Analysis of First,Second, and Fourth Sound Modes in a Helium 4 Superfluid

Christopher Spencer
UCLA Physics and Astronomy
Professor Gary Williams
11 June 2014


First,second,and fourth sound were successfully found and plotted. Graphs show a lack of steepness in decay as sound modes approach \(T_{\lambda}\), this could be due to refilling liquid helium later than recommended leaving less medium for the sound modes to propogate through. Scattering factor,n, was found to be n=1.239 ± .007 and porosity,P, was found to be 0.46 ± 0.02 close to the theoretical value of \(\approx 40 \%\) porosity.


Acoustics can be used to further investigate properties of material. For example the acoustics of sending sound waves in a cyldrical and rectangular geometry can be used to see if and how the speed of sound changes in air. In the same way, the properties of a liquid can investigated using acoustics. In quantum fluids, apparent attenuation and dispersion of sound occurs.Helium 4 is a bose liquid that exhibits these superfluid properties, which supports wave motions we can measure. In this experiment we want to investage sound modes in a superfluid helium 4 liquid called 1st,2nd, and 4th sound.


In this lab, the acoustic effects of superfluid helium 4 are measured. What is meant by saying superfluid is that a phase transition occurs in which a portion of the fluid is able to flow without friction or zero viscosity [1]. The superfluid component of liquid helium can then lead to new sound modes, ones that are investigated in this lab. The superfluid transition of helium 4 has a characteristic transition temperature called \(T_{\lambda}=2.172 K\) at saturated vapor pressure. The phase diagram of helium 4,shown in figure 1, and it shows that if the liquid helium can be cooled down to the \(\lambda\) line then superfluid transition can occur. This is done by pumping out vapor above the liquid which in turn lowers the temperature [1]. When pressure is applied to the liquid the \(T_{\lambda}\) is reduced to \(T_{\lambda}=1.81 K\).

Two-Fluid Model

Consider the liquid to be two fluids that are spread throughout the bulk liquid: a superfluid component with denisty \(\rho_{s}\) and normal fluid component with density \(\rho_{n}\). The sum of the two give the total measured density of the liquid \(\rho\).\(\rho_{s}\) is 0 at \(T_{\lambda}\),\(\frac{\rho_{s}}{\rho}=1\) at T=0 and the normal fluid has property that \(\frac{\rho_{n}}{\rho}=1\) at \(T_{\lambda}\) and then zero at T=0, the ratios are shown in figure 2 [1]. Since Helium 4 is bosonic, it has the property that a finite number of of helium 4 atoms falls into the lowest ground state, also known as bose condensation, giving the property that viscosity and entropy of the superfluid component are zero. The normal component being like a normal fluid has viscosity \(\eta\) and entropy \(s\). Both superfluid and normal components have their own independent velocity fields and these make up the net mass current \(J=\rho_s v_s + \rho_n v_n\). With these in hand, the continuity equation becomes \[\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho_sv_s+\rho_nv_n)=0\] Entropy is only carried by the normal fluid so conservation of entropy yields \[\frac{\partial \rho s}{\partial t} + \nabla \cdot (\rho s v_n)=0\] The superfluid component is accelerated by gradients in the chemical potential of the liquid [1]. The acceleration is given by \[\frac{ \partial v_s}{\partial t}=-\nabla \mu=-\frac{\nabla p}{\rho}+s\nabla T\] When taking thermal motions into account in a fluid, a pressure term comes into play and the net mass current is accelerated by the gradient in this pressure and is related by \[\frac{ \partial}{\partial t} (\rho_sv_s+\rho_nv_n)=\nabla p\] Further linearizing the density, pressure, and temperature give rise to phase velocities of two sound modes \[\left( 1-\frac{c^2}{c_1^2}\right) \left(1-\frac{c^2}{c_2^2}\right)=1-\frac{C_p}{C_v}\] \(C_p=T\frac{\partial s}{\partial T}_p\) and \(C_v=T\frac{\partial s}{\partial T}_v\) are specific heats. \(c_1\) and \(c_2\) are the phase velocities of first and second sound respectively given by \(c_1^2=\frac{\partial p}{\partial \rho}_s\) and \(c_2^2=\frac{\rho_s T s^2}{\rho_n C_p}\) and these are two of the sounds investigated in the lab.