ULTRASONIC PROPERTIES OF THE A AND B PHASES OF 3He

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ULTRASONIC PROPERTIES OF THE A AND B PHASES OF 3He

P. Wölfle

To cite this version:

P. Wölfle. ULTRASONIC PROPERTIES OF THE A AND B PHASES OF 3He. Journal de Physique

Colloques, 1978, 39 (C6), pp.C6-1278-C6-1282. �10.1051/jphyscol:19786557�. �jpa-00218046�

JOURNAL D E PHYSIQUE Colloque C6, suppliment au no 8, Tome 39, aoDt 1978, page C6-1278

U L T R A S O N I C P R O P E R T I E S O F THE A AND B PHASES O F 3 ~ e

La, Temperature Laboratory, He l s i n k i U n i v e r s i t y o f Technology, Espoo 15, Finland

R6sumd.- Nous prcsentons une revue des dtudes thdoriques et expdrimentales de la propagation des ul- tra-sons dans lt3He superfluide. Nous traitons d'une manisre semi-phGnomdnologique les mdcanismes d'absorption par rupture de paires et les excitations des modes collectifs d'oscillation du params- tre d'ordre. Nous discutons les rdsultats d'une th6orie microscopique et les comparons ii l'expd- rience.

Abstract.- A review of theoretical and experimental work on ultra sound propagation in superfluid 3 ~ e is given. The appearance of new absorption mechanisms by pair breaking and excitation of order parameter collective modes isconsidered in a semiphenomenological way. The results of a microscopic theory are discussed and compared with experiment.

I. INTRODUCTION.- Among the first experiments to be carried out in the superfluid phases of 3 ~ e were ul- trasound propagation studies (Lawson, Gully,

Goldstein, Richardson and Lee / I / , Paulson, Johnson and Wheatley / 2 / ) , demonstrating the existence of weakly damped sound waves at frequencies w much greater than the typical quasiparticle relaxation frequency 1 / ~ . This form of sound propagation in the collisionless regime, called zero sound, had been predicted successfully for normal liquid 3 ~ e by Landau's Fermi liquid theory. Zero sound is essen- tially a wave-like distortion of the density polari- zation field experienced by a quasiparticle as a re- sult of the interaction with the medium. Since the BCS-type pair correlations that are believed to be responsible for the superfluid transition in 3 ~ e /3/ do not change the density distribution in lowest approximation, one should expect zero sound to be a well-defined collective excitation in the superfluid state, too. However, quite unexpectedly, the experi- ments shor d a narrow, strongly frequency dependent peak in the sound attenuation closely below T (fi- gure 1) and a simultaneous drop in the sound veloci-

Fig. 1 : Attenuation of 10 MHz sound in liquid ' ~ e on the melting curve as a function of temperature according to Lawson et al. (1973)

(1) breaking up of Cooper pairs subsequent to the absorption of a sufficiently energetic phonon

(2) excitation of high-frequency collective modes of the tensor order parameter (pair vibrations).

The coherence of the sound wave is also destroyed ty, indicating the appearance of new excitation me-

by quasiparticle collisions, which is the damping chanisms in the superfluid phases 141.

mechanism operative in the normal state.

2. PAIR-BREAKING AND ORDER PARAMETER COLLECTIVE MO- The first process takes place if the energy conservation requirement

DES.- It was soon realized that the sound absorp-

tion peak could be explained on the basis of the Ih = 2Ek + O(q/kf) (1) pairing hypothesis as being caused by types can be met. Here Mu and Hq are the phonon frequency excitation processes /5,6,7/ :

and momentum, and Ek =

(5;

+ 1+12)'/2 is the single particle energy, with

Ck

= vf(k-k ) the normal qua-

"

Permanent address : Institut fiir Theoretische Phy- f

sik, Technische Universitzt Miinchen, D-8046 siparticle energy, vf and k the Fermi velocity and

Garching, F.R. Germany f

wave vector and the modulus of the anisotropic

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786557

gap parameter. It follows from equation (I) that pair breaking occurs for temperatures such that

< ~ A ~ ( T )

I

> ( Mw/2, where the brackets denote an ave- rage over the Fermi sphere. For typical experimental frequencies of about 10 MHz this corresponds to a temperature regime of (I-T/T ) $, lo-', which is in accordance with experiment. However, a detailed calculation yields that pair breaking can only ac- count for a small fraction of the observed excess attenuation.

By far more important is the absorption into pair vibration modes

-

excitations of the internal structure of the tensor order parameter. The exis- tence of such modes is a consequence of the two- or three-dimensional structure of the order parameter.

For example, the Anderson-Brinkman-Morel (ABM) p-wa- ve pairing state believed to describe the A phase is characterized by the tensor order parameter dia = Ao(T)

" ^(Gla

⁺ i2a), where i = 1. 2. 3 refers to spin space and a = 1 , 2, 3 refers to orbit space and

a, il

^and

G2

specify spontaneous preferred di- rections. The pair condensation energy is a strong function of the angle $I between

G,

and G 2 , ^{being ma-} ximumfor +=n/2,but isindependent of the relative orientation of

a

^and

G I , G2

apart from a small spin- orbit energy. In a simplified picture the dynamics of the collective variable 4 is governed by the mo- del Hamiltonian

where I is an effective moment of inertia, which may be expected to be roughly proportional to the density of the superfluid component ps (% A'(T) near T ) and ~ c o s ~ @ is the term in the pot. energy dependent on 4. In the Ginzburg-Landau regime V % (T). It is important to note that the quasi- particle energy Ek does not depend on $. One may therefore expect the motion of $I to be relatively decoupled from the gas of thermal excitations.

The effective Hamiltonian ( 2 ) describes os- cillations of @ with frequency w = dm% A(T) (at small amplitudes $I). Microscopic theory indeed pre- dicts such a mode with.frequency

MU

= 1.23 A,(T), 0 < T < T 171. Apart from this socalled "clapping mode" thereexists another pair vibration mode in the A phase (see below) and two in the B phase. The contribution of these modes to the sound absorption will now be considered in more detail.

3. SOUND PROPAGATION IN THE B PHASE.- The microsco- pic theory that describes the ultrasonic properties of superfluid '~e is rather complex. The results of this theory will be compared with experiment below.

However, the basic physics involved may be discus- sed in rather simple terms as follows /3c/.

Sound waves are a consequence of the conser- vation of mass and momentum, as expressed by the conservation laws

* + -+-

at

^g + VP + V a n =

o

where p, g and P are the mass and momentum density +

and the pressure, respectively. In the hydrodynamic regime the stress tensor nh associated with a sound wave is given by

(G

is the direction of sound propa- gation)

where q = - 1 p <(?k~k)2>r,, is the shear viscosity of 5 n

the normal component, and pn and T are the density

rl

and viscosity relaxation time of the normal compo-.

nent

.

At higher frequencies w I/T the normal com- ponent is no longer in a state of local equilibrium, but rather relaxes towards it. It is plausible that the normal (longitudinal) component of the stress tensor relaxes at a rate given by

A very important simplifying feature of sound pro- pagation in liquid ' ~ e comes from the fact that the sound velocity c is much larger than the Fermi velo- city vf, due to the strong repulsive interaction between the He atoms. For this reason the term in- volving the current + j n in equation (6) may be ne-

glected, because it is of order (vf/d2 small.

At still larger frequencies $w A there ari- ses a contribution to the stress tensor from high- frequency motions of the order parameter. In the B phase, which is believed to be a realization of the Balian-Werthamer (BW) state with order parameter dja = A(T) dja (assuming the spin coordinates to be rotated appropriately) the stress tensor receives a contribution rrs from quadrupolar distortions of the order parameter, 6d ₌

(^qj2a - -

1 ^6. )

ja 3 J U

C6- 1280 ^{JOURNAL DE PHYSIQUE}

Ifi a n i s o t r o p y i s o b s e r v e d i n e x p e r i m e n t s t r o n g l y c o r -

ns = An-

a

6 b S q .

A t (') r o b o r a t e s t h e i d e n t i f i c a t i o n o f t h e B p h a s e w i t h

Here n i s t h e number d e n s i t y a n d A i s a dimension- the pseudoisotropic BW state.

l e s s , t e m p e r a t u r e d e p e n d e n t complex c o u p l i n g cons- t a n t ( t h e i m a g i n a r y p a r t b e i n g r e l a t e d t o p a i r b r e a - k i n g ) , w i t h h ^a i A(T) n e a r T

.

The dynamics o f t h e o r d e r p a r a m e t e r component 6asq i s governed b y a H a m i l t o n i a n o f t h e f o r m o f e q u a t i o n ( 2 ) . More p r e c i s e l y , 6 ~ ~ ~ may b e shown from t h e m i c r o s c o p i c t h e o r y t o obey ( a p p r o x i m a t e l y ) t h e time-dependent Ginzburg-Landau e q u a t i o n / 7 /

where r i s t h e q u a s i p a r t i c l e l i f e time and Nf i s t h e d e n s i t y of s t a t e s a t t h e Fermi s u r f a c e . E q u a t i o n (8) d e s c r i b e s c o l l e c t i v e o s c i l l a t i o n s o f t h e o r d e r p a r a m e t e r component 6hsq, damped by q u a s i p a r t i c l e c o l l i s i o n s , w i t h f r e q u e n c y $w = ( 1 2 / 5 ) l / ~ A (T)

.

s q

T h i s s o c a l l e d " s q u a s h i n g mode" s h o u l d b e undampedin t h e l i m i t T + 0, t h e f r e q u e n c y b e i n g o u t s i d e t h e p a i r - b r e a k i n g continuum.

Subs t i t u t i n g t h e l o n g i t u d i n a l p a r t of t h e s t r e s s t e n s o r i n e q u a t i o n ( 4 ) w i t h a n + a s , employ- i n g 6P = (aP/ap)ap and F o u r i e r t r a n s f o r m i n g equa- t i o n s ( 3

-

⁸⁾ o n e o b t a i n s f o r t h e a t t e n u a t i o n o f sound

(9) I n t h e z e r o sound regime (WT >> 1) t h e f i r s t t e r m

rl

d e s c r i b e s a f r e q u e n c y i n d e p e n d e n t background a t t e - n u a t i o n due t o q u a s i - p a r t i c l e c o l l i s i o n s . The second term y i e l d s c o n t r i b u t i o n s from t h e i m a g i n a r y p a r t of A , d e s c r i b i n g p a i r - b r e a k i n g p r o c e s s e s , and from t h e c o l l e c t i v e p o l e . The l a t t e r c o n t r i b u t i o n g i v e s r i s e t o a peak i n t h e sound a t t e n u a t i o n a t a t e m p e r a t u r e Tcoll, where t h e r e s o n a n c e c o n d i t i o n

A(Tcoll) = ( 5 / 1 2 1 ~ / ~ ^MU i s met, o f h e i g h t

%(w2.r/c1) ( v f / c l ) ' (m /m) and w i d t h AT, d e t e r m i n e d x by t h e c o n d i t i o n A(Tcoll- AT)

-

A(Tcoll+ AT) 2 HIT.

The sound v e l o c i t y is found t o a p p r o a c h t h e f i r s t sound v e l o c i t y f o r T ^-+ 0.

I n f i g u r e 2 t h e a t t e n u a t i o n and v e l o c i t y s h i f t o f 15 MHz s o u a d i n 3 ~ e - ~ measured b y P a u l s o n e t a l . / 2 / i s compared w i t h t h e r e s u l t o f a m i c r o s - c o p i c c a l c u l a t i o n 171, u s i n g v a l u e s f o r A(T), T and T c o n s i s t e n t w i t h s p e c i f i c h e a t , s p i n r e l a x a t i o n

9

and v i s c o s i t y e x p e r i m e n t s . The e x c e l l e n t a g r e e m e n t o b t a i n e d t o g e t h e r w i t h t h e f a c t t h a t no t r a c e o f

P i g . 2a

196 bar I*-

F i g . 2b

F i g . 2 : A t t e n u a t i o n ( a ) and v e l o c i t y s h i f t ( b ) o f 15 MHz sound i n 3 ~ e - ~ v e r s u s r e d u c e d t e m p e r a t u r e a c c o r d i n g t o P a u l s o n e t a l . (1 973). The s o l i d c u r - v e i s t h e t h e o r e t i c a l r e s u l t from r e f e r e n c e /7/

Very r e c e n t l y t h e s e measurements have b e e n e x t e n d e d t o l o w e r p r e s s u r e s and lower t e m p e r a t u r e s

( P a u l s o n and Wheatley 1 8 1 ) . For 5 MHz sound and a t p r e s s u r e s below 5 b a r an e x t e n s i v e s a t e l l i t e peak s t r u c t u r e i s o b s e r v e d , w i t h s p a c i n g c o r r e s p o n d i n g r o u g h l y t o i n t e g e r m u l t i p l e s of A(T). T h i s e f f e c t i s n o t u n d e r s t o o d a t p r e s e n t , b u t i t seems l i k e l y t h a t i t i s a s s o c i a t e d w i t h ' n o n l i n e a r m o t i o n s o f t h e o r d e r p a r a m e t e r .

4. SOUND PROPAGATION I N THE A PHASE.- The u l t r a s o - n i c p r o p e r t i e s o f t h e A p h a s e a r e much more complex due t o t h e i n t r i n s i c a n i s o t r o p y o f t h e ABM s t a t e in o r b i t a l s p a c e . The s i n g l e p a r t i c l e e x c i t a t i o n spec- trum i s s t r o n g l y a n i s o t r o p i c , Ek = (5; - A:(T) s i n 2 8,.)'/', where €I1; i s t h e a n g l e of momentum k +

with respect to the axis of the gap, 2=6,x62. The

A

energy gap even vanishes along R, allowing pair- breaking to occur at any temperature, if only for

A

a small fraction of pairs with momenta along R.

The dependence of the quasiparticle spectrum

h

on the collective variable R gives rise to high- frequency collective oscillations of

2.

^{While the}

free energy is invariant against slow rotations of

2,

allowing the quasiparticle system to adjust to

A

the new instantaneous value of R, a rapid deflec-

h

tion of R increases the energy of the quasiparticle system for a time of the order of the quasiparticle lifetime. Thus the energy of the normal component

A

forms a potential well for rapid oscillations of R.

The frequency of this socalled "flapping mode" is obtained from the micro,scopic theory 171 as Haf = (415)

*

(T/T~)A~(T). uf tends to zero for T + 0 as the normal component vanishes. The flap- ping mode as well as the clapping mode discussed before are damped by pair-breaking processes even for T + 0.

The anisotropy of the sound attenuation in the ABM state is characterized by three components /7,9/ :

a = a

_{I I}

cos48 + 2ac cos28 sin20 + a sin40

₁ ,

⁽¹⁰⁾

where 0 is the angle of the sound propagation direc- tion with respect to the preferred direction

i .

It is found in the microscopic theory that the cou- pling of the pair vibration modes depends strongly on 0 171. The clapping mode only contributes to a the flapping mode to a,. a has only contribu-

I' I I

tlons from pair breaking and quasiparticle colli- sions. The difference of the sound velocity and the first sound velocity, c - c ~ , may be expressed in a form analogous to equation (10).

Experimentally a clean measurement of the anisotropic sound attenuation or velocity is diffi- cult due to the limited controllability of the

2-

textures that are unavoidably present in a sample of 3 ~ e - ~ . Nevertheless the anisotropy has been ob- served by Lawson, Bozler and Lee /lo/, Ketterson, Roach, Abraham and Roach 1111, and Paulson, Krusius and Wheatley 1121. In figure 3 attenuation data of Paulson et al. 1121 are compared with theory 1131, assuming perfect orientation of the R-field by the

,-.

combined effect of a magnetic field and a super- fluid-normalfluid counterflow. The collective mode peaks are clearly seen and overall agreement with theory is reasonable.

The strong anisotropy of the sound attenua-

I I I

I ^{3 ~ c - ~}

^{24.1 bar 15 MHz}

I

Fig. 3 : The three components of the anisotropic sound attenuation in 3 ~ e - ~ after Paulson et al.

(reference /12/). The curves are the theoretical result from reference 1131.

tion near T makes it an ideal probe of monitoring

h

the orientation of the Il-vector. The method has been successfully employed to study the subtle

A

orientation effects on Il subject to magnetic fields, heat flow, an intrinsic ferromagnetic polarization and electric fields as well as the dynamics of the

;-field 141.

5. CONCLUSION.- The ultrasonic properties of super- fluid 3 ~ e provide detailed information on the spec- trum of elementary excitations as well as the struc ture and the dynamics of correlations in the liquid.

In particular the sound absorption into pair vibra- tion modes may be used to measure the temperature dependence of the energy gap and the quasiparticle lifetime from the position and width of the clap- ping mode peak in the A phase and the squashing mo- de peak in the B phase 141. There is some uncer-

tainty due to the unknown Fermi liquid parameter F: and due to strong coupling effects, which how- ever should amount to no more than 10 % correction.

C6- 1282 JOURNAL DE PHYSIQUE

The experimental verification of the predicted col- /14/ Paulson,D.N. and Wheatley,J.C., preprint lective mode structure at lower temperatures, in 1151 Maki,K., preprint

particular the re-entrance of the flapping mode in 1161 Samalam,V.K. and Serene,J.W., preprint the A phase, would be of considerable interest.

The extreme sensitivity of the sound atte- nuation in the A phase to changes in the R-field makes ultrasound an ideal probe for studying the

A

dynamics of R.

Above Tc a precursor effect of the transi- tion has been observed in the sound absorption /14/. The effect appears to be well explained by pair fluctuations in the normal state, enhancing the quasiparticle scattering rate near T /15,16/.

The recently observed and as yet unexplained satellite peak structure in the B phase zero sound attenuation at low frequency and low pressure shows that this system may be rich enough to surprise us with further unexpected properties.

References

/I/ Lawson,D.T., Gully,W.J., Goldstein,S., Richard- son,R.C. and Lee,D.M., Phys. Rev. Lett.

30

(19731541 and J. Low Temp. Pnys.

2

(1974) 169 /2/ Paulson,D.N., Johnson,R.T. and Wheatley,;.C.,

Phys. Rev. Lett. 21 (1973) 746 /3/ For theoretical reviews see

(a) Leggett,A.J., Rev. Mod. Phys.

47

(1975) 331 (b) Anderson,P.W. and Brinkman,W.F., in "The

Helium Liquids", ed. J.G.M. Armitage a. I.

E. Farqhar (Academic Press, London) 1975 (c) ~Glfle,~., Rep. Progr. Phys. in print.

/4/ For an experimental review of sound propagation see Wheatley,J.C., in "Progress in Low Tempe- rature Physics" vo1.7, ed. D.F. Brewer (North Holland, Amsterdam) 1978

/5/ ~Glfle,P., Phys. Rev. Lett.

30

(1973) 1169 161 Ebisawa,H. and Maki,K., Progr. Theor. Phys.

2

(1974) 337

/7/ For a theoretical review of sound propagation see WGlfle,P., in "Progress in Low Temperature Physics", vol. 7, ed. D.F. Brewer (North Hol- land, Amsterdam) 1978

181

Paulson,D.N. and Wheatley,J.C., preprint /9/ Serene,J.W., thesis (Cornell University) 1973 /lo/ Lawson,D.T., Bozler,H.M. and Lee,D.M., Phys.

Rev. Lett.

2

(1975) 121

/11/ Ketterson,J.B., Roach,P.R., Abraham,B.M. and Roach,P.D., in "Quantum Statistics and the Many- Body Problem", eds. S.B. Trickey, W.P. Kirk and J.W. Dufty (Plenum, New York) 1975

1121 Paulson,D.N., Krusius,M. and Wheatley,J.C., 3.

Low Temp. Phys.

3

(1977) 73

1131 ~Elfle,P. and Koch,V.E., J. Low Temp. Phys.

30

(1978) 61