i.v. tanatarov et al- a new ripplon branch in he ii

8/3/2019 I.V. Tanatarov et al- A new ripplon branch in He II

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A new ripplon branch in He II

I.V. Tanatarov

National Science Center Kharkov Institute of Physics and Technology,Academicheskaya St. 1, Kharkov, 61108, Ukraine

I.N. Adamenko and K.E. NemchenkoKarazin Kharkov National University, Svobody Sq. 4, Kharkov, 61077, Ukraine

A.F.G. WyattSchool of Physics, University of Exeter, Exeter EX4 4QL, UK

(Dated: March 25, 2010)

We analyse the dispersion relation of ripplons, on the surface of superfluid helium, using thedispersive hydrodynamics approach and find a new ripplon branch. We obtain analytical equationfor the dispersion relation and analytic expressions for the limiting cases. We discuss where ripplonscan exist in the energy-wavenumber plane. A numerical solution for the ripplon dispersion curveis obtained in the allowed regions. The new ripplon branch is found at energies just below theinstability point.

I. INTRODUCTION

Ripplons are quantised capillary waves on the free sur-face of superfluid 4He. At low frequencies the dispersionlaw gives a good way of measuring the surface tension ofliquid helium [1, 2]. The temperature dependence of thesurface tension is due to ripplons [3]. Ripplons can be de-tected by neutron scattering, in a similar way to the bulkmodes, and have been shown to exist up to wavenumber

of 1.5 A1

, where the ripplon energy is equal to that ofthe roton minimum, 8.6 K, [4, 5]. Ripplons are the dom-inant scatterer of surface state electrons on liquid helium(see [6, 7] and references therein). Ripplons play a signif-icant role in the condensation [8], evaporation and reflec-tion of atoms from liquid 4He, [9, 10]. It has been sug-

gested that ripplons are the most favourable excitationsfor the simulation of general-relativistic effects related tohorizons of white holes [11].

Ripplons are well defined excitations and so have areasonably long lifetime, but they can be scattered byphonons and rotons, and by other ripplons. At low ener-gies there can be three ripplon scattering, in an analagousprocess to three phonon scattering. However for low en-ergy ripplons at low temperatures, the ripplon lifetimeis dominated to phonon scattering rather than ripplon-ripplon scattering [12].

In this paper we present a theoretical model of rip-plons which is good enough to account for the measured

dispersion curve (k) but is simple enough to expose theunderlying physics. It explains why ripplons only existin certain parts of the (, k) plane, and why the disper-sion curve approaches the line = rot at the top ofa parabola, where rot is the energy of the roton mini-mum. Moreover the model predicts a new roton branchat energies


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start from the dispersion relation that approximates theexperimental data for the spectrum of superfluid helium.The chosen (k) gives explicit expression for the kernelh(r), which determines the equation of state and the non-local wave equation.

In order to derive the dispersion relation of ripplons,we look for the solution of the nonlocal wave equation(1) in half-space, equipped with usual boundary condi-

tions (2). The use of simple model of superfluid heliumallows us to fully solve the problem of ripplons spec-trum. The ripplons dispersion equation is obtained inalgebraic form, and its analytic solutions are derived inlimiting cases. The analytic and numerical results of thepaper in the region below the roton gap are consistentwith the experiments, numerical computations and qual-itative estimates of other authors [4],[17].

A preliminary and partial account of this analysis isgiven in [19], where a very approximate function for thedispersion curve for bulk phonons and rotons was used.This was useful to show that the method had promisebut it could not show the new ripplon branch or otherfeatures. Here we use a much more detailed function forthe phonon-roton dispersion (k), which gives an excel-lent approximation of the experimentally measured curveover the whole wave vector range. This allows us to de-scribe the ripplons in detail in the whole energy interval.This range includes the energy of the Pitaevskii instabil-ity at twice the roton minimum energy [20]. It is justbelow this energy where we find a new ripplon branch.

II. EQUATIONS AND BOUNDARY

CONDITIONS

Let us consider the half-space z > 0 filled by superfluid

helium. Following the approach in Ref. [14], the liquidobeys the ordinary linearized equations of an ideal liquid,but the relation between the deviations P and of pres-sure and density from the respective equilibrium valuesis nonlocal, with some difference kernel h(r) [14, 15].

P(r, t) =

z1>0

d3r1 h(|rr1|)P(r1, t), z (0, ). (1)

Here the dots denote derivatives by time.We assume that the interface is sharp enough so that

the kernel h(r) is the same near the interface as it is inthe bulk medium. The kernel is related to the dispersionrelation of the bulk excitations (k) through its Fouriertransform (see [14]) h(k) = k2/2(k).

The dispersion relation (k), is for the continuousmedium which fills the half-space z > 0. It is nonlinearand has the distinctive form of the dispersion of phononsand rotons in superfluid helium (see Fig.1), and the char-acteristic length of the kernel is of the order of averageinteratomic separation.

The equation (1) is supplemented by the boundarycondition at the free surface. The pressure at the surface

with surface tension , must include the Laplace term,which for small deviations of the surface from equilibriumis

P =

2

x2+

2

y2

, (2)

where is the z-coordinate of the points of the surface(see for example [13] or [21]).

All the variables have the form exp(ikrit). Fora given frequency and the projection of wave vector kon the plane of the free surface (x, y), k, this expressionbecomes P = k2. The z-component of velocity ofthe surface is vz = i. We can therefore write theboundary condition in the form

vz|z=0 =i

k2P|z=0 . (3)

III. EQUATION FOR THE DISPERSION

RELATION OF RIPPLONS AND ANALYTICAL

SOLUTIONS

The dispersion relation, for undamped waves, can ingeneral be written 2 = 2(k2). We express the function2(k2) as a polynomial of degree S in powers of k2, suchthat the only real zero of 2(k2) is at k2 = 0. Near thispoint 2 k2. After some analysis, the Fourier imageby r and t of the solution of equation (1) can be writtenin the form (see [15] and references cited there):

P(kz;k, ) =Cout(,k)

kz k1 z(, k)

kizC+

kz ki z(0,k)kz ki z(,k) . (4)

Herek

=k

+e

zkz is the wave vector. The product istaken over all the roots kz = ki z of equation 2(k2 =

k2 + k2z ) =

2 in the upper half-plane C+ of the complexvariable kz. The real roots are assumed to be shifted upfrom the real axis. The root k1 z(, k) is the phononroot, i.e. the one in C+, which continuously goes to zeroat = 0 and k = 0. The prime superscript on the productdesignates that k1 z is excluded from the product. Thefunction Cout(, k) is the amplitude of the wave.

Ripplons have solutions which are damped with dis-tance from the liquids surface. These occur for all theki z, that are not real. For a given value of , the imag-inary ki z occur when k

2 is greater than any of the real

roots k2

i

() of equation 2(k2) = 2.The dispersion relation for superfluid helium is shown

in Fig.1. We assign subscripts to the positive roots ofequation 2(k2) = 2 in the ascending order of their ab-solute values, i.e. k1, k2 and k3 are phonons, R and R+

rotons respectively.The inverse Fourier transform of (4) is calculated from

the residues in C+, and velocity is obtained from the re-lation v/t =P/0 where 0 is the equilibrium den-sity. Then the values of P and vz, on the surface z = 0,can be expressed in terms of the residues of P(kz) and


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kzP(kz) at infinity. These are found directly by expand-ing P from (4). Then we obtain

vz|z=0 =1

0

k1 z +

Si=2

[ki zki z( =0)]

P|z=0 . (5)

Comparing (5) with the boundary condition (3), we fi-nally obtain the equation for the dispersion relation (k)

of ripplons:

2=

0 k

2

i

k1 z(, k)+

Si=2

[ki z(, k)ki z(0, k)]

.

(6)This equation is written in terms of ki z(, k) =

k2i () k2 C+, where k2i () in turn are the rootsof the polynomial equation 2(k2) = 2 with real coeffi-cients. Then taking into account the condition that allthe ki z are non-real, we can show, that the right partof the equation (6) is in fact real. Indeed, let us con-sider the equation 2(k2 = k2z +k

2) =

2 with regard tok2z . This equation is also a polynomial equation with real

coefficients. So its roots with regard to k2z are either neg-ative (no positive roots exist or there would be real ki z)or break up into complex-conjugate pairs. The negativeroots give imaginary ki z and complex-conjugate pairsk2i z, k

2j z give pairs ki z, kj z in C+, such that ki z = kj z ,

and for such a pair ki z + kj z = 2iIm ki z. Therefore theexpression inside of the braces in (6) is imaginary andthe right hand part is real.

Further insight is obtained if we search in Eq. (6) for2 in the form of a series expansion in k. On substi-tuting the expansions of all quantities by small k and into the equation (6), we obtain within O(k4)

2=

0 k3

1 k

20s2 +2 k2

820s

4 +

0

k2

0s2 k3

,

(7)

where = 12iS

i=2 k1i z (0, 0)

dk2i()

d2

=0

.

In the limit of small k and , we replace the braces inEq. (7) by unity, and so obtain the well-known disper-sion of capillary waves. The second and third terms inthe braces, take into account compressibility. The fourthterm, proportional to , expresses the influence of the(S 1) roots ki with i > 1, which include the rotons.The latter give only small correction at small , butwith increasing frequency they become of the order ofthe phonon terms and, as it will be shown below, they

determine the asymptotic behavior of the curve (k) inthe proximity of = rot.

Of particular interest is the behaviour of the dispersioncurve close to the roton gap = rot. In this region thefunctions k2,3 z(, k) contain terms

rot, which

leads to the same square-root singularity in the equation(6): (k2 z + k3 z)

rot. Therefore it can be shown

that the asymptote of (k) at = rot0 has the form

kc k = ba + c

rot , (8)

where

a = 2 02rot

k4c

; b = 2 krotkc

2 h /(k

2rotk2c );

ic = k11 z (rot, kc)+S

i=4k1i z (rot, kc)

Si=2

k1i z (0, kc);

krot is the momentum of a roton with energy rot, isthe roton mass and kc is determined by the condition

(k = kc) = rot. The numerical solution below, showsthat this point indeed exists, is unique and kc < krot, soa,b,c are all real. We see that the curve (k) approachesthe value = rot at the top of a parabola, and ends inthe adhesion point, with zero derivative. There is nodispersion curve below rot with k > kc. In the theoryof Ref. [17] the asymptote was obtained qualitativelyas one of the possible variants, from general quantum-mechanical considerations.

In Fig.1 we show the dispersion curve for bulk exci-tations. The areas bounded by the dispersion curve andthe lines at constant energy at rot, max, and 2rot arelabelled A to G. We consider the helium surface to have

a sinusoidal perturbation with wavenumber k and thensee if a stable ripplon solution is possible in each of theareas. A ripplon solution requires k1z to be imaginary.In region A, k < k1 and k

21z = k

21k2 so k1z is real, and

for k1z real, Eq. (6) gives k complex which means theripplon decays, so no ripplon can occur in this area. Theimposed perturbation would decay into phonons. In re-gion B, k1z is not real and therefore Eq. (6) gives k real,so ripplons can exist. They will lie on a ripplon disper-sion curve in this area. In region C, k1z k2z and k3z arereal and the perturbation decays into phonons, Rrotonsand R+ rotons. In region D, k2z and k3z are real and theperturbation decays into R rotons and R+ rotons. Inregion E, k

3zis real and the perturbation decays into R+

rotons. So there are no ripplons in these regions. In re-gion F there are no solutions to Eq. (6) because of thehigh gradient of the dispersion curve, so there are no rip-plons. In region G, the gradient is smaller and there aresolutions to Eq. (6) so we predict that there are ripplonsin this area.

IV. NUMERICAL SOLUTION AND THE NEW

RIPPLON BRANCH

We find the ripplon dispersion curve numerically in therange ofk

(0, 3) A1. It is shown on Fig.1. We see that,

as predicted, the computed curve at small wave vectorsis close to the classical k

3/2 law, but deviates from it at

larger k and approaches the level of rot at the top ofan inverted parabola at kc = 1.27 A

1.There is no ripplon solution in the region to the right

(see Fig.1) of the R+ roton dispersion curve for (rot, max). However, if we further increase whilemoving along the curve k3(), as we get close to the re-gion where (k)/k reaches its maximum, the curve turnsdown and aims to the point of instability [20] in almost


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FIG. 1. The dots are experimental data for the dispersionrelation of bulk excitations (k) [23], and the thin lineshows its analytic approximation that is used, with S=18.The thick line shows the results of numerical solution ofEq. (6) for the ripplon dispersion curve (k). Largedots with error bars are experimental data for the ripplondispersion [4]. Two black dots in the high-energy part ofthe bulk spectrum designate the end points of the veryhigh energy ripplon solution, and the insert graph showsits behaviour schematically, where = 0.7 103A1 and = 1.6 103K. The different regions of the (,k) planeare labelled A to G. The usual ripplons are in region Band the new ripplons are in region G. In the other regions,ripplons cannot exist, as described in the text

a straight line, the left hand part of Eq. (6) starts tochange slowly. At the same time, some of the functionsk2i () for i > 3, which were slowly changing functionsbelow max, begin to change fast and give significantcontribution to right-hand part of Eq. (6), so here theripplon solution is determined by the functions ki z.

Therefore, when we search for the solutions on the R+

roton curve, we find two common points above max.The first point is at 2.52 A1 and 16.65 K, which is almostexactly at the maximum of (k)/k, and the second pointis 2.75 A1 and 17.30 K. The ripplon dispersion curve(k) between these points lies near, but below, the bulkdispersion. The two common points are adhesion points.This is partly the reason for the deviation between thecurves being extremely small. At k = 2.6 A

1, midwaybetween the two end points, the deviation is 1.6 103 Kor 0.7 103 A1, which is too small to see on the scaleof the main graph, so in the inset to Fig.1 we show this

region expanded. Nevertheless the separation of the rip-plon dispersion curve from the roton dispersion curve, isgreatly exaggerated.

We now show that the common points of the ripplondispersion curve (k) and the R

+ roton dispersion curvek3() are adhesion points. The Eq. (6) can be rewrittenas F(, k)=0. There, the term k3 z in F(, k), goes to

zero as

kk3() near the R+ roton branch. So thegradient ofF(, k) in the plane (, k) tends to infinityon the curve k = k3() and is directed normal to the

curve. From the other side, the gradient of F(, k) isdirected normal to the curve (k), which is its levelcurve F = 0. Therefore at the common points of thetwo curves, the angle between them is zero, and so thesepoints are adhesion points.

We now consider the penetration depth of the ripplonsolution on this branch of the ripplon dispersion curve.At k = 2.6 A

1 the value of k3 z is close to its maximum

0.06 A1

. Hence the penetration depth of the ripplonsolution, determined by the term containing k3 z, hasa minimum value |k3 z|1 16A, and at the endpoints it tends to infinity. The deviation between thetwo curves is second order in the small parameter k3 zbecause |k3 z|

kk3.

The relatively large penetration depth of the solutionmeans that macroscopic films or the surface of bulk he-lium are needed to observe these ripplons. They will notbe seen on films of a few mononlayers. The high-energyripplons should exist and be observable on saturated filmsof He II, which have a typical thicknesses of 300A. In thesame way the penetration depth of the ripplons in region

B (see Fig.1) tends to infinity when tends to rot.So the ending point of the dispersion curve (rot, kc)can also be observed only in thick enough films of He II.Thus in the two most interesting regions of their disper-sion curve, the ripplon solutions have large penetrationdepth, much larger than the characteristic distances, ofseveral monolayers thickness, of the changes of the den-sity profile at the free surface. The major part of the en-ergy of the wave is stored outside of the transition layer,it can be neglected, and the surface can be consideredsharp (i.e. the kernel h(r) in Eq. (1) is the same as inthe bulk fluid).

The numerical calculations were carried out by repre-senting the function 2(k2) as a polynomial of degree S,

with S= 18 and S= 21. The high-energy ripplons solu-tion exists in both cases, and the end points do not differsubstantially. The deviation between the curves k3()and (k) remains very small, and is much less than thedifference between the two polynomials.

We have also checked for the stability of the solutionwith regard to changes in surface tension , as at highcurvatures of the surface its curvature dependence mightbe significant. If we decrease , the end points of theripplon dispersion curve slide apart along k3(). At thevalue = 1.75 N/m ( = 3.544 N/m at zero temperature[23]), the lower adhesion point reaches the point 2.44A1,16 K. If we increase , then the end points slide towards

each other and the solution disappears at = 8.7 N/m.This shows that any realistic dependence (k) can beinserted in Eq. (6) without complicating the numericalsolution.

V. CONCLUSION

We have developed a non-local hydrodynamical modelof ripplons on the free surface of superfluid 4He, which


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uses the measured dispersion curve for phonons and ro-tons to include the relevant characteristics of the liquid.The model gives a very good description of the measuredripplon dispersion curve, and it accounts for the parabolicapproach of the dispersion curve to the line = rot.The model gives the correct classical dispersion of capil-lary waves at long wavelengths. These good agreementsvalidate the model. Furthermore the model is transpar-

ent enough to expose the physical reasons why the lowenergy ripplon branch lies below the phonon dispersioncurve and ends at the energy rot. It is explained, withthe aid of Fig.1, that surface perturbations, on the lowk side of the phonon-roton dispersion curve, just decayto phonons or rotons and so stable ripplons can only befound on the high k side of the phonon-roton dispersion

curve. For energies > max, and on the high k side ofthe phonon-roton dispersion curve, ripplon solutions ofEq.(6) are possible at energies just below 2rot. It canbe shown that these ripplons are stable against decay intotwo ripplons with energy < rot. That such ripplons so-lutions exist is unexpected and is a striking predictionof this analysis. We hope this prediction stimulates newexperiments to detect these ripplons.

ACKNOWLEDGMENTS

We are grateful to EPSRC of the UK (grant EP/F019157/1) for support of this work.

[1] P. J. King and A. F. G. Wyatt, Proc. Roy. Soc. Lond. A,322, 355 (1971).

[2] P. Roche, G. Delville, N.J Appleyard and F.I. B.

Williams, J. Low Temp. Phys. 106, 565 (1997).[3] K. R. Atkins, Can. J. Phys. 31, 1165 (1953).[4] H. J. Lauter, H. Godfrin, V. L. P. Frank and P. Leiderer,

Phys. Rev. Lett. 68, 2484 (1992).[5] H.J. Lauter, H. Godfrin and P. Leiderer, J. Low Temp.

Phys. 87, N 3-4, 425 (1992).[6] C. C. Grimes and G. Adams, Phys. Rev. Lett. 36, 145

(1976).[7] D. Coimbra, S.S. Sokolov, J.-P. Rino and Nelson Studart,

Phys. Rev. B 74, 035411 (2006).[8] M. Brown and A. F. G. Wyatt, J. Phys. Condens. Mat

15, 4717 (2003).[9] V. U. Nayak, D. O. Edwards and N. Masuhara, Phys.

Rev. Lett. 50, 990 (1983).[10] A. F. G. Wyatt, M.A. H. Tucker and R. F. Cregan, Phys.

Rev. Lett. 74, 5236 (1995).[11] G. E. Volovik, J. Low Temp. Phys. 145, 337 (2006).[12] P. Roche, G. Delville, K.O. Keshishev, N.J Appleyard

and F.I.B. Williams, Phys. Rev. Lett. 75, 3316 (1995).

[13] L. Pitaevskii and S. Stringari, Phys. Rev. B 45, 13133(1992).

[14] I. N. Adamenko, K. E. Nemchenko and I. V. Tanatarov,

Phys. Rev. B 67, 104513 (2003).[15] I. N. Adamenko, K. E. Nemchenko and I. V. Tanatarov,

Phys. Rev. B 77, 174510 (2008).[16] K.R. Atkins, Phys. Rev. 116, 1339 (1959).[17] A. Lastri, F. Dalfovo, L. Pitaevskii and S. Stringari, J.

Low Temp. Phys. 98, Nos. 3/4, 227 (1995).[18] V.D. Natsik, Low Temp. Phys. 33, 999 (2007) [Fiz. Nizk.

Temp. 33, 1319, (2007)].[19] I. N. Adamenko, K. E. Nemchenko and I. V. Tanatarov,

J. Phys.: Conf. Series 150, No.3, 032107 (2009).[20] L. Pitaevskii, Zh. Eksp. Teor. Fiz. 36, 1168 (1959) [Sov.

Phys. JETP 9, 830 (1959)].[21] L.D. Landau, E.M. Lifshitz, Fluid Mechanics (Course of

Theoretical Physics, Vol. 6), Pergamon Press, London(1987).

[22] I.N. Adamenko, K.E. Nemchenko, I.V. Tanatarov andA.F.G. Wyatt, J. Phys.: Cond. Mat. 20, 245103 (2008).

[23] R. J. Donnelly, J. A. Donnelly and R. N. Hills, J. LowTemp. Phys. 44, 471 (1981).

i.v. tanatarov et al- a new ripplon branch in he ii

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