Quantum-Mechanical Free Electron Model

CLASSICAL FREE ELECTRON (CFE) MODEL

Many of the difficulties encountered by the classical free electron model were removed with the Quantum-Mechanical Free Electron Model. the classical free electron model. The first successful attempt to understand the electrical properties of metals was presented by P. Drude in 1900 and was extended by H. A. Lorentz in 1909. The results of these two investigators, as well as the work of others, is now called the classical free electron model (CFE).

Assumptions of the Model

1. The main assumption of the CFE model is that metal is composed of an array of ions with "valence electrons that are free to roam through the ionic array with the only restriction that they remain confined within the boundaries of the solid. Because these valence electrons are responsible for the electrical conduction of the solid, they are called conduction electrons.
2. The mutual repulsion between the negatively charged electrons is neglected. In addition, the potential energy due to the ions is assumed to be Paul Drude (1863-1906). everywhere constant. These free electrons are basically treated as an ideal neutral gas that obeys classical Maxwell-Boltzmann statistics
3. In the absence of an electric field, these electrons are moving with their random thermal velocities given by the ideal gas result

½ mv-2= (3/2)kT

where v-2 is the average of the square of the thermal speeds; k is Boltzmann's constant, 1.38 x 10- 23 J/K; and T is the absolute temperature of the solid and therefore of the electron gas.

1. When an electric field 'jg is applied to the solid, the free electrons acquire an average drift velocity in the direction opposite to the electric field, thus producing an electric current. Despite the many simplifying assumptions, this CFE model was able to explain many properties of metals in a quantitative and satisfactory way. As we will show, it was able to predict Ohm's law, as well as the Wiedemann Franz law, an empirical formula relating the electrical and thermal conductivities in metals, which will be discussed later. The CFE model was not totally adequate; subsequent models started with some of the basic assumptions of the Drude-Lorentz model but modified them with new concepts.

 

Quantum-Mechanical Free Electron Model (QMFE)

Many of the difficulties encountered by the classical free electron model were removed with the advent of quantum mechanics. In 1928, A. Sommerfeld modified the free electron model in two important ways:

1. The electrons must be treated quantum mechanically. This will quantize the energy spectrum of the electron gas.
2. The electrons must obey Pauli's exclusion principle; that is, no two electrons can have the same set of quantum numbers.

As a result of these modifications, when we put an electron gas in a solid, we begin by putting the electrons in the lowest energy states available, while obeying the exclusion principle, until we have used all the available electrons. This is to be contrasted with the classical free electron gas in which Arnold Sommerfeld (1868-1951). the electrons can assume continuous energy values, with many electrons having the same energy. This has profound implications for the statistical distribution of energies that the electrons can have. Thus, whereas a classical gas will obey Maxwell-Boltzmann statistics (Supplement 9-1, Chapter 9), the quantum mechanical gas will follow a new type of statistical distribution known as the Fermi-Dirac distribution (Supplement 23-2 at the end of this chapter). This in turn will affect the way the electron gas can absorb energy from an external source, such as a heat source, and the way it responds to an electric field. Aside from these two key modifications, Sommerfeld kept most of the assumptions of the Drude model:

1. The valence electrons are free to move through the solid.
2. Aside from collisions with the ions, the electrostatic interaction between the electrons and the lattice ions is ignored.
3. The interaction between the electrons is also neglected.