Structure of $^{128,129,130}$Xe through multi-reference energy density functional calculations

Abstract

Recently, values for the Kumar quadrupole deformation parameters of the nucleus $^{130}$Xe have been computed from the results of a Coulomb excitation experiment, indicating that this xenon isotope has a prominent triaxial ground state. Within a different context, it was recently argued that the analysis of particle correlations in the final states of ultra-relativistic heavy-ion collisions performed at the Large Hadron Collider (LHC) points to a similar structure for the adjacent isotope, $^{129}$Xe. In the present work, we report on state-of-the-art multi-reference energy density functional calculations that combine projection on proton and neutron number as well as angular momentum with shape mixing for the three isotopes $^{128,129,130}$Xe using the Skyrme-type pseudo-potential SLyMR1. Exploring the triaxial degree of freedom, we demonstrate that the ground states of all three isotopes display a very pronounced triaxial structure. Moreover, comparison with experimental results shows that the calculations reproduce fairly well the low-energy excitation spectrum of the two even-mass isotopes. By contrast, the calculation of $^{129}$Xe reveals some deficiencies of the effective interaction.

Expressions for the matrix elements of \({\hat{E}}^{(n)}_2\)

Let us first define the electric quadrupole operator ${\hat{E}}_2$ as the rank-2 tensor with components

$$\begin{aligned} {\hat{E}}_{2\mu } = \sum _{i=1}^{A} {\hat{e}}_i \, {\hat{r}}_i^2 \, Y_{2\mu } ({\hat{\theta }}_i, {\hat{\phi }}_i), \end{aligned}$$

(24)

where $\mu \in \llbracket -2, 2 \rrbracket$, ${\hat{e}}_i$ and (${\hat{r}}_i, {\hat{\theta }}_i, {\hat{\phi }}_i$) are the electric charge and spherical coordinates of the i-th particle, respectively, and $Y_{2\mu }$ are the usual spherical harmonics.

When considering the electric quadrupole moments of a classical ellipsoid, the finite sum of the above quantum mechanical operator is replaced by an integral over the volume of the ellipsoid and the charge operator by a charge density [4].

In the present work, we consider for the products ${\hat{E}}^{(n)}_2$ the standard expressions given in the literature [6, 77]

$$\begin{aligned} {\hat{E}}^{(2)}_2&= \left[ {\hat{E}}_2 \times {\hat{E}}_2 \right] _0 , \end{aligned}$$

(25)

$$\begin{aligned} {\hat{E}}^{(3)}_2&= \left[ \left[ {\hat{E}}_2 \times {\hat{E}}_2 \right] _2 \times {\hat{E}}_2 \right] _0 , \end{aligned}$$

(26)

$$\begin{aligned} {\hat{E}}^{(4)}_2&= \left[ {\hat{E}}^{(2)}_2 \times {\hat{E}}^{(2)}_2 \right] _0 = {\hat{E}}^{(2)}_2 {\hat{E}}^{(2)}_2 , \end{aligned}$$

(27)

$$\begin{aligned} {\hat{E}}^{(5)}_2&= \left[ {\hat{E}}^{(2)}_2 \times {\hat{E}}^{(3)}_2 \right] _0 ={\hat{E}}^{(2)}_2 {\hat{E}}^{(3)}_2 , \end{aligned}$$

(28)

$$\begin{aligned} {\hat{E}}^{(6)}_2&= \left[ {\hat{E}}^{(3)}_2 \times {\hat{E}}^{(3)}_2 \right] _0 = {\hat{E}}^{(3)}_2 {\hat{E}}^{(3)}_2 . \end{aligned}$$

(29)

With these definitions, by inserting the identity operator $\sum _{\varGamma ,\sigma } |\varPsi ^\varGamma _\sigma \rangle \langle \varPsi ^\varGamma _\sigma |$ between subparts of the operators coupled to zero angular momentum as well as using the expressions of matrix elements of irreducible tensor products [78], we obtain

$$\begin{aligned} \langle \varPsi ^\varGamma _\sigma |{\hat{E}}^{(2)}_2|\varPsi ^\varGamma _\sigma \rangle&= \frac{1}{\sqrt{5}(2J+1)} \sum _{J_1 \sigma _1} (-1)^{-J+J_1} \nonumber \\&\times \langle \varPsi ^{\varGamma }_{\sigma }||{\hat{E}}_2||\varPsi ^{\varGamma _1}_{\sigma _1}\rangle \langle \varPsi ^{\varGamma _1}_{\sigma _1}||{\hat{E}}_2||\varPsi ^{\varGamma }_{\sigma }\rangle , \end{aligned}$$

(30)

$$\begin{aligned} \langle \varPsi ^\varGamma _\sigma |{\hat{E}}^{(3)}_2|\varPsi ^\varGamma _\sigma \rangle&= \frac{(-1)^{2J}}{(2J+1)} \sum _{J_1 \sigma _1} \sum _{J_2 \sigma _2} \left\{ \begin{array}{ccc} 2 &{} 2 &{} 2 \\ J_2 &{} J &{} J_1 \end{array}\right\} \nonumber \\&\times \langle \varPsi ^{\varGamma }_{\sigma }||{\hat{E}}_2||\varPsi ^{\varGamma _1}_{\sigma _1}\rangle \langle \varPsi ^{\varGamma _1}_{\sigma _1}||{\hat{E}}_2||\varPsi ^{\varGamma _2}_{\sigma _2}\rangle \nonumber \\&\times \langle \varPsi ^{\varGamma _2}_{\sigma _2}||{\hat{E}}_2||\varPsi ^{\varGamma }_{\sigma }\rangle , \end{aligned}$$

(31)

$$\begin{aligned} \langle \varPsi ^\varGamma _\sigma |{\hat{E}}^{(4)}_2|\varPsi ^\varGamma _\sigma \rangle&= \sum _{\sigma _1} | \langle \varPsi ^\varGamma _\sigma |{\hat{E}}^{(2)}_2|\varPsi ^\varGamma _{\sigma _1}\rangle |^2 ,\end{aligned}$$

(32)

$$\begin{aligned} \langle \varPsi ^\varGamma _\sigma |{\hat{E}}^{(5)}_2|\varPsi ^\varGamma _\sigma \rangle&= \sum _{\sigma _1} \langle \varPsi ^\varGamma _\sigma |{\hat{E}}^{(2)}_2|\varPsi ^\varGamma _{\sigma _1}\rangle \langle \varPsi ^\varGamma _{\sigma _1}|{\hat{E}}^{(3)}_2|\varPsi ^\varGamma _{\sigma }\rangle , \end{aligned}$$

(33)

$$\begin{aligned} \langle \varPsi ^\varGamma _\sigma |{\hat{E}}^{(6)}_2|\varPsi ^\varGamma _\sigma \rangle&= \sum _{\sigma _1} | \langle \varPsi ^\varGamma _\sigma |{\hat{E}}^{(3)}_2|\varPsi ^\varGamma _{\sigma _1}\rangle |^2 , \end{aligned}$$

(34)

where we have used the shorthand notation $\varGamma _i \equiv (J_i MNZ\pi )$ for the set of quantum numbers that differ only by their angular momentum and the fact that the eletric quadrupole operator does neither change the parity nor the number of particles of the system such that $\sum _{\varGamma _i} \rightarrow \sum _{J_i}$. The expressions (30–34) are of course equalities only when summing over all possible states with relevant combinations of quantum numbers. In practice, however, one is limited to a finite set of states such that the Kumar deformation parameters determined from these sums have some truncation error.