The Rigid-Band Model (or RBM) is one of the models used to describe the behavior of metal alloys. In some cases the model is even used for non-metal alloys such as Si alloys. According to the RBM the shape of the constant energy surfaces (hence the Fermi surface as well) and curve of density of states of the alloy are the same as those of the solvent metal under the following conditions: The excess charge of the solute atoms localizes around them. The mean free path of the electrons is much greater than the lattice spacing of the alloy. The electron states of interest in the pure solvent are all in one energy band, which is greatly separated in energy from the other bands. The only effect of the addition of the solute, given that its valence is greater than that of the solvent, is the addition of electrons to the valence band. This results to swelling the Fermi surface and filling the density of states curve to a higher energy. Theory In a pure metal, because of the periodicity of the lattice, the features of its electronic structure are well known. The single-particle states can be described in terms of Bloch states, the energy structure is characterized by Brillouin zone boundaries, energy gaps and energy bands. In reality though no metal is perfectly pure. When the amount of the foreign element is dilute, the added atoms may be treated as impurities. But when its concentration exceeds several atomic % , an alloy is formed and the interaction among the added atoms can no longer be neglected. Before giving a more mathematical outline of the RBM it is convenient to give somewhat of a visualization of what happens to a metal upon alloying it. In a pure metal, we'll take silver as an example, all lattice sites are occupied by silver atoms. When different kind of atoms are dissolved into it, for example 10% of copper, some random lattice sites become occupied by copper atoms. Since silver has a valence of 1 and copper has a valence of 2, the alloy will now have a valence of 1.1. Most lattice sites however are still occupied by silver atoms and consequently the changes in electronic structure are minimal. Basic concepts behind the Rigid-Band model In a pure metal of valence Z1, all atoms become positive ions with the valence +Z1 by