Attenuation and dispersion of first sound near the superfluid transition of pressurizedHe4

The attenuation $\alpha$, the velocity $u$, and the dispersion $D=u\left(\omega \right)-u\left(0\right)\mathrm{}$ of first sound have been measured in pressurized liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ ($P=0.06, 5.01, 9.21, 15.24, 20.38, 25.46, 29.33$ bar) near the superfluid transition. The frequency range was $4.6 \mathrm{kHz}\le \frac{\omega }{2\pi }\le 1.0 \mathrm{MHz}$, the temperature range was $1 \mu \mathrm{K}\le |T-T_{\lambda }|\le 3 \mathrm{mK}$. From the measured velocities we calculate the thermodynamic velocity $u\left(0\right)\mathrm{}$, as well as ${\left(\frac{\partial S}{\partial P}\right)}_{\lambda }$ and ${\left(\frac{\partial V}{\partial P}\right)}_{\lambda }$. The attenuation and the dispersion at constant $T_{\lambda }-T$ are only weakly pressure dependent. They are interpreted as arising from a relaxation process occurring only below $T_{\lambda }$, and a fluctuation process occurring on both sides of the $\lambda$ transition; both contributions have about equal strength. The strength $A_R$ of the relaxation process and the amplitude ${\tau }_0^{\prime }$ of the relaxation time ${\tau }^{\prime }={\tau }_0^{\prime }{t}^{-x^{\prime }}$ are independent of pressure to within 10%; ($t=\frac{|T-T_{\lambda }|}{T_{\lambda }}$). The latter result seems to be inconsistent with the pressure independence of the correlation-length amplitude ${\xi }_0^{\prime }=1.0\mathrm{}\pm 0.05$ Å (which was confirmed in this work), and the known pressure dependence of the amplitude $u_{2,0}$ of second-sound velocity, if the relation ${\tau }^{\prime }=\frac{{\xi }^{\prime }}{u_2}$ is correct. For $T>\mathrm{}{T}_{\lambda }$, where only critical fluctuations contribute, our absorption and dispersion data for all $\omega$ and $P$ can be scaled with functions of $\omega \tau$ for ${10}^2>\omega \tau >{10}^{-2\mathrm{}}$. This scaling analysis shows that the time $\tau$ characterizing the critical order-parameter fluctuations at $T>\mathrm{}{T}_{\lambda }$ has the same temperature and pressure dependence as the relaxation time ${\tau }^{\prime }$ at $T<\mathrm{}{T}_{\lambda }$; these two times differ at most by a constant multiplicative factor. Below $T_{\lambda }$, the data are represented by the sum of the contribution represented by the scaling function plus the contribution from order-parameter relaxation. The weak pressure dependence of $\alpha$ and $D$, and the pressure independence of $A_R$, ${\xi }_0^{\prime }$, ${\tau }_0^{\prime }$, and ${\tau }_0$ contrasts with the strong concentration dependence of these quantities in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixtures. The scaling functions determined from our data are identical in form to those determined earlier from the mixture data. The frequency dependences of the attenuation and of the dispersion for $\omega \tau \gtrsim 1$ scale as $\alpha \propto {\omega }^{1+y}$ and $D\propto {\omega }^y$, respectively, with $y=0.15\mathrm{}\pm 0.03$.