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**Can a solid be superfluid?**

Symposium on Quantum Phenomena and Devices at Low Temperatures, 2008 Helsinki Univ. of Technology Moses Chan - Penn State

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**Outline Introduction Torsional oscillator measurements.**

Origin of supersolidity : zero point vacancies, grain boundaries glassy regions or dislocations? Specific heat and dc flow experiments Conclusions

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**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).

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**R I(T)=Iclassical[1-fs(T)]**

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) 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 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

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**Torsional Oscillator Technique is ideal for the detection of superfluidity**

Amp Drive Detection 3.5 cm Torsion rod Torsion cell 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 f~ 1kHz

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**Torsional oscillator studies of superfluid films**

Vycor Δ 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. Berthold,Bishop, Reppy, PRL 39,348(1977)

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**Bulk solid helium in annulus**

Torsion cell with helium in annulus Filling line Channel OD=10mm Width=0.9mm Torsion rod 3.5 cm Mg disk Torsion cell Al shell Detection Solid helium in annular channel Drive

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

Bulk solid helium in annulus f0=912Hz 51bar Total mass loading = 3012ns Measured decoupling, -o=41ns NCRIF = 1.4% E. Kim & M.H.W. Chan, Science 305, 1941 (2004)

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**Non-Classical Rotational Inertia Fraction**

|v|max Total mass loading =3012ns at 51 bars

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**Irrotational Flow Superfluids exhibit potential (irrotational) flow**

For our exact dimensions, NCRIF in the blocked cell should be about 1% that of the annulus* *E. Mueller, private communication.

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**Reversibly blocked annulus Rittner & Reppy, Cornell**

Top View: Blocked Top View: Open solid inertia Nonclassical rotational inertia Gap 73 mm, NCRI = 16 %

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**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.

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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 ?

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

Solid helium in Vycor glass 7nm f0=1024Hz 62bar Total mass loading = 4260ns Measured decoupling, -o=17ns NCRI fraction, or NCRIF = 0.4% (with tortuosity, 2% ) -*[ns] E. Kim & M.H.W. Chan, Nature 427, 225 (2004). *=971,000ns

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Crystal Growth High quality single crystals have been grown under constant temperature1 and pressure2 Best crystals grown in zero temperature limit 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).

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**Constant T/P growth from superfluid (1ppb 3He) Heat in**

Heat out Q ~ 500,000 Tony Clark and Josh West

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

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

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

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

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

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

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**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.

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**High temperature tail of NCRI**

Transition broadened in BC samples (probably “polycrystalline”) and by 3He impurities

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**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?

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**Dislocations Two of the common types: edge & screw In solid 4He**

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

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**3He impurities at “high” T**

Network pinning at the nodes 3He impurities are not ‘attached’ to the dislocation lines.

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**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)

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**3He Effect ( bulk samples)**

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

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**Onset of NCRI tracks the expected temp**

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

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**Direct measurement of the Shear Modulus**

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

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**Shear Modulus mirrors NCRI**

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

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**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.

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**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)

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**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

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**Heat Capacity measurements in a Silicon cell**

Reasons for Si: Low heat capacity: High thermal conductivity: Helium Volume= 0.926cc Si Al Glass Capillary Stycast 2850 Heater Thermometer 0.1mm ID 1.5 cm Lattice Materials CorP., MT, USA Optical Silicon purity : >99.999% Heat capacity of solid 4He is more than 10 times larger than the Si cell over the entire temp. range

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**AC Calorimetry KΘ Kb Sample Kh Internal time constant << 1/ω**

External time constant >> 1/ω Kb Sample Thermal Bath Thermometer KΘ Heater Kh Paul F. Sullivan, G. Seidel, Phys Rev. 173, 679 (1968). Yaakov Kraftmakher, Physics Reports, 356 (2002)

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

log C vs. log T Constant volume technique. 20 hours to finish solidification. No long time constant. No hysteresis No annealing effect No thermal cycle effect

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

log C vs. log T

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**C/T vs. T2 Data above 140 mK extrapolate to zero**

the data seem to diverge in the low-temperature limit. This behaviour, which is more prominent in the 30 p.p.m. sample, suggests the presence of a temperature-independent term in addition to the Debye contribution. Data above 140 mK extrapolate to zero

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**‘Linear T’ term from Pressure measurements ?**

PHYSICAL REVIEW B 76, V. N. Grigor'ev et al Freshly grown sample Annealed sample (19.83cc/mole) — Grüneisen constant ρ — molar volume CV~Tn P~Tn+1

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**Specific heat with Debye phonon term subtracted**

T3 subtracted ~20 mJ mol−1 K−1 ~2.5×10-6 kB per 4He atom Explain the Scatter.

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**Higher resolution in 2008 sample cell**

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. Solid sample grown under constant pressure condition has the smallest peak. T3 subtracted T3 subtracted

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**Comparison with the Helsinki melting curve measurement.**

Constant volume 1ppb 2008 Constant pressure 1ppb 2008 No anomaly in melting curve after correction for instrumental effect. Discrepancy we don’t understand Todoschenko et al., PRL 97, (2006); JETP Lett. 85, 555 (2007).

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**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.

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DC superflow reported by ‘pushing’ on superfluid in contact with solid : ‘UMass Sandwich’ (R.B. Hallock) 3 1 2 R, reservoirs V, Vycor S, sample C1 C2

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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.

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Xi, Tony, Eunseong, Josh

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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)) total energy zero-point energy Zero-point Energy Inter-atomic potential

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**Solid 4He at 62 bars in Vycor glass**

Period shifted by ns due to mass loading of solid helium *=966,000ns

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**Strong and ‘universal’ velocity dependence**

in all annular samples ω R vC~ 10µm/s =3.16µm/s for n=1 Vortices are important

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**Blocked capillary (BC) method of growing solid samples**

heat drain solid blocks fill-line Be-Cu torsion rod and fill-line gravity

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**Supersolid response of helium in Vycor glass**

Period drops at 175mK appearance of non-classical rotational inertia (NCRI) size of period drop - ~17ns *=971,000ns

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**3He impurity effect in Vycor**

-*[ns] *=971,000ns

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**Solid helium in porous gold**

490nm f0=359Hz 27bar Total mass loading = 1625ns Measured decoupling, -o=13ns NCRIF = 0.8% (with tortuosity, 1.2% ) E. Kim & M.H.W. Chan, JLTP 138, 859 (2005).

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**Specific heat with the temperature independent constant term subtracted**

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**Previous solid 4He heat capacity measurements below 1K**

Year Low temperature limit Swenson1 1962, K Edwards K Gardner K Adams K Hebral K Clark K 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. E. C. Heltemes and C. A. Swenson, Phys. Rev. 128, 1512 (1962); H. H. Sample and C. A. Swenson, Phys. Rev. 158, 188 (1967). D. O. Edwards and R. C. Pandorf, Phys. Rev. 140, A816 (1965). W. R. Gardner et al., Phys. Rev. A 7, 1029 (1973). S. H. Castles and E. D. Adams, J. Low Temp. Phys. 19, 397 (1975). B. Hébral et al., Phonons in Condensed Matter, edited by H. J. Maris (Plenum, New York, 1980), pg. 169. A. C. Clark and M. H. W. Chan, J. Low Temp. Phys. 138, 853 (2005).

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

Constant volume technique No long time constant. No hysteresis No change due to annealing No thermal cycle effect

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**C vs T3 10ppm sample 0.7+/-0.2 kB per 3He 30ppm sample**

Constant contribution from 3He impurity 10ppm sample 0.7+/-0.2 kB per 3He 30ppm sample 1.7+/-0.3 kB per 3He

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Is there a linear T term? For 0.3ppm, T>0.14K T3 relation only

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**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).

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

Ln C vs. Ln T What is the deviation?

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**Previous solid 4He heat capacity measurements below 1K**

Year Low temperature limit Swenson1 1962, K Edwards K Gardner K Adams K Hebral K Clark K 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. E. C. Heltemes and C. A. Swenson, Phys. Rev. 128, 1512 (1962); H. H. Sample and C. A. Swenson, Phys. Rev. 158, 188 (1967). D. O. Edwards and R. C. Pandorf, Phys. Rev. 140, A816 (1965). W. R. Gardner et al., Phys. Rev. A 7, 1029 (1973). S. H. Castles and E. D. Adams, J. Low Temp. Phys. 19, 397 (1975). B. Hébral et al., Phonons in Condensed Matter, edited by H. J. Maris (Plenum, New York, 1980), pg. 169. A. C. Clark and M. H. W. Chan, J. Low Temp. Phys. 138, 853 (2005).

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**Results from Edwards and Hebral**

The background problem Edwards Hebral Effect of changing 1% of the empty cell

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**Temperature scale based on**

Results: pure 4He (0.3ppm) The heat capacity of the empty cell at 0.2 K and 0.05 mK are respectively less than 3 % and 10 % of the heat capacity of solid helium at 27 bar. 0.3ppm Minimum temperature 40mK Temperature scale based on 3He melting curve

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**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)

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**BC samples can also be grown**

Heat out Q ~ 500,000

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**Granato-Lucke applied to 4He**

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

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Ln C vs. Ln T

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**Compare with torsional oscillator**

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Solid 4He at various pressures show similar temperature dependence, but the measured supersolid fraction shows scatter with no obvious pressure dependence NCRIF NCRIF NCRIF NCRIF

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**Pressure dependence of supersolid fraction**

Blue data points were obtained by seeding the solid helium samples from the bottom of the annulus. NCRIF What are the causes of the scatter in NCRIF?

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**Pressure dependence of supersolid fraction**

Blue data points were obtained by seeding the solid helium samples from the bottom of the annulus. NCRIF What are the causes of the scatter in NCRIF?

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**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 2 1 3 4

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End

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**Specific heat with T3 term subtracted**

Specific heat peak is independent of 3He concentrations. Assuming 3D-xy universality class (same as the lambda transition in liquid 4He). Use two-scale-factor universality hypothesis, ρs ~0.06%. 1ppb study of TO found this number lays between 0.03% and 0.3%. 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)

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**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).

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**Dislocations Two of the common types: edge & screw**

Dislocation density, L = ~5 <1010 cm-2 3-d network, LN ~ 1 to 10 mm (L~105 to 107) [LN ~ 0.1 to 1 mm (L~109)]

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**Granato-Lucke applied to 4He**

Dislocations intersect on a characteristic length scale of LN ~ 1 5mm Dislocations can also be pinned by 3He impurities Distance between 3He atoms (if uniformly distributed): 1ppb 1000a ~ 0.3mm 0.3ppm 150a ~ 45nm 1% a ~ 15nm

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**3He-dislocation interaction**

Actual 3He concentration on dislocation line is thermally activated *Typical binding energy, W0 is 0.3K to 0.7K

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**3He-dislocations interaction**

Line considered as crossover from network pinning to 3He impurity pinning of dislocations (LNetwork ~ L3He spacing) Average length LNetwork ~ 1 to 10mm For L ~ 105 to 106cm-2 Smaller lengths (< 1mm) are expected for larger dislocation densities

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**Solid helium in Vycor glass**

Weak pressure dependence… from 40 to 65bar Strong velocity dependence

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3He-4He mixtures -*[ns] *=971,000ns Question: If there is a transition between the normal and supersolid phases, where is the transition temperature?

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**3He-4He mixtures ( inside Vycor)**

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**Frequency dependence -TO increases with frequency**

-Low temperature NCRIF unchanged ~150mK ~220mK Aoki, Graves & Kojima, PRL 99, (2007).

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