Examples Including Disorder—Configurational Entropy

Each of the materials we looked at above is an ordered crystal, that is, there is only one way to arrange the set of atoms we are interested in to give the crystal structure. This is of course not true for all materials. Suppose we wanted to add to our class of possible materials a sample of Cu20 with a small number of O vacancies. Specifically, imagine we used a sample of Cu20 containing 2n Cu atoms and with k O atoms removed. There are 

This quantity is called the configurational entropy. Using Stirling s approximation, this entropy can be written as 

When we developed expressions for the grand potential above [Eqs. (7.1)— 

the last term is constant for all materials with the same number of Cu atoms within the approximation that entropic contributions from atomic vibrations are equivalent in each material. 

Once we have determined crystal and Evac in this way, we can write the energy for a sample of the material with N formula units and m vacancies as 

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