Pressure-Density-Temperature Surface ofHe4near the Critical Point

The density $\rho$ of ${\mathrm{He}}^4$ as a function of temperature and pressure has been measured along several different paths in the vicinity of the critical point. The results of the measurements are interpreted in terms of power-law descriptions near the critical point: For the coexistence curve, $|\rho -{\rho }_{\mathrm{c}}|\propto {(T_{\mathrm{c}}-T)}^{\beta }$; for the critical isotherm, $|P-P_c|\propto {|\rho -{\rho }_c|}^{\delta }$; for the critical isobar, $|T-T_c|\propto {|\rho -{\rho }_c|}^{\pi }$; for the isothermal compressibility along the coexistence curve, $\left(\frac{1}{\rho }\right){\left(\frac{\partial \rho }{\partial P}\right)}_T\propto {|T-T_c|}^{-{\gamma }^{\prime }T}$; and for the thermal expansion coefficient along the coexistence curve, $\left(\frac{1}{\rho }\right){\left(\frac{\partial \rho }{\partial T}\right)}_P\propto {|P-P_c|}^{-{\gamma }^{\prime }P}$. The subscript $c$ identifies the value of a quantity at the critical point. It is shown why simple power-law descriptions might be inadequate to describe all the behavior at the critical point and how this problem can be partly solved by the use of additional, non-singular factors. The values of the exponents are found to be $\beta =0.354\mathrm{}\pm 0.010$, $3.8\le \delta \le 4.1$, $3.8\le \pi \le 4.2$, ${{\gamma }^{\prime }}_T=1.1\mathrm{}\pm 0.1$, and ${{\gamma }^{\prime }}_P=1.5\mathrm{}\pm 0.2$. Combined with Moldover's specific-heat measurements, which give for the exponent ${\alpha }^{\prime }$ of the expression $C_{\mathrm{V}}\propto {|T-T_c|}^{-{\alpha }^{\prime }}$ a value of ${\alpha }^{\prime }=0.017\mathrm{}\pm 0.008$, we can test various inequalities proposed for the exponents. For Rushbrooke's and Fisher's inequality, ${{\gamma }^{\prime }}_T+2\mathrm{}\beta +{\alpha }^{\prime }\ge 2$, we obtain 2.0 for the maximum value of the left side; for Griffiths's inequality, $\frac{{{\gamma }^{\prime }}_T(\delta +1)}{(\delta -1)}+{\alpha }^{\prime }\ge 2$, we obtain 2.2 for the maximum value of the left side; but for Griffiths's inequality, $\beta (\delta +1)+{\alpha }^{\prime }\ge 2$, we obtain only 1.9 for the maximum value of the left side. Possible reasons for this discrepancy are discussed.