Consider the thermal conductivity K of a pure synthetic sapphire crystal that has a diameter of 3mm. (For sapphire delta D -1000 K , speed of sound - 10^4ms^-1, and for T <<delta D, cv - 10^-1 T^8 J M K^-1, delta D is the Debye temperature and T is the temperature).
b. Discuss which scattering mechanism dominates the thermal conductivity K of the sapphire crystal at 1000 K.
c. Given that K - 1/3 c, vl 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.
See attachment for better formula representation.
For a metal the collision or scattering mechanisms affects the thermal conductivities. these collissions can be classified as 1. Collision of a phonon with other phonons. 2. Collision of a phonon with imperfections of a crystal. 3. Collision of the phonon with external boundaries of the ...
This solution looks at the scattering mechanism in metals and uses concepts of thermal conductivity, specific heat capacity and thermal range