chapter 4 of introduction to thermal physics by shroeder 1vacancies and free energy in a crystal

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Chapter 4 of Introduction to Thermal Physics by Schroeder 1. Vacancies and Free Energy. In a crystalline solid, thermal fluctuations can cause an atom to spontaneously jump out of its lattice site and cram itself into an interstitial (I) site, leaving a vacancy (V) behind. The number of possible sites for an interstitial (or vacancy) is about equal to the number N of atoms in the solid. An activation energy E is required to form a single I-V pair, regardless of the final I and V positions. At a given temperature, n vacancy-interstitial (V-I) pairs are created among the N lattice sites of a crystalline solid. Assume N >> n >> 1. (A) Derive an expression for the entropy of n interstitial-vacancy pairs among N lattice sites. Be aware that once created, the members of the pair can wander anywhere (the I-V pair does not stay together and so their locations are random). Use the Stirling Approximation, with N >> n >> 1, to simplify your answer. [Hint: The multiplicity of accessible microstates is given by S = k ln Î©, where Î© is the multiplicity of interstitials and vacancies respectively. Remember to use N >> n >> 1. As an intermediate step, you should get ln Î© = n + n ln(N) - n ln(n).] (B) Write an expression for the free energy of the system at a temperature T as a function of the number n of vacancies. (C) Minimize the free energy with respect to n to derive an expression for the equilibrium number of I-V pairs n = n(T). As always, simplify your expression as much as possible (recall N >> n >> 1).

          Chapter 4 of Introduction to Thermal Physics by Schroeder

1. Vacancies and Free Energy. In a crystalline solid, thermal fluctuations can cause an atom to spontaneously jump out of its lattice site and cram itself into an interstitial (I) site, leaving a vacancy (V) behind. The number of possible sites for an interstitial (or vacancy) is about equal to the number N of atoms in the solid. An activation energy E is required to form a single I-V pair, regardless of the final I and V positions. At a given temperature, n vacancy-interstitial (V-I) pairs are created among the N lattice sites of a crystalline solid. Assume N >> n >> 1.

(A) Derive an expression for the entropy of n interstitial-vacancy pairs among N lattice sites. Be aware that once created, the members of the pair can wander anywhere (the I-V pair does not stay together and so their locations are random). Use the Stirling Approximation, with N >> n >> 1, to simplify your answer. [Hint: The multiplicity of accessible microstates is given by S = k ln Î©, where Î© is the multiplicity of interstitials and vacancies respectively. Remember to use N >> n >> 1. As an intermediate step, you should get ln Î© = n + n ln(N) - n ln(n).]

(B) Write an expression for the free energy of the system at a temperature T as a function of the number n of vacancies.

(C) Minimize the free energy with respect to n to derive an expression for the equilibrium number of I-V pairs n = n(T). As always, simplify your expression as much as possible (recall N >> n >> 1).

chapter 4 of introduction to thermal physics by shroeder 1vacancies and free energy in a crystalline solid thermal fluctuations can cause an atom to spontaneously jump out of its lattice sit 96206

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