Phase Transitions and Scaling Theory

  • David A. Lavis
  • George M. Bell
Part of the Texts and Monographs in Physics book series (TMP)


Most physical systems can exist in a number of different phases, distinguished by their different types of molecular or atomic order. This order may be in the spatial configurations of one or more kinds of microsystems or it may be in the orientations or conformations of the microsystems themselves. (See Ziman (1979) for a discussion of the wide variety of possible types of order.) In the case of the vapour (gas), liquid and solid phases of, for example, water or hydrogen, the order is spatial with no order in the vapour, a short-range clustering type of order in the liquid and long-range lattice order in the solid. For water this is not the complete picture. There are at least nine different ice phases distinguished by their lattice structures and proton configurations (Volume 1, Appendix A.3, Eisenberg and Kauzmann 1969). The most well-known example of orientational order occurs in magnetic systems where ferromagnetism corresponds to the alignment of the magnetic dipoles of the microsystems. Although a simple magnetic system may possess just one type of ferromagnetic phase, more complex ferrimagnetic systems can have a large number of different magnetic phases. In the case, for example, of cerium antimonide fourteen different phases have been identified by neutron diffraction experiments (Fischer et al. 1978) and specific heat analysis (Rossat-Mignod et al. 1980).


Transition Region Critical Exponent Critical Curve Tricritical Point Coexistence Curve 
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  1. 2.
    This was first presented in his doctoral thesis (van der Waals 1873). For accounts of the way that it stimulated the development of the theory of phase transitions see Rigby (1970), Rowlinson (1973), de Boer (1974) and Levelt Sengers (1974).Google Scholar
  2. 3.
    As we shall see later in this section, this analysis applies both to first-order transitions and to those for which some derivative of the potential diverges in the transition region. The term ‘singular’ is taken, therefore, to include all types of non-smooth behaviour, including finite discontinuities in derivatives of the potential.Google Scholar
  3. 8.
    In this section the coupling κ has the opposite sign to that in Sect. 2.8 since it is give in terms of the antiferromagnetic interaction between sublattices.Google Scholar
  4. 9.
    This assumption is supported by mean-field calculations with the Hamiltonian (2.205) (see, for example, Volume 1, Sect. 4.4).Google Scholar
  5. 10.
    The situation of connected transition regions has been considered in Sect. 2.4.Google Scholar
  6. 11.
    For a review, see Binder (1983).Google Scholar
  7. 12.
    For a review see Barber (1983) and, for a collection of papers on finite-size scaling, Cardy (1988).Google Scholar
  8. 14.
    For more detailed discussions of conformai invariance the reader is referred to the review of Cardy (1987) and the book of Christe and Henkel (1993).Google Scholar
  9. 15.
    For a full explanation of this and all the following discussion see Cardy (1987).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • David A. Lavis
    • 1
  • George M. Bell
    • 1
  1. 1.Department of Mathematics, King’s CollegeUniversity of London StrandLondonUK

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