91Ó°ÊÓ

Consider the formula for the relation between the energy of the particle in the box as follows:

$\mathbf{E}\mathbf{=}\frac{{\mathbf{n}}^{\mathbf{2}}{\mathbf{h}}^{\mathbf{2}}}{\mathbf{8}{\mathbf{mL}}^{\mathbf{2}}}$

Here, E is the energy, n is the number of antinodes, h is the planck’s constant and m is the mass.

Consider the crystal is considered as the box and the length L is taken as the length of the whole crystal. As the length L covers the all the n antinodes.

Consider the equation that determines the energy as:

$E=\frac{{\left(nN\right)}^2{\mathrm{Ï€³ó}}^2}{2m{d}^2}$

Resolve the equation as:

$$\begin{gathered} E=\frac{{\left(n\right)}^2{\mathrm{Ï€³ó}}^2}{2m{\left({\displaystyle \frac{W}{N}}\right)}^2} \\ E=\frac{{\left(n\right)}^2{\mathrm{Ï€³ó}}^2}{2m{a}^2} \end{gathered}$$

Hence, for n to be the band the use of L is the length of the single atom as the n corresponds to the number of antinodes in the single atom.

Consider that the band energy is clustered around the corresponding single atom energy and it do not depend on the number of atoms in the crystal this is why the band energy is independent of the size of the crystal.