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Attenuation of Second Sound in Superfluid 3He-A1
This experiment aimed at measuring the attenuation of spin-entropy (second sound) wave in the superfluid A1 phase of 3He. The measured attenuation is proportional to the square of frequency as expected for bulk, and the extracted attenuation coefficient is compared with existing theories and dissipative coefficients. Theory
In the low frequency limit, the attenuation of the second sound wave mode may be expressed by
There are seven dissipative coefficients. From the normal fluid hydrodynamics are the spin diffusion coefficient D, the shear viscosity h, the bulk viscosity z2, and thermal conductivity k. The viscosity coefficients z1 and z3 enter as “friction” coefficients in response to the gradients of the normal component velocity and of the relative velocity between superfluid and normal components, respectively. The coefficient z4, unique to A1 hydrodynamics, comes in as the dissipative spin current response to the temperature gradients. Method
We detect the resonances of second sound in a cylindrical tube (r = 3.8 mm and l = 12.5 mm), with oscillating superleak transducers placed at the two ends. The flexible superleak membranes (both 1 µm and 3 µm pore sizes) of which the transducers are made are clamped at the inner wall with a tension ~ 2 x 104 dyn/cm at room temperature. A small vibrating wire viscometer placed inside or outside of this cell is used for monitoring the temperature and the fluid viscosity. In the middle of the cell wall are one or two 1 mm diameter fluid inlets, which also provide thermal contact. The cell is filled with liquid 3He at 22.9 bars, and cooled through adiabatic demagnetization. The axis of the cell is aligned with a solenoid superconducting magnet. Magnetic fields up to 150 kOe may be applied. The magnetic field produces A1 phase temperature widths, Tc1 – Tc2, up to 0.78 mK. As the temperature drifts up at ~ 30 µK/h, the frequency response spectra of different modes are recorded. The resulting spectra are the superposition of positive and negative going plane waves with the wave vector k = ± (2pf/c + ia) within the cell. The solution to the wave equation is used to fit the spectra to obtain the attenuation coefficient a and the resonance frequency f. (Figure 1)
Results
Extracting the bulk attenuation ab requires subtracting the surface viscous loss of the normal fluid component occurring at the cell wall. (Figure 2) Note the quadratic dependence of the bulk attenuation on the frequency, as expected from theory, Equation (1).
The least square fit, ab = bf2, to the bulk attenuation at each reduced temperature r gives proportionality constant b, which is compared to the sum of the dissipative coefficients in Equation (1). When plotted against the temperature t = 1 – T / Tc1, redefined so that Tc1 – Tc = 32H (µK/T) and Tc1 – Tc2 = 52 H (µK/T), and Tc = 2.315 mK, the magnetic field dependence disappears. (Figure 3)
The terms involving the spin diffusion coefficient D, the viscosity coefficient z3, and the shear viscosity h in Equation (1) contribute significantly to the bulk attenuation. The h and z3 terms may be estimated reasonably from our measurement and from the previous calculation done for the B phase, respectively. The spin diffusion coefficient D, although treated in theory for the B phase, has not been calculated theoretically for the A1 phase, and therefore an estimate from the normal state (= 0.25/T2 cm2 mK2 /s) with T = Tc1 is used. The measured attenuation is greater than the theoretical value estimated in this manner. If the excess attenuation is attributed to the spin diffusion, one can extract the spin diffusion coefficient. The spin diffusion coefficient extracted in this manner is shown in Figure 4. The estimates of the viscosity coefficient z3 and the shear viscosity h can be subtracted from our b measurements, and extract the difference as the measured spin diffusion coefficient.
This represents the first measure of the spin diffusion coefficient in any of the superfluid phases of 3He. From "Attenuation of Second Sound in Superfluid 3He-A1,"
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