Spin-orbit coupling induced broadening of the spin crossover transition in divalent cobalt probed by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>K</mml:mi><mml:mi>β</mml:mi></mml:msub></mml:math> x-ray emission

The pressure-induced spin crossover in divalent cobalt is studied. It is shown that the change in spin as a function of pressure and its effects on ${K}_{\ensuremath{\beta}}$ x-ray emission depend on the $3d$ spin-orbit interaction and local exchange fields, leading to a broadening of the spin-crossover transition as a function of octahedral crystal field or pressure. A detailed comparison is made with experimental data on ${\mathrm{Na}}_{3}{\mathrm{Co}}_{2}{\mathrm{SbO}}_{6}$ as a function of the octahedral field tuned by pressure. Additionally, the spectral weight of the ${K}_{{\ensuremath{\beta}}^{\ensuremath{'}}}$ feature is studied for transition metals, and the Tsutsumi rules [K. Tsutsumi, J. Phys. Soc. Jpn. 14, 1696 (1959); K. Tsutsumi and H. Nakamori, J. Phys. Soc. Jpn. 25, 1418 (1968); K. Tsutsumi et al., Phys. Rev. B 13, 929 (1976)] are reexamined.