Shell Effects Erice 1 Magic Numbers of Boson
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) 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 4 He Clusters: 4 He. N N = 4, 5, 6…. from J. P. Toennies
from J. P. Toennies
Effective Slit Widths vs Particle Velocity He Atom versus He Dimer 64 V (particle-wall) = C 3 =0. 12 me. V 62 X 3 D Seff nm 3 =2. 5 nm He 61 60 He 2 59 <R> = 52. 0 + 0. 4 A 58 Eb ~ - 4 m <R> 2 -3 =1. 21. 10 -3 -3 K 10 K 57 56 2 =1. 1 10 K 0 500 1000 1500 2000 o o Effective Slit Width seff [nm] 63 C 3 Particle Velocity v [m/s] ° Scattering length a = 2 <R> = 97 104 AA Grisenti; Schöllkopf, Toennies, Hegerfeldt, Köhler and Stoll, Phys. Rev. Lett. 85 2284 (2000) Grisenti, Schöllkopf, Toennies Hegerfeldt, Köhler and Stoll Phys. Rev. Lett. 85 2284 (2000)
The 4 He dimer: the world‘s weakest bound and largest ground state molecule <R> Since <R> is much greater than Rout the dimer is a classically forbidden molecule A frail GIANT! 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! end of lecture 6
b) Magic numbers (or stability regions) Classical noble gas (van der Waals) clusters: - geometrical constraints only - magic numbers = highest point symmetry Quantum Bose clusters (4 He)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 T 0= 6. 7 K P 0 ≥ 20 bar - formed in nozzle beam vacuum expansion - stabilized through evaporative cooling clusters are superfluid! Shell Effects – Erice 3 clusters T= 0. 37 K
Theory (QMC): no magic numbers predicted for 4 He 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 (4 Ne)N clusters show instead stability regions! Shell Effects – Erice 5
Magic numbers, excitation levels, and other properties of small neutral 4 He 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
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 4 He 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
Single-particle excitation theory of evaporation and cluster stability spherical box model Magic numbers! Shell Effects – Erice 8
Atomic radial distributions 4 He n 3 He Barranco et al (2006) n
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 - V 0(N) = μB of 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 (4 He)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 C 6 = 1. 461 a. u. d 0 < 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 m* = 3. 2 4 a. u. - Auger-evaporation probability Shell Effects – Erice 14 μ( ) = 7. 3 K
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 4 He cluster size distribution at different temperatures Shell Effects – Erice 16
Comparison to experiment I
Comparison to experiment II
Guardiola et al thermodynamic approach Formation-evaporation equilibrium: He. N-1 + He ↔ He. N 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 4 He clusters Experimental evidence for the stability of the 4 He dimer and the existence of magic numbers in 4 He 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 Si. Nx 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 T 0 ≤ 35 K P 0 ≥ 20 bar Droplets are cooled by evaporation to =0. 37 K (4 He), =0. 15 K (3 He) Brink and Stringari, Z. Phys. D 15, 257 (1990)
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?
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?
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 3 He 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 4 He/3 He Droplets: Two Production Methods Small 4 He Clusters: N< 100 Large 4 He Clusters: 100< N< 5000
4 He / 3 He phase separation Barranco et al (2006) 4 He 3 He N
Stable 4 He + 3 He mixed clusters Barranco et al (2006) 4 3 2 1 0 1 3
Aggregation of 4 He Atoms Around an OCS Molecule Inside a 3 He Droplet 3 He OCS surrounded by a cage of 4 He
IR Spectra of OCS in 3 He Droplets with Increasing Numbers of 4 He 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 Relative Depletion [%] maxon Wavenumber [cm-1] According to this criterium 90 4 He Atoms are needed for Superfluidity! Pörtner, Toennies and Vilesov, in preparation
rotons: in 4 He only maxons: in both 4 He and 3 He
Localized phonon in 3 He 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: non-interacting atoms Collective excitations: E = E(q)
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
para-Hydrogen Decrease in the moment of inertia indicates superfluidity The reduced coordination In small droplets favors superfluid response
Aggregation of p-H 2 molecules around an OCS molecule inside a mixed 4 He/3 He droplet
(5 -6 H 2) (3 -4 H 2)
Average Moments of Inertia Ia Ib Ic 840 1590 55 1590 880 2500 This is the first evidence for superfluidity of p-H 2
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