Solid state assignment

1. Bulk modulus:

The thermodynamic expression for bulk modulus leads to the formula k = vd/dv(du/dv) where where u, v are the internal energy and volume per lattice point.

Show that this gives

K = √2/9r₀ ∂²u/∂r²|r =a

for fcc, and evaluate the corresponding geometric factor for sc and bcc lattices (where ro, is the nearest neighbour disiance, i.e the potential minimum at nn separation r ≡ r₀. { Note: in the general notation using primitive vectore  etc. the lattice constants' a, b,...... are not necessarity the nearrest neighbour separation.