$\mathrm{\Delta }K=0$ $M1$ Excitation Strength of the Well-Deformed Nucleus ${}^{164}\mathrm{Dy}$ from $K$ Mixing

The size of a $\mathrm{\Delta }K=0$ $M1$ excitation strength has been determined for the first time in a predominantly axially deformed even-even nucleus. It has been obtained from the observation of a rare $K$-mixing situation between two close-lying $J^{\pi }=1^{+}$ states of the nucleus ${}^{164}\mathrm{Dy}$ with components characterized by intrinsic projection quantum numbers $K=0$ and $K=1$. Nuclear resonance fluorescence induced by quasimonochromatic linearly polarized $\gamma$-ray beams provided evidence for $K$ mixing of the $1^{+}$ states at 3159.1(3) and 3173.6(3) keV in excitation energy from their $\gamma$-decay branching ratios into the ground-state band. The $\mathrm{\Delta }K=0$ transition strength of $B(M1;0_1^{+}\to 1_{K=0}^{+})=0.008(1){\mu }_N^2$ was inferred from a mixing analysis of their $M1$ transition rates into the ground-state band. It is in agreement with predictions from the quasiparticle phonon nuclear model. This determination represents first experimental information on the $M1$ excitation strength of a nuclear quantum state with a negative $\mathcal{R}$-symmetry quantum number.

Figure 1

Schematic representation of the interplay of $\mathcal{P}$ symmetry and $\mathcal{R}$ symmetry. The symmetry axis of the nuclei in the body-fixed coordinate frame is denoted by 1 and a generic axis perpendicular to the symmetry axis by 2. The signs denote the spatial distribution of the phases of the collective wave function over an axially deformed nucleus. With respect to a rotation of 180° about 2, the nuclei on the left-hand side are symmetric. Thus, they are $\mathcal{R}$ invariant with positive eigenvalue $r$ whereas the nuclei on the right-hand side carry $r=-1$.

Figure 2

$\gamma$-ray spectra from the ${}^{164}\mathrm{Dy}$$(\overrightarrow{\gamma },{\gamma }^{\prime })$ reaction taken at the $\mathrm{HI}\gamma \mathrm{S}$ facility [12] with beam energies $E_{\mathrm{beam}}=3075(50)\mathrm{keV}$ [panels (a) and (b)] and $E_{\mathrm{beam}}=3180(52)\mathrm{keV}$ [panels (c) and (d)]. Detectors were placed at a polar angle of $\vartheta =90°$ and azimuthally in the horizontal polarization plane (blue) of the incident $\gamma$-ray beam or perpendicular to it (red). The luminosity profile of the $\overrightarrow{\gamma }$-ray beam in arbitrary units is indicated in gray by the dashed Gaussian curve. In the spectrum shown in panel (a) the $1_3^{+}$ state is not excited. Hence, the peak observed at 3100 keV is the ground-state decay of hitherto unknown state(s).h Their transitions to the $2_1^{+}$ state are located at 3027 keV [cf. panels (a) and (c)] and decays to the $2_{\gamma }^{+}$ state are visible in the coincidence spectrum [panel (b)]. The spectra shown in panel (c) are dominated by the $1_i^{+}\to {0_1}^{+}$ ($i=1–3$) transitions at 3111.0(4), 3159.1(4), and 3173.6(4) keV and the transitions to the $2_1^{+}$ state at 3037.8(4), 3085.3(4), and 3100.1(4) keV, respectively. Already from these spectra the significant deviation from the Alaga prediction of 0.5 for the reduced relative decay-intensity ratios becomes evident for the $1_3^{+}$ state (cf. Table 1). The peak stemming from the $1_3^{+}\to 2_{\gamma }^{+}$ transition is visible in the coincidence spectrum shown in panel (d).

Figure 3

Representation of experimental constraints in the $Z-\gamma$ plane of the TSM parameters. The solid green and blue dash-dotted lines indicate experimental constraints along with their gray uncertainty bands obtained from the measured relative decay-intensity ratios $R_{1_2\to 2/0}^{\exp }$ and $R_{1_3\to 2/0}^{\exp }$, respectively. The occurrence of two blue lines originates from an undefined phase in the initial problem. The intersections of these constraints describe potential solutions ($Z$,$\gamma$) to the TSM scenario. Additional information on the ratio of excitation strengths of the involved states $B_{2/3}$, is shown by a red dashed line. The inlay depicts the compatibility of parameter tuples ($Z$,$\gamma$) in terms of ${\chi }^2$ values. The most probable solution, which corresponds to minimum ${\chi }^2$, is indicated by a red dot along with its $1\sigma$ confidence interval. It exhibits a 46 times smaller ${\chi }^2$ value than the solution with little mixing.

Figure 4

Comparison of level schemes obtained from experimental data [panel (a)] and the TSM analysis [panel (b)]. Absolute transition strengths $B(M1;1_{2,3}^{+}\to 0_1^{+})$ are taken from Ref. [18]. All other transition strengths were calculated from the decay-intensity ratios which were determined in the present work. Based on the given uncertainties, the agreement is excellent for each individual transition. The $E2$ transition between the $1_3^{+}$ and $2_{\gamma }^{+}$ states is shown in red. This assignment was possible only due to the identification of a prevailing $K=0$ component in the wave function of the $1_3^{+}$ state.