Section 13.2 minimised and reached

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction · Crystal structure, bonding and diffraction

Condensed Matter Physics, Ch Real crystals: vacancies, dislocations and the strength of metals

Problem

Section 13.2 minimised F(Nvac)=Nvacεv−kBTln⁡WF(N_{\rm vac}) = N_{\rm vac}\varepsilon_v - k_BT\ln W and reached, before any approximation, Nvac/(N−Nvac)=e−εv/kBTN_{\rm vac}/(N-N_{\rm vac}) = e^{-\varepsilon_v/k_BT}. Write K=e−εv/kBTK = e^{-\varepsilon_v/k_BT} and n=Nvac/Nn = N_{\rm vac}/N. ((a)) Solve n/(1−n)=Kn/(1-n) = K exactly for nn in terms of KK. ((b)) For copper at its melting point, Section 13.2 gives K=1.8×10−5K = 1.8\times10^{-5}. Evaluate the exact nn from part (a) and find the relative difference (K−n)/n(K-n)/n between it and the approximate value nv=Kn_v = K. ((c)) Show algebraically that (K−n)/n=K(K-n)/n = K exactly, for any KK, and explain in one sentence why this guarantees that Nvac≪NN_{\rm vac} \ll N whenever εv≫kBT\varepsilon_v \gg k_BT, without checking the assumption case by case.

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