Can a solid be superfluid?

Symposium on Quantum Phenomena and Devices

at Low Temperatures, 2008

Helsinki Univ. of Technology

Moses Chan - Penn State

Outline

• Introduction

• Torsional oscillator measurements.

• Origin of supersolidity :

zero point vacancies, grain boundaries

glassy regions or dislocations?

• Specific heat and dc flow experiments

• Conclusions

• Theoretical ‘consensus’ in 1970s: Superfluidity in solid is 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).

fs(T) is the supersolid or nonclassical rotational inertia (NCRI) fraction.Its upper limit is estimated by different theorists to range from 10-6 to 0.4; Leggett: 10-4

Solid Helium

RI(T)=Iclassical[1-fs(T)]

Quantum exchange of particles arranged in an annulus under rotation leads to a measured moment of inertia that is smaller than the classical value

The ideal method of detection of superfluidity is to subject solid to dc or ac rotation.

Leggett, “Can a solid be superfluid?” PRL 25, 1543(1970)

Torsional Oscillator Technique is ideal for the detection of superfluidity

DriveDetection

3.5 cmTorsion rod

Torsion cell f0

f

Am p Quality Factor

Q= f0 / f ~106

Stability in the period is ~0.1 ns

Frequency resolution of 1 part in 107

Mass sensitivity of ~10-7 g

K

Io 2 f~ 1kHz

Torsional oscillator studies of superfluid films

I total= I cell+ I helium film,

Above Tc the adsorbed normal liquid film behaves as solid and oscillates with the cell. In the superfluid phase, helium film decouples from oscillation.Hence Itotal and drops.

Vycor

Berthold,Bishop, Reppy, PRL 39,348(1977)

Δ

K

Io 2

Bulk solid helium in annulus

Torsion cell with helium in annulus

Mg disk

Filling line

Solid helium in annular channel

Al shell

Channel OD=10mm

Width=0.9mm

DriveDetection

3.5 cmTorsion rod

Torsion cell

E. Kim & M.H.W. Chan, Science 305, 1941 (2004)

f0=912Hz

Bulk solid helium in annulus

51bar

Total mass loading = 3012ns

Measured decoupling, -o=41ns

NCRIF = 1.4%

loading mass total

NCRIF

Total mass loading=3012ns at 51 bars

ρS/ρ |v|max

Non-Classical Rotational Inertia Fraction

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

NC

RIF

Temperature [K]

Open Annulus: 51bar, 4m/s

Blocked Annulus: 36bar, 3m/s

• Superfluids exhibit potential (irrotational) flow– For our exact dimensions, NCRIF in the blocked cell shou

ld be about 1% that of the annulus*

*E. Mueller, private communication.

Irrotational Flow

| |e

mv

i

S

0.0 0.1 0.2 0.3 0.4 0.5

-12

-10

-8

-6

-4

-2

0

2

4

Blocked annulus, gap 73 m Open annulus, gap 73 m

Per

iod

[ns]

Temperature [K]

Gap 73 m, NCRI = 16 %

Top View: Blocked

Top View: Open

solid inertia

Nonclassicalrotational inertia

Reversibly blocked annulusRittner & Reppy, Cornell

• Nonclassical rotational inertia results have been replicated in five other labs.

• The temperature dependence of NCRI is reproduced.

• However, the magnitude of NCRIF varies from 0.015% up to 16%.

• Simulations studies indicate NCRI is not possible in a ‘perfect’ crystal and that quantum zero point vacancies concentration is too low to account for the observations.

(this conclusion not universally accepted).• This leaves other defects, e.g. grain

boundaries, glassy regions and dislocations as possible origins or at least ‘enhancer’ for NCRI.

Grain boundaries

• It has been suggested that the observed NCRI are due to atomically thin superfluid (liquid) films along the grain boundaries, if true, NCRI should scales with total grain boundary area.

• Grain size of solid 4He inside Vycor glass is ~ 7nm and in bulk it is ~0.1 mm. This means the grain boundary surface differs by many orders of magnitude, but observed NCRI is about the same, ~1%.

• Nevertheless, most bulk solid samples grown by the blocked capillary method is polycrystalline. Does NCRI exists in a single crystal ?

Solid helium in Vycor glass

-*

[ns]

*=971,000ns

62bar

Total mass loading = 4260ns

Measured decoupling, -o=17ns

NCRI fraction, or NCRIF = 0.4%

(with tortuosity, 2% )

f0=1024Hz7nm

E. Kim & M.H.W. Chan, Nature 427, 225 (2004).

0.0 0.5 1.0 1.5 2.0 2.520

25

30

35

40

45

50

55

60

bcc He I

hcp

Pre

ssur

e [b

ar]

Temperature [K]

He II

• High quality single crystals have been grown under constant temperature1 and pressure2

• Best crystals grownin zero temperaturelimit

Crystal Growth

1. O.W. Heybey & D.M. Lee, PRL 19, 106 (1967); S. Balibar, H. Alles & A. Ya Parshin, Rev. Mod. Phys. 77, 317 (2005).2. L.P. Mezhov-Deglin, Sov. Phys. JETP 22, 47 (1966); D.S. Greywall, PRA 3, 2106 (1971).

Constant T/P growth fromsuperfluid (1ppb 3He) Heat in

Heat outQ ~ 500,000

Tony Clark and Josh West

Solid 4He (1ppb 3He) grown under constant P/T from superfluid

0.00 0.05 0.10 0.15 0.20

0.0

0.1

0.2

0.3

0.4

NC

RIF

[%

]

Temperature [K]

TF = 1.38K

0.00 0.05 0.10 0.15 0.20

0.0

0.1

0.2

0.3

0.4

NC

RIF

[%

]

Temperature [K]

TF = 1.38K

TF = 1.18K (CP)

Solid 4He (1ppb 3He) grown under constant P/T from superfluid

0.00 0.05 0.10 0.15 0.20

0.0

0.1

0.2

0.3

0.4

NC

RIF

[%

]

Temperature [K]

TF = 1.38K

TF = 1.18K (CP)

TF = 1.14K

Solid 4He (1ppb 3He) grown under constant P/T from superfluid

0.00 0.05 0.10 0.15 0.20

0.0

0.1

0.2

0.3

0.4

NC

RIF

[%

]

Temperature [K]

TF = 1.24K

TF = 1.38K

TF = 1.18K (CP)

TF = 1.14K

Solid 4He (1ppb 3He) grown under constant P/T from superfluid

0.00 0.05 0.10 0.15 0.20

0.0

0.1

0.2

0.3

0.4

NC

RIF

[%

]

Temperature [K]

TF = 1.24K

TF = 1.38K

TF = 1.18K (CP)

TF = 1.14K

TF = 1.32K

Solid 4He (1ppb 3He) grown under constant P/T from superfluid

0.00 0.05 0.10 0.15 0.20

0.0

0.1

0.2

0.3

0.4

NC

RIF

[%

]

Temperature [K]

TF = 1.24K

TF = 1.38K

TF = 1.18K (CP)

TF = 1.14K

TF = 1.32K

TF = 1.30K

TF = 1.15K

TF = 1.14K

Solid 4He (1ppb 3He) grown under constant P/T from superfluid

Samples grown under constant P/T from superfluid collapse onto one curve for T > 40mK and share common onset temperature, TC ~ 80mK. Superfluid film flow along grain boundary cannot be the ‘origin’ of NCRI.

0.02 0.05 0.1 0.25

0.0

0.1

0.2

0.3

0.4

NC

RIF

[%

]

Temperature [K]

TF = 1.24K

TF = 1.38K

TF = 1.18K (CP)

TF = 1.14K

TF = 1.32K

TF = 1.30K

TF = 1.15K

TF = 1.14K

High temperature tail of NCRITransition broadened in BC samples (probably “polycrystalline”) and by 3He impurities

0.02 0.06 0.10 0.14 0.18 0.22 0.26

0.00.10.20.30.40.50.60.70.80.91.0 BC

300 ppb [1] 300 ppb 1 ppb

CT or CP, , 300 ppb, , ,, , ,, 1 ppb

Temperature [K]

Nor

mal

ized

NC

RIF

• Superfluid film along the grain boundaries cannot be the mechanism for NCRI.

• What is the most likely origin for the large variation in NCRI from cell to cell?

Dislocation lines density in solid 4He ranges from ~10 to 109 cm-2 , sensitive to how the solid is nucleated and grown. But how does dislocations enable or enhance NCRI?

• Two of the common types: edge & screw

In solid 4He

• Dislocation density, = ~5 <1010 cm-2

Dislocations

3He impurities at “high” T

•Network pinning at the nodes

•3He impurities are not ‘attached’ to the dislocation lines.

Increasing fraction of 3He atoms condense onto and stiffen dislocation network with

decreasing T

Iwasa et.al., J.Phys. Soc. Jpn. 46, 1119( 1979);

Paalanen, Bishop and Dail,PRL 46, 664(1981)

3He Effect ( bulk samples)

E. Kim, J. S. Xia, J. T. West, X. Lin, A. C Clark, and M. H. W. Chan, , PRL 100, 065301( 2008)

Onset of NCRI tracks the expected temp. of 3He condensation onto dislocation

Direct measurement of the Shear Modulus

James Day and John Beamish, Nature (London) 450, 853 (2007).

Shear Modulus mirrors NCRI

James Day and John Beamish, Nature (London) 450, 853 (2007).

• The increase in shear modulus and the onset of NCRI at low T must be related, but how?

• An increase in shear modulus of solid 4He in TO can result an increase in its resonant freq., ie. the same sign as the observed NCRI. But simulation studies shows the magnitude of the ‘Beamish’ effect is typically orders of magnitudes smaller than that observed in most TOs.

• The ‘Beamish’ effect or shear modulus stiffening cannot explain the blocked annulus experiments.

• It has been suggested (Anderson) that the onset of supersolidity stiffens the shear modulus.

• Simulations show the dislocation lines are ‘superfluid’ and it has been suggested that the onset of NCRI requires a ‘rigid’ dislocation network.

• What about the superglass model?

• Since glass has no crystalline order, it is thought superfluidity is more likely to occur.

• If this is the case there should be thermodynamic signatures ( e.g. thermal hysteresis, and linear specific heat)

Heat capacity

Is there a thermodynamic signature of the transition?

There is no specific heat information on solid 4He below 200 mK. The heat capacity of metal container swamps the contribution of solid 4He

Heat Capacity measurements in a Silicon cell

Si

Al

Glass Capillary

Stycast 2850

Heater

Thermometer

• Reasons for Si:

Low heat capacity:

High thermal conductivity:

Helium Volume= 0.926cc

1.5 cm

0.1mm ID

Heat capacity of solid 4He is more than 10 times largerthan the Si cell over the entire temp. range

AC Calorimetry

1. Paul F. Sullivan, G. Seidel, Phys Rev. 173, 679 (1968).2. Yaakov Kraftmakher, Physics Reports, 356 (2002) 1-117.

221

0 )(cos tQQ 2

1

3

211

2, 2

int2

220

s

b

ssac C

QtLT

ac

ac

T

QC

C

QT

2

2

Internal time constant << 1/ω

External time constant >> 1/ω

Kb

Sample

Thermal Bath

Thermometer

KΘ

Heater

Kh

0.01 0.1 11

10

100

1000

C [J

/K]

f [Hz]

Specific heat of commercially pure 4He (0.3ppm 3He)

0.03 0.05 0.1 0.50.01

0.1

1

10

A*T3

0.3ppm 33bar solid 4He

C [m

J m

ol-1 K

-1]

T [K]

Constant volume technique.20 hours to finish solidification.

No long time constant.

No hysteresis

No annealing effect

No thermal cycle effect

log C vs. log T

Specific heat of solid 4He (0.3ppm & 1ppb)

0.03 0.04 0.1 0.2 0.3 0.4

0.01

0.1

1

10 0.3 ppm 1 ppb

C

n [m

J m

ol-1 K

-1]

T [K]

log C vs. log T

C/T vs. T2

0.00 0.05 0.10 0.15 0.200

2

4

6

8

10

12

14

16

T [K]

0.450.380.310.22

Cn

/T [m

J m

ol-1 K

-2]

T 2 [K2]

0.3 ppm

0.00

0.00 0.02 0.040

1

2

3

4

0.2

0.14

Data above 140 mK extrapolate to zero

0.00 0.02 0.040

1

2

3

4

0.2

1ppb

0.14

0.0 0.1 0.2

0.000

0.004

0.008

0.012

(P-P

0)/T

2 [bar

/K2 ]

T2 [K2]

freshly grown sample anneal sample 0.3ppm from HC

‘Linear T’ term from Pressure measurements ?

Freshly grown sample

Annealed sample (19.83cc/mole)

v

V

C

T

P

)(

— — Grüneisen constantGrüneisen constant ρρ

— molar volume— molar volume CCVV~T~Tnn P~T P~Tn+1n+1

PHYSICAL REVIEW B 76, 224524 2007V. N. Grigor'ev et al

Specific heat with Debye phonon term subtracted

0.00 0.05 0.10 0.15 0.20-10

-5

0

5

10

15

20

25

1 ppb 0.3 ppm

T [K]

C [

J m

ol-1

K-1]

T3 subtracted

~20 J mol−1 K−1

~2.5×10-6 kB per 4He atom

0.00 0.05 0.10 0.15 0.20 0.25-30

-20

-10

0

10

20

30

40

50

Cp

ea

k [J

mo

l-1 K

-1]

T [K]

2008 BC 2007 BC 2008 CP

Solid sample grown under constant pressure condition has the smallest peak.

0.00 0.05 0.10 0.15 0.20 0.25-30

-20

-10

0

10

20

30

40

50

Cpe

ak [

J m

ol-1 K

-1]

T [K]

1ppb 20081ppb 2007 0.3ppm 2007

T3 subtracted

Higher resolution in 2008 sample cell

0.00 0.05 0.10 0.15 0.20 0.25-30

-20

-10

0

10

20

30

40

50

Cpe

ak [

J m

ol-1 K

-1]

T [K]

New 1ppb 33bar solid

T3 subtracted

0.02 0.06 0.2 0.4

0.01

0.1

1

A*T3

New 1ppb 33bar solid

C [m

J m

ol-1

K-1

]

T [K]

Minimum Temperature 25mK

New preliminary result with a more sensitive thermometer. Disadvantage of Si?

8 out of 10 cells leaks under high pressure

4 hours to solidify a sample.

Comparison with the Helsinki melting curve measurement.

• No anomaly in melting curve after correction for instrumental effect.

• Discrepancy we don’t understand

0.0000 0.0005 0.0010 0.0015 0.0020-80

-70

-60

-50

-40

-30

-20

-10

0

10

Pre

ssur

e D

evia

tion

[b

ar]

T 4 [K4]

From 0.3ppm [PSU 2007] Pressure [HUT]

dP

dT

S S

V V

SC

TdT

V Vliq so l

liq so l

liqn

liq so l

Todoschenko et al., PRL 97, 165302 (2006); JETP Lett. 85, 555 (2007).

Constant pressure 1ppb 2008

Constant volume 1ppb 2008

dc ( superfluid-like ) flow?

• Greywall and Beamish carried out experiments by increasing pressure on one end of a solid ( bulk or in Vycor glass) sample. No evidence of flow.

• Balibar and collaborators saw flow in liquid-solid coexistence samples. The observed phenomenon is likely a consequence of liquid film flow through the ‘crack’ in the solid.

DC superflow reported by ‘pushing’ on super

fluid in contact with solid : ‘UMass Sandwich’ (R.B. Hallock)

31 2

R, reservoirs

V, Vycor

S, sample

C1 C2

Conclusions• NCRI or superfluid like behavior seen in solid 4He in 7

labs.• Shear modulus increases at low temperature, the

temperature dependence of the increases mimics that of NCRI

• It is possible that either (1) NCRI ‘requires’ a rigid dislocation network, or (2) onset of NCRI stiffens the solid lattice.

• No evidence of linear T term in specific heat, but a peak near NCRI onset is seen, size of peak is smaller for a ‘higher’ quality sample.

• Preliminary dc superfluid flow through solid helium reported.

Xi, Tony, Eunseong, Josh

• 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. B 55, 5767(1997))

26.02

a

L d

u

Zero-point Energy

Inter-atomic potential

total energy

zero-point energy

Solid 4He at 62 bars in Vycor glass

Period shifted by 4260ns due to mass loading of solid helium

*=966,000ns

Strong and ‘universal’ velocity dependence in all annular samples

vC~ 10µm/s

=3.16µm/s for n=1

nRm

hv

nm

hdlv

s

s

2

ω

R

Vortices are important

Blocked capillary (BC) method of growing solid samples

0.0 0.5 1.0 1.5 2.0 2.520

25

30

35

40

45

50

55

60

bcc

He I

hcp

Blocked Capillary (constant volume)

Pre

ssur

e [b

ar]

Temperature [K]

He II

heat drain

Be-Cu torsion rodand fill-line

solidblocks fill-line

gravity

Supersolid response of helium in Vycor glassSupersolid response of helium in Vycor glass• Period drops at 175mK appearance of non-classical rotational inertia (NCRI)

• size of period drop - ~17ns

*=971,000ns

-

*[ns

]

*=971,000ns

3He impurity effect in Vycor

Solid helium in porous gold

E. Kim & M.H.W. Chan, JLTP 138, 859 (2005).

f0=359Hz

27bar

Total mass loading = 1625ns

Measured decoupling, -o=13ns

NCRIF = 0.8%

(with tortuosity, 1.2% )

490nm

0.03 0.04 0.1 0.2 0.3 0.4

0.01

0.1

1

10

30 ppm 10 ppm 0.3 ppm 1 ppb

Cn

[mJ

mol

-1 K

-1]

T [K]

Specific heat with the temperature independent

constant term subtracted

Previous solid 4He heat capacity measurements below 1K

Year Low temperature limit

Swenson1 1962,1967 0.2K

Edwards2 1965 0.3K

Gardner3 1973 0.35K

Adams4 1975 0.13K

Hebral5 1980 0.1K

Clark6 2005 0.08K

1. E. C. Heltemes and C. A. Swenson, Phys. Rev. 128, 1512 (1962); H. H. Sample and C. A. Swenson, Phys. Rev. 158, 188 (1967).

2. D. O. Edwards and R. C. Pandorf, Phys. Rev. 140, A816 (1965).3. W. R. Gardner et al., Phys. Rev. A 7, 1029 (1973).4. S. H. Castles and E. D. Adams, J. Low Temp. Phys. 19, 397 (1975). 5. B. Hébral et al., Phonons in Condensed Matter, edited by H. J. Maris (Plenum, New York, 1980), pg. 169. 6. A. C. Clark and M. H. W. Chan, J. Low Temp. Phys. 138, 853 (2005).

They all observed T3 phonon contribution.

Their sample cells used in these experiments were all constructed with heavy wall metal or epoxy which contribute significantly to the heat capacity at low temperature.

0.03 0.05 0.1 0.25 0.5

0.1

1

10

100

1000

30 ppm 10 ppm 0.3 ppm 1 ppb Empty Cell

He

at

Ca

pa

city

, C

[J

K-1

]

Temperature, T [K]

Results: 4He at different 3He concentrations in glass capillary cell

No long time constant.

No hysteresis

No change due to annealing

No thermal cycle effect

Constant volume technique

0.00 0.02 0.04 0.06

0

1

2

3

4

5

6

70

30 ppm 10 ppm 0.3 ppm 1 ppb

0.390.34

Cn [

mJ

mo

l-1 K

-1]

T 3 [K3]

0.27T [K]

0.000 0.002 0.004

0.0

0.1

0.2

0.3

0.4

0.5

0.16

0.13

C vs T3

Constant contribution from 3He impurity

10ppm sample

0.7+/-0.2 kB per 3He

30ppm sample

1.7+/-0.3 kB per 3He

Is there a linear T term?

0.00 0.05 0.10 0.15 0.200

2

4

6

8

10

12

14

16

T [K]

0.450.380.310.22

Cn

/T [

mJ

mo

l-1 K

-2]

T 2 [K2]

30 ppm 10 ppm 0.3 ppm 1 ppb

0.00

0.00 0.02 0.040

1

2

3

4

5

6

0.2

0.3ppm

1ppb

0.14

For 0.3ppm, T>0.14K T3 relation only

NMR measurement of spin (3He) diffusion:

A. R. Allen, M. G. Richards & J. Schratter J. Low Temp. Phys. 47, 289 (1982).

M. G. Richards, J. Pope & A. Widom, Phys. Rev. Lett. 29, 708 (1972).

Results: pure 4He (0.3ppm & 1ppb)

Ln C vs. Ln T

What is the deviation?

0.03 0.04 0.1 0.2 0.3 0.4

0.01

0.1

1

10 0.3 ppm 1 ppb

Cn

[mJ

mol

-1 K

-1]

T [K]

Previous solid 4He heat capacity measurements below 1K

Year Low temperature limit

Swenson1 1962,1967 0.2K

Edwards2 1965 0.3K

Gardner3 1973 0.35K

Adams4 1975 0.13K

Hebral5 1980 0.1K

Clark6 2005 0.08K

1. E. C. Heltemes and C. A. Swenson, Phys. Rev. 128, 1512 (1962); H. H. Sample and C. A. Swenson, Phys. Rev. 158, 188 (1967).

2. D. O. Edwards and R. C. Pandorf, Phys. Rev. 140, A816 (1965).3. W. R. Gardner et al., Phys. Rev. A 7, 1029 (1973).4. S. H. Castles and E. D. Adams, J. Low Temp. Phys. 19, 397 (1975). 5. B. Hébral et al., Phonons in Condensed Matter, edited by H. J. Maris (Plenum, New York, 1980), pg. 169. 6. A. C. Clark and M. H. W. Chan, J. Low Temp. Phys. 138, 853 (2005).

They all observed T3 phonon contribution.

Their sample cells used in these experiments were all constructed with heavy wall metal or epoxy which contribute significantly to the heat capacity at low temperature.

Results from Edwards and Hebral

0 1 2 3 425

30

35

40

45

50

55

20. 93 cc/ mol 19. 68 cc/ mol 19. 18 cc/ mol 18. 22 cc/ mol 17. 87 cc/ mol 16. 90 cc/ mol

de

g. K

]

T [K]

0 1 2 3 425

30

35

40

45

50

55

20. 93 cc/ mol 19. 68 cc/ mol 19. 18 cc/ mol 18. 22 cc/ mol 17. 87 cc/ mol 16. 90 cc/ mol

de

g. K

]

T [K]

The background problem

0.01 0.1 11E-5

1E-4

1E-3

0.01

0.1

C [J

K-1

]

T [K]

solid 4He empty cell background

Edwards Hebral

34

v θ

T

5

π12C

baKN

Effect of changing 1% of the empty cell

0.03 0.05 0.1 0.51E-7

1E-6

1E-5

1E-4

1E-3 Background

0.3ppm Solid 4He

C [J

K-1

]

Temperature [K]

Glass capillary Cell Cu-Ni capillary Cell

Results: pure 4He (0.3ppm)

Temperature scale based on3He melting curve

Minimum temperature 40mK

0.00 0.05 0.10 0.15 0.20 0.25

-30

-20

-10

0

10

20

30 10 ppm 0.3 ppm 1 ppb

Cpe

ak [J

mo

l-1 K

-1]

T [K]

0.00 0.05 0.10 0.15 0.20

0

20

40

60

80

Specific heat peak is found when T3 term subtracted The peak is independent of 3He concentration

Peak height: 20 μJ mol-1 K-1

(2.5 x 10-6 kB per 4He atom)

Excess entropy: 28 μ J mol-1 K-1

(3.5 x 10-6 kB per 4He atom)

BC samples can also be grown

Heat outQ ~ 500,000

Granato-Lucke applied to 4He

• Dislocations pinned by network, 3He, jogs?• Typical LN in pure 4He crystals ~ 5m

0.03 0.04 0.1 0.2 0.3 0.4

0.01

0.1

1

10 0.3 ppm 1 ppb

Cn

[mJ

mol

-1 K

-1]

T [K]

Ln C vs. Ln T

Compare with torsional oscillator

0.02 0.06 0.10 0.14 0.18

0.00.10.20.30.40.50.60.70.80.91.0

0

5

10

15

20

25

CT or CP, , ,, , ,, 1 ppb

Temperature [K]

N

orm

aliz

ed N

CR

IF

Solid 4He at various pressures show similar temperature dependence, but the measured supersolid fraction shows

scatter with no obvious pressure dependenceN

CR

IF

NC

RIF

NC

RIF

NC

RIF

Pressure dependence of supersolid fraction

Blue data points were obtained by seeding the solid helium samples from the bottom of the annulus.N

CR

IF

What are the causes ofthe scatter in NCRIF?

Pressure dependence of supersolid fraction

Blue data points were obtained by seeding the solid helium samples from the bottom of the annulus.N

CR

IF

What are the causes ofthe scatter in NCRIF?

Thermal history of 1ppb samples

Velocity changes at low temperature lead to interesting behavior…

Protocol followed below: (1) cooling, (2) velocity increase, (3) warming, (4) cooling

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

29.5bar, 11m/s 29.5bar, 150m/s

NC

RIF

Temperature [K]

1

3

4

2

End

0.00 0.05 0.10 0.15 0.20 0.25-30

-20

-10

0

10

20

30 10 ppm 0.3 ppm 1 ppb

Cpe

ak [J

mo

l-1 K

-1]

T [K]

0.00 0.05 0.10 0.15 0.20

0

20

40

60

80

Specific heat with T3 term subtracted

Peak height: 20 μJ mol-1 K-1 (2.5 x 10-6 kB per 4He atom)

Excess entropy: 28 μ J mol-1 K-1 (3.5 x 10-6 kB per 4He atom)

1. Specific heat peak is independent of 3He concentrations.

2. Assuming 3D-xy universality class (same as the lambda transition in liquid 4He).

3. Use two-scale-factor universality hypothesis, ρs ~0.06%. 1ppb study of TO found this number lays between 0.03% and 0.3%.

Hysteresis in Pressure measurement of phase separation

A.N.Gan'shin, V.N.Grigor'ev, V.A.Maidanov, N.F.Omelaenko, A.A.Penzev, É.Ya.Rudavskii, A.S.Rybalko., Low Temp. Phys. 26, 869 (2000).

1E-6 1E-5 1E-4 1E-3 0.01

0.1

0.2

Standard theory model Ganshin warming Ganshin cooling

TP

S [

K]

X3

• Two of the common types: edge & screw

• Dislocation density, = ~5 <1010 cm-2

– 3-d network, LN ~ 1 to 10 m (~105 to 107)

[LN ~ 0.1 to 1 m (~109)]

Dislocations

• Dislocations intersect on a characteristic length scale of LN ~ 1 5m

• Dislocations can also be pinned by 3He impurities– Distance between 3He atoms (if uniformly distributed):

– 1ppb 1000a ~ 0.3m

– 0.3ppm 150a ~ 45nm

– 1% 5a ~ 15nm

Granato-Lucke applied to 4He

3He-dislocation interactionActual 3He concentration on dislocation line is thermally activated

x xW

T3 00

ex p

*Typical binding energy, W0 is 0.3K to 0.7K

3He-dislocations interaction

1 10 100 1000 10000 10000020

40

6080

100

200

400

600800

1000

2000

LC ~ L

N ~ 1m

Binding energy, W0 = 1K

TC [

mK

]

X3 [ppb]

open cell (PSU) open cell (UF) annular cel Vycor

Line considered as crossover from network pinning to 3He impurity pinning of dislocations (LNetwork ~ L3

He spacing)

Average lengthLNetwork ~ 1 to 10mFor ~ 105 to 106cm-2

Smaller lengths (< 1m) are expected for larger dislocation densities

L A W xW

TH e3 01 3

02 3 02

3

/ / ex p

Solid helium in Vycor glass

Weak pressure dependence…

from 40 to 65bar

Strong velocity dependence

-

*[ns

]

*=971,000ns

Question: If there is a transition between

the normal and supersolid phases,

where is the transition

temperature?

3He-4He mixtures

3He-4He mixtures ( inside Vycor)

Frequency dependence-TO increases with frequency-Low temperature NCRIF unchanged

Aoki, Graves & Kojima, PRL 99, 015301 (2007).

~150mK ~220mK