1. Cluster Magic Numbers

2. Recent highly accurate diffusion Monte Carlo (T=0) calculation rules out existence of magic numbers due to stabilities: R. Guardiola,O. Kornilov, J. Navarro and J. P. Toennies, J. Chem Phys, 2006 Cluster Number Size N

3. He2+He2+ from J. P. Toennies He N

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

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

6. 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. To explain Magic numbers recall that clusters are formed in early „hot“ stages of the expansion from J. P. Toennies

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

8. 2006 Thermalization via evaporation (DFT)

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

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

11. one-particle states

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

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

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

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

16. In quest of 4 He 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 ?  Vacancy diffusivity and solid 4 He Poisson ratio The Geyser effect in solid 4 He vacuum expansion  Bernoulli flow of a nominal 4 He solid  Suppression of flow anomalies by 1% 3 He  4 He vacuum expansion from low -T sources 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. no trend ? Kim & Chan 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 4 He Firenze 2005 - 6

22. continuity Bernoulli Firenze 2005 - 7

23. 4 He phase diagram Firenze 2005 - 8

24. The Geyser effect

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

26. Period versus P 0 at constant temperature Bernoulli  Firenze 2005 - 11

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

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

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

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

33.  V a = V* - V a V a = 35.15 Å 3 (atomic volume) V*  0.45V a (vacancy isobaric formation volume) A0A0 A s/l L Virtual volume to be filled by vacancies in the time L/u 0 u0u0 The vacancy mechanism Firenze 2005 - 16

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

35. Data on vacancy diffusivity and concentration in 4 He

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

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

38. - 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 X 0 V a u v /u 0 and is of the order of a few percent (as seen in experiment) fitting  reduced form:

39. Theory vs. experiment D v = 1.3·10 -5 cm 2 /s  v = 5.4·10 10 s/g u v = 2.0·10 -3 cm/s u s = 2u v  s = 60 s  v = 13 s  * = 10.7 s  0 = 82 s P 0 = 31 bar T 0 = 1.74 K best fit with  = 4  = 1.214

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

41. Period  0 vs. diffusivity finite L  approximate solution by Green’s function method X c = 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 3 He on the anomalies from R. Richardson et al Firenze 2005 - 27

46. small amounts of 3 He remove the anomaly!

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

49. CONCLUSIONS 1.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 4 He 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 4 He. 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. 3.A 3 He concentration of 0.1% is shown to suppress the flow anomalies, suggesting a quantum nature of the superflow.

50. Miklos Gyulassy, 2004 „ 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“ end of lecture 8