HELIUM IN AEROGELAn important theme that

threads through manyareas of current interest incondensed matter physics isthe effect of randomness anddisorder. Prior to the 1960s,disorder and impurities wereoften viewed as unavoidablenuisances that masked thetrue behavior of ideal sys-tems. We have since learnedthat disorder itself can bringforth fascinating and oftenunexpected new phenomena in condensed phases of mat-ter. (See the December 1988 special issue of PHYSICSTODAY, dedicated to disordered solids.)

One example is Anderson localization, in which ir-regularities in a metal cause localization of the wavefunction and render the metal insulating at low tempera-tures. The theoretical framework of this phenomenon,introduced by Philip Anderson some 38 years ago, hasbeen extended to explain a host of elegant experimentson quantum transport in submicron structures and inelectronic systems of reduced dimensionality.

A second example is found in glasses. Systematicstudies in the 1970s and 1980s revealed that all glasseshave common characteristic thermodynamic, elastic anddielectric properties that are different from those of theirchemically identical, crystalline counterparts. This char-acteristic glassy behavior is related to the structural dis-order inherent in glasses.

A third example is the Kondo effect, in which a verysmall quantity of magnetic impurity in a nonmagneticmetal causes a logarithmic increase in the metal's electri-cal resistivity with decreasing temperature, and anoma-lous temperature dependence in the metal's susceptibility,heat capacity and thermoelectric power.

In the statistical physics community, there is consid-erable interest in what are called random magnets. (Seethe article by Daniel Fisher, Geoffrey Grinstein and AnilKhurana in the December 1988 issue of PHYSICS TODAY,page 56.) In simple terms, random magnets are systemsin which different kinds of magnetic ions or nonmagneticions are placed as neighbors on a lattice. Dependent onthe nature of the exchange interactions between neigh-boring pairs of spins, and on whether there are localrandom magnetic fields at different sites, there are threedifferent classes of random magnets: random exchange,

MOSES CHAN is a professor of physics, and NORBERT MULDERSis a research associate, at Pennsylvania State University inUniversity Park. JOHN REPPY is a professor of physics at CornellUniversity in Ithaca, New York.

minute amount of appropriately placed random field and spin, r • • cc i iheoreticaJ models have

impurity has fascinating effects on the been quite successful in de-nature of the superfluid transitions in

helium-4 and helium-3, and on theordering of He- He, mixtures.

Moses Chan, Norbert Muldersand John Reppy

scribing the phases andphase transitions of a num-ber of real magnetic systems.For example, a disorderedIsing antiferromagnet suchas FeJfZn1_JCF2 in a uniformmagnetic field appears to bean experimental realizationof the random-field Isingmodel. In addition to mod-

eling magnetic systems, the random-magnet models alsoprovide a theoretical framework for understanding collec-tive phenomena in a number of other systems withquenched disorder. (By "quenched disorder," we mean thatthe positions of the impurities are fixed.) These systemsand phenomena include structural phase transitions ofrandom alloys, commensurate charge-density-wave sys-tems with impurity pinning, melting of intercalates and"dirty" superconductors. In the case of fluids, quencheddisorder can be introduced into a pure fluid system byplacing the fluid in a porous medium.1

This article focuses on the effects of disorder on thesuperfluids helium-4 and helium-3, as well as on the phaseseparation in 3He-4He mixtures, that is caused by thepresence of aerogel, a low-density porous material.- Thishighly porous silica glass is unique in that the quencheddisorder it provides, created by a network of fibrousstrands, exhibits correlations over long length scales.These long-range correlations appear to be responsible formost of the intriguing results that we describe below.

AerogelAerogels were first made and given their name by StevenKistler of Stanford University in the 1930s. They arehighly porous solids, formed by a sol-gel process.2 Afterthe formation of the gel network, the solvent is extractedby hypercritical drying to preserve the delicate structure.The porosity is controlled by the ratio between solid andsolvent used in the gelation process, and can range from85% to as much as 99.8%. In an aerogel of 99.8% porosity,only 0.2% of the volume is occupied by silica. The densityof such a sample is only four times that of air.

The initial stages of the gelation process consist ofhydrolysis and polycondensation of (in our case) tetra-methoxysilane. The result is the formation of small silicaparticles, typically 1-2 nm in diameter. Gelation isthought to be the diffusion-limited aggregation of theseparticles. Aggregation leads to the formation of a veryopen structure made up of silica strands that are connectedat random sites. The interconnectivity produces a solid thatis highly compressible and yet mechanically robust.

30 AUGUST 1996 PHYSICS TODAY

AEROGEL of 95% porosity,photographed under whitelight. The bluish tint of thesample and the yellowish tintof the transmitted light areconsequences of Rayleighscattering. FIGURE 1

Figure 1 is a photograph of an aerogel sample of 95%porosity The sample is translucent. Under white lightit appears bluish, while the transmitted light is yellow.This is a consequence of Rayleigh scattering, the phenome-non responsible for blue sky and red sunsets.

Figure 2 shows a representative transmission electronmicroscope picture, taken by George Ruben of DartmouthCollege, of a 98.5% porous aerogel grown by LawrenceHrubesh and Thomas Tillotson of Lawrence LivermoreNational Laboratory. The corresponding small-angle x-rayscattering (SAXS) data were gathered by Laurence Lurioand Mulders at the National Synchrotron Light Source atBrookhaven National Laboratory. The SAXS data showfractal-like correlations in the mass distribution, on lengthscales up to 65 nm. In lighter aerogels, this correlationis extended to longer length scales. However, SAXS datado not tell us much about the connectivity of the silicanetwork.

Recent interest in the material has been promoted byits utilization as the refractive medium in Cerenkov de-tectors. Aerogel is also commercially produced as a thermalinsulating material. Other applications include catalysis,catalyst support, gas storage, gas filtering and as a mediumto capture interstellar particles in space missions.

Superfluid transition of He in aerogelThe superfluid density and heat capacity of 4He in a 94%porous aerogel, measured by Gane Wong while he was atCornell University,3 are shown in figure 3b. As in thecase of pure 4He (figure 3a), the superfluid density van-ishes at the same temperature at which one finds amaximum in the heat capacity. (See box 1 for a definitionof superfluid density and a discussion of the superfluidtransition in pure 4He.) This property strengthens theinterpretation that the transition in the 4He/aerogel sys-tem is indeed a genuine phase transition. Remarkably,in spite of the random environment, the transition remains

exceedingly sharp. It is indeed one of the sharpest tran-sitions observed. The sharpness is demonstrated moreclearly in figure 4, which shows that ps for helium inaerogel, as well as for pure helium and helium in Vycor,has a simple power-law dependence in the reduced tem-perature t. (See box 1.) (Vycor is the trade name of aporous silica glass manufactured by Corning Glass-works; its porosity is 30%. Transmission electron micros-copy shows that the pore structure can be visualized asinterconnected wormlike channels approximately 7 nm indiameter.) In the case of helium in aerogel, this simplepower-law behavior extends down to t = 10"4. The expo-nent characterizing the power law, deduced from databetween t = 10~4 and 10~2, however, is found to be 0.81,significantly larger than the value for pure 4He.

In pure 4He the critical exponents and amplituderatios at elevated pressures are identical to those atsaturated vapor pressure if one includes pressure-depend-ent corrections. To investigate whether or not this uni-versal (that is, pressure-independent) behavior appliesequally in the 4He/aerogel system, second-sound measure-ments were made at the University of California, SantaBarbara,4 for a range of pressures up to 2.5 megapascals.Using a model for sound propagation in superfluids in ahighly compliant matrix, developed by Julian Maynard atPennsylvania State University, it was possible to deter-mine the superfluid density from the second-sound speed.The results at saturated vapor pressure confirmed theCornell results—that the dependence of ps on t can bedescribed by a simple power law with an exponent closeto 0.8. With increasing pressure, however, the exponentbecomes larger, and systematic deviations from a simplepower law become more apparent. It thus becomes clearthat, as in the case of pure 4He, pressure-dependentcorrections to scaling terms have to be included in theanalysis of the data. When correction is done, a universalexponent, £, is recovered. This exponent remains signifi-

AUGUST 1996 PHYSICS TODAY 31

TRANSMISSION ELECTRONMICROGRAPH (a) and a plot of

small-angle x-ray scattering data (b) oian aerogel of 98.5% porosity. The

micrograph shows that silica nodules ofabout 1 nm connect and form a randomfibrous network. The numbers indicate

approximate distances in nanometers.Scattering at a wavevector Q originates

from density-density correlations at alength scale of 2ir/Q. The data show

fractal-like correlations in the massdistribution in the sample up to alength scale of 65 nm. FIGURE 2

PL)

H

U

2H<

1000

100

10

1

0.1

o o<g

-

-

\

o

o

. o0.01 0.1

WAVEVECTOR Q (nm"1)

cantly larger than that for pure 4He: £aero = 0.75 versusipure ~ 0.67. Experiments at Cornell and Santa Barbaraon aerogels of lower porosity—for example, of 91% poros-ity—found faero similar to that for 94% aerogel.

The effect of quenched disorder on the superfluidtransition can be treated as analogous to random exchangein a magnetic system. Nonmagnetic impurities in anotherwise pure magnetic sample lead to spatial inhomo-geneities in the locally averaged strength of the magneticinteraction. Because the transition temperature dependson the strength of this averaged interaction, the effect ofimpurities can be modeled as a distribution of locallydefined transition temperatures, Tc(r). At first glance, onemight expect that this effect would lead to a rounded orsmeared transition. In 1974, A. Brooks Harris of theUniversity of Pennsylvania showed that such a result isnot necessarily the case, provided there is no long-rangecorrelation in the positions of the impurities. This argu-ment was later confirmed by more rigorous renormaliza-tion-group calculations.5

Imagine that the sample is divided into uncorrelatedblocks of £d spins, in which (as noted in box 1) £ is thecorrelation length and d is the dimensionality of thesystem. The difference in Tc between different blocks isexpected to go to zero as the correlation length grows toinfinity. In particular, Harris showed that the transitionwill be stable against the smearing effect of the disorderif the fluctuations in the local Tc(r) go to zero faster thanIT - Tc I as £ —> oo. This stability criterion, named afterHarris, can be written in terms of the critical exponentsfor the pure system:

2 - dvpure = apure < 0

Accordingly, a pure system that satisfies the Harris crite-rion would, after impurity dilution, still undergo a sharpphase transition with the same critical exponents as thepure system. Because a = -0.013 for the superfluid tran-sition of pure 4He, one would expect the Harris criterionto be satisfied for 4He in porous media. In fact, superfluiddensity measurements of 4He in porous Vycor glass founda sharp transition,1-3 as is shown in figure 4, with a criticalexponent, f, that matches that of bulk 4He. This agree-ment is all the more remarkable if one remembers thatin Vycor glass, silica constitutes 70% of the volume. Thisagreement also leaves the result of 4He in aerogel evenmore intriguing.

There is as yet no quantitative understanding of thedifference in the critical behavior in these two systems.However, it is most likely related to the pore structure.As noted above, transmission electron microscope picturesof Vycor show an interconnected network of pores with awell-defined diameter. This network configuration is con-firmed by small-angle neutron scattering measurements,which reveal a dominant length scale of 27 nm. Thus,there is no correlation in the position of the silica overlength scales longer than 27 nm. In contrast, there is nosuch characteristic length scale in aerogel. Fractal-likecorrelations appear to extend up to approximately 100 nm.Beyond 100 nm, SAXS data imply a uniform structure(see figure 2). Assuming the Josephson relation (see box1) to be applicable in the aerogel system, the correlationlength is approximately 120 nm at t = 10"3 and 600 nmat t = 10"4. It is therefore surprising that for t < 10"3 thecritical behavior is still different from that of pure 4He.It was noted by Onuttom Narayan of the University of

32 AUGUST 1996 PHYSICS TODAY

Box 1. The superfluid transition in 4Hewing to its small mass and correspondingly large zero-point motion, as well as the very weak attraction be-

tween the highly symmetric atoms, helium remains liquiddown to absolute zero. To produce the solid phase requiresa pressure of 2.5 megapascals. At low temperature, the liquiddisplays one of the most striking of macroscopic quantumphenomena—namely, superfluidity. (See also John Bardeen'sarticle in the December 1990 issue of PHYSICS TODAY, page25, for a discussion of macroscopic quantum phenomena, andRussell Donnelly's article in the July 1995 issue, page 30, foran account of the discovery of superfluidity.) In 4He the phasetransition between the normal and superfluid phases takesplace at a somewhat pressure-dependent temperature 7C ofabout 2.17 K, below which the liquid shows the astonishingability to flow without friction through narrow channels.

Many aspects of superfluid behavior can be understoodwithin the context of the two-fluid model, first suggested byLaszlo Tisza in 1938. This model assumes that the superfluidconsists of two completely interpenetrating components—anormal fluid with properties very similar to those of anyordinary fluid, and a superfluid that carries no entropy andhas no viscosity. The total density of the liquid is the sum ofthe superfluid and normal fluid densities: p = ps -I- pn. A con-venient parameter in the description of this system is thesuperfluid fraction ps /p, which grows from zero at Tc to oneat T = 0.

The superfluid density can be conveniently determined intwo ways. In 1946, Elevter Andronikashvili used a torsionalpendulum to determine ps in bulk 4He. Suspended from atorsional fiber, a cylindrical stack of disks, sufficiently closelyspaced to viscously drag the normal fluid, was immersed inliquid helium. When the helium was cooled below the tran-sition, the superfluid part of the liquid decoupled from themotion of the pendulum, causing a change in its resonancefrequency. This technique has been improved upon consider-ably, mainly by replacing the fiber with a stiff, hollow torsionrod. This torsion rod, typically on the order of 1 mm indiameter and 1 cm in length, is normally made of a beryllium-copper alloy. It is attached to a thin-walled cell that containsa porous substrate, and that also allows helium to be intro-duced into the cell. The cell and torsion rod form a pendulum,which is electrostatically driven at its torsional mode. Theresonance frequency will depend on the ratio of the stiffnessof the torsion rod and the total moment of inertia of the cell.Above the superfluid transition the helium is viscously lockedto the substrate and contributes its entire moment of inertia,usually less than 1% of the total, to the pendulum bob.However, when the temperature is lowered below Ta thiscontribution is reduced, because the superfluid partially decou-ples from the substrate. This decoupling from the motion ofthe oscillator results in a shift in resonance frequency propor-tional to the superfluid density. The torsional oscillator tech-

nique is particularly suitable for the detection of superfluidityin porous media.

One of the predictions of the two-fluid model is that, inaddition to a normal pressure-density sound mode, the super-fluid also supports a propagating temperature/entropy mode.The velocity of this mode, which has become known as secondsound, is primarily a function of the superfluid density. Thussecond-sound measurements are an alternative, and often pre-ferred, method to determine ps. The two-fluid model can beextended straightforwardly to take into account the viscouscoupling between a porous medium and the normal fluid.Thus, acoustic measurements can also be used to obtain thesuperfluid density in porous media.

The superfluid transition in 4He, also called the A-transitionbecause of the characteristic shape of the accompanying heatcapacity peak (figure 3a), provides the most convincing experi-mental confirmation of the scaling and renormalization grouptheory of critical phenomena.17 Such an assessment is basedon the excellent agreement between the measured and pre-dicted heat capacity, C, and superfluid density, ps, in thevicinity of the transition. According to modern theory ofcritical phenomena, very close to the transition, ps and C areexpected to exhibit power-law dependence on the reducedtemperature t = \ T- Tc\ /7C, of the form

(i)

and

Qt) =A/a r

for T < Tc

for T > Tr

for 7 > 7C

C(t)=A'/a' ra> for7< 7C

(2)

(3)

The parameters f and a are called critical exponents; p^ andA are the critical amplitudes. The superfluid density is relatedto the correlation length, f, through the Josephson relation

where m is the mass of the 4He atom and k is the Boltzmannconstant. This relationship implies that v, the critical expo-nent characterizing the divergence of the correlation length,is equal to £. Critical exponents are universal in the sense thatthey depend only on the spatial dimension, d, of the systemand the number of degrees of freedom of the order parametern that characterizes the symmetry of the system. In particular,the superfluid transition in 4He is thought to belong to theuniversality class of d = 3 and n = 2, the three-dimensional xymodel. The most recent experimental values for the exponentsare £ = 0,6705 + 0.0006 and a = -0.01285 ± 0.00038. Theseare in excellent agreement with the most reliable theoreticalvalues18 of 0.672 ± 0.002 and -0.016 ± 0.006.

California, Santa Cruz, and Daniel Fisher of HarvardUniversity that because a is so close to zero, the superfluidtransition satisfies the Harris criterion only marginallyand the recovery to bulklike behavior is logarithmicallyslow.6 Another possible explanation is that small-anglescattering data, which measure only two-point correla-tions, do not capture long-length-scale correlations in theconnectivity of the gel network.7

Even more surprising are the results of the heatcapacity measurements. As can be seen in figure 3, theheat capacity peak at the superfluid transition in aerogeldiffers dramatically from the one in pure 4He. Whereas

the 4He one is essentially symmetrical, with identicalexponents and similar amplitudes above and below thetransition, this symmetry is absent in aerogel. On thehigh-temperature side, one still observes a significantsingular contribution, whereas on the low-temperatureside, one finds a linear dependence on temperature. Ther-mal expansion coefficient measurements (thermodynami-cally equivalent to heat capacity measurements) show onceagain that this behavior persists at elevated pressures.8

Given that an aerogel in which silica occupies 6% ofthe volume has a dramatic effect on the superfluid's criticalbehavior, an interesting question is how this evolves to-

AUGUST 1996 PHYSICS TODAY 33

gPu<tuQ•—>

0.08

0.06

0.04

PH 0.02

0.00

zS 0.04

U

| 0.03

93 0.02

0.01

Pure He

He in aerogel

40

30

n

2.160 2.164 2.168 2.172

TEMPERATURE (K)

202.176

PHASE TRANSITION. Plotted are the superfluid fraction ps/p(black) and heat capacity (red) of 4He near the superfluidtransition of pure 4He (a) and of 4He in aerogel of 94%porosity (b). In both systems the superfluid fraction vanishesat the temperature at which one finds a maximum in the heatcapacity. This suggests that, in the 4He/aerogel system, we areobserving a genuine phase transition. The heat capacity peakin the 4He/aerogel system at 2.172 K is due to helium in largevoids inside the aerogel. The shape of the heat capacity peakin pure 4He has lead to the name aA-transition." FIGURE 3

ward bulk behavior when the aerogel density is furtherreduced. Torsional oscillator measurements on 4He inaerogel that is 98% porous, made at Perm State by JianMa (currently at Amherst College), find a well-definedsuperfluid density exponent £ = 0.80 (figure 4), not verydifferent from that found for 94% aerogel.

He- He mixtures in aerogelTorsional oscillator experiments on 3He-4He mixtures in98% aerogel produce a striking result.9 They indicate thatin this system the superfluid transition line in the phasediagram, instead of terminating at the tricritical pointnear 0.87 K, extends down to absolute zero. (See box 2for a discussion of bulk 3He-4He mixtures, "bulk" meaningwithout aerogel.) As figure 5 shows, the coexistence regionis completely contained within the superfluid part of thephase diagram. Phase separation is now a transitionbetween the usual 4He-rich superfluid and a new 3He-richsuperfluid. The tricritical point is replaced by a regularcritical point at the top of the coexistence curve.

This phase diagram is deduced from the followingexperimental observations. The frequency and amplitudeof the torsional oscillator, which is operated at constantdrive, are monitored as a function of temperature. Aftera temperature scan at a particular 3He concentration iscompleted, the concentration is changed and the nexttemperature scan is made. The superfluid transition insuch a scan is marked by the decoupling of the superfluid

from the motion of the oscillator. At high 3He concentra-tion, this decoupling is the only feature that is detected."High concentration" means that the ratio N% / (N3 + N4)is greater than 0.8, where N:i and N4 are the numbers,respectively, of 3He and 4He atoms in the mixture. Atlower 3He concentration, one finds at a temperature belowthe superfluid transition a second feature—namely, asharp dip in the amplitude of the oscillator. This dip istaken as the signature of the miscible mixture phaseseparation into 3He-rich and 4He-rich phases. This inter-pretation is reinforced by the observation of hysteresis inthe oscillator frequency inside the coexistence region.

An interesting question is how the system organizesitself to be able to support superfluidity at high 3Heconcentrations, whereas this is normally not possible inmixtures without the aerogel (see box 2). Because 4Hehas a smaller zero-point motion than 3He, it can profitmore from the attractive van der Waals interaction be-tween helium and silica. At high 3He concentration anda temperature far below the 3He-4He critical point, the4He atoms are plated out of the mixture onto the silicastrands and form a solidlike layer, similar to that foundin experiments of adsorbed 4He films. After completionof the localized solid layer, the 4He atoms are still expectedto accumulate near the silica strands, instead of beinghomogeneously distributed among the 3He. At low tem-peratures, where the solubility of 4He in 3He is low, thisprocess should lead to the formation of a well-defined4He-rich film. The superfluid transition can now bethought of as taking place purely in this firm.10

The main argument against this picture for mixturesin aerogel comes from heat capacity measurements. Arecent experiment used torsional oscillator and heat ca-pacity techniques to study superfluid transitions of 4Hefilms adsorbed from vapor onto the silica strands inaerogel.11 It was found that, based on the disappearenceof the superfluid density, the transition is quite sharp, butnot accompanied by any measurable peak in the heatcapacity. With the mixtures in aerogel, the situation isquite different.9 At low 3He concentrations, one finds aheat capacity peak at the superfluid transition, verysimiliar to that in bulk mixtures. But whereas in bulkthis peak decreases in size with increasing 3He concen-tration and vanishes at the tricritical point, in aerogel itpersists and can be observed all along the superfluidtransition line. Indeed, the phase diagram for mixturesin 98% aerogel based on heat capacity results is consistentwith that shown in figure 5.

Recent density functional calculations and path inte-gral Monte Carlo simulations of bulk mixtures found12

that the size of the 3He-4He interface is on the order of1 nm. Such a diffuse interface increases substantially theeffective thickness of the 4He film and thus enhances thethree-dimensional connectivity established by the aerogel.This mechanism is most likely responsible for the three-dimensional-like superfluidity observed at low tempera-

34 AUGUST 1996 PHYSICS TODAY

10

10" 10" 10" 10

REDUCED TEMPERATURE10

tures. In such a scenario, the coexistence region at lowtemperatures is likely to correspond to "capillary conden-sation" of the 4He film from neighboring silica strandsinto 4He-rich domains. With increasing 4He concentration,increasingly larger pockets are formed. On the 4He-richside of the phase diagram, coexistence ends when the last3He-rich domain, presumably situated farthest from anysilica strand, shrinks out of existence. With increasingtemperature, 4He atoms will progressively ''evaporate" intothe 3He-rich domains, giving rise to a more homogeneous

SUPERFLUID FRACTION pjp as a function of the reducedtemperature / = (7C - T)/Ta for pure 4He, for 4He in Vycorand for 4He in aerogels of 94% and 98% porosity. For thesake of clarity, the value of ps Ip for the 98% aerogel has beenmultiplied by 0.1 and for 4He in Vycor by 0.01. The Vycordata are parallel to those of pure 4He, implying identicalsuperfluid density exponents. The exponent for 4He inaerogel is clearly larger. FIGURE 4

solution. The :jHe-4He critical point marks the maximumtemperature where 3He- and 4He-rich domains are found.

These results have stimulated a number of recenttheoretical studies. Amos Maritan and his collaboratorsat the University of Padua, in Italy, and at Penn Statetake the silica strands as a source of quenched randomnessthat favors 4He atoms.13 Within the Blume-Emery-Grif-fiths model, the tricritical point was found to be unstable,which can lead to a phase diagram that resembles theone shown in figure 5. By constructing a correlatedrandom chemical potential model to mimic the connectiv-ity of the aerogel, Alexis Falicov and Nihat Berker of MITobtained a mLxture phase diagram remarkably similar towhat is deduced from experiment (figure 5).u

Interestingly, the new superfluid phase at high jHeconcentrations appears even in aerogels of much lowerdensity. This is illustrated in figure 6, which shows acomparison between the heat capacity of bulk mixturesand that for mixtures in 99.5% aerogel, on the yHe-richside of the phase diagram. The discontinuity associatedwith the phase separation in the bulk, although somewhatbroadened, is still clearly visible in the aerogel. In addi-

P4

H

wa,

MEASURED AND CALCULATED (inset)

phase diagrams of 3He-4He mixtures inaerogel of 98% porosity. In contrast towhat one observes in bulk mixtures (thatis, without aerogel), there is no tricriticalpoint. (See box 2.) The transition linebetween the normal (red) and superfluid(blue) phases is detached from thecoexistence curve, and the coexistenceregion resides entirely within thesuperfluid part of the phase diagram.The top of the coexistence curve nowcorresponds to a regular critical point.Alexis Falicov and Nihat Berker of MITdid the calculation. FIGURE 5

0.2 0.4 0.63He CONCENTRATION

0.8

AUGUST 1996 PHYSICS TODAY 35

tion, there is a peak at higher temperature related to thesuperfluid transition, which is conspicuously absent in thebulk mixtures.

Superfluid transition of 'He in aerogelHaving seen these remarkable new phenomena associatedwith the introduction of aerogel into both liquid 1He and3He-4He mixtures, one can ask what the consequencesmight be in the case of the 3He/aerogel system. It isknown that 3He is the paradigm of exotically pairedsuperfluids, in which the Cooper pairs exhibit a pairingother than the usual s-wave symmetry of ordinary super-conductors. The 3He undergoes a superfluid transition attemperatures between 2.5 mK (at high pressure) and 0.9mK (at vapor pressure). In the bulk, the A phase (withspin pairing I I I) + 111)) is favored over the B phase(I 11) + l l l )+ I 1 I + 11 >) only at high pressure and closeto the transition temperature. The pairing length is onthe order of several tens of nanometers. Because diffusequasiparticle scattering at boundaries breaks Cooper pairsand suppresses the superfluidity over a length scale onthe order of a coherence length, confinement of the super-fluid within the pore structure of conventional porousmaterials having micrometer-sized pores can reduce Tc ap-preciably Confinement in pores with diameters less than afew tens of nanometers destroys the superfluidity altogether.

In the 3He/aerogel system, every point in the fluid isless than a coherence length away from a silica strand.Nevertheless, recent torsional oscillator and nuclear mag-

netic resonance (NMR) experiments at Cornell Univer-sity15 and Northwestern University16 found that in 98%aerogel, the superfluid phase is stable. Although thesuperfluid density is significantly reduced (compared topure 3He), the transition is sharp and well defined. Thetemperature dependence of the superfluid density is dis-tinctly different from that in pure 3He. The NMR experi-ment identified the superfluid to be in an equal spin-pairedstate, similar to the A phase, based on the observation ofa temperature-independent susceptibility. The NMR spindynamics of the superfluid was found to be completelyaltered inside the aerogel.

The fact that the superfluid phase is stable in aerogelis probably related to the fact that quasiparticle scatteringby aerogel is qualitatively different from what happens inmore conventional porous media. The size of the gelstrands is on the order of 5 nm, much less than thesuperfluid coherence length, and the mean free path forthe 3He quasiparticles within the aerogel is significantlylonger than the mean distance between strands. Thus itseems that the scattering of the quasiparticles by theaerogel is more appropriately described by impurity scat-tering than by scattering from solid boundaries.

Some key aspects of this problem are being addressedin ongoing theoretical work at Northwestern University,Helsinki University of Technology and the Landau Insti-tute of Theoretical Physics in Moscow. Attempts to modelthe aerogel as an isotropic scattering medium, however,found that the B phase is more stable than the A phase,

Normal

Box 2. Mixtures of He and Hehe accompanying figure shows the phase diagram in the concentration/temperature plane for a 3He-4He mixture. Above0.872 K, liquid 3He and 4He are miscible in all proportions. Below this temperature a miscibility gap opens up in the phase

diagram; this gap is conventionally called the coexistence region (white). The boundary that separates this region from themiscible superfluid (blue) and normal (red) phases is called the coexistence curve. Owing to quantum statistics, in the 7 = 0limit the 4He-rich fluid can still contain up to 6.4% 3He. In contrast, 4He atoms are completely excluded from the 3He-rich

solution. Above 0.872 K the transition between the superfluid andnormal phases is similar in nature to that in pure 4He. Thesuperfluid transition temperature is depressed with increasing 3Heconcentration, tracing a line of superfluid transitions in the phasediagram. This transition line terminates at the tricritical point atT = 0.872 K and A"3 = 0.669, at the top of the coexistence curve.Here X} is the molar 3He concentration, or N}/(N3 + N^ where 7V3

and NA are the numbers, respectively, of 3He and 4He atoms in themixture.

The thermodynamic cooling path indicated in the phase diagramstarts with a normal fluid mixture, at point O. This mixturebecomes superfluid on crossing the transition line at A. Thetransition is accompanied by a A-like peak in the heat capacity.Subsequently, on meeting the coexistence curve at point B, themixture phase separates, and a 3He-rich phase nucleates with X5 asdefined by B'. Here, the heat capacity shows a discontinuity. Onfurther reduction of the temperature, the 3He concentrations of thetwo phases are specified by the coexistence curve—along the pathsB -> C and B' -> C . Starting with a mixture with a 3He concentra-tion of Xy > 0.669, the superfluid transition is absent and only phaseseparation is observed. (See figure 6b.)

Tricritical point

I I I L

0.2 0.4 0.6 0.83He CONCENTRATION

36 AUGUST 1996 PHYSICS TODAY

0.3

"c 0.2

6

£ 0.1<a

p 0.2

HU

Phase separation

Superfluid transition

Mixtures in99.5%-porous aerogel

^3 = 0.84

Bulk mixtures

J I I0.2 0.4 0.6

TEMPERATURE (K)

0.8

HEAT CAPACITY of 3He-rich 3He-4He mixturesin aerogel of 99.5% porosity (a), and withoutaerogel (b). The discontinuity observed in thebulk corresponds to phase separation. Inside theaerogel, this signature is broadened. In addition,one finds a peak in the heat capacity associatedwith the superfluid transition, which at theseconcentrations is absent in the bulk. Theparameter Ar

3 is the molar 3He concentration.FIGURE 6

in contrast to what is found experimentally. Anisotropicscattering induced by the geometry of the strand may leadto stabilization of the A phase, but as yet the details ofboth NMR and torsional oscillator experiments have notbeen explained by the theoretical models. Nevertheless,the existence of a stable 3He superfluid phase in aerogel,and the extension of the superfluid transition line toT = 0 as shown in figure 5, opens the possibility of findinga system of two interpenetrating superfluids.

We thank Lawrence Hrubesh for introducing us to the fine art ofaerogel making. We gratefully acknowledge support provided bythe National Science Foundation.

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