5$ gives better agreement than $\Lambda = 4$.
52	Te¹²⁹	A	β⁻	72 min. (S7) (A5)	I L	1.8 (R6) spect.
52	Te¹³¹	A	e⁻	30 hours (S7) (A5)	I U		0.177 (H8) spect.	$\dfrac{a_K}{a_L} \sim 2$. Probably E and M. $l = 4$ or 5.
52	Te¹³¹	A	β⁻	25 min. (S7) (A5)	I L
54	Xe¹²⁷	B	e⁻, γ	75 sec. (C5)	I U		0.175 or 0.125 (C5) spect.	No L-conversion of the 0.125 γ-ray was observed; 0.175 γ-ray is probably responsible for conversion; $\Lambda = 4$ gives $\tau \sim 500$ sec.
54	Xe¹³⁷	B	e⁻, γ, K (?)	34 days (C5)	I L		0.9 (C5) abs. e⁻

Z	Isotope	Type	Radiation	Half-life					Notes
	Xe¹³⁵	A	e⁻, γ	10 min. (W22); 15.6 min. (R7); 12 min. (G7)	I	U		0.52 (P4) spect.; 0.6 (S24) abs.	Probably, Λ = 5; theory gives τ ∼ 4 hours.
	Xe¹³⁵	A	β⁻, γ	9.4 hours (S15) (W22)	I	L	0.93 (P4)	0.247 (P4) spect.
55	Cs¹³⁴	A	β⁻, e⁻, γ	3.15 hours (S25) (K5)	I, II	U	2.4 (S25) abs.	0.150 (P7) spect.; 0.7 (S25) abs.	Probably, Λ = 4. It is necessary to know the branching ratio.
	Cs¹³⁴	A	β⁻, γ	1.7 years (K5)	I, II	L	0.03 (25%), 0.65 (75%) (E4) (D15) spect.	0.568 25%, 0.602, 0.791 (E4) (D15) spect.
56	Ba¹³³	A	e⁻	38–39 hours (K9) (W18) (Y1)	I	U		0.276 (C12) spect.	$\dfrac{a_K}{a_L}=3.2$. Λ = 5 gives τ ∼ 100 days. The formulas for $a$ and $\dfrac{a_K}{a_L}$ are inapplicable in this region.
	Ba¹³³	A	e⁻, K, γ	20 years (K9)	I	L		0.32, 0.085 (Y1) abs.
	Ba¹³⁵	C	e⁻, γ	28.7 years (Y2)	I	U		0.29 (Y2) abs.	The choice of isotopes is unreliable. If L-electrons are observed, then probably Λ = 5.
	Ba¹³⁵	A		stable	I	L
	Ba¹³⁷	A	e⁻, γ	158 sec. (T9); 156 sec. (M18)	I	U		0.663 (T4) (M18) spect.	$a=0.14$; Λ = 5 gives 1200 sec. Probably, spin $\dfrac{11}{2}\pm$. (M18) points to difficulties in the theory of β-decay in the case of the decay of Cs¹³⁷.
	Ba¹³⁷	A		stable	I	L			Spin $\dfrac{3}{2}\mp$.

Continuation

Z	A	Class	Type of radiation	Half-life	Type of isomer	Radiation energy, particles	Radiation energy, γ-rays	Notes
63	Eu\(^{153}\)	A	\(\beta^{-}, \gamma, e^{-}, K\) (R8)	9.2 hours (P9)	II	1.88 (T6) spect.	0.123, 0.163, 0.925 (T6) spect.
63	Eu\(^{153}\)	A	\(\beta^{-}, \gamma, e^{-}\) (S32)	\(\sim 5\) years (I2)	II	0.751 (S32) spect.	Several (S32)	Known as an isomer with a period of 9 hours as a result of mass-spectrographic investigation (I2). It is difficult to determine the energy of the γ-rays owing to the activity of Eu\(^{154}\).
65	Tb\(^{160}\)	A	\(\beta^{-}\)	3.9 hours (H12) (M7)	II
65	Tb\(^{160}\)	A	\(\beta^{-}, \gamma\)	72–73.5 days (B9) (B21) (I2)	II	0.516, 0.882 (C18) spect.	0.086, 0.195, 0.212, 0.297, 1.15 (C18) spect.	Data on the γ-rays are doubtful.
66	Dy\(^{165}\)	B	\(e^{-}\)	1.25 min. (I3) (F11)	I \(U\)		0.102 (H38) spect.	The mass value is probably from mass-spectrographic investigations. \(\dfrac{a_K}{a_L}\) is small. Probably, \(\Lambda = 4\).
66	Dy\(^{165}\)	A	\(\beta^{-}, \gamma\)	2.5 hours (P9) (H13) (M7)	I \(L\)	1.18 (D16) spect.; 0.42, 0.88, 1.25 (S27) spect.	1.0, 0.37 (M13) spect.; 0.091, 0.37, 0.78 (S27) spect.
68	Er	C	\(e^{-}\)	2.5 sec. (D18)	I \(U\)		0.2 (D18) abs.	Probably, \(\Lambda = 4\).

No.	Isotope	Class	Decay	Half-life					Remarks
69	Tm\(^{169}\)	A	\(e^-, \gamma\)	\(1\cdot10^{-6}\) sec. (D22)	I	\(U\)		0.19 (D22) abs.	\(\Lambda=2\) and \(\Lambda=3\) give values of \(\tau\) differing from the observed ones by a factor of 1000. Knowledge of the exact values of the energy and \(a\) is necessary.
	Tm\(^{169}\)	A		stable
	Tm\(^{171}\)	B	\(e^-, \gamma\)	\(2.5\cdot10^{-6}\) sec. (D22)	I	\(U\)		0.113 (K10) spect.	\(a=1.3;\ \Lambda=2\) gives \(\tau\sim10^{-8}\) sec.
	Tm\(^{171}\)	B		\(\sim 500\) days (K10)	I	\(L\)	0.1 abs.
70	Yb	C	\(e^-\)	6 sec. (D18)	I	\(U\)		0.23 abs. (D18)	Obtained by \((n,\gamma)\)-reactions.
	Yb	C	\(e^-\)	50 sec. (D18)	I	\(U\)		0.02 abs. (D18)	Obtained by \((n,\gamma)\)-reactions.
71	Lu\(^{176}\)	A	\(\beta^-\)	3.4 hours (D17) (F9)	II	\(U\)	1.15 abs.	no \(\gamma\).	Obtained by excitation with X-rays (D17).
	Lu\(^{176}\)	A	\(\beta^-, \gamma\)	\(7.3\cdot10^{10}\) years (N14) (L13)	II	\(L\)	0.215 (L13) abs. 0.40 (F9) abs.	0.260 (F9) abs.
72	Hf\(^{177,179}\)	C	\(e^-\)	19 sec. (F11)	I	\(U\)		0.20 (F11) abs.	Must be isomeric. Isotope determined on the basis of the “Mattauch rule” (F11). \(\Lambda=4\) gives \(\tau\sim50\) sec.
	Hf\(^{177,79}\)	A		stable
73	Ta\(^{181}\)	A	\(e^-, \gamma\)	\(2.2\cdot10^{-5}\) sec. (D22) (B22); \(2.0\cdot10^{-5}\) sec. (B5)	I	\(U\)		0.133, 0.145, 0.478 (C17) spect., 0.128, 0.472 (B22) spect.	From (B22) a 0.342 \(\gamma\)-ray predominantly isomeric. \(a\sim0.6\). Probably \(\Lambda=2\). According to (B5), \(\Lambda=3\); for the 0.478 \(\gamma\), \(\Lambda=2\). Their scheme does not forbid transition to the ground state.

Z	A	Class	Type of radiation	Half-life	Type of isomer	Radiation energy: particles	Radiation energy: γ-rays	Notes
73	Ta\(^{181}\)	A		stable	I \(L\)
73	Ta\(^{182}\)	A	\(e^-\), \(\gamma\)	16.2 min. (S14)	I \(U\)		0.22 (S14) abs.	The value 0.22 was obtained on the assumption that \(L\)-electrons are observed (S14). Probably, \(\Lambda = 4\).
73	Ta\(^{182}\)	A	\(\beta^-\), \(\gamma\), \(e^-\)	117 days (Z1)	I \(L\)	0.499 (J2) spectrum	0.15, 0.22, 1.13, 1.22 (R6) spectrum
74	W\(^{183}\)	D	\(e^-\)(D18)	5.5 sec. (D18)	I \(U\)		0.100 abs. (D18)	The electrons have a maximum energy of 0.08. Assuming that the electrons are \(L\)-electrons, probably \(\Lambda = 4\).
75	W\(^{185}\)			stable	I \(L\)
75	Re\(^{187}\)	A	\(e^-\), \(\gamma\)	\(0.65 \cdot 10^{-6}\) sec. (D19) (D22)	I \(U\)		0.135, 0.086, 0.101 (V3) spectrum	The 0.135 γ-rays are most strongly converted. The two other γ-rays are probably inessential. \(\Lambda = 2\) or 3. Spin, probably, \(\frac{1}{2}\).
75	Re\(^{187}\)	A		stable				Spin \(\frac{5}{2}\).

*8	No.	Isotope	Class	Radiation	Half-life				Notes
*8	77	Ir\(^{192}\)	A	\(e^{-},\ \gamma\)	1.5 min. (G19)	I	\(U\)	\(0.06\) (G19)	Conversion only on the \(L\)-shell, \(\Lambda=3\) or 4.
*8	77	Ir\(^{193}\)	A	\(\beta^{-},\ \gamma\)	75 days (G20); 60 days (M6) (F4)	I	\(L\)	0.59 (G20) abs.; 0.307, 0.467, 0.603 (D10) spec.	In (L16) more \(\gamma\)-rays were observed.
*8	78	Pt\(^{197}\)	B	\(\beta^{-}\)	18 hr. (M6)	II		0.65 (S16) abs.; 0.72 (K7) abs.
*8	78	Pt\(^{197}\)	B	\(\beta^{-},\ \gamma\) (K7)	3.3 days (M6)	II
*8	79	Au\(^{196}\)	B	\(\beta^{-}\) (30%), \(K\) (70%), \(\gamma\), \(e^{-}\) (S33)	5.55 days (W27) (S33)	II (?)		\(\sim 0.27,\ \sim 0.43\) (S33) spec.; 0.139, 0.358 with \(K\), 0.173, 0.334 with \(\beta^{-}\) (S33) spec.
*8	79	Au\(^{196}\)	B	\(\beta^{-}, K\) (W28)	14.0 hr. (W28) (M6)	II (?)
*8	79	Au\(^{197}\)	A	\(e^{-}\)	7.4 sec. (F16) (W15)	I	\(U\)	0.250 (F16) abs.; 0.077 (F16) abs.	The \(\gamma\)-rays are cascades. For the 0.07 \(\gamma\)-ray, \(\tau<10^{-6}\) sec. \(\Lambda=4\) gives \(\tau\sim 10\) sec.
*8	79	Au\(^{197}\)	A		stable	I	\(L\)
*8	80	Hg\(^{197}\)	A	\(K,\ \gamma,\ e^{-}\)	23 hr. (F17) (F16) (W12)	II	\(U\)	0.125, 0.157 (V2) spec.	The half-life of the 0.157 \(\gamma\)-ray is \(<10^{-7}\) sec. (F16).
*8	80	Hg\(^{197}\)	A	\(K,\ \gamma,\ e^{-}\)	64 hr. (W12)	II	\(L\)	0.075 (H26) spec.; 0.077 (F16) abs.
*8	80	Hg\(^*\)	C	\(e^{-}\)	43 min. (M6) (F17)	I	\(U\)	0.222 or 0.362 (H24) spec.	Both \(\gamma\)-rays are strongly converted. Probably \(\Lambda=5\). Isomer of stable Hg.

Continuation

\(Z\)	\(A\)	Class	Type of radiation	Half-life	Type of isomer	Energy of radiation: particles	Energy of radiation: \(\gamma\)-rays	Notes
82	\(\mathrm{Pb}^{204}\)	B	\(e^{-},\ \gamma\)	65–68 min. (M3) (F18) (T7)	I \(U\)		1.1 (F18) thresh.; \(e^{-},\gamma;\ 0.9\) (M3) thresh.	Exception to “Mattauch’s rule.”
82	\(\mathrm{Pb}^{204}\)	A		stable	I′ \(L\)
82	\(\mathrm{Pb}^{*}\)	C	\(e^{-}\)	1.6 min. (W11)	I \(U\)		0.15–0.25 (W11)	Probably, \(\Lambda=4\). Isomer of the stable isotope Pb.
91	\(\mathrm{UX}_{2}^{231}\)	A	\(\beta^{-},\ \gamma\) (M10)	1.14 min. (C14)	I, II \(U\)	2.32 (98%), 1.4 (1.7%) (B7) spect.	0.394, 0.822, 0.782, 0.95 (B7) spect.	From the values of \(\tau\) and \(\alpha\) for the \(\gamma\)-ray \(\Lambda=5\) (B7). The isomeric \(\gamma\)-ray is observed only in 0.12% of cases.
91	\(\mathrm{UZ}^{231}\)	A	\(\beta^{-},\ \gamma\)	6.7 hours (C14) (F2)	I, II \(L\)	0.45 (90%), 1.2 (10%) (B7) spect.	0.85 (B7) spect.
93	\(\mathrm{Am}^{242}\)	A	\(\beta^{-}\)	16 hours (S26)	II	1.0 (S26) thresh.
93	\(\mathrm{Am}^{242}\)	A	\(\beta^{-},\ \alpha\) (0.2%)	400 years (S26)	II	0.5 (S26) thresh.

In the column “Radiation Energy” only references to the most recent works, containing the most accurate measurements, are included. In the column “Particles” only \(\beta^-\)- and \(\beta^+\)-particles are included. In the column “\(\gamma\)-rays” the transition energy from the upper to the lower isomeric state is given in italics. Doubtful cases are noted in “Remarks.” The abbreviations and symbols used have the following meanings:

abs. — absorption;

chamb. — Wilson chamber (with a magnetic field in the case of beta particles);

spect. — magnetic deflection (magnetic spectrograph or spectrometer or counter with a magnetic field); no distinction is made between conversion electrons and secondary photoelectrons;

D-\(\gamma\)-n reaction — measurement of neutron energy from a D-\(\gamma\)-n reaction.

A semicolon separates quantities determined independently of one another, for example, independent determinations of the half-life or of the radiation energy.

The “Remarks” include corresponding information on the values of \(\alpha\), \(\dfrac{\alpha_K}{\alpha_L}\), etc. (references are given in the columns “Radiation Energy”); values of \(\Lambda\) are also indicated here. As stated above, for example, \(\Lambda=4\) may mean electric \(2^4\)-pole (\(E2^4\)) or magnetic \(2^3\)-pole (\(M2^3\)) radiation, or both together. The half-life is denoted by the letter \(\tau\). In some cases the spins of states are accompanied by an indication of parity. For example, \(\dfrac{7}{2}+\) and \(\dfrac{1}{2}+\) denote spin \(\dfrac{7}{2}\hbar\) and \(-\dfrac{1}{2}\hbar\) and opposite parity.

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