The surface of The surface of helium crystals:helium crystals:

review and open questionsreview and open questions

Sébastien BalibarSébastien Balibar

Laboratoire de Physique StatistiqueLaboratoire de Physique Statistique

de l ’ENS (Paris, France)de l ’ENS (Paris, France)

CC2004, Wroclaw, sept. 2004CC2004, Wroclaw, sept. 2004

for references and files, including video sequences,for references and files, including video sequences,go to go to http://www.lps.ens.fr/~balibar/

to appear into appear in Rev. Mod. Phys. (jan. 05) Rev. Mod. Phys. (jan. 05)

download from:download from:http://www.lps.ens.fr/~balibar/

44He and He and 33He crystals:He crystals:model crystals with bothmodel crystals with both

universaluniversalandand

exotic quantum propertiesexotic quantum properties

static and dynamic static and dynamic properties: properties:

roughening and roughening and growth mechanismsgrowth mechanisms

open problemsopen problems

hcp-helium 4 hcp-helium 4 crystalscrystals

helium 4 crystals growing from helium 4 crystals growing from superfluid helium 4 superfluid helium 4

photographs by S.Balibar, photographs by S.Balibar, C. Guthmann and E. Rolley,C. Guthmann and E. Rolley,

ENS, 1994ENS, 1994

hexagonal close packed structurehexagonal close packed structure

just like any other crystal,just like any other crystal,more facets at low T : more facets at low T :

successive successive "roughening transitions""roughening transitions"

1.4 K1.4 K

1.1 K1.1 K

0.5 K0.5 K

0.1 K0.1 K

crystal shapes: lead crystallitescrystal shapes: lead crystallites

growth shapesgrowth shapes

the growth reveals facetted the growth reveals facetted directionsdirections

more facets at low Tmore facets at low T

electron microscopeelectron microscopephotographs by photographs by JJ Metois and JC HeyraudJJ Metois and JC Heyraud(CRMC2 - Marseille, (CRMC2 - Marseille, France)France)

T > 120 °CT > 120 °C T > 120 °CT > 120 °C

50 °C < T < 120 °C50 °C < T < 120 °C T < 50 °CT < 50 °C

indiumindium

more facets at low Tmore facets at low T

photographs by photographs by JJ Metois and JC HeyraudJJ Metois and JC HeyraudCRMC2 Marseille CRMC2 Marseille

T > 100 °CT > 100 °C

40 < T < 100 °C40 < T < 100 °C 20 < T < 40 °C20 < T < 40 °C

10 < T < 20 °C10 < T < 20 °C T < 10 °CT < 10 °C

video sequencevideo sequence

crystallization waves

bcc - helium 3bcc - helium 3crystalscrystals

helium 3 atoms are lighterhelium 3 atoms are lighter

larger quantum fluctuationslarger quantum fluctuations

in the solid in the solid

larger zero point energylarger zero point energy

smaller surface tensionsmaller surface tension

facetting at lower Tfacetting at lower T

eq. shape at 320 mK; eq. shape at 320 mK; = 0.060 erg.cm = 0.060 erg.cm-2-2

1 mm1 mm

(110) facets at 80 mK(110) facets at 80 mK

E. Rolley, S. Balibar, F. Gallet, F. Graner E. Rolley, S. Balibar, F. Gallet, F. Graner and C. Guthmann, Europhys. Lett. 8, 523 (1989) and C. Guthmann, Europhys. Lett. 8, 523 (1989)

E. Rolley, S. Balibar and F. Gallet,E. Rolley, S. Balibar and F. Gallet,Europhys. Lett. 2, 247 (1986) Europhys. Lett. 2, 247 (1986)

coalescence of coalescence of 33He crystals at 320 mKHe crystals at 320 mKR. Ishiguro and S. Balibar, submitted to PRL (2004)R. Ishiguro and S. Balibar, submitted to PRL (2004)

the neck radius varies asthe neck radius varies ast t 1/31/3 after contact after contact instead of t ln(t) or t instead of t ln(t) or t 1/21/2 for viscous liquid dropsfor viscous liquid drops

facet sizes are enlarged by a slow growth

facets facets grow and meltgrow and melt

much more slowlymuch more slowlythan rough cornersthan rough corners

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

the roughening the roughening transitiontransition

at T = 0at T = 0 atoms minimize their potential atoms minimize their potential energyenergythe surface is localized near a lattice the surface is localized near a lattice plane, i.e. plane, i.e. "smooth""smooth"Landau 1949: crystal surfaces are Landau 1949: crystal surfaces are smooth in all rational directions smooth in all rational directions (n,p,q) at T=0(n,p,q) at T=0

at T > 0at T > 0 , fluctuations: , fluctuations:adatoms, vacancies, steps with kinks, adatoms, vacancies, steps with kinks, terraces...terraces...the surfaces are the surfaces are "rough""rough" above above a roughening temperature Ta roughening temperature TRR

the crystal surface is free from the the crystal surface is free from the influence of the latticeinfluence of the lattice

numerical simulations by Leamy and Gilmer 1975numerical simulations by Leamy and Gilmer 1975solid on solid model, bond energy J per atomsolid on solid model, bond energy J per atom

TTRR= 0.63 J= 0.63 J

roughening and facetting: roughening and facetting:

coupling of the surface to the lattice vs thermal fluctuationscoupling of the surface to the lattice vs thermal fluctuations

weak coupling: weak coupling: wide steps wide steps

with a small energy with a small energy << << d d

((: surface tension): surface tension)ex: liq-sol interfaceex: liq-sol interface

helium 4, liquid crystalshelium 4, liquid crystals

strong coupling: strong coupling: narrow steps narrow steps

with a large energy with a large energy ~ ~ dd

((: surface tension): surface tension)ex: metal-vacuum interfaceex: metal-vacuum interface

helium 3 :helium 3 : weak for 60 < T< 100 mK ? strong below 1 mK ? weak for 60 < T< 100 mK ? strong below 1 mK ?

dd

the roughening transition the roughening transition of soft crystalsof soft 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)

First estimates of the step energy on (110) 3He facets: Rolley et al., Paris, 1986

Measurement of Measurement of the surface tension from the surface tension from the equilibrium shape of the equilibrium shape of large crystals:large crystals: = 0.060 +/- 0.011 erg/cm= 0.060 +/- 0.011 erg/cm22

eq. shape at 320 mK; eq. shape at 320 mK; = 0.060 erg.cm = 0.060 erg.cm-2-2

1 mm1 mm

The roughening temperature of (110) facets should be The roughening temperature of (110) facets should be TTRR = (2/ = (2/dd22 = 260 mK = 260 mK

Why no visible facets above 100 mK ?Why no visible facets above 100 mK ?

dynamic rougheningdynamic roughening

(110) facets at 80 mK(110) facets at 80 mK

dynamic roughening

the critical radius rthe critical radius rcc for for

the nucleation of terraces:the nucleation of terraces:rrcc = = ddccwhere where

LLC C : :

chemical potential differencechemical potential differencerrcc

the correlation length the correlation length = 2= 2dd2 2 / (/ (22

the surface is dynamically rough is rthe surface is dynamically rough is rcc < < , ,

i.e. if i.e. if < 2 < 2c c dd33 / / 22

in in 33He (Rolley et al. 1986), if He (Rolley et al. 1986), if <  10 <  10-11-11 erg/cm above 100 mK erg/cm above 100 mK

dynamic roughening dynamic roughening in helium 4in helium 4

grow a crystal grow a crystal through a holethrough a hole

watch the watch the relaxation of relaxation of the surface to the surface to its equilibrium its equilibrium

height height (Wolf, Gallet, (Wolf, Gallet, Balibar et al. Balibar et al.

(1983-87)(1983-87)

HH vv

helium crystalhelium crystal

liquidliquid

from linear from linear to to

non-linearnon-lineargrowthgrowthiin iin 44HeHe

T < TT < TRR : non-linear growth (v is quadratic or exponential in the applied force) : non-linear growth (v is quadratic or exponential in the applied force)

( spiral growth due to step motion around dislocations( spiral growth due to step motion around dislocations or nucleation of terraces)or nucleation of terraces)T > TT > TRR : linear growth : linear growth

v = k v = k (sticking of atoms one by one) (sticking of atoms one by one)

0

10

20

30

40

50

0 200 400 600 800 1000

1.205K1.218K1.234K1.252K1.285K

(height difference )m

closer to the closer to the roughening roughening temperaturetemperature

rrcc < <

critical critical behaviour behaviour

of of the growth the growth

raterate

Nozières's RG calculation also describes Nozières's RG calculation also describes the the evolution of the growth rate evolution of the growth rate (i.e. the surface mobility) (i.e. the surface mobility)fits with the same values of fits with the same values of the parameters as for the step energy (Tthe parameters as for the step energy (TRR = 1.30 K ; t = 1.30 K ; tcc = 0.58 ; L = 0.58 ; L00 = 4 a ) = 4 a )

dynamic rougheningdynamic roughening : facets are destroyed by a fast growth : facets are destroyed by a fast growth ( ( a"finite size effect"a"finite size effect" in the renormalization calculation) in the renormalization calculation)

comparison with experiments in helium:comparison with experiments in helium:the step free energythe step free energy

the step free energy is the step free energy is calculated from the relationcalculated from the relation

=(4a/=(4a/) ) [[ (L (Lmaxmax)/V(L)/V(Lmaxmax))]]1/21/2

where Lwhere Lmaxmax is the max scale at is the max scale at

which the renormalization is which the renormalization is stopped ("truncated")stopped ("truncated")it vanishes exponentially as:it vanishes exponentially as: ~ exp [ -~ exp [ -/2(tt/2(ttcc))1/21/2]]

where t = 1 - Twhere t = 1 - TRR/T is the reduced /T is the reduced

temperaturetemperatureand tand tcc measures the strength of measures the strength of

the coupling to the latticethe coupling to the latticea measurement in heliuma measurement in helium(ENS group 1983-92) :(ENS group 1983-92) :TTRR = 1.30 K = 1.30 K

ttcc = 0.58 (weak coupling) = 0.58 (weak coupling)

0.0

0.5

1.0

1.5

1.1 1.15 1.2 1.25 1.3 1.35

( )Temperature K

the universal relationthe universal relation

kkBBTTR R = (2/ = (2/ (T (TRR) a) a22

the surface stiffness tends to the surface stiffness tends to (T(TRR) = ) = k kBBTTRR / 2 a / 2 a22 = 0.315 erg.cm= 0.315 erg.cm-2-2 at zero tilt angle at zero tilt angle if Tif TRR = 1.30 K and t = 1.30 K and tcc = 0.58 = 0.58

agreement with the agreement with the curvature measurements curvature measurements by Wolf et al. (ENS-Paris)by Wolf et al. (ENS-Paris)and by Babkin et al. (Moscow)and by Babkin et al. (Moscow)

universaluniversal : no dependence on microscopic quantities (lattice potential ...) : no dependence on microscopic quantities (lattice potential ...)Nozières's theory also predicts Nozières's theory also predicts the angular variation of the angular variation of , as another finite , as another finite size effectsize effect

Nozières’RG-theory of rougheningNozières’RG-theory of roughening

The sine - Gordon model The sine - Gordon model an effective hamiltonian for a surface deformation z(r):an effective hamiltonian for a surface deformation z(r):

€

H = d2∫ r1

2γ ∇z( )

2+ V cos

2πz

d

⎡ ⎣ ⎢

⎤ ⎦ ⎥

= = + d + d22/d/d22 : surface stiffness : surface stiffnesssurface tensionsurface tensionV : lattice potentialV : lattice potentialnear Tnear TR R , assumptions :, assumptions :

small height zsmall height zweak coupling to the lattice weak coupling to the lattice

'' ''

we use the renormalization calculation by Nozières who revisited this problem we use the renormalization calculation by Nozières who revisited this problem in 1985-94, using several previous works, in 1985-94, using several previous works, in particular Knops and den Ouden Physica A103, 579, 1980)in particular Knops and den Ouden Physica A103, 579, 1980)=> the renormalization trajectories => the renormalization trajectories L) , V(L)]L) , V(L)]

the coupling strength in Nozières’s theory

€

H = d2∫ r1

2γ ∇z( )

2+ V cos

2πz

d

⎡ ⎣ ⎢

⎤ ⎦ ⎥

principle of the calculation: principle of the calculation: a coarse graining at variable scale La coarse graining at variable scale Lassume that assume that (L) and V(L) depend on scale L(L) and V(L) depend on scale Lstart start at the microscopic scale at the microscopic scale (L(L00) = ) = 00 ; V (L ; V (L00) = V) = V00

inject fluctuations at larger and larger scale, inject fluctuations at larger and larger scale, calculate the free energy of the surface for each coarse grainingcalculate the free energy of the surface for each coarse grainingdeduce the L dependence of deduce the L dependence of and V and V

the « microscopic scale » :the « microscopic scale » :where the surface starts feeling thermal fluctuations where the surface starts feeling thermal fluctuations

the parameter tthe parameter tcc ~ V ~ V00//00 measures the coupling strength measures the coupling strength

the T-variationthe T-variationof the step energyof the step energy

A. Hazareesing and J.P. Bouchaud Eur. A. Hazareesing and J.P. Bouchaud Eur.

Phys. J. B 14, 713 (2000)Phys. J. B 14, 713 (2000)::functional renormalization functional renormalization

calculation of the step energy calculation of the step energy

the coupling strength :the coupling strength :Nozieres' parameter tNozieres' parameter tcc ≈ 13 V ≈ 13 V0 0 //00

helium 4 : thelium 4 : tcc = 0.58 medium = 0.58 medium

strength at microscopic scalestrength at microscopic scalehelium 3 : dynamic roughening at helium 3 : dynamic roughening at

100 mK ~ 0.4 T100 mK ~ 0.4 TRR

implies timplies tcc << 1 << 1

ttcc ≈ 1 ≈ 1

strong couplingstrong coupling

ttcc ≈ 0.01 ≈ 0.01

weak couplingweak coupling

helium 3 : weak coupling

at high T

Todoshchenko et al.Todoshchenko et al.(Helsinki, aug. 2004)(Helsinki, aug. 2004)step energy fromstep energy fromv (v (p) (spiral growth)p) (spiral growth)in the range 60 -110 mKin the range 60 -110 mKweak coupling weak coupling compatible with upper compatible with upper bound by Rolley et al. bound by Rolley et al. and universal relationand universal relationTTRR = 260 mK = 260 mK

V. Tsepelin et al. (Helsinki + Leiden):strong coupling at 0.55 mK

at 0.55 mKat 0.55 mKthe step energy the step energy is comparable with is comparable with the surface energy the surface energy d: d: ~ 0.3 ~ 0.3 d dstrong coupling ?strong coupling ?

a possible explanation : quantum fluctuations(Todoshchenko et al. , preprint aug. 2004)

due to quantum fluctuations, due to quantum fluctuations, the solid - liquid interface is thick the solid - liquid interface is thick compared to the lattice spacingcompared to the lattice spacingthis implies weak coupling of the surface to the latticethis implies weak coupling of the surface to the lattice

according to Puech et al. 1983 , the growth rate k = v/according to Puech et al. 1983 , the growth rate k = v/ is proportional to is proportional to the sticking probabilitythe sticking probability of of 33He atoms : He atoms : ~ (S~ (SCC - S - SLL)/S)/SL L ~ 1/T~ 1/T

at low T where Sat low T where SCC = k ln2 and S = k ln2 and SLL~T << S~T << SCC

but above the superfluid transition at Tbut above the superfluid transition at Tcc=2mK=2mK

and the antiferromagnetic transition at Tand the antiferromagnetic transition at TNN = 1 mK = 1 mK

Todoshchenko et al. : Todoshchenko et al. : in in 33He , quantum fluctuations are damped at low T, not at high THe , quantum fluctuations are damped at low T, not at high T

Todoshchenko et al.extend Nozières’ renormalization theory

In Nozières’ theory, the effect of quantum fluctuations is included In Nozières’ theory, the effect of quantum fluctuations is included in the value of the lattice potential Vin the value of the lattice potential V00 at the atomic scale L at the atomic scale L00

no problem in no problem in 44He, the quantum fluctuations are always there He, the quantum fluctuations are always there and make the liquid-solid interface rather thick at the scale Land make the liquid-solid interface rather thick at the scale L00

Todoshchenko et al. start the renormalization procedureTodoshchenko et al. start the renormalization procedure at the atomic scale d but include quantum effectsat the atomic scale d but include quantum effects in the renormalization treatment of surface fluctuationsin the renormalization treatment of surface fluctuationsThis allows them to caculate the case of This allows them to caculate the case of 33He where He where the amplitude of quantum fluctuations strongly depends on Tthe amplitude of quantum fluctuations strongly depends on T

new fit of the step energy by Todoshchenko’s RG-theory

good agreeement but:good agreeement but:1- the theory is valid only1- the theory is valid only

for weak coupling for weak coupling 2- only for 2 < T < 100 mK2- only for 2 < T < 100 mK

where Swhere SLL ~T <<S ~T <<SCC

needed : measurements needed : measurements of of and and

accross Taccross TNN and T and Tcc

also as a function of also as a function of magnetic field magnetic field

Todoshchenko’s theoryTodoshchenko’s theory

Nozières’ theoryNozières’ theory

two-dimensional two-dimensional nucleation of nucleation of

terracesterraces

interferometric interferometric measurement of measurement of the relaxation the relaxation of a crystal surface of a crystal surface to its equilibrium to its equilibrium heightheight

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

1.13K

1.145K

1.155K

1.173K

1.178K

1.19K

1.23K

1/H (mm-1)

10-1

10-2

10-3

10-4

10-5

experimental evidence :experimental evidence :velocity: velocity: v =k v =k exp[- exp[-22/(3a/(3aCC k kBBT)]T)]

difference in chemical potential:difference in chemical potential: = H (= H (CC--LL)/)/CCLL

slope -> step energy slope -> step energy

some results of the some results of the renormalization calculationrenormalization calculation

as first predicted by several groups in the late 70's ,as first predicted by several groups in the late 70's ,the roughening transition is the roughening transition is a "Kosterlitz - Thouless transition" a "Kosterlitz - Thouless transition" like the superfluid transition in 2D, the 2D-crystallization, XY model...like the superfluid transition in 2D, the 2D-crystallization, XY model...(H. van Beijeren, H.J.F. Knops, S.T. Chui and J.D. Weeks...) (H. van Beijeren, H.J.F. Knops, S.T. Chui and J.D. Weeks...)

infinite orderinfinite order : the step free energy vanishes exponentially : the step free energy vanishes exponentiallythe surface stiffness shows a "the surface stiffness shows a "universal jumpuniversal jump" and a " and a square root cuspsquare root cusp::

T < TT < TRR : infinite surface stiffness (the facet is flat) : infinite surface stiffness (the facet is flat)

T = TT = TRR : : (T (TRR) = ) = TTRR / 2a / 2a2 2

T > TT > TRR : : (T) = (T) = (T (TRR) ) [[ 1 - (tt 1 - (ttcc))1/21/2] ] where t = T/Twhere t = T/TRR - 1 is the reduced temperature - 1 is the reduced temperature

the remarkable the remarkable growth growth

dynamics of dynamics of helium crystalshelium crystals

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 effects-> no bulk resistance to the growth, the growth velocity is limited by surface effectssmooth surfaces : step motionsmooth surfaces : step motionrough surfaces : collisisions with phononsrough surfaces : collisisions with phonons (cf. electron mobility in metals) (cf. electron mobility in metals)v = k v = k with k ~ T with k ~ T -4-4 : the growth rate is very large at low T: the growth rate is very large 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.

solidsolid

superfluidsuperfluid

normal liquidnormal liquid

gasgas

pres

sure

(ba

r)pr

essu

re (

bar)

temperature (K)temperature (K)00

2525

2211

the dispersion relation of the dispersion relation of crystallization wavescrystallization waves

2 restoring forces2 restoring forces : : - surface stiffness - surface stiffness (at high frequency or short wavelength) (at high frequency or short wavelength)-gravity g ( at low frequency or large wavelength) gravity g ( at low frequency or large wavelength)

inertia : mass flow in the liquidinertia : mass flow in the liquid ( ( CC > > LL))

-> experimental measurement of the stiffness -> experimental measurement of the stiffness

€

ω 2 =ρ L

ρC − ρ L( )2 γq3 + ρC − ρ L( )gq[ ]

crystalcrystal

superfluidsuperfluid

surface surface stiffness stiffness

measurementsmeasurements

Rolley et al. (ENS - Paris)Rolley et al. (ENS - Paris) PRL 72, 872 (1994)PRL 72, 872 (1994)J. Low Temp. Phys. 99, 851 J. Low Temp. Phys. 99, 851 (1995)(1995)

the anisotropy of stepped surfacesthe anisotropy of stepped surfaces

for a stepped surface: for a stepped surface: small tilt angle small tilt angle with respect to a facetwith respect to a facet

two stiffness componentstwo stiffness components

: step energy: step energy

: interaction between steps: interaction between steps

€

⊥ =aφ

€

// =6δ

a3φ

aa

wide steps : crossover to rough wide steps : crossover to rough at at ≈ a/6L ≈ a/6L00 ≈ 1/24 rad ≈ 1/24 rad

step-step interactionsstep-step interactions

entropic interaction: entropic interaction: steps do not cross (no overhangs)steps do not cross (no overhangs)steps are confined by their steps are confined by their neighboursneighboursentropy reductionentropy reduction

entropic repulsionentropic repulsion

elastic interaction:elastic interaction:overlap of strain fieldsoverlap of strain fields elel/l/l2 2 ~ ~ 22/El/El22

(E : Young modulus)(E : Young modulus)

elastic repulsionelastic repulsion

€

S

l2=

π 2

6

(kBT)2

β l2

€

el

l2≈

γ 2

E l2

ll

ll

elastic + entropic interactionselastic + entropic interactions

solid line: solid line: prediction for thin stepsprediction for thin stepsbut, in helium,but, in helium, the steps are very wide the steps are very wide (weak coupling to the (weak coupling to the lattice) lattice) the measurement needs the measurement needs to be done at very small to be done at very small tilt angletilt angleor calculate a correction or calculate a correction due to the finite step due to the finite step widthwidth

terrace width terrace width distributionsdistributions

ononSi surfacesSi surfaces

E.D. Williams and N.C. Bartelt, E.D. Williams and N.C. Bartelt, Science 251, 393 (1991)Science 251, 393 (1991)Schartzentruber et al. Schartzentruber et al.

PRL 65, 1913 (1990)PRL 65, 1913 (1990)

the step energy in helium 3the step energy in helium 3

the T variation of the step the T variation of the step energy energy agrees with RG- agrees with RG-theory and very weak theory and very weak coupling (tcoupling (tcc ≈ 0.01), ≈ 0.01),

but but (T=0) ≈ 0.3 (T=0) ≈ 0.3 dd is much too largeis much too large(Tsepelin et al. Helsinki 2002) (Tsepelin et al. Helsinki 2002)

a change in coupling a change in coupling strength between strength between 0.55 mK and 100 mK ?0.55 mK and 100 mK ?- Fermi liquid- Fermi liquid- superfluid transition- superfluid transition- magnetic ordering in the - magnetic ordering in the solidsolid

the truncation of the the truncation of the renormalizationrenormalization

Our analysis was done by integrating the RG Our analysis was done by integrating the RG trajectories up to a max scale such that the lattice trajectories up to a max scale such that the lattice potential U = VLpotential U = VL22

maxmax ≈ k ≈ kBBTT

However, in his 1992 lectures at Beg Rohu, However, in his 1992 lectures at Beg Rohu, Nozieres explains that the criterion for weak Nozieres explains that the criterion for weak coupling is U < kcoupling is U < kBBT/4T/4Should one stop using the theory where it fails ?Should one stop using the theory where it fails ?the values of the fitting parameters depend on this the values of the fitting parameters depend on this One would like to doOne would like to doan independant measurement of both an independant measurement of both = (a/2= (a/2/V)/V)1/21/2 and and = (4a/= (4a/V)V)1/21/2

a possible measurementa possible measurementof the correlation lengthof the correlation length

X ray scattering on a solid 4He filmX ray scattering on a solid 4He film grown by epitaxy on a Si(111) substrate ? grown by epitaxy on a Si(111) substrate ? hcp 4He crystals grow by epitaxy on hcp 4He crystals grow by epitaxy on graphite, why not on Si(111) ?graphite, why not on Si(111) ?

study the continuous evolution of critical study the continuous evolution of critical layering transitions towards the roughening layering transitions towards the roughening transition as a function of film thickness transition as a function of film thickness TTcc(n) ≈ T(n) ≈ TRR [1 - c/ln [1 - c/ln22(n)](n)]

(Huse 1984, Nightingale, Saam and Schick (Huse 1984, Nightingale, Saam and Schick 1984)1984)

Ramesh and Maynard, PRL 1982-84Ramesh and Maynard, PRL 1982-84

heliumheliumcrystalcrystal

glassglassprismprism

imaging lensimaging lens

whitewhitelightlight maskmask

color strioscopy