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//
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
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
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
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
video sequence
video sequence
crystallization waves
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)
coalescence of
coalescence of
33He 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
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
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
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
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
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
2a
n2≈ γa
2n
2experiments: 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 roughening
dynamic roughening
(110) facets at 80 mK (110) facets at 80 mK
dynamic roughening
the critical radius r
the critical radius rcc for for the nucleation of terraces:
the nucleation of terraces:
rrcc = = ddccwhere where
LLC C : :
chemical potential difference chemical potential difference
r r
ccthe correlation length the correlation length
= 2= 2dd2 2 / (/ (22
the surface is dynamically rough is r
the 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 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
from linear from linear
to to
non-linear non-linear
growth growth iin iin
44He 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
heihtiffeence(μ
closer to the closer to the roughening roughening temperature temperature
rrcc < <
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)
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
Teeatue(K
the universal relation the universal relation
k k
BBT T
R R= (2/ = (2/ (T (T
RR) a ) a
22the 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
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 =
∫
d2r⎡⎣⎢12 γ(∇z)2 +V cos 2πzd ⎤⎦⎥
= = aa + d + d22aa/d/dff22 : surface stiffness : surface stiffness aasurface 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)]
the coupling strength in Nozières’s theory
€
H =
∫
d2r⎡⎣⎢12 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/ /
00measures the coupling strength measures the coupling strength
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
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
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 ?
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
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
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
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
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
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
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 ρC − ρL
( )2 [γq3 + (ρC − ρL)gq] crystal
crystal superfluid superfluid
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)
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 aφ
€
γ
//= 6δ a
3φ
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
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
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
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)
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
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