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Unformatted text preview: Physics 212: Statistical mechanics II, Fall 2006 Lecture XXVII-XXVIII Many physical systems have a unique ground state in the limit of zero temperature, and hence zero entropy. However, there are a number of systems where either because of a degeneracy of ground states or because of dynamical considerations, a nonzero residual entropy is measured down to zero temperature (typically by measuring the specific heat over a range of temperatures up to some well-understood high-temperature limit, then integrating). We first discuss ice models, where a residual entropy results from a geometric degeneracy of configurations, and then glasses, in which a unique ground state may exist. Reference: Ziman, Models of Disorder Water ice is a surprisingly complicated material. At atmospheric pressure, there is a range of temperatures where the Ice I structure appears: this is a wurzite structure in which each oxygen atom is tetrahedrally coordinated with four other oxygen atoms. There is exactly one hydrogen atom on each oxygen-oxygen bond, but the hydrogen atom sits closer to one oxygen atom than the other, in such a way that each oxygen has two hydrogen atoms close to it. This Pauling model of ice has an entropy resulting from the different ways in which the hydrogen atoms can be arranged. To estimate the entropy, we can start from assuming that each bond has 2 configurations, depending on whether the hydrogen atom is closer to one oxygen or the other (giving 2 2 N configurations, where N is the number of oxygen atoms), then noting that for each oxygen atom, only 6 of the 16 possible configurations of its four bonds will have exactly 2 hydrogens next to the oxygen. This leads to the estimate S N = k B log 2 2 N (6 / 16) N N = k B log 3 2 (1) which is within about 1 percent of the correct answer. Note that we ignored some correlations between the bond configurations in this argument, so this agreement is a bit surprising. Similar geometric degeneracies appear in many magnetic and other models when interactions are frustrated: not every interaction energy can be simultaneously minimized. An example is the Ising antiferromagnet on the triangular lattice, which has a macroscopic entropy of ground states. Glasses: violate both periodicity (simplifying assumption of solid-state physics) and ergodicity (simplifying assumption of statistical mechanics). Simplest example of a glass, but very hard to study theoretically: glass of hard spheres or configurational glass. Other types include covalently bonded glasses (random bond networks) and disorder-driven glasses, which turn out to be simpler theoretically. The radial distribution function of a configurational glass is liquid-like, but has a shear modulus like a solid: instantaneous snapshot looks like a liquid, but time-domain study reveals no long-distance flow of a tracer particle....
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