Question: Why do solids expand when heated, assuming no change in the lattice structure (no phase change)? This problem and the character of the underlying interatomic potentials provide a microscopic answer. Note that a purely harmonic solid would not expand, a result which follows from (c). Your numerical results in (c) should seem reasonable, and show how microscopic parameters can be determined from macroscopic measurements, once we understand the statistical connection.
		Note: The reference given to the CRC Handbook of Chemistry and Physics is specific to one student edition/printing. Look under "thermal expansion" in the index to get the proper page in the edition you use. Comments: The initial harmonic expression for the potential, generalized to three dimensions, would be the normal starting point for the description of a cubic solid. The Hamiltonian can be simplified in terms of a set of uncoupled normal mode oscillators. The relation given in (c) between the speed of sound, the fundamental frequency , and the lattice spacing a given in (c) follows from the solution for long wavelength oscillations in the solid. The spectrum of the normal mode frequencies, not considered here, is crucial in the Debye theory of the specific heats in quantum statistical mechanics. The classical specific heats are simply given by the equipartition theorem and the counting of modes.

Comment: This practical problem, much in the news lately, illustrates why the possession of certain types of ultracentrifuge is of interest to the International Atomic Energy Agency.

		Comments: Classical statistical mechanics is formulated in terms of phase space distributions and canonical variables. The canonical desciption is crucial for the foundations of the theory, but we may well want on occasion to use noncanonical variables in actual calculations. The result you get in (a) in going from canonical to noncanonical variables should not be a surprise. Landau and Lifshitz simply use the result without derivation in their treatment of rotating molecules in § 29, Prob. 5 - they already know the answer!
		In case you have forgotten, the Jacobian is the determinant of the matrix of partial derivatives of the old variables with respect to the new, or, using properties of determinants, the inverse of the matrix of derivatives of the new variables with respect to the old. Use whichever is easier to calculate.
		The last part of the problem gives an answer to the question "How do we know, or how can we measure, the permanent dipole moments of molecules?" The function that appears in the result you will obtain for p(x) is called the "Langevin function." It will reappear in the equivalent problem of a collection of magnetic dipoles in a magnetic field, i.e., in the discussion of classical paramagnetism. And again: get used to Gaussian units. There are no funny factors around, so definitions of moments, etc., and theoretical calculations are simpler. You can always convert to SI units at the end using Jackson's table.

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