Effective medium approaches to the alloy problem

Early on, the Virtual Crystal Approximation (VCA) was tried. Here one just puts 

Another attempt was made by Korringa [86], who tried to average the scattering amplitudes, which are equivalent to the t matrices, on each site  

The CPA can be understood by thinking of the scattering in an alloy and what we want the effective potential to actually do. We want to have an effective potential that on average differs as little as possible from the alloy, that is, an electron moving through the effective medium should scatter from the effective potential as close as possible to what it would experience in a random alloy. This can be formulated as  

We can now solve self-consistently for the effective medium From the electronic structure calculation we get the tls for the different atomic kinds and the S matrix. Then we solve Eq.(5.6) to get the tc for the effective medium, which we use in Eq.(5.4) to get the scattering path operator for the effective medium. This then gives the new scattering path operators for the different atomic kinds by using Eq.(3.44) for the impurity problem. These are what is needed to restart the calculations for the next iteration. 

In the EMTO-CPA method, the total energy for an alloy is gotten from the expression  

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