IGNOU PHE 13 2018 - Copper has a Debye temperature of 340 K and a Fermi temperature of 810

Question

Copper has a Debye temperature of 340 K and a Fermi temperature of 81000 K. At what temperature will the contribution of the vibrations of the crystal lattice (within the Debye theory) and that of the conduction electrons (within the free electron theory) to the specific heat become equal?

27 Jan 2018

Answer :

Word Count : 426

Let's solve this step by step, carefully, using manual calculation. We are asked: > Copper has a Debye temperature $\Theta_D = 340 \, \text{K}$ and a Fermi temperature $T_F = 81000 \, \text{K}$. At what temperature will the lattice vibrations contribution (Debye theory) and conduction electron contribution (free electron theory) to the specific heat become equal? --- ### Step 1: Write expressions for the specific heats 1. Lattice vibrations (Debye model): For $T \ll \Theta_D$ (low temperature), the specific heat of the lattice: $$ C_\text{lattice} = \frac{12 \pi^4}{5} N k_B \left(\frac{T}{\Theta_D}\right)^3 $$ where $N$ = number of atoms, $k_B$ = Boltzmann constant. --- 2. Conduction electrons (free electron model): For $T \ll T_F$, the electronic specific heat: $$ C_\text{electrons} = \frac{\pi^2}{2} N k_B \frac{T}{T_F} $$ --- ### Step 2: Equate the two contributions $$ C_\text{lattice} = C_\text{electrons} $$ $$ \frac{12 \pi^4}{5} N k_B \left(\frac{T}{\Theta_D}\right)^3 = \frac{\pi^2}{2} N k_B \frac{T}{T_F} $$ Cancel $N k_B$ on both sides: $$ \frac{12 \pi^4}{5} \left(\frac{T}{\Theta_D}\right)^3 = \frac{\pi^2}{2}

___ _________ _________ __________ ____ ___.
_____ _____ ________ _____ ___ ______ __________ __________ _________ __________ ___ __________.
__________ _______ ___ _________ ____ __________ _____ _______ _________ ______ _____ ___.
__________ ______ _______ _________ ____ ____ ______ ____ ___.
_______ _______ __________ ___ _____ _______ ______.
______ __________ ________ __________ _____ ___ _______ __________ _____ _________.
_________ __________ ______ ___ __________ ____ ____ ________ ________.
________ ____ _______ _______ ______ _______ ___ ________.
_____ ________ _________ ____ _____ __________ _______ ___ __________ ___.
________ _______ _________ ___ ________ _________ _________.
____ ______ _______ _________ ___ ______ ______ _______ ___ ____ _______.
________ __________ __________ ____ __________ __________ _________ ____ _______ ________ ________.
___ ___ _______ _____ _____ ____ _________.
___ __________ _______ ______ _____ ____ ______ ______ _________ ___.
_____ ____ _______ _______ ________.
__________ _____ ________ ___ ____.
_______ ____ ________ __________ _________ ____ ______ _________ ______.
____ ____ _________ ________ _____ ______ ________ ________ ______ ___ _______ ________.
_____ ______ ________ _____ __________ ____ ___ _______.
_______ __________ _________ _________ ____ ___ ____ _________ __________ _________.
_______ ________ ___ ______ _____ _________ ____ __________ _____.
________ ___ ______ __________ ____ ________ ________ ________.
_____ ________ ______ ______ _____ ___ ___ ______ ______ ________ ____ _________.
_____ _______ ________ ______ ______ ________ ____ ____ __________ _________.
______ __________ ________ ____ _________ ____ ________ _______ ___.
______ _____ ______ __________ _____.
_____ ______ _____ ______ ________ ___ ________.
_______ _______ __________ __________ _________ _____ _______ ____ ____ __________.
________ ___ ___ ____ _______ _______ __________.
___ _____ ____ _____ ______ _______.
________ __________ ___.

IGNOU ACADEMY

Your IGNOU journey, made easier.

Solved Assignments
Study Notes
Guess Papers
Question Papers

Call Us WhatsApp

STUDY • PREPARE • SUCCEED

Skip in 10