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From previous section we learned that:

- 1.
- Free energy
*A*is an*analytical*function of magnetization*M* - 2.
- At
*T*>*T*_{c}it has one minimum, at*T*<*T*_{c}--two

Landau idea: let us use *only* these facts!

Let us expand free energy *A* in powers of *M*.

First, a *symmetric* system: *H*=0. Then *M*=0 is
either a minimum (*T*>*T*_{c}) or a maximum (*T*<*T*_{c}) of the free
energy. Due to symmetry only *even powers* are present:

**Assumption:**- We neglect all powers higher than 4.

(9) |

Differentiate (9):

Solutions:(10) |

Equation of binodal is (10):

(11) |

(12) |

(13) |

Entropy is

Differentiating (13): Specific heat:
There is a *jump* in specific heat near *T*_{c}:

Now we add a small *external field* *H*:

(14) |

(15) |

Two cases:

- 1.
- If
*T*>*T*_{c}, then*M*(*H*=0)=0. For small*H*we neglect*M*, and Susceptibility This^{3}*diverges*at . - 2.
- If
*T*<*T*_{c}, then with . Let Expanding (15), we obtain and This also diverges

We obtained a -like curve:

© 1997

Tue Oct 14 22:58:59 EDT 1997