Could a quantum solid flow Could a quantum solid flow like a superfluid ? like a superfluid ?

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Could a quantum solid flow Could a quantum solid flow

like a superfluid ? like a superfluid ?

S. Sasaki, R. Ishiguro , F. Caupin, H.J. Maris*

and and S. Balibar S. Balibar

Laboratoire de Physique Statistique (ENS-Paris) Laboratoire de Physique Statistique (ENS-Paris)

* Brown University, Providence (RI, USA)

Oxford, 25 jan 2007 Oxford, 25 jan 2007

A reference: Science 313, 1098 (25 aug. 2006) A reference: Science 313, 1098 (25 aug. 2006)

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Evangelista Torricelli (1608-1647)

Galileos friend Galileos friend

invented the first barometer invented the first barometer

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liquid Hg liquid Hg 1 atm = 760 mmHg

1 atm = 760 mmHg vacuum vacuum

two communicating vessels (inside and outside the tube) two communicating vessels (inside and outside the tube) hydrostatic equilibrium

hydrostatic equilibrium

the weight of the liquid column is compensated by the atmospheric pressure the weight of the liquid column is compensated by the atmospheric pressure

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under vacuum: same level

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when Torricelli pumped through E:

liquid-gas equilibrium in A and B liquid-gas equilibrium in A and B

same temperature same temperature same vapor pressure same vapor pressure

same levels same levels

because a liquid allows the mass flow because a liquid allows the mass flow

which is necessary to achieve hydrostatic equilibrium which is necessary to achieve hydrostatic equilibrium

we did the same experiment we did the same experiment with solid

with solid ⁴⁴He in eq. with liquid He in eq. with liquid ⁴⁴HeHe

E. Torricelli, Florence 1644 E. Torricelli, Florence 1644

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Motivation : is solid

⁴

He « supersolid »?

E. Kim and M. Chan E. Kim and M. Chan (Penn. State U. 2004):

(Penn. State U. 2004):

a torsional oscillator (

a torsional oscillator ( ~1 kHz) ~1 kHz) a change in the period of

a change in the period of oscillation

oscillation

below 200 mK below 200 mK

1 % of the solid mass decouples 1 % of the solid mass decouples from the oscillating walls ?

from the oscillating walls ?

Be-Cu

Torsion Rod Torsion Bob containing helium

Drive

Detection

K I

o

π

τ = 2

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1% superfluid

density in solid 4He ?

NCRI NCRI

(non classical rotational inertia) (non classical rotational inertia)

~1% at 51 bar

~1% at 51 bar no effect in

no effect in

³³

He He

the effect is strongly reduced the effect is strongly reduced

with a barrier in the rotating annulus

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early theoretical ideas

Penrose and Onsager 1956:

BEC is impossible in a solid BEC is impossible in a solid

(but they used non-symetrized wave fonctions) (but they used non-symetrized wave fonctions) Andreev and Lifshitz 1969:

Andreev and Lifshitz 1969:

delocalized defects (vacancies) could exist at T=0 delocalized defects (vacancies) could exist at T=0 ( the crystal would be « incommensurate »)

( the crystal would be « incommensurate »)

BEC => superplasticity at low velocity or long times BEC => superplasticity at low velocity or long times Reatto, Chester and Leggett 1969-70:

Reatto, Chester and Leggett 1969-70:

NCRI is possible if atoms are delocalized NCRI is possible if atoms are delocalized (if there are free vacancies ?)

(if there are free vacancies ?) Imry and Schwartz (1975):

Imry and Schwartz (1975):

no supersolidity in a true crystal without free vacancies no supersolidity in a true crystal without free vacancies (a lattice gas is different)

(a lattice gas is different) ......

EE₀₀

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recent theoretical ideas

Prokofev and Svistunov 2005: no BEC in crystals without free vacancies Prokofev and Svistunov 2005: no BEC in crystals without free vacancies (commensurate crystal, vacancy-interstitial pairs); BEC in a

(commensurate crystal, vacancy-interstitial pairs); BEC in a ⁴⁴He glass He glass (Boninsegni et al. PRL 2006)

(Boninsegni et al. PRL 2006)

Galli and Reatto 2006: superfluidity in simulations with trial functions Galli and Reatto 2006: superfluidity in simulations with trial functions

(« SWF ») which reproduce the properties of solid 4He (« SWF ») which reproduce the properties of solid 4He

Clark and Ceperley (2006) : superfluidity depends on the trial functions Clark and Ceperley (2006) : superfluidity depends on the trial functions

not found in quantum Monte Carlo simulations;

the crystal is commensurate, no vacancies at T =0 the crystal is commensurate, no vacancies at T =0

Anderson Brinkman and Huse 2005: a new analysis of the T variation of the Anderson Brinkman and Huse 2005: a new analysis of the T variation of the

lattice spacing (old experiments by Simmons) lattice spacing (old experiments by Simmons)

and the specific heat C

and the specific heat C_v_v(T) = AT(T) = AT³³ + BT + BT⁷⁷

 a low density of zero-point vacancies (< 10a low density of zero-point vacancies (< 10^-3^-3 ?); T ?); T_BEC_BEC ~ a few mK ; ~ a few mK ; _s_s ? ?

PG de Gennes (CR-Physique 2006): quantum dislocations are mobile at low T PG de Gennes (CR-Physique 2006): quantum dislocations are mobile at low T ......

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puzzling

experimental results

Kim and Chan: the critical velocity Kim and Chan: the critical velocity is 10 is 10 m/s, independent of P m/s, independent of P

The critical temperature is The critical temperature is also independent of Palso independent of P

the superfluid fraction increases the superfluid fraction increases before decreasing as a fct of P before decreasing as a fct of P

although atoms should be less mobile although atoms should be less mobile and vacancies should disappear

and vacancies should disappear as P increases

as P increases

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annealing the crystals, adding ³ He

Rittner and Reppy (Cornell, 2006): annealing destroys supersolid behavior Rittner and Reppy (Cornell, 2006): annealing destroys supersolid behavior Kim and Chan (Penn State, 2006): annealing enhances supersolid behavior ! Kim and Chan (Penn State, 2006): annealing enhances supersolid behavior !

Shirahama et al. (Tokyo, 2006):

no effect of annealing but the supersolid density

no effect of annealing but the supersolid density _s_s = 0.1%, not 1% ... = 0.1%, not 1% ...

Kim and Chan (Penn State, 2006):

33He impurities increase Tc but decrease He impurities increase Tc but decrease _s_s but ultrapure 4He shows very small

but ultrapure 4He shows very small _s_s thermodynamic quantities :

thermodynamic quantities :

very small change in the specific heat (Kim and Chan) very small change in the specific heat (Kim and Chan)

no singularity in the melting curve (Todoshchenko et al. Helsinki 2006) no singularity in the melting curve (Todoshchenko et al. Helsinki 2006)

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two previous experiments on superflow

. . . . . .

Day, Herman and Beamish (PRL 2005):

no flow in Vycor glass no flow in Vycor glass

the lattice is probably pinned at low T, the lattice is probably pinned at low T, mass flow requires motion of the lattice mass flow requires motion of the lattice But probably not in the new expt through But probably not in the new expt through capillaries (PRL 2006)

capillaries (PRL 2006)

crystal crystal liquid liquid Bonfait, Godfrin and Castaing (J. Physique 1989)

Bonfait, Godfrin and Castaing (J. Physique 1989) growth inside a thin capacitor at T < 20 mK

growth inside a thin capacitor at T < 20 mK blocked by a facet at the entrance ?

blocked by a facet at the entrance ?

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ENS 2006: experimental setup

Fill a test tube (1 cm

Fill a test tube (1 cm ) at 1.3 K) at 1.3 K lower T down to 50 mK

lower T down to 50 mK melt the outside melt the outside follow the level inside follow the level inside

any change in the level inside any change in the level inside requires a mass flow through the requires a mass flow through the

solid (

solid (_C_C = 1.1 = 1.1 _L_L))

melting velocity V = 3 mm/h melting velocity V = 3 mm/h if critical velocity 10

if critical velocity 10 m/s and m/s and superfluid density

superfluid density _s_s / / _C_C = 10 = 10^-2^-2

V V

liquid liquid

solid solid

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Ishiguro’s tube

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the ENS fridge the ENS fridge with optical access with optical access

large optical access large optical access through sets of windows through sets of windows

down to 30 mK down to 30 mK

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filling the tube filling the tube

with solid 4He with solid 4He

makes defects makes defects

liquid liquid

solid solid

the inside crystallizes the inside crystallizes only if a substantial only if a substantial stress is applied.

stress is applied.

For example if the For example if the outside is warmed up outside is warmed up to 1.4K for a few

to 1.4K for a few seconds while the seconds while the inside is at 1.3K inside is at 1.3K

P P

_m_m

( 1.4 K) - P ( 1.4 K) - P

_m_m

( 1.3 K) = 0.3 bar ( 1.3 K) = 0.3 bar

fast growth under inhomogeneous stress creates defects fast growth under inhomogeneous stress creates defects

liquid

liquid liquidliquid

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cusps and grain boundaries

crystal 1 crystal 1 crystal 2

crystal 2

grain boundary grain boundary liquid

liquid mechanical equilibrium

mechanical equilibrium of surface tensions

of surface tensions

at the liquid-solid interface:

each cusp signals the existence each cusp signals the existence

of an emerging grain boundary (GB) of an emerging grain boundary (GB) most cusps move away in

most cusps move away in a few hours a few hours (melting-crystallization + pinning) (melting-crystallization + pinning) some GBs stay pinned

some GBs stay pinned

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no flow in good quality crystals

for 10 crystals with no for 10 crystals with no or very few cusps the tube or very few cusps the tube we could see no flow

we could see no flow

no mass leak along the glass wall no mass leak along the glass wall if supersolidity were due to a if supersolidity were due to a 1% superfluid density in the bulk 1% superfluid density in the bulk with a critical velocity v

with a critical velocity v_c_c = 10 = 10 m/sm/s the interface should relax at

the interface should relax at V = [

V = [_s_s/(/(_C_C - - _L_L)]v)]v_c_c = 1 = 1 m/sm/s that is 3.6 mm in 1 hour that is 3.6 mm in 1 hour

Instead, we see no flow within Instead, we see no flow within 50 50 m in 4 hours,m in 4 hours,

meaning 300 times less meaning 300 times less

=> supersolidity is not due to the superfluidity of

=> supersolidity is not due to the superfluidity of a 1% equilibrium density of vacancies

a 1% equilibrium density of vacancies

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mass flow in crystals with enough grain boudaries

for 3 crystals with some cusps inside the tube we observed a mass flow for 3 crystals with some cusps inside the tube we observed a mass flow If the cusps disappear, the mass flow stops (see crystal #1)

If the cusps disappear, the mass flow stops (see crystal #1) Mass flows along grain boudaries

Mass flows along grain boudaries

Solids with grain boudaries may be supersolid Solids with grain boudaries may be supersolid

(polycrystals) but not single crystals (polycrystals) but not single crystals

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crystal 1 relaxed 1 mm down and stopped

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crystal 1

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crystal 2 had many defects

Many grain boundaries Many grain boundaries more in the lower part more in the lower part

faster

faster flow down to flow down to equilibrium at h = 0 equilibrium at h = 0

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crystal 2 relaxed down to eq. (h = 0)

time

time x x 250 250

5 s = 20 min

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crystal 2:

relaxation at 50 mK

0.0 2.0 4.0 6.0 8.0

0 500 1000 1500 2000

crystal #2

time t (seconds)

relaxation is not exponential but linear

relaxation is not exponential but linear with two successive regimes, with two successive regimes, constant velocity : 6

constant velocity : 6 m/s for 0 < t < 500 sm/s for 0 < t < 500 s 11 11 m/s for 500 < t < 1000 sm/s for 500 < t < 1000 s more defects in the lower part of crystal 2

more defects in the lower part of crystal 2 typical of superfluid flow at a critical velocity typical of superfluid flow at a critical velocity

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crystal 1 : a single grain boundary

7.6 7.8 8.0 8.2 8.4

0 500 1000 1500 2000

height h(t) (mm) crystal #1

time (seconds)

The relaxation at V = 0.6

The relaxation at V = 0.6   m/s stops when the cusp disappears m/s stops when the cusp disappears (the grain boundary moves away, unpinning from the wall

(the grain boundary moves away, unpinning from the wall somewhere)

somewhere)

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grain boundaries at P

_m

are comparable to liquid films with atomic thickness

If we assume the existence of a single grain boundary with If we assume the existence of a single grain boundary with thickness e , width w ,

thickness e , width w , the critical velocity inside isthe critical velocity inside is

vv_c_c^GB^GB= (= (DD²²/4ew/4ew_s_s)()(_C_C--_L_L)V = 1.5 (a/e)(D/w)()V = 1.5 (a/e)(D/w)(_C_C / /_s_s) ) m/sm/s comparable to 2

comparable to 2 m/sm/s measured by Telschow et al. (1974) measured by Telschow et al. (1974) on free adsorbed films of liquid He

on free adsorbed films of liquid He

agreement with the prediction by Burovski, Prokof’ev and agreement with the prediction by Burovski, Prokof’ev and

Svistunov (PRL 2005) Svistunov (PRL 2005) in a general model.

in a general model.

simulations of GBs in solid helium 4 are in progress simulations of GBs in solid helium 4 are in progress

in their group (U. Mass. Amherst) and at Urbana (Ceperley and Clark) in their group (U. Mass. Amherst) and at Urbana (Ceperley and Clark)

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Numerical simulation of grain boundaries

Nature 21 octobre 2006

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crystal 4 at 1.13 K

0.0 1.0 2.0 3.0

1500 2000 2500 3000 3500 4000 4500 height h(t) (mm)

time t (seconds) crystal #4

a highly distorted crystal ; final relaxation at 0.9

a highly distorted crystal ; final relaxation at 0.9   m/s m/s grain boundaries are superfluid up to 1.13 K at least grain boundaries are superfluid up to 1.13 K at least

consistent with e

consistent with e ~ 2 ~ 2 a and a and  

_s_s

~ ~  

_C_C

at P = P at P = P

_m_m

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have we seen the same effect as Kim and Chan ?

the effect of annealing:

Rittner and Reppy (2006) vs Kim and Chan (2004) Rittner and Reppy (2006) vs Kim and Chan (2004) large scatter of data

large scatter of data

evidence for the importance of quenched disorder evidence for the importance of quenched disorder not an intrinsic property of He crystals

not an intrinsic property of He crystals most natural defect: grain boundaries most natural defect: grain boundaries

increase of

increase of  

_s_s

(P) : more (P) : more grain boundaries ?

grain boundaries ?

decrease of at large P:

superfluidity disappears superfluidity disappears

at high density

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T _c and v _c are different

at P = P

at P = P_m_m, equilibrium with the liquid: , equilibrium with the liquid:

Partial wetting of grain boundaries by the liquid phase Partial wetting of grain boundaries by the liquid phase (long range van der Waals forces)(long range van der Waals forces)

The thickness is microscopic (a few times a) The thickness is microscopic (a few times a) Out of equilibrium at high P:

Out of equilibrium at high P:

prewetting near P prewetting near P_m_m, , e(P) should decrease,

e(P) should decrease, T T_c_c et v et v_c_cas well below one layeras well below one layer

PP ee

( or( or T T_c_c)) ( or( or v v_c_c) )

PP_m_m

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1% superfluid density is large

In torsional oscillator In torsional oscillator

experiments, crystallization at experiments, crystallization at constant V

constant V

from the normal liquid from the normal liquid At variable T and P At variable T and P

=> polycrystals => polycrystals grain boundaries grain boundaries every 100 à 200 a every 100 à 200 a , , about 50nm ??

about 50nm ??

1% vacancies would be very large too

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crystals grown from the normal liquid at 1.9 K

dendritic growth dendritic growth

strong light scattering by a high density of defects

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work in progress

The research is now focusing on the effect of disorder, The research is now focusing on the effect of disorder, especially grain boundaires (GB):

especially grain boundaires (GB):

calculate the thickness e and superfluid transition temperature T

calculate the thickness e and superfluid transition temperature T_c_c of GBs of GBs measure the T

measure the T_c_c of GBs with variable misorientation of GBs with variable misorientation measure v

measure v_c_c in fixed GBs, find a model for it in fixed GBs, find a model for it measure GBs at P > P

measure GBs at P > P_m_m : thinner ? lower T : thinner ? lower T_c_c ? lower v ? lower v_c_c ? ? measure the adsorption of 3He on GBs

measure the adsorption of 3He on GBs

characterize the density of GBs in crystals grown at cst V : X rays, light scattering characterize the density of GBs in crystals grown at cst V : X rays, light scattering study the pinning of GBs on different walls

study the pinning of GBs on different walls

torsional oscillator experiments in good quality crystals grown at cst T and P torsional oscillator experiments in good quality crystals grown at cst T and P supersolidity under rotation

supersolidity under rotation

reproduce the measurement of the vacaqncy density vs T reproduce the measurement of the vacaqncy density vs T change the frequency of torsional oscillator measurements change the frequency of torsional oscillator measurements ......

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1% superfluid density is large

In torsional oscillator experiments, all crystals In torsional oscillator experiments, all crystals

have been grown at constant V have been grown at constant V

from the normal liquid phase from the normal liquid phase

variable T and P variable T and P

=> polycrystals

grain boundaries every 100 to 200 a

grain boundaries every 100 to 200 a ~ 50 nm ? ~ 50 nm ? a very high density

a very high density

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facets block the growth

no growth if the crystal no growth if the crystal level is raised again outside level is raised again outside

except if a large

except if a large P is P is applied:

applied:

facets are easily pinned to facets are easily pinned to

wall defects wall defects

facets disappear during facets disappear during melting ( a geometrical melting ( a geometrical

effect) => no pinning effect) => no pinning

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Could a quantum solid flow Could a quantum solid flow like a superfluid ? like a superfluid ?