Thermal Expansion and Heat Capacity

a) Find the temperature dependence of the thermal expansion coefficient if the interaction between atoms is described by a potential

where λ is a small parameter.

b) Derive the anharmonic corrections to the Dulong–Petit law for a potential

where η is a small parameter.

SOLUTION

a) First solution: We can calculate the average displacement of an oscillator:

(1)

Since the anharmonic term is small, mλx³/3 << τ, we can expand the exponent in the integral:

(2)

where we set α ≡ K₀/2τ. So,

(3)

Note that, in this approximation, the next term in the potential would not have introduced any additional shift (only antisymmetric terms do).

Second solution: We can solve the equation of motion for the nonlinear harmonic oscillator corresponding to the potential V₀(x):

(4)

where is the principal frequency. The solution gives

(5)

where A' is defined from the initial conditions and A is the amplitude of oscillations of the linear equation. The average ⟨x⟩ over a period T = 2π/ω₀ is

(6)

We need to calculate the thermodynamic average of ⟨x⟩:

(7)

Substituting A²(ε) =2ε/K₀, we obtain

(8)

the same as before.

b) The partition function of a single oscillator associated with this potential energy is

(9)

So, the free energy F per oscillator is given by

(10)

where we approximated ln (1 – x) ≈ - x. The energy per oscillator may be found from

(11)

The heat capacity is then

(12)

The anharmonic correction to the heat capacity is negative.