8 - Critical phenomena in systems with disorder

Summary

Harris criterion

In studies of the phase-transition phenomena, the systems considered before were assumed to be perfectly homogeneous. In real physical systems, however, some defects or impurities are always present. Therefore, it is natural to consider what effect impurities might have on the phase-transition phenomena. As we have seen in the previous chapter, the thermodynamics of the second-order phase transition is dominated by large-scale fluctuations. The dominant scale, or the correlation length, R_c ∼ |T/T_C – l|^–v grows as T approaches the critical temperature T_c, where it becomes infinite. The large-scale fluctuations lead to singularities in the thermodynamical functions as |τ| ≡ |T/T_c – 1| → 0. These singularities are the main subject of the theory.

If the concentration of impurities is small, their effect on the critical behavior remains negligible so long as R_c is not too large, i.e. for T not too close to T_c. In this regime the critical behavior will be essentially the same as in the perfect system. However, as |τ| → 0 (T → T_c) and R_c becomes larger than the average distance between impurities, their influence can become crucial.

As T_c is approached the following change of length scale takes place. First, the correlation length of the fluctuations becomes much larger than the lattice spacing, and the system ‘forgets’ about the lattice. The only relevant scale that remains in the system in this regime is the correlation length R_c(τ).