The Particle-Rotor-Quadrupole-Coupling Model for Transitional Odd-A Nuclei ^†

Abstract

The particle-rotor-quadrupole-coupling model, in which the quadrupole–quadrupole interaction of the even-even core is described by a triaxial rotor with a single-j particle, is adopted to describe low-lying spectra of odd-A nuclei within the vibrational to triaxial transition region. In contrast to the particle-plus-rotor-model, the quadrupole–quadrupole interaction introduced in the particle-rotor-quadrupole-coupling model keeps the rotational symmetry in the collective model framework without approximation. To demonstrate the usability, low-lying level energies, reduced E2 transition probabilities, and ground-state quadrupole moments of ¹³⁵Ba and ¹³¹Xe are fit by the model, of which the results are compared with the experimental data and those of other models. It is shown that the fitting results of the particle-rotor-quadrupole-coupling model to the low-lying level energies, reduced E2 transition probabilities, and ground-state electric quadrupole moments of ¹³⁵Ba and ¹³¹Xe are the best, of which the model parameters of the even-even core are determined by the triaxial rotor model in fitting the low-lying spectra of ¹³⁴Ba and ¹³⁰Xe. In comparison with the E(5/4) model results of ¹³⁵Ba, it is also shown that the quadrupole–quadrupole interaction of the even-even core with the single particle adopted can indeed reproduce the E(5/4) critical point behavior. The fitting quality of the reduced E2 transition probabilities among low-lying states by the particle-rotor-quadrupole-coupling model is also noticeably improved. Thus, it can be concluded that the particle-rotor-quadrupole-coupling model is suitable to describe low-lying properties of odd-A nuclei within the transitional region.

1. Introduction

Until now, it has been shown that both the collective model [1] and the interacting boson model [2] successfully describe low-lying spectra of medium and heavy mass nuclei. In the well-deformed mass region, shell model description based on the pseudo SU(3) model [3,4,5] has also been proven to be successful [6,7,8]. Shape (phase) evolution in these nuclei has been extensively studied [9,10,11,12,13,14,15].

For odd-mass nuclei, besides the pseudo SU(3) model [16,17], the quadrupole–quadrupole coupling of the even-even core with an odd-particle is adopted in the interacting boson–fermion model (IBFM) [18], while it is treated approximately in the particle-plus-rotor-model (PRM) by using the Nilsson deformed shell model basis for the odd-particle considered [1]. The PRM has been widely used to describe deformed odd-mass nuclei in the collective model framework [19,20,21,22,23,24,25,26]. Nevertheless, though less attention has been paid, multipole interactions of the even-even core with an odd-particle, including the quadrupole–quadrupole interaction, can be considered explicitly without approximation in the collective model framework [27,28,29,30,31,32,33].

In this work, the particle-rotor-quadrupole-coupling model (PRQCM)—involving quadrupole–quadrupole interaction of the even-even core with a single-j particle—is adopted to describe low-lying spectra of odd-A nuclei within the vibrational to triaxial transition region. As an example of the model application, low-lying level energies, reduced E2 transition probabilities, and the ground-state quadrupole moments of ¹³⁵Ba and ¹³¹Xe are calculated and compared with the experimental data and the results of the IBFM and the PRM.

2. The Particle-Rotor-Quadrupole-Coupling and Other Models

2.1. The PRQCM Model

Similar to the prescription shown in [27], the Hamiltonian of the PRQCM is expressed as

$$\widehat{H}={\widehat{H}}_{\mathrm{rot}}-\kappa Q·Q^{\mathrm{F}}+b{\widehat{J}}^2,$$

(1)

where ${\widehat{H}}_{\mathrm{rot}}$ is the triaxial rotor Hamiltonian of the even-even core; the second term is the quadrupole–quadrupole interaction of the core with the single particle concerned; and the third term is proportional to the scalar product of the total angular momentum, which is equivalent to the dipole–dipole interaction of the core with the particle introduced in [27]. Since only one particle in a j-orbit is considered, the single-particle energy contribution obtained from the spherical shell model to (1) is merely a constant and not included. The ${\widehat{H}}_{\mathrm{rot}}$ is given by

$${\widehat{H}}_{\mathrm{rot}}=\sum _{k=1}^3\frac{{{\widehat{L}}_k^{\prime }}^2}{2{\Im }_k},$$

(2)

where ${\widehat{L}}_k^{\prime }$ ($k=1,2,3$) are the angular momentum operators of the core in the core-fixed intrinsic frame. The three moments of inertia are given by [1]

$${\Im }_k=B{\beta }^2{sin}^2(\gamma -\frac{2\pi k}{3})$$

(3)

for $k=1,2,3$, where B is the inertia parameter, and $\beta$ and $\gamma$ are the deformation and the triaxiality of the rotor, respectively. The quadrupole operator of the core in the laboratory frame can be expressed as

$$Q_{\mu }=\beta cos\gamma D_{\mu 0}^{2*}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)+\frac{\beta sin\gamma }{\sqrt{2}}(D_{\mu 2}^{2*}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)+D_{\mu -2}^{2*}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)),$$

(4)

where $D_{\mu \nu }^{2*}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)$ is the Wigner D-function of the Euler angles ${\vartheta }_1,{\vartheta }_2,{\vartheta }_3$ of the rotating core. In the single-j shell model basis, the quadrupole operator of the particle is given by

$$Q_{\mu }^{\mathrm{F}}={\gamma }_j{(a_j^{†}\times {\tilde{a}}_j)}_{\mu }^2$$

(5)

with

$${\gamma }_j=(N_{\mathrm{o}}+\frac{3}{2}){(-1)}^{j-\frac{1}{2}}\frac{(2j+1)}{\sqrt{20\pi }}\langle j\frac{1}{2},j-\frac{1}{2}|20\rangle ,$$

(6)

where $a_{jm}^{†}$ ($a_{jm}$) is the creation (annihilation) operator of the particle in the single-j orbit with ${\tilde{a}}_{jm}={(-)}^{j+m}{a}_{j-m}$; $N_{\mathrm{o}}$ is the number of phonons of the major oscillator shell; and ${(a_j^{†}\times {\tilde{a}}_j)}_{\mu }^2={\sum }_{m_1{m}_2}\langle j{m}_1,j{m}_2|2\mu \rangle a_{j{m}_1}^{†}{\tilde{a}}_{j{m}_2}$, in which $\langle j{m}_1,j{m}_2|2\mu \rangle$ is the SU(2) Clebsch–Gordan (CG) coefficient.

In the calculation, the Hamiltonian (1) is diagonalized in the coupled basis $\left\{\right|(L,j)J{M}_{J}K\rangle \}$, of which the core part is the basis vectors of the triaxial rotor, where $K=0,2,4,\cdots$ is the quantum number of the core angular momentum projected onto the third axis of the intrinsic frame, L is the angular momentum quantum number of the triaxial rotor with

$$\begin{array}{cc}& \hfill L=\left\{\begin{array}{c}\hfill K,K+1,\cdots ,\mathrm{for}K\ne 0,\hfill \\ \hfill 0,2,4,\cdots ,\mathrm{for}K=0,\hfill \end{array}\right.\hfill \end{array}$$

(7)

and J and $M_J$ are the quantum number of the total angular momentum and that of the third component in the laboratory frame, respectively. In the representation of the Euler angles of the core, the corresponding wave function can be written as

$$\begin{array}{cc}\hfill & \langle {\vartheta }_1,{\vartheta }_2,{\vartheta }_3|(L,j)J{M}_{J}K\rangle =\sqrt{\frac{2L+1}{16{\pi }^2}}\sqrt{\frac{1}{1+{\delta }_{K0}}}{\sum }_{M_{L}m}\langle L{M}_L,jm|J{M}_J\rangle \times \\ & \left(D_{M_{L}K}^{L*}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)+{(-1)}^L{D}_{M_L-K}^{L*}({\vartheta }_1,{\vartheta }_2,{\vartheta }_3)\right)a_{jm}^{†}|0\rangle ,\end{array}$$

(8)

where $|0\rangle$ is the vacuum state of the single-j particle. After diagonalization of the Hamiltonian (1), the eigenstates of (1) are expressed as

$$\begin{array}{c}\hfill |J_{\alpha }{M}_J\rangle =\sum _{L,K}{c}_{LK}^{\alpha }|\left(Lj\right)J{M}_{J}K\rangle ,\end{array}$$

where the expansion coefficients $c_{LK}^{\alpha }$ are determined after the diagonalization, and the additional quantum number $\alpha$ labels the $\alpha$-th eigenstate for given J.

The E2 transition operator is defined as

$$T_{\mu }=q_2(Q_{\mu }+\xi Q_{\mu }^{\mathrm{F}}),$$

(9)

where $q_2$ is the effective-charge-related parameter, and the dimensionless parameter $\xi$ measures the contribution from the single particle. Thus, the reduced E2 transition probability for the transition $J_{\alpha }\to J{{}^{\prime }}_{{\alpha }^{\prime }}$ and the electric quadrupole moment of the $\alpha$-th J state $|J_{\alpha }{M}_J=J\rangle$ are given by

$$\begin{array}{cc}\hfill & \hfill B(E2,J_{\alpha }\to J{{}^{\prime }}_{{\alpha }^{\prime }})=\frac{2J^{\prime }+1}{2J+1}{q}_2^2\left|\right(\langle J{{}^{\prime }}_{{\alpha }^{\prime }}\parallel Q\parallel J_{\alpha }\rangle +\xi \langle J{{}^{\prime }}_{{\alpha }^{\prime }}\parallel Q^{\mathrm{F}}\parallel J_{\alpha }{\rangle \left)\right|}^2\hfill \end{array}$$

(10)

and

$$\begin{array}{cc}\hfill & \hfill Q\left(J_{\alpha }\right)=\sqrt{\frac{16\pi }{5}}\langle JJ,20|JJ\rangle q_2(\langle J_{\alpha }\parallel Q\parallel J_{\alpha }\rangle +\xi \langle J_{\alpha }\parallel Q^{\mathrm{F}}\parallel J_{\alpha }\rangle ),\hfill \end{array}$$

(11)

respectively, in which the reduced matrix element $\langle J{{}^{\prime }}_{{\alpha }^{\prime }}\parallel T\parallel J_{\alpha }\rangle$ is defined in terms of the SU(2) CG coefficient according to the Wigner–Eckart theorem.

2.2. The Particle-Plus-Rotor Model

In contrast to the PRQCM, the particle-plus-rotor model (PRM) has been widely used to describe odd-mass nuclei [19,20,21,22,23,24,25,26]. Actually, the PRM Hamiltonian is the same as that shown in (1) with $\kappa =b=0$ but with the additional deformed single-particle energy term described in the intrinsic frame:

$$\widehat{H}={\widehat{H}}_{\mathrm{rot}}+{\widehat{H}}_{\mathrm{sp}}$$

(12)

with

$$\begin{array}{cc}\hfill & \hfill {\widehat{H}}_{\mathrm{sp}}=\pm \frac{1}{2}C\{cos\gamma ({{\widehat{j}}_0^{\prime }}^2-\frac{j(j+1)}{3})+\frac{sin\gamma }{2\sqrt{3}}({{\widehat{j}}_{+}^{\prime }}^2+{{\widehat{j}}_{-}^{\prime }}^2)\},\hfill \end{array}$$

(13)

where the plus sign refers to a particle and the minus sign to a hole; C is a parameter proportional to the deformation $\beta$ as adopted in [34,35,36,37]; and ${\widehat{j}}_{\mu }^{\prime }$ ($\mu =+,-,0$) are the angular momentum operators of the single particle, which are equivalent to the spherical single-particle energy term plus the quadrupole-coupling of the core with the single-particle term being treated approximately. The single-particle Hamiltonian (13) is digonalized with the eigenstates

$$\mid \nu \rangle =\sum _{\mathsf{\Omega }}{c}_{\mathsf{\Omega }}^{\nu }|j,\mathsf{\Omega }\rangle ,\mid \overline{\nu }\rangle =\sum _{\mathsf{\Omega }}{(-1)}^{j-\mathsf{\Omega }}{c}_{\mathsf{\Omega }}^{\nu }|j,-\mathsf{\Omega }\rangle ,$$

(14)

where $\mathsf{\Omega }$ is the quantum number of the angular momentum of the single particle projected onto the third axis of the intrinsic frame, $c_{\mathsf{\Omega }}^{\nu }$ is determined by the diagonalization of ${\widehat{H}}_{\mathrm{sp}}$, and $\mid \overline{\nu }\rangle$ is the time reversal state of $\mid \nu \rangle$. Then, the eigenstates of (12) are expressed in terms of the eigenstates of the total angular momentum coupled to the intrinsic single-particle states as

$$\begin{array}{c}\hfill |J_{\alpha }{M}_J\rangle =\sum _{K,\nu }{\mathsf{\Lambda }}_{K,\nu }^{\alpha }{|J{M}_{J}K;\nu \rangle }_{\mathrm{S}},\end{array}$$

(15)

where $\{{\mathsf{\Lambda }}_{K,\nu }^{\alpha }\}$ are the expansion coefficients determined by the diagonalization of (12) and

$$\begin{array}{c}\hfill |J{M}_J{K;\nu \rangle }_{\mathrm{S}}=\sqrt{\frac{1}{2}}\left(|J{M}_{J}K\rangle |\nu \rangle +{(-1)}^{J-K}|J{M}_J-K\rangle |\overline{\nu }\rangle \right).\end{array}$$

(16)

In the diagonalization process, the angular momentum operators of the core in the core-fixed intrinsic frame ${\widehat{L}}_k^{\prime }$ are expressed as ${\widehat{L}}_k^{\prime }={\widehat{J}}_k^{\prime }-{\widehat{j}}_k^{\prime }$, where ${\widehat{J}}_k^{\prime }$ is the total angular momentum operator in the core-fixed intrinsic frame. Since contribution of the quadrupole moment of the single particle to E2 transitions are very small, the formula of B(E2) values and that of the quadrupole moments shown in (10) and (11) apply to the PRM as well, in which the parameter $\xi$ is simply taken to be zero in the PRM.

2.3. The IBFM Formalism

By comparison with the results of the PRQCM, the IBFM Hamiltonian suitable to describe odd-mass nuclei in the transitional region [38] is also considered, which is expressed in terms of the s- and d-boson and the single-particle operators as

$${\widehat{H}}_{\mathrm{IBFM}}=ϵ{\widehat{n}}_d-{\kappa }_0({\widehat{Q}}^B+{\alpha }_0{q}^{\mathrm{F}})·({\widehat{Q}}^B+{\alpha }_0{q}^{\mathrm{F}})+f{\widehat{L}}^2+g{\widehat{J}}^2,$$

(17)

where ${\widehat{n}}_d={\sum }_{\mu }{d}_{\mu }^{†}{d}_{\mu }$ is the d-boson number operator; ${\widehat{Q}}_{\mu }^B=d_{\mu }^{†}s+s^{†}{\tilde{d}}_{\mu }$ is the U(5)-O(6) quadrupole operator of the IBM core; ${\widehat{q}}_{\mu }^{\mathrm{F}}={(a_j^{†}\times {\tilde{a}}_j)}_{\mu }^2$; ${\alpha }_0$ is a dimensionless parameter; and $ϵ$, ${\kappa }_0$, f, and g are real model parameters.

The IBFM Hamiltonian (17) is diagonalized in the (U(6) ⊃ O(6) ⊃ SO(5) ⊃ SO(3)) ⊗ SU_j2)⊃SU_J(2) coupled basis

$$|N\sigma \tau \beta L;j;JM\rangle =\sum _{M_{L}m}\langle L{M}_L,jm|JM\rangle a_{jm}^{†}|N\sigma \tau \beta L{M}_L\rangle ,$$

(18)

where $|N\sigma \tau \beta L{M}_L\rangle$ is the basis vector of the IBM core labeled under the group chain U(6) ⊃O(6)⊃O(5)⊃SO(3) ⊃SO(2), where N is the total number of bosons, $\sigma$ is the O(6) seniority number, $\tau$ is the d-boson seniority number, and $\beta$ is an additional quantum number needed in the reduction O(5)↓O(3).

After diagonalization of (17), the eigenstates of the IBFM Hamiltonian can be expressed as

$$|J_{\alpha }{M}_J{\rangle }_{\mathrm{IBFM}}\equiv |N,j;J_{\alpha }{M}_J\rangle =\sum _{\sigma \tau \beta L}{C}_{\sigma \tau \beta L}^{\alpha ,J}|N\sigma \tau \beta L;j;J{M}_J\rangle ,$$

(19)

where $C_{\sigma \tau \beta L}^{\alpha ,J}$ is determined by the diagonalization. Similar to (9), the E2 operator is defined as

$$T_{\mu }^{\mathrm{IBFM}}={\tilde{q}}_2({\widehat{Q}}_{\mu }^B+{\alpha }_0{\widehat{q}}_{\mu }^F).$$

(20)

Therefore, the reduced E2 transition probability for the transition $J_{\alpha }\to J{{}^{\prime }}_{{\alpha }^{\prime }}$ and the electric quadrupole moment of the state $|J_{\alpha }{M}_J{=J\rangle }_{\mathrm{IBFM}}$ are still given by (10) and (11) with replacement of the matrix element of $T_{\mu }$ by the matrix element of $T_{\mu }^{\mathrm{IBFM}}$ between the eigenstates shown in (19).

3. Model Fit to ¹³⁵Ba and ¹³¹Xe

To demonstrate the applicability of the PRQCM, low-lying level energies, reduced E2 transition probabilities, and the ground-state quadrupole moments of ¹³⁵Ba and ¹³¹Xe are fit. Since there is no $\beta$- and $\gamma$-vibration in the triaxial rotor model, the core parameters of ¹³⁵Ba and ¹³¹Xe are determined in fitting to the ground-state band and $\gamma$-band level energies of the adjacent even-even nuclei ¹³⁴Ba and ¹³⁰Xe, respectively. Specifically, the core parameters of both the PRQCM (1) and the PRM Hamiltonian (17) are $\frac{{\hslash }^2}{B}$ = 65 keV, $\gamma$ = ${30}^{\circ }$, $\beta$ = −0.127, and b = 15 keV for ¹³⁴Ba, while $\frac{{\hslash }^2}{B}$ = 77 keV, $\gamma$ = ${40}^{\circ }$, $\beta$ = −0.127, and b = 5 keV for ¹³⁰Xe, where the deformation $\beta$ is determined by $\beta =\frac{4}{3}\sqrt{\frac{\pi }{5}}{\epsilon }_2$ with ${\epsilon }_2$ = −0.12 taken from [39]. The fitting results of the level energies of ¹³⁴Ba and ¹³⁰Xe are shown in

Table 1. Low-lying level energies (in MeV) below 2.211 (2.172) MeV in the ground-state band and $\gamma$-band of ¹³⁴Ba (¹³⁰Xe) fitted by the triaxial rotor model (this work) and the IBM, where the spin and parity of the 7-th level of ¹³⁴Ba (¹³⁰Xe) is $4^{+}$ ($5^{+}$).

${\mathit{L}}_{\mathit{\alpha }}^{\mathit{\pi }}$	134Ba [40]	This Work	IBM	130Xe [41]	This Work	IBM
$0_g^{+}$	0	0	0	0	0	0
$2_1^{+}$	0.605	0.504	0.665	0.536	0.385	0.672
$2_2^{+}$	1.168	0.801	1.070	1.122	0.925	1.074
$3_1^{+}$	1.643	1.305	1.525	1.643	1.305	1.525
$4_1^{+}$	1.401	1.479	1.227	1.205	1.155	1.261
$4_2^{+}$	1.970	2.302	1.613	1.808	1.921	1.645
$4_3^{+}$($5_1^{+}$)	2.118	2.744	1.993	2.172	2.192	2.224
$6_1^{+}$	2.211	2.871	1.868	1.944	2.263	1.947

, in which the IBM results are also shown. The parameters of the IBM Hamiltonian are $ϵ$ = 420.00 keV, ${\kappa }_0$ = 2.65 keV, $f+g$ = 14.69 keV for ¹³⁴Ba, and $ϵ$ = 415.00 keV, ${\kappa }_0$ = 2.30 keV, $f+g$ = 20.43 keV for ¹³⁰Xe, where the values of f, g, and ${\alpha }_0$ are further determined in fitting to the level energies of ¹³⁵Ba or ¹³¹Xe.

After the model parameters of the core are fixed, the effective-charge-related parameter $q_2$ in the triaxial rotor model and ${\tilde{q}}_2$ in the IBM are determined in fitting to B(E2;$2_1^{+}\to 0_{\mathrm{g}}^{+}$), which yields $q_2$ = 109.4234 $\sqrt{\mathrm{W}.\mathrm{u}.}$ and 138.1106$\sqrt{\mathrm{W}.\mathrm{u}.}$ for ¹³⁴Ba and ¹³⁰Xe, respectively, and yields ${\tilde{q}}_2$ = 5.0584 $\sqrt{\mathrm{W}.\mathrm{u}.}$ and 5.287 $\sqrt{\mathrm{W}.\mathrm{u}.}$ for ¹³⁴Ba and ¹³⁰Xe, respectively, in the IBM.

Table 2. Some B(E2;$L_{\alpha }\to L_{\alpha }^{\prime }$) values (in $\mathrm{W}.\mathrm{u}.$) of ¹³⁴Ba and ¹³⁰Xe, where only experimentally known values of the intra- and inter-band transitions of the ground-state band and $\gamma$ band fitted by the models are shown, and * indicates that the corresponding experimental value is exactly fit in determining the effective-charge-related parameter $q_2$ or ${\tilde{q}}_2$.

	134Ba [40]	This Work	IBM		130Xe [41]	This Work	IBM
$2_1^{+}\to 0_g^{+}$	33.6(6)	33.6 *	33.6 *	$2_1^{+}\to 0_g^{+}$	38(5)	38 *	38 *
$4_1^{+}\to 2_1^{+}$	52(6)	47.02	47.32	$4_1^{+}\to 2_1^{+}$	>0.054	71.7	34.09
$4_1^{+}\to 2_2^{+}$	0.16(16)	0.42	0.32	$6_1^{+}\to 4_1^{+}$	>0.033	84.6	47.70
$2_2^{+}\to 0_g^{+}$	0.42(13)	0.10	0.11	$2_2^{+}\to 2_1^{+}$	>0.056	19.6	1.77
$2_2^{+}\to 2_1^{+}$	73(22)	44.03	43.41	$5_2^{+}\to 6_1^{+}$	13(3)	1.90	15.22
$3_1^{+}\to 2_2^{+}$	4.3(12)	1.97	3.51	$5_2^{+}\to 3_1^{+}$	0.0017(13)	3.92	8.20
$3_1^{+}\to 2_1^{+}$	0.024(8)	0.18	0.02

shows some B(E2) values of ¹³⁴Ba and ¹³⁰Xe with the effective-charge-related parameter fixed, in which only experimentally known values fitted by the models are provided.

We then use the model Hamiltonian (1) with the above parameters of the core to fit level energies of ¹³⁵Ba and ¹³¹Xe to determine the quadrupole–quadrupole interaction strength $\kappa$, of which the fitting results are presented in

Table 3. Low-lying level energies (in MeV) below 2.283 MeV of ¹³⁵Ba and below 1.621 MeV of ¹³¹Xe fitted by the PRQCM (this work), where the spin and parity of the levels with the level energy underlined is not determined in experiments [44,45], and “–” indicates the level is not calculated in the E(5/4) model [42].

${\mathit{J}}_{\mathit{\alpha }}^{\mathit{\pi }}$	135Ba [45]	This Work	IBFM	PRM	E(5/4) [42]	131Xe [44]	This Work	IBFM	PRM
$3/2_g^{+}$	0	0	0	0	0	0	0	0	0
$3/2_2^{+}$	0.588	0.414	0.600	0.480	0.613	0.405	0.355	0.595	0.281
$3/2_3^{+}$	0.855	0.711	1.005	0.750	0.683	0.700	0.895	0.996	0.986
$3/2_4^{+}$	1.214	1.125	1.374	1.230	0.989	0.994	1.250	1.357	1.267
$1/2_1^{+}$	0.221	0.334	0.567	0.375	0.228	0.080	0.332	0.556	0.315
$1/2_2^{+}$	0.910	0.701	0.972	0.855	0.901	0.565	0.888	0.958	0.952
$5/2_1^{+}$	0.481	0.350	0.654	0.331	0.569	0.364	0.386	0.659	0.293
$5/2_2^{+}$	0.980	0.925	1.060	0.749	0.901	0.723	0.914	1.060	1.021
$5/2_3^{+}$	1.238	1.200	1.063	0.974	1.208	1.034	1.076	1.066	1.220
$5/2_4^{+}$	1.557	1.366	1.449	1.185	1.251	1.245	1.275	1.449	1.510
$5/2_5^{+}$	1.941	2.054	1.737	2.094	1.663
$5/2_6^{+}$	2.075	2.430	1.816	2.542	–
$7/2_1^{+}$	0.875	0.625	0.730	0.427	0.875	0.637	0.413	0.750	0.281
$7/2_2^{+}$	1.008	0.860	1.135	0.802	1.199	0.973	0.957	1.151	0.826
$7/2_3^{+}$	1.130	1.305	1.140	1.053	1.514	0.952	1.117	1.158	0.977
$7/2_4^{+}$	1.165	1.311	1.525	1.178	1.558	1.621	1.310	1.543	1.262
$9/2_1^{+}$	1.200	1.384	1.238	1.001	1.593	0.971	1.164	1.274	0.808
$9/2_2^{+}$	2.283	1.440	1.622	1.197	1.899	1.456	1.355	1.660	1.256
$11/2_1^{+}$	1.955	1.718	1.742	1.028	2.065	1.397	1.213	1.414	0.841
$13/2_1^{+}$						1.584	2.266	2.018	1.594
$\sigma \left(E\right)$ (MeV)		0.271	0.251	0.413	0.276		0.254	0.294	0.277

for the level energies below 2.283 MeV for ¹³⁵Ba and

Table 4. Some B(E2;$J_{\alpha }\to J_{\alpha }^{\prime }$) (in $\mathrm{W}.\mathrm{u}.$) values of ¹³⁵Ba and ¹³¹Xe, where “–” indicates the corresponding value is not available experimentally or is not calculated in the E(5/4) model [42].

	135Ba [45]	This Work	IBFM	PRM	E(5/4) [42]		131Xe [44]	This Work	IBFM	PRM
$1/2_1^{+}\to 3/2_g^{+}$	4.6(2)	14.80	19.09	38.52	17.43	$1/2_1^{+}\to 3/2_g^{+}$	<37	36.70	17.82	41.26
$5/2_1^{+}\to 1/2_1^{+}$	2.6(5)	3.80	0.19	0.94	–	$5/2_1^{+}\to 1/2_1^{+}$	7.64(24)	7.68	0.39	4.55
$5/2_1^{+}\to 3/2_g^{+}$	28.3(10)	32.60	13.02	38.21	18.81	$5/2_1^{+}\to 3/2_g^{+}$	27.8(9)	38.80	14.10	68.30
$7/2_1^{+}\to 3/2_2^{+}$	<1.0	1.47	0.11	0.44	–	$7/2_1^{+}\to 3/2_2^{+}$	1.52(25)	4.39	0.35	6.78
$7/2_1^{+}\to 5/2_1^{+}$	12.8(12)	9.40	0.32	1.96	–	$7/2_1^{+}\to 5/2_1^{+}$	1.6(13)	11.50	0.68	26.39
$7/2_1^{+}\to 3/2_g^{+}$	19.9(8)	31.00	2.60	37.01	19.9	$7/2_1^{+}\to 3/2_g^{+}$	22.2(19)	37.70	3.10	52.31
$1/2_2^{+}\to 3/2_g^{+}$	11.7(10)	1.62	0.41	0	–	$1/2_2^{+}\to 3/2_g^{+}$	10(6)	12.00	1.08	6.28
$3/2_2^{+}\to 1/2_1^{+}$	–	2.48	0.01	0	–	$3/2_2^{+}\to 1/2_1^{+}$	24(+26 −24)	20.50	0.02	30.80
$3/2_2^{+}\to 3/2_g^{+}$	18.0(10)	33.60	0.18	23.49	–	$3/2_2^{+}\to 3/2_g^{+}$	30	38.00	0.42	50.22
$5/2_2^{+}\to 1/2_1^{+}$	–	9.30	0.77	13.40	14.8	$5/2_2^{+}\to 1/2_1^{+}$	25.7(25)	1.61	0.62	52.63
$5/2_2^{+}\to 5/2_1^{+}$	–	2.01	1.32	6.93	4.0	$5/2_2^{+}\to 5/2_1^{+}$	–	0.88	1.32	4.81
$5/2_2^{+}\to 7/2_1^{+}$	–	8.90	3.83	20.48	6.69	$5/2_2^{+}\to 7/2_1^{+}$	–	10.01	3.24	0.43
$9/2_1^{+}\to 7/2_1^{+}$	–	9.85	4.39	10.91	6.63	$9/2_1^{+}\to 7/2_1^{+}$	–	15.40	3.70	19.13
$3/2_3^{+}\to 1/2_1^{+}$	–	15.30	4.90	19.26	8.40	$5/2_2^{+}\to 3/2_2^{+}$	4(+6 −4)	7.97	3.00	25.30
$3/2_3^{+}\to 5/2_1^{+}$	–	15.11	5.10	29.18	13.47	$5/2_2^{+}\to 3/2_g^{+}$	4.8(5)	9.86	0.04	0.34
$3/2_3^{+}\to 7/2_1^{+}$	–	9.04	0.88	6.17	2.95	$7/2_2^{+}\to 3/2_g^{+}$	1.6(7)	11.0	0.17	0.56
$3/2_3^{+}\to 3/2_g^{+}$	7.0(10)	0.10	0.31	0	–
$\sigma \left(\mathrm{B}\left(\mathrm{E}2\right)\right)$		9.02	12.45	14.44	16.34			10.24	15.59	19.39

for the level energies below 1.621 MeV for ¹³¹Xe, respectively. The best fit yields $\kappa =3.115\left(0.257\right)$ MeV for ¹³⁵Ba (¹³¹Xe). It should be noted that $\kappa \left|\beta \right|cos\gamma$ and $q_2\left|\beta \right|cos\gamma$ are compatible with the IBFM parameters ${\kappa }_0$ and ${\tilde{q}}_2$, respectively, although a relatively larger value of $\kappa$ is needed for ¹³⁵Ba. In Table 3 and Table 4, the IBFM and the PRM results are also presented for comparison. For the IBFM, the best fit yields ${\alpha }_0=0.8$ ($1.3$), $f=3.85$ ($7.54$) keV, and $g=10.84\left(12.89\right)$ keV for ¹³⁵Ba (¹³¹Xe). The PRM parameter $C=0.3\left(0.1\right)$ MeV for ¹³⁵Ba (¹³¹Xe) is taken. In Table 3, the E(5/4) model fitting results [42] of ¹³⁵Ba are also shown for comparison since ¹³⁵Ba is often considered to be the E(5/4) critical point candidate of the U(5) (vibration) to O(6) ($\gamma$-soft) shape phase transition [38,43].

It should be pointed out that spin and parity of several levels in both ¹³⁵Ba and ¹³¹Xe with the level energy underlined shown in Table 3 have not been determined in experiments. If the spin and parity of the levels with the underlined level energy shown in Table 3 are indeed as predicted in all three models, there is one-to-one correspondence of the experimental level energy below 2.283 MeV in ¹³⁵Ba and 1.621 MeV in ¹³¹Xe, respectively, to that predicted by the three models.

The fitting quality is measured by the rms deviation of the level energies defined as

$$\begin{array}{c}\hfill \sigma \left(E\right)=\sqrt{\frac{1}{n_0-n_m}\sum _{i=1}^{n_0}{|E_{\mathrm{th}}^i-E_{\exp }^i|}^2},\end{array}$$

(21)

where $n_0$ is the total number of level energies fitted with $n_0=18$ (17) for ¹³⁵Ba (¹³¹Xe); $n_m$ is the number of model parameters with $n_m=1$ in the PRQCM and the PRM; $n_m=2$ in the IBFM; $n_m=5$ in the E(5/4) model [42] for ¹³⁵Ba, because other model parameters have already been determined in the fitting to the level energies of the even-even nuclei; and $E_{\exp }^i$ and $E_{\mathrm{th}}^i$ are experimental and theoretical level energies. It is clearly shown that the fitting quality of the three models is quite the same. Nevertheless, as far as the level energies of ¹³⁵Ba and ¹³¹Xe shown in Table 3 are concerned, the average values of the rms deviation of ¹³⁵Ba and that of ¹³¹Xe in the PRQCM, the IBFM, and the PRM fit are $\overline{\sigma }\left(E\right)=0.263$, $0.273$, and $0.345$, respectively, indicating that the fitting quality of the PRQCM is the best. The average level energy deviation in the PRM is the largest, which may be due to the fact that the quadrupole-coupling is approximately treated.

By using the model parameters shown above and the effective charge parameter $q_2$ in the PRQCM or the PRM and ${\tilde{q}}_2$ in the IBFM from fitting to the B(E2;$2_1^{+}\to 0_{\mathrm{g}}^{+}$) of ¹³⁴Ba and ¹³⁰Xe, the B(E2) values of the transitions between the states with the level energies shown in Table 3 are calculated. The best fit yields the additional parameter $\xi$ = −0.0104 (−0.0025) of the PRQCM for ¹³⁵Ba (¹³¹Xe), which shows that the contributions of the quadrupole moment of the single particle to the total E2 transition operators are indeed small. The E(5/4) model results presented in [42] are also included in Table 3 for comparison. The fitting deviation from the experimental B(E2) values is defined the same as that shown in (21), with the level energies replaced by the B(E2) values also being calculated. Since the effective charge $q_2$ or ${\tilde{q}}_2$ has already been determined in fitting the B(E2) values of the even-even nuclei, $n_m=1$ in the PRQCM, $n_m=2$ in the E(5/4) model [42], and $n_m=0$ in both the PRM and the IBFM in calculating the corresponding rms deviations. It is clearly shown in Table 4 that the fitting quality of the PRQCM is the best. The average values of the rms deviation in B(E2) values of ¹³⁵Ba and that of ¹³¹Xe in the PRQCM, the IBFM, and the PRM fit are $\overline{\sigma }\left(\mathrm{B}\left(\mathrm{E}2\right)\right)=9.63$, $14.02$, $16.92$, respectively. The calculated results of the low-lying level energies of ¹³⁵Ba and ¹³¹Xe with some reduced E2 transition rates in comparison to the experimental ones are shown in Figure 1 and Figure 2, respectively.

Since the electric quadrupole moments of ¹³⁴Ba and ¹³⁰Xe, and those of excited states of ¹³⁵Ba and ¹³¹Xe are not available experimentally, only the ground-state electric quadrupole moment $Q(3/2_g^{+})$ of ¹³⁵Ba and ¹³¹Xe are calculated, which are provided in

Table 5. The ground-state electric quadrupole moment $Q(3/2_g^{+})$ (in eb) of ¹³⁵Ba and ¹³¹Xe.

135Ba [45]	This work	IBFM	PRM
+0.160(3)	+0.187	+0.075	+0.412
131Xe [44]	This work	IBFM	PRM
−0.114(1)	+0.055	+0.126	+0.276

. It can be seen that the PRQCM results of the ground-state electric quadrupole moments of ¹³⁵Ba and ¹³¹Xe are closest to the experimental values, although the sign of the ground-state electric quadrupole moment of ¹³¹Xe obtained from the PRQCM is still incorrect.

4. Summary

In this work, the particle-rotor-quadrupole-coupling model—involving the quadrupole–quadrupole interaction of the even-even core described by a triaxial rotor with a single-j particle—is adopted to describe low-lying spectra of odd-A nuclei within the vibrational to triaxial transition region. To demonstrate the usability, the low-lying level energies, reduced E2 transition probabilities, and ground-state quadrupole moments of ¹³⁵Ba and ¹³¹Xe are fit by the model, of which the results are compared with the experimental data and those of other models. It is shown that the fitting results of the particle-rotor-quadrupole-coupling model to the low-lying level energies, reduced E2 transition probabilities, and ground-state electric quadrupole moments of ¹³⁵Ba and ¹³¹Xe are the best, of which the model parameters of the even-even core are determined by the triaxial rotor model in fitting to the low-lying spectra of ¹³⁴Ba and ¹³⁰Xe. In comparison with the E(5/4) model results of ¹³⁵Ba, it is also shown that the quadrupole–quadrupole interaction of the even-even core with the single particle adopted can indeed reproduce the E(5/4) critical point behavior. The fitting quality of the reduced E2 transition probabilities among the low-lying states by the particle-rotor-quadrupole-coupling model is also noticeably improved. Thus, it can be concluded that the particle-rotor-quadrupole-coupling model is suitable to describe low-lying properties of odd-A nuclei within the transitional region.

To improve the present model for odd-mass nuclei, a model Hamiltonian including the $\gamma$-unstable motion initiated by Wilets and Jean for transitional nuclei or that of the rotor–vibrator with an axially deformed shape for well-deformed nuclei may be adopted to replace the triaxial rotor for the even-even core in the collective model framework. Extension of the model to inclusion of multi-j, single-particle orbits is also possible. These possible extensions and improvements may be considered in our future work.