thermal resistance of 4he below but very near the super ...thermal resistance of 4he below but very...

Thermal resistance of 4He below but very near the

superfluid transition∗

Hanan Baddar, Guenter Ahlers, Kerry Kuehn, and Haiying Fu

Department of Physics and Quantum Institute,University of California, Santa Barbara, CA 93106, USA

We report high-resolution measurements at saturated vapor pressure of thethermal resistivity R of superfluid 4He over the reduced-temperature range3 × 10−7 < t ≡ 1 − T/Tλ < 3 × 10−5 (Tλ is the transition temperature forQ = 0) and heat-current-density range 4 < Q < 200µW/cm2. For smallerQ and t, no thermal resistance was detectable below Tc(Q) < Tλ. For Q >

∼10µW/cm2 we find that the results can be described well by R = (t/t0)

−2.8

Kcm/W with t0 = (Q/Q0)0.904 and Q0 = 393 W/cm2. Thus R has an

incipient divergence at Tλ which is, however, supplanted by Tc(Q) where Rremains finite. The results imply R ∝ Q(m−1) with m = 3.53 ± 0.02. Thisdiffers from the original assumption m = 3 of Gorter and Mellink, andfrom experimental results obtained well below Tλ. However, it agrees withmeasurements by Leiderer and Pobell at larger currents and further belowbut still close to Tλ. Our measurements could not resolve a critical heatcurrent for the onset of resistance.

∗These measurements were to be the main project of and were started by Hanan Baddar,

who was killed by an automobile on July 15, 1996.

H. Baddar, G. Ahlers, K. Kuehn, and H. Fu

Contents

1. Introduction 3

2. Experimental Apparatus 5

3. Experimental Procedure and Data Analysis 7

3.1. General Procedure . . . . . . . . . . . . . . . . . . . . . . . . 73.2. Temperature Profile and Heat-current Loss at Finite Ramp

Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3. Determination of Tλ(Q = 0) . . . . . . . . . . . . . . . . . . . 13

4. Results 15

4.1. The behavior of R near Tc(Q) . . . . . . . . . . . . . . . . . . 154.2. Results for R and an Analysis with wc = 0 . . . . . . . . . . . 184.3. The resistivity at small Q . . . . . . . . . . . . . . . . . . . . 214.4. Analysis with wc > 0 . . . . . . . . . . . . . . . . . . . . . . . 22

5. Comparison with prior results 22

6. Summary and discussion 25

7. Acknowledgment 27

Thermal resistance of 4He below the superfluid transition

1. Introduction

At sufficiently small heat currents, superfluid 4He has an effectively infi-nite thermal conductivity because of the superfluid and normalfluid counter-flow. According to Landau two-fluid hydrodynamics1,2 there are two velocityfields vs and vn, with the associated currents js = vsρs and jn = vnρn. Hereρs + ρn = ρ is the total density of the fluid, and ρs and ρn are referred to asthe superfluid and normalfluid densities. The current jn is associated withentropy transport, and leads to a heat current

Q = ρSTvn . (1)

When there is no net mass transport, the total current j = js + jn = 0. Thisleads to

Q = ρsSTw (2)

where w = vn−vs is the counterflow velocity (for simplicity we are assuminga one-dimensional geometry where the velocities are signed scalars).

For superfluid confined in restricted geometries, like packed powders, itis well established experimentally3 that there is a critical counterflow veloc-ity wc below which the superflow is essentially dissipation free except for theclassical viscous dissipation due to any normalfluid flow through the finitegeometry. If we neglect this viscous dissipation, the state with a finite veloc-ity may then be regarded as a thermodynamic equilibrium state which hasno chemical-potential and thermal gradients. For bulk helium it is less clearwhether there is a non-zero critical velocity, or whether dissipation actuallystarts as soon as w exceeds zero.4–6 When dissipation does occur in this sys-tem, it is believed to be caused by the interaction between quantized vorticesin the superfluid component and the excitations which comprise the normalcomponent. This interaction is known as “mutual friction”. In order toprovide a phenomenological model of this interaction, Gorter and Mellink7

added a mutual-friction force to the Landau two-fluid equations and on thebasis of this derived the relation

∇T = Awmρn/S (3)

for the associated temperature gradient. Eqations 2 and 3 lead to

∇T = A(Q/ρsST )mρn/S . (4)

The exponent m was found to be equal to three on the basis of early exper-iments. The amplitude A in general would be expected to be temperaturedependent. We shall write it as

A = A0t−α (5)


where t = 1 − T/Tλ. Although in this model superfluid dissipation beginsas soon as Q exceeds zero, the thermal resistivity

R ≡ −∇T/Q ∝ Qm−1 (6)

vanishes as Q goes to zero. Since all parameters in Eq. 4 except A and ρs

have constant finite values at Tλ, we can (for sufficiently small t) write R as

R = (t/t0)−(mν+α) (7)

where ν = 0.6705 is the exponent8 of the superfluid density.Although the Gorter-Mellink model has been remarkably successful, a

sizable body of experimental data9–20 showed deviations from it. In order todescribe the experiments, it became necessary to treat the exponent m as anadjustable parameter. The experimental values ranged from the anticipatedvalue m = 3 at low temperatures to values near four as the superfluid tran-sition at Tλ was approached. Initially it seemed difficult to accept the largervalues. Thus, Leiderer and Pobell 20 (LP) proposed that a critical velocitywc had to be invoked even for the large samples with diameters of about 1cm used by them. They proposed to fit the data for ∇T (Q) to the equation

∇T = Awm(1−wc/w)2ρn/S for w > wc (8)

and to∇T = 0 for w < wc (9)

with m fixed at three and wc adjusted to fit the data. In terms of knownthermo-hydrodynamic properties, Eq. 8 can be written as

∇T = A(Q/ρsST )m(1− wcρsST/Q)2ρn/S . (10)

LP state that an analysis of their measurements over the temperature range9 × 10−5 < Tλ − T < 0.016 K and the heat-current range 1.3 < Q < 8.84mW/cm2 was consistent with m = 3 and wc ' 0.3±0.15 cm/s for axial heatflow in a cylindrical helium sample of diameter 1.38 cm.21

In the present paper we report new measurements of R much closer toTλ which cover the range Q <

∼ 200µW/cm2. We find that they can be repre-sented very well by the powerlaw Eq. 7, with the coefficient t0 proportionalto a power of Q. This leads to a simple representation of R(Q, t) by Eq. 7,with Eqs. 16, 17, and 18 given below, over our entire experimental range.An analysis of the data in terms of Eqs. 4 and 10 is presented as well. Ourmeasurements could not resolve a critical counterflow velocity wc. If we as-sume the conventional value m = 3 for the Gorter-Mellink exponent, we findthat the amplitude A0 of the mutual friction coefficient A is Q-dependent.


1.3 K

1.7 K 1µK +

1.8 K 1nK

2.2 K 1µK

-

+ -

+ - Heat Leak

Vacuum Can

Sample

overflow

Cell

Heat Leak

Heat Leak

A

B

C E

F G

D

Fig. 1. Left: Schematic diagram of the apparatus. Right: Schematic diagramof the cell. The cell hangs from a mounting stage A. It consists of two OFHCcopper endplates B and F, a stainless-steel side wall E, and two side planes D.High-resolution thermometers (not shown) are mounted on the wings C andG of the endplates and on the side planes to allow a four-probe measurement.

A Q-independent A is obtained only if m = 3.53 ± 0.02. We note here thata non-integer value of m leads to complicated units for A0. In cgs-unitsthese are cm4−msm−2/g. An extrapolation of the fit of Eq. 4 to our data tothe heat-current and temperature range used by Leiderer and Pobell agreeswell with the measurements of those authors.21 Thus we now have consistentresults and a closed-form expression for the mutual friction which cover therange of 10µW/cm2 <

∼ Q <∼ 10mW/cm2 and 8× 10−7 <

∼ t <∼ 10−3.

In the next Section we describe our apparatus and the sample cell. Theexperimental procedure and data analysis are discussed in Sect. 3.. Therewe analyze in some detail the effect of finite ramp rates on the data, anddiscuss the determination of Tλ(Q = 0). The results for the resistance of thesuperfluid and their analysis are presented in Sect. 4.. A brief summary anddiscussion is given in Sect. 6..

2. Experimental Apparatus

The low-temperature portion of the apparatus is shown schematically inFig. 1. It is a modified version of one described earlier 22. A vacuum can isimmersed in liquid helium at 4.2 K. Three DC SQUIDS and one AC SQUID


(not shown in Fig. 1) which were used for high-resolution susceptibility ther-mometry were located inside the vacuum can and were attached to the topplate of this can. Thus the SQUIDS operated at the bath temperature ofabout 4.2 K. In future designs, it will be preferable to provide a separatedtemperature-regulated platform for the SQUIDs since the bath-temperaturedrift seemed to have a detrimental effect on the thermometer stability 23.

Inside the vacuum can, a continuous 4He evaporator 24 provided a basetemperature near 1.3 K. Electrical leads and the superconducting shields forthermometer leads were thermally attached to two thermal isolation stageswhich typically were held at constant temperatures near 1.7 and 1.8 K. Thetemperature stability of these stages was about 10−6 and 10−8 K for theupper and the lower stage respectively. We will refer to them as the micro-and the nano-stage respectively. A separate shield, held a few mK abovethe sample temperature of nominally 2.176 K, completely surounded thesample cell and provided radiation shielding. It was cooled by the lowerisolation stage, and its temperature was controlled to about ±1µK. Thethermal attachment of the sample to the shield was weak. Most of thecooling of the sample was provided by a heat leak of about 250 µW to thenano-stage. The relatively strong thermal attachment of the sample to itssurroundings was required to enable measurements at relatively large heatcurrents, but limited the sample-temperature stability to a few nK 23.

The fill capillary came directly from 4.2 K to an overflow reservoir lo-cated on the shield. Ultra-pure 4He (0.5 ppb 3He) was used for the sample.The liquid-vapor interface was maintained in the reservoir, and thus washeld at a constant temperature within ±1µK. The associated vapor pressurefluctuations are about 10−7 bar, and correspond to a stability of Tλ of aboutone nK. A 0.02 cm diameter stainless steel capillary connected the overflowvolume to the sample.

A schematic diagram of the sample cell is also shown in Fig. 1. Ithad a cylindrical cross section, and was suspended within the shield fromits mounting plate A. Its top (B) and bottom (F) ends consisted of OFHCcopper. The heat leak to the nano-stage left the cell at the top as shownby the arrow in the figure. Heaters for controlling the cell temperature werelocated on the mounting plate. High-resolution susceptibility thermometersas well as germanium thermometers were located on plates C and G whichwere integral parts of the cell ends.

The cell had a 0.012 cm thick stainless-steel sidewall (E) of 1.27 cminside diameter (1.267 cm2 cross sectional area) which overlapped the copperanvils of the cell ends. The gap between the copper anvils and the sidewallwas eliminated by filling it with epoxy 22. The total vertical length of thehelium sample was H = 0.46 cm.


Two sideplanes made of OFHC copper were clamped to the sidewall.One was located a vertical distance of 0.08 cm above the cell bottom. Theother was positioned 0.30 cm above the first. They were used to probe thelocal temperatures along the helium sample. The sideplanes had a circularcutout in their interior which had a diameter slightly (about 0.01 cm) smallerthan the outer diameter of the cell wall. They were split into two sectionswhich could be compression fit around the cell. The circular part contactingthe cell wall was beveled at a 45◦ angle. When installed, the contact occurredover a vertical length of approximately 0.01 cm. Thus the precise effectivelength of the sample between the two sideplanes was somewhat uncertain.It was determined experimentally (see Sect. 3.3.) to be h = 0.281±0.005 cmby measuring the superfluid-transition temperatures at the two positions inthe presence of gravity. High-resolution susceptibility thermometers 23 weremounted on the top and bottom endplates and on both sideplanes. Thusa four-probe measurement of the thermal resistance, unencumbered by theboundary resistance at the copper-helium interface, could be made.

3. Experimental Procedure and Data Analysis

3.1. General Procedure

Measurements were made by increasing the temperature of the cell toplinearly in time at a ramp rate β of typically 3.3 nK/s. Slower ramp rateswere sometimes used, but not required because of the extremely high ef-fective thermal conductivity and short relaxation time of the system in thesuperfluid phase. The top, top side, bottom side, and bottom cell tempera-tures were recorded at intervals of about one second. They will be referredto as Tt, Tst, Tsb, and Tb respectively. The working temperature scales of thefour thermometers were shifted relative to each other so as to cause them toagree well below Tλ and with Q = 0.

Even when no heat current was applied deliberately to the cell bottom,a small “dark current” Qd (which could be positive or negative) enteredthe cell. This current depended on both the shield temperature and thenano-stage temperature. After proper adjustment of these temperatures,Qd was typically of order 10−9 W or less. It was measured by increasingthe cell temperature to several µK above Tλ and measuring the resultingTb − Tt. The applied currents were corrected for Qd when this correctionwas significant.

When a heat current was turned on well below Tλ, a temperature differ-ence of about 2 K cm2/W developed between the top and bottom thermome-ters because of the Kapitza resistance between the helium and the copper


0 1000 2000 3000 4000time (sec, separate arb. orig. for each ramp )

-4

-2

0

2

T (

µK, s

ame

arb.

orig

. for

ea.

ram

p)

Fig. 2. A typical example of Tsb and Tst for successive ramps at a small (15.8nW/cm2), a large (33.46 µW/cm2), and again at a small current. Only partof each ramp is shown, and for display purposes the three ramps are shiftedcloser to each other. The small horizontal bars indicate the locations of Tλ,sb

or Tc,sb. The shift of the transition due to the finite Q is easily seen. BelowTc,sb, the difference between Tsb and Tst for the large-current ramp is causedby mutual friction.

and the thermal gradients in the copper. A small temperature difference ofabout 2 × 10−3 K cm2/W developed also between the two sideplane ther-mometers even at temperatures well below Tλ where the superfluid is knownto have no measurable thermal resistance. We believe this sideplane offsetto be the result of two-dimensional heat flow involving the copper bottomand top, the Kapitza resistance, and the stainless-steel sidewall. The effectwas nearly negligible for our measurements of the thermal resistance of thesuperfluid, but we applied an appropriate correction.

During the ramps, the superfluid transition occurred first at the cell bot-tom at Tλ,b because of the hydrostatic pressure variation with height whichwas caused by Earth’s gravity.26 When the temperature was increased fur-ther, an interface between a superfluid (top) and normal (bottom) phasedeveloped and moved vertically upwards. The interface position is deter-mined by

dTλ

dz=

dTλ

dPρg (11)

which, at saturated vapor pressure and for g equal to the Earth’s gravita-tional acceleration, is 1.276 µK/cm. The transition temperature

Tλ ≡ Tλ,sb(Q = 0) (12)


at the bottom sideplane and in a negligible heat current was used as areference temperature. Ramps were carried out alternatingly at a smallreference current of 16 nW/cm2 and then at a larger one. A typical exampleis shown in Fig. 2. The small-current ramp served to determine the locationof Tλ on the working temperature scale. During the large-current ramps, thetemperature differences ∆T = Tsb − Tst, after correction for the sideplaneoffset, yielded the average thermal resistance

R̄ = ∆T/hQ (13)

of the superfluid over the temperature interval from Tst to Tsb. The mea-surement was assigned to the mean temperature

T̄ = (Tsb + Tst)/2 . (14)

We examined the need to apply a curvature correction 25 to the datato convert them from R̄(t̄) to the true thermal resistance R(t̄). For thispurpose we used the powerlaw Eq. 7 with x = mν + α = 2.8 (see below).We integrated Eq. 6 from tst ≡ 1− Tst/Tλ to tsb ≡ 1− Tsb/Tλ to obtain Q.Dividing Q into (tsb − tst)Tλ/h gives an estimate of R̄. A curvature factor

f ≡R(t̄)

R̄=

t̄−x(t1+xst − t1+x

sb )

(1 + x)(tst − tsb)(15)

can then be used to obtain R(t̄) = fR̄(t̄) from the measured R̄. However, wefound that f was within 0.6 % of unity over the entire experimental rangeR ≤ 0.04 K cm / W used in our data analysis. Thus we did not apply acorrection to the data.

3.2. Temperature Profile and Heat-current Loss at Finite Ramp

Rates

For a number of purposes, including the determination of Tλ,sb fromexperimental ramp data like those in Fig. 2, we computed the expectedtemperature profile in the sample as a function of time. We assumed a locallyquasi-static temperature profile and used Fourier’s law; but we allowed forthe variation of the heat current with vertical position. For the purpose ofthis section we neglected any shift in the transition temperature due to anyfinite Q from Tλ(Q = 0) to Tc(Q). We used the known thermal conductivity22 λ(t) and heat capacity Cp(t) of helium in the zero-current limit and tookthe gravity effect 26 into account. Even for an axially uniform heat currentQ0 this had to be done by numerical integration of Fourier’s law since the


-0.2 0 0.2 0.4 0.6Tt - Tλ,b ( µK )

0

5

10

15

Q0,

Q(H

), o

r ∆Q

(H)

( n

W/c

m2 )

Fig. 3. Heat currents entering (Q0, solid line), leaving (Q(H), dashed lines),and absorbed by (∆Q(H), dotted line) a sample of length 0.5 cm. The threedashed lines are Q(H) for ramp rates (from bottom to top) of 3.34, 1.67,and 0.42 nK/s. The dotted line is ∆Q(H) for a ramp rate of 3.34 nK/s. Thetwo vertical lines indicate the location of Tλ,b and Tλ,t.

conductivity depends not only on the local temperature T (z), but throughthe z-dependence of Tλ(z) also on the vertical position z. However, theproblem is more complicated.

When the top temperature of the cell is ramped at a finite rate β, onlysome of the current Q0 applied at the bottom actually leaves the top of thesample. The remainder is used to heat the helium. Thus Q depends on z,and the total absorbed current depends on the sample length as well as onthe ramp rate. The problem is numerically difficult because it is non-localin the sense that the temperature gradient at a particular vertical positiondepends on the gradients at all positions below it. In order to calculatethe temperature profile for a uniform ramp of the helium temperature atthe sample top, we computed two profiles for closely spaced top tempera-tures, starting with the assumption of a uniform current Q0. From the localtemperature ramp rates and the heat capacity 27 we computed the locallyabsorbed current per unit sample length δQ(z ′), and from it an estimateof Q(z) = Q0 − ∆Q(z) where ∆Q(z) =

∫ z0 δQ(z′)dz′ (we take the origin

of the z-axis at the bottom of the sample; the sample top is at z = H).The process had to be iterated. For an applied current Q0 = 15.8 nW/cm2,β = 3.34 × 10−9 K/s, and H = 0.5 cm, which corresponds closely to ourreference ramps (first and last ramp in Fig. 2), the total absorbed current isshown as a dotted line in Fig. 3 as a function of Tt−Tλ,b. The current leaving


-0.2 0 0.2 0.4 0.6Tt - Tλ,b ( µK )

0

10

20

30∆Q

(H

) (

nW

/cm

2 )

Fig. 4. Heat currents ∆Q(H) absorbed by a sample of length 0.5 cm for aramp rate of 1 nK/s. The three lines are for applied currents (from bottomto top) of Q0 = 39.5, 78.9, and 158 nW/cm2. The two vertical lines indicatethe location of Tλ,b and Tλ,t.

the top of the sample is shown by the dashed lines for the three ramp rates(from bottom to top) 3.34, 1.67, and 0.42 nK/s. The horizontal solid lineis Q0. The absorbed current increases in the two-phase region because es-tablishing the relatively large thermal gradient in the normal phase requiressignificant enthalpy. Figure 4 shows this more clearly for larger currentswhere the thermal gradients in the normal phase, and thus the absorbedenthalpy, are larger. In this figure we show the current absorbed by thesample for β = 1× 10−9 K/s and (from bottom to top) Q0 = 39.5, 78.9, and158 nW/cm2. The noise in the data is numerical noise associated with thedifficult iterative solution to this non-local heatflow problem.

It is interesting to note that the increase with increasing T of the ab-sorbed current in the two-phase region (see Fig. 4) is altered dramatically bya change in the gravitational acceleration. This is illustrated in Fig. 5, wherewe show ∆Q(H)/Q0 for H = 0.8cm,Q0 = 70nW/cm2, and β = 10−10K/sfor accelerations equal to 1, 0.3, 0.1, and 0.03 times g (we took g = 982cm/s2 as the Earth’s gravitational acceleration). The numerical results canbe understood as folows. For small acceleration the two-phase region spans arelatively narrow temperature interval (see Eq. 11) and for a given ramp ratethe system spends a relatively short time in it. During this time the tem-


-0.2 0 0.2 0.4 0.6 0.8 1 1.2T - Tλ,b ( µK )

0

0.1

0.2

0.3

∆Q (

H)

/ Q0

Fig. 5. Heat currents ∆Q(H) absorbed by a sample of length 0.8 cm for aramp rate of 0.1 nK/s and an applied current Q0 = 70nW/cm2. The fourlines are (from top left to bottom right) for accelerations equal to 0.03, 0.1,0.3, and 1 times the Earth’s gravity.

perature profile characteristic of the imposed heat current and single-phasenormal helium must be established. Since the enthalpy required to do sois nearly independent of gravity, the relatively short time available at smallacceleration implies a relatively large ∆Q(H). Indeed, in our model and forzero acceleration we would find ∆Q(H) = Q0 at T = Tλ. We remark that inthe physical system very near Tλ the phenomena described above would bealtered somewhat because of finite-current effects which influence the con-ductivity 28 and the heat capacity 29 (these effects have been neglected inour model).

There is an interesting side effect associated with the heat loss in thesample which may be important if (as in our experiment) the temperature ofthe cell top (rather than that of the helium at the sample top) is ramped lin-early in time. Because in the two-phase region the current flowing across thetop helium-copper boundary decreases in time, the temperature drop due tothe Kapitza resistance decreases also. This effectively reduces the ramp rateof the superfluid helium in the top portion of the cell. The phenomenon canbe seen clearly in the middle ramp of Fig. 2, where the temperature ramprates are dramatically reduced as Tλ,b is exceeded even though the ramprate of the top temperature Tt (not shown) remains essentially unaltered.


-0.2 0 0.2 0.4 0.6Tst - Tλ,sb ( µK )

-0.2

0

0.2

0.4

0.6

0.8

1

Tsb

- T

λ,sb

( µ

K )

Fig. 6. The temperature Tsb as a function of Tst for a ramp rate of 3.34nK/s, H = 0.5 cm, and Q0 = 15.8 nW/cm2. The origin of both axes is Tλ,sb

as determined from a least-squares fit of a calculated temperature profile(dashed line) to the data (solid line). The dotted line (Tsb = Tst) is a guideto the eye. The dash-dotted line corresponds to the erroneous assumptionthat the heat current is uniform and equal to Q0.

We have not yet developed a successful algorithm for treating this case nu-merically, and so far all our simulations have involved a uniform ramp rateof the helium temperature at the sample top.

3.3. Determination of Tλ(Q = 0)

It is well known that the experimentally observed transition tempera-ture Tc(Q) is depressed below Tλ(Q = 0) by the heat current Q.30–33 Thisphenomenon is very apparent in Fig. 2, where the small horizontal bars indi-cate the transitions. We determined the (effectively) zero-current transitiontemperature Tλ by ramping the system from the superfluid phase into thetwo-phase region with a net current Q0 = 15.8 nW/cm2 entering the cellbottom (see Fig. 2). From previous measurements 31,32 and from theory 33

we estimate that the transition-temperature shift due to this current is be-tween 4 and 6 nK, which is sufficiently small to be neglected in the presentwork.

In the experiment, the origin of the temperature axis initially is arbi-trary. The computation described in the previous section yields the tem-perature profile T (z) relative to Tλ,sb as a function of time τ . From this


0 2 4 6 8 10 12 14Point Number

-0.01

0

0.01

Tλ,

bs (

µK

, ar

b. o

rig. )

Fig. 7. Successive determinations of Tλ,sb. In this experiment ramps werecarried outalternatingly at the reference current Q0 = 15.8 nW/cm2 and ata higher current. The total duration of the run was about 26 hours. Theopen circles are actual measurements at the reference current. The solidones are an average of the two adjacent measurements and were used as theorigin of the temperature scale for the high-current ramps.

we use Tsb(τ) and Tst(τ) to get the dependence of δTsb ≡ (Tsb − Tλ,sb) onδTst ≡ (Tst − Tλ,sb). We fitted the computed δTsb(δTst) to the experimen-tal data, least-squares adjusting the origin Tλ,sb. In the fit we excluded theregion within 50 nK of an initial estimate of the transition point in orderto avoid the influence of possible small rounding effects. Aside from the ex-cluded region, we used the range from one µK below to 0.6 µK above Tλ,sb.In Fig. 6 we show the experimental data (solid line) and the fit of the com-puted data (dashed line) after a shift along both axes by the least-squaresvalue of Tλ,sb. Clearly the agreement is quite good over the entire temper-ature range shown. From the quality of the fit we estimate that errors inTλ,sb may be expected to be about 5 nK. As a guide to the eye, the dottedline is an extrapolation of Tsb = Tst which applies in the superfluid phase.

To illustrate the importance of using the temperature profile based onQ(z), we show as a dash-dotted line the result for δTsb(δTst) based on a uni-form current equal to Q0. If this profile were used in the determination of thelambda point, a poor fit would be obtained and Tλ would be systematicallytoo high.

Close inspection of the data in Fig. 6 shows a discontinuity in the slopeat Tst−Tλ,sb = 0.358µK. This point is indicated by the small vertical bar, andcorresponds to the superfluid-normalfluid interface passing the top sideplanethermometer. It yields an effective length h = 0.281± 0.005 cm between thetwo sideplanes, as mentioned in Sect. 2..

Results of successive determinations of Tλ,sb are shown in Fig. 7. In


-6 -5 -4 -3Tst - Tλ,sb(0) ( µK )

0

0.2

0.4

0.6

0.8

1

δT =

Tsb

- T

st (

µK

)1.4 1.6 1.8 2 2.2 2.4

10-3 time (sec, arb. orig. )

-6

-4

-2

0

2

Tλ(

Q=

0) -

T

( µK

)(a) (b)

Fig. 8. (a) Expanded view of the middle ramp from Fig. 2 (33.46 µW/cm2).The difference between Tsb and Tst is shown in (b) . Below Tc,sb(Q) (hori-zontal bar) it is due to mutual friction.

this experiment a sequence of ramps, alternatingly at the reference currentQ0 = 15.8 nW/cm2 and at a higher current, were carried out. The actualduration of this run was about 26 hours. The open circles are actual resultsfor Tλ,sb(Q0). The solid points are the averages of two adjacent points, andwere used as the origins of the temperature scale of the high-current ramp.The data in Fig. 7 suggest a small long-term drift of the thermometers, asreported previously 23. The rms deviation from a smooth curve is about 4nK.

4. Results

4.1. The behavior of R near Tc(Q)

Having determined the reference temperature Tλ,sb(Q = 0), we cannow proceed to analyze data like those shown in the middle of Fig. 2 forQ0 = 33.46µW/cm2. An expanded view of those data is shown in Fig. 8a.The horizontal bar indicates the approximate location of Tc,sb(Q). BelowTc,sb(Q), the difference δT = Tsb−Tst is due to mutual friction in the super-fluid phase. For Tsb > Tc,sb(Q) there is an additional, generally much larger,contribution to δT from the thermal gradient in the layer of normal fluid


10-1 100 101

Tλ,sb-Tst or Tc,sb-Tst ( µK )

10-2

10-1

100

101

δ T

= T

sb -

Tst

( µ

K )

Fig. 9. The difference between Tsb and Tst for Q = 33.46µW/cm2 as afunction of Tc,sb(Q) − Tst (dashed line) and Tλ,sb(0) − Tst (solid line) onlogarithmic scales. The figure shows that the resistivity is finite at Tc(Q).

which is growing as the time and Tst increase. In Fig. 8b we show δT as afunction of Tst. Here the origin of the temperature scale is at Tλ,sb(0), andthe dramatic increase of δT near Tst− Tλ,sb(0) ' −2.7µK signals the arrivalof the normalfluid-superfluid interface at Tc,sb. Both Figs. 8a and b suggestthat R is finite at Tc(Q). This can be seen more clearly on the logarithmicscale of Fig. 9. Here the solid line gives δT as a function of Tλ,sb(Q = 0)−Tst.For sufficiently large Tλ,sb(0)− Tst (to the right of the small horizontal bar),the data can be described well by a powerlaw which diverges at Tλ(0) butwhich has a finite value at Tc(Q). Alternatively, if the data are plotted as afunction of Tc,sb(Q)−Tst (dashed line), then they do not fall on a straight lineand thus a powerlaw is not suitable for their representation. In that case thecurvature of the data suggests that the results tend to a finite limit at Tc(Q)(−∞ in the figure). Although there is no explicit theoretical prediction forR(t,Q), the experimental observation of a finite R[Tc(Q)] is consistent withpredictions for other properties 28,29, especially with that for ρs

28.


10-7 10-6 10-5

( <Tλ(0)> - <T> ) / Tλ

10-2

10-1

R (

K c

m /

W )

Fig. 10. The resistivity R of superfluid helium for various heat currents asa function of (< Tλ(0) > − < T >)/Tλ on logarithmic scales. Here < T >≡(Tsb + Tst)/2 and < Tλ(0) >≡ (Tλ,sb(0) + Tλ,st(0))/2. The vertical dashedline is the location of Tλ,sb(0). The dotted line corresponds to Tsb−Tst = 10nK. From left to right, the data are for Q0 = 2.87, 3.59, 4.49, 5.61, 7.02,8.77, 10.97, 13.70, 17.14, 21.41, 26.76, 33.46, 41.82, 97.08, 121.4, 151.7, and196.6 µW/cm2.

0 50 100 150 200Q ( µW / cm2 )

2.5

3

3.5

mν

+ α

Fig. 11. The exponents mν + α obtained from a fit of Eq. 7 to the datain Fig. 10. Open circles: Fits for 0.001 < R < 0.015. Plusses: Fits forR > 0.001 and below the plusses in Fig. 10. Dashed line: mν + α = 2.8.


10-5 10-4

Q ( W / cm2 )

10-7

10-6t 0

Fig. 12. The results for t0 (see Eq. 7) obtained from a fit of the data belowthe plusses in fig. 10 to the powerlaw R = (t/t0)

−2.8. The solid line is a fit tot0 = (Q/Q0)

x for Q > 10−5 W/cm2. It gives x = 0.904 and Q0 = 393W/cm2.

4.2. Results for R and an Analysis with wc = 0

In Fig. 10 we show some of the results for R as a function of t̄ ≡(T̄λ(0) − T̄ )/Tλ on logarithmic scales. Here T̄λ(0) ≡ [Tλ,sb(0) + Tλ,st(0)]/2.The vertical dashed line is the location of Tλ,sb(0) and illustrates that thelimit set by gravity is not important for the experiment. The dotted linecorresponds to Tsb− Tst = 10 nK; below it the data are subject to relativelylarge random and systematic errors due to thermometer resolution and drift.

Below the plusses in Fig. 10, the data for Q >∼ 5µW/cm2 fall on straight

lines and thus can be represented by the powerlaw Eq. 7 which diverges atT̄λ(0). For Q >

∼ 10µW/cm2, a fit of Eq. 7 to the data gave the exponentsmν + α shown in Fig. 11. The open circles correspond to fits over the range0.001 < R < 0.015, and the plusses are for R > 0.001 but less than indicatedby the plusses in Fig. 10. There is no significant dependence of mν + α onthe range of the fit. For Q >

∼ 30µW/cm2, the exponent is independent of Qwithin our resolution and has a value close to

mν + α = 2.8± 0.1 (16)

as shown by the horizontal dashed line. Fixing the exponent at 2.8 gives theresults for t0 shown in Fig. 12 when R has the units Kcm/W. For Q >

∼ 10−5


101 102

Q ( µW / cm2 )

0

20

40

60

100

A3

and

A3.

53 (

cgs

)

A3A3.53

Fig. 13. Amplitudes A0 (see Eq. 5) of the mutual friction coefficient A inEq. 4. Open circles: 100A0 for m = 3. Here the units are cm s/g. Solidcircles: A0 for m = 3.53. In this case the units are cm(4−m)s(m−2)/g.

W/cm2, the results are well represented by the straight line in the figurewhich corresponds to

t0 = (Q/Q0)y (17)

with

y = 0.904; Q0 = 393W/cm2 . (18)

Combining Eqs. 7 and 17, and recognizing from Eq. 4 that R ∝ Qm−1, wehave

m = 1 + y(mν + α) = 3.53 ± 0.02; α = 0.433 ± 0.11 . (19)

Here the uncertainties are based on the values obtained by fixing mν + αat 2.9 and 2.7. The results for m and α are very close to those obtainedby Leiderer and Pobell 20 at temperatures much further below Tλ and usingcurrents which were two orders of magnitude larger than ours.

Using the Q = 0 entropy 34 and normal-fluid density 8, we examinedexplicitly whether the very weak variations of S, ρn, and T in Eq. 4 affectthe values of the exponents determined from the data. We found that a fitof Eq. 7 on the one hand or of Eq. 6 with Eq. 4 on the other to a given dataset gave the same values of mν + α within 0.001, as we had expected. Sincedata at one value of Q determine only the combination mν + α and not mand α separately, the results for mν+α obtained from fits with m fixed wereindependent of the value used for m.

As will be discussed below, our measurements were unable to resolvea finite critical velocity wc. Thus, we determined the amplitude A0 of themutual-friction coefficient A from fits of Eq. 6 with Eqs. 4 and 5 to the data.The results obtained by fixing m at the experimental value 3.53 are given


0

0.02

0.04

0

0.02

0.04

0

0.02

0.04

-5 -4 -3 -2 -1 0Tst - Tλ,sb ( µK )

0

0.02

0.040

0.02

0.04

5.61

7.02

8.77

10.97

13.70

17.14

0

0.02

0.04

Tsb

- T

st (

µK

) 0

0.02

0.04

4.49 µW/cm2

Fig. 14. The temperature differences between the two sideplanes as a func-tion of the top sideplane temperature. The smooth lines are calculated fromthe powerlaw Eq. 7.


0 50 100 150 200Q ( µW / cm2 )

-0.05

0

0.05

wc1

( c

m /

s )

Fig. 15. Results for the critical velocity wc obtained by a fit of Eq. 10 tosome of our data.

as solid circles in Fig. 13. As expected, the values are independent of Q forQ >∼ 10µW/cm2. Within their uncertainty they are equal to 54 in cgs units.

This value is given by the horizontal line in the figure. For smaller Q, A0

decreases with decreasing Q. This Q-dependence is another manifestationof the Q-dependence of t0 and mν + α seen already in Figs. 12 and 11 atsmall Q.

When m was fixed at its classical value m = 3, the fit described aboveyielded the open circles in Fig. 13. As expected for this m, A0 is Q-dependentover the entire range of Q.

4.3. The resistivity at small Q

For Q <∼ 10µW/cm2, the resistivity R of the superfluid is smaller than

expected on the basis of powerlaw extrapolations from larger Q. This isreflected in the values for mν + α, t0, and A0 given in Figs. 11, 12, and 13.However, in the relevant parameter range the measured temperature differ-ences are quite small. Thus, in order to see how definitive the measurementsare, we show in Fig. 14 the original data as well as the temperature differencecalculated from the fit Eqs. 7, 16, 17, and 18. For Q <

∼ 9µW/cm2 the datafall systematically below the lines, suggesting that the departures from Eq. 7are real. However, the differences are only a few nano-K, and we can notrule out that they are caused by systematic errors of unknown origin. Thusmeasurements with even better temperature resolution and thermal stabilitythan the present ones would be desirable. This could be attained in a futureexperiment specifically designed for the relatively small heat currents whichare involved.


4.4. Analysis with wc > 0

We searched for evidence for a critical counterflow velocity wc by fittingEq. 10 to some of our data sets. It turns out that wc is very sensitive tosmall systematic offsets between the temperature scales of the two sideplanethermometers. Thus systematic errors of wc are relatively large and difficultto estimate. We show the results, with random errors only, in Fig. 15. Ourinterpretation is that the data are unable to resolve a critical velocity, andthat in any case wc < 0.05 cm/s. This upper limit is significantly smallerthan the value 0.3 cm/s deduced by LP from their data further below Tλ,but significantly larger than typical values which one might expect4 for bulkhelium samples with a characteristic size of order one cm.

5. Comparison with prior results

In Fig. 16 we show as open squares and circles some of the results for Rwhich we derived from the graphs published by LP 21. Also shown there, assolid dots, are some of our data. The solid lines represent the fit of Eqs. 4,5, and 6 to our data (m = 3.53, α = 0.433, A0 = 54), evaluated for the Q-values of the experimental runs.35 Similarly, the dashed lines are the sameequations with the parameters (m = 3.49, α = 0.41, A0 = 80) quoted by LP.One sees that there is excellent overall consistency between the data over theentire range of Q. Our parameters and those of LP yield values of R withessentially the same Q- and t-dependence. The predicted values of R basedon our parameters are systematically lower than the LP values by about 20percent.

LP also fitted Eq. 10 to their data, and obtained m = 3 and a criticalvelocity wc = 0.3 cm/s. An evaluation of this fit for some of the experimentalvalues of Q is given by the dotted lines in the figure. Whereas this fit isreasonably consistent with the LP results at relatively large Q, it does notfit our data at much smaller Q. This agrees with our conclusion in Sect. 4.4.that our data can not resolve a critical velocity.


10-6 10-5 10-4 10-3

( Tλ - T ) / Tλ

10-3

10-2

R (

K c

m /

W )

Fig. 16. Comparison of some of our results (solid dots, from left to right21.41, 41.8, 97.08, and 196.6 µW/cm2) with those of LP 20 (squares: 2200µW/cm2, open circles: 8840 µW/cm2). The solid lines are Eqs. 4 and 5 withm = 3.53, mν + α = 2.8, and A0 = 5.4 × 108 from the fit to our data. Thedashed lines are equivalent fits of these equations to their data quoted byLP (m = 3.49, α = 0.41, A0 = 8.0 × 108). The dotted lines are the fits totheir data by LP of a function with a critical velocity, evaluated with (fromleft to right) Q = 97.08, 196.6, 2200, and 8840 µW/cm2.

In early investigations the heat current Qλ necessary to maintain oneend of several samples of various cross sections and lengths at Tλ and theother end at a reduced temperature tλ below Tλ was determined.18,19 Theresults could be represented by

Qλ = Btxλ . (20)

The value of x was found to be 1.077 ± 0.014 in Ref. 18 and 1.07 ± 0.01 inRef. 19. These measurements were interpreted in terms of the integral ofEqs. 4 and 5. Near Tλ where ρn, T, and S can be regarded as constant, thisintegral yields

x = (1 + α + mν)/m (21)


100 101

L ( cm )

102

B (

W/c

m2 )

Fig. 17. The mutual-friction coefficient B determined in Refs. 18 and 19. Thesolid line was calculated from Eq. 22, with A0 = 54,m = 3.53, α = 0.433from the present measurements. The solid and open circles are from Ref. 19.They are for a cylindrical tube of diameter 1 cm, and for an annulus ofdiameter 1 cm and width 0.1 cm, respectively. The open square is fromRef. 18, and is for a capillary of diameter 0.02 cm.

and

B = kρST

[

TλS

(α + mν)LA0ρn

]1/m

(22)

when we write ρs = ρktν . With our new results α + mν = 2.80, m = 3.53we find x = 1.077, in excellent agreement with Refs. 18 and 19. Usingour new value A0 = 54 cgs and 8,34 k = 2.380, ρn = ρ = 0.1462 g/cm3,S = 1.574 × 107erg/g K, we find B = 96.7 W/cm2 for our sample lengthL = 0.281 cm. This is shown as a plus in Fig. 17 . As a function of Lthe same calculation yields the solid line in the figure. The solid circles arefrom Ref. 19 for a tube of 1 cm diameter, which is similar to the diameterof our sample. The open circles19 are for an annulus of diameter 1 cm andwidth 0.1 cm. There is no noticeable dependence on the geometry, and theagreement between the older and our new measurements is excellent over thislarge range of diameters and channel lengths. The result from Ref. 18 (opensquare), however, is a factor of 2.2 larger than the solid line. It was obtainedin a thin capillary of diameter 0.02 cm. In order to establish whether thereis a genuine dependence of B on the channel diameter, it would be desirableto have additional results for capillary diameters in the range 0.1 to 0.01 cm.

It is worth noting that the excellent consistency between our new mea-surements and those of Refs. 19 and 20 displayed in Figs. 17 and 16 respec-


tively suggests that our present measurements for a relatively short sampleare not significantly influenced by an entrance-length effect, and insteadpertain to bulk helium unencumbered by the sample ends.

Recently, the mutual friction coefficient A (based on the assumptionthat m = 3) was calculated by Haussmann, using renormalization-groupmethods.36 We show some of his results as dashed lines in Fig. 18. Alsoshown are some of the experimental data. If m truly had the value m = 3,then there should be no Q-dependence of A. Both the theoretical and theexperimental results are dependent on Q, although the Q-dependence ofthe theoretical result is weaker than that of the experimental data. Theexperimental values are a factor of 25 (at small Q) to 10 (at the larger Q)smaller than the theoretical calculation. Possible reasons for this differencewere discussed by Haussmann.36.

6. Summary and discussion

We presented new experimental results for the thermal resistance R ofsuperfluid He4 over the reduced-temperature range 3× 10−7 <

∼ t <∼ 3× 10−5

and heat-current range 4 <∼ Q <

∼ 200µW/cm2. The temperature depen-dence of R is consistent with a divergence at Tλ(Q = 0), but since thefinite-current transition intervenes at Tc(Q) < Tλ(Q = 0),31 the resistanceremains finite over the entire superfluid range. For Q >

∼ 10µW/cm2 the datacan be fit very well by the simple powerlaw in t and Q given by Eqs. 7 and17, with the parameters given by Eqs. 16 and 18. The data can also be rep-resented by a fit of Eq. 4 which explicitly includes the theoretically expecteddependence on the entropy and the normalfluid density, but the fit is nobetter over our parameter range. A fit of Eq. 10 to the data could not re-solve a critical heat current or counterflow velocity. The exponents mν + αand y determined from the powerlaw fits yield m = 3.53 and α = 0.433.The value of m is larger than the classically expected value m = 3, butagrees with earlier experiments.18,20. When the fit to our data is extrap-olated to larger Q, it agrees very well with the measurements by Leidererand Pobell20,21 for Q = 2.2 and 8.84 mW/cm2. Thus the simple power lawprovides a good representation of the superfluid resistance over the currentrange from10 µW/cm2 to 10 mW/cm2 and reduced-temperature range fromabout 10−6 to 10−3. Our results also agree with early measurements18,19 ofthe heat current carried by samples with one end just above and the otherbelow Tλ. The good agreement between all of those data suggests that ourpresent measurements for our relatively short sample are not significantlyinfluenced by an entrance-length effect, and instead pertain to bulk helium


10-6 10-5 10-4 10-3

t

104

105

A (

cm

s /

g )

Fig. 18. The mutual-friction coefficient A derived from the experimentalvalues of R and Eq. 4 by assuming that m = 3. The solid lines are our datafor (from left to right) 10.97, 26.76, 41.82, 97.08, 121.4, 151.7, 1nd 196.6µW/cm2. The squares and circles are the results of LP 20 for 2200 and 8840µW/cm2. The dashed lines are the calculations of Haussmann36 for (fromleft to right) 10, 100, and 1000 µW/cm2.


unencumbered by the sample ends.At currents below 10 µW/cm2 our results for R are systematically

smaller than the powerlaw fit to the data at larger currents. This provoca-tive observation deserves further experimental study with greater thermalstability and resolution than we were able to achieve in our apparatus whichwas designed for relatively large currents.

It has long been known that there can be two distinct regimes of dissipa-tion in the superfluid phase, with an experimentally sharp transition betweenthem.37 For a sample with uniform properties (i.e. when ∆T << Tλ − T ),the regime at lower heat currents is known as the TI regime, and the high-current one is called TII. It is believed that the TI regime corresponds toturbulent superfluid and laminar normalfluid flow, and that the transitionto the TII regime occurs when the normalfluid becomes turbulent as well.Recently a stability analysis of the laminar normalfluid flow, subjected tothe interaction with the turbulent superfluid via the mutual-friction force,was carried out by Melotte and Barenghi (MB).6 In distinction to classicalpipe flow (which is stable to infinitesimal perturbations at all Reynolds num-bers), they found that the normalfluid flow in a tube of circular cross sectionbecomes unstable to infinitesimal perturbations at a distinct finite value ofa control parameter β (β is proportional to the square of the counterflowvelocity w2). Near Tλ their predictions yield a critical heat current for theTI-TII transition Qc2 proportional to

√

ρs/B where B is a mutual frictioncoefficient. Experiments suggest that B ∝ t−ζ with38 ζ ' 0.35. 39,40 TheMB stability analysis then yields Qc2 ∝ t0.5. Thus the critical heat currentvanishes at Tλ, and sufficiently close to Tλ only the TII state should be ob-served at finite Q. Therefore we assume that our measurements pertain tothe TII state. This is consistent with the integral measurements18 of Qλ dis-cussed above. These measurements reveal a regime close to Tλ with a uniquegeometry-independent exponent x = 1.077 = (1 + α + mν)/m, and a tran-sition further away from Tλ to a regime with a smaller geometry-dependentexponent. It seems likely that the geometry-dependent regime exists whenthe cold end of the sample is at a sufficiently low temperature (and thus atsufficiently small w2) to have entered the TI state. The geometry-dependentlarge-t regime in the integral experiments would then correspond to a two-phase system with the TII state at the warmer and the TI state at the colderend of the sample.

7. Acknowledgment

This work was supported by NASA through Grant No. NAG3-1847.


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21. Leiderer and Pobell 20 made measurements for Q ≤ 8.84 mW/cm2 in a tubeof 1.38 cm diameter, with thermometers placed a distance of 1.6 cm apart tomeasure Tsb and Tst. They report only Tsb − Tst and Tsb − Tλ (gravity effectsare negligible in their parameter range). Because of concerns about curvaturecorrections, they fit these data to an integral of Eq. 4 and never reported actualexperimental values of R. We read some of their data points from their graphsand obtained the results for R displayed in Figs. 16 and 18. LP also mademeasurements for Q up to 158 mW/cm2 in an annulus of length 1mm, diameter1 cm, and width 1 mm. The values for R which we obtained from those data fallabout a factor of four below the curves for R(Q, t) which we compute from Eq. 4with their or with our parameters. We have no explanation for this discrepancy.Because of it we only consider the LP data from the wider geometry and forQ ≤ 8.84 mW/cm2.

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40. R.J. Donnelly, Quantized vortices in helium II (Cambridge University Press,1991).

thermal resistance of 4he below but very near the super ...thermal resistance of 4he below but very...

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