Relationship between the static dipole polarizability and the static dipole magnetizability of the relativistic hydrogen-like atom

We show that the ground-state static dipole polarizability αd(Z) and the ground-state static dipole magnetizability χd(Z) of the relativistic hydrogen-like atom, with a spinless and point-like nucleus of charge +Ze, are related through the formula α d ( Z ) = - 3 ( γ 1 - 2 ) 2 ( α Z ) - 2 χ d ( Z ) - a 0 3 Z - 4 ( γ 1 + 1 ) ( 2 γ 1 + 1 ) ( 14 γ 1 3 - 63 γ 1 2 + 55 γ 1 - 24 ) / 36 , where α and a0 are, respectively, the Sommerfeld fine-structure constant and the Bohr radius, while γ 1 = 1 - ( α Z ) 2 .