p. roche, m. roger and f. i. b. williams- interpretation of the low damping of subthermal capillary...

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Interpretation of the low damping of subthermal capillary waves „ripplons… on superfluid 4He

P. Roche, M. Roger, and F. I. B. WilliamsService de Physique de l’Etat Condense , Commissariat a l’Energie Atomique,

Centre d’Etudes de Saclay, 91191 Gif-sur-Yvette Cedex, France

Received 11 August 1995

The problem of ripplon damping on superfluid 4He is reexamined to show how imposing the boundary

condition constraint at the perturbed rather than unperturbed surface position results in strong cancellation of the previous result for the three-ripplon interaction. This has orders of magnitude consequences for the relax-ation rate of subthermal ripplons, which is shown to be dominated instead by ripplon-phonon scatteringqT 4, in excellent quantitative agreement with recent experiments.

I. INTRODUCTION

Atkins1 first proposed the idea of treating the long-wavelength excitations at the free surface of superfluid he-lium as quantized capillary waves, or ripplons. It was verysuccessful in explaining the temperature dependence of thesurface tension and also accounts well for the frequencies.2– 4

The idea was further developed by Saam to examine thestability of ripplons to decay by ripplon-ripplon5 andripplon-phonon6 scattering as a result of hydrodynamic non-linearity. Saam’s formalism was consequently used to esti-mate finite temperature lifetimes7,8 where it was concludedthat subthermal ripplon decay should be dominated byripplon-ripplon scattering. Experiment however shows rip-plons to be much longer lived than these predictions. Theupper bound on the damping deduced from observed life-times of coupled surface charge-ripplon modes3,4 was al-ready 2 orders of magnitude lower, but was conceivablywithin the range of error introduced by approximations.8

Now that measurements of ripplon damping9 have revealed asix orders of magnitude discrepancy, a bolder explanation iscalled for. Either there is an error in the logic relating thecalculated result to the model or the model itself is wrong.

We have therefore reexamined the formalism. The con-straint imposed by the surface boundary condition relates thevelocity potential to the surface deformation and thereby de-termines the form of the Lagrangian in terms of the surfacevariables. The correction to the ripplon-ripplon scatteringHamiltonian, introduced by applying the constraint at theperturbed rather than the unperturbed surface configuration,is of the same order as the term obtained by Saam using theunperturbed constraint. The sign, however, is opposite and

the two terms almost exactly cancel in the limit of subther-mal ripplons. Indeed, the cancellation is so strong(1012) that the mechanism of ripplon-phonon scattering,previously thought to be unimportant, becomes dominant. Anexplicit calculation shows that this process accounts well forthe new experimental results.

In what follows, we dissert the problem in such a way asto try to understand why taking account of the curvature onapplying the surface boundary condition—itself a well-known procedure10—leads to such a dramatic cancellation inthe damping rate.

In a coordinate system defined by the three unit vectors

(x ˆ ,y ˆ ,z ˆ), let the liquid be located, at time t , in the half space z ( x , y , t ). In the absence of any excitation, 0. The su-perfluid is described by an irrotational velocity fieldv ( x, y , z ,t ) subject to the boundary condition that therebe no particle flow though the free surface. For an elementds of the boundary surface with normal vector n ˆ ,z ˆ–n ˆ ˙ dsv–n ˆ ds. Since n ˆ is proportional to (z ˆ ),( / x)x ˆ( / y)y ˆ, the boundary condition becomes10

˙ / z z – z . 1

Expanding Eq. 1 to second order in ( , ),

˙

z

z0

2

z 2 z0

– z0 . 2

This is the constraint which was approximated to the firstterm in Ref. 5. We define Fourier transforms in the ( x , y)plane r xx ˆ yy ˆ

r , z q

q z e i q–r, rq

qe i q–r. 3

Incompressible fluid: the three-ripplon interaction. The equa-tion of continuity in the incompressible liquid 0 leadsto q( z) qeqz , with qq. The boundary condition 1imposes

q q ˙q

q

q–q

q ˙

q qq

, 4

where the last term corresponds to the sum of the second-order terms in Eq. 2. The Lagrangian of a nonviscous in-

compressible fluid is L T V , where

T 0

2

dxdy

x , y v2 x, y , z dz 5

represents the kinetic energy11 and

V 0

dxdy12 1/ 2

1 6

is the potential energy from surface-tension forces.12 0 rep-resents the bulk 4He density and 0 the surface tension at

PHYSICAL REVIEW B 1 FEBRUARY 1996-IVOLUME 53, NUMBER 5

530163-1829/96/535 /22254 /$06.00 2225 © 1996 The American Physical Society



zero temperature. Expanding T and V to order three in( , ), one obtains the same expression as Saam:5

L 0

2 q q q2

0

2 q q2 q2

0

2 q,q

q–qqq q qq q. 7

The variables ( q , q) are coupled through Eq. 4, fromwhich L L 0L 1 with

L 0 0

2q1

q ˙

q2

0

2 q q2 q2,

L 1 0

2 q,q

q–qqq

qq ˙

q qq ˙

q8

and the classical Hamiltonian, correct to order 3, isH H 0H 1 with

H 0 12 0

qq q2

02 q q2 q2,

H 11

2 0q,q

q–qqq q qq q, 9

where q L / ˙ q is the momentum conjugate to q . Thequantum version is obtained through the usual transforma-tion of canonical variables into Bose operators.5

The interaction Hamiltonian H 1 differs from Eq. 6 of Ref. 5 only in the sign of the qq term. This seeminglyinnocent remark is crucial. First of all, the sign ensures theinvariance of the physics to a z translation of the coordinate

system. In a new set of coordinates defined by ( x x, y y , z z ) the new Fourier components are related tothe old by q q , q0, and 0 0 and the Hamil-tonian is transformed into

H H q q , q qq

2 0

qq–qqqqq q2. 10

Only if the term in curly brackets is null are the form and thephysical results invariant; otherwise an arbitrary effectivemass term is added to H resulting in arbitrary ripplon fre-

quencies depending on the choice of coordinate reference.By the same token, the present form changes radically theprevious result for the damping of subthermal ripplons. Forripplons of wave vector Q, the dominant effect arises fromterms involving Q . Taking account of the vector additionQqq, these now become

1

2 0

q

1

2qq2

Q2 q Q qQ 11

instead of

1

2 0

q

1

2qq2

Q2 q Q qQ . 12

The new coefficients are bounded by Q2 /2, whereas previ-ously the only bound was imposed by the availability of thermally excited ripplons q ,qq T . Very roughly, the ef-fect on the damping is to introduce a factor(Q / q)4( Q / k BT )8/ 31013 in the typical conditions of the experiment of Ref. 9. In fact, in contrast with the previ-ous result, the principal contribution now arises from qQ

and not from qq T , so that the decay rate becomes propor-tional to T . A good approximation to the numerically inte-grated result, accurate to 15% for 105

Q / k BT 103 isgiven by

Q Q1

1.2

18 aQ 2

k BT

0a2 . 13

If a is taken to be the Wigner-Seitz radius (2.2 Å), 0a2 / k B1.4 K. It is related to the previous estimate7 by afactor 0.05( Q / k BT )7/ 3 (1012 for 10 m wavelength at0.5 K. In view of the dramatic reduction factor for thismechanism, it becomes important to estimate other mecha-

nisms and in particular the ripplon-phonon interaction.Compressible fluid: the one-ripplon–two-phonon interac-

tion. Saam6 also set up a formalism for the compressiblefluid using linearized continuity equations. We shall see how-ever that the second-order terms in the continuity equationshave only a small effect on the ripplon-phonon interactionfor subthermal ripplons.

The Lagrangian of the ideal compressible fluid isL T V V , where

T 1

2

dxdy

dz 0 v2 14

represents the kinetic energy, V is given by Eq. 6 and

V s2

2 0

dxdy

dz 2 15

is the contribution to the energy due to the compression of the liquid. represents the variation of the density 0 from the equilibrium value 0 and s is the firstsound velocity. is coupled to by Eq. 2 and iscoupled to v through the equation of continuity in thebulk12

˙ 0

–

0. 16

We write as the sum of two terms, 0 1, where 0 satisfies the continuity equation to first order

˙ 0 00 17

and 1 is a second-order term satisfying

0 1 0 0– 0. 18

The quadratic part of the Lagrangian, L 0T 0V 0V 0 , isobtained by expanding T , V , and V to second order,

2226 53BRIEF REPORTS



T 0 0

2

dxdy

0dz 02,

V 0 0

2

dxdy 2,

V 0

s 2

2 0

dxdy0

dz

2

, 19

and restricting the continuity equations 2 and 16 to linearterms. The normal modes of L 0 are expressed in Ref. 6. Weshall however reformulate the problem somewhat differently.The key point is that it is only possible to eliminate variablesthrough the constraints of Eqs. 2 and 16 taken to secondorder if we choose and rather than and Ref. 6 asindependent generalized coordinates. In addition to the Fou-rier transforms of Eq. 3, we introduce q,k defined by

x , y , z q,k

q,k eiq–rkz , q,k

q,k * . 20

0 must satisfy Eq. 17, the general solution for which is asum over

q0 z q

Reqz

1

0

k

˙ q,k

q2k 2

e ik z. 21

The coefficient q R is fixed by the boundary condition 2 at

the free surface. 1 is determined by Eq. 18 to be of order2.

To first order in the surface constraint

q q R ˙ q with q q

1

0

k

ik

q2k 2

q,k . 22

Setting q,k R q ,k q,k , where

R q,k q2k 22ibqk

q2k 22ibqk

with b q 0q2

2 0s2 23

represents the reflection coefficient for phonons impingingon the surface, the Lagrangian L 0 takes the diagonal form of Ref. 6

L 0 0

2q ˙ q2

q2b q

21/ 2 0

2 qq2 q q2

q2b q

21/ 2

1

4 0q,k

8 q,k 2

q2k 2

s2

4 0q,k ˜ q,k 2

, 24

where qb q(q2b q

2)1/ 2 and

˜ q,k q,k 4ibqk 0

q2k 22ib qk

q . 25

The eigenfrequencies for ripplons are q( 0 qq2 / 0)1/ 2

and for phonons q ,k s(q2k 2)1/ 2.6

The next higher order introduces the interaction termswhich give rise to damping. We shall limit ourselves here13

to the terms of interest for damping of subthermal ripplons( qk BT ). Only three-ripplon and one-ripplon–two-phonon scattering is allowed by the kinematics. Futhermore

we retain only the lowest order in b q / q (0.015 for thermalphonons at 1 K. Within this approximation, R q,k 1, z00, and q,k ˜ q,k . Hence, we neglect the third-orderterm order (b q / q)2 in the expansion of V . After an inte-gration by parts with the help of Eq. 18, the kinetic energymay be written T T 0T 1 where

T 1

dxdy 0

2 0 z02 0

dz

2 02

dxdy 0 0 1

z 0

0

z z0

. 26

( 1 / z) z0 can be expressed in terms of its Fourier trans-form in the ( x , y) plane, after integration of Eq. 18

q1

z

z0i

q ,k

k 22q2qqqqqk 2

0kq k 24q2 ˙ q

˜ qq,k .

As T 1 is already of order 3, the variable q0 may be elimi-

nated from it by using the first-order surface boundary con-dition of Eq. 22 and Eq. 21. The resulting contribution tothe three-ripplon interaction is

L 1a 0

2 q,q

q–qqq

qq ˙ q qq

˙ q. 27

Neglecting terms of order ˙ ˜ 8 compared with 8 8 , thelowest order one-ripplon–two-phonon interaction from T 1 isdescribed by

L 11

2 0q,k

q,k

kk

q2k 2q

2k 2

8 q,k qq 8 q,k

.

28Additional interaction terms, those omitted in Ref. 6, arisefrom T 0 on expressing q

R in terms of the generalized coor-dinates by means of the surface boundary condition of Eq.2 taken to second order. Equation 22 is then replaced by

q q R ˙ q

q

q–q

q ˙ q

qq

q1

z

z0

1

0q,k

ik q–q ˙ q ˜ qq ,k

qqq2k 2

. 29

The first second-order term in Eq. 29 contributes to thethree-ripplon interaction

L 1b 0

q,q

q–q

qq ˙ q qq

˙ q. 30

The sum L 1L 1aL 1

b is identical to the three-ripplon termof Eq. 8 for the incompressible fluid. The other second-order terms induce two-ripplon–one-phonon interactions thatare kinematically forbidden in the regime that we considerhere. The relevant Lagrangian is then L L 0L 1L 1 . Interms of the conjugate momenta q L / ˙ q and q,k L / 8 q,k , the Hamiltonian is H H 0H 1H

1

with




H 0q

q2b q

21/ 2

2 0 q2

0q2 q

2q2b q

21/ 2 q2

q,k

0q2k 2 q,k

2

s 2

4 0 ˜ q,k

2 ,

H 1

1

2 0q,q q–q

qq q qq q,

H 12 0q,k

q,k

kk q,k qq q,k

. 31

The three-ripplon term H 1 is identical to Eq. 9 and, asalready pointed out, differs from the previously accepted re-sult. Our leading term H 1 for the one-ripplon–two-phononinteraction, on the other hand, is identical to that derived inRef 6; the higher order terms in the continuity equationsmake no significant contribution here. The ripplon dampingfactor associated with H 1 for Qk BT is

23 Q Q

1 23 2

60Q 0 Q k BT

s 4

7.7 W 0

0a212 T

D

4

Qa1/ 2. 32

The Stefan-Boltzmann like part of this result arises from theenergy momentum loss associated with the Doppler shift of phonons reflected from the moving surface and the Q / 0 partfrom the ripplon effective mass. Introducing the Wigner-Seitz radius, a, highlights the more evidently dimensionlesssecond form where W 0

2 / ma22.5K (m is the 4He atommass is a measure of the zero point energy and the Debye

temperature D is defined in the usual way.The result is compared in Fig. 1 with recent measure-

ments of the damping factor for ripplons with wavelength inthe range 3–20 m.9 The agreement with this ab initio cal-culation is remarkable.

We conclude that the basic physics of the quantum hydro-dynamical model of Atkins and Saam is not put into questionby the new experiments. The problem stems rather from aninconsistency in the formalism. Once this is put right, themodel, which contains no adjustable parameters, gives anextremely good quantitative account of the laboratory resultson ripplon lifetimes.

It is a pleasure to thank G. Deville, S. Balibar, and N. J.Appleyard for many stimulating discussions.

1 K.R. Atkins, Can. J. Phys. 31, 1165 1953.2 K.A. Pickar and K.R. Atkins, Phys. Rev. 178, 399 1969.3 C.C. Grimes and G. Adams, Phys. Rev. Lett. 42, 745 1979.4 G. Deville, J. Low. Temp. Phys. 72, 135 1988.5 W.F. Saam, Phys. Rev. A 8, 1918 1973.6 W.F. Saam, Phys. Rev. B 12, 163 1975.7 H. Gould and V.K. Wong, Phys. Rev. B 18, 2124 1978.8 R.C. Jones, A.J.E. Williams, W.F. Vinen, and P.A. Ewbank, J.

Low Temp. Phys. 92, 239 1993.9 P. Roche, G. Deville, K.O. Keshishev, N.J. Appleyard, and F.I.B.

Williams, Phys. Rev. Lett. 75, 3316 1995.

10 H. Lamb, Hydrodynamics, 6th ed. Cambridge University Press,Cambridge, 1932, Chap. I.

11Applying Green’s theorem with 0, the kinetic energy can berepresented as an integral over the free surface S :T S v–n ˆ ds . Using the exact boundary condition,T dxdy ˙ z . Although this leads to a more elegant for-malism, we have preferred to follow Saam.

12 L.D. Landau and E.M. Lifshitz, Fluid Mechanics Pergamon,London, 1959, Chap. VII.

13 The other terms appear in P. Roche, Ph.D. thesis, University of Paris VI, France, 1995, and will be published elsewhere.

FIG. 1. Theoretical damping factor straight lines for wave-

length 3.3 and 20 m compared to experimental data Ref. 9.


p. roche, m. roger and f. i. b. williams- interpretation of the low damping of subthermal capillary...

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