F.E.G. fermi sphere radius problem

[lordkelvin]

Homework Statement

Calculate the electron concentrations (# electrons/atom) needed for the fermi sphere to contact the zone faces (first bril. zone edges) in BCC and FCC structures.

Homework Equations

k_f = (3*pi^2*n)^(1/3) where n is # electrons per atom.

For cubic structures, d_hkl=a*(h^2+k^2+l^2)^(-1/2), and the reciprocal magnitude of d is 2*pi/d_hkl

The Attempt at a Solution

In BCC structure, the brill. zone edge is at the {110} planes, so that d_hkl = a*2^(-1/2) and this distance in reciprocal space is 2^(3/2)*pi/a. (where a is the lattice parameter). Setting this reciprocal distance equal to the fermi vector,
(2^(3/2)*pi/a)^3/(3*pi^2) = n

I'm just unsure about this, maybe someone could give me a shove in the right direction. For the FCC structure, the edges of the FBZ are at the {110} and {200} planes.

 

[King Jama]

bump