  Core-shell nanowires made of Si and Ge can be grown experimentally with
excellent control for different sizes of both core and shell. We have studied
the structural properties of Si/Ge and Ge/Si core-shell nanowires aligned along
the $[110]$ direction, with diameters up to 10.2~nm and varying core to shell
ratios, using linear scaling Density Functional Theory (DFT). We show that
Vegard's law, which is often used to predict the axial lattice constant, can
lead to an error of up to 1\%, underlining the need for a detailed \emph{ab
initio} atomistic treatment of the nanowire structure. We analyse the character
of the intrinsic strain distribution and show that, regardless of the
composition or bond direction, the Si core or shell always expands. In
contrast, the strain patterns in the Ge shell or core are highly sensitive to
the location, composition and bond direction. The highest strains are found at
heterojunction interfaces and the surfaces of the nanowires. This detailed
understanding of the atomistic structure and strain paves the way for studies
of the electronic properties of core-shell nanowires and investigations of
doping and structure defects.
