### 3.H.1. Definition Of Critical Exponents

The "distance" from a critical point is usually measured in terms of the reduced parameter

ε= T T C T C                                (3.80)

The critical exponent λ of a thermodynamic function f is defined as

λ= lim ε0 ln| f( ε ) | ln| ε |                       (3.82)

which implies f can be written in the form

f( ε )=A ε λ ( 1+B ε y + )                y>0                 (3.81)

since

lnfλlnε            as ε0

Thus, as ε0 ,

1.         λ<0         Þ    | f |

2.         λ>0         Þ    | f |0

3.         λ=0 .  A modified exponent λ' is defined as

λ'=j+ lim ε0 ln| f ( j ) ( ε ) | ln| ε |                                    (3.83)

where j is the smallest integer such that d j f d ε j = f ( j )  diverges.

For example, if   f( ε )=A+B ε y , we have

λ= lim ε0 ln| A | ln| ε | = lim x ln| A | x =0

If y is a positive fraction, we have j=1  and f'=By ε y1  so that

λ'=1+ lim ε0 ln | ε | y1 ln| ε | =y

Another example is the logarithmic divergence   f( ε )=Alnε+B   which gives

λ= lim ε0 ln| lnε | ln| ε | = lim x lnx x = lim x x 1 1 =0

where we've used the L'Hospital rule.  Since f'= A ε , we have j=1  and

λ'=1+ lim ε0 ln | ε | 1 ln| ε | =0

Typical plots of f vs ε can be found in Fig.3.26.