Understanding Superfluid $^{3}$He by Determining \textit{$\beta $}-Coefficients of Ginzburg-Landau Theory

The Ginzburg-Landau (GL) theory is a phenomenological theory that is used to characterize thermodynamic properties of a system near a phase transition. The free energy is expressed as an expansion of the order parameter and for superfluid $^{3}$He there is one second order term and five fourth order terms. Since the GL theory is a phenomenological theory, one can determine the coefficients to these terms empirically; however, existing experiments are unable to determine all five fourth order coefficients, the \textit{$\beta $}'s. To date, only four different combinations of \textit{$\beta $}'s are known [1]. In the case of supeprfluid $^{3}$He, using quasiclassical theory, the coefficients have been calculated [2]. We used the calculation as a guide to construct a model to define all five \textit{$\beta $}'s independently. The model provides us with the full understanding of the GL theory for $^{3}$He, which is useful in understanding various superfluid phases of both bulk $^{3}$He and disordered $^{3}$He in aerogel. \newline [1] H. Choi \textit{et al}., J. Low Temp. Phys., submitted; http://arxiv.org/abs/cond-mat/0606786. \newline [2] J.A. Sauls and J.W. Serene, Phys. Rev. B \textbf{24}, 183 (1981).