First-principles relativistic theory of the magnetic response of paramagnetic metals: Application to yttrium and scandium

We describe a first-principles theoretical formalism for the magnetic response of paramagnetic metals in which all relativistic effects such as spin-orbit coupling are included. In particular, the easy axis and dependence upon wave vector q of the paramagnetic spin susceptibility $\chi \left(\mathbf{q}\right)$ can be calculated. To illustrate we apply the method to two transition metals, yttrium and scandium. In each case we find $\chi \left(\mathbf{q}\right)$ to peak at a wave vector $\mathbf{q}=(0,0,0.57)\mathrm{}\pi /c,$ coincident with a Fermi surface nesting vector, and to have an easy axis perpendicular to $\mathbf{q}.$ Since $\chi \left(\mathbf{q}\right)$ plays a key role in determining the interaction between magnetic impurities in these metals, these results are consistent with the helical antiferromagnetic order found in many dilute rare-earth Y alloys. Conversely, the easy axis for the response to a uniform magnetic field, $\mathbf{q}=0,$ lies along the c axis.