Shell Effects – Erice 1. Magic Numbers of Boson Clusters. a) He cluster mass selection via diffraction. b) The magic 4 He dimer. c) Magic numbers in larger 4 He clusters? The Auger evaporation picture. Giorgio Benedek with J. Peter Toennies (MPIDSO, Göttingen)
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Shell Effects – Erice 1
Magic Numbers of Boson Clusters
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 with
J. Peter Toennies (MPIDSO, Göttingen)
Oleg Kornilov (UCB, Berkeley)
Elena Spreafico (UNIMIB, Milano)
Can discriminate against atoms with mass spectrometer set at mass 8 and larger
from J. P. Toennies
from J. P. Toennies
At Low Source Temperatures New Diffraction Peaks Appear
Searching for Large 4He Clusters: 4HeN
N = 4,5,6….
from J. P. Toennies
He2+
from J. P. Toennies
Effective Slit Widths vs Particle Velocity
He Atom
versus
He Dimer
64
C
3

V (particlewall) =
3
X
63
D
S
eff
3
C =0.12 meV nm
=2.5
3
]
He
62
m
nm
n
[
f
f
e
s
61
h
t
d
i
W
60
t
He
i
l
2
S
e
59
v
i
t
c
e
<R> = 52.0
+
f
f
E
58
2
~
E

2
b
4m
<R>
57
.
3
=1.2 10 K
=1.1
103 K
56
o
o
0
500
1000
1500
2000
Particle Velocity v [m/s]
Scattering length a = 2 <R> = 97 A
Grisenti, Schöllkopf, Toennies Hegerfeldt, Köhler and Stoll
Phys. Rev. Lett. 85 2284 (2000)
20030702T1Schr.
0.4 A
103 K
1
°
104 A
Grisenti; Schöllkopf, Toennies, Hegerfeldt, Köhler and Stoll, Phys. Rev. Lett. 85 2284 (2000)
The 4He dimer: the world‘s weakest bound and
largest ground state molecule
Since <R> is much greater than Rout the dimer
is a classically forbidden molecule
<R>
A frail
GIANT!
High SR
from J. P. Toennies
To Further Study the Dimer it is Interesting
to Scatter from an Object Smaller than the Dimer: an Atom!
A.Kalinin, O. Kornilov, L. Rusin, J. P. Toennies, and G. Vladimirov, Phys. Rev. Lett. 93, 163402 (2004)
from J. P. Toennies
The Kr atom can pass through
the middle of the molecule
without its being affected
The dimer is nearly invisible:
magic!
trim
end of lecture 6
b) Magic numbers (or stability regions)
Quantum Bose clusters (4He)N are superfluid
 no apparent geometrical constraint
 no shellclosure argument
are there magic numbers or stability regions for boson clusters?
Shell Effects – Erice 2
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
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
More recent highly accurate diffusion Monte Carlo (T=0) calculation
rules out existence of magic numbers due to stabilities:
Cluster Number Size N
R. Guardiola,O. Kornilov, J. Navarro and J. P. Toennies, J. Chem Phys, 2006
Diffraction experiments with neutral (4Ne)N clusters
show instead stability regions!
Shell Effects – Erice 5
Magic numbers, excitation levels, and other properties of small neutral
4He clusters
Rafael Guardiola
Departamento de Física Atómica y Nuclear, Facultad de Fisica, Universidad de Valencia, 46100 Burjassot,
Spain
Oleg Kornilov
MaxPlanckInstitut fur Dynamik und Selbstorganisation, Bunsenstrasse 10, 37073 Gottingen, Germany
Jesús Navarro
IFIC (CSICUniversidad de Valencia), Apartado 22085, 46071 Valencia, Spain
J. Peter Toennies
MaxPlanckInstitut fur Dynamik und Selbstorganisation, Bunsenstrasse 10, 37073 Gottingen, Germany
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
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
Singleparticle excitation theory of evaporation and cluster stability
spherical box model
Magic numbers!
Shell Effects – Erice 8
Atomic radial distributions
4Hen
3Hen
Barranco et al (2006)
Fitting a sphericalbox model (SBM) to QMC calculations
Condition: same number of quantum singleparticle levels
this can be achieved with:
 a(N) = QMC average radius
 V0(N) = μBof bulk liquid
 a constant effective mass m*
From:
Shell Effects – Erice 12
QMC (Pandharipande et al 1988)
the linear fit of QMC shell energies () for (4He)70 rescaled to the bulk liquid μB gives
m*~ 3.2 m
this m*/m value works well for all N since
Shell Effects – Erice 13
The Augerevaporation mechanism
exchangesymmetric twoatom wavefunction
612 LennardJones potential
= 40 Å3
C6 = 1.461 a.u.
d0<r <R(N)
Integration volume
R(N) = cluster radius
Shell Effects – Erice 10
TangToennies potential
Replaced by covolume (excluded volume)
Shell Effects – Erice 11
 Centerofmass reference
total L = even
μ() = 7.3 K
m* = 3.2 4 a.u.
 Augerevaporation probability
Shell Effects – Erice 14
Shell Effects – Erice 15
 Cluster kinetics in a supersonic beam
stationary
fission and coalescence neglected:
cluster relative velocity very small
 Cluster size distribution:
 Comparison to experiment:
Jacobian factor
Gaussian spread (s 0.002)
Ionisation efficiency
Calculated 4He cluster size distribution at different temperatures
Shell Effects – Erice 16
Comparison to experiment I
Comparison to experiment II
at each insertion of a new bound state
Guardiola et al thermodynamic approach
HeN1 + He ↔ HeN
Formationevaporation equilibrium:
Equilibrium constant:
ZN = partition function:
Magic Numbers
Guardiola et al., JCP (2006)
SIF 2008 Genova  14
In conclusion we have seen that…
Highresolution grating diffraction experiments allow to study the
stability of 4He clusters
Experimental evidence for the stability of the4He dimer and the existence of magic numbers in 4He boson clusters
Substantial agreement with Guardiola et al thermodynamic approach:
magic numbers related to the insertion of new bound states with increasing N
Electron Microscope Picture of the SiNx Transmission Gratings
Courtesy of Prof. H. Smith and Dr. Tim Savas, M. I. T.
Lecture 2: Helium Droplets
Grebenev, Toennies & Vilesov
Science 279, 2083 (1998)
Helium Droplets
T0 ≤ 35 K
P0 ≥ 20 bar
Droplets are cooled
by evaporation to
=0.37 K (4He),
=0.15 K (3He)
Brink and Stringari,
Z. Phys. D 15, 257 (1990)
Some Microscopic Manifestations of Superfluidity
How many atoms are needed for superfluidity?
How will this number depend on the observed
property?
Laser Depletion Spectroscopy
OCS
Sharp spectral features indicate
that the molecule rotates without
friction
The closer spacing of the lines
indicates a factor 2.7 larger
moment of inertia
Is this a new microscopic manifestation of superfluidity?
2.Evidence for Superfluidity in Pure 4He Droplets:
Near UV Spectrum of the S1 S0 Transition of Glyoxal
Since IR absorption lines are so sharp, what about electronic transitions?
The experimental sideband reflects the DOS of Elementary Excitations
Roton gap:
signature of superfluidity
rotational lines
Magic number in fermionic 3He clusters
(Barranco et al, 2006)
(p + 1)(p + 2)(p + 3)/3
= 2, 8, 20, 40, 70, 112, 168, 240, 330, ...
stable for N > 30
Mixed 4He/3He Droplets: Two Production Methods
Small 4He Clusters: N< 100
Large 4He Clusters: 100< N< 5000
4He / 3He phase separation
Barranco et al (2006)
4HeN3He
Stable 4He + 3He mixed clusters
Barranco et al (2006)
4
3
2
1
0
1
3
Aggregation of 4He Atoms Around an OCS Molecule
Inside a 3He Droplet
3He
OCS surrounded by a cage of 4He
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)
The Appearance of a Phonon Wing Heralds the Opening up of the Roton Gap
roton
maxon
Relative Depletion [%]
Wavenumber [cm1]
According to this criterium 90 4He Atoms are needed for Superfluidity!
Pörtner, Toennies and Vilesov, in preparation
maxons: in both 4He and 3He
rotons: in 4He only
Localized phonon in 3He at the impurity molecule
Space localization spectral localization!
The localized phonon (LP) is much sharper than the bulk phonon width!
electron – collective excitation coupling
molecule
He atoms
spatial decay of molecule electronic wavefunctions
Inelastic part of dipolar matrix element:
Sideband absorption coefficient:
Dynamic form factor:
interatomic potential
Response function:
noninteracting atoms
E = E(q)
Collective excitations:
Barranco et al
“Shell” model for dynamics
n
n +1
particlehole excitation spectrum
collective excitation (phonon) spectrum
ParaHydrogen Has Long Been A Candidate for Superfluidity
Noncondensed
Bose condensed
paraHydrogen
Decrease in the moment
of inertia indicates
superfluidity
The reduced coordination
In small droplets favors
superfluid response
cartoon H2 on OCS
Aggregation of pH2 molecules around an OCS molecule inside a mixed 4He/3He droplet
(56 H2)
(56 H2)
(34 H2)
(3 H2)
Average Moments of Inertia
IaIb Ic
840 1590 1590
55 1590 1590
880 2500 2500
This is the first evidence
for superfluidity of pH2