Cluster Magic Numbers
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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
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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
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He2+
from J. P. Toennies
HeN
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Magic Numbers in Large 4He Clusters
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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)
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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
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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
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2006
Thermalization via evaporation (DFT)
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Binding energy per atom
Barranco et al (2006)
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Atomic radial distributions
3Hen
4Hen
Barranco et al (2006)
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Barranco et al (2006)
one-particle states
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3He in 4Hen
Barranco et al (2006)
l
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4He / 3He phase separation
Barranco et al (2006)
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Stable 4He + 3He mixed clusters
Barranco et al (2006)
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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
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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
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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
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Kim & Chan
2004
measurements of non-classical rotational inertia
Firenze 2005 - 3
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no trend ?
Kim
& C
han
Firenze 2005 - 4
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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
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Vacuum expansion of solid 4He
)/(4 2dmkTSPu detdet
Firenze 2005 - 6
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2/1/
200
)/2(
)/4(
s
s
P
dAuu
continuity
Bernoulli
Firenze 2005 - 7
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4He phase diagram
Firenze 2005 - 8
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The Geyser effect
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Period vs. T at constant pressure
TTm 032.0 bar
35.0 bar
40.7 bar
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Period versus P0 at constant temperature
3
2
2
1
)(0
mPP
Bernoulli
Firenze 2005 - 11
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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
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Poisson ratio of solid 4He
Firenze 2005 - 13
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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
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PVa DF
kTD vvvv mm 0uFu
v
P
u
v
P
u
Vacancy drift
solid 4He p-type SC
Firenze 2005 - 15
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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
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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
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Data on vacancy diffusivity and concentration in 4He
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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
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]'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
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- 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
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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
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better fits are obtained with finite
L (one more parameter)
large means fast recombination
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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
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mm3.00
2
L
LvD
64.05.0
Firenze 2005 - 24
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Anomalies below the ’
point!
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a sharp transition in the flow regime at 1.58 K !
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Effects of 3He
on the anomalies
from R. Richardson et al
Firenze 2005 - 27
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small amounts of 3He remove the anomaly!
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normal behaviour induced by less than
1% 3He !
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normal behaviour induced by less than
1% 3He !
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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.
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Miklos Gyulassy, 2004
„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“
end of lecture 8