513.001 Molecular and Solid State Physics

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MIT 8.231 Physics of Solids, 4.3

Consider a two-dimensional solid with one atom per primitive unit cell. Use a Debye model to find an approximation for the lattice heat capacity.

a) Derive expressions for the Debye wavevector kD and the Debye temperature Θ in terms of the number of atoms per unit area N/A, the sound velocity, and Boltzmann's constant kB.

b) Find the density of states as a function of energy for this model. Draw a carefully labeled picture of this density of states.

c) Set up, but do not evaluate, an expression for the lattice contribution to the energy of the solid.

d) Find an expression for the heat capacity. Leave your answer in terms of a dimensionless (but temperature-dependent) integral. What is the temperature dependence of the heat capacity in the low temperature limit?

e) Without doing any calculations, explain what one should expect for the temperature dependence of the heat capacity of a one-dimensional lattice in the low temperature limit.