Source Terms for the Boltzmann-Peierls Equation for Phonons

Abstract

Consider pulses of long-wavelength acoustic phonons impingent upon a detector. These pulses are produced by point or extended sources of monochromatic or Planckian phonons. The sources are located at the origin of a Cartesian coordinate frame with the z-axis perpendicular to the specimen surface and directed into the medium. Assume that the macroscopic sample at temperature T much lower than the Debye temperature θ _D is a perfect crystal, then the generated phonons move ballistically towards the detector. The phonons are characterized by the direction $\hat q$ and the length q of the wave vector, polarization j (j = 0,1, 2), frequency ω(Q) $(Q = (\hat q,q,j))$ phase $c(\hat Q)$ and the group v(Q) velocities $(here{\text{ }}\hat Q \equiv (\hat q,j))$ A nonequilibrium state of the phonon gas is described by the distribution function N(Q; r, t). At the described conditions the deviation $\delta N\left( {Q;r,t} \right) \equiv N\left( {Q;r,t} \right) - N_0 [\hbar \omega \left( Q \right)/k_B T]$ (N ₀ is the Planck function) obeys the BPE, which in place of a collision term contains the source term (cf. [1]) $\zeta \left( {{\text{Q;r,t}}} \right)$

$$\partial /\partial t\delta N(Q;r,t) + v(\hat Q)\nabla \delta N(Q;r,t) = \dot \zeta (Q;r,t).$$

(1)