Rotons, superfluidity, and He crystals Sébastien Balibar Laboratoire de physique statistique Ecole Normale Supérieure, Paris (France) LT 24, Orlando, aug.

Rotons, superfluidity, and He crystals

Sébastien BalibarLaboratoire de physique statistique

Ecole Normale Supérieure, Paris (France)

LT 24, Orlando, aug. 2005

Laszlo Tisza, june 17, 2005

From: [email protected]: 17 juin 2005 17:55:40 GMT+02:00ﾋ o [email protected] Sebastien,Dear Sebastien,I am delighted to read in Physics Today that you are to receive the Fritz London I am delighted to read in Physics Today that you are to receive the Fritz London Prize. Prize. This is wonderful! Please receive my warmest congratulations.This is wonderful! Please receive my warmest congratulations.Yesterday I was leafing through old correspondence and I found a letter in whichYesterday I was leafing through old correspondence and I found a letter in whichI nominated Landau for the Prize. I am sure I was not alone. I nominated Landau for the Prize. I am sure I was not alone. I was actually at LT-7 in Toronto when the Prize was announced.I was actually at LT-7 in Toronto when the Prize was announced. It is actually unconscionable of Landau not to have taken note of the remarkableIt is actually unconscionable of Landau not to have taken note of the remarkable Simon - London work on helium […]Simon - London work on helium […] All he said that London was not a good physicist.All he said that London was not a good physicist. I am looking forward to your book to straighten out matters.I am looking forward to your book to straighten out matters.With warmest regards,With warmest regards,LaszloLaszlo

Outline

BEC and rotons: the London-Tisza-Landau controversy

Quantum evaporation

The surface of He crystals

The metastability limits of liquid helium

Looking back to the history of superfluidity

1928-38 : discovery of superfluidity at Leiden, Toronto, Cambridge, Moscow…

J.F. Allen and A.D. Misener (Cambridge, jan 1938): J.F. Allen and A.D. Misener (Cambridge, jan 1938):

flow rate Q in a capillary (radius R)flow rate Q in a capillary (radius R)

instead of Poiseuille’s lawinstead of Poiseuille’s law Q = Q = R R44 P / (8 P / (8 l) l)

Q is nearly independent of Q is nearly independent of P and of R (10 to 500 P and of R (10 to 500 m)m)

« the observed type of flow cannot be treated as « the observed type of flow cannot be treated as laminar nor turbulent »laminar nor turbulent »

The hydrodynamics of helium II is non classicalThe hydrodynamics of helium II is non classical

P. Kapitza rediscovers superleaks and P. Kapitza rediscovers superleaks and introduces the word « superfluid », introduces the word « superfluid », in analogy with « superconductor »in analogy with « superconductor »

P. Kapitza (Moscow, dec. 1937) :P. Kapitza (Moscow, dec. 1937) :

below Tbelow T , the viscosity of helium is very , the viscosity of helium is very

small small **... ... 

« it is perhaps sufficient to suggest, by « it is perhaps sufficient to suggest, by analogy with superconductorsanalogy with superconductors, that the , that the helium below the helium below the -point enters a special -point enters a special state which might be called state which might be called a ‘a ‘superfluidsuperfluid’’  »»

* this had already been observed by Keesom * this had already been observed by Keesom and van den Ende, Proc. Roy. Acad. and van den Ende, Proc. Roy. Acad. Amsterdam 33, 243, 1930)Amsterdam 33, 243, 1930)

5 march 1938, 5 march 1938, Institut Henri Poincaré (Paris) :Institut Henri Poincaré (Paris) :

Fritz London:Fritz London:superfluidity has to be connected superfluidity has to be connected with Bose-Einstein condensationwith Bose-Einstein condensation

Paris 1938: Laszlo Tisza introduces Paris 1938: Laszlo Tisza introduces the « two-fluid model »the « two-fluid model »

two parts:two parts: a superfluid with zero entropy and viscosity a superfluid with zero entropy and viscosity a « normal fluid » with non zero entropy and non zero a « normal fluid » with non zero entropy and non zero

viscosityviscositytwo independent velocity fields: vtwo independent velocity fields: vss and v and vnn

predicts thermomechanic effects:predicts thermomechanic effects:• the fountain effect observed by Allen and Jones, and the reverse effectthe fountain effect observed by Allen and Jones, and the reverse effect

• thermal waves (second sound)thermal waves (second sound)

Lev D. Landau Moscow 1941 - 47Lev D. Landau Moscow 1941 - 47

19381938: Landau comes out of prison thanks : Landau comes out of prison thanks to Kapitzato Kapitza19411941: in view of Kapitza’s results on : in view of Kapitza’s results on thermal waves, Landau introduces a more thermal waves, Landau introduces a more rigorous version of Tisza’s two fluid model, rigorous version of Tisza’s two fluid model, but ignoresbut ignores Fritz London and BEC : Fritz London and BEC :« the explanation advanced by Tisza (!) not only has no foundations in his suggestions but is in direct contradiction with them »The normal fluid is made of quantum The normal fluid is made of quantum elementary excitations (elementary excitations (quasiparticlesquasiparticles): ): phonons et phonons et rotonsrotons ( elementary vortices ??) ( elementary vortices ??)Calculates the thermodynamic propertiesCalculates the thermodynamic propertiesprédicts the existence of a critical velocityprédicts the existence of a critical velocityand thermal waves (« second sound »and thermal waves (« second sound » in in agreement with Kaptiza’s resultsagreement with Kaptiza’s results

The London-Tisza-Landau controversy

LT0 at Cambridge (1946), opening talk: Fritz London criticizes Landau‘s « theory based on the shaky grounds of imaginary rotons »:

« The quantization of hydrodynamics [by Landau] « The quantization of hydrodynamics [by Landau] is a very interesting attempt… is a very interesting attempt… however however quite unconvincingquite unconvincing as far as it is based on a representation of the states as far as it is based on a representation of the states of the liquid by phonons and what he calls « rotons ». There is unfortunately no of the liquid by phonons and what he calls « rotons ». There is unfortunately no indication that there exists anything like a « roton »; at least one searches in vain indication that there exists anything like a « roton »; at least one searches in vain for a definition of this word…for a definition of this word…nor any reason given why one of these two fluids should have a zero entropy nor any reason given why one of these two fluids should have a zero entropy

(inevitably taken by Landau from Tisza) …(inevitably taken by Landau from Tisza) …

Despite their rather strong disagreement, Landau was awarded the London prize in 1960, six years after London's death

BEC in 4He

BEC takes place : the : the condensate has been measured condensate has been measured and calculated:and calculated:at 0 bar: from 7 to 9%at 0 bar: from 7 to 9%at 25 bar: from 2 to 4 %at 25 bar: from 2 to 4 %33He behaves differentlyHe behaves differently

and rotons exist they are not elementary quantum vortices, but a consequence of they are not elementary quantum vortices, but a consequence of local order in the liquidlocal order in the liquid

Moroni and Boninsegni (J. Low Temp. Phys. 136, 129, 2004)

London and Landau died too London and Landau died too early to realize that they both early to realize that they both had found part of the truthhad found part of the truth

neutron scattering: rotons exist

R+ and R- rotons have opposite group velocitiesThe roton gap decreases with pressure

0

2

4

6

8

10

12

14

0 5 10 15 20 25

Wavenumber (nm-1)

20 bar

svp

phonons

rotons

RR + +RR - -

rotons : a consequence of local orderF. London, LT0, Cambridge (1946) :  F. London, LT0, Cambridge (1946) :  « …there has to be some short range order in liquid helium. »« …there has to be some short range order in liquid helium. »

A liquid-solid instability (Schneider and Enz 1971):A liquid-solid instability (Schneider and Enz 1971): As the roton minimum As the roton minimum decreases, order extends to larger and larger decreases, order extends to larger and larger distances and the liquid structure gets closer to that of a crystal. distances and the liquid structure gets closer to that of a crystal. An instability when An instability when =0 ; some information from acoustic crystallization ? =0 ; some information from acoustic crystallization ?

R. Feynman, Prog. in LT Phys. 1955 :R. Feynman, Prog. in LT Phys. 1955 : A vortex ring ?A vortex ring ?the dispersion relation of elementary excitations is: the dispersion relation of elementary excitations is:

hhqq = h = h22qq22/ 2mS(q)/ 2mS(q)P. Nozières J. Low Temp. Phys. 137, 45, 2004:P. Nozières J. Low Temp. Phys. 137, 45, 2004: « rotons are ghosts of a Bragg peak » « rotons are ghosts of a Bragg peak » The roton minimum is a consequence of a maximum in the struture factor S(q), The roton minimum is a consequence of a maximum in the struture factor S(q), i.e. a large probability to find atoms at the average interatomic distance from i.e. a large probability to find atoms at the average interatomic distance from their neighbors.their neighbors.

P.W. Anderson 1966:P.W. Anderson 1966: analogy with the photoelectric effectanalogy with the photoelectric effect1 photon hv ejects 1 electron with a kinetic 1 photon hv ejects 1 electron with a kinetic energy energy EEkinkin = hv - E = hv - E0 0 (E(E00 : binding energy) : binding energy)

1 roton with a energy E > 1 roton with a energy E > = 8.65 K evaporates = 8.65 K evaporates 1 atom with a kinetic energy 1 atom with a kinetic energy

EEkinkin - 7.15 = 1.5 K - 7.15 = 1.5 K v > 79 m/s v > 79 m/s

Quantum evaporation

R R --R R ++

rotons (E > 8.65K)rotons (E > 8.65K)

evaporated atomsevaporated atomsEEkinkin > 1.5K > 1.5K

gasgas

liquidliquid

S. Balibar et al. (PhD thesis 1976 and Phys. Rev. B18, 3096, 1978) :S. Balibar et al. (PhD thesis 1976 and Phys. Rev. B18, 3096, 1978) : heat pulses at T < 100 mK heat pulses at T < 100 mK ballistic rotons and phonons ballistic rotons and phononsatoms  evaporated by rotons travel with a minimum velocity 79 m/satoms  evaporated by rotons travel with a minimum velocity 79 m/sdirect evidence for the existence of rotons and the quantization of heat at low Tdirect evidence for the existence of rotons and the quantization of heat at low T

For a quantitative study and the evidence for R For a quantitative study and the evidence for R ++ and R and R -- rotons, see rotons, see M.A.H. Tucker, G.M. Wyborn et A.F.G. Wyatt , Exeter (1990-99)M.A.H. Tucker, G.M. Wyborn et A.F.G. Wyatt , Exeter (1990-99)

The surface of helium The surface of helium crystalscrystals

For a detailed review, seeFor a detailed review, seeS. Balibar, H. Alles, and A. Ya. Parshin,S. Balibar, H. Alles, and A. Ya. Parshin,

Rev. Mod. Phys. 77, 317 (2005)Rev. Mod. Phys. 77, 317 (2005)

The roughening transitions.The roughening transitions.

Helium crystals are model systems whose Helium crystals are model systems whose static static propertiesproperties can be generalized to all classical crystalscan be generalized to all classical crystalsCrystallization waves and dynamic properties.Crystallization waves and dynamic properties.

Helium crystals are also exceptional systems whose Helium crystals are also exceptional systems whose dynamic dynamic propertiesproperties are quantum and surprising: are quantum and surprising: at 100 mK at 100 mK 44He crystals grow 10He crystals grow 101111 times faster than times faster than 33He crystalsHe crystals

the roughening transitions

As T decreases, the surface is coveredwith more and more facets.Successive roughening transitions in high symmetry directions:rough above TR smooth below TR

large scale fluctuations disappear(no difference at the atomic scale)

detailed study of critical behaviorsstep energy, step width, growth rate, curvature… as a function of T and orientation quantitative comparison with RG theory (P. Nozières 1987-92)a Kosterlitz-Thouless transition

1.4 K1.4 K

1 K1 K

0.4 K0.4 K

0.1 K0.1 K

roughening transitions in He 4

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the universal relationthe universal relation

D.S. Fisher and J.D. Weeks, PRL 1983D.S. Fisher and J.D. Weeks, PRL 1983C. Jayaprakash, W.F. Saam and S. Teitel, PRL 1983 :C. Jayaprakash, W.F. Saam and S. Teitel, PRL 1983 :

kkBBTTRR = (2/ = (2/) ) RRdd22

TTRR : roughening transition temperature : roughening transition temperature

= = + ∂ + ∂22 /∂ /∂ 22 : surface stiffness : surface stiffness (( : surface tension, : surface tension, : angle) : angle)

R R = = ( T( TRR))(0001) or « c » facets in (0001) or « c » facets in 44He: the universal relation isHe: the universal relation is precisely satisfied with precisely satisfied with R R = 0.315 cgs and T= 0.315 cgs and TRR= 1.30K = 1.30K

other facets in other facets in 44He are anisotropic : checking the universal relation He are anisotropic : checking the universal relation is more difficult since kis more difficult since kBBTTRR = (2/ = (2/) () (1 1 22))1/21/2dd22

up to 11 different facets on helium 3 crystalsup to 11 different facets on helium 3 crystals

(110)(110)(110)(110) (110)(110)

(100)(100)

(100)(100)

Wagner et al., Leiden 1996 :Wagner et al., Leiden 1996 :(100) and (211) facets(100) and (211) facets

Alles et al. , Helsinki 2001 :Alles et al. , Helsinki 2001 :up to 11 different facetsup to 11 different facets

0.55 mK0.55 mK2.2 mK2.2 mK

quantum fluctuations quantum fluctuations and coupling strength and coupling strength

in in 33HeHe

(110) facets can be seen only below (110) facets can be seen only below ~100 mK ~100 mK E. Rolley , S. Balibar, and F. Gallet, E. Rolley , S. Balibar, and F. Gallet, EuroPhys. Lett. 1986 and 1989 :EuroPhys. Lett. 1986 and 1989 :due to a very weak coupling of the due to a very weak coupling of the crystal surface to the lattice, facets crystal surface to the lattice, facets are too small to be seen near Tare too small to be seen near TRR = =

260 mK (known from 260 mK (known from = 0.06 = 0.06 erg/cmerg/cm22))

I. Todoshchenko et al. Phys. Rev. Lett. 93, 175301 (2004) and LT24 :I. Todoshchenko et al. Phys. Rev. Lett. 93, 175301 (2004) and LT24 :quantum fluctuations are responsible for the weak coupling at high T but quantum fluctuations are responsible for the weak coupling at high T but

damped at low T where the coupling is strong and many facets visible.damped at low T where the coupling is strong and many facets visible.

growth shapes below 100 mK

eq. shape at 320 mK = 0.06 erg/cm2

up to 60 different facets up to 60 different facets in liquid crystalsin liquid crystals

shear modulus << surface tension : shear modulus << surface tension : a << a << steps penetrate as edge dislocations below the steps penetrate as edge dislocations below the crystal surfacecrystal surface-> the step energy -> the step energy ~ ~ aa22/4/4 is very small is very smallsteps are very broad but steps are very broad but their interaction their interaction ~ (~ (a)a)22 / / ll22 is large is large and and compensate each other compensate each otherthe roughening temperature for (1,n,0) the roughening temperature for (1,n,0) surfaces issurfaces is

in the end, many facets because the unit cell in the end, many facets because the unit cell a ~ 50 Angström is largea ~ 50 Angström is largefor (1,1,2) surfaces Tfor (1,1,2) surfaces TRR ~ 27000 K ! ~ 27000 K !

for (9,8,15) surfaces Tfor (9,8,15) surfaces TRR ~ 360 K ~ 360 K€

TRn =2

πγ⊥γ // an

2=2

π

6βδ

a2an

2 ≈γa2

n2

experiments: Pieranski et al. PRL 84, experiments: Pieranski et al. PRL 84, 2409 (2000); Eur. Phys. J. E5, 317 (2001)2409 (2000); Eur. Phys. J. E5, 317 (2001)theory: P. Nozières, F. Pistolesi and theory: P. Nozières, F. Pistolesi and S. Balibar Eur. Phys. J. B24, 387 (2001)S. Balibar Eur. Phys. J. B24, 387 (2001)

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3He crystals at 320 mK: coalescence without viscosity

no facetsH.J. Maris: a purely geometrical problemdR/dt ≈ k/R2

neck radius:R ~ t 1/3

(as for superfluid drops)

inertia: t1/2

viscosity: t ln(t)

R. Ishiguro, F. Graner, E. Rolley and S. Balibar,PRL 93, 235301 (2004)

Crystallization waves

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melting and freezing

2 restoring forces2 restoring forces : : -surface tension surface tension (more precisely the "surface stiffness" (more precisely the "surface stiffness" ) )- gravity g- gravity ginertia : mass flow in the liquidinertia : mass flow in the liquid ( ( CC > > LL))

€

2 =ρ L

ρC − ρ L( )2 γq

3 + ρC − ρ L( )gq[ ]

helium 4 crystals grow from a superfluid (no viscosity, large thermal conductivity)helium 4 crystals grow from a superfluid (no viscosity, large thermal conductivity)the latent heat is very small (see phase diagram)the latent heat is very small (see phase diagram) the crystals are very pure wih a high thermal conductivitythe crystals are very pure wih a high thermal conductivity no bulk resistance to the growth, the growth velocity is limited by surface effectsno bulk resistance to the growth, the growth velocity is limited by surface effectssmooth surfaces : step motionsmooth surfaces : step motionrough surfaces : collisisions with phonons (no thermal rotons below ~0.6K)rough surfaces : collisisions with phonons (no thermal rotons below ~0.6K) (cf. electron (cf. electron mobility in metals)mobility in metals)v = k v = k with k ~ T with k ~ T -4-4 : the growth rate diverges at low T: the growth rate diverges at low Thelium crystals can grow and melt so fast that helium crystals can grow and melt so fast that crystallization wavescrystallization waves propagate at their propagate at their surfaces as if they were liquids.surfaces as if they were liquids.

crystalcrystal

superfluidsuperfluid

experimental measurement of the surface stiffness experimental measurement of the surface stiffness

surface stiffness measurementssurface stiffness measurementsthe the surface tension surface tension is anisotropic is anisotropicthe anisotropy of the anisotropy of the surface stiffnessthe surface stiffness is even larger, especially is even larger, especially for stepped surfaces close to facets.for stepped surfaces close to facets. /d/d//// 6 6dd

step width, energy step width, energy , interactions , interactions

E. Rolley, S. Balibar and C. GuthmannE. Rolley, S. Balibar and C. GuthmannPRL 72, 872, 1994 and J. Low Temp. Phys. 99, 851, 1995PRL 72, 872, 1994 and J. Low Temp. Phys. 99, 851, 1995

the metastability limits of liquid He Liquid-gas and liquid-solid : 1st

order transitionssuppress impurities and wallsliquid helium can be observed in a metastable state for a finite timefollowing J. Nissen (Oregon) and H.J. Maris (Brown Univ.),we use high amplitude, focused acoustic wavesthe tensile strength of liquid He: how much can one stress liquid He without bubble nucleation ? a similar question: how far can one pressurize liquid He without crystal nucleation ?

a 1.3 MHz transducera 1.3 MHz transducer

spherical geometryspherical geometry

high amplitude high amplitude acoustic wavesacoustic waves

At the focal point:

P = Pstat + P cos (2 .t)

f ~1 MHz

large pressure oscillations away from any wall

(here : ± 35 bar)

during ~ T/10 ~ 100 ns

in a volume (/10)3 ~ 15 m3

-50

0

50

0 5 10 15 20 25 30 35

Time (microseconds)

cavitation at Pm

= 25.3 bar

flight time (22 )s

G.Beaume, S. Nascimbene, A. Hobeika, F. Werner, G.Beaume, S. Nascimbene, A. Hobeika, F. Werner, F. Caupin and S. Balibar (2002 - 2003) F. Caupin and S. Balibar (2002 - 2003)

The tensile strength of liquid helium The tensile strength of liquid helium F. Caupin , S. Balibar et al.F. Caupin , S. Balibar et al.

see Phys. Rev. B 64, 064507 (2001) and J. Low Temp. Phys. 129, 363 (2002)see Phys. Rev. B 64, 064507 (2001) and J. Low Temp. Phys. 129, 363 (2002)

A singularity A singularity at 2.2K and at 2.2K and

-7 bar in -7 bar in agreement agreement

with with predictions of predictions of TT at negative at negative

pressurepressure

-15

-12

-9

-6

-3

0

3

0 1 2 3 4 5 6

Caupin 2001

Caupin 2001

Hall 1995

Pettersen 1994

Nissen 1989

Nissen 1989

Sinha 1982

Temperature (K)

liquid-gas equilibrium

nucleation line(Barcelona)

standard theory

(Vτ=.10 -16cm

)s

spinodal limit( )Barcelona

criticalpoint

acoustic cristallization acoustic cristallization on a glass wallon a glass wall

X. Chavanne, S. Balibar and F. CaupinX. Chavanne, S. Balibar and F. CaupinPhys. Rev. Lett. 86, 5506 (2001) Phys. Rev. Lett. 86, 5506 (2001)

amplitude of the acoustic wave at the nucleation threshold :

± 4.3 bar

0.168

0.170

0.172

0.174

0.176

0.178

20 22 24 26 28 30 32

temps (microsecondes)

transmission

reflexion

the extended phase diagram of liquid 4He

no homogeneous nucleation solid 4He up to 160 bar

superfluidity at high P ?Nozieres JLTP 137, 45 (2004).

an instability where = 0 ? L. Vranjes, J.Boronat et al. (preprint 2005) : P > 200 bar ?

R. Ishiguro, F. Caupin and S. Balibar, LT24

HeNe laser

lens

sphericaltransducer

experimental cell

a spherical transducer:

larger amplitudelarger non-linear effects

calibration of the acoustic pressure : Brillouin scattering inside the acoustic wave (in progress)

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0 20 40 60 80 100

time ( )s

5.bar

0bar

Possible observation of homogeneous crystallization

cavitation

no nucleation

crystallization ? We observe 2 nucleation regimes:

at high P: crystallization ?

at low P : cavitation

Intensity and time delay

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0 5 10 15 20 25

Pressure (bar)

1.00

1.50

2.00

2.50

3.00

3.50

0 5 10 15 20 25

Pressure (bar)

The signal intensity increases when approaching Pm = 25.3 barnucleation at high P is delayed by 1/2 period compared to low P crystallization at high P ?calibration of the nucleation pressure :

Brillouin scattering inside the wave

R. Ishiguro, F. Caupin, and S. Balibar, this conference

with many thanks to the co-authors of my papers: students, postdocs, visitors, hosts and collaborators

(chronological order)

B. Perrin, A. Libchaber, D. Lhuillier, J. Buechner, B. Castaing, C. Laroche, D.O. Edwards, P.E. Wolf, F. Gallet, E. Rolley, P. Nozières, C. Guthmann, F. Graner, R.M.Bowley,

W.F. Saam, J.P. Bouchaud, M. Thiel, A. Willibald, P. Evers, A. Levchenko, P. Leiderer, R.H. Torii, H.J.Maris, S.C.Hall,

M.S.Pettersen, C. Naud, E.Chevalier, J.C.Sutra Fourcade, H. Lambaré, P. Roche, O.A.Andreeva, K.O.  Keshishev,

D. Lacoste, J. Dupont-Roc, F. Caupin, S. Marchand, T. Mizusaki, Y. Sasaki, F. Pistolesi, X. Chavanne, T. Ueno,

M. Fechner, C. Appert, C. Tenaud, D. d'Humières, F. Werner,  G. Beaume, A. Hobeika, S. Nascimbene,

C. Herrmann, R. Ishiguro, H. Alles and A.Ya. Parshin

Dripping of helium 3 crystals

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