Excess entropy for a vacancy in a crystal

The following topics are discussed: (1) Within the harmonic theory a general expression is given for excess entropy of a localized detect in any crystal with cyclic boundary conditions and in the small perturbation limit. For a single vacancy defect Stripp and Kirkwood's formula is obtained, ΔS/k_B = d/2,d being the dimension of the space. For other localized defects the perturbation formula requires calculation of the unperturbed Green's matrix. (2) It is argued that the effect of free surface boundary conditions, as well as other surface changes, leaves ΔS effectively invariant. (3) Using the vacancy perturbation result for ΔS and Lawson's formula, a value for the vacancy formation volume is obtained only in terms of known measured parameters; this result is compared to experimental and other calculated values of ΔV_f. (4) If localized vibrational modes are present due to the vacancy, what effect these might have on ΔS, and, from this, what materials might be candidates for observing localized modes.