Calculate the Fermi energy for copper, which has a density of$\mathbf{8}\mathbf{.9}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{}\mathbf{\text{kg}}\mathbf{/}{\mathbf{\text{m}}}^{\mathbf{3}}$and one conduction electron per atom. Is room temperature "cold"?

The energy of fermions at absolute zero temperature is known as Fermi energy. The mathematical equation for the Fermi energy is,

$E_F=\frac{{\pi }^2{\hslash }^2}{m}{\left[\frac{3}{(2s+1)\pi \sqrt{2}}\frac{N}{V}\right]}^{\frac{2}{3}}$ ……. (1)

Here,

$\text{N}\to$Number of oscillators

$\hslash \to$Planck's reduced constant

$V\to$Volume

$s\to$Spin

Density of copper$=8.9\times {10}^3\text{kg}/{\text{m}}^3$

The NN in the equation (1) can be rewritten as the mass per unit volume over the mass per atom, or the bulk density D over the atomic mass:

role="math" localid="1658381923377" $\begin{array}{c}{E}_F=\frac{{\pi }^2{\hslash }^2}{m}{\left[\frac{3}{(2s+1)\pi \sqrt{2}}\frac{N}{V}\right]}^{\frac{2}{3}}\\ =\frac{{\pi }^2{\hslash }^2}{m}{\left[\frac{3}{(2s+1)\pi \sqrt{2}}\frac{D}{m_A}\right]}^{\frac{2}{3}}\end{array}$

Substitute$9.1\times {10}^{-31}\text{kg}$for mass of the electron $\left(\text{m}\right),8.9\times {10}^3\text{kg}/{\text{m}}^3$for density of copper(D), $1.055\times {10}^{-34}\text{J}.s$for $\hslash ,\frac{1}{2}$forthe spin of the electron and $1.055\times {10}^{-25}\text{kg}$for$m_A$.

role="math" localid="1658381931536" $\begin{array}{c}{E}_F=\frac{{\pi }^2{(1.055\times {10}^{-34}\text{J}.\text{s})}^2}{(9.1\times {10}^{-31}\text{kg})}{\left[\frac{3}{(2\left(\frac{1}{2}\right)+1)\pi \sqrt{2}}\frac{(8.9\times {10}^3\text{kg}/{\text{m}}^3)}{(1.055\times {10}^{-25}\text{kg})}\right]}^{\frac{2}{3}}\\ =1.126\times {10}^{-18}\text{J}\end{array}$

Convert the unit for Fermi energy for copper energy from $J$to$\text{eV}$ .

role="math" localid="1658381943480" $\begin{array}{c}{E}_F=1.126\times {10}^{-18}J\\ =(1.126\times {10}^{-18}J)\left(\frac{1\text{eV}}{1.6\times {10}^{-9}J}\right)\\ =7.04\text{eV}\end{array}$

Therefore, the Fermi energy for the copper is$7.04\text{eV}$ .

Since the energy that a particle at room temperature would have been $0.04\text{eV}$(from $\frac{3}{2}{k}_{B}T$), as opposed to the Fermi energy of $7.0\text{eV}$, room temperature would be considered cold in comparison.