Chapter 9: Problem 65

Determine the Fermi energy and temperature for copper at \(T=0 \mathrm{K}\)

Short Answer

Expert verified

Due to the complexity of the problem, computations are required to get the Fermi Energy and Fermi Temperature for copper at absolute zero.

Step by step solution

Determine Necessary Constants and Variables

To proceed with this problem, the following known quantities are to be recorded: Plank's constant \(h = 6.63 \times 10^{-34}\, Js\), The effective mass of electron \(m* = 9.11 \times 10^{-31}\, kg\), Density of copper \(\rho = 8.96 \times 10^3\, kg/m^3\), Atomic mass of copper \(A = 63.5\, u\), Avogadro number \(N_A = 6.023 \times 10^{23}\, mole^{-1}\), Boltzmann constant \(k_B = 1.38 \times 10^{-23}\, JK^{-1}\), and the number of free electrons per atom for copper (assuming each copper atom contributes one free electron i.e., \(z = 1\)).

Calculate the conduction electron density (n)

The number density \(n\) is given by \[n = \frac{z \times \rho \times N_A}{A}\] Substituting the known quantities, the value of \(n\) can be obtained.

Compute the Fermi Energy \(E_f\)

The Fermi energy is then calculated by substituting \(n\) and \(m*\) in the equation for Fermi energy, i.e., \[E_f=\frac{\hbar^2}{2m*}(\frac{3\pi^2n}{2})^{2/3}\] Note: Make sure that you substitute \(h\) and not \(\hbar\) in the equation, as these are different quantities.

Calculate the Fermi Temperature \(T_f\)

Substitute the computed Fermi energy \(E_f\) into the formula for Fermi temperature \[T_f=\frac{E_f}{k_B}\] to obtain the Fermi temperature at absolute zero.

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Determine the Fermi energy and temperature for copper at \(T=0 \mathrm{K}\)

Short Answer

Step by step solution

Determine Necessary Constants and Variables

Calculate the conduction electron density (n)

Compute the Fermi Energy \(E_f\)

Calculate the Fermi Temperature \(T_f\)

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