Magic Numbers of Boson Clusters
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Magic Numbers of Boson Clusters
Shell Effects – Erice 1
a) He cluster mass selection via diffraction
b) The magic 4He dimer
c) Magic numbers in larger 4He clusters? The Auger evaporation picture
Giorgio Benedek withJ. Peter Toennies (MPI-DSO, Göttingen)
Oleg Kornilov (UCB, Berkeley)Elena Spreafico (UNIMIB, Milano)
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Low temp.cluster source
T0
0
-
-
v
v
40 K
1 barP
Non - destructive Diffraction Grating “Mass Spectrometer”
Previous: Na atoms, Pritchard et al (1988); He*, Mlynek et al (1991)
m m 5 5slit slit
80 cm
Mass spectrometerdetector
m 20~~slit +
detect
He atoms at mass 4 4
2003-01-24-T1-Ka
J
He clusters at mass 8 4
Can discriminate against atoms with mass spectrometer set at mass 8 and larger
from J. P. Toennies
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2002-07-24-T2-WK
He Atom Diffraction Pattern for 300 K Beam
22 22
n=
15 15
8 8
5 5
105
10 4
10 3
102
10 1
Mas
s 4
Io
n S
ign
al [
cts/
sec]
-12 -8 -4 0 4 8 12
Deflection Angle [mrad]
T = 294 K0
P = 140 bar0
= 0.56 A°
-1 1
Bragg: A°0.561000 A°
nd
= (n=1) = 0.56 10-3 rad..
= 150 radmDJ
from J. P. Toennies
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At Low Source Temperatures New Diffraction Peaks Appear
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He
He
2
3
4 5678
He
N=
Deflection Angle [mrad]
-4 -3 -2 -1 00
5
10
15
20
He
Sig
nal [
cts/
sec]
+
T =6.7 KoP =1.5 bar
=4.0 Alo
Magic Numbers in He Clusters: He4 4N
Angular resolution 20 10 rad.-6DJ .
x0.03
2003-08-11-T1a-Schr.
Searching for Large 4He Clusters: 4HeN
He2+
from J. P. Toennies
N = 4,5,6….
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Measure Size of Dimer from Cross Sectionon Scattering from Grating Bars
<R>2
s0 seff- :
s0 seff
He (1s)2
He<R>
Break-up reduces effective slit width
Hegerfeldt and Köhler, PRL 84 (2000)
2003-07-10-T1-Schr.
from J. P. Toennies
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0 500 1000 1500 200056
57
58
59
60
61
62
63
64
Effe
ctiv
e S
l it W
idth
s
[nm
]e
ff
Particle Velocity v [m/s]
Effective Slit Widths vs Particle VelocityHe Atom versus He Dimer
Scattering length a = 2 <R> = 97 A
C =0.12 meV nm33
He
He2
Grisenti, Schöllkopf, Toennies Hegerfeldt, Köhler and StollPhys. Rev. Lett. 85 2284 (2000)
=2.5nm
SeffD
oo
V (particle-wall) = 33C
X-
<R> = 52.0 +
Eb -~4m 2
2
<R>
=1.2 10 K-3.1 10-3 K
104 A°
=1.1 10-3 K
0.4 A
Grisenti; Schöllkopf, Toennies, Hegerfeldt, Köhler and Stoll, Phys. Rev. Lett. 85 2284 (2000)
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Since <R> is much greater than Rout the dimeris a classically forbidden molecule
<R>
The 4He dimer: the world‘s weakest bound and largest ground state molecule
A frail GIANT!
from J. P. Toennies
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He
He
He
2
23
+
He, He ,He
Cluster beam
Kr
l
3
n
Cluster Size Resolved Integral Cross Sections
0 2.0 4.0 6.0 8.0 10.0103
104
Pea
k A
rea
[arb
. uni
ts]
He4
He3
He2
12.0
Pressure Krypton Gas [10 mbar]-5
He
7 10 4.
2003-06-26-T1-Schr.
I=I exp (- n l)s. .o
See Monday poster No 172
of He Clusters in Scattering from Kr Atoms
A.Kalinin, O. Kornilov, L. Rusin, J. P. Toennies, and G. Vladimirov, Phys. Rev. Lett. 93, 163402 (2004)
To Further Study the Dimer it is Interestingto Scatter from an Object Smaller than the Dimer: an Atom!
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The Kr atom can pass through the middle of the molecule without its being affected
The dimer is nearly invisible:
magic!
from J. P. Toennies
end of lecture 6
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b) Magic numbers (or stability regions)
Classical noble gas (van der Waals) clusters:
- geometrical constraints only
- magic numbers = highest point symmetry
Quantum Bose clusters (4He)N are superfluid
- no apparent geometrical constraint
- no shell-closure argument
are there magic numbers or stability regions for boson
clusters?Shell Effects – Erice 2
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4He clusters
T0= 6.7K
P0 ≥ 20bar
T= 0.37K
- formed in nozzle beam vacuum expansion
- stabilized through evaporative cooling
clusters are superfluid!
Shell Effects – Erice 3
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Theory (QMC): no magic numbers
predicted for 4He clusters!- R. Melzer and J. G. Zabolitzky (1984)- M. Barranco, R. Guardiola, S. Hernàndez, R. Mayol, J. Navarro, and M. Pi. (2006)
Binding energy per atom vs. N:
a monotonous slope, with
no peaks nor regions of
larger stability!
Shell Effects – Erice 4
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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
More 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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Diffraction experiments with neutral (4Ne)N
clusters show instead stability regions!
Shell Effects – Erice 5
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Magic numbers, excitation levels, and other properties of small neutral
4He clusters
Rafael GuardiolaDepartamento de Física Atómica y Nuclear, Facultad de Fisica, Universidad de Valencia, 46100 Burjassot,
Spain
Oleg KornilovMax-Planck-Institut fur Dynamik und
Selbstorganisation, Bunsenstrasse 10, 37073 Gottingen, Germany
Jesús NavarroIFIC (CSIC-Universidad de Valencia), Apartado 22085,
46071 Valencia, Spain
J. Peter ToenniesMax-Planck-Institut fur Dynamik und
Selbstorganisation, Bunsenstrasse 10, 37073 Gottingen, Germany
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R. Brühl, R. Guardiola, A. Kalinin, O. Kornilov, J. Navarro, T. Savas and J. P. Toennies,
Phys. Rev. Lett. 92, 185301 (2004)
Shell Effects – Erice 6
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The size of 4He clusters
QMC (V. R. Pandharipande, J.G. Zabolitzky, S. C. Pieper, R. B.
Wiringa, and U. Helmbrecht, Phys. Rev. Lett. 50, 1676 (1973)
R(N) = (1.88Å) N 1/3 + (1.13 Å) / (N 1/3 1)
Shell Effects – Erice 7
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0n
0)( ,01 ndRkj
Single-particle excitation theory of evaporation and cluster stability
Magic numbers!
200 /2 MVk
spherical box model
Shell Effects – Erice 8
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Atomic radial distributions
3Hen
4Hen
Barranco et al (2006)
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Fitting a spherical-box model (SBM) to QMC calculations
Condition: same number of quantum single-particle levels
this can be achieved with:
- a(N) = QMC average radius
- V0(N) = μB of bulk liquid
- a constant effective mass
m*
)(
)12(max
8
*2
222
Na
n
mm
m
B
m
)(
)12(
*81
2
222
Na
n
m
E
BB
n
m
mFrom:
Shell Effects – Erice 12
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Bm
the linear fit of QMC shell
energies () for
(4He)70 rescaled
to the bulk liquid
μB gives
m*~ 3.2 m
constant)(
)12(max2
2
Na
n
QMC (Pandharipande et al 1988)
this m*/m value works well for all N since
Shell Effects – Erice 13
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The Auger-evaporation mechanism
2/12/3
22
2
4)( E
mVEg
)'(),()(),(2
)',',( 121
2
11''2121 2121EEEgrEEEEW EEijEE 22 rrrr
v
exchange-symmetric two-atom wavefunction
2,1,)12()]([*8
)(),( 22
22
ilnNRm
NlnE iiii
m
![Page 24: Magic Numbers of Boson Clusters](https://reader035.fdocuments.net/reader035/viewer/2022081506/56813928550346895da0d05b/html5/thumbnails/24.jpg)
6
60
6
60
60
6
21)(
r
R
r
R
R
Crv
3/4 30db = 40 Å3
C6 = 1.461 a.u.
d0 < r < R(N)
R(N) = cluster radius
6-12 Lennard-Jones potential
Integration volumeShell Effects – Erice 10
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322
2
)()()(
nnn
nr-
attrep
r
Cr)( feA
rrr VVV
n
k
kx
n k
xexf
2
02 !
1)(
Tang-Toennies potential
Replaced by co-volume (excluded volume)
Shell Effects – Erice 11
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)]'(1)[()()',,()( 221',,
221221
EnEnEnEEEEWdENP BBBEEE
)()()12()1(8
),(),( 0,2/12/
,121 LL
LLkEE YkrjLr
N2rr
)](2[*2 2121 NEEmk m
- Auger-evaporation probability
- Center-of-mass reference
total L = even
1/))(( ]1[)( TkNEB
BeEn m
m* = 3.2 4 a.u. μ() = 7.3 K
Shell Effects – Erice 14
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- Cluster size distribution:
- Cluster kinetics in a supersonic beam
stationaryfission and coalescence neglected:cluster relative velocity very small
)(/1)( NPNn
' 42
2
2
)'(exp)'(
1)(*
N Ns
NNNn
sNNn
- Comparison to experiment:
N
1
Jacobian factor
Gaussian spread (s 0.002)
Ionisation efficiency
Shell Effects – Erice 15
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Calculated 4He cluster size distribution at different temperatures
Shell Effects – Erice 16
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Comparison to experiment I
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Comparison to experiment II
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Guardiola et al thermodynamic approach
Guardiola et al., JCP (2006)
HeN-1 + He ↔ HeN Formation-evaporation equilibrium:
Equilibrium constant:
ZN = partition function:
at each insertion of a new bound
state
Magic Numbers
SIF 2008 Genova - 14
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In conclusion we have seen that…
Experimental evidence for the stability of the 4He dimer and the existence of magic numbers in 4He boson clusters
A kinetic theory based on the Auger evaporation mechanism for a spherical-box model qualitatively accounts for the experimental cluster size distributions
Substantial agreement with Guardiola et al thermodynamic approach: magic numbers related to the insertion of new bound states with increasing N
High-resolution grating diffraction experiments allow to study the stability of 4He clusters
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Electron Microscope Picture of the SiNx Transmission Gratings
Courtesy of Prof. H. Smith and Dr. Tim Savas, M. I. T.
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Lecture 2: Helium Droplets
Grebenev, Toennies & VilesovScience 279, 2083 (1998)
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Helium Droplets
T0 ≤ 35 KP0 ≥ 20 bar
Droplets are cooledby evaporation to=0.37 K (4He),=0.15 K (3He)
Brink and Stringari,Z. Phys. D 15, 257 (1990)
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Some Microscopic Manifestations of Superfluidity
1. Free Rotations of Molecules
2. The Roton Gap (Phonon Wing)
3. Anomalously Small Moments of Inertia
How many atoms are needed for superfluidity?
How will this number depend on the observed property?
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2002-03-01-T3a-Ka
Low temp.nozzle
Scatteringchamber
Photon absorptionand
Evaporation
Ionizer
Massspectrometer
Mirror
La
ser
be
am
T0
0
-
-
v
v
20 K
20 bar
d=5 mm
P
Apparatus for Laser Depletion Spectoscopy
Mass.Spect.Signal
Laser Frequency n
none IR photon evaporates
4
DN ~ ~~ ~h
7.2K-7%
400 atoms
For an N=6000 He dropletthis leads to a 7%signal depletion
+
Laser Depletion Spectroscopy
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Sharp spectral features indicate that the molecule rotates without friction
The closer spacing of the lines indicates a factor 2.7 largermoment of inertia
Is this a new microscopic manifestation of superfluidity?
OCS
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Since IR absorption lines are so sharp, what about electronic transitions?
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The experimental sideband reflects the DOS of Elementary Excitations
Roton gap:signature of superfluidity
rotational lines
![Page 42: Magic Numbers of Boson Clusters](https://reader035.fdocuments.net/reader035/viewer/2022081506/56813928550346895da0d05b/html5/thumbnails/42.jpg)
stable for N > 30
(p + 1)(p + 2)(p + 3)/3
Magic number in fermionic 3He clusters (Barranco et al, 2006)
= 2, 8, 20, 40, 70, 112, 168, 240, 330, ...
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Large 4He Clusters: 100< N< 5000
Small 4He Clusters: N< 100
Mixed 4He/3He Droplets: Two Production Methods
![Page 44: Magic Numbers of Boson Clusters](https://reader035.fdocuments.net/reader035/viewer/2022081506/56813928550346895da0d05b/html5/thumbnails/44.jpg)
4He / 3He phase separation
Barranco et al (2006)
4HeN3He
![Page 45: Magic Numbers of Boson Clusters](https://reader035.fdocuments.net/reader035/viewer/2022081506/56813928550346895da0d05b/html5/thumbnails/45.jpg)
Stable 4He + 3He mixed clusters
Barranco et al (2006)
1
2
3
4
1 3
0
![Page 46: Magic Numbers of Boson Clusters](https://reader035.fdocuments.net/reader035/viewer/2022081506/56813928550346895da0d05b/html5/thumbnails/46.jpg)
Aggregation of 4He Atoms Around an OCS Molecule Inside a 3He Droplet
3He
OCS surrounded by a cage of 4He
![Page 47: Magic Numbers of Boson Clusters](https://reader035.fdocuments.net/reader035/viewer/2022081506/56813928550346895da0d05b/html5/thumbnails/47.jpg)
IR Spectra of OCS in 3He Droplets
with Increasing Numbers of 4He
Atoms
~ 60 He atoms are needed to restore free rotations:
Number needed for superfluidity? Grebenev Toennies and Vilesov Science, 279, 2083 (1998)
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Wavenumber [cm-1]
Rel
ativ
e D
eple
tion
[%]
The Appearance of a Phonon Wing Heralds the Opening up of the Roton Gap
Pörtner, Toennies and Vilesov, in preparation
According to this criterium 90 4He Atoms are needed for Superfluidity!
maxon
roton
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rotons: in 4He only
maxons: in both 4He and 3He
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Localized phonon in 3He at the impurity molecule
Space localization spectral localization!
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The localized phonon (LP)
is much sharper than the
bulk phonon width!
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)(),(!
1),( ,, ijiegieg
NRRrRr
]|)([1
),( o,
o,, ii iegegieg rRr
N
]exp[),(1
),( ,,, j j
egjiegNieg i
VRkRrRr
0,oo3
0,o
.]c.cˆ[1
ΨˆΨ
j
ii
i
iji
giiei V i
ieggeeg
eψψeRdV
e
kRk
k
r
r
DD
D
mm
E
E
),()exp()/()(,,,
2/13,, J
egeg mlegego
eg Yrr
)/exp()/()( 02/3
02/1
1 arZaZ eeis Rr
electron – collective excitation coupling
spatial decay of molecule electronic
wavefunctions
molecule
He atoms
![Page 53: Magic Numbers of Boson Clusters](https://reader035.fdocuments.net/reader035/viewer/2022081506/56813928550346895da0d05b/html5/thumbnails/53.jpg)
4
2/33
01
)(
)(46)(
iqZ
aeq
ge
ge
e
ineleg
m
),()(4)(22 EqSqnE
qinelegm mn
)]/exp(1/[),(Im),( 1 kTEEqEqS
)],()(1/[),(),( 00 EqqEqEq v
iΓqEE
AEqph
)(),(
Inelastic part of dipolar matrix element:
Sideband absorption coefficient:
Dynamic form factor:
Response function:
non-interacting atoms
interatomic potential
0),(Re)(1 0 Eqq vCollective excitations: E = E(q)
100 ]/Re)[(Im E
100 ]/Reln[ E
![Page 54: Magic Numbers of Boson Clusters](https://reader035.fdocuments.net/reader035/viewer/2022081506/56813928550346895da0d05b/html5/thumbnails/54.jpg)
)()2( 112
20
nnn uufun
ff m
D 01 EEEΛ m 2/))(/(Δ 2212
0
11111/11 )](1)[(Im
)1(
1)(
EΛEe
ES phphkTE
),(|,1)(2
11 EqqE phq
ph
)]cos(/)[sin(|0,1 11121
0 RqRqRqq /- N
“Shell” model for dynamics n
n +1
Barranco et al
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2211)(
1)(
LPLP
LP
ΓEE
ΓES
111 )Re(
1
2
1 phmLP EE
111 )Im(
1 phLPΓ
particle-hole excitation spectrum collective excitation (phonon) spectrum
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EEEΛ m 2/))(/(Δ 2212
0
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SEARCH FOR SUPERFLUIDITY INPARA- H ( pH ) CLUSTERS2 2
(Ginzburg and Sobyanin, JETP Lett. , 242 (1972))15
pH has no total nuclear spin, I = 0at T = 0 all molecules are in j = 0
pH are spinless Bosons like He indistinguishable
The superfluid transition temperature is given by
T = n 3.31 g Mk
c
T = 6.0 K c
22/3
2/3B
for pH g = 12
but H solidifies at
T = 13.8 K !m
2
2
2
T = 1.4 K
For ortho - H (oH ), I = 1 and j = 1, g = 9. 2 2
c
Para-Hydrogen Has Long Been A Candidate for Superfluidity
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Bose condensed
Non-condensed
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The reduced coordinationIn small droplets favorssuperfluid response
Decrease in the moment of inertia indicatessuperfluidity
para-Hydrogen
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5.
24 3
2001-06-13-t2-kus
4.
3.
2.
1.OCS in largemixed droplet
Capture of firstH molecule2
Capture of secondH molecule2
H molecule movesfreely in liq. He andbinds at OCS replacinga He atom
24
After many H capturesOCS is surrounded by rings of H
2
2
H2
Aggregation of p-H2 molecules around an OCS molecule inside a
mixed 4He/3He droplet
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(5-6 H2)
(3-4 H2)
(5-6 H2)
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Average Moments of Inertia
Ia Ib Ic
840 1590 1590
55 1590 1590
880 2500 2500
This is the first evidencefor superfluidity of p-H2