Rotons, superfluidity, and He crystals Sébastien Balibar Laboratoire de physique statistique Ecole Normale Supérieure, Paris (France) LT 24, Orlando, aug. 2005

Laszlo Tisza, june 17, 2005 From: Date: 17 juin :55:40 GMT+02:00 ﾋ o Dear Sebastien, I am delighted to read in Physics Today that you are to receive the Fritz London Prize. This is wonderful! Please receive my warmest congratulations. Yesterday I was leafing through old correspondence and I found a letter in which 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. It is actually unconscionable of Landau not to have taken note of the remarkable It 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, Laszlo

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 : discovery of superfluidity at Leiden, Toronto, Cambridge, Moscow… J.F. Allen and A.D. Misener (Cambridge, jan 1938): flow rate Q in a capillary (radius R) instead of Poiseuille’s law Q =  R 4  P / (8  l) Q is nearly independent of  P and of R (10 to 500  m) « the observed type of flow cannot be treated as laminar nor turbulent » The hydrodynamics of helium II is non classical

P. Kapitza rediscovers superleaks and introduces the word « superfluid », in analogy with « superconductor » P. Kapitza (Moscow, dec. 1937) : P. Kapitza (Moscow, dec. 1937) : below T, the viscosity of helium is very small *... below T, the viscosity of helium is very small *... « it is perhaps sufficient to suggest, by analogy with superconductors, that the helium below the -point enters a special state which might be called a ‘superfluid’ » * this had already been observed by Keesom and van den Ende, Proc. Roy. Acad. Amsterdam 33, 243, 1930)

5 march 1938, Institut Henri Poincaré (Paris) : Fritz London: superfluidity has to be connected with Bose-Einstein condensation

Paris 1938: Laszlo Tisza introduces the « two-fluid model » two parts: a superfluid with zero entropy and viscosity two parts: a superfluid with zero entropy and viscosity a « normal fluid » with non zero entropy and non zero viscosity a « normal fluid » with non zero entropy and non zero viscosity two independent velocity fields: v s and v n predicts thermomechanic effects: the fountain effect observed by Allen and Jones, and the reverse effect the fountain effect observed by Allen and Jones, and the reverse effect thermal waves (second sound) thermal waves (second sound)

Lev D. Landau Moscow : Landau comes out of prison thanks to Kapitza 1941: in view of Kapitza’s results on thermal waves, Landau introduces a more rigorous version of Tisza’s two fluid model, but ignores 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 elementary excitations (quasiparticles): phonons et rotons ( elementary vortices ??) Calculates the thermodynamic properties prédicts the existence of a critical velocity and thermal waves (« second sound » in agreement 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] is a very interesting attempt… however quite unconvincing 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 indication that there exists anything like a « roton »; at least one searches in vain for a definition of this word… nor any reason given why one of these two fluids should have a zero entropy (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 4 He : the condensate has been measured and calculated: BEC takes place : the condensate has been measured and calculated: at 0 bar: from 7 to 9% at 25 bar: from 2 to 4 % 3 He behaves differently and rotons exist they are not elementary quantum vortices, but a consequence of local order in the liquid Moroni and Boninsegni (J. Low Temp. Phys. 136, 129, 2004) London and Landau died too early to realize that they both had found part of the truth

neutron scattering: rotons exist R+ and R- rotons have opposite group velocities The roton gap decreases with pressure R +R +R +R + R -R -R -R -

rotons : a consequence of local order F. London, LT0, Cambridge (1946) : F. London, LT0, Cambridge (1946) : « …there has to be some short range order in liquid helium. » A liquid-solid instability (Schneider and Enz 1971): As the roton minimum  decreases, order extends to larger and larger distances and the liquid structure gets closer to that of a crystal. An instability when  =0 ; some information from acoustic crystallization ? R. Feynman, Prog. in LT Phys : A vortex ring ? the dispersion relation of elementary excitations is: h  q = h 2 q 2 / 2mS(q) P. Nozières J. Low Temp. Phys. 137, 45, 2004: « rotons are ghosts of a Bragg peak » 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 their neighbors.

P.W. Anderson 1966: analogy with the photoelectric effect 1 photon hv ejects 1 electron with a kinetic energy E kin = hv - E 0 (E 0 : binding energy) 1 roton with a energy E >  = 8.65 K evaporates 1 atom with a kinetic energy E kin  = 1.5 K  v > 79 m/s Quantum evaporation R - R + rotons (E > 8.65K) evaporated atoms E kin > 1.5K gas liquid S. Balibar et al. (PhD thesis 1976 and Phys. Rev. B18, 3096, 1978) : heat pulses at T < 100 mK  ballistic rotons and phonons S. Balibar et al. (PhD thesis 1976 and Phys. Rev. B18, 3096, 1978) : heat pulses at T < 100 mK  ballistic rotons and phonons atoms evaporated by rotons travel with a minimum velocity 79 m/s direct evidence for the existence of rotons and the quantization of heat at low T For a quantitative study and the evidence for R + and R - rotons, see M.A.H. Tucker, G.M. Wyborn et A.F.G. Wyatt, Exeter ( )

The surface of helium crystals For a detailed review, see S. Balibar, H. Alles, and A. Ya. Parshin, Rev. Mod. Phys. 77, 317 (2005) Rev. Mod. Phys. 77, 317 (2005) The roughening transitions. Helium crystals are model systems whose static properties can be generalized to all classical crystals can be generalized to all classical crystals Crystallization waves and dynamic properties. Helium crystals are also exceptional systems whose dynamic properties are quantum and surprising: at 100 mK 4 He crystals grow times faster than 3 He crystals

the roughening transitions As T decreases, the surface is covered with more and more facets. Successive roughening transitions in high symmetry directions: rough above T R  smooth below T R large scale fluctuations disappear (no difference at the atomic scale) detailed study of critical behaviors step energy, step width, growth rate, curvature… as a function of T and orientation quantitative comparison with RG theory (P. Nozières ) a Kosterlitz-Thouless transition 1.4 K 1 K 0.4 K 0.1 K

roughening transitions in He 4

the universal relation D.S. Fisher and J.D. Weeks, PRL 1983 C. Jayaprakash, W.F. Saam and S. Teitel, PRL 1983 : k B T R = (2/  )  R  d 2 T R : roughening transition temperature  =  + ∂ 2  /∂  2 : surface stiffness (  : surface tension,  : angle)  R =  ( T R ) (0001) or « c » facets in 4 He: the universal relation is precisely satisfied with  R = cgs and T R = 1.30K precisely satisfied with  R = cgs and T R = 1.30K other facets in 4 He are anisotropic : checking the universal relation is more difficult since k B T R = (2/  ) (  1  2 ) 1/2  d 2

up to 11 different facets on helium 3 crystals (110) (110) (110) (100) (100) Wagner et al., Leiden 1996 : (100) and (211) facets Alles et al., Helsinki 2001 : up to 11 different facets 0.55 mK 2.2 mK

quantum fluctuations and coupling strength in 3 He (110) facets can be seen only below ~100 mK E. Rolley, S. Balibar, and F. Gallet, EuroPhys. Lett and 1989 : due to a very weak coupling of the crystal surface to the lattice, facets are too small to be seen near T R = 260 mK (known from  = 0.06 erg/cm 2 ) I. Todoshchenko et al. Phys. Rev. Lett. 93, (2004) and LT24 : 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. growth shapes below 100 mK eq. shape at 320 mK  = 0.06 erg/cm 2

up to 60 different facets in liquid crystals up to 60 different facets in liquid crystals shear modulus << surface tension :  a <<  steps penetrate as edge dislocations below the crystal surface -> the step energy  ~  a 2 /4  is very small steps are very broad but their interaction  ~ (  a) 2 /  l 2 is large  and  compensate each other the roughening temperature for (1,n,0) surfaces is in the end, many facets because the unit cell a ~ 50 Angström is large for (1,1,2) surfaces T R ~ K ! for (9,8,15) surfaces T R ~ 360 K experiments: Pieranski et al. PRL 84, 2409 (2000); Eur. Phys. J. E5, 317 (2001) theory: P. Nozières, F. Pistolesi and S. Balibar Eur. Phys. J. B24, 387 (2001)

3 He crystals at 320 mK: coalescence without viscosity no facets H.J. Maris: a purely geometrical problem dR/dt ≈ k/R 2 neck radius: R ~ t 1/3 (as for superfluid drops) inertia: t 1/2 viscosity: t ln(  t) R. Ishiguro, F. Graner, E. Rolley and S. Balibar, PRL 93, (2004)

Crystallization waves

melting and freezing 2 restoring forces : -surface tension  (more precisely the "surface stiffness"  ) - gravity g inertia : mass flow in the liquid (  C >  L ) helium 4 crystals grow from a superfluid (no viscosity, large thermal conductivity) the latent heat is very small (see phase diagram) the crystals are very pure wih a high thermal conductivity the crystals are very pure wih a high thermal conductivity  no bulk resistance to the growth, the growth velocity is limited by surface effects smooth surfaces : step motion rough surfaces : collisisions with phonons (no thermal rotons below ~0.6K) (cf. electron mobility in metals) v = k  with k ~ T -4 : the growth rate diverges at low T helium crystals can grow and melt so fast that crystallization waves propagate at their surfaces as if they were liquids. crystal superfluid  experimental measurement of the surface stiffness 

surface stiffness measurements the surface tension  is anisotropic the anisotropy of the surface stiffness     is even larger, especially for stepped surfaces close to facets.     /d   //  6  d   step width, energy , interactions  E. Rolley, S. Balibar and C. Guthmann PRL 72, 872, 1994 and J. Low Temp. Phys. 99, 851, 1995

the metastability limits of liquid He Liquid-gas and liquid-solid : 1st order transitions suppress impurities and walls liquid helium can be observed in a metastable state for a finite time following J. Nissen (Oregon) and H.J. Maris (Brown Univ.), we use high amplitude, focused acoustic waves the 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 transducer spherical geometry

high amplitude acoustic waves At the focal point: P = P stat +  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  m 3 G.Beaume, S. Nascimbene, A. Hobeika, F. Werner, F. Caupin and S. Balibar ( )

The tensile strength of liquid helium F. Caupin, S. Balibar et al. see Phys. Rev. B 64, (2001) and J. Low Temp. Phys. 129, 363 (2002) A singularity at 2.2K and -7 bar in agreement with predictions of T at negative pressure

acoustic cristallization on a glass wall X. Chavanne, S. Balibar and F. Caupin Phys. Rev. Lett. 86, 5506 (2001) amplitude of the acoustic wave at the nucleation threshold : ± 4.3 bar

the extended phase diagram of liquid 4 He no homogeneous nucleation solid 4 He 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 spherical transducer experimental cell a spherical transducer: larger amplitude larger non-linear effects calibration of the acoustic pressure : Brillouin scattering inside the acoustic wave (in progress)

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 The signal intensity increases when approaching P m = 25.3 bar nucleation 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