Question

On which of the following does the interval between adjacent energy levels in the highest occupied band of a metal depend: (a) the material of which the sample is made, (b) the size of the sample, (c) the position of the level in the band, (d) the temperature of the sample, (e) the Fermi energy of the metal?

Short answer

The interval between the adjacent energy levels in the highest occupied band of a metal depends on:

(c) The position of the level in the band.

(d) The temperature of the sample.

(e) The Fermi energy of the metal.

Step-by-step solution

The given data

Here, the highest occupied band of the metal represents the valence band of the band structure.

Understanding the concept of the highest occupied band

The highest occupied band of metal is given by the valence band of a metal that is further divided into sublevels as the orbital subshells fill up with the electrons from the innermost shell to the outermost shell. This helps in studying the nature of the band structure. Thus, the position of each band level is vital in a transition process during the excitation of the electrons as photons to the higher band state. Again, from this concept, we also get to know that temperature plays a major role in excitation. Thus, at absolute zero temperature, the occupancy is a unity to all the levels and for higher temperatures, the energies with a lower value than that of Fermi energy have low occupancy, and the band just above the Fermi level has high occupancy.

Calculation of the dependence factors

From the given concept, we can clearly say that options (a) and (b) are not valid.

Now, the position of a level highly matters in determining the occupancy level. The energy levels are occupied by electrons starting from lower energy levels and then occupying the higher ones. Thus, the position of energy level plays a significant role in determining the occupancy of the level, and option (c) is valid.

At higher temperatures, the electrons get excited and jump to higher energy levels. Hence, the occupancy of higher levels is comparatively higher than the valence band. Thus, temperature affects occupancy, and option (d) is valid.

Lastly, the energy bands that are present below the fermi level have a higher occupancy probability than those present above it. Thus, option (e) is also valid.

Hence, the dependence is found to be on the position of level in the band, the temperature of the sample, and the Fermi energy of the metal.