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Unformatted text preview: Physics 927 E.Y.Tsymbal 1 Section 7: Free electron model A free electron model is the simplest way to represent the electronic structure of metals. Although the free electron model is a great oversimplification of the reality, surprisingly in many cases it works pretty well, so that it is able to describe many important properties of metals. According to this model, the valence electrons of the constituent atoms of the crystal become conduction electrons and travel freely throughout the crystal. Therefore, within this model we neglect the interaction of conduction electrons with ions of the lattice and the interaction between the conduction electrons. In this sense we are talking about a free electron gas . However, there is a principle difference between the free electron gas and ordinary gas of molecules. First, electrons are charged particles. Therefore, in order to maintain the charge neutrality of the whole crystal, we need to include positive ions. This is done within the jelly model , according to which the positive charge of ions is smeared out uniformly throughout the crystal. This positive background maintains the charge neutrality but does not exert any field on the electrons. Ions form a uniform jelly into which electrons move. Second important property of the free electron gas is that it should meet the Pauli exclusion principle, which leads to important consequences. One dimension We consider first a free electron gas in one dimension. We assume that an electron of mass m is confined to a length L by infinite potential barriers. The wavefunction ( ) n x of the electron is a solution of the Schrdinger equation ( ) ( ) n n n H x E x = , where E n is the energy of electron orbital. Since w can assume that the potential lies at zero, the Hamiltonian H includes only the kinetic energy so that 2 2 2 2 ( ) ( ) ( ) ( ) 2 2 n n n n n p d H x x x E x m m dx = = = . (7.1) Note that this is a one-electron equation, which means that we neglect the electron-electron interactions. We use the term orbital to describe the solution of this equation. Since the ( ) n x is a continuous function and is equal to zero beyond the length L , the boundary conditions for the wave function are (0) ( ) n n L = = . The solution of Eq.(7.1) is therefore ( ) sin n n x A x L = , (7.2) where A is a constant and n is an integer. Substituting (7.2) into (7.1) we obtain for the eigenvalues 2 2 2 n n E m L = . (7.3) These solutions correspond to standing waves with a different number of nodes within the potential well as is shown in Fig.1. Physics 927 E.Y.Tsymbal 2 Fig.1 First three energy levels and wave-functions of a free electron of mass m confined to a line of length L . The energy levels are labeled according to the quantum number n which gives the number of half-wavelengths in the wavefunction. The wavelengths are indicated on the wavefunctions....
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- Fall '11