= Oscillatory spacer-mediated exchange
{title2=$E_{\rm ex}/A=-J(t)\hat{\mathbf m}_1\cdot\hat{\mathbf m}_2$}
Electrons reflected by two magnetic interfaces can mediate an interaction across a nonmagnetic metal spacer. The phase accumulated on traversing the spacer varies with its thickness, producing oscillatory contributions to $J(t)$ from <Fermi surface> spanning vectors. In the sign convention in the title, positive $J$ favours parallel moments and negative $J$ antiparallel moments. Multiple periods, interface phase shifts, disorder and finite temperature prevent a universal sign rule based on thickness alone.
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