= Hard-sphere atomic volume in cubic lattices
For identical touching atoms represented by <hard spheres> of radius $r$, a <face-centered cubic lattice> has conventional side $a_F=2\sqrt2r$ and four atoms per conventional cell. A <body-centered cubic lattice> has side $a_B=4r/\sqrt3$ and two atoms per conventional cell. Their <primitive unit cell> volumes are respectively
$$
\Omega_F=4\sqrt2r^3,\qquad \Omega_B=\frac{32}{3\sqrt3}r^3.
$$
Each primitive cell contains one atom in the monatomic model, so its <mass density> is atomic <mass> divided by $\Omega$. These are contact-geometry estimates; a fixed hard-sphere radius does not encode <thermal expansion> or phase-dependent electronic bonding.
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