9.4 Free electron model of metals

Metals, such as copper and aluminum, are held together by bonds that are very different from those of molecules. Rather than sharing and exchanging electrons, a metal is essentially held together by a system of free electrons that wander throughout the solid. The simplest model of a metal is the free electron model    . This model views electrons as a gas. We first consider the simple one-dimensional case in which electrons move freely along a line, such as through a very thin metal rod. The potential function U ( x ) for this case is a one-dimensional infinite square well where the walls of the well correspond to the edges of the rod. This model ignores the interactions between the electrons but respects the exclusion principle. For the special case of $T=0\text{K},$ N electrons fill up the energy levels, from lowest to highest, two at a time (spin up and spin down), until the highest energy level is filled. The highest energy filled is called the Fermi energy    .

The one-dimensional free electron model can be improved by considering the three-dimensional case: electrons moving freely in a three-dimensional metal block. This system is modeled by a three-dimensional infinite square well. Determining the allowed energy states requires us to solve the time-independent Schrödinger equation

where we assume that the potential energy inside the box is zero and infinity otherwise. The allowed wave functions describing the electron’s quantum states can be written as

where $n_x,n_y,$ and $n_z$ are positive integers representing quantum numbers corresponding to the motion in the x -, y -, and z -directions, respectively, and $L_x,L_y,\text{and}{L}_z$ are the dimensions of the box in those directions. [link] is simply the product of three one-dimensional wave functions. The allowed energies of an electron in a cube $(L=L_x=L_y=L_z)$ are

Associated with each set of quantum numbers $(n_x,n_y,n_z)$ are two quantum states, spin up and spin down. In a real material, the number of filled states is enormous. For example, in a cubic centimeter of metal, this number is on the order of ${10}^{22}.$ Counting how many particles are in which state is difficult work, which often requires the help of a powerful computer. The effort is worthwhile, however, because this information is often an effective way to check the model.