Relativistic many-body calculation of energies, lifetimes, hyperfine constants, multipole polarizabilities, and black-body radiation shift in $^{137}$Ba~II

Excitation energies of the [Xe]$ns_{1/2}$ [Xe]$np_j$, and [Xe]$nd_j$ ($n \leq$ 12 and [Xe]=$1s^22s^22p^63s^23p^63d^{10}4s^24p^634d^{10}5s^25p^6$) in Ba~II are evaluated. First-, second-, third-order, and all-order Coulomb energies and first- and second-order Coulomb-Breit energies are calculated. Electric-dipole ($6s_{1/2}\ -np_j$, $n$ = 6--26), electric-quadrupole ($6s_{1/2}\ -nd_j$, $n$ = 5--26), and electric-octupole ($6s_{1/2}\ -nf_j$, $n$ = 4--26) matrix elements are calculated to obtain the ground state E1, E2, and E3 static polarizabilities. Scalar polarizabilities of the $ns_{1/2}$, $np_j$and $nd_j$ states, and tensor polarizabilities of the $np_{3/2}$ and $nd_j$ excited states of Ba$^+$ are evaluated. All above-mentioned matrix elements are determined using all-order methods. We investigate the hyperfine structure in $^{137} $Ba~II. The hyperfine $A$- and B-values are determined for the first the first low-lying levels up to $n$ = 9. The quadratic Stark effect on hyperfine structure levels of $^{137}$Ba~II ground state is investigates. The calculated shift for the ($F$ = 2, $M$ =0) $\leftrightarrow$ ($F$ = 1, $M$ =0) transition is -0.2931~Hz/(kV/cm)$^2$, in agreement with previous theoretical result -0.284(3). These calculations provide a theoretical benchmark for comparison with experiment and theory.