Magnetic birefringence near the lyotropic isotropic-nematic Landau point

The influence of an isolated critical point (Landau point) on the first-order lyotropic isotropicnematic transition lines is studied with regard to magnetic birefringence $\Delta n$ in the disordered (isotropic) phase. Near the Landau point $\Delta n\sim {\chi }_{Q}h[1+fh+O(h^2\left)\right]$, where ${\chi }_Q$ is the zero field ($h=0\mathrm{}$) order-parameter susceptibility and $f=\frac{1}{2}\left[\frac{\partial \left(\ln {\chi }_Q\right)}{\partial h}\right]$ is the $h=0\mathrm{}$ "curvature" correction. Critical exponents and universal scaling functions for ${\chi }_Q$ and $f$ are obtained through renormalization-group $\epsilon$-expansion methods ($d=4-\epsilon$). It is suggested that magnetic birefringence experiments in the pretransitional region be analyzed in terms of the effective exponents for the susceptibility ${\gamma }_{\mathrm{eff}}$ and curvature ${\left({\phi }_f\right)}_{\mathrm{eff}}$. To $O\left(\epsilon \right)$, the results are ${\gamma }_{\mathrm{eff}}=1+\left[\frac{7\epsilon }{26(1+Y)\mathrm{}}\right]$ and ${\left({\phi }_f\right)}_{\mathrm{eff}}=2+\left[\frac{4\epsilon }{13(1+Y)\mathrm{}}\right]$ where $Y=g_4{g}_r^{\left|{\phi }_4\right|}$; $g_r$ is the thermal scaling field, ${\phi }_4=\frac{-\epsilon }{2}+O\left({\epsilon }^2\right)$ is the universal correction to scaling exponent, and $g_4$ is a nonuniversal amplitude ("slow transient" scaling field).