Origin of the electric hexadecapole isomer in $^{93}$Mo

TL;DR

The paper tackles the origin of the $21/2^+$ electric hexadecapole ($E4$) isomer in $^{93}$Mo, a long-lived state decaying via a slow $E4$ transition because the fast $E2$ path is energetically forbidden. Using a shell-model framework in the $\nu 1d_{5/2} \otimes \pi 0g_{9/2}^2$ space with empirically tuned $\nu\pi$ and like-particle interactions, the authors reveal that a strong attractive $\nu\pi$ interaction in the maximally aligned $J_\pi=7$ configuration inverts the $21/2^+$ and $17/2^+$ levels, enabling the $E4$ decay to $13/2^+$. The $E4$ transition is found to be proton-dominated, with the matrix element $B(E4;21/2^+\to 13/2^+) = (0.83\,e_\nu + 3.53\,e_\pi)^2 \, b^8$, where constructive interference with the neutron component enhances the amplitude; the calculated values agree well with experimental data and show a ~40% reduction in the $B(E2;17/2^+\to21/2^+)$ relative to earlier estimates. A comparative analysis with $^{99}$Cd, where no inversion occurs, underscores the central role of the $\nu\pi$ interaction in maintaining the $E4$ isomer, with implications for NEEC and the study of similar high-multipole isomers in other nuclei.

Abstract

We present a shell-model analysis of $^{93}$Mo to investigate the unusual behavior of its ${21/2}^+$ isomer -- a prominent candidate for nuclear excitation by electronic capture. This state is unique as its decay is dominated by a slow electric hexadecapole $E4$ transition, while the typically much faster electric quadrupole $E2$ decay path is energetically forbidden. We investigate the microscopic origin of this phenomenon by examining in detail the structure of the wave functions of the initial and final states, and the $E4$ transition matrix elements. This analysis of $^{93}$Mo is contrasted with that of its particle-hole conjugate, $^{99}$Cd, where such an $E4$ transition is absent.

Figures (5)

• Figure 1: (Color online) A part of the experimental level scheme in (a) $^{93}$Mo and (b) $^{99}$Cd. The ${21/2}^+$ state in $^{93}$Mo can only decay via possible $E4$ transition to the lower-lying ${13/2}^+$ state.
• Figure 2: (Color online) Calculated wave-function probability for the lowest-lying (a) ${13/2}^+$ and (b) ${17/2}^+$ states. $J_\pi$ denotes the angular momentum of two protons in the $0g_{9/2}$ orbital which couples to the odd-neutron in the $1d_{5/2}$ orbital to generate the respective states.
• Figure 3: (Color online) Schematic representation of the real part of the rank-4 spherical harmonics which govern the angular properties of an electric hexadecapole $E4$ transition. The color indicates the sign of the function; red for positive and blue for negative.
• Figure 4: (Color online) Variation of normalized radial integrands with radial distance involved in $E2$ and $E4$ radial overlaps compared with the proton density shown for the $0g_{9/2}$ orbital in $^{93}$Mo. $R$ denotes the radial harmonic oscillator wave function.
• Figure 5: (Color online) Decomposition of the proton contribution to the $E4$ transition matrix element (in units of $b^4$) for the ${21/2}^+ \rightarrow {13/2}^+$ transition in $^{93}$Mo. The bars show the contribution from each two-proton coupling $(J_\pi=4,6,8)$ in the final $13/2^+$ state. The neutron part of the matrix element from the $1d_{5/2} \longleftrightarrow 1d_{5/2}$ transition is calculated to be 23.32 (in units of $b^4$). The constructive interference between proton and neutron contributions leads to a total reduced transition probability of $B(E4;{21/2}^+ \rightarrow {13/2}^+)= (0.83 e_\nu+3.53 e_\pi)^2b^8$.