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The clusters in a d = 2 diluted system with 50% dilution. [From Birgeneau et ~ f . ~ ] 18. 43 PHASE TRANSITIONS only isolated clusters, and in addition shows that the larger clusters are very ramified, that is, there are a large number of one-dimensional links between the distant parts of the cluster. At the percolation point, x = x,,, T = 0, it is possible to destroy the long-range order, either geometrically by decreasing x or thermally by raising the temperature. A study of percolation is then the study of the behavior of the long-range order and critical fluctuations as a function of temperature and concentration around the percolation point.
In the case of three-dimensional systems with isotropic symmetry, as discussed in the previous section, the transition is believed to be similar to that in short-range systems but with slightly different exponents. The uncertainty is that real systems are cubic rather than isotropic, and the role of the cubic anisotropy with dipolar interactions is still not known. The behavior of uniaxial dipolar systems is well understood,52and the dipolar interactions have a large effect on the critical fluctuations and exponents.
These results were used to obtain the exponents as listed in Table I. The treatment of the transverse scattering is clearly somewhat ad hoe in this approach. 1, 26 R . A . COWLEY 0 OlI 1 I i I I i I i I L LLI . ii. U. 1 4 i- it is most probably adequate, and as indicated in Table I the exponents are those of a d = 2 Ising model. The transverse fluctuations have, however, had a large effect in decreasing the amplitude KO’ [Eq. 14)l by a factor of 4 in K2NiF4and a factor of 2 in KZMnF4, as shown in Fig.