# Scattering Mechanism in Metals

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).

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, 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.

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#### Solution Preview

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 ...

#### Solution Summary

This solution looks at the scattering mechanism in metals and uses concepts of thermal conductivity, specific heat capacity and thermal range