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.

مواضيع ذات صلة

مصطلحات الفيزياء النووية

إزاحة حمراء REDSHIFT

المركبات النانوية (Nanocomposites)

مولد فان دي جراف الالكتروستاتي

التطويرات التقنية الحديثة في المعجلات الالكتروستاتية

معجلات الالكترونات الالكتروستاتية

معجلات الفان دي جراف الترادفية

الحد من زيادة الطاقة في معجل فان دي جراف

أنبوبة التعجيل قي معجلات الجسيمات من نوع فان دي جراف

ارتكاز معجل فان دي جراف

مبدأ تشغيل معجل الفان دي جراف

القضايا الأخلاقية المتعلقة بتكنولوجيا النانو