Fingerprints of the triaxial deformation from energies and B(E2) transition probabilities of $\gamma$-bands in transitional and deformed nuclei

Abstract

The energies and B(E2) transitions involving the states of the ground- and $\gamma$-bands in thirty transitional and deformed nuclei are calculated using the triaxial projected shell model (TPSM) approach. Systematic good agreement with the existing data substantiates the reliability of the model predictions. The Gamma-rotor version of the collective Bohr Hamiltonian is discussed in order to quantify the classification with respect to the triaxial shape degree of freedom. The pertaining criteria are applied to the TPSM results and the staggering of the energies of the $\gamma$-bands is analyzed in detail. An analog staggering of the intra-$\gamma$ $B(E2, I \rightarrow I-2)$ is introduced for the first time. The emergence of the staggering phenomena in the transitions is explained in the terms of interactions between the bands.

Appendix: Detailed discussion of the Gamma-rotor model

Section 2 presented the results of the Gamma-rotor model for a selection collective potentials, which were used to classify the nature of the triaxiality of the nuclear shape. A summary of characteristic relations between energies and transition probabilities between of the lowest collective excitations of the quadrupole type was given. Here we provide the details of how these characteristic come about.

The potential $\chi -\kappa =200-0$ represents a rigid prolate nucleus with a harmonic $\gamma$-vibration far above the $4^+_1$ level of the ground-band and at about twice the energy are the two-phonon bands. The $K=4$ band labeled by $4^+_3$ in Fig. 1 represents a traveling wave generated by adding a second $K=2$ on top of the first with the same angular momentum along the symmetry axis. Transforming the ratio $B(E2,4^+_3\rightarrow 2^+_2)/B(E2,2^+_2\rightarrow 0^+_1)=0.094/0.033$ in Table 2 into the ratio of the matrix elements for transitions between states in an axial symmetric potential (see [1, 40]), one obtains 1.42, which is approximately the ratio expected for the two- and one-phonon states of a harmonic vibration. The $K=0$ two-phonon state is generated by putting the second phonon with opposite angular momentum on top of the first, which represents a pulsating wave. Transforming the ratio $B(E2,0^+_2\rightarrow 2^+_2)/B(E2,2^+_2\rightarrow 0^+_1)=0.195/0.033$ in Table 2 into the ratio of the matrix elements for transitions between the states, one obtains 1.08, which is close to 1 expected for a harmonic vibration. The potential $\chi (1-\cos {3\gamma })=3/2\gamma ^2+...$ is not exactly harmonic. The distances between the bands are large, such that the couplings of the harmonic bands with higher terms is small, which is reflected by the small value of ${\bar{S}}(6)=-0.03$ in Table 1.

The difference of the harmonic ratios can be understood by the following classical consideration. The $K=2$ one-phonon state correspond to a traveling wave in the plane perpendicular to the symmetry axis with $x=A\cos {\omega t}$ and $y=A\sin {\omega t}$. The average radiation power is $\propto \overline{ x^2} +\overline{ y^2}+2\overline{xy}= A^2$. The $K=4$ two-phonon state correspond to a traveling wave with the amplitude $\sqrt{2}A$. The radiation power of the $K=4$ two-phonon state is 2$A^2$. The $K=0$ two-phonon state correspond to a pulsating wave $x=\sqrt{2} A\cos {\omega t}$, which gives a radiation power $\propto \overline{x^2}=A^2$, corresponding to a ratio of 1.

The second limiting case is the $\gamma$-independent potential (0–0) of the Wilet-Jean model [26], which is discussed in detail in the textbook by Rowe and Wood [40] (p. 117, 222 ff.) The states carry the seniority quantum number v. The quadrupole operator changes the seniority by 1, which is reflected by the B(E2) values in Table 2. The static quadrupole moments vanish because they are matrix elements between states with the same v, and the $B(E2,2^+_2\rightarrow 0^+_1)=B(E2,4^+_3\rightarrow 2^+_2)=0$ because v changes by 2. The states organize into $\varDelta I=2$ bands of good signature, which are connected by strong $B(E2,I\rightarrow I-2)$ values (cf. also Fig. 2.3 of Ref. [40]). The band based on the $2^+_2$ state has $v=2+I/2$ and the one on $3^+_1$ has $v=3+I/2$. As the energy of the states is $E(v)=v(v+3)$, the two $\gamma$-band branches have a pronounced even-I-down staggering

$$\begin{aligned} S(I)=\frac{1}{8}\mp \left( \frac{I}{4}+\frac{9}{8}\right) ,~~I=\begin{array}{c} even\\ odd\end{array}. \end{aligned}$$

(24)

The $\gamma$-band head has the same energy as the $4^+_1$ state of the ground-band, $E(2^+_2)=E(4^+_1)$, which is another signature of the $\gamma$-independent potential.

The shallow potential (10–0) and the soft potentials 20–0 and 50–0 illustrate the transition to rigid prolate limit (for more details see Ref. [39]). The static quadrupole moments quickly approach the limit for prolate shape as a consequence of the seniority mixing. The $B(E2,2^+_2\rightarrow 0^+_1)$ and $B(E2,4^+_3\rightarrow 2^+_2)$ first increase because of the seniority mixing, and then they decrease because the amplitude of the $\gamma$-vibration decreases with $\chi$. The $B(E2, 2^+_2\rightarrow 2^+_1)$ values decrease due to the seniority mixing. The potential term $\cos 3\gamma$ couples states of $v=\pm 1$ [40]. The coupling of the even-I branch of the $\gamma$-band with the ground-band pushes them up. The even-I band on top of the $0^+_2$ band does not couple because v differs by 2, which reduces the staggering. For large values of the $\chi$ the higher order $\varDelta v=\pm 1$ couplings quench the staggering. The repulsion caused by the coupling pushes the $2^+_2$ band head above the $4^+_1$ state of the ground-band.

The prolate harmonic limit provides another perspective that is instructive for interpreting the TPSM results. First inspect the probability density $P(\gamma )$ of the wave functions of the lowest bands obtained by integration over the angle degrees of freedom. Figure 9 of Ref. [39] shows the case of potential 50–0. The density $P(\gamma )$ of the ground band is centered at $\gamma =0^\circ$ (after dividing out the volume element $\sin 3\gamma$). For the $\gamma$-band built on the $2^+_2$, the density has a maximum around 20$^\circ$, which indicates that the state represents a wave that travels around the symmetry axis. The double $\gamma$-band ($\gamma \gamma 4$) built on the $4^+_3$ state has a maximum at $27^\circ$, which correspond to a traveling wave with a larger amplitude. (The concept of traveling (tidal) waves has been developed to microscopically calculate unharmonic many phonon excitation (see e.g. Ref. [2]).) For the double $\gamma$-band ($\gamma \gamma 0$) built on the 0$^+_2$ state the density $P(\gamma )$ has two maxima with a zero at $15^\circ$ in between. The $0^+_2$ state represents a vibration of the triaxial shape between the prolate and oblate turning points (see [1]). The structure of the wave functions for other $\chi$ values is qualitatively the same. For larger $\chi$ the distributions $P(\gamma )$ are squeezed such that they fit into the potential (see 200–0 in Fig. 9 of [39]). For smaller $\chi$ the distributions are shifted to larger $\gamma$ until they become symmetric about $30^\circ$ for $\chi =0$.

The deviation of “$1-\cos 3\gamma$” from the leading order term $3\gamma ^2/2$ couples the even-I-states of the $\gamma$-band with the states of the ground-band and of the $0^+_2$ band, which, respectively, shifts them up or down. The down shift by the repulsion from the upper level prevails because it has a larger coupling matrix element. The reason is that the difference $\vert 1-\cos 3\gamma -3\gamma ^2/2\vert$ becomes larger with increasing $\gamma$, and the probability density $P(\gamma )$ of the ground-band is closer to zero localized than the one of the $0^+_2$ band, which has a zero and reaches further out (compare the cases g, $\gamma$ and $\gamma \gamma 0$) for the potential 200–0 in Fig. 9 of [39]). With decreasing $\chi$, the potentials become shallower, which reduces distance between the bands. The level shift become larger because the energy differences decrease and the coupling matrix elements increase. The even-I-down staggering pattern of the $\gamma$-band evolves.

The potentials with $\chi$=0 represent the cases of maximal triaxiality. The term $\cos ^23\gamma$ of the potential is symmetric with respect to $\gamma \rightarrow 60^\circ \gamma$. It changes the seniority by $\varDelta v=0, \pm 2$. The states have good $\gamma$-parity, which is reflected by $Q(2^+_1)=Q(2^+_2)=0$, $B(E2, 2^+_2\rightarrow 0^+_1 )=0$ and $B(E2, 4^+_3\rightarrow 2^+_2 )=0$ in Table 2. For small $\kappa$ the diagonal term $\langle Iv\vert \cos ^2 3\gamma \vert Iv\rangle$ dominates, which is larger for even I ($\gamma$ parity even) than for odd I ($\gamma$-parity odd). With increasing $\kappa$ level shifts decrease the even-I-down staggering of the $\gamma$-band until it disappears and changes into the even-I-up pattern (see 0–20 and 0–100 in Fig. 2 and Table 2). In the case of $\kappa <0$ with a barrier at $\gamma =30^\circ$ the shifts increase the staggering (see 0- (-20) in Fig. 2 and Table 2). With increasing $\kappa$ the $\varDelta v=\pm 2$ terms become important and the structure of the bands based on the $0^+_1,~2^+_2,~4^+_3$ state quickly approach the ones of the triaxial rotor with $\gamma =30^\circ$.

The case is discussed as the Meyer-ter-Vehn limit in Ref. [40] and is known as the symmetric top in molecular physics. The ratios of the moments of inertia are 4:1:1 for the medium, short, long axes, respectively, which means the eigenfunctions are the ones of a symmetric rotor, where K is the angular momentum projection on the medium axis with the largest moment of inertia. The energy ratios are

$$\begin{aligned} \frac{E(I,K)}{E(2^+_1)}=\frac{4I(I+1)-3K^2}{12}. \end{aligned}$$

(25)

The corresponding ratio $E(2^+_2)/E(4^+_1)=0.75$ represent the limit of maximal triaxiality. For the potential 0–200 the ratio is 0.81 and for the potential 0–20 it is 0.97. The Meyer-ter-Vehn limit of the staggering parameter is

$$\begin{aligned} S(I)=\frac{1}{6}\pm \left( I-\frac{5}{2}\right) ,~~I=\begin{array}{c} even\\ odd\end{array}. \end{aligned}$$

(26)

The value ${\bar{S}}(6)=4$ is to be compared with 3.87 for the potential 0–200 in Table 2. The quadruple operator is $(Q_2+Q_{-2})/\sqrt{2}$ with respect to the quantization axis “m”. The ratios of reduced transition probabilities are given by the corresponding ratios of the squares of the Clebsch-Gordan coefficients, (quoted e.g., Fig. 2.6 of Ref. [40]). For the lowest transitions, the B(E2) values divided by the $B(E2, 2^+_1\rightarrow 0^+_1)$ are $B(E2, 2^+_2\rightarrow 2^+_1)=1.43$ (1.42, 1.42), $B(E2, 4^+_2\rightarrow 2^+_2)=0.60$ (0.58, 0.73), $B(E2, 4^+_1\rightarrow 2^+_1)=1.39$ (1.43, 1.42), $B(E2, 3^+_1\rightarrow 2^+_2)=1.79$ (1.71, 1.45), $B(E2, 4^+_3\rightarrow 3^+_1)=0.56$ (0.60, 0.86), where the ratios for the potentials 0–200 and 0–20 are quoted in parenthesis. The B(E2) ratios for the Meyer-ter-Vehn limit are more rapidly approached than for the energies.

The transition from the symmetric triaxial to the prolate potentials involves the competition between the two potential terms discussed. Consider the soft potentials 0–50, 50–50 and 50–0. The symmetry with respect to $\gamma \rightarrow 60^\circ -\gamma$ of 0–50 is progressively broken in 50–50 and 50–0. As a consequence, the $B(E2, 2^+_2\rightarrow 0^+_1 )$ and $B(E2, 4^+_3\rightarrow 2^+_2 )$ become quickly large and the $B(E2, 2^+_2\rightarrow 2^+_1 )$ decrease. The even-I-up staggering of the $\gamma$-band is reduced, and it reverses to the even-I-down pattern for the potentials. Along the sequence the state $2^+_2$ moves away from the state $4^+_1$ to larger energies. Both trends can be taken as signatures of decreasing triaxiality. The other cases between the prolate and symmetric triaxial limits exemplify the changes. Hence, a small staggering parameter indicates either prolate potential or a transitional one, where the former has a large and the latter a small ratio $E(2^+_2)/E(4^+_1)$. For all the cases, the $0^+_2$ states represent a pulsating $\gamma$ vibration with a zero in the $P(\gamma )$ probability density, the position of which depends on the potential parameters (see Fig. 9 of [39]). The $B(E2, 0^+_2\rightarrow 2^+_2 )$ values are always substantial, being larger for more flatter potentials.

The reversal of the phase of S(I) with increasing $\kappa$ is understood as follows: The term symmetric $(\cos 3\gamma )^2$ couples the $\gamma$-band strongly to the ground-band because both the states do not have a zero in the $\gamma$ degree of freedom. The coupling to the $0^+_2$ band is weaker because its states have a zero in the $\gamma$ degree of freedom. Hence the repulsion between the even-I-states of the ground- and $\gamma$-band prevails which generates even-I-up staggering pattern. As discussed above, the “$\cos 3\gamma$” term generates the even-I-down staggering pattern, and the competition of both terms determine the phase of S(I).

The potential 20–50 represents a case incipient of prolate-oblate shape coexistence. There is a regular rotational ground-band with $P(\gamma )$ localized on the prolate side. The $0^+_2$ state comes low in energy. The second bump of $P(\gamma )$ on the oblate side is larger than the first on the prolate side. The $\varDelta I=2$ sequence built on the $0^+_2$ state can be interpreted as the “oblate” rotational band. However, one can alternatively interpret the sequence as a band built on a large-amplitude $\gamma$-vibration, which applies to all potentials (see discussion of the WJ limit related to Fig. 4.15 of Ref. [40]).

Now the mixing of the even-I members of the harmonic ground-band (g), $\gamma$-band ($\gamma$) and double-$\gamma$-band ($\gamma \gamma 0$) is considered in detail,

$$\begin{aligned} \psi (I,M,\varOmega ,\gamma )= & {} c_g\psi _g(I,M,\varOmega ,\gamma )\nonumber \\{} & {} +c_{\gamma \gamma 0}\psi _{\gamma \gamma 0}(I,M,\varOmega ,\gamma )\nonumber \\{} & {} +\sqrt{1-c_g^2-c_{\gamma \gamma 0}^2}\psi _\gamma (I,M,\varOmega ,\gamma ),\nonumber \\ \end{aligned}$$

(27)

where $c_g$ and $c_{\gamma \gamma 0}$ are the mixing amplitudes. The basis states are eigenfunctions of the Gamma-rotor Hamiltonian (3) with the harmonic prolate potential $\chi 3/2\gamma ^2$,

$$\begin{aligned} \psi _g(I,M,\varOmega ,\gamma )=N(I)D^I_{M0}(\varOmega )\phi _g(\gamma ), \end{aligned}$$

(28)

$$\begin{aligned} \psi _\gamma (I,M,\varOmega ,\gamma )=N(I)D^I_{M2}(\varOmega )\phi _{\gamma }(\gamma ), \end{aligned}$$

(29)

$$\begin{aligned} \psi _{\gamma \gamma 0}(I,M,\varOmega ,\gamma )=N(I)D^I_{M0}(\varOmega )\phi _{\gamma \gamma 0}(\gamma ), \end{aligned}$$

(30)

where N(I) are the normalization factors of the D functions. The $\psi (\gamma )$ are the normalized wave functions in the $\gamma$-degree of freedom, the probability distributions are shown in Fig. of Ref. [39]. Their phases are chosen such that they are positive for small $\gamma$. The reduced transition matrix elements for the $I\rightarrow I-2$ transitions in units of $3ZR_0^2\beta /4\pi$ are the following (cf. Ref. [1] Eq. (4–91)).

$$\begin{aligned}{} & {} \langle I-2,\gamma \vert \vert \mathscr {M}(E2)\vert \vert I, \gamma \rangle \nonumber \\{} & {} \quad = \sqrt{2I+1}\langle I~2~2~0\vert I-2~2\rangle \langle \psi _\gamma \vert \cos \gamma \vert \psi _\gamma \rangle ~>0. \end{aligned}$$

(31)

$$\begin{aligned}{} & {} \langle I-2,g\vert \vert \mathscr {M}(E2)\vert \vert I,\gamma \rangle \nonumber \\{} & {} \quad =\sqrt{2I+1}\langle I~2~2~-2\vert I-2~0\rangle \langle \psi _g\vert \sin \gamma \vert \psi _\gamma \rangle /\sqrt{2}~>0.\nonumber \\ \end{aligned}$$

(32)

The matrix element is positive because both $\psi _g(\gamma )$ and $\psi _\gamma (\gamma )$ do have a zero and are positive over the whole $\gamma$ range.

$$\begin{aligned}{} & {} \langle I-2,\gamma \gamma 0\vert \vert \mathscr {M}(E2)\vert \vert I,\gamma \rangle \nonumber \\{} & {} \quad =\sqrt{2I+1}\langle I~2~2~-2\vert I-2~0\rangle \langle \psi _{\gamma \gamma 0}\vert \sin \gamma \vert \psi _\gamma \rangle /\sqrt{2}~<0.\nonumber \\ \end{aligned}$$

(33)

The function $\psi _{\gamma \gamma 0}(\gamma )$ is positive for small $\gamma$ and negative for large $\gamma$, while $\psi _g(\gamma )$ is positive over the whole $\gamma$ range. Integrating the $\sin \gamma$ function results in a negative value because it weights more the large $\gamma$ values. As discussed for the prolate regime, $\vert \langle I-2,\gamma \gamma 0\vert \vert \mathscr {M}(E2)\vert \vert I,\gamma \rangle \vert =\langle I-2,g\vert \vert \mathscr {M}(E2)\vert \vert I,\gamma \rangle$ in the harmonic limit.

The coupling is generated by the difference of the potential from its prolate quadratic approximation

$$\begin{aligned} V_c(\gamma )=\chi (1-\cos 3\gamma )+\kappa (\cos ^23\gamma -1)-\chi \frac{3\gamma ^2}{2}. \end{aligned}$$

(34)

For a prolate potential ($\kappa =0$) $V_c(\gamma )$ is weakly negative for small $\gamma$ and strongly negative for large $\gamma$. The coupling matrix element $V_{g\gamma }=\langle \psi _g\vert V_c(\gamma )\vert \psi _\gamma \rangle <0$. The matrix element $V_{\gamma \gamma 0\gamma }=\langle \psi _{\gamma \gamma 0}\vert V_c(\gamma )\vert \psi _\gamma \rangle >0$ because $\psi _{\gamma \gamma 0}(\gamma )<0$ for large $\gamma$ where $\vert V_c\vert$ is large. For $\psi _{g}(\gamma )$ is predominantly localized in the small-$\gamma$ region and $\psi _{\gamma \gamma 0}(\gamma )$ is predominantly localized in the large-$\gamma$ region $\vert V_{g\gamma }\vert <V_{\gamma \gamma 0\gamma }$. With $E_\gamma (I)-E_g(I)= E_{\gamma \gamma 0}-E_\gamma (I)$ the diagonalization of the 3 x 3 matrix gives a down-shift of the middle eigenvalue, which represents the $\gamma$-band with admixtures. As expected, the repulsion from the $\gamma \gamma 0$ state prevails over the one from the g-state because of the larger coupling matrix element of the former. The mixing amplitudes are both negative, where $\vert c_{\gamma \gamma 0}\vert >\vert c_g\vert$, which increases the reduced matrix element by $\vert c_{\gamma \gamma 0}-c_g\vert \langle I-2,g\vert \vert \mathscr {M}(E2)\vert \vert I, \gamma \rangle$.

Now consider a triaxial potential by increasing $\kappa$ while keeping $\chi$ constant (see 50–0, 50–50, 50–100 in Fig. 1). The term $\kappa (\cos ^23\gamma -1)$, which decreases $V_c$, is symmetric about $30^\circ$. As a cosequence,the coupling matrix element $V_{g\gamma }=\langle \psi _g\vert V_c(\gamma )\vert \psi _\gamma \rangle$ becomes more and more negative. The matrix element $V_{\gamma \gamma 0\gamma }=\langle \psi _{\gamma \gamma 0}\vert V_c(\gamma )\vert \psi _\gamma \rangle$ does not change much because $\psi _{\gamma \gamma 0}(\gamma )$ is antisymmetric about $30^\circ$. Hence for a certain value of $\kappa$ coupling matrix element reverses order to $\vert V_{g\gamma }\vert >V_{\gamma \gamma 0\gamma }$ and the staggering of the energies and of the B(E2) values change to the even-I-up pattern of triaxial potentials.

For the limiting case of the WJ potential the staggering of the inband B(E2) values reflects the seniority conservation. The B(E2) transition operator (5) has the selection rule $\varDelta v=\pm \,1$. As seen in Fig. 2, the transitions $I_1\rightarrow I_1-2, ~I$-odd are allowed while the transitions $I_2\rightarrow I_2-2, ~I$-even are forbidden. This results in a staggering pattern of SE22(I) with the phase opposite to the one of S(I). In analogous conjecture holds for SBE21(I). For the symmetric top limit the reversed staggering pattern appears as a consequence of the fact that the ratios of B(E2) values follow the Alaga rules.