The Fermi wavevector ( k_F ) in a free electron model is related to electron density ( n ) by ( k_F = (3\pi^2 n)^1/3 ). In a periodic lattice, this changes subtly but topologically.
$$V_total(\mathbfr) = V_0(\mathbfr) + \delta V(\mathbfr, t)$$ ziman principles of the theory of solids 13