Dimensional crossover in dilute magnets

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Abstract
A discussion is given of the critical curves (transition temperature Tc versus concentration) of diluted 'low dimensional' magnets (layer magnets, and chain magnets). A model based on real-space renormalisation-group calculations and certain exact considerations is constructed for the Ising case, and analysed. Resulting critical curves are given for various exchange ratios for various types of layer and chain ferro- and antiferromagnet. For the case of an Ising layer magnet, as the magnetic concentration is reduced towards the two-dimensional percolation concentration, KTc falls until it becomes comparable with the weak exchange between layers. This weak exchange raises the effective dimensionality to three so that the critical curve flattens off and heads towards the smaller limiting concentration for the three-dimensional system. For a quasi-one-dimensional (chain) Ising magnet, KTc is only non-zero by virtue of the interchain coupling. Dilution causes KTc to fall very rapidly to values smaller by a factor roughly equal to the ratio of interchain to intrachain couplings, and eventually to vanish at the percolation concentration for the higher dimensionality corresponding to coupled chains.