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arXiv:cond-mat/06

10240v1

[cond-mat.other]9Oct200

6

Search for Superfluidity in Solid Hydrogen

A. C. Clark, X. Lin, and M. H. W. ChanDepartment of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802

(Dated: February 6, 2008)

A torsional oscillator study of solid para-hydrogen has been carried out down to 20 mK in searchfor evidence of superfluidity. This work was inspired by the observation of the supersolid phase insolid 4He. We found evidence of a possible phase transition, marked by an abrupt increase in the

resonant period of oscillation and onset of extremely long relaxation times as the temperature wasraised above 60 mK. The change in the period for para-hydrogen, in contrast to solid 4He, is notrelated to superfluidity as it fails a crucial test showing that it is not a consequence of irrotationalflow. The long relaxation times observed suggest the effect is related to the motion of residualortho-hydrogen molecules in the solid.

PACS numbers: 66.35.+a, 66.30.-h, 67.80.-s, 67.80.Mg

A supersolid is a solid that possesses an element of su-perfluidity. The existence of supersolid 4He was first sug-gested 67 years ago [1], while possible underlying mecha-nisms were discussed 30 years later [2]. Experimental ev-idence of supersolid 4He was reported by Kim and Chan

(KC) only two years ago in a series of torsional oscillatorexperiments on bulk solid helium [3, 4] and solid heliumconfined in porous media [5, 6]. KC demonstrated thatthe torsional oscillator is an ideal technique for the detec-tion of solid 4He acquiring non-classical rotatonal inertia(NCRI) [7]. NCRI is indicated by a drop in the resonantperiod of the oscillator below the transition temperature.The decrement in the period is proportional to the su-persolid fraction in the limit of low oscillation speed [3].These observations have since been replicated in threeother laboratories [8]. This prompted us to address thequestion of whether or not the supersolid phase is presentin any system other than 4He.

Helium is the most quantum mechanical solid of allthe elements. This is due to the combination of largezero-point energy of individual atoms and weak attrac-tive interactions among them. A measure of the quantumnature of a substance in the condensed phases, i.e. liq-uid or solid, is given by the de Boer parameter, =h/

m, where and are fitting parameters for the

inter-particle potential energy [9]. Only helium and hy-drogen isotopes have greater than unity (3He = 3.0,4He = 2.6, H2 = 1.7, HD = 1.4, D2 = 1.2). There-fore H2, after

4He, is the most likely system to exhibitsuperfluidity. Indeed the zero-point motion of solid H2

at 5.4 K under zero pressure is 18% of the nearest neigh-bor distance [10], while in low density solid 4He it is 26%[11]. Both of these values are significantly higher thanthe Lindemann (melting) criterion of approximately 10%for classical solids. The fact that the supersolid phasein 4He persists up to at least 135 bar [4], where it isexpected to be less quantum mechanical than at lowerpressures, was further incentive to investigate solid H2.

We have carried out a series of torsional oscillator mea-surements on solid hydrogen. Our experimental configu-

ration is similar to that used by KC [3]. A sample cell issuspended from the cooling stage of a 3He-4He dilutionrefrigerator by an annealed beryllium-copper torsion rod.The rod provides a restoring force to keep the cell in tor-sional motion and doubles as a filling line to the cell. The

H2 space in the cell is comprised of an open annulus anda central filling line, which are connected by three chan-nels that span the diameter of the cell. The annulus hasan average radius, width, and height of 6.5 mm, 2.3 mm,and 4.8 mm, respectively. The radial channels and fillingline have a diameter of 1 mm, a combined volume thatconstitutes 7% of the total cell volume, and contributeabout 2% of the rotational inertia of all the H2 in thecell. Electrodes on the exterior of the torsion cell areused to capacitively drive and detect oscillatory motion.The resonant period (0) of the oscillator, is determinedby the moment of inertia (I) of the torsion cell and thetorsion spring constant () of the rod, such that 0 =

2I/. The mechanical quality factor (Q) of the os-

cillator at low temperature is 1.1 x 106, which allows forthe detection of changes in 0 of one part in 107.

Solid H2 samples were grown at saturated vapor pres-sure from high purity H2 gas containing very low levels ofisotopic impurities: less than 10 ppm of both hydrogendeuteride (HD) and deuterium (D2). However, for pureH2 crystals there exist impurities of a second kind. Thetwo lowest rotational energy states of each molecule arethe para- (p-H2) and ortho- (o-H2) states, having angularmomentum J = 0 and J = 1, respectively. The equilib-rium concentrations of p-H2 (1 - x) and o-H2 (x) at 300 K

are respectively 0.25 and 0.75. At temperatures (T) be-low 4 K the equilibrium value ofx is approximately zeroand the solid is essentially pure p-H2. However, the con-version rate from o-H2 to p-H2 in the solid is very slowfor T > 500 mK (eg. about three weeks of conversionare required for x to drop from 0.01 to 0.009). This ratecan be increased substantially by keeping H2 in the liquidphase (with T> 13.8 K, the triple point temperature) inthe presence of a high surface area, magnetic material. Inthis way we attain x < 0.01. This process is irreversible

http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1http://arxiv.org/abs/cond-mat/0610240v1

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so that the average x of each sample can only decreasewith time. Concentrations are determined by keepingthe mixing chamber at 20 mK and measuring the tem-perature difference between the torsion cell and mixingchamber. Based on the temperature difference and thethermal conductance of the Be-Cu torsion rod, we caninfer the amount of o-p conversion heat being releasedand hence x within the H2 sample. The smallest mea-

surable temperature gradient corresponds to x = 0.01.All H2 samples discussed in this work have x < 0.01 (i.e.no observed gradient). To limit the uncertainty in theabsolute value ofx, we measured the thermal conductiv-ity of our Be-Cu torsion rod and found good agreementwith extrapolated values from high temperature (T > 1K) data previously reported [12].

In Fig. 1a we have plotted the resonant period as afunction of temperature for several samples. The emptycell period at 1 K (1K) is 709,700 ns, which decreasesby only 3 ns, or five parts in 106, as the cell is cooled to20 mK. This change is determined by the T-dependence,if any, of I and . Based on earlier work [13], the smallchange in 0 depicted in Fig. 1a is due to small changesin I and/or rather than large, nearly compensatingchanges of each. For the H2 sample in Fig. 1, the pe-riod increases from the empty cell value by 2682 ns uponsolidification, indicating that the cell is 88% full. Themass loading due to the heavier and denser isotope, HD,is 4675 ns, corresponding to 91% filling. Each data sethas been shifted vertically for easy comparison. The cor-responding values of1K for each are noted in the figure.There is a slight difference in the slope of each curveshown. This modification of the background when thesample is introduced into the cell is always observed in

torsional oscillator studies of liquid helium films [13], aswell as in the solid helium work [3, 5].

The most striking feature in the data of Fig. 1a is theclear difference between H2 and HD. The abrupt drop in0 below 180 mK for the cell containing H2 is absent forHD. This discrepancy is similar to that found between4He and 3He, and indicates that the drop in the periodcould be a signature ofNCRI, as it is seen only in bosonicsolids. The net change in 0 is obtained by subtractingthe measured period from values extrapolated from hightemperature, as indicated in the figure. While the pe-riod drop is more than an order of magnitude smallerthan that found in solid 4He in a similar geometry [3], its

temperature dependence is similar [Fig. 1b]. The changein 0 first becomes measurable below 180 mK, and in-creases first gradually with decreasing T and then muchmore rapidly prior to saturation near 60 mK. This simi-larity in the T-dependence also suggests that the perioddrop in H2 could be a signature ofNCRI.

However, there are four puzzling features in the H2data. First, for solid 4He the drop in the period is ac-companied by a minimum in the amplitude of oscillationat the temperature where 0 changes most rapidly [3].

There is no such minimum in oscillation amplitude nearthe transition region in the H2 experiment. Second, for4He the measured NCRIF decreases once the maximumoscillation speed of the sample exceeds 10 m/s [3]. Thisspeed has been interpreted as the critical velocity of su-perflow. There is no evidence of a critical velocity in solidH2. We found the period drop to be independent of os-cillation speed up to 500 m/s, the maximum within the

linear response range of the oscillator.

Third, the apparent supersolid onset temperature (180mK) and saturation temperature (60 mK) are remark-ably close to the 230 mK and 50 mK values found forsolid 4He [3]. The similarity in these characteristic tem-peratures is inconsistent with the substantial differencein 4He and H2. For solid

4He in Vycor glass, very low3He impurity levels can significantly enhance the onsettemperature [5]. While the TC for a sample containing0.2 ppm of3He is about 175 mK, a sample with 10 ppmhas an onset temperature around 370 mK. This effect hasalso been observed in bulk 4He [14], thus a small amount

of HD impurities may alter TC for solid H2. However,we do not know the exact HD impurity level in our solidH2 samples. The reason is that HD is heavier than H2and hence preferentially adsorbs onto surfaces, which are

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-12

-10

-8

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-2

0

"Open Annulus Cell":

Empty, 1K

= 709,700 ns

HD, 1K

= 714,375 ns

H2 [x< 1%], 1K

= 712,382 ns

"Blocked Annulus Cell":

H2 [x< 1%], 1K

= 783,629 ns

a.)

b.)

0-1K

[ns]

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.00

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0.08

0.09 "Open Annulus Cell":

H2 [88% full, x < 1%]

"Blocked Annulus Cell":

H2 [80% full, x < 1%]

ApparentNCRIF[%]

Temperature [K]

FIG. 1: a.) T-dependence of the resonant period for boththe open and blocked cells. All of the data shown wereobtained at maximum rim velocities less than 40 m/s. Forthe open annulus, data is plotted for the empty cell, cell con-taining HD, and cell containing H2. H2 data taken using theblocked cell exhibits the same qualitative features, includinglong relaxation times. b.) T-dependence of the absolute pe-riod change from the high-T background, normalized by thetotal H2 mass loading (NCRIF, the apparent NCRI frac-tion). The T-dependence is similar to that observed in solid4He. However, the same effect is seen in the blocked cell.

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abundant in the H2 gas handling system leading into thetorsion cell. Specifically, the capillary system contains achamber filled with a high surface area, FeO(OH) com-pound. Thus, we expect that the actual HD concentra-tion in each sample is much less than the quoted value of10 ppm. Since we do not know with certainty the exactHD concentration we cannot validate the 180 mK onsetand 60 mK saturation temperatures ofNCRI.

The final and most unusual feature of the H2 data,compared to 4He, is the exceedingly long relaxation time(t) of the period reading [Fig. 2]. It is expected that theoscillator comes into thermal and mechanical equilibriumin the following manner. The e1 time constant requiredfor thermal equilibration increases with decreasing T andtypically reaches 15 minutes below 50 mK. Since the me-chanical Q is on the order of 106, the necessary timeto allow for mechanical equilibrium is about 10 minutes.Therefore it is necessary to wait at least 75 minutes af-ter each temperature step before collecting 0 data. Thisprotocol was used in the solid helium experiments to en-sure reproducible data upon warming and cooling. How-ever, in gathering data on H2 we observe t-values as longas 36 hours just above 60 mK. There is no such evidenceof relaxation in the HD sample or for the empty cell, forwhich 0 equilibrates within 75 minutes at all T. We willreturn to the issue of t-values below.

We have carried out a crucial control experiment toascertain whether the observed effect in solid H2 is a sig-nature of superfluidity. We constructed another samplecell, identical to the first except for the addition of a bar-rier at one point in the annulus. Its purpose is to makethe path for the irrotational superflow much more tortu-ous, leading to a predictable decrement in NCRI [15, 16],

thus reducing the drop in the period. This control exper-iment provides a convincing case for the supersolid phaseof4He. It was found that when the annulus was blocked,the drop in 0 was reduced by a factor of 70 from theopen annulus value [3]. This reduction is in agreementwith what is expected for irrotational superflow [16]. Theresult of this control experiment for solid H2 is plottedalongside the data of the open annulus cell in Fig. 1a.The period drop below 180 mK is still present for theblocked annulus cell. In fact, we find the magnitudeis larger in the latter. One possible reason for this isthat, while each sample has x < 0.01, the exact ortho-concentrations of the two samples may differ. Regardless,

the result of the control experiment clearly indicates thatthe H2 result is completely different from that found insolid 4He and is not related to superfluidity. We notethat, as in the open cell, long relaxation times are ob-served just above the saturation temperature of 60 mK.

Long equilibration times are typical in studies of o-H2diffusion in solid p-H2 [17]. In the dilute limit where xis less than several percent, isolated o-H2 molecules (sin-gles) cluster together below 1 K even though thermallyactivated diffusion is severely limited. The o-H2 impuri-

ties effectively propagate throughout the lattice via thetransferral of J from molecule to molecule [18], ratherthan by particle exchange. The equilibrium state ofthe solid thus involves a temperature dependent distri-bution of singles, pairs (two neighboring o-H2 molecules)and larger clusters of o-H2. At several kelvin, all of theo-H2 molecules are randomly distributed as singles. Asthe temperature is lowered the concentration of isolated

singles decreases in favor of the formation of pairs andlarger clusters. Since o-H2 and p-H2 have different den-sities, in the limit of zero temperature the system prefersto be in a macroscopically phase-separated state.

Clustering was first observed by monitoring the timeevolution of the nuclear magnetic resonance (nmr) ab-sorption spectra for o-H2 singles and pairs after rapidwarming or cooling of solid H2 crystals [19]. Representa-tive growth (above 1 K) and decay (below 1 K) rates ofthe singles spectrum [20] are presented in Fig. 2. The sin-gles decay rate exhibits a non-monotonic T-dependence,decreasing dramatically between 300 mK and 100 mK.Well below 100 mK the decay of singles is on a timescale of minutes rather than hours, rendering the mea-surement impossible. The pair spectrum also shows anenhancement in its decay rate at low T but time con-stants are at least one order of magnitude longer, eg. onthe order of 10 hours at 25 mK [21]. Also included inFig. 2 are relaxation times obtained from simultaneousthermal conductivity measurements [20]. The time evo-lution of the thermal conductivity depends strongly onT. For T > 300 mK the time dependence is describedby a simple exponential. However, contrary to the sin-gles decay rate, a sum of two exponentials best fits theconductivity data below 300 mK. This difference may be

related to the presence of o-H2 clusters at lower temper-ature. Since it is difficult to resolve the spectra of triplesor larger clusters [22], nmr studies cannot quantify thedegree of mobility of larger o-H2 clusters that form at lowT. Their presence, however, has been detected for solidH2 crystals held at 25 mK for only 24 hours [23].

Our torsional oscillator data are obtained using the fol-lowing protocol. Upon cooling H2 samples to 20 mK, oneto two weeks are allowed for equilibration, during whichtime 0 drops smoothly until finally stabilizing to within0.05 ns, our limiting resolution. We note that the rate ofirreversible o-p conversion is enhanced for T < 200 mKsince local x-values are higher in clustered regions [24].

However, upon stabilizing at 20 mK, we observe no longterm drift in the 0-reading. After complete equilibrationthe temperature is raised in successive steps, for each ofwhich 0 is measured as a function of time [Fig. 2 inset].We find for all samples that 0 equilibrates quickly forany T < 60 mK. However, further increase in T leadsto extremely slow relaxation. Near and above 180 mKwe find that t shortens, returning to a value of about15 minutes, the usual e1 equilibration time. When thetemperature sweep is complete the system is returned

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0.05 0.1 0.5 1 50.25

0.5

0.751

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5

7.510

25

50

75

0 4 8 12 16 20 240.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

115 mK

82 mK

157 mK

0

[ns]

Time [hrs]

57 mK

nmr [0.4 - 3% o-H2]

, th cond [0 .4 - 3% o-H2]

tors osc [0.5% o-H2]

RelaxationT

ime,t[hours]

Temperature [K]

FIG. 2: T-dependence of measured relaxation times, includ-ing data from the literature [20]. The sharp increase in tcoincides with the observed period shift. At T 180 mK, aminimum in t is observed for both the torsional oscillator and

thermal conductivity measurements. Inset: Time dependenceof0 at several different temperatures. The curves have beenshifted vertically for clarity. The longest relaxation times areobserved at T 80 mK.

to 20 mK and allowed to re-equilibrate. Our measuredt-values are presented in Fig. 2.

Associating the motion of o-H2 impurities with the ob-served time dependence of0 in the present work explainsthe absence of this unusually long relaxation in HD, 4He,and 3He. In our experiment we allow clustering to en-sue for up to two weeks at 20 mK. Based on measured

decay times in the literature, no isolated o-H2

singles orpairs remain in our samples at the time we begin eachtemperature sweep. Thus, we are able to follow the un-clustering process upon gradual warming of the torsioncell.

The moment of inertia and hence 0 are extremely sen-sitive to the radial density profile of the sample. Changesin the period require the rearrangement of o-H2 withinthe torsion cell in such a way as to reduce or enhancethe density at large radii. We think the most likely ex-planation for the observed drop in 0 below 180 mK isthe formation of large clusters of nearly pure o-H2 in thecenter of the cell (fill line and radial channels), increasing

the purity of p-H2 in the annulus. Since the density ofthe o-H2 is 1.7% higher than that of p-H2, the clusteringof o-H2 toward the center has the net effect of reducingI. An extreme case is where all of the o-H2 moleculescluster to the center of the torsion cell (the filling line).In this situation, the reduction of0 is 1.7 x 104 of thetotal H2 mass loading. This is on the same order of ourobservations [Fig. 1], i.e. 5.5 x 104 and 9.0 x 104 inthe open and blocked cells, respectively. It is known thatsolid H2 does not wet many metal substrates [25]. If the

contact angle of the sample to the cell walls is differentfor the mixed and phase-separated phases (eg. p-H2), Iwill show a change upon phase separation. This is onepossible explanation for larger than expected change in0 that we observe.

In conclusion, a resonant period shift in our torsionaloscillator containing solid H2 is observed. While the phe-nomenon shares some features with the supersolid tran-

sition in 4He, there are several dissimilarities. Most no-table is the presence of the period drop in both the openand blocked cells, proving that the effect is not related tosuperfluidity. We believe the abrupt rise in 0 accompa-nied by a sudden increase in t near 60 mK is consistentwith a transition from a phase-separated configuration toone that is un-clustered at high temperature.

We thank E. Kim and W. N. Hardy for their advice.Special thanks to H. Meyer for many enlightening discus-sions. Support comes from NSF Grant DMR 0207071.

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Penzyev and M. Kubota, private communication.[9] J. de Boer, Physica 14, 139 (1948).[10] M. Nielsen, Phys. Rev. B 7, 1626 (1973).[11] C. A. Burns and E. D. Isaacs, Phys. Rev. B 55, 5767

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Rev. B 39, 8934 (1989).[14] E. Kim and M. H. W. Chan, private communication.[15] A. L. Fetter, J. Low Temp. Phys. 16, 533 (1974).[16] E. J. Mueller, private communication (2005).[17] H. Meyer, Can. J. Phys. 65, 1453 (1987); Low Temp.

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26, 646 (1971).

[19] L. I. Amstutz, J. R. Thompson, and H. Meyer, Phys.Rev. Lett. 21, 1175 (1968).

[20] X. Li, D. Clarkson, and H. Meyer, J. Low Temp. Phys.78, 335 (1990).

[21] S. Washburn, R. Schweizer, and H. Meyer, J. Low Temp.Phys. 40, 187 (1980).

[22] A. B. Harris et al., Phys. Rev. 175, 603 (1968).[23] R. Schweizer et al., J. Low Temp. Phys. 37, 309 (1979).[24] R. Schweizer et al., J. Low Temp. Phys. 37, 289 (1979).[25] M. Sohaili, J. Klier, and P. Leiderer, J. Phys.: Condens.

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mailto:bg20339@hotmail.comhttp://arxiv.org/abs/cond-mat/0604528http://arxiv.org/abs/cond-mat/0607032http://arxiv.org/abs/cond-mat/0607032http://arxiv.org/abs/cond-mat/0604528mailto:bg20339@hotmail.com