Discuss which scattering mechanism dominates the thermal

Question

Consider the thermal conductivity K of a pure synthetic sapphire crystal that has a diameter of 3mm.

(For sapphire θ_D -1000 K , speed of sound - 10^4ms^-1, and for T <<θ_D, c_v - 10^-1 T⁸ J m^-3K^-1, delta D is the Debye temperature and T is the temperature).

a. Discuss which scattering mechanism dominates the thermal conductivity K of the sapphire crystal at 20 k

b. Discuss which scattering mechanism dominates the thermal conductivity K of the sapphire crystal at 1000 K.

c. Given that K - 1/3 c_vvl where Cv is the specific heat capacity per unit volume, v is the speed of sound and l is the mean free path, and that the thermal conductivity K reaches a sharp maximum at a temperature of 30 K, estimate the maximum value of K.

d. Sketch K versus T for the temperature range 2K to 100 K.