Fermi liquid theory applied to a film on an oscillating substrate
FFermi liquid theory applied to a film on an oscillating substrate
J. A. Kuorelahti, J. A. Tuorila and E. V. Thuneberg
Department of Physics, P.O.Box 3000, FI-90014 University of Oulu, Finland (Dated: June 15, 2018)We consider a film of a normal-state Fermi liquid on a planar substrate. Landau’s Fermi liquidtheory is applied to calculate the linear response of the film to transverse oscillation of the substrate.The response consists of a collective transverse zero sound mode, as well as incoherent quasiparticleexcitations of the degenerate fermions. We calculate numerically the acoustic impedance of the filmunder a wide range of conditions relevant to normal state He at millikelvin temperatures. Somecases of known experiments are studied but most of the parameter range has not yet been testedexperimentally. a r X i v : . [ c ond - m a t . o t h e r] N ov partially specular reflectionspecular reflection FIG. 1. Simplified diagram of the problem of a liquid film on an oscillating substrate, and its solution in the space formedby momentum projection µ = p z /p F = cos θ and the dimensionless vertical coordinate ζ = z/d . The boundary conditions aregiven in Eq. (17). I. INTRODUCTION
Let us consider a layer of liquid on a planar substrate. Assume the substrate oscillates harmonically in its plane. Theliquid is dragged into motion by the moving substrate. A measurable quantity is the transverse acoustic impedanceof the liquid layer. It is defined as the ratio of the force on the liquid to the velocity amplitude of the substrate.The impedance consists of a dissipative real part and a reactive imaginary part. The latter can be interpreted asthe amount of mass of the liquid that is coupled to the oscillation of the substrate. For an ordinary liquid theNavier-Stokes equations reduce to a diffusion equation and the transverse acoustic impedance can straightforwardlybe calculated. A schemantic of the set up is presented in Fig. 1.The purpose of this paper is to calculate the transverse acoustic impedance of a layer of a Fermi liquid. ByFermi liquid we mean that the fluid is described by Landau’s Fermi liquid theory . Landau’s theory is a paradigmof what can be the state of an interacting many-body system. The central idea is that although the particles arestrongly interacting, the low-energy properties of the system can be described by weakly interacting excitations calledquasiparticles. Similar to molecules in a rarefied gas, the quasiparticles can have a long mean free path, but thereis an essential difference that even in the absence of collisions, interactions between the quasiparticles remain. Thishas important effects. For example, it allows the propagation of density oscillations even in the absence of collisions,so called zero sound. Also transverse oscillations can propagate as a wave, in contrast to Navier-Stokes fluid wheresuch motion obeys a diffusion equation. Landau’s theory is explained in many articles and textbooks . OriginallyLandau formulated the theory for liquid He, but it also forms the basis for understanding the behavior of conductionelectrons in metals. Extension of the Fermi liquid theory to include paring correlations gives an accurate descriptionof the superfluid or superconducting state of a Fermi liquid .The calculation of the impedance requires solution of the transport equation, the Landau-Boltzmann equation inappropriate geometry. This was first done by Bekarevich and Khalatnikov . Their solution was extended by Flowersand Richardson . These solutions are basically analytic but they are very complicated. Simpler approximate resultswere derived by Richardson . All these assume a thick liquid layer, in principle filling a half space. Our purposeis to generalize these calculations to a liquid layer of finite thickness. Instead of an analytic approach, we solve theLandau-Boltzmann equation by discretization and numerical inversion of the resulting large matrix.There are a several motivations for the present work. First, in previous work the response of a Fermi liquid on avibrating cylinder was studied . Such a calculation is computationally demanding and therefore it is of interest tostudy similar phenomena in the simpler planar geometry. Second, the behavior in a finite layer is much more diversethan in the thick layer limit. An indication of this is already given in the torsional oscillator calculations . Third, theexperiments by Casey et al and Dimov et al show unexpected decoupling of the liquid from the substrate. Previousanalysis of these experiments neglected the Fermi-liquid interactions , and therefore we wanted to check if these havean effect. Fourth, we were interested to check if finite thickness effects could have affected previous experiments,and to predict the outcome of possible future experiments. Fifth, understanding the Fermi-liquid interactions in thenormal state could form a useful step for properly incorporating the Fermi-liquid effects in the acoustic impedance ofthe superfluid state .We use Fermi-liquid equations in the relaxation-time approximation and including interaction effects up to secondorder in spherical harmonics. The approach includes the effect of the collective transverse zero-sound mode as wellas incoherent quasiparticles. In the limit of short mean free path of the quasiparticles, the flow of a Fermi liquidobeys Navier-Stokes equations. The leading correction to this hydrodynamic limit arises as “slip” in the boundaryconditions . In our numerical solution no such expansion is made and thus the slip effect is included in all orders.The calculations are mainly aimed for experiments in liquid He. However, the results apply also to simultaneouspresence of a Bose condensate , and thus can also be applied to mixtures of He and He. Reviews of the acousticimpedance studies in both normal and superfluid He has been given by Halperin and Varoquaux and Okuda andNomura . A theoretical review is given by Nagai et al . Recently experiments using a planar micromechanicaloscillator in normal He have been made by Gonzalez et al and are analyzed using slip theory .This paper is structured in the following way. In Sec. II we state the basic Fermi-liquid equations and transformthem to a form suitable for numerical solution. In Sec. III we take a look at different limiting cases. In Sec. IV weexplain the numerical method and finally in Sec. V we present and comment the results. II. FERMI LIQUID EQUATIONSA. Equations of motion
We study the linear response of a Fermi liquid film to the transverse oscillations of a planar substrate. We derive anexpression for the acoustic impedance Z = F/u , where F is the force on the liquid per unit area of the film, and u thevelocity amplitude of the substrate, which is assumed to oscillate at angular frequency ω . In this section we transformanalytically the equations of a Fermi liquid to a form that then can be solved numerically. The transformation couldbe done by making only slight modification to the derivation by Flowers and Richardson . Here we present a moregeneral derivation that directly utilizes the distribution function defined in terms of momentum direction and energyinstead of momentum. The two distributions appear in several works , and their relation is clearly pointed outin Ref. 6. We use the notation of a recent work that also includes the effect of condensed bosons . We start byconsidering a pure fermion system, and postpone the minor effect of the bosons to Sec. II C.Fermi liquid theory deals with quasiparticles with momenta p close to the Fermi surface, p ≈ p F . Assuming nospin dependence, the quasiparticle distribution function and energy can be written as n p ( r , t ) = n ( (cid:15) p ( r , t )) + δ ¯ n ( ˆ p , (cid:15) p ( r , t ) , r , t ) , (1) (cid:15) p ( r , t ) = (cid:15) (0) p + δ(cid:15) ˆ p ( r , t ) . (2)Here ˆ p = p /p is the momentum direction, n ( (cid:15) ) = 1 / ( e (cid:15)/T + 1) the Fermi function, (cid:15) (0) p = v F ( p − p F ) the unperturbedquasiparticle energy, T the temperature, v F = p F /m ∗ the Fermi velocity and m ∗ the effective mass. We also definethe energy-integrated distribution function ψ ˆ p ( r , t ) = (cid:90) δ ¯ n ( ˆ p , (cid:15), r , t ) d(cid:15). (3)In the relaxation-time approximation, the linearized kinetic equation can be written in a closed form for ψ ˆ p . Onegets the equations ∂∂t ( ψ ˆ p − δ(cid:15) ˆ p ) + v F ˆ p · ∇ ψ ˆ p = − τ [ ψ ˆ p − (cid:104) ψ ˆ p (cid:48) (cid:105) ˆ p (cid:48) − (cid:104) P ( ˆ p · ˆ p (cid:48) ) ψ ˆ p (cid:48) (cid:105) ˆ p (cid:48) + 5( ξ − (cid:104) P ( ˆ p · ˆ p (cid:48) ) ψ ˆ p (cid:48) (cid:105) ˆ p (cid:48) ] , (4) δ(cid:15) ˆ p = ∞ (cid:88) l =0 F l F l / (2 l + 1) (cid:104) P l ( ˆ p · ˆ p (cid:48) ) ψ ˆ p (cid:48) (cid:105) ˆ p (cid:48) . (5)Here F l with l = 0 , , . . . are the interaction parameters, P l the Legendre polynomials [ P ( x ) = 1, P ( x ) = x , P ( x ) = (3 x − (cid:104) . . . (cid:105) ˆ p the average over the unit sphere of momentum directions. In a pure fermionsystem F is related to the ratio of effective and particle masses, m ∗ /m = 1 + F . Equations (4)-(5) are the sameas derived in Ref. 20 except the following generalization. We have allowed two relaxation times, τ = τ /ξ for aquadrupole deformation of the Fermi surface and τ for all higher order deformations.In the following we neglect Fermi-liquid interaction coefficients beyond second order, F l = 0 for l >
2. We alsoassume harmonic time dependence ∝ exp( − iωt ). These allow to write Eqs. (4)-(5) to the form τ v F a ˆ p · ∇ ψ ˆ p + ψ ˆ p − /a + F F (cid:104) P ( ˆ p · ˆ p (cid:48) ) ψ ˆ p (cid:48) (cid:105) ˆ p (cid:48) − b (cid:104) P ( ˆ p · ˆ p (cid:48) ) ψ ˆ p (cid:48) (cid:105) ˆ p (cid:48) − c (cid:104) P ( ˆ p · ˆ p (cid:48) ) ψ ˆ p (cid:48) (cid:105) ˆ p (cid:48) = 0 . (6)We have defined dimensionless complex constants a = 1 − iωτ, (7) b = 1 /a + F /
31 + F / , (8) c = 5 /a + F F / − ξ a , (9)in accordance with Ref. 8.We choose z axis perpendicular to the liquid film and assume homogeneity in the x - y plane. The x axis is chosenparallel to the oscillation of the wall. With these assumptions the most general distribution allowed in linear responsecan be written as ψ ˆ p ( r ) = ˆ p x ψ (ˆ p z , ζ ) . (10)We also have defined ζ = z/d as the dimensionless z coordinate, where d is the thickness of the liquid film. The form(10) allows to simplify the averages (cid:104) P ( ˆ p · ˆ p (cid:48) ) ψ ˆ p (cid:48) (cid:105) ˆ p (cid:48) = 0 , (cid:104) P ( ˆ p · ˆ p (cid:48) ) ψ ˆ p (cid:48) (cid:105) ˆ p (cid:48) = ˆ p x g ( ζ ) , (cid:104) P ( ˆ p · ˆ p (cid:48) ) ψ ˆ p (cid:48) (cid:105) ˆ p (cid:48) = ˆ p x ˆ p z g ( ζ ) . (11)Here the first average vanishes because transverse oscillations do not change the density of the liquid. The latter twoaverages depend on the integrals g ( ζ ) = (cid:90) − dµ (1 − µ ) ψ ( µ, ζ ) , (12a) g ( ζ ) = (cid:90) − dµ µ (1 − µ ) ψ ( µ, ζ ) . (12b)Inserting these into the kinetic equation (6) gives µh ∂∂ζ ψ ( µ, ζ ) + ψ ( µ, ζ ) − bg ( ζ ) − cµg ( ζ ) = 0 . (13)We have abbreviated the equation by defining µ = ˆ p z and one more complex coefficient h = adv F τ = dlξ − i Ω(1 + F / . (14)The latter form expresses h using the dimensionless parameterΩ = ωdv F (1 + F /
3) (15)and the mean free path l . Since τ is the effective relaxation time in the hydrodynamic limit, the quasiparticle meanfree path is defined as l = v F τ = v F τ /ξ . A convenient set of dimensionless parameters that define the problem isformed by Ω, l/d , ξ , F and F .We may further solve for ψ by integrating the kinetic equation (13) from ζ to ζ : ψ ( µ, ζ ) = ψ ( µ, ζ ) e hµ ( ζ − ζ ) + 34 hµ (cid:90) ζζ e hµ ( ζ (cid:48) − ζ ) [ bg ( ζ (cid:48) ) + cµg ( ζ (cid:48) )] dζ (cid:48) . (16)We use this equation to integrate in the direction of particle propagation. That is, we integrate in the direction ofincreasing ζ for angles pointing up ( µ >
0) and decreasing ζ for angles pointing down ( µ < z = d , that is ζ = 1. We assume that the quasiparticle scattering at this surface is diffusive except for a fraction s of quasiparticles, which is scattered specularly. The case s = 1 then mimics a free surface of a liquid. We set theoscillating wall at z = ζ = 0. Its velocity is u ˆ x e − iωt . We assume that the quasiparticle scattering at this wall isdiffusive except for a fraction s of quasiparticles, which is scattered specularly. These imply the following boundaryconditions for the distribution function ψ ( µ < ,
1) = s ψ ( − µ, , (17a) ψ ( µ > ,
0) = s ψ ( − µ,
0) + (1 − s ) p F u. (17b)Note that because of symmetry, it is also possible to consider the liquid between two equally oscillating walls bysetting s = 1 and 2 d being the distance between the walls.We see from the boundary conditions (17) that the distribution function has to be proportional to p F u . We canthen factor out p F u for the sake of numerical convenience by defining the effective fields ψ e = ψp F u , g e = g p F u , g e = g p F u . (18)This simplifies the boundary conditions (17) but the bulk equations (12) and (16) remain the same for the effectivefields. Equations (12), (16) and (17) constitute the set of integral equations and boundary conditions that we cansolve numerically. B. Observables
The macroscopic forces acting in the liquid are obtained by calculating the stress tensorΠ ij = 3 n (cid:104) ˆ p i ˆ p j ψ ˆ p (cid:105) ˆ p , (19)where n = p F / π (cid:126) is the number density of the fermions. The shear force, the xz component of the stress tensor,can be evaluated using (10) and (12b), which givesΠ xz ( ζ ) = 34 ng ( ζ ) = 34 np F ug e ( ζ ) . (20)The acoustic impedance of the liquid film is then Z = Π xz ( ζ = 0) u = 34 np F g e ( ζ = 0) . (21)The mass current in the liquid is J = mp F π (cid:126) (cid:104) ˆ p ψ ˆ p (cid:105) ˆ p . (22)Evaluating this using (10) and (12a) gives that the current is in the x direction and its magnitude J ( ζ ) = mp F π (cid:126) g ( ζ ) = 34 ρug e1 ( ζ ) , (23)where the liquid density ρ = mn . Thus g e1 ( ζ ) can be interpreted as the average velocity normalized by the substratevelocity u . In the hydrodynamic limit this should approach unity at the substrate ( ζ = 0). This is the velocity fieldin the transverse wave and should not be confused with the velocity of the wave itself. C. Bose-Fermi liquid
The theory above can straightforwardly be generalized to the simultaneous presence of condensed bosons . Intransverse oscillations the superfluid component remains at rest. In the notation of Ref. 20, v s = 0 and δµ B = 0. Theequations, in particular Eqs. (34) and (35), of Ref. 20 reduce to those ones in the present paper. The only difference isthat Eq. (23) gives only the fermionic contribution to the current. In order to get the total mass current, the fermiondensity ρ should be replaced by the normal fluid density ρ n = m ∗ n/ (1 + F / III. LIMITING CASES
There are limiting cases that are worth of studying separately. In some cases analytic solutions are known.
A. Hydrodynamic limit
At high temperatures quasiparticle collisions become frequent and thus the mean free path is short. In the hydrody-namic regime l is short compared to other length scales, l (cid:28) d and l (cid:28) v F /ω . In this regime the Fermi-liquid theoryimplies equations of motion that are the well known hydrodynamic, or Navier-Stokes equations. Solving these forlaminar flow between two parallel planes is a common exercise in books on hydrodynamics . We need to considertwo boundary conditions for the top surface; a free liquid surface ( s = 1) and an unmoving solid surface, i.e. Couetteflow ( s = 0). For the oscillating wall, we assume no slip ( s = 0). We use the Navier-Stokes equation to calculatethe force at the boundary of the liquid and the oscillating surface. This way one gets the acoustic impedance Z = Fu = ρωδ − i ) e − i ) d/δ ∓ e − i ) d/δ ± . (24)The upper signs stand for a free liquid surface and the lower signs for Couette flow. These are identical when d/δ islarge. Here δ is the viscous penetration depth, which is related to other parameters by equations δ = d (cid:114) l d = (cid:114) v F τ (1 + F / ω . (25)We see that the essential dimensionless parameter in (24) is δ/d . B. Ballistic limit
At very low temperatures quasiparticle collisions become so infrequent that they may be neglected. This is theballistic regime, where the quasiparticles travel between the oscillating plane and the liquid surface without encoun-tering each other. In the ballistic limit we take l → ∞ . In the general case, this does not lead to any simplificationof the kinetic equation (16), as the limit a → ∞ still leaves b (8) and c (9) finite. However, if we also set the Fermiliquid interactions F and F to zero, then b = c = 0. We call this the ballistic gas limit . This leads to essentialsimplification of the kinetic equation (16), which reduces to the form ψ e ( µ, ζ ) = e hµ ( ζ − ζ ) ψ e ( µ, ζ ) . (26)By using this equation and the boundary conditions (17) to traverse rectangular paths in the ( µ, ζ ) space (Fig. 1), weobtain the quasiparticle distribution µ > ψ e ( µ,
0) = 1 − s − s s e − h/µ , (27) µ < ψ e ( µ,
0) = s (1 − s ) e h/µ − s s e h/µ . (28)We can now compute g e at the oscillating wall, which gives the acoustic impedance Z = 34 np F (1 − s ) (cid:90) dµ µ (1 − µ ) 1 − s e i Ω /µ − s s e i Ω /µ . (29)We see that this depends essentially on Ω (15). Note that this result is valid only in the case of F = F = 0. C. Thick film limit
Let us consider the case of a very thick film, d → ∞ . This was first studied by Bekarevich and Khalatnikov andmore generally by Flowers and Richardson . They found an analytic solution. Since the result is not simple, we willnot reproduce it here. Instead we point out that there are two variational ansatz solutions given in equations (3.32)and (3.38) of Ref. 9. The essential dimensionless parameter in the thick film limit is Ω l/d = ωτ / (1 + F ). IV. NUMERICAL SOLUTION
The search for the numerical solution begins with the discretization of the ( µ, ζ ) space. Generally the ζ axis isdivided into segments of equal length so that ζ j = ( j − / ( n − j = 1 , ..., n . For high temperatures, thewave emanating from the oscillating wall will not penetrate deep into the liquid layer. In this case, instead of equallyspaced lattice points, it is more efficient to use a discretization that places lattice points more densely in the vicinityof the oscillating wall.Different discretization schemes may be employed for the µ axis. We approximate integrals over µ (12) with (cid:90) − f ( µ ) dµ ≈ m (cid:88) i =1 w i f ( µ i ) . (30)The values µ i and the weights w i are selected using Gaussian quadrature. We use an even number m of µ i values inorder to avoid the µ = 0 point, which could require special treatment.The integration over ζ (16) is made only between neighboring discretized points. Instead of a simple trapezoidalformula we use (cid:90) ζζ e αζ (cid:48) f ( ζ (cid:48) ) dζ (cid:48) ≈ w f ( ζ ) + w f ( ζ ) ,w = 1 α ( ζ − ζ ) [ e αζ − (1 + α ( ζ − ζ )) e αζ ] ,w = 1 α ( ζ − ζ ) [ e αζ − (1 − α ( ζ − ζ )) e αζ ] , (31)where α = h/µ . This method allows better accuracy if the exponential factor inside the integral in equation (16)varies rapidly.The discrete versions of the integral equations (12), (16) and (17) provide a network of linear dependencies between ψ ( µ i , ζ j ), g ( ζ j ) and g ( ζ j ). We now form a vector Ψ that holds all these variables in the following fashion:Ψ = ( ψ e ( µ , ζ ) , ..., ψ e ( µ m , ν n ); g e ( ζ ) , ..., g e ( ζ n ); g e ( ζ ) , ..., g e ( ζ n )) . (32)The length of this vector is d = mn + 2 n . The network of linear dependencies may now be represented in the formΨ = D Ψ + B. (33)Here the left hand side represents the left hand side of equations (12), (16) and (17), and correspondingly for the righthand sides. The matrix D is of dimension d × d . The vector B is the inhomogeneity term arising from the non-specularscattering at the oscillating wall in the last term of Eq. (17b). Equation (33) is a system of linear equations for Ψ,whose solution can be written as Ψ = ( I − D ) − B. (34)The task is now to first assemble the D matrix and then solve the linear system (33) to get Ψ. The acousticimpedance (21) is then obtained by picking out the element that corresponds to g e (0). We can also pick out g e at any ζ to get the stress tensor (20) within the liquid or g e to get the transverse velocity field (23). While the dimension d × d may be very large, the D matrix only has 7 mn − m elements that can be non-zero. We use sparse matrixmethods for solving the the inverse (33). This requires specifying those elements of the matrix D that can be nonzeroso that the zero elements never need to be addressed. Since B is sparse and only a few elements of Ψ are of interest,there is no need to calculate the whole inverse matrix ( I − D ) − . V. RESULTS
Before presenting the results of the numerical calculations, we outline the parameter values that define experimen-tally relevant conditions. Foremost there are the Fermi-liquid interaction parameters F and F that describe theforces between the quasiparticles. In pure He the parameter F is pressure dependent and its value has been deter-mined experimentally . Some notable values are F = 5 . F = 13 . FIG. 2. Parametric plot of Z = Z (cid:48) + iZ (cid:48)(cid:48) with l/d as a variable for a small Ω = 10 − (15). Different curves correspond todifferent values of the Fermi-liquid parameter F and the specularity s . The liquid has a free top surface ( s = 1) and theparameter s controls the specularity of the oscillating bottom wall. The arrows point out the direction of increasing l/d , thevarious analytical limiting cases and the approximate point where l/d ∼
1. For solid curves F = 0 and for dashed curves F = 5 .
4. The hydrodynamic limit (24) is shown by black dashed line. The ballistic gas limit (29), represented by the blacksolid line, was computed using s = 0 and F = 0. Other parameters are F = 0 and ξ = 1. There are no generally accepted values for F , but it is thought to range between − . A requirement forthe existence of transverse zero sound is expected to be F + 3 F F > F n with n > ξ = τ /τ of the two relaxationtimes, for which the value ξ = 0 .
35 has been suggested . Other dimensionless parameters are Ω = ωd/v F (1 + F )(15) and l/d . The former depends essentially on the film thickness d and frequency ω whereas the latter dependsessentially on the temperature as the mean free path l ∝ T − in the Fermi liquid regime. We use the coefficient lT based on viscosity measurements as given in Tables III and IV of Ref. 32 and m ∗ /m from Ref. 26. The specularityparameters s and s define conditions at the two liquid boundaries.We show plots of transverse acoustic impedance Z = Z (cid:48) + iZ (cid:48)(cid:48) . The real part of the impedance corresponds todissipation and the imaginary part to reactance. We display parametric plots of Z as well as separate plots of Z (cid:48) and Z (cid:48)(cid:48) as functions temperature or l/d , for which we use a logarithmic scale.We consider first the case of small Ω (15), Ω (cid:28)
1. Supposing there are waves whose speed is on the order of theFermi velocity v F , the condition Ω (cid:28) l/d . The origin, Z = 0, corresponds to the stationaryfilm limit, where the liquid remains at rest in spite of the oscillation of the substrate. For small but finite l thediffusive waves generated at the oscillating surface penetrate to depth δ (25) into the liquid. These give rise to Z on a straight line in the direction 1 − i (24). The penetration depth increases with growing l . When δ ∼ d thewave starts to feel the liquid surface, and the path in the Z plane starts to curve. For small s the curves still stayclose to the hydrodynamic limit for some range of l . The hydrodynamic limit curve (24) continues towards the point Z/p F n Ω = − i , which corresponds to rigid body motion of the liquid with the substrate. Before reaching this point,the curves develop a cusp at l ≈ d . With further increase of l the system enters the ballistic, or Knudsen, regime.With l → ∞ they reach the ballistic limit point. A set of ballistic limit points (29) in the noninteracting, s = 0 caseis shown by the black solid line in Fig. 2.We see that with the scaling of Figure 2, the effect of the interaction parameter F is limited to mean free paths l > d . The same holds for F as well. Also, the effect of the interaction parameters is an order of magnitude smallerthan the whole scale of Z (cid:48)(cid:48) in the figure. With increasing specularity of the oscillating wall, the film becomes more FIG. 3. Acoustic impedance in the absence of Fermi-liquid interactions at different Ω (15). Displayed here are, from left toright, a parametric plot of the impedance and the real and the imaginary parts of the impedance as functions of l/d . Similarto Fig. 2, the black solid curve represents the ballistic gas limit (29). The dashed black curves represent the thick film limitaccording to Ref. 9. In the parametric plot (left panel) the dashed lines collapse to a single curve but in the real part vs. l/d (center panel) they are shifted from each other. For clarity, the thick film limit is omitted in the imaginary part vs. l/d (rightpanel). The results for the finite film differ from the thick film limit when the liquid surface is felt. From the middle panel wesee that this is the case when l/d > .
1. Other parameters are s = 0, s = 1, and ξ = 0 . decoupled. However, rather high specularity ( > .
9) is required to have a half of the liquid film mass decoupled in theballistic limit and there still remains strong dissipation that shows no sign of decreasing with increasing specularity.The results above may be applied to the experiments by Casey et al and Dimov et al , which report decouplingof the liquid from the substrate with decreasing temperature. Our motivation was to check if Fermi-liquid interactionscould be responsible for the decoupling. Since we see only minor decoupling, we have to conclude that our Fermi-liquidmodel is not capable to explain these experimental observations.Let us next consider the case of large Ω. In this case, when waves develop, there is room for several wave lengthsin the film. In this case the rigid body limit cannot be reached, and it is more convenient to analyze Z withoutscaling with Ω. An example of curves up to Ω = 2 is shown in Fig. 3. We again follow one curve in the order ofincreasing l/d . Initially the curve starts from the origin along a straight line in the direction Z (cid:48)(cid:48) = − Z (cid:48) , similarly asin the case of small Ω. However, for large Ω we exit the hydrodynamic regime before reaching δ ∼ d . This happensbecause ωτ ∼ Ω l/d approaches unity. Thus, the impedance deviates from the hydrodynamic limit and follows thecurve calculated in the thick film limit (Sec. III C). This curve is shown by dashed lines in Fig. 3. This continues aslong as the waves generated at the oscillating wall start to feel the surface and are reflected back. At this point thecurve deviates from the dashed line, as will be analyzed shortly.Figure 3 depicts the special case that the Fermi liquid interactions vanish, F = F = 0. This case is not realizedin pure He. Experimentally a close case could be studied in mixtures of He and He, where the Fermi liquidinteractions are weaker than in pure He [Ref. 11]. Interestingly, no waves are expected according to criterion (35).Still we see waves, the end points of the curves lie on the spiral, not at the end point of the dashed line. The spiralingdown indicates damping of these waves. Note that the ballistic limit curves (black solid lines) in Figs. 2 and 3 are thesame, the difference comes only from the different scalings used.We note that we did not succeed to compute numerically the exact analytic result of the thick film limit in the caseof Fig. 3. This apparently has to do with some numerical problem when the inequality (35) is not satisfied. Instead,we use the simpler of the approximate formulas, Eq. (3.32), of Ref. 9. We see that the curves initially follow the thickfilm behavior, until the effect of the liquid surface appears. Comparing the real parts of the two solutions gives thatthis takes place around l/d ∼ . Z plotted separately as functionsof l/d . We see that both the real and imaginary parts of the impedance fully plateau as l/d → F = 13 . . Similar to the noninteracting case,the curves initially follow the thick film behavior, until the effect of the liquid surface appears.0 FIG. 4. The same as Fig. 3 except that F = 13 .
3. The ballistic gas line is the same as in Fig. 3 to allow easier comparison ofthe figures. Because of interactions, the curves do not end on this line. The dashed black curves represent the exact thick filmlimit . Other parameters are F = 0, s = 0, s = 1, and ξ = 0 . F = 1. There is significant difference between the figures at large Ω. Figure 5 is again similar to the previous two, but this time also the second interaction parameter has a nonzerovalue, F = 1. Comparing this to Fig. 4, we see that the effect of F strongly increases for increasing Ω.We can understand the behavior of the ends of the curves in figures 4 and 5 as follows. With the condition (35)satisfied, the damping of the waves is weak. This means that with increasing Ω the end points nearly circle arounda point in the complex Z plane without any apparent damping. The waves emanating from the oscillating wall arereflected back from the top surface. By changing the layer thickness, sound velocity or the oscillation frequency, wepotentially alter the phase at which the waves return back to the oscillating surface. If the returning wave is in-phasewith the wall oscillations, then the oscillations are amplified. Conversely, a returning wave in opposite phase leads todestructive interference. Changing the liquid layer thickness by a quarter of the wavelength results in the oppositephase and a deviation in the opposite direction from the thick film limit. An implication of this is that in order tosee finite thickness effects, the boundaries of the liquid need to be accurately parallel.In Fig. 6 the layer thickness is fixed and the mean free path changes as a function of temperature. We haveused parameter values that correspond to the experiment by Roach and Ketterson . In the calculation the liquid isconfined between two diffusely reflecting plates spaced d = 25 µ m apart. We see that in this setting the presence ofthe top plate, which is seen as the bifurcation of the three curves, is only felt at temperatures below the superfluidtransition temperature T c . The liquid layer is simply too thick for the transverse sound wave to penetrate all the wayto the other wall and back at temperatures above T c . This can be confirmed in Fig. 7, where the transverse velocity1 FIG. 6. Acoustic impedance as a function of temperature for three different frequencies at the pressure of 23 bar. The solidcurves are for a liquid confined between two diffusely reflecting plates spaced 25 µ m apart, and correspond to Ω in the rangefrom 31 to 93. The dashed curves give the thick film limit. The two cases differ only at low temperatures, the slight differenceat high temperatures is due to numerical error. The black solid lines give the hydrodynamic solution (24). The vertical arrowdenotes the superfluid transition point, T c = 2 . F = 11 . F = 0, s = 0, s = 0, and ξ = 0 . field (23) is plotted as a function of temperature.The obvious thing to do is to repeat the computation in Fig. 6 for a thinner film. By selecting d = 2 . µ m andusing the smallest oscillation frequency 36 MHz, we conveniently have Ω ≈ F . The results are shown in Fig. 8. We have used two differentvalues of F . Both the thick film and finite film solutions show sensitivity to F . The bifurcation between these twosolutions happens well above the superfluid transition temperature. For both solutions an increase in F results inan initially identical shift in the impedance but, in addition to this, the thin film solution is influenced by the topplate once the temperature gets sufficiently low. This is especially apparent for Z (cid:48)(cid:48) , for which the thick film solutionsconverge as T → VI. SUMMARY
We have formulated how to calculate the transverse acoustic impedance of a Fermi-liquid film. We have built up ascheme for numerical evaluation. Some example results are presented in this paper aiming to clarify the case of a fewknown experiments and stimulate new ones. In the future we plan to extend the calculations to more general boundaryconditions, to transmission of transverse waves, and to separation of bulk and surface contributions. A generalizationof the present calculation to take Fermi-liquid effect into account in the superfluid state is under consideration.2
FIG. 8. The effect of F for a film of thickness d = 2 . µ m. Other parameter values are pressure 23 bar, frequency 36 MHz, F = 11 . s = s = 0, and ξ = 0 .
35. Solid curves depict the finite film solution, dashed curves the thick film limit. The blacksolid line represents the hydrodynamic solution.
ACKNOWLEDGMENTS
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