Measurements of the acoustic impedance on superfluid 3HeB

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Physica 107B (1981) 685-686 North-Holland Publishing Company

MEASUREMENTS OF THE ACOUSTIC IMPEDANCEON SUPERFLUID 3He-B D. B. Mast, J. R. Owers-Bradley, W. P. Halperin I. D. Calder, Bimal K. Sarma and J. B. Ketterson Department of Physics and Astronomy Northwestern University Evanston, I l l i n o i s 60201 The locations of the squashing and real squashing (new) modes in superfluid 3He-B have been found using an acoustic impedance technique for frequencies of 12.1, 36.3 and 60.5 MHz at pressures between 1.74 and 4.88 bar.

In recent studies I-3 of the attenuation and velocity of sound in 3He-B, a new mode was discovered in addition to the well known squashing mode; the nature and coupling of ultrasound to this mode (henceforth referred to as the real squashing mode) was c l a r i f i e d by Koch and Wolfle ~ and by the experimental work of Avenel et al 3. Studies of the squashing mode are hampered by the enormous attenuation occuring at higher frequencies. To overcome this d i f f i culty we have applied a continuous wave acoustic impedance method which is p a r t i c u l a r l y powerful for this situation. In the hydrodynamic regime the acoustic impedance, Z, is given by Z=R+iX=pV where V=jk is the phase velocity, and k=B+ia is the complex propagation constant. In the zero sound regime this relation is incomplete in that a coupling to the quasi particle spectrum must also be included; however i t is s u f f i c i e n t l y accurate for our purposes. From the above i t follows that R=pB~/(~2+B2) and X=-pam/(~2+B2). Clearly this method is very useful when the attenuation a is large (where propagation experiments become impractical), or when B is a strong function of T or m. In our method a frequency modulated r f signal is applied to a X-cut quartz transducer in contact with the 3He; the f.m. bandwidth is small relative to the transducer bandwidth. The frequency dependence of the electrical impedance of the transducer near a desired overtone produces an a.m. component in the resulting r f signal. Changes in the frequency dependent electrical impedance resulting from sweeping (via temperature) the collective mode frequency through the o s c i l l a t o r frequency cause readily detectable changes in the demodulated a.m. signal. A quantitative description is complicated and involves a knowledge of the combined frequency response of the 3He loaded transducer, the transmission l i n e , and the detector, and w i l l not be discussed here. Our immediate interest is to determine the temperature at which the anamoly, caused by collective mode passage, occurs. In Fig. 1 the results of such measurements are compared with data obtained previouslyl, 3 using 0378-4363/81/0000-0000/$02.50

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transmission acoustics (including attenuation and group velocity measurements); all impedance measurements are shown as closed symbols. The data taken at a pressure P in the range up to 4.88 bar are normalized to a common pressure of 2.00 bar through multiplication of the measured mode frequency by A+(2.00)/A*(P) where A+ is the energy gap function in the weak-couplingplus model of Rainer and Serene.s This factor is 1.4% for the single datum at 4.88 bar and is less than I% for a l l other pressures.

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1.8

*

~1.6

I

I

I

1

x 5VS fit NU 13.1 bar o OC fif NU 13.1 bar ISQ • Z NU 2.0 bar TX TX Z

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ORSAY 2.0 bar "1 NU 13.1 bar }REAL SQ NU 2.0 bar I ~..X O

--

-

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~ O

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1.2

1.0

I

I

.2

.4

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.6 T/Tc

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Fig. 1. The temperature dependence of ~sq and ~rsq for pressures near 2.0 bar and 13.1 bar. The values shown are determined for both modes from acoustic impedance measurements (Z), for from pulsed transmission attenuation (a) ~ group velocity (GVS) p r o f i l e s and for Wrsq from pulsed transmission experiments (Tx). The squashing and real-squashing mode frequencies are exhibited in the figure expressed as a function of the weak coupling BCS gap function determined numerically. The results normalized 685

686

to 13.1 bar were reported e a r l i e r and measurements from the Orsay group3 on the real squashing mode obtained at low pressures are included. I t is immediately clear from this figure that the pressure dependence of the squashing mode frequencies is substantially greater than that of the real-squashing mode. The frequencies for the squashing and realsquashing modes are 2 12 Wsq : A2(T) (I + 2~ ~sq FS) ~rsq =

(T)

3 rsq (I + 2~

and

(1)

(2)

where Xsa, Xrsq are coupling constants and F~(a) are secodd order Fermi l i q u i d parameters. These expressions can account for strong coupling effects of the sort which Rainer and Serenes have incorporated in their weak-couplingplus model for the B-phase gap functions A(T). In principle there may be other strong coupling effects4, 7 and we discuss our data in this context. We find that the squashing mode frequency increases with pressure from 2 bar to 13 bar by 7%. This is exactly the same increase in the square root of the heat capacity jump7 (AC/C)½ which sets the scale of the gap function ACT). We know independently that there is essentially no pressure dependence of F~. Measurements1,9 of the velocity difference Between f i r s t and zero sound determine F~ provided the effective mass m*/m is known. T~e recent measurementsm of m*/m indicate that F~ - I whereas e a r l i e r data I0 imply that F~= 0~8. Consequentlywe argue that the pressure dependence of ~ can be accounted for e n t i r e l y through t ~ gap functions A(T). Equation ( I ) can numerically account for a l l of our squashing mode data i f F~ = -1. S a ~ a n d Serene7 point out that mso c~n be reduced i f there is a s i g n i f i c a n t att -~ ractive ~=3 pairing interaction. I f the 3He effective mass is chosen11 such as to force F~ to be above - I i t may be that higher order pairing can account for the observed mode fre~ which is s i g n i f i c a n t l y less than ACT). On the other hand we find very minimal pressure dependence in the real squashing mode frequency which is reduced by about 20% below ~ A ( T ) at 13 bar. We have very l i t t l e information about F~ derived from B-phase s u s c e p t i b i l i t y measurements12, however, i t appears that i t too should be negative using an effective mass from Alvesalo et al. 8 This could reduce rsn to the observed value 7 i f F~ = -1.6. Sauls ana Serene7 have found that a reduction in this mode frequency is also possible from an attractive ~=3 interaction and that i t is not possible to distinguish between these two effects. From the experimental point of view however i t is clear that the squashing mode frequency is more strongly pressure dependent than that of the

real-squashing mode. Consequently, i t is unl i k e l y that ~=3 pairing can account for the reduction in frequency of both modes. More accurate measurements of the Fermi l i q u i d parameters are called for in order to extract further information from this collective mode spectroscopy. Of course the comparison between theoretical and experimental frequencies is imprecise to the same extent as is the temperature scale (~4%). Finally some experiments were also performed for transverse fields of up to 400G. The real squashing mode was observed to s p l i t in agreement with findings of Ref. 3. No observable s p l i t t i n g was seen in the squashing mode for these fields. For the temperatures and frequencies studied to date the expected s p l i t t ing 13 i s o f o r d e r o f the l i n e w i d t h . Support for this research was provided by the National Science Foundation through Grants No. DMR-78-11771 and No. DMR-78-11660. (i)

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D. B. Mast, Bimal K. Sarma, J. R. OwersBradley, I. D. Calder, J. B. Ketterson and W. P. Halperin, Phys. Rev. Lett. 45, 266 (1980); I. D. Calder, D. B. Mast,-Bimal K. Sarma, J. R. Owers-Bradley, J. B. Ketterson, and W. P. Halperin, Phys. Rev. Lett. 45, 1866 (1980). R. W. ~'i-annetta, A. Ahonen, E. Polturak, J. Saunders, E. K. Zeise, R. C. Richardson and D. M. Lee, Phys. Rev. Lett. 45, 262 (lg80). O. Avenel, E. Varoquaux, and H. Ebisawa, Phys. Rev. Lett. 45, 1952 (1980). V. E. Koch and P . ~ 6 1 f l e , Phys. Rev. Lett. 46, 486 (1981). D__ Rainer and J. W. Serene, Phys. Rev. B13 4745 (1976). P. W61fle, Physica 90B, 96 (1977). J. A. Sauls and J. W. Serene, to be published. T. A. Alvesalo, T. Haavasoja, M. T. Manninen, and A. T. Soinne, Phys. Rev. Lett. 44, 1076 (1980). J. B. K~tterson and P. Roach. Sanibel I. J. C. Wheatley, Rev. Mod. Phys. 47, 415 (1975). W. P. Halperin, F. B. Rasmussen, C. N. Archie, and R. C. Richardson, Jour, Low Temp. Phys. 31, 617 (1978).. A. I. Ahonen~--M. Krusius, and M. A. Paalanen, J. Low Temp. Phys. (1976). N. Schopohl and L. Tewordt (preprint).