Third-order relativistic many-body calculations of energies, transition rates, hyperfine constants, and black-body radiation shift in $^{171}$Yb$^+$

Relativistic many-body perturbation theory is applied to study properties of singly ionized ytterbium, Yb$^+$. Specifically, energies of the [Xe]$4f^{14}ns_{1/2}$, [Xe]$4f^{14}np_j$, and [Xe]$4f^{14}nd_j$ ($n \leq$9) are calculated through third order. Reduced matrix elements, oscillator strengths, and transition rates are determined for electric-dipole transitions including the $6s$, $7s$, $8s$, $6p$, $7p$, $5d$ and $6d$ states. Lifetime values are determined for the $6p$ states. Electric-dipole ($6s_{1/2}\ -np_j$, $n$ = 6--12) matrix elements are calculated to obtain the ground state E1 polarisabilitie. The hyperfine $A$-values are determined for the low-lying levels up to $n = 7$ of $^{171}$Yb~II. The quadratic Stark effect on hyperfine structure levels of $^{171}$Yb~II ground state is investigated. The calculated shift for the ($F$ = 1, $M$ =0) $\leftrightarrow$ ($F$ = 0, $M$ =0) transition is -0.1796~Hz/(kV/cm)$^2$, in agreement with previous theoretical result -0.171$\pm$0.009. These calculations provide a theoretical benchmark for comparison with experiment and theory.