Isomeric decays with multipolarity $\lambda \ge$ 3

Abstract

Isomeric decays with multipolarity $\lambda \ge 3$ are discussed. Parity changing E3, M4, E5 decays are more widespread within the nuclide chart when compared to parity conserving M3, E4 and M5 transitions, which is the consequence of the properties of the nuclear shell model. The E3 transitions around $^{208}$Pb are classified depending on their origins. Systematics of M4 and E5 transitions are presented. These provide an excellent test to the study of shell evolution in different parts of the nuclide chart as well as the nature of the effective charges.

1 Introduction

Isomeric (or metastable) states are long-lived nuclear excitations. There is no a clear definition of what constitute an isomer. Often states with half-lives longer than a limit of 10 ns or 100 ns are considered isomeric. For example, the latest Atlas of nuclear isomers [1] published in 2023 uses the $T_{1/2}$ = 10 ns limit. The definition of isomerism does not require any hindrance of the decay related to the structure of the states involved. For a detailed discussion of isomerism in nuclei, including types of isomers, see [2].

The Weisskopf $\gamma$-ray half-life (which is the partial half-life) estimate is based on an extreme single-particle approximation. It depends very strongly on the multipolarity $\lambda$, e.g. the angular momentum carried by the photon: $T^W_{1/2}\sim E^{-(2\lambda +1)}$ (see Fig. 1), where E is the transition energy. Consequently, transitions with large multipolarity have large partial half-lives. They are relatively rare, as low multipolarity transitions usually dominate. When an excited state cannot decay by low multiplicity photons due to lack of lower lying state with similar spin, large multipolarity decays are the only internal decays possible. This leads to long-lived isomeric states. As a general rule, the excited nuclear states decaying by transitions of multipolarity $\lambda \ge$ 4 are isomeric. This is valid also for magnetic $\lambda$ = 3 M3 transitions. In case of E3 decays, the situation is more complex: the isomerism of the state depends on the collectivity of the decay, and strongly on its energy (and the definition of isomer). Isomeric states can often decay also by charged particle decays, not only internally. Consequently, the majority of internally decaying long-lived isomers are in nuclei close to stability.

The Weisskopf hindrance factor can be expressed as: $F_W = T_{1/2}^\gamma /T_{1/2}^W$, where $T_{1/2}^\gamma$ is the partial $\gamma$-ray half-life, which is determined experimentally. Not only the size of the angular momentum, but also its orientation is conserved, which is relevant in the case of deformed nuclei. Here, the K quantum number is the projection of the total angular momentum on the symmetry axis. For K isomers (where in the isomeric state the angular momentum is generated by unpaired nucleons, therefore I=K), the reduced hindrance is defined as $f_\nu = F_W ^{1/\nu }$, where the degree of K forbiddenness is $\nu = \Delta K - \lambda$.

Excited states usually decay with $\gamma$-ray transitions of low multipolarity. As the multipole order increases, the number of known cases decreases rapidly. Throughout this article, we use a lot of experimental nuclear data. As a rule, these are taken from the ENSDF compilation (reference [3]), and this is not explicitly referenced at each place. However, the newer articles not in the compilations are explicitly referenced.

Fig. 1

Weisskopf single-particle partial $\gamma$-ray half-live estimates for mass A = 100. Note that the electron conversion is not considered

Fig. 2

Shell-model single-particle orbitals between magic numbers 8 and 126. Positive parity orbitals are indicated in black, whilst negative-parity ones are in red. The stretched $\Delta j$ = $\lambda$ orbital pairs for M3, E4, M4, E5, M5, E6, M6 are connected with arrows, red for parity changing transitions and black for parity conserving ones. Arrows with dashed lines indicate l-forbidden transitions. See text for details

Internal $\gamma$ decays are governed by one-body operators. Consequently only the quantum number of a single nucleon can change, whilst the others act as spectators. The spherical shell-model single-particle orbitals are shown in Fig. 2. The strongest transitions (i.e. largest reduced transition strengths) with multiplicity $\lambda$ are expected between orbitals with $\Delta l$ = $\Delta j$ = $\lambda$. Transitions are allowed also for other orbital pairs, but generally these are weaker due to the values of the angular momentum coupling constants. In Fig. 2, the stretched $\Delta j = \lambda$ orbital pairs for M3, E4, M4, E5, M5, E6, M6 transitions are indicated with arrows. One notes that parity changing M4 and E5 transitions can be found within the same shell (therefore the transitions are of low energy), which can result in long-lived isomeric states. In contrast, parity conserving stretched M3, E4 and M5 transitions usually connects far lying orbitals, and consequently there are many other decay channels open and these will not be isomeric. Or when they are in the same shell, the transitions are l forbidden. The consequence of this is that parity changing $\lambda \ge$3 are more abundant than partiy conserving ones. The ENSDF compilation [3] lists about $\sim$1000 stretched E3 compared to $\sim$100 M3 decays. Similarly, the numbers for observed M4/E4 and E5/M5 decays are 121/34 and 17/1, respectively.

2 E3 decays around $^{208}$Pb

There is a large number of E3 transitions observed. They are not all isomeric, depending mainly on their energy, but also on whether they are of collective nature. The discovery of isomerism [4] itself involves an E3 decaying isomer in $^{234}$Pa, and it is attributed to Otto Hahn [5]. Isomeric E3 transitions were observed over a large part of the nuclide chart, as shown in Fig. 10 of [1]. Here we will focus on the E3 transitions around the doubly magic $^{208}$Pb nucleus. Similar concepts and logic could be applied to other parts of the nuclide chart.

Fig. 3

Single-particle orbitals around the doubly magic nucleus $^{208}$Pb. In each shell, orbital pairs which can give rise to E3 transitions are indicated, separately for $\Delta l$ = 3 (in red) and $\Delta l$ = 1 (in blue) pairs. In addition, $\Delta j = \Delta l$ = 3 pairs from adjacent shells are connected with black arrows

Fig. 4

Collective E3 decay in doubly magic $^{208}$Pb, as well as single-particle governed ones in one-valence particle nuclei $^{209}$Bi and $^{209}$Pb

The single-particle proton and neutron orbitals around $^{208}_{82}$Pb$_{126}$ are shown in Fig. 3. The first excited state in $^{208}$Pb at 2.614 MeV has spin-parity 3$^-$ and a large $B(E3; 3^-\rightarrow 0^+)$= 33.8(6) W.u. transition strength to the ground state. It is, therefore, a collective state, interpreted to be the consequence of a large number of $\Delta l$=$\Delta j$=3 orbital pairs across the Z = 82 and N = 126 shell gaps, shown by arrows in Fig. 3. Other orbital pairs also contribute for the composition of the collective octupole phonon see [6]. Consequently, shell model calculations incorporating full shells around Z = 82 (covering 50–126) and N = 126 (82–184) reproduce well the properties of this 3$^-$ state and other collective states in $^{208}$Pb [7] and neighbouring [8,9,10] nuclei.

Nuclei in the vicinity of $^{208}$Pb exhibit also non-collective octupole decays which are linked to individual orbital pairs within the same shell. As a consequence of the spin–orbit coupling, un-natural parity high-j orbitals are present in each shell around Z = 82 and N = 126, and these form pairs with opposite parity orbitals which can give rise to E3 transitions. These are indicated for both $\Delta l = \Delta j$ = 3 (in red), and $\Delta l$=1, $\Delta j$ = 2 (in blue) pairs in Fig. 3. First, we examine the one-valence particle/hole nuclei. In the one proton-particle $^{209}$Bi, the $\pi i_{13/2}$ state is yrast and can decay into the $\pi f_{7/2}$ and $\pi h_{9/2}$ states via stretched and un-stretched E3 transitions, respectively. Due to the energy factor, only the $i_{13/2}\rightarrow h_{9/2}$ transition was observed and its E3 component has a transition strength of B(E3) = 4.6(9) W.u. [11]. This relatively modest strength is a consequence of being an un-stretched transition, but it is also boosted by the $\pi h_{9/2}\times 3^-$ component of the wave function of the predominantly $\pi i_{13/2}$ state (3$^-$ is the collective octupole phonon), as shown in Fig. 4. In the one neutron-particle $^{209}$Pb nucleus, the situation is slightly different. The $\nu j_{15/2}$ state is yrast and can decay into the $\nu g_{9/2}$ and $\nu i_{11/2}$ states via stretched and un-stretched E3 decays, respectively. In this case, the energy factor favours the stretched E3 and only this one was observed. Its strength is rather large at 26(7) W.u. This is due to its $\Delta j = \Delta l$ = 3 nature and the octupole phonon admixture into the $\nu j_{15/2}$ dominated state (see Fig. 4). In the shells below Z = 82 and N = 126, E3 transitions are mediated by the $\pi h_{11/2}$ and $\nu i_{13/2}$ orbitals, respectively, paired with orbitals which are not yrast (Fig. 3). As a consequence in $^{207}$Tl and $^{207}$Pb, no E3 transitions were observed connecting the relevant single-particle states.

Fig. 5

Systematics of E3 transitions in $(top)$ Z = 82 and $(bottom)$ N = 126 nuclei. The origins of the transitions are indicated. The pictures side by side present the same data, using $(left)$ logarithmic and $(right)$ linear y axes. See the text for details

For a better understanding of the origin of the E3 decays in the region, Fig. 5 presents the sytematics of these decays for the Z=82 isotopes and N = 126 isotones. In lead isotopes with $N<$ 126, all observed E3 decays have transition strengths of $B(E3)<$ 1 W.u. These are mediated by the $i_{13/2}-f_{7/2}$ neutron orbitals. The low strengths are the consequence of the non-yrastness of the $\nu f_{7/2}$ orbital, meaning that the isomeric states (which are based on the high-j $i_{13/2}$ orbital) decay into a level where the $f_{7/2}$ orbital provides a minor part of the wave function. In contrast, the B(E3) values for $N>$ 126 lead isotopes are large [12]. These are mediated by the $j_{15/2}-g_{9/2}$ neutron-orbital pair. $^{208}$Pb itself has an $I^{\pi }$ = 23$^+$ isomer mediated by the same transition [13], in addition to the (non-isomeric) collective 3$^-$. One notes that the largest B(E3) value for neutron-deficient Pb nuclei is the $B(E3;7^- \rightarrow 4^+)$ = 0.7(3) W.u. in $^{202}$Pb. The large uncertainty relates to the branching ratio and this would merit revisiting.

N = 126 isotones below $^{208}$Pb exhibit isomeric E3 decays governed by the $\pi h_{11/2}-d_{5/2}$ transition (Fig. 5). These were observed in $^{206}$Hg, $^{205}$Au and $^{204}$Pt with $B(E3)<$1 W.u. [14]. The low value is because the $\pi d_{5/2}$ is not yrast, and therefore this orbital provides a minority component into the wave functions of the daughter states. In contrast, the neutron-deficient N = 126 isotones exhibit isomeric states decaying with $B(E3)>$ 1 W.u. Their origins are more diverse. The proton $i _{13/2}-h_{9/2}$ mediated transitions have strengths in order of few W.u., similar to that already mentioned in $^{209}$Bi. In contrast, the ones mediated by the $\Delta j$ = 3 proton $i _{13/2}-f_{7/2}$ and neutron $j_{15/2}-g_{9/2}$ pairs are stronger. In some cases, small mixing can increase considerably the transition strength. This is quite common at high spins, where configuration mixing is more prevalent. For the 16$^+$ isomer in $^{210}$Po and 39/2$^-$ in $^{211}$At, the main configurations suggest a $\nu i_{13/2}-h_{9/2}$ decay. However, the measured B(E3) values are large, quantitatively explained [15] by wave-function admixtures which allow $\pi i_{13/2}-f_{7/2}$ and $\nu j_{15/2}-g_{9/2}$ transitions. The situation is similar for the 25$^-$ and 30$^+$ isomeric states in $^{212}$Rn [16]. These cases, when the large strength is determined by minority components of the wave function are labeled as mixed in Fig. 5. There is also one transition labeled as proton $s_{1/2}-f_{7/2}$, observed in $^{209}$Bi. The $\pi s_{1/2}$ is an intruder from below Z = 82. Therefore, the 1/2$^-$ state configuration is actually $\pi s_{1/2}^{-1}(h_{9/2}^2)_0$, and decays into the $\pi f_{7/2}(h_{9/2}^2)_0$ component of the predominantly $\pi f_{7/2}$ state. Alternatively the decay is due to the $\pi f_{7/2}\times 3^-$ component of the 1/2$^-$ state. In addition to single-orbital pair mediated transitions, Fig. 5 also indicates the collective B(E3) values for $^{206}$Hg [8] and $^{208}$Pb (these are not isomeric due to their large energy). One should note that it is not always possible to divide the E3 decays so neatly based on their strengths. For example, in $^{208}$Pb, the 8$^+$ isomeric state decays into two 5$^-$ states. Both E3 decays are governed by the neutron $j_{15/2}-g_{9/2}$ transition; however because of the mixing between the 5$^-$ states, the strengths are lower than would be otherwise, at 12.3(2) W.u. and 0.71(17) W.u. [13]. Also a 28$^-$ isomer was identified in $^{208}$Pb decaying with B(E3) = 56(7) W.u. The large value is likely due to two $j_{15/2}$ neutrons in the structure of the isomer [13]. These are not shown in Fig. 5. For discussions of E3 transitions in the vicinity of $^{208}$Pb, see also [15,16,17]. The above exercise, namely connecting the use of the measured transition strengths to the underlying origin of the transitions, can be extended to the whole region, beyond the semi-magic nuclei discussed here.

3 M3 and E4 decays

There is no concentration of M3 (magnetic octupole) and E4 (electric hexadecapole) isomeric decays in any region of the nuclide chart (see Fig. 10 of [1]). This is a consequence of the nuclear shell model, where the low number of M3 decays is attributed to the fact that the $\Delta j$ = 3 same parity orbital pairs are in different shells. The exceptions are the $h_{9/2}-p_{3/2}$ and $g_{7/2}-s_{1/2}$ pairs; however, in both cases, the M3 transitions are l forbidden. The situation is somewhat similar for E4 transitions, as shown in Fig. 2, in the sense that the orbitals are far in energy from each other. Therefore, M3 and E4 transitions are generally of very high energy and other, much faster, decay branches dominate.

The only stretched M3 transition observed connecting single particle states is in $^{113}$Sn$_{63}$. Here the first excited state with 7/2$^+$ $\nu g_{7/2}$ decays into the 1/2$^+$ $\nu s_{1/2}$ ground state. The B(M3) = 0.0281(15) W.u. transition strength is rather low, which probably is the consequence of being l forbidden ($\Delta l$ = 4). Note that this transition is believed to have an E4 component, with the mixing ratio determined from the $\alpha _K$/$\alpha _L$ ratio of [18]. The mixing should be remeasured as the deduced value of B(E4) = 140(50) W.u. is too large [19], beyond the recommended upper limit [20].

M3 transitions were identified in several deformed nuclei, and they are not discussed here. There are much fewer E4 isomeric decays oberved. Odd-mass $^{179}$W, $^{189}$Os and $^{191}$Os have such one-quasiparticle states. In the odd-odd $^{174}$Lu, a two-quasiparticle $K^{\pi }$ = $I^{\pi }$ = 6$^-$ isomer with configuration $\pi 7/2[404]\nu 5/2[512]$ decays into the 2$^-$ and 3$^-$ members of the $K^{\pi }$ = 1$^-$ rotational band built on the same, but anti-aligned configuration, with B(M3) = 6.4(6)$\times$10$^{-7}$ W.u. and B(E4) = 0.00011(8) W.u., respectively. These correspond to very large reduced hindrance value of $f_{\nu }\sim$1250 for the M3 and $f_{\nu }$ $\sim$10$^{4}$ for the E4 transition. In $^{177}$Lu, a three-quasiparticle $K^{\pi }$=$I^{\pi }$ = 23/2$^-$ isomer with $\pi 7/2[404]\nu 7/2[514]9/2[624]$ configuration decays by B(M3) = 2.3(6)$\times$10$^{-11}$ W.u and B(E4) = 1.7(4)$\times$10$^{-9}$ W.u. into members of the $K^{\pi }$ = 9/2$^-$ $\pi 9/2[514]$ band [21, 22]. The reduced hindrance values are $f_{\nu }$ = 460(3) for the M3 and $f_{\nu }$ = 830(6) for the E4 transition [23], which are also rather large.

4 M4 decays

The orbital pairs which can give rise to stretched M4 (magnetic hexadecapole) transitions are $g_{9/2}-p_{1/2}$, $h_{11/2}-d_{3/2}$, $i_{13/2}-f_{5/2}$ (see Fig. 2). These are all $\Delta l$ = 3, $\Delta j$ = 4 pairs. First, we examine isotopic/isotonic chains at or in the immediate vicinity of magic numbers where the wave functions are expected to be the purest.

The first orbitals below the magic number 50 are $g_{9/2}$ and $p_{1/2}$. Consequently the ground state of the Z = 49 odd-mass indium nuclei is 9/2$^+$, whilst the first excited state has spin-parity 1/2$^-$. The 1/2$^-$ state is isomeric, and it can decay internally through an M4 transition. Figure 6 shows the energy and reduced B(M4) transition strength systematics. In neutron-rich indium nuclei, the isomeric state $\beta ^-$ decays. $\beta$ decays dominate over internal decay starting from $^{115}$In, whilst from $^{123}$In, no internal decay was observed. On the proton-rich side of stability, in $^{103}$In, $\beta ^++EC$ decay dominates over internal decay. In $^{101}$In [24] and $^{99}$In [25], the 1/2$^-$ states were identified in mass measurements. The transition strength and transition energy change smoothly with neutron number. Interestingly they seem to be correlated. There is one B(M4) data point which seems to break the smooth changing behaviour, that at $^{117}$In$_{68}$. Possibly the internal decay to $\beta$ decay ratio is incorrect and it should be remeasured.

Fig. 6

Systematics of M4 transitions in Z = 49, N = 49, N = 81 and Z = 82 nuclei. The transition energies and transition strengths are shown. The orbital pairs mediating the M4 transitions are indicated

Similarly, the ground state of the N = 49 odd-mass isotones is 9/2$^+$ based on a neutron $g_{9/2}$ hole, and they have a 1/2$^-$ first excited state based on $\nu p_{1/2}$. Internal M4 decays were observed in $^{85}$Kr, $^{87}$Sr, $^{89}$Zr, $^{91}$Mo and $^{93}$Ru (see Fig. 6). On the neutron-rich side, the 1/2$^+$ isomers were identified in $^{83}$Se and $^{81}$Ge and they $\beta ^-$ decay, without observed internal decay branches. On the neutron-deficient side, the energies of the 1/2$^-$ isomer in $^{95}$Pd, $^{97}$Cd are not known. Surprisingly, its decay was observed and its lifetime measured in the more exotic $^{97}$Cd [26], but not in $^{95}$Pd. Similarly to the proton-rich Z = 49 case, these are of great interest as they give information on the evolution of single-particle states approaching doubly magic $^{100}$Sn. The deduced B(M4) values are constant for $Z\le$ 40 nuclei, with lower values beyond the Z = 40 sub-shell closure.

The N = 81 odd-mass isotones are characterised by a long-lived 11/2$^-$ $\nu h_{11/2}$ state. This is usually the second excited state (except in neutron-rich $^{131}$Sn where it is the first one), with 3/2$^+$ $\nu d_{3/2}$ being the ground state (except on the proton-rich side in $^{147}$Dy, $^{149}$Er where 1/2$^+$ $\nu s_{1/2}$ is the ground-state). The systematics of these $\nu h_{11/2}-d_{3/2}$-mediated M4 transitions are shown in Fig. 6. The B(M4) transition strengths decrease smoothly as proton number increases. The reduction in the B(M4) strength whilst departing from Z = 50 is most likely due to configuration mixing. In contrast to Z = 49 nuclei, it seems that there is an anti-correlation between the energies and B(M4) values. The change in the energy difference is shell evolution in action, and can be explained as the effect of the tensor force [27]. Above Z = 50, toward the heavier elements, the proton orbitals $g_{7/2}$, $d_{3/2}$, $s_{1/2}$ and $h_{11/2}$ are filled in. Filling the $j = l-1/2$ $\pi g_{7/2}$ orbital lowers the $\nu h_{11/2}$ $j = l+1/2$ orbital due to the attractive tensor force, and thus increasing the $h_{11/2}-d_{3/2}$ gap. Filling the $\pi h_{11/2}$ raises the $\nu h_{11/2}$ orbital due to the repulsive tensor force, thus decreasing the $h_{11/2}-d_{3/2}$ gap.

In odd-mass Z = 82 Pb isotopes, 13/2$^+$ $\nu i_{13/2}$ isomeric states decay into the 5/2$^-$ $\nu f_{5/2}$ first excited state. The systematics of these M4 decays are shown in Fig. 6. The B(M4) values stay constant with neutron number, whilst the transition energies change significantly.

Now, we look into the wider regions, first focussing on the $\nu h_{11/2}-d_{3/2}$ transitions. In addition to N = 81, such transitions were observed also in the N = 79 and N = 77 isotonic chains. The systematics of the B(M4) values are shown in Fig. 7. The values seem to be rather independent from the neutron number, they change in the same way with Z. There is one data point which seems to break this trend, that in $^{133}_{56}$Ba$_{77}$, where the B(M4) value is lower than in the other barium isotopes. Possibly this is related to deformation and mixing in the 3/2$^+$ $d_{3/2}$ dominated state (the second 3/2$^+$ is only 290 keV away, much closer than in $^{135}$Ba and $^{137}$Ba at 588 keV and 1464 keV, respectively).

Similarly, $\nu i_{13/2}-f_{5/2}$-mediated M4 transitions were identified in several even Z nuclei around lead, with the systematics of the B(M4) values presented in Fig. 7. The values are independent of neutron number for a given Z. In $_{84}$Po and $_{82}$Pb isotopes, the transitions seem to exhibit the same strength. In contrast, below the Z = 82 shell closure, the B(M4) values decrease for $_{80}$Hg and even more for $_{78}$Pt. One data point seems to break the trend, at $^{201}_{84}$Po$_{117}$, with a very low B(M4). There are no error bars associated to this data point in the ENSDF evaluation [28], due to the missing uncertainty on the internal decay branch. This branch should be remeasured.

A very different case is that of $^{199}$Bi and $^{201}$Bi. They have a $\pi h_{9/2}$ 9/2$^-$ ground state and 1/2$^+$ first excited state. The 1/2$^+$ state has dominant intruder $\pi s_{1/2}$ configuration. The experimental B(M4) values are < 0.017 W.u. (from the internal decay branch limit of 2%) in $^{199}$Bi, and 0.00057(7) W.u. in $^{201}$Bi. These B(M4) values are much lower than any other discussed so far. This, at least partially, can be attributed to the fact that the $h_{9/2}-s_{1/2}$ M4 decay is l forbidden ($\Delta l$ = 5). A detailed discussion can be found in [29].

M4 transitions were identified in deformed nuclei as well. The $T_{1/2}$ = 31 y isomer in $^{178}$Hf has such a decay branch. The K forbidden decay has a very low strength of 3.7 $\times$ (5) $\times$ 10$^{-8}$ W.u., corresponding to a reduced hindrance of $f_{\nu }$ = 72(3) [23].

The transition strength of 95 M4 transitions is presented in the compilation of [30]. It was shown that all M4 transitions are hindered relative to the Moszkowsky estimates [31], and it advocated that for magnetic transitions, these estimates are preferable to Weisskopf units.

Fig. 7

Systematics of M4 transitions in N = 77, 79, 81 and Z = 78, 80, 82, 84 nuclei. The transition strengths are shown. The orbital pairs mediating the M4 transitions are indicated

5 E5 decays

In the spherical shell model, the pairs of orbitals which can give rise to E5 (electric triacontadipole) transitions are $h_{11/2} - s_{1/2}$ in the 50–82 shell and $i_{13/2} - p_{3/2}$ in the 82–126 one. At higher nucleon numbers, the $j_{15/2} -f_{5/2}$ pair within the N = 126–184 shell could also result in E5 transitions; however, no such decay has been observed so far.

E5 transitions mediated by the neutron $h_{11/2} -s_{1/2}$ orbital pair were observed in a number of nuclei: $^{113}$Cd$_{65}$, $^{117}$Sn$_{67}$, $^{123,125}$Te$_{71,73}$ and $^{137}$Ba$_{81}$. Here the low-energy (140–320 keV) 11/2$^-$ state was observed to decay into the 1/2$^+$ ground state.The B(E5) transition strengths are very low, in order of 0.04-$-$0.05 W.u. The only exception seems to be the case of $^{133}_{56}$Ba$_{77}$, where a high value was reported at 10(7) W.u. [32]. This value is possibly incorrect, and it might be a consequence of the difficulty of determining the very low branching ratio at <0.1%. There are several other 11/2$^-$ low-energy long-lived isomeric states in this region. However, the 3/2$^+$ $\nu d_{3/2}$ state is often the ground state, consequently these isomers decay by M4 transitions.

Proton $h_{11/2} - s_{1/2}$-mediated E5 transitions were observed below Z = 82, in $^{207}$Tl and $^{206}$Tl nuclei. The transition strengths are much higher than in the case of the neutron-mediated one, at 5.0(13) W.u and 16.6(21) W.u., respectively.

Interestingly, both $^{202}$Pb and $^{204}$Pb have two E5 decays each, mediated by the $\nu i_{13/2} - p_{3/2}$ neutron-orbital pair. These are from the 9$^-$ isomeric states with $\nu f_{5/2} i_{13/2}$ configuration. In both nuclei, the E5 transitions connect the isomer to the two lowest energy 4$^+$ levels, with transition strengths of $B(E5; 9^- \rightarrow 4^+_1)\sim$3–4 W.u. and $B(E5; 9^- \rightarrow 4^+_2)\sim$0.5-$-$0.6 W.u. This ratio as well as the B(E4) from the same isomer in $^{202}$Pb provides a possibility to test and refine the proton–proton residual interaction in this region.

Intriguingly, E5 transitions were observed in medium mass nuclei $^{90}$Y and $^{90}$Zr. The relatively large transition strengths of 1.66(15) W.u. and 8.74(33) W.u., respectively, are puzzling as there are no $\Delta j$ = 5 orbital pairs around Z, N = 50. It was suggested that the 5$^-$ isomer in $^{90}$Zr as well as other 5$^-$ states in Zr, Mo, Ru, Pd, Cd isotopes are collective states corresponding to one-phonon excitations [33]. Evidence for this comes for significant population of these states in one- and two-neutron-transfer reactions, not expected for proton-dominated states (the dominant configuration of the 5$^-$ isomer in $^{90}$Zr is $\pi g_{9/2} p_{1/2}$). Near constancy of their excitation energies and large $\beta _5$ deformation parameters are also supporting this interpretation [34].

There are E5 decays observed also in well-deformed nuclei. These are originating from K isomers. E5 decays from low-energy (< 200 keV) two-quasiparticle isomers were observed in odd-odd $^{184,186}$Re and $^{192}$Ir nuclei. More interestingly, there is also an E5 decay from the four quasiparticle high-energy 2446 keV long-lived $T_{1/2}$ = 31 y metastable state in $^{178}$Hf. Here the very low transition strength of 1.22(10) $\times$ 10$^{-4}$ W.u. is a consequence of K hindrance, and it corresponds to a reduced hindrance of $f_{\nu }$ = 175(11) [23].

6 M5, E6 and higher multipolarity decays

Only one M5 (magnetic triacontadipole) and one E6 ( electric hexacontatetrapole) transition was observed to date, both in $^{53}$Fe. The 19/2$^-$ $T_{1/2}$ = 2.54(2) min. isomer at 3040 keV can decay by an E6, an M5 and an E4 transition, with energies of 3040, 1713 and 701 keV, respectively. The first direct confirmation of the E6 $\gamma$ decay was claimed [35] very recently, in 2023. Its definitive observation needed the quantification of the sum contributions (simultaneous observation of several $\gamma$ rays in the same Ge crystal adding up to the energy of 3040 keV). The relative $\gamma$-ray intensities of the E6, M5 and E4 transitions depopulating the isomer were determined as 0.056(17), 1.05(5) and 100, respectively. These correspond to reduced transition strengths of B(E6) = 0.45(8), B(M5) = 5.4(4) and B(E4) = 0.2587(21) W.u. All states involved have predominantly three-valence particle character $\pi f_{7/2}^{-2} \nu f_{7/2}^{-1}$ and with the isomer at 19/2$^-$, they are fully aligned. Shell-model calculations in the full fp shell were performed, and effective charges were determined. Strikingly the proton effective charges $e_{\pi }$ for both E4 and E6 are around 0.6e, much lower than unity. In contrast, theoretical effective charges are expected to be larger than 1e [36]. Note that the E4, M5 and E6 decays are all mediated by the $f_{7/2}\rightarrow f_{7/2}$ transition. The large E6 strength is the result of the large number of particles, n = 13, in the $f_{7/2}$ orbital. Interestingly, the corresponding 19/2$^-$ state in the mirror nucleus $^{53}$Co has a much shorter half-life at 247(12) ms. This is because particle decay channels are open. The first ever observation of proton radioactivity [37, 38] is from this isomeric state (the proton branch is 1.5%, with the remaining 98.5% being $\beta ^+ +EC$ decay [3]).

No internal decay with multiplicity larger than six was ever observed. But they could exist. They would have enormously long partial half-lives, and they have to be in nuclei very close to stability. As indicated in Fig. 2, M6 transitions could be mediated by the intra-shell $i_{13/2}$-$p_{1/2}$ neutron transition. Therefore, in $^{207}$Pb, the 13/2$^+$ $\nu i_{13/2}$ state could decay to the 1/2$^-$ $\nu p_{1/2}$ ground state by an 1633 keV M6 transition. This would need to compete with a 569.7 keV M4 to the 5/2$^-$ $\nu f_{5/2}$ state. The expected branching is very small, in the order of 10$^{-7}$.

The most intriguing case is related to the only naturally occurring isomeric state, that is in $^{180}$Ta. Whilst the 1$^+$ ground-state $\beta$ decays with $T_{1/2}\approx$8 h, the 77 keV 9$^-$ isomer has an enormously long lifetime [3]. The 9$^+$ K isomer could decay internally by 38 keV E7 and 77 keV M8 transitions (these would be very highly electron converted), and also by both $\beta ^-$ and electron capture. Theoretical calculations suggest that internal E7 decay dominates, for which a half-life of $T_{1/2}$ = 8 $\times$ 10$^{18}$ years was predicted (since the decay is mainly via internal conversion electron, the partial $\gamma$-decay half-life would be much larger at 1.4 $\times$ 10$^{31}$ y) [39]. The corresponding experimental limit for internal decay is $T_{1/2}>$ 0.63(8)$\times$10$^{18}$ yr (90% confidence interval) [40], still one order of magnitude shorter than the theoretically predicted value.

7 Conclusion

Isomers decaying by high multipolarity internal transitions were discussed, focussing mainly on spherical nuclei, although examples from K isomeric states were also given. The higher the multipolarity of the decay, the longer the lifetime tends to be. Therefore, $\lambda \ge$ 4 decays are concentrated on and around the line of stability, as in more exotic nuclei charged particle channels would dominate. E3 decays in the vicinity of $^{208}$Pb were discussed and they were classified depending on their single-particle origins. The chains of M4 and E5 decays next to closed shells show different behaviour in different regions. Therefore, their study within the framework of the large-scale shell model is of great interest, addressing shell evolution. In addition, comparing effective charges needed in different regions of the nuclide chart could lead to a deeper understanding of them. The highest multipolarity decays so far observed are M5 and E6, both identified from the decay of an isomeric state in $^{53}$Fe. The prospect of identifying higher multipolarity decays was discussed.