Fermi liquid theory applied to vibrating wire measurements in 3He-4He mixtures

2.6 MB·English

by  Timo H. Virtanen

Fermi liquid theory applied to vibrating wire measurements in 3He-4He mixtures
T. H. Virtanen1 and E. V. Thuneberg1
1Department of Physics, P.O.Box 3000, FI-90014 University of Oulu, Finland
(Dated: January 19, 2011)
We use Fermi liquid theory to study the mechanical impedance of 3He-4He mixtures at low
temperatures. The theory is applied to the case of vibrating wires, immersed in the liquid. We
present numerical results based on a direct solution of Landau-Boltzmann equation for the 3He
quasiparticle distribution for the full scale of the quasiparticle mean-free-path (cid:96). The two-fluid
nature of mixtures is taken into account in the theory, and the effect of Fermi liquid interactions
and boundary conditions are studied in detail. The results are in fair quantitative agreement with
experimental data. In particular, we can reproduce the anomalous decrease in inertia, observed
in vibrating wire experiments reaching the ballistic limit. The essential effect of the experimental
container and second-sound resonances is demonstrated.
PACSnumbers: 67.60.G-,67.10.Db
1
1 I. INTRODUCTION
0
2 Vibrating wire resonators have extensively been used to study the properties of quantum fluids. Different regimes

n canbeobservedinFermisystemsatlowtemperatures. Atrelativelyhightemperaturesthefluidobeyshydrodynamic
a description and the vibrating wire can be used to extract the density and the coefficient of viscosity of the liquid. As
J
the temperature is reduced below the Fermi temperature, the collisions between the quasiparticles, the elementary

8 excitations of the liquid, get rare as the final states of a collision are limited by the Pauli principle. This leads to
1
increasing mean-free-path of the quasiparticles, and to increasing viscosity. With increasing viscosity the vibrating

wire becomes more sensitive to its surroundings since more liquid is dragged to move with the wire, and this becomes
]
l limitedbytheexperimentalcontainer. Theincreasingmean-free-pathalsoleadstodeviationsfromthehydrodynamic
e
behavior, which are first visible as a slip on the walls: the velocity of the fluid does not vanish at the wall but
-
r extrapolates to zero a slip-distance behind the wall (in the rest frame of the wall). Lowering the temperature further,
t
s the mean-free-path becomes comparable to the radius of the wire. In this regime the dynamics of the quasiparticle
.
t gas needs to be described by the Fermi liquid theory, which is similar as the theory of rarified gases but uses Fermi
a
distributionandhasremaininginteractionsbetweenquasiparticles. Inthelimitoflowesttemperaturesoneentersthe
m
ballistic limit where the mean-free-path exceeds the size of the experimental container.
-
d Inthispaperweprovidetheoreticalcalculationsforavibratingwireinthefullrangeofmean-free-pathsinanormal
n Fermi system. The theory is applicable to pure liquid 3He, but the range of applicability is limited by the superfluid
o transition. Therefore the main application is in 3He-4He mixtures, where the fermions remain in the normal state
c in full range that has been measured. Several vibrating wire experiments have been made in 3He-4He mixtures. We
[ compareourcalculationstothemeasurementsbyMartikainenet al.1,2,whichextenddeepestintotheballisticregime.
1 In order to incorporate the 4He-component, we use Landau’s Fermi-liquid theory3 generalized to the simultaneous
v presence of condensed bosons, as formulated by Khalatnikov4. A formulation of this theory adapted to the present
8 workisgiveninRef.5. AshortaccountofthepresentcalculationsisgiveninRef.6. ThisletterhighlightstheLandau
2
force, themacroscopicforcecausedbytheFermiliquidinteractions. HerewecalculatetheLandauforceamongother
5
effects and compare to experiments. We find strong effect of the experimental container on the resonance frequency
3
of the vibrating wire. We also study sound resonances that affect the measurements at high frequencies or in large
.
1 containers.
0
Our calculation is the first two-dimensional solution of normal-state Fermi-liquid equations for general mean-free-
1
path. Even the cylindrically symmetric case we study, although effectively one-dimesional, is far more sophisticated
1
than has been done before. Previous work has considered uniform bulk liquid3,7–9 or one-dimensional geometries
:
v such as an oscillating planar wall in semi-infinte liquid10–12 or Poiseulle flow between parallel planes13,14. For more
i
X general geometries calculations have been done only in limiting cases of small mean-free-path15–17 and in the ballistic
limit16,18–20.
r
a WestartinSec.IIbystatingmorepreciselytheproblemwestudy. Webrieflypresenttheequationsweneed,which
are derived in detail in Ref. 5. The symmetries of the problem are studied in Sec. III. In Secs. IV and V we study
the special cases of vanishing mean-free-path in the hydrodynamic limit, and infinite mean-free-path in the ballistic
limit. The numerical method is introduced in Sec. VI and the parameter values used are discussed in Sec. VII. The
results of the numerical calculations are presented and discussed in Sec. VIII.
2
II. FORMULATION OF THE PROBLEM
A. Main assumptions
We study the force that a vibrating wire exerts on a Fermi liquid surrounding the wire. The wire is modeled as an
infinitecylinder. Weassumethatboththewireandtheliquidareinastationarycontainerthatalsohastheshapeof
an infinite cylinder. These assumptions are not essential in principle, but they simplify the numerical calculations by
allowing translation symmetry in the direction of the cylinder axis. The cross section of the wire is taken as a circle
of radius a. The oscillation direction of the wire (chosen as the x axis) is perpendicular to wire axis (chosen as the z
axis).
Forthecontainerweconsidertwodifferentshapes. Bycylindricalcontainerwemeanacircularcylindricalcontainer
of radius b that is concentric with the wire. By slab container we mean that the wire is in the middle between two
parallelplanewallsspacedby2h,andtheoscillationdirectionofthewireisnormaltothewalls. Becauseofthelatter
case, circular symmetry is not valid in general.
TheforceexertedontheliquidperunitlengthofthewireisdenotedbyF. Weareinterestedinthelinearresponse
of the fluid. For that we express the force in terms of the mechanical impedance of the fluid, Z, as
F =Zu, (1)
where u is the velocity of the wire. In general Z is a tensor, but we concentrate on symmetric cases, where Z is
a scalar. We assume harmonic time dependence exp(−iωt) with angular frequency ω. Thus Z is complex valued,
Z =Z(cid:48)+iZ(cid:48)(cid:48). Theimpedanceisdirectlyrelatedtotheresonancefrequencyf andthelinewidth∆f oftheresonator
0
by
Z(cid:48) = 2π2a2ρ ∆f,
w
Z(cid:48)(cid:48) = 4π2a2ρ (f −f ). (2)
w 0 vac
Hereρ isthedensityofthewirematerialandf theresonancefrequencyofthewireinvacuum. Therestrictionto
w vac
linear response implies that the wire velocity is small compared to the Fermi velocity, u(cid:28)v , or any sound velocity
F
in the fluid. Because of limitation to linear response, we systematically write all equations below in their linearized
form.
The plan is to calculate Z in the full range of the quasiparticle mean-free-path (cid:96), from the hydrodynamic regime
(cid:96)(cid:28)atotheballisticlimit(cid:96)(cid:29)a. Theliquidisassumedtobemixtureofbosonsandfermions, butthecorresponding
number densities n and n are arbitrary. Thus the theory applies to pure Fermi liquid as well (n =0).
4 3 4
Besides linearity, another simplification is that we neglect the coupling between the normal and superfluid compo-
nents. This can be justified in two different limits: at small frequencies and at small concentrations. In the former
caseweaimtocalculateZ toaccuracythatiscorrecttofirstorderinaω/v . ThecouplingcontributestoZ insecond
F
order in ω. Assuming that the relevant dimensionless parameter is aω/v , the coupling indeed is a small correction
F
in the experimental case we study, where aω/v ∼0.02 (Table I). In a finite container there appears a second length
F
scale, h=8a. Since hω/v is not very small, the small-frequency approximation is not necessarily accurate. For the
F
superfluid component the low frequency approximation is well satisfied, hω/c(cid:28)1, due to the large sound velocity c
of liquid 4He. We note that one more frequency dependent parameter, a2ω/v (cid:96), occurs in the theory, but it can have
F
any magnitude (see Sec. IV).
Alternatively, the normal-superfluid coupling can also be neglected in the limit of small 3He concentration. This
is because the response of the superfluid to the motion of 3He is proportional to the ratio of their number densities,
n /n [5]. Thus, irrespective of frequency, the coupling can be neglected for small n /n .
3 4 3 4
B. Bulk Fermi-liquid theory
The Fermi liquid theory was formulated by Landau to describe the low energy states of interacting Fermi liquids3.
It was generalized by Khalatnikov to simultaneous presence of condensed bosons4. A new presentation of the Fermi-
Bose-liquid theory is given in Ref. 5. Here we briefly review the equations of Ref. 5 that are necessary for the
numerical calculation.
The central quantity in the theory is the quasiparticle distribution function ψ (r,t). It depends on the momentum
pˆ
p only through its direction pˆ=p/p. In addition it depends on the location r and on time t. The distribution ψ is
pˆ
obtained from the more common quasiparticle distribution n by integration over the magnitude of the momentum
p
and making a transformation that partly decouples the normal and superfluid components5. The state of the bosons
3
is described by the deviation of the chemical potential δµ (r,t)=µ (r,t)−µ(0) from its equilibrium value µ(0) and
4 4 4 4
by the superfluid velocity v (r,t).
s
One more important quantity is the quasiparticle energy shift on the Fermi surface, δ(cid:15) (r,t). It depends on ψ
pˆ pˆ
and on δµ as
4
∞
δ(cid:15) = K δµ +(cid:88) Fl (cid:104)P (pˆ·pˆ(cid:48))ψ (cid:105) . (3)
pˆ 1+F 4 1+ 1 F l pˆ(cid:48) pˆ(cid:48)
0 l=0 2l+1 l
Here K is a parameter describing coupling to the superfluid component, F with l = 0, 1, 2, etc. are the Landau
l
parameters describing interactions between quasiparticles, P are Legendre polynomials, and (cid:104)...(cid:105) denotes average
l pˆ
overtheunitsphereofpˆ. ItfollowsfromEqs.(5)and(15)belowthatthecouplingtermcontributestoZ proportional
to ω2, andwe neglectit. Sincethere areno experimentaldeterminations ofF for l≥2, weneglectthecorresponding
l
terms. Therefore δ(cid:15) simplifies to
pˆ
F F
δ(cid:15) = 0 (cid:104)ψ (cid:105) + 1 pˆ·(cid:104)pˆ(cid:48)ψ (cid:105) . (4)
pˆ 1+F pˆ(cid:48) pˆ(cid:48) 1+F /3 pˆ(cid:48) pˆ(cid:48)
0 1
The kinetic equation in the relaxation-time approximation takes the form
∂ 1
(ψ −δ(cid:15) )+v pˆ·∇ψ =− (ψ −ψle). (5)
∂t pˆ pˆ F pˆ τ pˆ pˆ
Herev istheFermivelocitythatisrelatedtotheFermimomentump andtotheeffectivemassm∗ byv =p /m∗.
F F F F
In the collision term τ =(cid:96)/v is the relaxation time and the local-equilibrium distribution
F
ψle =(cid:104)ψ (cid:105) +3pˆ·(cid:104)pˆ(cid:48)ψ (cid:105) . (6)
pˆ pˆ(cid:48) pˆ(cid:48) pˆ(cid:48) pˆ(cid:48)
Assumingtimedependenceexp(−iωt)andparameterizingthequasiparticletrajectoriesbyr =r +spˆ,thetransport
0
equation can be integrated in the form
(cid:90) 0 (cid:20)1 ω (cid:21)
ψpˆ(r0)=ψpˆ(r0+s0pˆ)eks0 + ds (cid:96)ψplˆe(r0+spˆ)−iv δ(cid:15)pˆ(r0+spˆ) eks, (7)
s0 F
where k = 1/(cid:96)−iω/v . For convergence of the integral s < 0, and the limit s → −∞ is approached. When a
F 0 0
boundary is hit, the solution should be constructed piecewise, and the boundary conditions applied in between.
C. Boundary Conditions
The boundary condition appropriate for specular scattering from a surface moving with velocity u is
ψ =ψ +2p (nˆ ·pˆ)(nˆ ·u), (8)
pˆ pˆ−2nˆ(nˆ·pˆ) F
wherenˆ istheunitsurfacenormalpointedtotheliquid. Inspecularscattering,theincomingandreflectedquasiparticle
trajectories make the same angle to the surface normal on the wire. The boundary condition for diffusive scattering
is that for outgoing quasiparticles (nˆ ·pˆ >0)
out
2
ψ =−2(cid:104)nˆ ·pˆ ψ (cid:105) +p (pˆ + nˆ)·u, (9)
pˆout in pˆin pˆin F out 3
where (cid:104)...(cid:105) is an average over half of the unit sphere (nˆ ·pˆ < 0). In diffuse scattering the reflected (outgoing)
pˆin in
quasiparticlesonthesurfaceofthewireareinequilibriumevaluatedatthequasiparticleenergy(cid:15) =v (p−p )+δ(cid:15) .
p F F pˆ
The distribution of reflected quasiparticles depends on the incoming quasiparticles only on the average. We can also
consider mixed boundary conditions where fraction S of incoming quasiparticles is scattered specularly and fraction
1−S diffusely. Wenotethatinusingthelinearizedboundaryconditions(8)and(9),thedisplacementofthewireleads
to a second order correction, which is neglected, and thus the boundary conditions can be applied at the equilibrium
location of the wire surface.
At the container wall, one possiblity is to assume diffuse scattering, i.e. to use Eq. (9) with u=0. An alternative,
simpler boundary condition is an absorbing wall. This reflects no quasiparticles and is described by the condition
ψ = 0, where nˆ ·pˆ > 0. Such a boundary condition could be a reasonable model for experimental cells that
pˆout out
4
have walls made of sintered silver1,2. The absorbing boundary condition is also a theoretical tool to suppress sound
resonances (Sec. VIII).
WedemonstratethegeneralizationofEq.(7)toincludemixedboundarycondition. BydefiningX (s)=ψle(s)/(cid:96)−
pˆ pˆ
iωδ(cid:15) (s)/v we write
pˆ F
ψpˆ(s=0)=Sgceksc +(cid:8)2SpF(nˆw·pˆ)(nˆw·u)+(1−S)[pF(pˆ+ 32nˆw)·u+gw](cid:9)eksw
(cid:90) 0 (cid:90) sw
+ dsX (s)eks+S dsX (s)eks (10)
pˆ pˆ(cid:48)
sw sc
for a trajectory that is reflected from the wire (surface normal nˆ ) at s=s , and hits the container (surface normal
w w
nˆ ) at s=s . In the latter integral the prime in direction pˆ(cid:48) stands for the specularly reflected trajectory. We have
c c
defined two boundary condition terms
g (r )=−2nˆ ·(cid:104)pˆ ψ (r )(cid:105) ,
w w w in pˆin w pˆin
g (r )=−2nˆ ·(cid:104)pˆ ψ (r )(cid:105) , (11)
c c c in pˆin c pˆin
which depend on the locations r and r on the wire and container surfaces, respectively. For a trajectory with no
w c
collision with the wire, we get simply
(cid:90) 0
ψpˆ(0)=gceksc + dsXpˆ(s)eks. (12)
sc
D. Force on the liquid
The momentum flux tensor is5
↔Π=P(0)↔1 + ρs δµ ↔1 +3n (cid:104)pˆpˆψ (cid:105) , (13)
m 4 3 pˆ pˆ
4
whereP(0) istheequilibriumpressure. Thesuperfluiddensityρ isdefinedasρ =m n −Dm∗n /(1+F /3),where
s s 4 4 3 1
m isthemassofa4Heatom, n the4Henumberdensity, D =1−(1+F /3)m /m∗ andn =p3/3π2(cid:126)3 thenumber
4 4 1 3 3 F
↔
density of 3He. The force per area exerted by a surface element of the wire on the fluid is nˆ ·Π evaluated at the wire
surface r =a(xˆcosθ+yˆsinθ) . Integrating this over the perimeter of the wire gives the force
w
aρ (cid:90) 2π (cid:90) 2π
F = s δµ nˆdθ+3an (cid:104)(nˆ ·pˆ)pˆψ (cid:105) dθ. (14)
m 4 3 pˆ pˆ
4 0 0
In linear theory both terms are proportional to u and thus this expression allows to determine Z (1).
The two terms appearing in Eq. (14) can be interpreted as superfluid and normal fluid contributions, respectively,
and correspondingly Z can be written as Z = Z +Z . We can calculate Z as follows. Because the superfluid
s n s
component is curl free, ∇×v = 0, the fluid flow can be described as potential flow, v = ∇χ. The ideal fluid
s s
equation of motion5 ∂v /∂t+∇δµ /m =0 then gives δµ /m =−∂χ/∂t. For small frequencies ω (cid:28)c/a, v /a we
s 4 4 4 4 F
can neglect the compressibility of the superfluid and the coupling to the normal component, and assume ∇·v =0,
s
or ∇2χ=0. This has to be solved using the boundary conditions nˆ·v =nˆ·u on the wire surface and nˆ·v =0 on
s s
the container walls. Thus the problem reduces to solving the Laplace equation. The solutions in several geometries
have been found, see Ref. 21. The result is
Z =−iωπa2ρ G, (15)
s s
where the factor G depends on the geometry of the container. For infinite fluid G = 1, for the cylindrical container
G=(b2+a2)/(b2−a2), and for the slab container G≈1+π2a2/12h2.
In the limit (cid:96) (cid:28) a the Fermi liquid theory reduces to the hydrodynamic theory (Sec. IV). In the extreme limit
(cid:96)/a→0 the viscosity is negligible, and the whole fluid behaves like an ideal fluid except thin boundary layers on the
walls. In this limit Z approaches
Z =−iωπa2ρG, (16)
ideal
where ρ=m n +m n is the total density of the fluid. It is convenient to present Z by its deviation from the ideal
4 4 3 3
fluid behavior. Simultaneously, we define a dimensionless impedance Z˜ by writing
Z =Z +an p Z˜. (17)
ideal 3 F
5
Using the symmetry assumption F (cid:107)u on equation (14) and comparing to (17) gives
iπG aω 3 (cid:90) 2π
Z˜ = + (cid:104)(nˆ ·pˆ)(pˆ·uˆ)ψ (cid:105) dθ, (18)
1+F /3 v p u pˆ pˆ
1 F F 0
where the second term remains to be calculated numerically. It follows from the equations above that Z˜ depends on
the dimensionless parameters aω/v (1+F /3), (cid:96)/a, S, F , F , and b/a in the cylindrical container or h/a in the slab
F 1 0 1
container. Note that choosing Ω/(1+F /3) as an independent variable instead of Ω ≡ aω/v is convenient because
1 F
this combination frequently occurs in the hydrodynamic region, an example being the first term in Eq. (18).
III. SYMMETRY
The problem stated in Sec. IIA has symmetries that can be used to simplify the numerical calculation. We need to
consider scalar and vector functions of location r. Because of the translational symmetry, there is no dependence on
z, and therefore effectively r =xˆx+yˆy, or using polar coordinates r =r(xˆcosθ+yˆsinθ). Because we are studying
linear response, these functions depend linearly on the wire velocity u. In the slab container we have reflection
symmetries in the planes perpendicular to both symmetry directions x and y. Choosing uˆ = xˆ we have for scalar
functions f and vector functions f =f rˆ+f θˆthe symmetries
r θ
f(r,θ) = f(r,−θ)=−f(r,π−θ)=−f(r,θ+π),
f (r,θ) = f (r,−θ)=−f (r,π−θ)=−f (r,θ+π),
r r r r
f (r,θ) = −f (r,−θ)=f (r,π−θ)=−f (r,θ+π). (19)
θ θ θ θ
For the cylindrical container we argue as follows. Because of rotational symmetry in simultaneous rotations of
r and u, a scalar function has the form f(r,u) = f(r,u,θ), where θ denotes the relative plane angle and r = |r|
and u = |u|. Assuming reflection symmetry with respect to the x-axis f(r,u,θ) = f(r,u,−θ). This allows to write
f(r,rˆ·uˆ,u) or f(r,rˆ·u,u). Requiring f to be linear in u limits this to the form
f(r,u)=f˜(r)rˆ·u (20)
with some function f˜(r). Note that the symmetry (20) often appears in solutions of the Laplace equation. Our
derivation shows that this symmetry remains valid in the case of an arbitrary linear and isotropic equation.
Next, we consider a vector field f(r,u) = rˆf (r,u)+θˆf (r,u). Rotational symmetry limits the two functions to
r θ
the forms f (r,u,θ) and f (r,u,θ). Assuming reflection symmetry f (r,u,θ) = f (r,u,−θ) allows to write this into
r θ r r
the form f (r,rˆ·u,u). Assuming reflection symmetry f (r,u,−θ)=f (r,u,π+θ) allows to write this into the form
r θ θ
f (r,θˆ·u,u). Finally, assuming linearity with respect to u implies
θ
f(r,u)=rˆrˆ·uf˜(r)+θˆθˆ·uf˜(r). (21)
r θ
The results above are applied to the bulk angular averages
c(r)=(cid:104)ψ (r)(cid:105) , b(r)=3(cid:104)pˆψ (r)(cid:105) (22)
pˆ pˆ pˆ pˆ
and to the boundary-condition averages g (r ) and g (r ) (11). The functions c and b are needed to calculate δ(cid:15)
w w c c pˆ
(4) and ψle (6).
pˆ
IV. HYDRODYNAMIC LIMIT
The Fermi-Bose liquid theory reduces to two-fluid hydrodynamic theory22 in the limit of short mean-free-path4,5.
The normal fluid component is described by a Navier-Stokes equation where the density is the normal fluid density
ρ =m∗n /(1+F /3)=ρ−ρ andthecoefficientofviscosityη = 1n p (cid:96). Thediffusiveboundaryconditionleadsto
n 3 1 s 5 3 F
no-slipboundarycondition,whichmeansthatthefluidvelocityatawallequalsthevelocityofthewall. Thespecular
boundary condition leads to perfect slip, where the transfer of transverse momentum between the liquid and the wall
vanishes. Applied to the wire surface this means Π = 0. The hydrodynamic equations have analytic solution in
rθ
some cases, which we discuss below.
In the low-frequency limit the normal and superfluid equations decouple and both components can be considered
asincompressible. Inthislimit,thesuperfluidcomponentZ (15)wasfoundabove. Analyticsolutionsforthenormal
s
6
0.8
Experiments, 3.6%
0.7 Slab h=8a
Cylinder b=8a
Cylinder b=80a AW
0.6
Cyl. b=8a S=1
Cyl. b=80a S=1 AW
0.5 HD cylinder b=8a
HD specular
0.4 HD no-slip
HD Jensen slip
0.3
0.2 pF
3
n
0.1 a
Z'/ (Z''-Z'' )/an p
0 ideal 3 F
-0.25 -0.2 -0.15 -0.1 -0.05 0
FIG.1: TheimpedanceZ =Z(cid:48)+iZ(cid:48)(cid:48)intheregionofshortmean-free-paths,(cid:96)(cid:46)a. Thecurvesareplottedusing(cid:96)asparameter.
The hydrodynamic results in unlimited fluid for diffusive wire (23) and for specular wire (24) are shown by thin dashed and
dotted lines, respectively. The slip correction to the former is shown by dash-dotted line13. The hydrodynamic result for a
diffuse wire in diffuse container of radius b=8a is shown by thick dashed line. Also shown are our numerical results: a large
cylindrical container with a diffusive wire (green) and specular wire (pink) with absorbing walls, a small container, b = 8a,
with a diffusive wire (light blue) and specular wire (light green) in diffusive container, and a slab of thickness 16a (blue) with
diffusivewalls. Theparametersareappropriatefora3.6%mixture,aω/v =0.0180,F =−0.25,andF =0.22. Experimental
F 0 1
data from Ref. 1 is shown for reference (yellow points).
component are known in the following cases. The no-slip case in unlimited liquid was solved by Stokes23. The result
is
4 H(1)(cid:48)(qa)
Z =−iωπa2ρ [1+ 0 ]. (23)
n n aq H(1)(qa)
0
Here, H(j)(x) are the Hankel functions with complex argument, prime indicates derivative, q = (1 + i)/δ, and
i
(cid:112)
δ = 2η/ρ ω is the viscous penetration depth. The no-slip case in a cylindrical container was calculated by Carless,
n
Hall, and Hook15. The case of perfect slip in unlimited liquid was calculated by Bowley and Owers-Bradley16, and
the result is
8H(1)(cid:48)(qa)
Z =−iωπa2ρ [1+ 0 ]. (24)
n n 2aqH(1)(qa)−a2q2H(1)(cid:48)(qa)
0 0
The hydrodynamic results are illustrated in Fig. 1.
The results (23) and (24) illustrate the argument made in Sec. IIA that Z˜ (17) is linear in aω/v but one should
F
allow a2/δ2 = 5a2ω/v (cid:96)(1+ 1F ) to have arbitrary magnitude. In the parameter space formed by aω/v and (cid:96)/a, Z˜
2 F 3 1 F
is singular at the point (0,0).
It is possible to go beyond the low-frequency approximation at the expense of neglecting viscosity. We study the
caseofthecylindricalcontainer. Atfrequencyωthevelocitypotentialχofidealfluidshouldsatisfythewaveequation,
ω2χ+c2∇2χ = 0, (25)
where c is the sound velocity. We make the ansatz
χ(r,θ)=[AJ (kr)+BY (kr)]cosθ, (26)
1 1
whereJ andY areBesselfunctionsandherek =ω/c. Theradialvelocityhastoequalucosθ atr =aandvanishat
1 1
r =b. This gives conditions from which A and B can be solved. Calculating the force similarly as for the superfluid
component in Eq. (14), we get
iπaρω J (ka)Y(cid:48)(kb)−Y (ka)J(cid:48)(kb)
Z = 1 1 1 1 , (27)
ideal k J(cid:48)(ka)Y(cid:48)(kb)−Y(cid:48)(ka)J(cid:48)(kb)
1 1 1 1
where ρ is the density of the fluid. To first order in ω this reduces to the low-frequency result Z (16). Sound
ideal
resonancesarefoundatfrequencieswherethedenominatorofZ (27)vanishes. Forlargeb/athisconditionreduces
ideal
to zeros of J(cid:48)(kb). The lowest zero appears at kb=1.84.
1
7
The two-fluid hydrodynamics allows two sound modes. The first sound is in-phase motion of the normal and
superfluid components. The corresponding sound velocity is close to the sound velocity of pure liquid 4He, which
is relatively high, and is not of interest here. Second sound is the counter motion of the normal and superfluid
components. In 3He-4He mixture it is essentially a compressional mode of the normal fluid as the superfluid motion
causes only a small correction to the sound velocity because of smallness of n /n [4]. In the present approximation
3 4
(cid:112)
(Sec. IIA) the sound velocity is v (1+F /3)(1+F )/3 ≈ 0.4v in the absence of dissipation. This result will be
F 1 0 F
applied below to identify resonances of second sound.
An extension of the hydrodynamic theory to slightly longer mean-free-path is known as slip theory14. The lead-
ing corrections appear in the boundary conditions. Considering the diffusive boundary condition (9), the outgoing
(scattered from the wall) quasiparticles are in equilibrium with the wall, but the incoming quasiparticles generally
are not. In the case of short (cid:96) these populations get rapidly mixed and therefore also the incoming quasiparticles are
in equilibrium with the wall, which leads to the no-slip boundary condition. With increasing (cid:96) this is no more the
case. In slip theory one uses bulk hydrodynamic theory together with boundary conditions that assume the velocity
of the fluid to extrapolate to the wall velocity the slip length ζ behind the wall. Microscopic calculation of ζ for
degenerate fermions on a planar wall was made by Jensen et al.13. The generalization of the slip boundary condition
to curved surfaces and partial specularity is discussed in Refs. 16,17,24. The slip increases the validity range of the
hydrodynamic theory to slightly larger (cid:96), see Fig. 1. Several extrapolations of the slip theory for vibrating wires have
been suggested, see Carless, Hall and Hook15, Gu´enault et al.25, and more recently by Bowley and Owers-Bradley16
together with Perisanu and Vermeulen17. None of these have attempted to include the effect of the container or the
Fermi-liquid interactions.
V. BALLISTIC LIMIT
At the lowest temperatures one enters the ballistic regime, where the mean-free-path of quasiparticles becomes
large compared to the experimental dimensions. In the extreme case one can neglect the collision term in the kinetic
equation(5). InthisballisticlimitthequasiparticlesstillinteractthroughtheFermi-liquidinteractions,whichappear
through δ(cid:15) . In spite of this, some results can be obtained analytically16,19,20.
pˆ
If, in addition to the collision term, one also neglects the time dependence, the kinetic equation (5) reduces to
pˆ·∇ψ (r) = 0. This implies that ψ is constant along trajectories, and changes only when the trajectory hits a
pˆ pˆ
wall. If the fluid far from the wire is in equilibrium, one finds ψ = 0 on incoming trajectories. The distribution
pˆ
for outgoing trajectories is then obtained from the boundary conditions (8) or (9). The impedance is obtained by
integrating over the wire surface, and it gives16,19
43π
Z = an p ,
diff 48 3 F
3π
Z = an p (28)
spec 4 3 F
corresponding to the diffusive and specular boundary conditions. These are purely dissipative since our assumption
of ω → 0. These results have been generalized to the cylindrical container by taking into account the reflection of
quasiparticles from the container wall back to the wire in Ref. 20.
Atfinitefrequencythemaincomplicationarisesfromtheinteractiontermδ(cid:15) inthekineticequation(5). Neglecting
pˆ
thistermitispossibletosolvetheballisticlimitatarbitraryfrequencyinthecylindricalcontainer. Weassumediffusive
boundary condition on both the wire and the container. Using definitions (11) and the symmetry (20), the boundary
conditions (9) can be written as
2
ψ (r ) = g˜ nˆ ·u+p (pˆ + nˆ)·u, (29)
pˆw,out w w F w,out 3
ψ (r ) = g˜ nˆ ·u, (30)
pˆc,out c c c
where g˜ and g˜ are constants. Tracing the different trajectorices one can now calculate the averages appearing in
w c
definitions (11). We get g˜ =Dg˜ and
w c
2 2
g˜ =Ag˜ +B(g˜ + p )+ Cp . (31)
c c w 3 F 3 F
8
The coefficients A, B, C, and D are expressed as integrals
2 (cid:90) π (cid:90) π/2 (cid:18)2iωbcosγ(cid:19)
A = dζsin2ζ dγcosγ(1−2cos2γ)exp ,
π v sinζ
0 arcsin(a/b) F
a2 (cid:90) π (cid:90) 1 (cid:32)a (cid:114) a2 (cid:112) (cid:33) (cid:32)iω(√b2−a2x2−a√1−x2)(cid:33) a
B = dζsin2ζ dx x2+ 1− x2 1−x2 exp = D,
bπ b b2 v sinζ b
0 0 F
a3 (cid:90) π (cid:90) 1 (cid:114) a2 (cid:32)iω(√b2−a2x2−a√1−x2)(cid:33)
C = dζsin3ζ dx 1− x2exp ,
bπ b2 v sinζ
0 0 F
2 (cid:90) π (cid:90) 1 (cid:32)a (cid:112) (cid:114) a2 (cid:33) (cid:32)iω(√b2−a2x2−a√1−x2)(cid:33)
D = dζsin2ζ dx x2+ 1−x2 1− x2 exp , (32)
π b b2 v sinζ
0 0 F
where ζ is the angle between the trajectory and the cylinder axis. We can now solve (31) and get
2(B+C)p
g˜ = 3 F. (33)
c 1−A−BD
Continuation to calculate the force gives
(cid:18)43p 1 b (cid:19) (cid:32)43 a (D+ bC)2 (cid:33)
Z =πan F + g˜ + Cg˜ =πan p + a . (34)
n 3 48 2 w 2a c 3 F 48 3b1−A− aD2
b
In the special case ω =0 the integrals (32) reduce to
1 a 2a3
A = − + − ,
3 b 3b3
(cid:32) (cid:114) (cid:33)
2 a a2 a
C = 1− +arcsin ,
π b b2 b
(cid:90) 1 (cid:32)a (cid:112) (cid:114) a2 (cid:33)
D = dx x2+ 1−x2 1− x2 . (35)
b b2
0
where D still needs to be calculated numerically. These reproduce the result given in Ref. 20.
The result (34) is illustrated in Fig. 2. We see that Z oscillates around its static value approximately like the
n
exponential∝exp(2iωb/v ). Thereasonisthatthequasiparticlesexcitedbythewirearereflectedfromthecontainer
F
wallbacktothewire,butdelayedbytime≈2b/v . Thefirstconstructiveinterference(afterthezerothoneatω =0)
F
corresponds to ω ≈ πv /b. This is approximately a factor 4 larger than the first second-sound resonance frequency
F
in the hydrodynamic limit (Sec. IV). The difference arises because the second sound velocity is smaller than the
quasiparticle velocity v , and because the first zero of J(cid:48)(kb) is at kb = 1.84, which is less than π. Second sound is
F 1
strongly damped in the ballistic regime because of excitation of quasiparticles, known as Landau damping.
The results above could be generalized to include the Fermi-liquid interactions. However, the integrals get very
cumbersome even in the case of unlimited liquid. Instead of partial analytic solution, we therefore prefer the full
numerical solution in the following.
VI. NUMERICAL METHOD
Wesolvethequasiparticledistributionψ (r)inadiscretegridaroundthewireandfordiscretemomentumdirections
pˆ
pˆ. The location r = xˆrcosθ+yˆrsinθ can be parameterized with cylindrical coordinates r and θ, where we have
n
fixed xˆ =uˆ. The momentum direction pˆ =sinζ (cosβ xˆ+sinβ yˆ)+cosζ zˆ is parameterized by angles ζ and β .
jl j l l j j l
For the cylindrical container we use a grid with exponentially increasing spacing ∆r in the radial direction, and a
fixed spacing ∆θ. In the slab container, we use a cylindrical lattice near the wire, and rectangular lattice near the
container walls, with partial overlap between the lattices. In y-direction the length of the slab is increased until the
results converge (absorbing walls are used).
The quasiparticle distribution ψ ≡ψ (r ) for each lattice point n and for each discrete trajectory direction pˆ
njl pˆjl n jl
can be written, using Eq. (10), as the sum
ψ =I +B +SH +(1−S)Q , (36)
njl njl njl njl njl
9
0.6 (a) nu(Zm'- rZe')/anp 0.4 (b) W =0.01
0 3 F
0.4 nuZm''n i/man3pF W=0.01 W =0.03
0.2 W=0.3
0.2
0.0
0.0
W=0.2
W =0.25
-0.2 -0.2 p3FW=0.5
an W =0.6 W =0.35
/
-0.4 -0.4 )0
-0.6 aw /vF -0.6 (Z'-Z' W =0.15 Zn''/aWn=30p.09F
0.0 0.2 0.4 0.6 0.8 1.0 -0.4 0.1 0.6

FIG. 2: The normal-fluid impedance Z = Z(cid:48)+iZ(cid:48)(cid:48) in the ballistic limit in the cylindrical container of radius b = 10a. The
n n
impedance Z is plotted relative to its static value Z =(43π/48)an p (28) in unlimited fluid, and the total impedance Z is
n 0 3 F
obtainedbyaddingZ (15). Thesolidlinesarefullnumericalresult,thedashedlinestheapproximation(34),andthedifference
s
between them is the Landau force6. (a) The real (red) and imaginary (blue) parts of impedance as a function of Ω=aω/v .
F
(b)ParametricplotofZ incomplexplanewithΩasaparameter. Inbothpanelsdiffusiveboundaryconditionsareusedand
n
the full calculation uses F =−0.44 and F =0.449.
0 1
whereI =(cid:82)0 dsX (s)eks+S(cid:82)swdsX (s)eks istheintegralpart,B =[2Sp (nˆ·pˆ)(nˆ·u)+(1−S)p (pˆ+2nˆ)·
njl sw pˆjl sc pˆ(cid:48)jl njl F F 3
u]eksw is the ψ-independent part of the boundary condition, Hnjl describes the ψ-dependent part of the boundary
condition on the container, and Q describes the boundary condition on the wire. The integral is calculated from
njl
r along the trajectory pˆ , |s | is the distance to the wire (if there is a collision with the wire at point r ), and |s |
n jl w w c
is the distance at which the container wall is hit, at point r . If there is no collision with the wire, we get from Eq.
c
(12) ψ =I +H , where I =(cid:82)0 dsX (s)eks. We write g =g (r ) and g =g (r ) at each lattice point
njl njl njl njl sc pˆjl c,t c t w,t w t
r on the surfaces, and define
t
(cid:88)
H = H g ,
njl ntjl c,t
t
(cid:88)
Q = Q g , (37)
njl ntjl w,t
t
where the sums are over all lattice points on the corresponding surface, Hntjl = wteksc and Qntjl = wteksw. The
weight factors w are obtained by interpolating from the lattice points closest to r or r on the surfaces. In the
t c w
following the averages (22) are denoted by Ψ(1) = c , Ψ(2) = b , and Ψ(3) = b . Now, the integral part I can
n n n r,n n θ,n njl
be conveniently presented as the sum
3
I =(cid:88) (cid:88) a(h) Ψ(h), (38)
njl nmjl m
h=1 m
wherethecoefficientsa(h) givetheweightswithwhichtheequilibriumdistributionatotherlatticepointsr affects
nmjl m
ψ . The coefficients are calculated by numerical integration, using a modified Simpson’s rule, which takes the
njl
exponential term exp(ks) into account exactly. It is necessary to use interpolation between lattice points. Taking the
numerical average over the trajectories, we can write the averages (22) as a sum
Ψ(i) =(cid:88)dih Ψ(h)+B(i)+(cid:88)H(i)g +(cid:88)Q(i)g , (39)
n nm m n nt c,t nt w,t
m,h t t
where
d1h = (cid:104)a(h) (cid:105) ,
nm nmjl pˆjl
d2h = 3rˆ ·(cid:104)pˆ a(h) (cid:105) ,
nm n jl nmjl pˆjl
d3h = 3θˆ ·(cid:104)pˆ a(h) (cid:105) , (40)
nm n jl nmjl pˆjl
and similar averaging gives B(i), H(i), and Q(i). The diffusive boundary condition terms can be written as sums over
n nt nt
10
1.8% 3.6% 5.6% 6.6% 7.0% 9.5% 1.8% 3.6% 5.6% 6.6% 7.0% 9.5%
m∗/m 2.26 2.31 2.34 2.35 2.60 2.63 2.45 2.50 2.53 2.54 2.85 2.88
3
v (m/s) 21.10 25.97 29.60 31.07 29.56 32.36 19.46 24.00 27.38 28.75 26.97 29.55
F
an p (kg/ms) 0.0058 0.0145 0.0259 0.0321 0.0394 0.0589 0.0058 0.0145 0.0259 0.0321 0.0394 0.0589
3 F
F 0.151 0.219 0.266 0.284 0.269 0.301 0.138 0.201 0.245 0.261 0.243 0.272
1
aω/v , wire 1 0.0222 0.0180 0.0158 0.0151 0.0159 0.0145 0.0241 0.0195 0.0171 0.0163 0.0174 0.0159
F
aω/v , wire 2 - - - 0.0233 0.0245 0.0224 - - - 0.0252 0.0268 0.0245
F
TABLE I: Parameter values used in calculations at different concentrations x . The first four concentrations are at saturated
3
vapor pressure, the last two at P = 10 atm. We use α = 0.284 at saturated vapor pressure and α = 0.207 at P = 10 atm26.
The frequency parameter Ω=aω/v is calculated for vacuum frequency f =1202.85 Hz (wire 1) and f =1857.7 Hz (wire 2).
F
On the left we have used the hydrodynamic mass m /m = 2.15 as a starting point, and on the right m /m = 2.34 (see
H 3 H 3
text).
lattice points m as well,
g = (cid:88)g(h) Ψ(h)+B +(cid:88)H g +(cid:88)Q g ,
c,t c,tm m c,t c,ts c,s c,ts w,s
m,h s s
g = (cid:88)g(h) Ψ(h)+(cid:88)H g , (41)
w,t w,tm m w,ts c,s
m,h s
where the new coefficients g(h) , B , H , Q , g(h) , and H are similar to the terms above, but the averages
c,tm c,t c,ts c,ts w,tm w,ts
are taken over the incoming trajectories only. Once the coefficients have been calculated, we can solve the averages
(22) from the matrix equation
Ψ=DΨ+B ⇔Ψ=(I−D)−1B. (42)
IfthetotalnumberoflatticepointsisN,thenumberofpointsoncontainersurfaceisV,andthenumberofpointson
wire surface is T, then Ψ is a 3N +V +T component vector Ψ=(Ψ(1),Ψ(1),...,Ψ(3),g ,...,g ,g ,...,g )T,
1 2 N c,1 c,V w,1 w,T
B = (B(1),B(1),...,B(3),B ,...,B ,0,...,0)T, and the (3N +V +T)×(3N +V +T) matrix D contains the
1 2 N c,1 c,V
terms dih , H(i), Q(i), g(h) , H , Q , g(h) , and H in appropriate order.
nm nt nt c,tm c,ts c,ts w,tm w,ts
When the averages are known, we can calculate the full distribution ψ on the wire surface using Eq. (36). The
njl
forceexertedbythewireonthefluidisthenreadilycalculatedfromEq.(18). Ifcylindricalsymmetryisassumed,the
treatment is similar, but since the dependence of the averages on θ is of a simple form [(20) and (21)], the calculation
becomes effectively one-dimensional.
VII. PARAMETER VALUES
In order to relate the numerical calculations to experiments we need to consider the parameters that characterize
3He-4He mixtures and the experimental setup.
The molar volume of a mixture of molar concentration x =N /(N +N ) at temperature T and pressure P is26
3 3 3 4
V (x ,P,T)=V (P,T)[1+α(x ,P,T)x ], (43)
m 3 4 3 3
where V (P,T) is the molar volume of pure 4He and α is the so-called BBP parameter27. For the molar volume of
4
pure 4He, we use the data of Tanaka et al.28, and for α(P) at zero temperature we use the results of Watson et al.26.
The number density of the 3He component is obtained from
N x
n = A 3 , (44)
3 V (1+αx )
4 3
where N is the Avogadro number. From n we can calculate the Fermi momentum
A 3
(cid:112)
p = 3 3π2(cid:126)3n . (45)
F 3
If we know the 3He quasiparticle effective mass m∗, we can also calculate the Fermi velocity v = p /m∗, which
F F
is needed in calculating the dimensionless frequency parameter Ω = aω/v . The effective mass m∗ is calculated
F