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whose detailed nature we need not specify. In other words, we assume that translational invariance is spontaneously broken. Let u( x) be ad-component displacement vector, which gives the deviation of the atoms in a volume element at x from their supposed equilibrium position. In the harmonic approximation we can decompose u( x) into normal modes:

u(x) = -

e ikox q;{k)

(16.60)

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where i labels various types of normal modes (e.g., longitudinal and transverse vibrations). The sum over k is cut off at the upper limit at some characteristic inverse length. The energy residing in a normal mode is given by

The picture is added to the file and the Picture tools appear on the Format tab. You may need to resize or reposition the picture to fit the space.

(16.61)

whose average value is kT, by the equipartition of energy. Hence the mean-square amplitude of a normal mode is given by

(16.62)

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and the mean-square total displacement is, in the infinite-volume limit,

(16.63)

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Since by assumption a continuous symmetry has been broken spontaneously, there must exist among the normal modes Goldstone excitations whose frequencies vanish as k --+ 0. Assuming w;( k) to be regular at k = 0, we expect wi(k) --+ cik, where ci is a constant. Thus the integral (16.63) behaves near the lower limit like

one made a crystal in dimensions d :s; 2, the spatial order will be destroyed by long wavelength density fluctuations. Similarly, spontaneous magnetization cannot happen in continuous media in dimensions d:s; 2, because the magnetic order would have been destroyed by spin waves. For such systems d = 2 is the "lower critical dimension." In the marginal case d = 2, the integral in (16.63) diverges only logarithmically. This means that, although the correlation length for long-ranged order is not infinite, as required of a theoretical crystal, it is large enough to be of macroscopic size. Thus, in practice, we should be able to make two-dimensional crystals of finite but macroscopic size.

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Another approach to the problem is to consider whether long-ranged order is stable against the formation of domain walls, which are boundary layers between regions with different values of the order parameter. Let us discuss this in the context of a magnetic system. * The argument is a generalization of that given in 14 to show that there can be no phase transition in the Ising model in d = l. To minimize the free energy U - TS, we want the internal energy U to be small, and the entropy S large. (We consider only T> 0.) Entropy considerations always favor the creation of domains. Hence the key is the energy cost of one domain. Suppose the system is completely magnetized except for one domain of linear dimension L, in which the magnetization points in the opposite direction. In the case of an Ising model, the spins are either up or down. Hence the energy cost of the domain is proportional to the number of spins that interact across the domain surface, and thus to the surface area L d - 1 For d :$ 1, the energy cost is nonmacroscopic. Creation of N domains will cost energy of order N, but the N log N. Hence lots of domains will be entropy will be of order log N! created, destroying the long-ranged order. The lower critical dimension for Ising models is therefore d = 1, in agreement with our findings in 14. F or a system with continuous order parameter m (x), the domain wall is of finite thickness, in which m(x) changes continuously from the value inside the domain to that outside. A crude estimate of the energy cost of the domain wall is

(Volume of wall)

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(16.64)

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