Superfluidity in solid helium and
solid hydrogen
E. Kim*, A. Clark, X. Lin, J. West,
M. H. W. Chan
Penn State University
* KAIST, Korea
Outline



Introduction
Theoretical background for supersolid
Experimental setup
- Torsional oscillator technique


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
Supersolid in porous media
Supersolid in bulk 4He
Heat Capacity of solid helium.
Results in para-hydrogen
Summary
Phase diagram of 4He
Solid
Normal
Liquid,
(He I)
Superfluid (He II)
Fritz London is the first
person to recognize that
superfluidity in liquid 4He
is a BEC phenomenon.
Condensation fraction
was predicted and
measured to be 10%
near T=0. Superfluid
fraction at T=0,
however is 100%.
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)

Lindemann Parameter
the ratio of the root mean square of the
displacement of atoms to the interatomic distance
(da)
total energy
zero-point energy
 u2 
L 
 0.26
da
Zero-point Energy
Inter-atomic potential
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. B 55, 5767(1997))

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).
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
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
Solid Helium
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
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)
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
Crystalline solid
Elbaum et al. Adams et al.
Brewer et al.
Torsional oscillator is ideal for
the detection of superfluidity
Be-Cu
Torsion Rod
Resolution
Torsion Bob
containing
helium
Resonant period (o) ~ 1 ms
Drive
Detection
I
 o  2
K
/o = 5×10-7
Amp
stability in  is 0.1ns
f
Mass sensitivity ~10-7g
Q=f0/f ~ 2×106
f0
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)
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
Solid 4He at 62 bars in Vycor glass
Period shifted by 4260ns
due to mass loading of
solid helium
*=966,000ns
Supersolid response of helium in Vycor glass
• Period drops at 175mK
 appearance of NCRI
• size of period drop
- ~17ns

*=971,000ns
Superfluid response
Total mass loading =4260ns
Measured decoupling
-o=17ns
-*[ns]
“ Apparent supersolid fraction”=
0.4%
(with tortuosity correction s/ =2%
Weak pressure dependence
*=971,000ns
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
Control experiment I : Solid 3He?
Nature 427,225(2004)
4He
solid diluted with low concentration of 3He
Effect of the addition of 3He impurities
At 0.3ppm, the separation of the
3He atoms is about 450Å
Search for the supersolid phase
in the bulk solid 4He.
Be-Cu Torsion Rod
Torsion cell
with helium in annulus
I D=0.4mm
Filling line
OD=2.2mm
Mg disk
Filling line
Channel
OD=10mm
Width=0.63mm
Al shell
Solid helium in
annulus channel
Detection
Drive
Torsional Oscillator (bulk solid helium-4)
Torsion rod
Torsion cell
3.5 cm
Detection
Drive
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)
Non-Classical Rotational Inertia Fraction
NCRIF
ρS/ρ
|v|max
NCRIF 

total mass loading
Total mass loading
=3012ns at 51 bars
Non-Classical Rotational Inertia Fraction
ρS/ρ
|v|max
NCRIF 

total mass loading
Total mass loading
=3012ns at 51 bars
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
Filling line
Mg barrier
Mg disk
Mg Disk
Al shell
Al shell
Solid helium
Channel OD=15mm
Width=1.5mm
Solid helium in
annulus channel
-*[ns]
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.
Superflow viewed
in the rotating frame
-*[ns]
* 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
Irrotational flow pattern
in a blocked annular channel
(viewed in the rotating frame)
A. L. Fetter, JLTP(1974)
[ns]
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.
Conclusion :
superflow in solid as in superfluid is irrotational.
Superflow persists up to at least 136 bars !
136bar
108bar
ρs/ρ
ρs/ρ
Strong and ‘universal’ velocity dependence
in all samples
ω
R
vC~ 10µm/s
h
 vs dl  m  n
h
vs 
n
2Rm
=3.16µm/s
for n=1
Pressure dependence

As a function of pressure the supersolid fraction shows a
maximum near 55bars. The supersolid fraction extrapolates
to zero near 170 bars.
Superfluidity in ultra-pure Solid Helium: 1ppb 3He



0 ~ 0.77ms [1300Hz]
4He - 0 = 3920ns
NCRIF ~ 1.25/3920 = 0.03%
4724.0
-* [ns]
4723.5
4723.0
Exp. done at
at U. of Florida
4722.5
4722.0
0
* = 770,000ns
50
100
150
Temperature [mK]
200
250
Phase
Diagram
of 4He
Heat Capacity signature?

No reliable heat capacity
measurement of solid 4He below
200mK because of large background
contribution due to the sample cell.
Experimental cell of Xi Lin and Tony Clark
Silicon!!
Results: pure 4He (0.3ppm 3He)
Results: pure 4He (0.3ppm 3He)
Results: pure 4He (0.3ppm 3He)
Results: pure 4He (0.3ppm 3He)
Results: pure 4He (0.3ppm 3He)
Heat capacity peak near the supersolid transition
Is the supersolid phase unique with 4He?
Apparently not!
Preliminary torsional oscillator data of Tony
Clark and Xi Lin indicate similar supersolidlike 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
Hydrogen in a cylindrical cell
Inside
Mg bob:
Oscillation Period vs. Temperature
Period Change [ns]
2.0
1.5
Hydrogen
space
BeCu wall
1.0
0.5
0.0
Empty Cell
-0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Temperature [K]
(PO = 560,400ns)
Q = 1.6million
P0 = 560,400ns
P ~0.05ns
Hydrogen in a cylindrical cell
Inside
Mg bob:
Oscillation Period vs. Temperature
Period Change [ns]
2.0
1.5
Hydrogen
space
BeCu wall
1.0
0.5
PHD = 4014ns
(93% filling)
0.0
Empty Cell
Cell Full of HD
HD
-0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Temperature [K]
(PO = 560,400ns)
Hydrogen in a cylindrical cell
Inside
Mg bob:
Oscillation Period vs. Temperature
Period Change [ns]
2.0
1.5
Hydrogen
space
BeCu wall
1.0
0.5
0.0
-0.5
0.0
PH2 = 1638ns
(64% filling)
Empty Cell
HDFull of HD
Cell
Cell
H2 Full of H2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Temperature [K]
(PO = 560,400ns)
Hydrogen in a cylindrical cell
Temperature below
50mK uncertain,
thermometer not
on the torsional
cell
Oscillation Period vs. Temperature
0.5
0.4
Period Change [ns]
0.3
0.2
0.1
Ortho
concentration is
most likely less
than 0.5%
0.0
-0.1
-0.2
Empty Cell
Cell Full of HD
HD
Cell Full of H2
H2
-0.3
-0.4
-0.5
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Temperature [K]
(PO = 560,400ns)
HD concentration
uncertain
NCRIF ~ 0.015%
Hydrogen in an annular cell
Samples contain < 50ppm HD
83% filling, x1~2.0%
0.05
68% filling, x1<0.5%
NCRIF [%]
0.04
Mg
88% filling, x1<0.5%
Hydrogen
space
BeCu (h=3.5mm
wall w=2.3mm)
0.03
X is the ortho conc,
0.02
0.01
0.00
0.0
Inside
Mg bob:
0.1
0.2
0.3
Temperature [K]
0.4
0.5
Q = 350,000
P0 = 709,700ns
P <0.1ns
Hydrogen in an annular cell
Comparison of 50 and 200ppm HD
0.05
88% filling, x1<0.5%
68% filling, x1<0.5%
NCRIF [%]
0.04
Mg
94% filling, x1<1.0%**
83% filling, x1~2.0%
0.03
0.02
0.01
0.00
0.0
Inside
Mg bob:
0.1
0.2
0.3
Temperature [K]
0.4
0.5
**200ppm HD
Hydrogen
space
BeCu wall
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%
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
Xi, Tony, Eunseong, Josh