1. Introduction:

It was demonstrated in the previous chapter that different thermodynamic quantities become singular as . They exhibit either branch point singularity or diverging singularity. The order parameter continuously goes to zero as and exhibits a branch point singularity since it becomes a double valued function. The response functions and correlation length diverge and exhibit diverging singularity. Long range order appears in density-density or spin-spin correlation and it decays with power law. The singular behaviour of thermodynamic quantities around the critical temperature can be described by power series. The leading singularity of the power series in the limit are characterized by certain exponents called critical exponents. The power series for different thermodynamic quantities and the associated critical exponents will be described below.

2. Critical exponents:

The power series describing the thermodynamic quantities in the critical regime are usually expressed in terms of the reduced temperature . In terms of the reduced temperature, the power series of different thermodynamic quantities and the associated critical exponents are given below.

The order parameter, density difference in fluid and spontaneous magnetization M in ferromagnets, below are given as

and

respectively with . Note that below , t is negative. The exponent β describes the leading singularity of these quantities and is called the critical exponent of the order parameter.

The specific heats at constant volume V or constant magnetic field H below and above are given as

where and α are the specific heat exponents below and above .

The isothermal compressibility and isothermal susceptibility below and above are given by

where and γ are the compressibility or susceptibility exponents below and above .