Gap exponents

matematikuvol

Gap exponents are denoted like critical exponents for higher derivatives of Gibbs potential.
[tex]\Delta_l'[/tex]
[tex](\frac{\partial G}{\partial H})_T=G^{(1)}\propto (1-\frac{T}{T_c})^{-\Delta_1'}G^{0}[/tex]

[tex](\frac{\partial^l G}{\partial H^l})_T=G^{(1)}\propto (1-\frac{T}{T_c})^{-\Delta_l'}G^{l-1}[/tex]

[tex]\alpha'[/tex] is critical exponent for heat capacity. People used that
[tex]G^{0}\propto (1-\frac{T}{T_c})^{2-\alpha'}[/tex]

How to get that? Why gap exponents are important?

matematikuvol

Because I still didn't find answer for my question I will write here more details.
So ##G(T,H)## is Gibbs thermodynamics potential. Derivatives of Gibbs potential are defined by
[tex](\frac{\partial G}{\partial H})_T \equiv G^{(1)}\propto (1-\frac{T}{T_c})^{-\Delta'_1}G^{(0)}[/tex]
[tex](\frac{\partial G^{2}}{\partial H^{2}})_T \equiv G^{(2)}\propto (1-\frac{T}{T_c})^{-\Delta'_2}G^{(1)}[/tex]
[tex]...[/tex]
[tex](\frac{\partial G^{l}}{\partial H^{l}})_T \equiv G^{(l)}\propto (1-\frac{T}{T_c})^{-\Delta'_l}G^{(l-1)}[/tex]

Quontities ##\Delta'_l## are called gap exponents. In eqns ##H=0, T\rightarrow T^{+}_c##.
[tex]G^{(1)} \propto M \propto (1-\frac{T}{T_c})^{\beta}[/tex]
but that is for ##H=0, T\rightarrow T_c^{-}##. That is first conceptual problem. ##T\rightarrow T_c^{+}## or ##T\rightarrow T_c^{-}##.