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In quest of 4He supersolid
a work with J. Peter Toennies (MPI-DSO Göttingen), Franco Dalfovo (Uni Trento),
Robert Grisenti & Manuel Käsz (Uni Frankfurt), Pablo Nieto (Automoma Madrid)
History of a conjecture: BEC in a quantum solid ?
Vacancy diffusivity and solid 4He Poisson ratio
The Geyser effect in solid 4He vacuum expansion
Bernoulli flow of a nominal 4He solid
Suppression of flow anomalies by 1% 3He
4He vacuum expansion from low -T sources
Firenze 2005 - 1
History of a conjecture:
BEC in a quantum solid?
1969
Andreev $ Lifshitz
1970
Chester Leggett
1977
Greywall
2004
Kim & Chan
2004
Ceperley & Bernu
Firenze 2005 - 2
Kim & Chan
2004
measurements of non-classical rotational inertia
Firenze 2005 - 3
no trend ?
Kim
& C
han
Firenze 2005 - 4
Galli & Reatto
2001
(a) no ground state vacancies but only thermal vacancies
(b-d) ground state + thermal vacancies (for different vacancy formation energies)
what about injected (non-equilibrium) vacancies?
Firenze 2005 - 5
Vacuum expansion of solid 4He
)/(4 2dmkTSPu detdet
Firenze 2005 - 6
2/1/
200
)/2(
)/4(
s
s
P
dAuu
continuity
Bernoulli
Firenze 2005 - 7
4He phase diagram
Firenze 2005 - 8
The Geyser effect
Firenze 2005 - 9
Period vs. T at constant pressure
40.7 bar
35.0 bar
32.0 bar
TTm 0
Firenze 2005 - 10
Period versus P0 at constant temperature
3
2
2
1
)(0
mPP
Bernoulli
Firenze 2005 - 11
1)( min,/0 lsPPP
Ps/l information on dynamical processes inside solid 4He
P information on Poisson ratio of solid 4He
Firenze 2005 - 12
Poisson ratio of solid 4He
Firenze 2005 - 13
Plastic flowmotion of dislocation
motion of vacancies dominant in solid He(high diffusivity!)
Polturak et al experiment (PRL 1998)
vacancy injection at s/l
interface + sweeping by
pressure gradient
Firenze 2005 - 14
PVa F
kTD vvvv 0uFu
v
P
u
v
P
u
Vacancy drift
solid 4He p-type SC
Firenze 2005 - 15
Va = V* - Va Va = 35.15 Å3 (atomic volume)
V* 0.45Va (vacancy isobaric formation
volume)
A0
As/l
L
Virtual volume to be filled by vacancies
in the time L/u0
u0
a
lsv
VX
AA
u
u
0
0/
0 2
/1
The vacancy mechanism
poise1016
8
0
/0
aav
lssolid VXV
AA
Firenze 2005 - 16
accumulation of vacancies up to a critical concentration Xc
drift + diffusion
diffusion
Pre
ssur
e
distance from s/l interface
0 L
COLLAPSE!
Geyser mechanism
vacancy bleaching &
resetting of initial conditions
Data on vacancy diffusivity and concentration in 4He
Firenze 2005 - 17
Transport theory
),(2
2txG
vvC
x
vu
x
vD
t
v
rvvv
PVC vvv2
),()()()()(),( 00 txuXxLxtXtxG s
0),(),( XtxXtxv
v
uvuvuu v
vvv
)(
vCvuvPVvu
vx
v
v
uvuvvu
vvvvvionlinearizat
vvv
')('
'')(
2
vreffrr C211,
1
Generation function
surface generation velocity
Firenze 2005 - 18
]'4
'4
erfc[),(0
'4/)'(/'/21
2
t
tDxtuts
to
vvrr eetvD
dtutvD
xtvueXtxv
*erf
*
2erf
),0(),0(')(
/21*/
41
0
t
u
utee
tuX
tvutvDtj
vv
s
v
ttvv
vvosc
v
)/(* rvrv
FukTuD vvv /4/4 2
Solution for L
Excess vacancies
Current at the s/l interface (x = 0) due to excess vacancies
rvs uu /2
= surface depletion layer thickness
Firenze 2005 - 19
- the shape of the current depends on 2 parameters (, )
- the time scale implies another parameter (v)
- the ratio of the oscillation amplitude to the constant
background is measured by X0Vauv/u0 and is of the order
of a few percent (as seen in experiment)
fitting
reduced form:
1*//2/ vvsv uuty
]erferf2[)( 041 yye
euXtj y
y
vosc
Firenze 2005 - 20
Theory vs. experimentDv = 1.3·10-5 cm2/s
v = 5.4·1010 s/g
uv = 2.0·10-3 cm/s
us = 2uv
s = 60 s
v = 13 s
* = 10.7 s
0 = 82 s
P0 = 31 bar T0 = 1.74 K
best fit with = 4 = 1.214
Firenze 2005 - 21
better fits are obtained with finite
L (one more parameter)
large means fast recombination
Firenze 2005 - 22
Period 0 vs. diffusivity
finite L approximate solution by Green’s function method
2
021
010 )*
(*
v
c
c
XX
XXerf Xc = critical concentration
v
vc
X
X
*1
*1)( 21
0
L
D
XX
v
c
)(0
0
2
LL
L
Firenze 2005 - 23
mm3.00
2
L
LvD
64.05.0
Firenze 2005 - 24
Anomalies below the ’
point!
Firenze 2005 - 25
a sharp transition in the flow regime at 1.58 K !
Firenze 2005 - 26
Effects of 3He
on the anomalies
from R. Richardson et al
Firenze 2005 - 27
3 He-vacancy binding energy
Firenze 2005 - 28
normal behaviour induced by less than
1% 3He !
Firenze 2005 - 29
CONCLUSIONS
1. The geyser effect indicates (via Bernoulli’s law) an oscillation of the s/l (quasi-)equilibrium pressure at a given T: vacancy concentration appears to be the only system variable which can give such effect.
2. Below the ’ temperature flow anomalies are observed:
(a) The most dramatic one is the occurrence of a Bernoulli flow corresponding to pressures > Pm, at which 4He should be solid. (b) Below 1.58 K a sharp drop of the geyser period signals a dramatic change in the flow properties of solid 4He. These anomalies, suggesting superflow conditions, are attributed to injected excess vacancies, and agree with Galli and Reatto predictions for a vacancy-induced (Andreev-Lifshitz) supersolid phase.
3. A 3He concentration of 0.1% is shown to suppress the flow anomalies, suggesting a quantum nature of the superflow.
Firenze 2005 - 30
gA
I
L
PP
Ag
I
LL
PP
mL
L
000
0
2 I = flow (current), assumed approximately constant over a period
A0 = tube section
A = average flow cross section in the s/l interface
region (A is slightly < A0)
g0 = conductivity far away from the s/l interface
due to the equilibrium concentration of
vacancies X0 : g0 = X0v where v is the
vacancy mobility
g = conductivity near the s/l interface: g = Xv
where X is the actual vacancy concentration
near the s/l interface. Immediately after the
collapse (brown and red lines in the figure)
X << X0 and g << g0 whereas just
before the collapse (green line) X >> X0 and
g >> g0 . When X = X0 (purple line) the
gradient is the same between 0 and L0.
The corresponding gradients are inversely
proportional (see figure)!
1 Pressure gradients:
3 Length L of the gradient near the s/l interface (solve the above system for PL and L):
AggA
Ag
I
gA
L
PP
L
L m
00
00
0
0
01
where the term in parenthesis is constant. For A A0 it appears that L grows with g/g0 = X/X0 as qualitatively shown in the figure. Thus the sensor during the period measures a pressure varying from P0 to Ps/l