Physics 6311: Statistical Mechanics - Homework 11

Problem 1: Phonons in liquid ${}^4$Helium (10 points)

The longitudinal phonons in ${}^4$He at low temperatures have a velocity of $c=238.3$ m/s. Transversal phonons do not exist in liquids. The density is 0.145 g/cm${}^3$.

a)

Calculate the Debye temperature (within the Debye model).

b)

Calculate the heat capacity and compare to the experimental value of $c_V$ = 0.0204 (T/K)${}^3$ J/gK.

Problem 2: Thermodynamics of Magnons (15 points)

Spin waves or magnons are elementary excitations of Bose type in ferromagnetic materials. Their dispersion relation is $\omega =\mathit{D }{k}^2$ for small frequencies $\omega \ll {\omega }_{\mathit{max}}$. Calculate the contribution of the magnons to the specific heat at low temperatures $k_{B}T\ll \hslash {\omega }_{\mathit{max}}$.

(Hints: There is no conservation law for the magnon number, the rest mass is zero. You do not have to evaluate dimensionless integrals if you have shown that they converge.)

Problem 3: Bose gas with internal degrees of freedom (15 points)

Consider a three-dimensional ideal Bose gas of particles of mass $m$. The particles have an internal degree of freedom which can be described as a two level system. Bosons whose internal degree of freedom is in the ground state have energy $p^2∕2m$ where $p$ is the momentum . If it is in the excited state, their energy is $p^2∕2m+\mathrm{\Delta }$. Compute the Bose-Einstein condensation temperature $T_c$. Is it changed by the existence of the internal degrees of freedom? (Assume $\mathrm{\Delta }\gg T_c$.)