  We apply homogenization theory to calculate the effective electric
conductivity and Hall coefficient tensor of passive three-dimensionally
periodic metamaterials subject to a weak external static homogeneous magnetic
field. We not only allow for variations of the conductivity and the Hall
coefficient of the constituent material(s) within the metamaterial unit cells,
but also for spatial variations of the magnetic permeability. We present four
results. First, our findings are consistent with previous numerical
calculations for finite-size structures as well as with recent experiments.
This provides a sound theoretical justification for describing such
metamaterials in terms of effective material parameters. Second, we visualize
the cofactor fields appearing in the homogenization integrals. Thereby, we
identify those parts of the metamaterial structures which are critical for the
observed effective metamaterial parameters, providing a unified view onto
various previously introduced single-constituent/multiple-constituent and
isotropic/anisotropic architectures, respectively. Third, we suggest a novel
three-dimensional non-magnetic metamaterial architecture exhibiting a sign
reversal of the effective isotropic Hall coefficient. It is conceptually
distinct from the original chainmail-like geometry, for which the sign reversal
is based on interlinked rings. Fourth, we discuss two examples for metamaterial
architectures comprising magnetic materials: Yet another possibility to reverse
the sign of the isotropic Hall coefficient and an approach to conceptually
break previous bounds for the effective mobility.
