Quantum Theory of Free Electrons

Now that we are better equipped to handle some of the quantum theory necessary to understand the behavior of electrons in metals, let us review some of the problems we encountered in the simple Drude theory. 

The drift velocity when an electric field is applied is still given by Vd = —erEjm, so the entire Fermi sphere is shifted by an amount 5p = —erE. In the Drude model, all of the electrons were assumed to have random velocities with an average speed but the average velocity would be zero in the absence of an applied field. An applied field would increase each electron s average velocity by Vd so the current is carried by all of the electrons with a velocity Vd- 

However, the electrons in a Fermi gas do not have random velocities. Those on the leading edge of the Fermi sphere have velocities Vp and those on the back edge have velocities —Vp. While it is true that the applied field increases each electron s velocity by Vd, it is not true that the current is carried by each electron moving at d- To illustrate this, consider a simple one-dimensional Fermi model with 11 particles in which the Fermi velocity is 5 m/s and quantum states are separated by 1 m/s as shown below. 

Now let an applied field increase each particle s velocity by 1 m/s. 

Note that the particle with velocities from 4 to —4 m/s cancel each other and carry no net charge. All the charge is carried by the two particles with velocities 5 and 6 m/s. So instead of 11 particles moving at 1 m/s, as would be the case with the Drude model, we have two particles moving with an average speed of 5.5 m/s to carry the current. The sum of the products of the charge carriers and their velocities is the same in either model. 

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This rule conforms with the principle of equipartition of energy, first enunciated by Maxwell, that the heat capacity of an elementary solid, which reflected the vibrational energy of a three-dimensional solid, should be equal to 3RJK-1 mol-1. The anomaly that the free electron theory of metals described a metal as having a three-dimensional structure of ion-cores with a three-dimensional gas of free electrons required that the electron gas should add another (3/2)R to the heat capacity if the electrons behaved like a normal gas as described in Maxwell s kinetic theory, whereas the quantum theory of free electrons shows that these quantum particles do not contribute to the heat capacity to the classical extent, and only add a very small component to the heat capacity. 

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