Attenuation and dispersion of first sound near the superfluid transition ofHe3-He4mixtures

The attenuation $\alpha$, the velocity $u$, and the dispersion $D=u\left(\omega \right)-u\left(0\right)\mathrm{}$ of first sound have been determined in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixtures (molar ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ concentrations $X_3=0, 0.070, 0.194, 0.377, \mathrm{and} 0.517$) at frequencies $2.3\le \frac{\omega }{2\pi }\le 627$ kHz, and in the temperature range 1 μK ≤ $|T-T_{\lambda }|$ ≤ 10 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 are considerably reduced when the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ concentration is increased. 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 decreases nearly three orders of magnitude with increasing $X_3$ in our concentration range. Using the obtained relaxation time ${\tau }^{\prime }=\tau _{}^{\prime }{}_0{}^{}{t}^{-x^{\prime }}$ (with $t=\frac{|T-T_{\lambda }|}{T_{\lambda }}$), and published data for the correlation length ${\xi }^{\prime }$, and for the second-sound velocity $u_2$, we find ${\tau }^{\prime }=\frac{{\xi }^{\prime }}{u_2}$ for $T<\mathrm{}{T}_{\lambda }$. The amplitude