ROUGH DRAFT authorea.com/7698

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

Abstract

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.

# Introduction

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.

# Theory

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.