Chapter 41, Problem 1Q

To determine

To select correct options:

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

Answer to Problem 1Q

Solution:

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

(b) the size of the sample

(c) the position of the level in the band

(d) the temperature of the sample.

Explanation of Solution

Explanations

1) Concept

An isolated atom can exist in only a discrete set of energy levels. As atoms come together to form a solid, the levels of the individual atoms merge to form the energy bands. An energy band is made up of an enormous number of very closely spaced levels.

The density of occupied states N_o(E) is given by the product of two quantities P(E) and N(E).

N_o(E) = P(E) x N(E)

P(E) is the probability of occupancy. And N(E) is the number of energy levels per unit volume of the sample per unit energy.

P(E) = $\frac{1}{e^{\frac{(E-E_f)}{kT}}+1}$

2) Given:

Five option to select from to answer the question regarding adjacent energy levels.

3) Equations

P(E) = $\frac{1}{e^{\frac{(E-E_f)}{kT}}+1}$

4) Calculations

The highest occupied band in a metal is the conduction band.

The interval between adjacent energy levels in a conduction band will depend on N(E) and the number of electrons – which means it will depend on the size of the sample. This implies that option (b) is correct.

Probability of occupancy P(E), and N(E) both depend on energy E – that is the position of the energy level in the band. This implies that option (c) is correct.

In addition, we see that the energy levels are dependent on temperature T. Also thermal expansion and contraction of metals, will increase or decrease the electron energy overlaps, and hence the adjacent energy levels.

Conclusion

The free electrons in a metal fill up the conduction band. But all energy levels including adjacent energy levels are closely spaced levels due to the Pauli Exclusion Principle.