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 ?

4He
vacuum expansion from low -T sources
 The Geyser effect in solid 4He vacuum expansion

Vacancy diffusivity and solid 4He Poisson ratio
 Bernoulli flow of a nominal 4He solid
 Suppression of flow anomalies by 1% 3He
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
Kim & Chan
no trend ?
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
u   4 Pdet S /( kTdet m  d 2 )
Firenze 2005 - 6
continuity
u  u0 (4 A0  s /  d 2   )
 (2 Ps /  /   )1 / 2
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
 0  Tm  T
32.0 bar
Firenze 2005 - 10
Period versus P0 at
constant temperature
 0  ( P  Pm )
1
2
 
2
3
Firenze 2005 - 11

Bernoulli

P  information on Poisson
ratio of solid 4He
P  ( P0  Ps / l , min )
Ps/l  information on
dynamical processes
inside solid 4He
Firenze 2005 - 12

1
Poisson ratio of solid 4He
Firenze 2005 - 13
Firenze 2005 - 14
Plastic flow
motion of dislocation
motion of vacancies
Polturak et al experiment
(PRL 1998)
dominant in solid He
(high diffusivity!)
vacancy injection at s/l
interface + sweeping by
pressure gradient
F  Va P
Vacancy drift
 P
 uv
uv  v F  u0
 P
uv 
Dv  v  kT
solid 4He
 p-type SC
Firenze 2005 - 15
The vacancy mechanism
uv 1 As / l / A0

u0
2 X 0Va
A0
L
As/l
Virtual volume to be
filled by vacancies
in the time L/u0
u0
solid
Va = V* - Va
A0  As / l

 108 poise
16v Va X 0Va
Va = 35.15 Å3 (atomic volume)
V*  0.45Va (vacancy isobaric formation
volume)
Firenze 2005 - 16
Geyser mechanism
accumulation of vacancies up
to a critical concentration Xc
diffusion
Pressure
COLLAPSE!
drift + diffusion
L
0
distance from s/l interface
vacancy bleaching &
resetting of initial conditions
Data on vacancy diffusivity and concentration in 4He
Firenze 2005 - 17
Firenze 2005 - 18
Transport theory
v
 2v
v
v
 Dv
 uv
 Cv v   G( x, t )
2
t
x
r
x
v( x, t )  X ( x, t )  X 0
 r1   r,1eff   r1  2Cv
Cv  v Vv2 P
uv
u v  u v (v )  u v  v
v
uv v
 uv (v)v'  uv v'
v
v x
ion
linearizat


 uv v'(  v Vv 2 P)v  uv v'Cv v
Generation function
G( x, t )   X 0  (t ) ( x) ( L  x)  X 0 us  ( x) (t ),
surface generation velocity
Firenze 2005 - 19
Solution for L
Excess vacancies
v( x, t )  X o [ 12 e
t /  r
t
u tx
t '/  r ( uvt ' x ) 2 / 4 Dvt '
v
dt
'
erfc
 us 
e
e
]
4D t
0 4 D t '
v
v
Current at the s/l interface (x = 0) due to excess vacancies
josc (t )   Dv v' (0, t )  uv v(0, t )
 1  v t /  * 1  t /  v

u
t

*
t
s

 X 0uv  4
e
 2e
erf

erf


t

2
u


*
v
v
v


 *   v r /( v   r )
us  2uv   /  r
 v  4Dv / uv2  4kT / Fu
 = surface depletion layer thickness
reduced form:
y  t / v
  2us / uv
e  y
   v / *  1

josc (t )  X 0uv [
 2e erf y 
erf  y ]


1
4
y
- 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 
Firenze 2005 - 20
Theory vs. experiment
P0 = 31 bar
T0 = 1.74 K
best fit with  = 4  = 1.214
Dv = 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
Firenze 2005 - 21
large  means fast
recombination
better fits are
obtained with finite
L (one more
parameter)
Firenze 2005 - 22
Period 0 vs. diffusivity
finite L  approximate solution by Green’s function method
 0  1 X c  X 0  v 
 erf (
)
1
* 
Xc  2 X0  * 
Dv 
L
2
Xc = critical concentration
X c () 1  12  *  v

X0
1  *  v
L2
0
Xc
 L( X )
0
L
Firenze 2005 - 23
Dv 
2
L
0
L  0.3 mm
  0.5  0.64
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
Firenze 2005 - 29
normal
behaviour
induced by
less than
1% 3He !
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
1
Pressure gradients:
P0  PL
I

L0  L
g0 A0
PL  Pm
I

L
gA
2
3
Length L of the gradient near the s/l interface (solve
the above system for PL and L):
L  P0  Pm A0 g0 
Ag
 
 1
L0  L0
I
 A0 g0  Ag
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
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 = X0v where v is the
vacancy mobility
g = conductivity near the s/l interface: g = Xv
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)!