N Independent Spins

Consider a system of N independent spin-1/2 particles. In a magnetic field H, in the ẑ direction, they can point either up or down with energy ±μH, where μ is the magnetic moment. Derive expressions for the

a) Partition function

b) Internal energy

c) Entropy

SOLUTION

a) The partition function is given by

(1)

where ξ = βμH. Each spin is independent, so one has the same result as for one spin, but raised to the Nth power

(2)

b) The internal energy is the derivative of ln Z with respect to β:

(3)

c) The entropy is derivative of ln Z with respect to τ:

(4)