Consider an idealization of a crystal which has N lattice...

Consider an idealization of a crystal which has $N$ lattice points and the same number of interstitial positions (places between the lattice points where atoms can reside). Let $E$ be the energy necessary to remove an atom from a lattice site to an interstitial position and let $n$ be the number of atoms occupying interstitial sites in equilibrium. (a) What is the internal energy of the system? (b) What is the entropy $S$ ? Give an asymptotic formula valid when $n \gg 1$ ? (c) In equilibrium at temperature $T$, how many such defects are there in the solid, i.e., what is $n$ ? (Assume $n \gg 1$.)

Consider an idealization of a crystal which has $N$ lattice points and the same number of interstitial positions (places between the lattice points where atoms can reside). Let $E$ be the energy necessary to remove an atom from a lattice site to an interstitial position and let $n$ be the number of atoms occupying interstitial sites in equilibrium.
(a) What is the internal energy of the system?
(b) What is the entropy $S$ ? Give an asymptotic formula valid when $n \gg 1$ ?
(c) In equilibrium at temperature $T$, how many such defects are there in the solid, i.e., what is $n$ ? (Assume $n \gg 1$.)

Key Concepts

Internal Energy of Defects

In a crystal, if an atom is removed from its lattice site and placed in an interstitial position, an energy cost E is incurred per defect. Since there are n such defects, the total internal energy due solely to the formation of these defects is U = nE. This concept is a specific application of the idea that the internal energy of a system with localized excitations (defects) is the sum of the energies of each excitation.

Configurational Entropy

The system’s entropy arises from the number of different ways one can choose which of the available positions are occupied by defects. Since there are N lattice sites (from which atoms may be removed creating vacancies) and N interstitial sites (into which atoms can be inserted), the total number of configurations (or microstates) is the product of the number of ways to choose n vacancies and the number of ways to choose n occupied interstitials, namely ? = (N choose n) × (N choose n). The entropy is then given by S = k ln ?. This combinatorial entropy is a key concept in statistical mechanics as it quantifies the disorder associated with defect distributions.

Stirling’s Approximation

When n is large (n ? 1) – although it typically remains small compared to N – one often uses Stirling’s approximation to simplify expressions for the logarithm of factorials. This approximation allows one to write expressions such as ln(N choose n) in a simpler asymptotic form; for example, ln(N choose n) ? n ln(N/n) + n (with further lower?order corrections). Such simplifications are crucial in obtaining tractable formulas for the entropy in systems with many particles.

Free Energy Minimization

The equilibrium concentration of defects is found by minimizing the free energy F = U ? TS with respect to the number of defects, n. Here U is the internal energy, and S is the entropy. By substituting the expressions U = nE and S (from the combinatorial counting with Stirling’s approximation) into the free energy and then minimizing F with respect to n, one obtains a condition that balances the energetic cost of creating defects with the entropic benefit of having many possible configurations.

Equilibrium Defect Concentration and Boltzmann Factor

The minimization of the free energy leads to an expression for n in equilibrium that involves an exponential factor. In this context, one finds that ln(N/n) is proportional to E divided by an appropriate multiple of kT; solving for n leads to an expression of the form n = N exp(?E/(2kT)). This resembles a Boltzmann factor, showing that the defect concentration decreases exponentially with the energy required to form a defect and inversely with temperature. It illustrates the role of thermal fluctuations in activating defect formation in solids.

Please give Ace some feedback