Exploring Gap Exponents and Their Significance

[matematikuvol]

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

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

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

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
$$(\frac{\partial G}{\partial H})_T \equiv G^{(1)}\propto (1-\frac{T}{T_c})^{-\Delta'_1}G^{(0)}$$
$$(\frac{\partial G^{2}}{\partial H^{2}})_T \equiv G^{(2)}\propto (1-\frac{T}{T_c})^{-\Delta'_2}G^{(1)}$$
$$...$$
$$(\frac{\partial G^{l}}{\partial H^{l}})_T \equiv G^{(l)}\propto (1-\frac{T}{T_c})^{-\Delta'_l}G^{(l-1)}$$

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