Abstract
This chapter presents the Ising model for ferromagnetism, which is a standard simple model of a phase transition. Using the approximation of mean-field theory , the free energy is minimized, and hence the magnetization is calculated, as a function of temperature and applied field. This calculation demonstrates some fundamental concepts in statistical mechanics, including spontaneous symmetry breaking , an order parameter with magnitude and direction, first- and second-order phase transitions, and a critical point characterized by critical exponents . In the rest of the book, these concepts will be applied to the theory of soft matter.
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Notes
- 1.
The parameter J is sometimes called the “exchange constant,” for reasons based on the quantum mechanics of magnetism.
- 2.
To be precise, h is proportional to the magnetic field H, scaled by the magnetic moment \(\mu \) per spin; people often disregard the factor of \(\mu \) and refer to h as a field.
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Selinger, J.V. (2016). Ising Model for Ferromagnetism. In: Introduction to the Theory of Soft Matter. Soft and Biological Matter. Springer, Cham. https://doi.org/10.1007/978-3-319-21054-4_2
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DOI: https://doi.org/10.1007/978-3-319-21054-4_2
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-21053-7
Online ISBN: 978-3-319-21054-4
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