Journal of Low Temperature Physics, Vot 101, Nos; 3,/4, 1995

H y d r o d y n a m i c Propert ies on Superfluid 3He in High Magnet ic Fields up to 12 Tesla

Hikota Akimoto , Tetsuji Okuda, and tt idehiko Ish imoto

fnstitute for Solid State Physics, University o/Tokyo Roppongi Miuato-ku Tokyo 106 Japan

Using a torsional oscillator, superfluid fraction and viscosity of superfluid 3He were measured in the A1 and the A2 phase up to 12 Tesla below 1 mK in the pressure of 29 bar. This is the first measurement in the high magnetic field region, in a well defined geometry and in a controlled texture. The A1 and the A2 transition were seen clearly in the torsional oscillator frequency and amplitude. Two GL parameters, /~245 and fl24, are independent of field up to 12 Tesla. fl245 is 1.18 from the A2 phase measurements, which agree with those from heat capacity measurement. While/~24 is 3.08 from the A1 phase, which is smaller than that from recent spin entropy wave measurement.

PACS numbers: 67.57.-z, 67.57.Bc, 67.57.De

1. I N T R O D U C T I O N

In the high magnetic field, there appears a peculiar superfluid 3He phase, A]. The existing temperature region of the phase is narrow (60 #K/Tesla at 29 bar) and experimental investigations are very few except a phase dia- gram. Hydrodynamic properties of 3I-Ie in high magnetic fields were mainly measured by the vibrating wire experiments so far. The qualitative hydro- dynamic properties were welI measured by Leiden group. 1 They found a viscosity minimum in the A1 phase above a certain magnetic field and an anomaly near A2 transition in the A2 phase. Although the origin of the anomaly is still unclear, it is supposed to be a texture transition and to be identical to those of deVegvar et al. 2 The A1 and the A2 phase are anisotropic and if we want to determine any parameters, it is desirable to measure un- der a well defined texture. However it is hard to control the texture and

7~.1

0022-2291/95/1100-0721507.50/0 �9 1995 Plenum Publishing Corporation

722 H. Akimoto~ T. Okuda, and H. Ishimoto

sometimes a texture transition may occur with vibrating wire measurement. Recently Bastea et al. 3 reported anomalous spin entropy wave (second

sound) attenuation near the TA2 in the A1 phase and they attributed the increasing of sound attenuation to the presence of a minority condensate population. They also deduced the superfiuid density in the AI phase~ 4 which is smaller than the estimated value by Tang et al. 5

A torsional oscillator is a useful tool to measure a superfluid density and also a viscosity in liquid 311e, and we can control the l texture easily with a

proper boundary condition and a direction of the magnetic field. We report here both superfluid fraction and viscosity with a torsional oscillator both in the AI and the A2 phase in the controlled I texture. This experiments give both thermodynamic and kinetic properties of superfiuid 3He.

2. E X P E R I M E N T A L

An experimental set up is the same as that in our previous work on solid 3He.6 Now the maximum field of second stage magnet was increased up to 12 Tesla. The measurement at high magnetic field and in the low tempera- ture region includes a lot of difficulties in refrigeration and also thermometry. For example a silver material, which is usually used for a sample cell, has a nuclear specific heat peak at 0.4 mK in the field of 12 Tesla. It is not so easy

to cool the sample cell across the peak. Therefore we chose titanium for the main part of the sample cell except the cap. Even a heat exchanger, made of a composition of platinum and silver powder (total surface area 180 m 2) was placed in the lower field region than the maximum field as shown in Fig.l.

In this way, the total nuclear heat capacity in the field region was reduced by a factor of five compared with that for the cell made of silver only at the lowest temperature.

Thermometry is another problem to be solved. Here thermal link be-

tween the sample cell and the first nuclear stage was designed to have as a high thermal conductance as possible. So the sample temperature can be measured with a Pt NMR thermometer located in the low field region. At the same time, vibrating wires in the liquid 3He also worked as an indicator to know the temperature of the liquid itself. Sample volume is about 15 cc and a liquid pressure of 29 bar was controlled within 5P/P ~_ 10 -4 by use of a feed back circuit.

Using a torsional oscillator in a high magnetic field causes problems due to an eddy current loss in its metallic parts and t.o the magnetic heat capacity. We developed a new torsional oscillator for this experiment. A beryllium copper alloy(BeCu 25) was used as a torsion rod for its reliable mechanical property and a quartz glass was for the head to decrease its own

Superfluid 3He in High Magnetic Fields

J of copper s t a g e

723

~rmometer

front view

cross view

F--

.=o o

-u {D

q=

._o

r 1 0 5 O3

E

hal link )0

rn case

~ilver and platinum [surface area 180 m z )

He(15 cc)

I wire viscometers

10 crn

oscillator

Fig. 1. Schematics of experimental space below nuclear stage.

heat capacity. Details will be presented elsewhere. 7 To get an appropriate I texture of superfluid 3He, the liquid was put in an annular flow channel with 100 #m width and 8 mm height. The magnetic field was applied in parallel with the oscillation axis. In this geometry, I vector is well defined and is perpendicular to a flow vector k, so the measured superfluid fraction tensor and viscosity correspond to Psi and ~• respectively. The torsional oscillation was driven and detected by capacitive electrodes. The quality factor of the torsional oscillator itself was about 5 • 104 at 12 Tesla and an accuracy of 6(Pn/P) is l x 10 -3. Here p~ and p are normal and total density, respectively. A computer controlled phase lock loop(PLL) circuit tracked the resonance frequency and kept the oscillation amplitude constant by changing the excitation voltage. The excitation frequency and driving voltage were monitored.

During cooling down, the thermal time constant above 3 mK was dom-

724 H. Akimoto , T. Okuda, and H. Ishimoto

inated by a heat capacity of the liquid in the torsion head and a thermal conductivity of liquid in the rod. Below 1 mK, it wa s determined by the nuclear heat capacity of the metal container and the thermal conductivity of the silver thermal l ink. It took about two hours to get an equiLibrium for a small temperature change between the transition temperature (2.5 inK) and 1 mK.

3. R E S U L T S a n d D I S C U S S I O N S

For temperatures below a few millikelvin, the viscous penetration depth is much larger than the width of Liquid 3He, and therefore the normal com- ponent is locked to the wall. Raw data of resonance frequency ( f (T ) ) and driving voltage are converted to the normal fluid fraction by using following relation, s

Pn.l_(T) _ {1 f ( T ) - f (Tc ) }. (1 - ~). p f(O) - f (Tc )

Here Tc is the superfluid transition temperature, f(0) - f (Tc) corresponds to total frequency shift due to the liquid, which was evaluated from low magnetic field measurement in the B phase, where we can cool the liquid sufficiently low to determine the frequency f(0) at zero temperature. 9 ~ is a small correction factor ( < 0.07 ) due to imperfect lock of the normal fluid component and depends on the frequency and the driving -voltage.

The normal fluid fraction and the viscosity at 12 Tesla are shown in Fig.2 in the whole temperature region. We can see two superfluid transitions A1 and A2 points indicated by arrows. As the temperature goes to zero, the normal component and viscosity seem to vanish. In the A! phase, the viscosity has a minimum, which was reported previously 1 and it decreases monotonously below TA2. In both the superfluid density a~ld the viscosity, we did not see any anomalous behaviors near TA2 reported in vibrating wire measurement 1 and in spin entropy wave measurement. 3

The bare normal fluid fraction (pn~ is given by

f in ~ i _ Phi I

P P 1 + -~- �9 ( 1 - P"• ' p

where F1 ~ is a Landau parameter and we employ the value given by Greywall. 1~ The result near the transition temperature is shown for five magnetic fields in Fig.3. "Appearance of the A1 phase and the magnetic field dependence are seen dearly. The splitting of the A1 phase is proportional to the magnetic field up to 12 Tesla and the superfluid fraction increases with decreasing temperature. The initial slope of (pn~ against reduced temperature (T/TA1) near A1 point is 1.30 independent of the magnetic

Superfluid aHe in High Magnetic Fields 725

e"

.9

m

O e-

1 . 2 . . . . i . . . . i . . . . i . . . . i . . . . i . . . .

A 1 p o i n t

A~ point

o o �9

o

o ~ e �9 � 9 1 4 9 1 4 9

~

l*

0 . 8

0 . 6

0 . 4

0 . 2

O

Q O

Q

O

O O

o

. . . . . I . . . . I , , l l l l l l t l l l l l

0.5 1 1.5 2 2.5

Temperature(inK)

1.2

<

O ).8 u~

N:

0 0 .6 -~

3

0.4 ~e~

0.2

Fig. 2. The normal fluid fraction (o) and the viscosity (o) at 29 bar and 12 Tesla in the whole temperature region.

field, which gives Ginzburg-Landau(GL) parameter/~24 = 3.08. The value is smaller than from a recent spin entropy wave measurement by Bastea et al, 4

but is in agreement with the extrapolated value of the pressure dependence reported by Tang et al. 5 Below the A2 transition, the increasing rate of su- perfluid fraction is about three times larger than in the AI phase. In the A2 phase, the values of pn~ are on an universal function, which gives the field independent GL parameter ~245 = 1.18, which agrees with those from the heat capacity measurement by Greywall I~ and the early superfluid fraction measurement by Hook et al. II

Now we get ~5 = -1.9, whose deviation from the weak coupling value (-2) is positive. This result is inconsistent with that of Bastea et al. 4 Main difference between our results and theirs is the value of the superfluid fraction at the A2 point. In the spin entropy wave measurement, the signal was detected only in the AI phase and disappeared at the A2 point. On the other hand, in this experiment, the superfluid density could be determined both in the AI phase and A2 phase, and the superfiuid fraction Ps/P can be obtained correctly from a total frequency shift due to the liquid. So our measurements have less ambiguity to specify the A2 transition and to determine the superfluid fraction at the A2 point. Further investigations are now under way to know the pressure dependence of these values.

726 H. Akimoto, T. Okuda, and H. Ishimoto

N,,,.

O

1

0.9

0.8

0.7

2

A1

T~ 0

�9 o A O z~

O az~ +/; A z "r

;+ ~e

a ~

29 bar 12 Tesla

o 9 Tesla

�9 6 Tesla

a 3 Tesla

�9 0.6 Tesla

2.2 2.4 2.6 2,8

Temperature(mK)

Fig. 3. The bare normal fraction near the transition temperatures for five magnetic fields.

R E F E R E N C E S

1. T.Hata, S.A.J.Wiegers, R.Jochemsen, and G.Frossati, Phys.Rev.Lett. 63, 2745 (1989).

2. P.G.N.deVegvar, R.Movshovich, E.L.Ziercher, and D.M.Lee, Phys.Rev.Lett. 57, 1028 (1986).

3. M.Bastea, Y.Okuda, V.LaBella, and H.Kojima, Phys.l~ev.Lett 73, 1126 (1994). 4. M.Bastea, Y.Okuda, and H.Kojima, Phys.Rev.Lett. 74, 2531 (1995). 5. Y.H.Tang, I.Hahn, H.M.Bozler, and C.M.Gould, Phys.Rev.Lett. 67~ 1775

(1991). 6. H.Ishimoto, H.Fukuyama, T.Fukuda, T.Okamoto~ T,Tzaki, K.Skayofi, and

S.Ogawa, in AIP conference proceedings 194, 281 (1989). 7. H.Akimoto, T.Okuda, and H.Ishimoto, in this volume. 8. For full analysis see C.N.Archie, T.A.Alvesalo, J.D.Reppy, and R.C.Rechardson,

J.Low Temp.Phys. 42, 295 (1985) and Ref.9. 9. J.M. Parpia, D. G. Wildes, J. Saunders, E. K. Zeise, J. D. Leppy, and 1%. C.

Rechardson, Y.Low Temp.Phys. 61, 45 (1985). 10. D.S.Greywall, Phys.Rev.B 33, 7520 (1986). 11. J.R.Hook, E.Faraj, S.G.Gould, and H.E.Hall, J.Low Temp,Phys. 74, 45 (1989).