We report a determination of the effective critical exponents ${\beta }_{\mathrm{eff}}$ and ${\gamma }_{\mathrm{eff}}$ in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ as a function of the distance $T-T_c$ from the liquid-vapor critical point. These exponents characterize respectively the singular behavior of the densities for the coexisting phases ($T<\mathrm{}{T}_c$) and of the compressibility along the critical isochore for $T>\mathrm{}{T}_c$. The density measurements use the dielectric-constant method and cover the range $2\times {10}^{-1\mathrm{}}>\left|t\right|>3\mathrm{}\times {10}^{-5\mathrm{}}$ where $t=\frac{(T-T_c)}{T_c}$. Far away from $T_c$ we find ${\beta }_{\mathrm{eff}}=0.36\mathrm{}$, but as $T_c$ is approached, ${\beta }_{\mathrm{eff}}$ decreases progressively. Although the error becomes large for $t\lesssim 5\times {10}^{-4\mathrm{}}$, ${\beta }_{\mathrm{eff}}$ tends into the direction of the limiting value $\beta =0.32\mathrm{}$ that is predicted by recent theories. For $2\times {10}^{-2\mathrm{}}>t>\mathrm{}5\mathrm{}\times {10}^{-4\mathrm{}}$, we find ${\gamma }_{\mathrm{eff}}=1.19\mathrm{}\pm 0.01$. Both the coexistence curve and the compressibility are fitted to power-law series that include correction-to-scaling terms. The amplitudes of these terms for the coexistence curve are compared with those for Xe and S${\mathrm{F}}_6$. The slope of the rectilinear diameter is found to be $d\frac{\rho }{{\rho }_c}dt=-0.022\mathrm{}$. For the 80%-${}_{}{}^3{}_{}{}^{}\mathrm{He}$—20%-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixture above $T_c$, the singular density gradient in the earth's gravity field is found to diverge strongly as $T_c$ is approached. The relevant exponent is again ${\gamma }_{\mathrm{eff}}\simeq 1.18\pm 0.02$ and the amplitude of the singular term is intermediate between those for ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and for ${}_{}{}^4{}_{}{}^{}\mathrm{He}$. These results are in good agreement with those calculated from the Leung-Griffiths model for ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixtures.

• Received 21 May 1979

DOI:https://doi.org/10.1103/PhysRevB.20.3678