cluster magic numbers. recent highly accurate diffusion monte carlo (t=0) calculation rules out...
TRANSCRIPT
2003-06-23-T2a-Schr.
Cluster Magic Numbers
Geometrical Electronic
Metal clusters
Fermi Level
Do liquid He clusters have magic numbers?
R. Melzer and J.G. Zabolitzky say No!
Ar55 C60
J. Phys. A: Math. Gen. 17 L565 (1984)
Cluster Magic Numbers
Det
achm
ent E
nerg
y [K
]
2004-08-16-T1-Schr.
Ground State Energies of He Clusters
Guardiola and Navarro, priv. comm.
Monte Carlo Calculations: Diffusion
0
1
2
3
4
5
0
0
10
10
20
20
30
30
40
40
50
50-150
-100
-50
0
Binding Energies
Bin
ding
Ene
rgy
E
[K]
b
Atom DetachmentEnergies
m = EN
DD
Recent highly accurate diffusion Monte Carlo (T=0) calculationrules out existence of magic numbers due to stabilities:
R. Guardiola,O. Kornilov, J. Navarro and J. P. Toennies, J. Chem Phys, 2006
Cluster Number Size N
10-4
10-3
10-2
10-1
100
G(N
)
Cluster Size Distributions G(N), N < 100
0 10 20 30 40 50 60 70 80 90 100Cluster Number Size N
0
1
2
3
4
5
Ge
xp(N
) / G
fit(N
)P0 = 1.33 bar
1.28 bar
1.22 bar
1.16 bar
1.10 bar
P0 = 1.33 bar
1.28 bar
1.22 bar
1.16 bar
1.10 bar
Brühl et al. Phys. Rev. Lett. 92 185301-1 (2004)
42
23
13,149,10
2004-01-21-T6-Schr.
T =6.7 K0
G (N) = I J( ) N
G (N) = I J J( ) ddN
-2
J N-1
26
Bruehl et al Phys. Rev. Lett. 92 185301 (2004)
2003-06-26-T1-Fu
C lus tergro wth
Evapo ra tive Co oling
d= 5 mm
Clusters Reach Final Sizes in Early,“ Hot “ Stage of Expansion
Growth reaction
Equilibrium constant
Abrupt changes in equilibrium constants areknown to affect size distributions
He + He HeN-1
N-1 1
N
NNK =
X
X
X X
S g j e-E j /kT
j
Where are partition functionsX
The K have sharp peaks whenever the N cluster has a new excited state. Then both Ξ and K will increase.
But for the N+1 cluster both Ξ will be about the same and K will fall back.
To explain Magic numbers recall that clusters
are formed in early „hot“ stages of the expansion
fro
m J
. P
. T
oe
nn
ies
0n 0)( ,01 ndRkj
)()()0(
2
)12(
)(
,,02
,
2
,
ndndnd
nd
RIRkjR
n
dS
Rd
P
)/(
,)()(
22
0
22
dB
xd
TRMk
dxxxjeRI
Single-particle excitation theory of evaporation and cluster stability
Magic numbers!
evaporation probability
200 /2 MVk
Electron bubbles in 4He droplets
R 1.7 nm
0.48 dyn/cm
E 0.26 eV
322
22
3
44
2PRR
RmE
e
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 ?
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
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
D 1)( min,/0 lsPPP
Ps/l information on dynamical processes inside solid 4He
DP information on Poisson ratio of solid 4He
Firenze 2005 - 12
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
DVa = 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
Dm
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
Transport theory
),(2
2txG
vvC
x
vu
x
vD
t
v
rvvv
PVC vvv2 Dm
),()()()()(),( 00 txuXxLxtXtxG s
0),(),( XtxXtxv
v
uvuvuu v
vvv
)(
vCvuvPVvu
vx
v
v
uvuvvu
vvvvvionlinearizat
vvv
')('
'')(
2Dm
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
Theory vs. experimentDv = 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
P0 = 31 bar T0 = 1.74 K
best fit with = 4 = 1.214
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
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.