Relativistic formulae for the wavevector-dependent magnetic susceptibilities and the numerical calculation of the orbital and spin-orbit magnetic susceptibilities in vanadium

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Abstract
From a relativistic treatment based on the Dirac equation, the wavevector-dependent magnetic susceptibility is expressed as the sum of three terms. Two of them, chi s(q) and chi o(q) (q being the wavevector), reduce to the well known expressions for the q-dependent spin and orbital susceptibilities, respectively, in the non-relativistic limit. The other term chi so(q) arises as a higher-order term and represents the q-dependent spin-orbit susceptibility. The uniform unenhanced orbital and spin-orbit susceptibilities chi o and chi so are derived as the q to 0 limit of chi o(q) and chi so(q), respectively. A self-consistent relativistic APW band-structure calculation is carried out in the relativistic local-density approximation for vanadium at the observed lattice constant 5.713 au. By making use of this result of the band calculation, chi o(q) and chi so(q) at small q are calculated and the values of chi o and chi so are obtained as 199.4 and -5.2*10-6 EMU mol-1, respectively. With a proper estimation of the enhanced spin susceptibility chi s', the total susceptibility chi =chi s'+chi so+ chi o and its volume dependence are estimated and the results are in reasonable agreement with the experimental results.