Transcript Diapositiva 1

Cluster Magic Numbers
Cluster Magic Numbers
Geometrical
Electronic
Fermi Level
Ar 55
C 60
Metal clusters
Do liquid He clusters have magic numbers?
R. Melzer and J.G. Zabolitzky say No!
J. Phys. A: Math. Gen. 17 L565 (1984)
2003-06-23-T2a-Schr.
Recent highly accurateDiffusion
diffusion
Monte
(T=0) calculation
Monte
CarloCarlo
Calculations:
rules out existence ofGround
magicState
numbers
dueof to
Energies
Hestabilities:
Clusters
Binding Energy E b [K]
0
Binding Energies
-50
-100
-150
0
10
20
30
40
50
Detachment Energy [K]
5
4
3
2
Atom Detachment
Energies
m = DE
DN
1
0
0
10
20
30
40
50
Guardiola
and N
Navarro, priv. comm.
Cluster Number
Size
R. Guardiola,O. Kornilov, J. Navarro and J. P. Toennies, J. Chem Phys, 2006
2004-08-16-T1-Schr.
HeN
from J. P. Toennies
Magic Numbers in
Large 4He Clusters
Cluster Size Distributions G(N), N < 100
dJ
G (N) = I (J) dN
J N-1
G (N) = I (J) N-2
23
0
10
42
-1
G(N)
10
10
P =1
0
.3 3 b
ar
1 .2 8 b
-2
13,14
9,10
1 .22
ar
ba r
26
-3
1 .16
10
T0 =6.7 K
1.1
0
-4
bar
ba
r
Gexp(N) / G fit (N)
10
P0 = 1.33 bar
5
1.28 bar
4
1.22 bar
3
1.16 bar
2
1.10 bar
1
0
0
10
20
30
40
50
60
70
80
90
100
Cluster Number Size N
2004-01-21-T6-Schr.
Brühl et al. Phys. Rev. Lett. 92
185301-1
Bruehl
et al(2004)
Phys. Rev. Lett. 92 185301 (2004)
To explain Magic numbers recall that clusters
Clusters Reach Final Sizes in Early,
are formed
in early
„hot“ stages
of the expansion
“ Hot
“ Stage
of Expansion
d=5 mm
Cluster
growth
Evaporative Cooling
Growth reaction
He N-1 + He
He N
Equilibrium constant
KN =
XN
X N-1 X1
X
S
j
gje
-E j /kT
The K have sharp peaks whenever the N cluster has a new excited state.
Abrupt
changes
equilibrium
constants are
Then
both Ξin
and
K will increase.
known to affect size distributions
But for the N+1 cluster both Ξ will be about the same and K will fall back.
from J. P. Toennies
Where X are partition functions
Single-particle excitation theory of
evaporation and cluster stability
j1(k0 Rd ,n )  0
 n  0
k 0  2MV0 /  2
dP ( Rd ,n )
dS

evaporation probability
(2  1) 2 n(0) 2
j (k0 Rd ,n ) I  ( Rd ,n )
2
Rd ,n

I  ( Rd )   e
x 2
j ( x) x 2 dx,
0
   2 /( Mk BTRd2 )
Magic numbers!
Thermalization via
evaporation (DFT)
2006
Binding energy per atom
Barranco et al (2006)
Atomic radial distributions
4He
n
3He
Barranco et al (2006)
n
one-particle states
Barranco et al (2006)
3He
in 4Hen
l
Barranco et al (2006)
4He
Barranco et al (2006)
/ 3He phase separation
Stable 4He + 3He mixed clusters
Barranco et al (2006)
Electron bubbles in 4He
droplets
 2 2
4
3
E

4


R

PR
3
2me R 2
2
R  1.7 nm
  0.48 dyn/cm
E  0.26 eV
dynamics?
end of lecture 7
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 /(kT det 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
Period vs. T at constant pressure
40.7 bar
35.0 bar
 0  Tm  T
32.0 bar
Period versus P0 at
constant temperature
 0  ( P  Pm )
1
2
 
2
3
Firenze 2005 - 11

Bernoulli

DP  information on Poisson
ratio of solid 4He
DP  ( P0  Ps / l , m in )
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  DVaP
Vacancy drift
 P
 uv
uv  mv F  u0
 P
uv 
Dv  mv  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
DVa = V* - Va
A0  As / l

 108 poise
16mv DVa 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 - 18
Transport theory
v
 2v
v
v
 Dv
 uv
 C v v   G ( x, t )
2
t
x
r
x
Cv  mv DVv2 P
v( x, t )  X ( x, t )  X 0
 r1   r,1eff   r1  2Cv
uv
u v  u v (v )  u v  v
v
uv v
 uv (v)v'  uv v'
v
v x
ion
linearizat


 uv v'( m v DVv 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 
Theory vs. experiment
P0 = 31 bar
T0 = 1.74 K
best fit with  = 4  = 1.214
Dv = 1.3·10-5 cm2/s
mv = 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
large  means fast
recombination
better fits are
obtained with finite
L (one more
parameter)
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!
a sharp transition
in the flow regime
at 1.58 K !
Effects of 3He
on the anomalies
from R. Richardson et al
Firenze 2005 - 27
small amounts of 3He remove the anomaly!
normal
behaviour
induced by
less than
1% 3He !
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.
„There is no end to this wonderful world
of experimental discovery and
mental constructions
of reality as new facts become known.
That is why physicists have more fun
than most people“
Miklos Gyulassy, 2004
end of lecture 8