Calculate the temperature at which the lattice contribution to the specific heat and the electronic contribution to the specific heat become equal in a metal which has a Debye temperature of and a Fermi energy
.
The lattice contribution to the specific heat at low temperatures is given by the Debye model: [ C_\text{lattice} = \frac{12 \pi^4}{5} N k_B \left(\frac{T}{\Theta_D}\right)^3 ] The electronic contribution to the specific heat is given by: [ C_\text{electronic} = \gamma T, \quad \gamma = \frac{\pi^2}{2} \frac{N k_B^2}{\epsilon_F} ] We are asked to find the temperature (T) at which (C_\text{lattice} = C_\text{electronic}). Setting them equal: [ \frac{12 \pi^4}{5} N k_B \left(\frac{T}{\Theta_D}\right)^3 = \frac{\pi^2}{2} \frac{N ________ __________ __________ __________ __________ ____ ________.
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