Paramagnetism at High Temperature

a) Show that for a system with a discrete, finite energy spectrum ε_n, the specific heat per particle at high temperatures (τ >> ε_n for all n) is

where σ is the spectrum variance

b) Use the result of (a) to derive the high-temperature specific heat for a paramagnetic solid treated both classically and quantum mechanically.

c) Compare your quantum mechanical result J = 1/2 for with the exact formula for c.

SOLUTION

a) The specific heat c of a system that has N energy states is given by

(1)

Using 1/τ = β, we may rewrite c:

(2)

where we have used In (1 + x) ≈ x – x²/2. Note that, in general, the parameter βE_n is not small (since it is proportional to the number of particles), but, subsequently, we obtain another parameter βε << 1.

b) For a classical paramagnetic solid:

so

and we have

(3)

where dz/2 is the probability density. Therefore,

(4)

For the quantum mechanical case, ⟨ε⟩ = 0; there is an equidistant energy spectrum: E_m = -gμ_BHm and

(5)

To calculate  we can use the following trick (assuming J integer):

(6)

From (6) we have

(7)

With the familiar sum

we arrive at

(8)

We wish to perform the sum from –J to J, so

and (5) gives

(9)

(10)

c) For J = 1/2,

and

(11)

For J = 1/2:

(12)

where y = gμ_BH. We then find

(13)

For τ → ∞, β → 0,

which coincides with (11).