Magic Numbers of Boson Clusters 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)

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 from J. P. Toennies He2+

from J. P. Toennies

Effective Slit Widths vs Particle Velocity He Atom versus He Dimer 500 1000 1500 2000 56 57 58 59 60 61 62 63 64 E f e c t i v S l W d h s [ n m ] Particle Velocity v [m/s] Effective Slit Widths vs Particle Velocity He Atom versus He Dimer Scattering length a = 2 <R> = 97 A C =0.12 meV nm 3 He 2 Grisenti, Schöllkopf, Toennies Hegerfeldt, Köhler and Stoll Phys. Rev. Lett. 85 2284 (2000) =2.5 nm eff D 2003-07-02-T1-Schr. o V (particle-wall) = C X - <R> = 52.0 + b ~ 4m <R> =1.2 10 K -3 . 0.4 A =1.1 10-3 K 10-3 K 1 ° 104 A Grisenti; Schöllkopf, Toennies, Hegerfeldt, Köhler and Stoll, Phys. Rev. Lett. 85 2284 (2000)

A frail GIANT! 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)

magic! The Kr atom can pass through the middle of the molecule 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

are there magic numbers or stability regions for boson clusters? 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

clusters are superfluid! 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

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 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

R. Brühl, R. Guardiola, A. Kalinin, O. Kornilov, J. Navarro, T 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 R(N) = (1.88Å) N 1/3 + (1.13 Å) / (N 1/3  1) 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

Single-particle 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 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* From: Shell Effects – Erice 12

this m*/m value works well for all N since 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 Auger-evaporation mechanism exchange-symmetric two-atom wavefunction

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

Tang-Toennies potential Replaced by co-volume (excluded volume) Shell Effects – Erice 11

- Center-of-mass reference total L = even μ() = 7.3 K m* = 3.2  4 a.u. - Auger-evaporation probability Shell Effects – Erice 14

Ionisation efficiency 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

Guardiola et al thermodynamic approach HeN-1 + He ↔ HeN Formation-evaporation equilibrium: Equilibrium constant: ZN = partition function: Magic Numbers at each insertion of a new bound state Guardiola et al., JCP (2006) SIF 2008 Genova - 14

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 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

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 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?

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?

signature of superfluidity 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 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)

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

maxons: in both 4He and 3He rotons: in 4He only

Space localization  spectral localization! 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

 E = E(q) Inelastic part of dipolar matrix element: Sideband absorption coefficient: Dynamic form factor: interatomic potential Response function: non-interacting atoms  E = E(q) Collective excitations:

“Shell” model for dynamics Barranco et al “Shell” model for dynamics n n +1

particle-hole excitation spectrum collective excitation (phonon) spectrum

Para-Hydrogen Has Long Been A Candidate for Superfluidity

Non-condensed Bose condensed

The reduced coordination In small droplets favors superfluid response para-Hydrogen Decrease in the moment of inertia indicates superfluidity The reduced coordination In small droplets favors superfluid response cartoon H2 on OCS

Aggregation of p-H2 molecules around an OCS molecule inside a mixed 4He/3He droplet

(5-6 H2) (5-6 H2) (3-4 H2) (3 H2)

Average Moments of Inertia Ia Ib Ic 840 1590 1590 55 1590 1590 880 2500 2500 This is the first evidence for superfluidity of p-H2