Question

Verify the numerical factor 0.121 in Eq. 41-9.

Short answer

The numerical factor ${\left(\frac{3}{16\sqrt{2\mathrm{\pi }}}\right)}^{2/3}$is equal to the numerical factor 0.121.

Step-by-step solution

The given data

The equation of Fermi energy according to Eq. 41-9 is

$E_F={\left[\frac{3}{16\sqrt{2\mathrm{\pi }}}\right]}^{2/3}\frac{h^2}{m}{n}^{2/3}$.

where, n is the number of conduction electrons per unit volume, m is the mass of an electron and h is the Planck’s constant.

Understanding the concept of Fermi energy

The highest level to be occupied by the electron in the valence band at 0 K is known as Fermi level.

Calculation of verifying the numerical factor 0.121

From the given Fermi energy equation, the numerical constant value is given by,

$$\begin{gathered} c={\left(\frac{3}{16\sqrt{2\mathrm{\pi }}}\right)}^{2/3} \\ ={\left(4.22\times {10}^{-2}\right)}^{2/3} \\ =0.121 \end{gathered}$$

Hence, the numerical factor 0.121 is verified.