Cluster Magic Numbers

1. Cluster Magic Numbers

2. 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

3. HeN from J. P. Toennies He2+

4. Magic Numbers in Large 4He Clusters 2nd cl

5. 26 Bruehl et al Phys. Rev. Lett. 92 185301 (2004)

6. To explain Magic numbers recall that clusters are formed in early „hot“ stages of the expansion The K have sharp peaks whenever the N cluster has a new excited state. Then both Ξ and K will increase. But for the N+1 cluster both Ξ will be about the same and K will fall back. from J. P. Toennies

7. Single-particle excitation theory of evaporation and cluster stability evaporation probability Magic numbers!

8. Thermalization via evaporation (DFT) 2006

9. Binding energy per atom Barranco et al (2006)

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

11. one-particle states Barranco et al (2006)

12. 3He in 4Hen l Barranco et al (2006)

13. 4He / 3He phase separation Barranco et al (2006)

14. Stable 4He + 3He mixed clusters Barranco et al (2006)

15. Electron bubbles in 4He droplets R 1.7 nm   0.48 dyn/cm E  0.26 eV dynamics? end of lecture 7

16. In quest of 4He supersolid a work with J. Peter Toennies (MPI-DSO Göttingen), Franco Dalfovo (Uni Trento), Robert Grisenti & Manuel Käsz (Uni Frankfurt), Pablo Nieto (Automoma Madrid) History of a conjecture: BEC in a quantum solid ?  4He vacuum expansion from low -T sources  The Geyser effect in solid 4He vacuum expansion  Vacancy diffusivity and solid 4He Poisson ratio  Bernoulli flow of a nominal 4He solid  Suppression of flow anomalies by 1% 3He Firenze 2005 - 1

17. History of a conjecture: BEC in a quantum solid? 1969 Andreev $ Lifshitz 1970 Chester  Leggett 1977 Greywall 2004 Kim & Chan 2004 Ceperley & Bernu Firenze 2005 - 2

18. Kim & Chan 2004 measurements of non-classical rotational inertia Firenze 2005 - 3

19. Kim & Chan no trend ? Firenze 2005 - 4

20. Galli & Reatto 2001 (a) no ground state vacancies but only thermal vacancies (b-d) ground state + thermal vacancies (for different vacancy formation energies) what about injected (non-equilibrium) vacancies? Firenze 2005 - 5

21. Vacuum expansion of solid 4He Firenze 2005 - 6

22. continuity Bernoulli Firenze 2005 - 7

23. 4He phase diagram Firenze 2005 - 8

24. The Geyser effect

25. Period vs. T at constant pressure 40.7 bar 35.0 bar 32.0 bar

26. Period versus P0 at constant temperature Bernoulli  Firenze 2005 - 11

27. P  information on Poisson ratio of solid 4He Ps/l  information on dynamical processes inside solid 4He Firenze 2005 - 12

30. Poisson ratio of solid 4He Firenze 2005 - 13

31. Firenze 2005 - 14 motion of dislocation motion of vacancies Plastic flow dominant in solid He (high diffusivity!) vacancy injection at s/l interface + sweeping by pressure gradient Polturak et al experiment (PRL 1998)

32. Vacancy drift solid 4He  p-type SC Firenze 2005 - 15

33. The vacancy mechanism A0 L As/l Virtual volume to be filled by vacancies in the time L/u0 u0 Va = V* - Va Va = 35.15 Å3(atomic volume) V*  0.45Va(vacancy isobaric formation volume) Firenze 2005 - 16

34. Geyser mechanism accumulation of vacancies up to a critical concentration Xc diffusion COLLAPSE! drift + diffusion Pressure L 0 distance from s/l interface vacancy bleaching & resetting of initial conditions

35. Data on vacancy diffusivity and concentration in 4He

36. Firenze 2005 - 18 Transport theory Generation function surface generation velocity

37. Firenze 2005 - 19 Solution for L Excess vacancies Current at the s/l interface (x = 0) due to excess vacancies  = surface depletion layer thickness

38. reduced form: • the shape of the current depends on 2 parameters (, ) • the time scale implies another parameter (v) • the ratio of the oscillation amplitude to the constant • background is measured by X0Vauv/u0 and is of the order • of a few percent (as seen in experiment) • fitting 

39. Dv = 1.3·10-5 cm2/s v = 5.4·1010 s/g uv = 2.0·10-3 cm/s us = 2uv s = 60 s v = 13 s * = 10.7 s 0 = 82 s Theory vs. experiment P0 = 31 bar T0 = 1.74 K best fit with = 4  = 1.214

40. large  means fast recombination better fits are obtained with finite L (one more parameter)

41. Period 0 vs. diffusivity finite L  approximate solution by Green’s function method Xc = critical concentration Firenze 2005 - 23

42. Firenze 2005 - 24

43. Anomalies below the ’ point!

44. a sharp transition in the flow regime at 1.58 K !

45. Effects of 3He on the anomalies from R. Richardson et al Firenze 2005 - 27

46. small amounts of 3He remove the anomaly!

47. normal behaviour induced by less than 1% 3He !

48. normal behaviour induced by less than 1% 3He !

49. CONCLUSIONS • The geyser effect indicates (via Bernoulli’s law) an oscillation of the s/l (quasi-)equilibrium pressure at a given T: vacancy concentration appears to be the only system variable which can give such effect. 2.Below the ’ temperature flow anomalies are observed: (a) The most dramatic one is the occurrence of a Bernoulli flow corresponding to pressures > Pm, at which 4He should be solid. (b) Below 1.58 K a sharp drop of the geyser period signals a dramatic change in the flow properties of solid 4He. These anomalies, suggesting superflow conditions, are attributed to injected excess vacancies, and agree with Galli and Reatto predictions for a vacancy-induced (Andreev-Lifshitz) supersolid phase. • A 3He concentration of 0.1% is shown to suppress the flow anomalies, suggesting a quantum nature of the superflow.

50. „There is no end to this wonderful world of experimental discovery and mental constructions of reality as new facts become known. That is why physicists have more fun than most people“ Miklos Gyulassy, 2004 end of lecture 8