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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 (MPI-DSO, Göttingen)

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slide1

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 (MPI-DSO, Göttingen)

Oleg Kornilov (UCB, Berkeley)

Elena Spreafico (UNIMIB, Milano)

slide5

Searching for Large 4He Clusters: 4HeN

N = 4,5,6….

from J. P. Toennies

He2+

slide7

Effective Slit Widths vs Particle Velocity

He Atom

versus

He Dimer

64

C

3

-

V (particle-wall) =

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

10-3 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)

2003-07-02-T1-Schr.

0.4 A

10-3 K

1

°

104 A

Grisenti; Schöllkopf, Toennies, Hegerfeldt, Köhler and Stoll, Phys. Rev. Lett. 85 2284 (2000)

slide8

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

slide9

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)

slide10

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

slide11

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

4 he clusters
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

slide13

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

slide14

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

slide15

Diffraction experiments with neutral (4Ne)N clusters

show instead stability regions!

Shell Effects – Erice 5

slide16

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

Max-Planck-Institut fur Dynamik und Selbstorganisation, Bunsenstrasse 10, 37073 Gottingen, Germany

Jesús Navarro

IFIC (CSIC-Universidad de Valencia), Apartado 22085, 46071 Valencia, Spain

J. Peter Toennies

Max-Planck-Institut fur Dynamik und Selbstorganisation, Bunsenstrasse 10, 37073 Gottingen, Germany

slide17

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

slide18

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

slide19

Single-particle excitation theory of evaporation and cluster stability

spherical box model

Magic numbers!

Shell Effects – Erice 8

slide20

Atomic radial distributions

4Hen

3Hen

Barranco et al (2006)

slide21

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) = μBof bulk liquid

- a constant effective mass m*

From:

Shell Effects – Erice 12

slide22

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

slide23

The Auger-evaporation mechanism

exchange-symmetric two-atom wavefunction

slide24

6-12 Lennard-Jones potential

= 40 Å3

C6 = 1.461 a.u.

d0<r <R(N)

Integration volume

R(N) = cluster radius

Shell Effects – Erice 10

slide25

Tang-Toennies potential

Replaced by co-volume (excluded volume)

Shell Effects – Erice 11

slide26

- Center-of-mass reference

total L = even

μ() = 7.3 K

m* = 3.2  4 a.u.

- Auger-evaporation probability

Shell Effects – Erice 14

slide27

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

slide28

Calculated 4He cluster size distribution at different temperatures

Shell Effects – Erice 16

slide31

at each insertion of a new bound state

Guardiola et al thermodynamic approach

HeN-1 + He ↔ HeN

Formation-evaporation equilibrium:

Equilibrium constant:

ZN = partition function:

Magic Numbers

Guardiola et al., JCP (2006)

SIF 2008 Genova - 14

slide33

In conclusion we have seen that…

 High-resolution 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

  • 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

slide34

Electron Microscope Picture of the SiNx Transmission Gratings

Courtesy of Prof. H. Smith and Dr. Tim Savas, M. I. T.

slide35

Lecture 2: Helium Droplets

Grebenev, Toennies & Vilesov

Science 279, 2083 (1998)

slide36

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)

slide37

Some Microscopic Manifestations of Superfluidity

  • Free Rotations of Molecules
  • The Roton Gap (Phonon Wing)
  • Anomalously Small Moments of Inertia

How many atoms are needed for superfluidity?

How will this number depend on the observed

property?

slide39

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?

slide40

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?

slide41

The experimental sideband reflects the DOS of Elementary Excitations

Roton gap:

signature of superfluidity

rotational lines

slide42

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

slide43

Mixed 4He/3He Droplets: Two Production Methods

Small 4He Clusters: N< 100

Large 4He Clusters: 100< N< 5000

slide44

4He / 3He phase separation

Barranco et al (2006)

4HeN3He

slide45

Stable 4He + 3He mixed clusters

Barranco et al (2006)

4

3

2

1

0

1

3

slide46

Aggregation of 4He Atoms Around an OCS Molecule

Inside a 3He Droplet

3He

OCS surrounded by a cage of 4He

slide47

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)

slide48

The Appearance of a Phonon Wing Heralds the Opening up of the Roton Gap

roton

maxon

Relative Depletion [%]

Wavenumber [cm-1]

According to this criterium 90 4He Atoms are needed for Superfluidity!

Pörtner, Toennies and Vilesov, in preparation

slide49

maxons: in both 4He and 3He

rotons: in 4He only

slide50

Localized phonon in 3He at the impurity molecule

Space localization  spectral localization!

slide52

electron – collective excitation coupling

molecule

He atoms

spatial decay of molecule electronic wavefunctions

slide53

Inelastic part of dipolar matrix element:

Sideband absorption coefficient:

Dynamic form factor:

interatomic potential

Response function:

non-interacting atoms

E = E(q)

Collective excitations:

slide54

Barranco et al

“Shell” model for dynamics

n

n +1

slide55

particle-hole excitation spectrum

collective excitation (phonon) spectrum

slide58

Non-condensed

Bose condensed

slide59

para-Hydrogen

Decrease in the moment

of inertia indicates

superfluidity

The reduced coordination

In small droplets favors

superfluid response

cartoon H2 on OCS

slide62

(5-6 H2)

(5-6 H2)

(3-4 H2)

(3 H2)

slide63

Average Moments of Inertia

IaIb Ic

840 1590 1590

55 1590 1590

880 2500 2500

This is the first evidence

for superfluidity of p-H2

ad