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Magic Numbers of Boson Clusters

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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Magic Numbers of Boson Clusters

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  1. 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)

  2. Can discriminate against atoms with mass spectrometer set at mass 8 and larger from J. P. Toennies

  3. from J. P. Toennies

  4. At Low Source Temperatures New Diffraction Peaks Appear

  5. Searching for Large 4He Clusters: 4HeN N = 4,5,6…. from J. P. Toennies He2+

  6. from J. P. Toennies

  7. 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)

  8. 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

  9. 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)

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. Diffraction experiments with neutral (4Ne)N clusters show instead stability regions! Shell Effects – Erice 5

  16. 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

  17. 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

  18. 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

  19. Single-particle excitation theory of evaporation and cluster stability spherical box model Magic numbers! Shell Effects – Erice 8

  20. Atomic radial distributions 4Hen 3Hen Barranco et al (2006)

  21. 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

  22. 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

  23. The Auger-evaporation mechanism exchange-symmetric two-atom wavefunction

  24. 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

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

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

  27. 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

  28. Calculated 4He cluster size distribution at different temperatures Shell Effects – Erice 16

  29. Comparison to experiment I

  30. Comparison to experiment II

  31. 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

  32. 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

  33. Electron Microscope Picture of the SiNx Transmission Gratings Courtesy of Prof. H. Smith and Dr. Tim Savas, M. I. T.

  34. Lecture 2: Helium Droplets Grebenev, Toennies & Vilesov Science 279, 2083 (1998)

  35. 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)

  36. 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?

  37. Laser Depletion Spectroscopy

  38. 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?

  39. 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?

  40. The experimental sideband reflects the DOS of Elementary Excitations Roton gap: signature of superfluidity rotational lines

  41. 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

  42. Mixed 4He/3He Droplets: Two Production Methods Small 4He Clusters: N< 100 Large 4He Clusters: 100< N< 5000

  43. 4He / 3He phase separation Barranco et al (2006) 4HeN3He

  44. Stable 4He + 3He mixed clusters Barranco et al (2006) 4 3 2 1 0 1 3

  45. Aggregation of 4He Atoms Around an OCS Molecule Inside a 3He Droplet 3He OCS surrounded by a cage of 4He

  46. 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)

  47. 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

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

  49. Localized phonon in 3He at the impurity molecule Space localization  spectral localization!

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