Thermal Expansion and Heat Capacity
المؤلف:
Sidney B. Cahn, Gerald D. Mahan And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 2 , p 45
6-9-2016
1738
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.
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