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Consider a piece of two-dimensional graphite where N carbon atoms form a honeycomb lattice. Assume that it costs energy Delta to remove a carbon atom from a lattice site and place it in the center of a hexagon to form a vacancy and an interstitial, as shown in the sketch. Show that there are N/2 possible locations for interstitials. Consider a microcanonical ensemble of the system, at given total energy E. For M interstitials, find the statistical entropy for large N and M. Find the most probable value M of M, using the method of Lagrange multipliers. Find the equilibrium entropy S, and express the Lagrange multiplier in term of the temperature, defined by T^-1 (partial differential S/partial differential E). Given M in the limits T rightarrow 0 and T rightarrow infinity.
Consider a piece of two-dimensional graphite where N carbon atoms form a honeycomb lattice. Assume that it costs energy Delta to remove a carbon atom from a lattice site and place it in the center of a hexagon to form a vacancy and an interstitial, as shown in the sketch. Show that there are N/2 possible locations for interstitials. Consider a microcanonical ensemble of the system, at given total energy E. For M interstitials, find the statistical entropy for large N and M. Find the most probable value M of M, using the method of Lagrange multipliers. Find the equilibrium entropy S, and express the Lagrange multiplier in term of the temperature, defined by T^-1 (partial differential S/partial differential E). Given M in the limits T rightarrow 0 and T rightarrow infinity.
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