A fully self-consistent three-dimensional tilted-axis-cranking covariant density functional (3DTAC-CDFT) study of the odd-odd nucleus 106^{106}Rh found two distinct chiral solutions, the negative-parity configuration πg9/2−1⊗νh11/2\pi g_{9/2}^{-1}\otimes\nu h_{11/2} (whose calculated spectrum and B(M1)/B(E2)B(M1)/B(E2) ratios match the observed doublet) and a predicted positive-parity configuration πg9/2−1⊗νh11/22(gd)1\pi g_{9/2}^{-1}\otimes\nu h_{11/2}^{2}(gd)^{1}. Being a static mean-field treatment, however, it can describe neither quantal chiral vibrations nor tunneling between the left- and right-handed sectors, so the energy splitting between chiral partner bands, the partner-band wave functions, and the tunneling amplitudes and lifetimes could not be computed, and the predicted positive-parity chiral band awaited experimental confirmation. The open problem is to go beyond that static mean field for 106^{106}Rh — quantizing the chiral motion on the 3DTAC-CDFT Routhian surfaces (collective Hamiltonian or RPA) or by an equivalent microscopic quantal treatment — to determine whether the multiple chirality in 106^{106}Rh is static or vibrational and to compute its observable consequences, and to specify the experimental measurement that would test the predicted positive-parity chiral band — the excitation energies, spins, and B(M1)B(M1)/B(E2)B(E2) values that would confirm or exclude it. A 2022 systematic 3DTAC-CDFT survey of 102−107^{102-107}Rh settled the configuration systematics at mean-field level, and quantal or dynamical microscopic descriptions now exist for other chiral nuclei (130^{130}Cs, 135^{135}Nd, 106^{106}Ag), but no quantal dynamical treatment exists for 106^{106}Rh, and its predicted positive-parity chiral band remains unobserved.

Background

In a triaxial nucleus — one whose three principal axes have unequal lengths — rotating at high spin, the angular momenta carried by the valence protons, the valence neutrons, and the collective core can align along the short, long, and intermediate axes, respectively. These three mutually perpendicular angular momenta form a screw that can be left- or right-handed: the intrinsic-frame chiral symmetry is spontaneously broken, as predicted by Frauendorf and Meng in 1997. Since the laboratory frame retains the symmetry, the observable signature is a pair of near-degenerate ΔI=1\Delta I=1 rotational bands of the same parity, called chiral doublet bands. Two idealized limits are distinguished: static chirality, where left- and right-handed intrinsic minima both exist and the two observed bands are the even and odd tunneling combinations of one chiral configuration; and chiral vibration, where the minimum is planar and the angular momentum oscillates between the left- and right-handed sectors, so the two bands are zero- and one-phonon vibrational states. The energy splitting between the partner bands is the observable that most directly discriminates the two, but it is governed by quantal tunneling and cannot be obtained from a static mean field, in which the left- and right-handed solutions are exactly degenerate.

The microscopic tool of the source study is covariant density functional theory (CDFT), a relativistic mean-field-type energy-density framework, combined with tilted-axis cranking (TAC), in which the rotational axis is free to orient away from the principal axes and the mean field is solved self-consistently in the rotating frame; the energy in this rotating frame is the Routhian. In 3DTAC-CDFT the orientation of the angular velocity is determined by minimizing the Routhian over the polar and azimuthal angles, and a nonzero azimuthal angle (deviation from the plane of the short and long axes) signals aplanar, chiral rotation. Applied to 106^{106}Rh, the method found, for the configurations πg9/2−1⊗νh11/2\pi g_{9/2}^{-1}\otimes\nu h_{11/2} (negative parity) and πg9/2−1⊗νh11/22(gd)1\pi g_{9/2}^{-1}\otimes\nu h_{11/2}^{2}(gd)^{1} (positive parity) — where π\pi and ν\nu label the valence proton and neutron quasiparticles, the subscript names the shell-model orbital, and superscripts count quasiparticles, with −1-1 denoting a hole — a transition from planar to chiral rotation with increasing rotational frequency, with the calculated spectrum and B(M1)/B(E2)B(M1)/B(E2) ratios (the ratio of magnetic-dipole to electric-quadrupole transition strengths) for the negative-parity configuration in good agreement with the pair of negative-parity chiral bands observed in 106^{106}Rh. Because the negative-parity partner is well reproduced, the prediction of a second, positive-parity chiral band — multiple chirality, i.e., more than one pair of chiral doublet bands in one nucleus — carries substantial weight. The Routhian barriers between the left- and right-handed minima, however, are soft (of order only tens of keV), implying substantial tunneling and, consistently, the observed partner-band separation of roughly 300 keV. The source paper states explicitly that the mean-field level cannot calculate chiral vibrations or tunneling, so the energy splitting between the partner bands cannot be computed there, and names two beyond-mean-field remedies: the random phase approximation (RPA, small-amplitude quantal oscillation theory on top of the mean field) or a collective Hamiltonian (quantizing the chiral coordinate in analogy with the Bohr Hamiltonian for shape vibrations) built on the 3DTAC-CDFT energy/Routhian surfaces. Before that work, a collective Hamiltonian for chiral modes had been constructed only on tilted-axis-cranking surfaces of a phenomenological triaxial rotor coupled to an h11/2h_{11/2} proton particle and neutron hole, which restores the broken chiral symmetry and reproduces particle-rotor-model spectra, but is not based on microscopic CDFT surfaces and had not been applied to 106^{106}Rh.

Problem Statement

Provide a beyond-mean-field, quantal description of the chiral motion associated with the two chiral configurations predicted for 106^{106}Rh, πg9/2−1⊗νh11/2\pi g_{9/2}^{-1}\otimes\nu h_{11/2} and πg9/2−1⊗νh11/22(gd)1\pi g_{9/2}^{-1}\otimes\nu h_{11/2}^{2}(gd)^{1} — the required deliverable, together with its falsifiable predictions — and specify the experimental measurement that would test it. On the theory side, this means quantizing the chiral degree of freedom on the 3DTAC-CDFT potential-energy/Routhian surfaces — for example by constructing a collective Hamiltonian with potentials and mass parameters extracted from those surfaces, or by an RPA treatment of the cranked solutions — or delivering an equivalent microscopic quantal treatment (angular-momentum projection, i.e., projecting the intrinsic mean-field states onto states of good total angular momentum, or configuration interaction, i.e., diagonalization within a basis of multi-quasiparticle configurations), such that one obtains all of the following: the tunneling-induced energy splitting between the chiral partner bands as a function of spin, the partner-band wave functions, quantitative tunneling amplitudes and lifetimes set by the soft Routhian barriers of order tens of keV, and the observable electromagnetic matrix elements, in particular B(M1)B(M1) and B(E2)B(E2), thereby deciding whether the multiple chirality in 106^{106}Rh is static or vibrational. On the experimental side, this means specifying the measurement that would test the predicted positive-parity chiral band in 106^{106}Rh — its excitation energies, spin and parity assignments, and B(M1)B(M1)/B(E2)B(E2) values, i.e., the observables that would confirm or exclude it — rather than performing that measurement. The common objective of these two linked components is to complete the establishment and characterize the mode of the predicted multiple chirality in 106^{106}Rh: the quantal treatment must confront the already observed negative-parity doublet (including its roughly 300 keV partner-band separation, which the static mean field cannot explain) and make falsifiable predictions for the positive-parity band, whose experimental identification or exclusion is the corresponding test.

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