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. Can a solid be ‘superfluid’?*
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)
*A.J.Leggett ,PRL 25, 1543 (1970)

5. 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
(Burns and Issacs, Phys. Rev. B 55, 5767(1997))
Zero-point Energy
Inter-atomic potential
total energy
zero-point energy

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. A 2, 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 I(T)=Iclassical[1-fs(T)]
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
Solid Helium R
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, (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 above 200mK.
 9.3 MHz
x 28MHz
 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, PRL 39,348(1977)

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

14. Solid 4He at 62 bars in Vycor glass
Period shifted by ns due to mass loading of solid helium
*=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. Superfluid response Total mass loading =4260ns Measured decoupling
“ Apparent supersolid fraction”= 0.4%
(with tortuosity correction s/ =2% )
Weak pressure dependence

17. Strong velocity dependence
Δ0[ns]
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?
Nature 427,225(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.
Torsion cell
with helium in annulus
Be-Cu Torsion Rod
OD=2.2mm
I D=0.4mm
Filling line
Mg disk
Filling line
Solid helium in annulus channel
Al shell
Channel OD=10mm
Width=0.63mm
Detection
Drive

22. Torsional Oscillator (bulk solid helium-4)
Torsion rod
Torsion cell
3.5 cm
Drive
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|max dependence
(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)
Science 305, 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
|v|max
Total mass loading
=3012ns at 51 bars

26. Torsion cell with blocked annulus
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
Mg disk
Filling line
Solid helium in annulus channel
Al shell
Mg barrier
Mg Disk
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. * If no block, the expected Δ=90ns
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. Superflow persists up to at least 136 bars !

31. Strong and ‘universal’ velocity dependence
in all samples ω R
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 Clark Silicon!!
Lattice Materials CorP., MT, USA
Optical Silicon purity : >99.999%

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. Hydrogen in a cylindrical cellHD H2
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

51. Xi, Tony, Eunseong, Josh