Superfluidity in solid helium and solid hydrogen

1. Superfluidity in solid helium and solid hydrogen E. Kim*, A. Clark, X. Lin, J. West, M. H. W. Chan Penn State University * KAIST, Korea

2. Outline • Introduction • Theoretical background for supersolid • Experimental setup - Torsional oscillator technique • Supersolid in porous media • Supersolid in bulk 4He • Heat Capacity of solid helium. • Results in para-hydrogen • Summary

3. Phase diagram of 4He Fritz London is the first person to recognize that superfluidity in liquid 4He is a BEC phenomenon. Solid Condensation fraction was predicted and measured to be 10% near T=0. Superfluid fraction at T=0, however is 100%. Normal Liquid, (He I) Superfluid (He II)

4. - In a perfect solid when each atom is localized at a specific lattice site and symmetry is ignored, then there is no BEC at T=0. Penrose and Onsager, Phys. Rev.104, 576 (1956) - Off Diagonal Long Range Order, or superfluidity, which is directly related to Bose-Einstein Condensation, may occur in a solid phase if particles are not localized. C. N. Yang,Rev. Mod. Phys.34, 694 (1962) Can a solid be ‘superfluid’?* *A.J.Leggett ,PRL 25, 1543 (1970)

5. Zero-point Energy total energy zero-point energy Inter-atomic potential • Lindemann Parameter the ratio of the root mean square of the displacement of atoms to the interatomic distance (da) A classical solid will melt if the Lindemann’s parameter exceeds the critical value of ~0.1 . • X-ray measurement of the Debye-Waller factor of solid helium at ~0.7K and near melting curve shows this ratio to be 0.262. (Burns and Issacs, Phys. Rev. B55, 5767(1997))

6. Superfluidity in solid: not impossible! - If solid 4He can be described by a Jastraw-type wavefunction that is commonly used to describe liquid helium then crystalline order (with finite fraction of vacancies) and BEC can coexist. G.V. Chester,Lectures in Theoretical Physics Vol XI-B(1969); Phys. Rev. A2, 256 (1970) J. Sarfatt, Phys. Lett.30A, 300 (1969) L. Reatto, Phys. Rev.183, 334 (1969) - Andreev and Liftshitz assume the specific scenario of zero-point vacancies and other defects ( e.g. interstitial atoms) undergoing BEC and exhibit superfluidity. Andreev & Liftshitz, Zh.Eksp.Teor.Fiz.56, 205 (1969).

7. R Solid Helium The ideal method to detect superflow would be to subject solid helium to undergo dc or ac rotation to look for evidence of ‘Non-Classical Rotational Inertia’. Leggett,Phys. Rev. Lett. 25, 1543 (1970) Quantum exchange of particles arranged in an annulus under rotation leads to a measured moment of inertia that is smaller than the classical value I(T)=Iclassical[1-fs(T)] fs(T) is the supersolid fraction Its upper limit is estimated by different theorists to range from 10-6 to 0.4; Leggett: 10-4

8. No experimental evidence of superfluidity in solid helium prior to 2004 • Plastic flow measurement • Andreev et al. Sov. Phys. JETP Lett 9,306(1969) • Suzuki J. Phys. Soc. Jpn. 35, 1472(1973) • Tsymbalenko Sov. Phys. JETP Lett. 23, 653(1976) • Dyumin et al.Sov. J. Low Temp. Phys. 15,295(1989); • Torsional oscillator • Bishop et al. Phys. Rev. B 24, 2844(1981) • Mass flow • Greywall Phys. Rev. B 16, 1291(1977) • Bonfait, Godfrin and Castaing, J. de Physique 50, 1997(1989) • Day, Herman and Beamish, Phys. Rev. Lett. 95, 035301 (2005) • PV(T) measurement • Adams et al. Bull. Am. Phys. Soc. 35,1080(1990) • Haar et al. J. low Temp. Phys. 86,349(1992) • However, interesting results are found in • Ultrasound Measurements at UCSD. • Goodkind Phys. Rev. Lett. 89,095301(2002) and • references therein

9. Ultrasound velocity and dissipation measurements in solid 4He with 27.5ppm of 3He The results are interpreted by the authors as showing BEC of thermally activated vacancies above200mK.  9.3 MHz x28MHz 46MHz P.C. Ho, I.P. Bindloss and J. M. Goodkind, J. Low Temp. Phys. 109, 409 (1997)

10. TEM of Vycor glass Solid helium in a porous medium should have more disorder and defects, which may facilitate the appearance of superflow in solid? Amorphous boundary layer Solidification proceeds in two different directions: 1) In the center of the pore a solid cluster has crystalline order identical to bulk 4He 2) On the wall of a pore amorphous solid layers are found due to the van der Waals force of the substrate Elbaum et al. Adams et al. Brewer et al. Crystalline solid

11. Torsional oscillator is ideal for the detection of superfluidity Be-Cu Torsion Rod Torsion Bob containing helium Drive Detection Resolution Resonant period (o) ~ 1 ms stability in  is 0.1ns /o = 5×10-7 Mass sensitivity ~10-7g f Amp f0 Q=f0/f ~ 2×106

12. Torsional oscillator studies of superfluid films Vycor Δ Above Tc the adsorbed normal liquid film behaves as solid and oscillates with the cell, since the viscous penetration depth at 1kHz is about 3m. Berthold,Bishop,Reppy, PRL39,348(1977)

13. Solid helium in Vycor glass f0 = 1024Hz Q ~ 1*106 0.38mm 2.2mm Torsion Rod Torsion Bob (vycor glass) 5cm Drive Detect A=A0sinωt v=|v|maxcosωt |v|max=rA0ω

14. Period shifted by 4260ns due to mass loading of solid helium Solid 4He at 62 bars in Vycor glass *=966,000ns

15.  Supersolid response of helium in Vycor glass • Period drops at 175mK •  appearance of NCRI • size of period drop • - ~17ns *=971,000ns

16. -*[ns] *=971,000ns Superfluid response Total mass loading =4260ns Measured decoupling -o=17ns “ Apparent supersolid fraction”= 0.4% (with tortuosity correction s/ =2% ) Weak pressure dependence

17. Δ0[ns] Strong velocity dependence • For liquid film adsorbed on Vycor glass • vc > 20cm/s • Chan et. al. Phys. Rev. Lett.32, 1347(1974). • For superflow in solid 4He • vc < 30 µm/s

18. Control experiment I : Solid 3He? Nature427,225(2004)

19. 4He solid diluted with low concentration of 3He

20. Effect of the addition of 3He impurities At 0.3ppm, the separation of the 3He atoms is about 450Å

21. Search for the supersolid phase in the bulk solid 4He. Be-Cu Torsion Rod Detection Filling line I D=0.4mm Channel OD=10mm Width=0.63mm Mg disk OD=2.2mm Drive Al shell Filling line Solid helium in annulus channel Torsion cell with helium in annulus

22. Drive Torsional Oscillator (bulk solid helium-4) Torsion rod Torsion cell 3.5 cm Detection

23. Porous media are not essential ! Solid 4He at 51 bars 4µm/s corresponds to amplitude of oscillation of 7Å NCRI appears below 0.25K Strong |v|maxdependence (above 14µm/s) Amplitude minimum, Tp 0= 1,096,465ns at 0 bar 1,099,477ns at 51 bars (total mass loading=3012ns due to filling with helium) Science305, 1941(2004)

24. Non-Classical Rotational Inertia Fraction |v|max ρS/ρ NCRIF Total mass loading =3012ns at 51 bars

25. Non-Classical Rotational Inertia Fraction ρS/ρ |v|max Total mass loading =3012ns at 51 bars

26. Filling line Mg barrier Mg disk Al shell Mg Disk Solid helium in annulus channel Control experiment II • With a barrier in the annulus, there should be NO simple superflow and the measured superfluid decoupling should be vastly reduced Torsion cell with blocked annulus Mg barrier Al shell Solid helium Channel OD=15mm Width=1.5mm

27. If there is no barrier, then the supersolid fraction appears to be stationary in the laboratory frame; with respect to the torsional oscillator it is executing oscillatory superflow. -*[ns] Superflow viewed in the rotating frame

28. With a block in the annulus, irrotational flow of the supersolid fraction contributes about 1% (Erich Mueller) of the barrier-free decoupling. Δ~1.5ns -*[ns] Irrotational flow pattern in a blocked annular channel (viewed in the rotating frame) * If no block, the expected Δ=90ns A. L. Fetter, JLTP(1974)

29. Similar reduction in superfluid response is seen in liquid helium at 19 bars in the same blocked cell measured superfluid decoupling in the blocked cell Δ(T=0)≈ 93ns. While the expected decoupling in unblocked cell is 5270ns. Hence the ratio is 1.7% similar to that for solid. [ns] Conclusion : superflow in solid as in superfluid is irrotational.

30. 108bar ρs/ρ 136bar ρs/ρ Superflow persists up to at least 136 bars !

31. ω R Strong and ‘universal’ velocity dependence in all samples vC~ 10µm/s =3.16µm/s for n=1

32. Pressure dependence • As a function of pressure the supersolid fraction shows a maximum near 55bars. The supersolid fraction extrapolates to zero near 170 bars.

33. Superfluidity in ultra-pure Solid Helium: 1ppb 3He • t0 ~ 0.77ms [1300Hz] • t4He - t0 = 3920ns • NCRIF ~ 1.25/3920 = 0.03% Exp. done at at U. of Florida

34. Phase Diagram of 4He

35. Heat Capacity signature? • No reliable heat capacity measurement of solid 4He below 200mK because of large background contribution due to the sample cell.

36. Experimental cell of Xi Lin and Tony ClarkSilicon!!

37. Results: pure 4He (0.3ppm 3He)

38. Results: pure 4He (0.3ppm 3He)

39. Results: pure 4He (0.3ppm 3He)

40. Results: pure 4He (0.3ppm 3He)

41. Results: pure 4He (0.3ppm 3He) Heat capacity peak near the supersolid transition

42. Is the supersolid phase unique with 4He? Apparently not! Preliminary torsional oscillator data of Tony Clark and Xi Lin indicate similar supersolid-like decoupling in solid H2. de Boer parameter • 3He  3.09 • 4He  2.68 • H2 1.73 • HD  1.41 • D2 1.22 More quantum mechanical

43. Hydrogen in a cylindrical cell Inside Mg bob: Hydrogen space BeCu wall Q = 1.6million P0 = 560,400ns dP ~0.05ns

44. Hydrogen in a cylindrical cell Inside Mg bob: Hydrogen space BeCu wall DPHD = 4014ns (93% filling) HD

45. Hydrogen in a cylindrical cell Inside Mg bob: Hydrogen space BeCu wall DPH2 = 1638ns (64% filling) HD H2

46. HD H2 Hydrogen in a cylindrical cell Temperature below 50mK uncertain, thermometer not on the torsional cell Ortho concentration is most likely less than 0.5% HD concentration uncertain NCRIF ~ 0.015%

47. Hydrogen in an annular cell Samples contain < 50ppm HD Inside Mg bob: Hydrogen space (h=3.5mm w=2.3mm) Mg X is the ortho conc, BeCu wall Q = 350,000 P0 = 709,700ns dP <0.1ns

48. Hydrogen in an annular cell Comparison of 50 and 200ppm HD Inside Mg bob: Hydrogen space BeCu wall Mg

49. Summary • Superflow is seen in solid helium confined in Vycor glass with pores diameter of 7nm and also in bulk. Results in bulk have been replicated in three other labs. • Supersolid fraction is on the order of 1% for He-4 sample with 0.3ppm of He-3 impurities. For ultra-high purity sample (1ppb He-3) the supersolid fraction is on the order of 0.03% and the transition temp. is depressed. • There is preliminary evidence of a heat capacity peak at the transition. • Superflow is also seen in para-hydrogen. The supersolid fraction is on the order of 0.05%

50. We are grateful for many informative discussions with many colleagues, too numerous to acknowledge all of them. P.W. Anderson J. R. Beamish, D. J. Bishop, D. M. Ceperley, J. M. Goodkind, T. L. Ho, J. K. Jain, A. J. Leggett, E. Mueller, M. A. Paalanen, J. D. Reppy, W. M. Saslow, D.S. Weiss