(Solved): Debye considered the vibrating atoms as coupled oscillators, which generate a wave which has disper ...

Debye considered the vibrating atoms as coupled oscillators, which generate a wave which has dispersion relation given by $?(k)=v_{s}k$. The average energy in this case is given by $[10]$ $\stackrel{?}{U}=\frac{3?V}{2{?}^2{\stackrel{?}{v}}_s^3}{\left(\frac{k_{B}T}{?}\right)}^4{?}_0^{x_D=?{?}_D/k_{B}T}\frac{x^{3}dx}{e^x?1}$ i. Approximate the average energy at very high temperatures. ii. Also find the low temperature limit of average energy. iii. Find and plot the specific heat as a function of temperature. iv. Discuss limitations of Debye theory and how it is modified in case of metals? [where ${?}_D$ is cut-off frequency, ${\stackrel{?}{v}}_s$ is the mean speed of sound]