Energy Distribution of Electrons in Metals

ENERGY DISTRIBUTION OF ELECTRONS IN METALS

We know according to Quantum free electron theory the energy levels are discrete (microscopically). But since the spacing between any two energy levels is very less (10^-6 eV), the distribution of energy levels seems to be continuous. (macroscopically).

Proof:

We know the energy eigen value in three dimension is

This equation represents the permissible energy values of the valence electrons in a cubical metal piece. If the cubical metal piece has a dimension say  1cm³, then the ground state energy is given by

Also, the maximum spacing between the consecutive energy level is very less, say in the order of 10^-6 eV.

If a plot is made between the number of energy levels N (E) that are filled with electrons per unit energy and Energy E, it is found that the number of energy levels N(E) increases parabolically with the increase of energy 'E' as shown in Fig. 1.8.

In figure 1.8, the dotted line shows the change in energy of electron at room temperature.

Here each energy level can provide only two states, one for spin up and the other for spin down. Hence only 2 electrons can be filled in a given energy state as shown in Fig. 1.9. Thus the Pauli's exclusion principle is satisfied.

At T=0, if there are 'N' number of electrons (N being even), then we have N/2 number of filled energy levels and other higher energy levels will be completely empty.

This (N/2)^th level is the highest filled energy level, called Fermi Energy level (E_F).

Fermi Energy and its importance

Fermi energy level (E_F): Fermi energy level is the maximum energy level upto which the electrons can be filled at OK.

Importance

(i) Thus it act as a reference level which seperates the vacant and filled states at OK.

(ii) It gives the information about the filled electrons states and the empty states.

(iii) At OK, below Fermi energy level electrons are filled at above Fermi energy level it will be empty.

(iv) When the temperature is increased, few electron gains the thermal energy and it goes to higher energy levels.

Conclusions

(i) In the quantum free electron theory, though the energy levels are decrete (microscopically), the spacing between consecutive energy level is very less and thus the distribution of energy levels seems to be continuous.

(ii) The number of energy levels N(E) that are filled with electrons per unit energy increases parabolically with the increase of energy E as shown in Fig. 1.8

(iii)Each energy level can provide only two states, namely, one for spin up and the other for spin down and hence only 2 electrons can be occupied in a given energy state i.e., paulis exclusion principle.

(iv) At T=0, If there are 'N' number of electrons (N being even), then we have N/2 number of filled energy levels and the other higher energy levels will be completely empty.

(v) This (N/2)^th level is the highest filled energy level is known as Fermi energy level (E_F0).

(vi) The electrons are filled in a given energy level, according to pauli's exclusion principle (i.e) No two electrons will have the same set of four quantum numbers, as shown in Fig. 1.9.

(vii) At Room temperature, the electrons within the range of K_B T [where K_B → Boltzmann constant] below the Fermi energy level will absorb ̴ Kg T and goes to higher energy states with energy + E_Fo + K_BT as shown in Fig. 1.8.