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The surface of The surface of helium crystals:

helium crystals:

review and open questions review and open questions

Sébastien Balibar Sébastien Balibar

Laboratoire de Physique Statistique Laboratoire de Physique Statistique

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

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

for references and files, including video sequences, for references and files, including video sequences,

go to

go to http://www.http://www.lpslps..ensens..frfr/~/~balibarbalibar//

(2)

to appear in to appear in

Rev. Mod. Phys. (jan. 05) Rev. Mod. Phys. (jan. 05)

download from:

download from:

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

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

44He and He and 33He crystals:He crystals:

model crystals with both model crystals with both

universal universal

andand

exotic quantum properties exotic quantum properties

static and dynamic static and dynamic

properties:

properties:

roughening and roughening and growth mechanisms growth mechanisms

open problems open problems

(3)

hcp-helium 4 hcp-helium 4

crystals crystals

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 structure hexagonal 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 K 1.4 K

1.1 K 1.1 K

0.5 K 0.5 K

0.1 K 0.1 K

(4)

crystal shapes: lead crystallites crystal shapes: lead crystallites

growth shapes growth shapes

the growth reveals facetted the growth reveals facetted directions

directions

more facets at low T more facets at low T electron microscope electron microscope photographs by photographs by

JJ Metois and JC Heyraud JJ Metois and JC Heyraud (CRMC2 - Marseille,

(CRMC2 - Marseille, France)

France)

T > 120 °C

T > 120 °C T > 120 °CT > 120 °C

50 °C < T < 120 °C

50 °C < T < 120 °C T < 50 °CT < 50 °C

(5)

indium indium

more facets at low T more facets at low T

photographs by photographs by

JJ Metois and JC Heyraud JJ Metois and JC Heyraud CRMC2 Marseille

CRMC2 Marseille

T > 100 °C T > 100 °C

40 < T < 100 °C

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

10 < T < 20 °C

10 < T < 20 °C T < 10 °CT < 10 °C

(6)

video sequence

video sequence

(7)

crystallization waves

(8)

bcc - helium 3 bcc - helium 3

crystals crystals

helium 3 atoms are lighter helium 3 atoms are lighter larger quantum fluctuations larger quantum fluctuations

in the solid in the solid

larger zero point energy larger zero point energy smaller surface tension smaller surface tension

facetting at lower T facetting 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)

(9)

coalescence of

coalescence of

33

He crystals at 320 mK He crystals at 320 mK

R. Ishiguro and S. Balibar, submitted to PRL (2004) R. Ishiguro and S. Balibar, submitted to PRL (2004)

the neck radius varies as the neck radius varies as t 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 drops for viscous liquid drops

(10)

facet sizes are enlarged by a slow growth

facets facets

grow and melt grow and melt much more slowly much more slowly than rough corners than rough corners

(11)

up to 11 different facets on helium 3 crystals up 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 facets up to 11 different facets 0.55 mK

0.55 mK 2.2 mK

2.2 mK

(12)

the roughening the roughening

transition transition

at T = 0 at T = 0

atoms minimize their potential atoms minimize their potential energy

energy

the 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 > 0

at 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 T a roughening temperature TRR

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

influence of the lattice

numerical simulations by Leamy and Gilmer 1975 numerical simulations by Leamy and Gilmer 1975

solid on solid model, bond energy J per atom solid on solid model, bond energy J per atom

TTRR= 0.63 J= 0.63 J

(13)

roughening and facetting:

roughening and facetting:

coupling of the surface to the lattice vs thermal fluctuations coupling 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 interface ex: liq-sol interface helium 4, liquid crystals helium 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 interface ex: 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

(14)

the roughening transition the roughening transition

of soft crystals of 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 surface

crystal surface -> the step energy

-> the step energy  ~ ~ aa22/4/4 is very small is very small steps 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 other

the roughening temperature for (1,n,0) the roughening temperature for (1,n,0) surfaces is

surfaces is

in the end, many facets because the unit cell in the end, many facets because the unit cell a ~ 50 Angström is large

a ~ 50 Angström is large for (1,1,2) surfaces T

for (1,1,2) surfaces TRR ~ 27000 K ! ~ 27000 K ! for (9,8,15) surfaces T

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

T

Rn

= 2

π γ

γ

//

a

n2

= 2 π

6βδ

a

2

a

n2

γa

2

n

2

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)

(15)

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 roughening

dynamic roughening

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

(16)

dynamic roughening

the critical radius r

the critical radius rcc for for the nucleation of terraces:

the nucleation of terraces:

rrcc = = ddccwhere where



LLC C : :

chemical potential difference chemical potential difference

r r

cc

the correlation length the correlation length

= 2= 2dd2 2 / (/ (22

the surface is dynamically rough is r

the surface is dynamically rough is rcc < < , , i.e. if

i.e. if  < 2 < 2c 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



(17)

dynamic roughening dynamic roughening

in helium 4 in helium 4

grow a crystal grow a crystal through a hole through 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 crystal helium crystal

liquid liquid

(18)

from linear from linear

to to

non-linear non-linear

growth growth iin iin

44

He He

T < T

T < 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 > T

T > 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

1.205K 1.218K 1.234K 1.252K 1.285K

heihtiffeence(μ

closer to the closer to the roughening roughening temperature temperature

rrcc < <

(19)

critical critical behaviour behaviour

of of

the growth the growth

rate rate

Nozières's RG calculation also describes Nozières's RG calculation also describes

the evolution of the growth rate the 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 (T

the 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 roughening

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

(20)

comparison with experiments in helium:

comparison with experiments in helium:

the step free energy the step free energy

the step free energy is the step free energy is

calculated from the relation calculated from the relation

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

where L

where 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 - T

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

temperature and t

and tcc measures the strength of measures the strength of the coupling to the lattice

the coupling to the lattice a measurement in helium a 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

Teeatue(K

(21)

the universal relation the universal relation

k k

BB

T T

R R

= (2/ = (2/     (T (T

RR

) a ) a

22

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)

universal

universal : 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 effect

size effect

(22)

Nozières’RG-theory of roughening Nozières’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 =

d2r12 γ(∇z)2 +V cos 2πzd

⎦⎥

= = aa + d + d22aa/d/dff22 : surface stiffness : surface stiffness aasurface tensionsurface tension

V : lattice potential V : lattice potential near T

near TR R , assumptions :, assumptions : small height z

small height z

weak coupling to the lattice weak coupling to the lattice

ff

aa aa

a'a' a'a'

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)]

(23)

the coupling strength in Nozières’s theory

H =

d2r12 γ(∇z)2 +V cos 2πzd

⎦⎥

principle of the calculation:

principle of the calculation: a coarse graining at variable scale La coarse graining at variable scale L assume that

assume that (L) and V(L) depend on scale L(L) and V(L) depend on scale L start

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 graining calculate the free energy of the surface for each coarse graining

deduce 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 t

the parameter t

cc

~ V ~ V

00

/ /  

00

measures the coupling strength measures the coupling strength

(24)

the T-variation the T-variation of the step energy of 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 t

Nozieres' parameter tcc ≈ 13 V ≈ 13 V0 0 //00 helium 4 : t

helium 4 : tcc = 0.58 medium = 0.58 medium

strength at microscopic scale strength at microscopic scale helium 3 : dynamic roughening at helium 3 : dynamic roughening at

100 mK ~ 0.4 T 100 mK ~ 0.4 TRR implies t

implies tcc << 1 << 1

ttcc ≈ 1 ≈ 1

strong coupling strong coupling

ttcc ≈ 0.01 ≈ 0.01

weak coupling weak coupling

(25)

helium 3 :

weak coupling at high T

Todoshchenko et al.

Todoshchenko et al.

(Helsinki, aug. 2004) (Helsinki, aug. 2004) step energy from

step energy from

v (v (p) (spiral growth)p) (spiral growth) in the range 60 -110 mK in the range 60 -110 mK weak coupling

weak coupling

compatible with upper compatible with upper bound by Rolley et al.

bound by Rolley et al.

and universal relation and universal relation TTRR = 260 mK = 260 mK

(26)

V. Tsepelin et al. (Helsinki + Leiden):

strong coupling at 0.55 mK

at 0.55 mK at 0.55 mK

the step energy the step energy  is comparable with is comparable with the surface energy

the surface energy  d: d:

 ~ 0.3 ~ 0.3  d d

strong coupling ? strong coupling ?

(27)

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 spacing

compared to the lattice spacing

this implies weak coupling of the surface to the lattice this 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 aa of of 33He atoms : He atoms : aa ~ (S~ (SCC - S - SLL)/S)/SL L ~ 1/T~ 1/T

at low T where S

at low T where SCC = k ln2 and S = k ln2 and SLL~T << S~T << SCC but above the superfluid transition at T

but above the superfluid transition at Tcc=2mK=2mK and the antiferromagnetic transition at T

and the antiferromagnetic transition at TN = 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

(28)

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 V

in 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 L and make the liquid-solid interface rather thick at the scale L00 Todoshchenko et al. start the renormalization procedure

Todoshchenko 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 fluctuations in the renormalization treatment of surface fluctuations This 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 T the amplitude of quantum fluctuations strongly depends on T

(29)

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

good agreeement but:

good agreeement but:

1- the theory is valid only 1- the theory is valid only

for weak coupling for weak coupling

2- only for 2 < T < 100 mK 2- only for 2 < T < 100 mK

where S

where SLL ~T <<S ~T <<SCC needed : measurements needed : measurements

of of and and  accross T

accross TNN and T and Tcc also as a function of also as a function of

magnetic field magnetic field Todoshchenko’s theory

Todoshchenko’s theory

Nozières’ theory Nozières’ theory

(30)
(31)

two-dimensional two-dimensional

nucleation of nucleation of

terraces terraces

interferometric interferometric measurement of measurement of the relaxation the relaxation

of a crystal surface of a crystal surface to its equilibrium to its equilibrium height

height

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

(32)

some results of the some results of the

renormalization calculation renormalization 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 "a "Kosterlitz - Thouless transition"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 order

infinite order : the step free energy vanishes exponentially : the step free energy vanishes exponentially the surface stiffness shows a "

the surface stiffness shows a "universal jump"universal jump" and a square root cusp and a square root cusp:: T < T

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

T = TRR : : (T (TRR) = ) = TTRR / 2a / 2a2 2 T > T

T > TRR : : (T) = (T) = (T (TRR) ) [[ 1 - (tt 1 - (ttcc))1/21/2] ]

where t = T/T

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

(33)

the remarkable the remarkable

growth growth

dynamics of dynamics of helium crystals helium 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 effects smooth surfaces : step motion

smooth surfaces : step motion

rough surfaces : collisisions with phonons

rough 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 T helium 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.

solid solid

superfluid superfluid

normal liquid normal liquid gasgas

pressure (bar)pressure (bar)

temperature (K) temperature (K)

00 2525

22 11

(34)

the dispersion relation of the dispersion relation of

crystallization waves crystallization waves

2 restoring forces 2 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 liquid

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

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

ω2 = ρL ρCρL

( )2 [γq3 + (ρCρL)gq] crystal

crystal superfluid superfluid

(35)

surface surface stiffness stiffness

measurements measurements

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)

(36)

the anisotropy of stepped surfaces the anisotropy of stepped surfaces

for a stepped surface:

for a stepped surface:

small tilt angle small tilt angle ff with respect to a facet with respect to a facet two stiffness components two stiffness components

: step energy: step energy

: interaction between steps: interaction between steps

γ  β

γ

//

= 6δ a

3

φ

aa ff

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

(37)

step-step interactions step-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 neighbours

neighbours

entropy reduction entropy reduction entropic repulsion entropic repulsion

elastic interaction:

elastic interaction:

overlap of strain fields overlap of strain fields elel/l/l2 2 ~ ~ 22/El/El22

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

elastic repulsion elastic repulsion

δS

l2 = π 2 6

(kBT)2 β l2

δel

l2 γ2 E l2

ll ll

(38)

elastic + entropic interactions elastic + entropic interactions

solid line:

solid line:

prediction for thin steps prediction for thin steps but, 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 angle

tilt angle

or calculate a correction or calculate a correction due to the finite step due to the finite step width

width

(39)

terrace width terrace width

distributions distributions

on on

Si surfaces Si 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)

(40)

the step energy in helium 3 the 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 (t

coupling (tcc ≈ 0.01), ≈ 0.01), but but (T=0) ≈ 0.3 (T=0) ≈ 0.3 dd

is much too large is 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 solid

solid

(41)

the truncation of the the truncation of the

renormalization renormalization

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 = VL

potential U = VL22maxmax ≈ 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 < k

coupling is U < kBBT/4T/4

Should 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 do

One would like to do

an 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

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