1. Indranil Mazumdar Dept. of Nuclear & Atomic Physics, Tata Institute of Fundamental Research, Mumbai 400 005 Bose Institute, Kolkata 26 th August, 2011 Halo World: The story according to Faddeev,Efimov and Fano

2. Halo Nuclei: Exotic Structures and exotic effects

3. Plan of the talk Introduction to Nuclear Halos Three-body model of 2-n Halo nucleus probing the structural properties of 11 Li Efimov effect in 2-n halo nuclei Fano resonances of Efimov states Probing few other candidates: The Experimental Angle Summary and future scope

4. Collaborators V.S. Bhasin Delhi Univ. V. Arora Delhi Univ. A.R.P. Rau Louisiana State Univ. Phys. Rev. Lett. 99, 269202 Nucl. Phys. A790, 257 Phys. Rev. Lett. 97, 062503 Phys. Rev. C69, 061301(R) Phys. Rev. C61, 051303(R) Phys. Rev. C56, R5 Phys. Rev. C50, R550 Few Body Systems, 2009 Pramana, 2010 Phys. Lett. B (In press) Phys. Rep 212 (1992) J.M. Richard Phys. Rep. 231 (1993) 151(Zhukov et al.) Phys. Rep. 347 (2001) 373 (Nielsen et al.) Prog. Part. Nucl. Phys. 47,517 (2001) (Brown) Rev. Mod. Phys. 76,(2004) 215(Jensen et al.) Phys. Rep. 428, (2006) 259(Braaten & Hammer) Ann Rev. Nucl. Part. Sci. 45, 591(Hansen et al.) Rev. Mod. Phys. 66 (1105)(K. Riisager)

5. terra incognita Stable Nuclei Known nuclei R = R O A 1/3

6. R = R 0 A 1/3 Shell Structure Magic Numbers 11 Li 208 Pb

7. Advent of Radioactive Ion Beams Interaction cross section measurements    e  t  I =  [R I ( P ) + R I ( T )] 2

8. R = R 0 A 1/3 Shell Structure Magic Numbers 11 Li 208 Pb

9. “ The neutron halo of extremely neutron rich nuclei ” Europhys.Lett. 4, 409 (1987) P.G.Hansen, B.Jonson Pygmy Resonance PDR in 68 Ni O. Wieland et. al. PRL 102, 2009

10. Exotic Structure of 2-n Halo Nuclei 11 Li Z=3 N=8 Radius ~3.2 fm

11. Typical experimental momentum distribution of halo nuclei from fragmentation reaction S 2n = 369.15 (0.65) keV S 2n = 12.2 MeV for 18 O Kobayashi et al., PRL 60, 2599 (1988) Accepted lifetime: 8.80 (0.14) ms J  = 3/2 -

12. RIBF, RIKEN JAEA, Tokai HIMAC, Chiba CYRIC, Tohoku RCNP, Osaka HIRF, Lanzhou CIAE, Beijing Vecc, Kolkata FRIB, MSU ATLAS, ANL HRIBF, Oak Ridge TRIUMF, Canada GSI, Darmstadt SPIRAL, Ganil FLNR-JINR, Dubna Production Mechanisms ISOL In-Flight projectile fragmentation Courtesy: V. Oberacker, Vanderbilt Univ. H. Sakurai, NIM-B (2008)

13. Neutron skin

14. Theoretical Models Shell Model Bertsch et al. (1990) PRC 41,42, Kuo et al. PRL 78,2708 (1997) 2 frequency shell model Brown (Prog. Part. Nucl. Physics 47 (2001) Ab initio no-core full configuration calculation of light nuclei Navratil, Vary, Barrett PRL84(2000), PRL87(2001) Quantum Monte Carlo Calculations of Light Nuclei: 4,6,8 He, 6,7 Li, 8,9,10 Be Pieper, Wiringa Cluster model Three-body model ( for 2n halo nuclei ) RMF model EFT Braaten & Hammer, Phys. Rep. 428 (2006) Talmi & Unna, PRL 4, 496 (1960) 11 Be S n = 820 keV S n = 504 keV Audi & Wapstra (2003)

15. Ludwig Dmitrievich Faddeev 2010: The Golden Jubilee year of Faddeev Equations. L.D. Faddeev, Zh. Eksperim, I Teor. Fiz. 39, 1459 (1960) S. Weinberg, Phys. Rev. 133, B232 (1964) C. Lovelace: Phys. Rev. 135, No. 5B B1225 (1964) J. G. Taylor, Nuovo Cimento (1964) L. Rosenberg, Phys. Rev 135, B715 (1964) A.N. Mitra, Nucl. Phys. 32, 529 (1962)

16. NN Interaction & Nuclear Many-Body Problem Nov. 17 – 26, 2010, TIFR AN Mitra LD Faddeev

17. Dasgupta, Mazumdar, Bhasin, Phys. Rev C50,550

21. We Calculate 2-n separation energy Momentum distribution of n & core Root mean square radius Inclusion of p-state in n-core interaction  -decay of 11 Li

22. matter = A c /A core + 1/A  2 = r 2 nn + r 2 nc Fedorov et al (1993) Garrido et al (2002) (3.2 fm) The rms radius r matter calculated is ~ 3.6 fm Dasgupta, Mazumdar, Bhasin PRC50, R550

23. Dasgupta, Mazumdar, Bhasin, PRC 50, R550 Data: Ieki et al, PRL 70,1993 2-particle correlations: Hagino et al, PRC, 2009 11 Li Vs 6 He

24. 11 Li 6 He 11 Li is different from 6 He Strong influence of virtual s-state n-core interaction in 11 Li

26. Efimov effect: “ From questionable to pathological to exotic to a hot topic …” Nature Physics 5, 533 (2009) Vitaly Efimov Univ. of Washington, Seattle To Efimov Physics 2010: The 40 th year of a remarkable discovery

27. Belyaev, Faddeev, Efimovs Nov. 2010, TIFR

29. Efimov, 1990 Ferlaino & Grimm 2010

30. V. Efimov: Sov. J. Nucl. Phys 12, 589 (1971) Phys. Lett. 33B (1970) Nucl. Phys A 210 (1973) Comments Nucl. Part. Phys.19 (1990) Amado & Noble : Phys. Lett. 33B (1971) Phys. Rev. D5 (1972) Fonseca et al. Nucl. PhysA320, (1979 ) Adhikari & Fonseca Phys. Rev D24 (1981) Theoretical searches in Atomic Systems T.K. Lim et al. PRL38 (1977) Cornelius & Glockle, J. Chem Phys. 85 (1986) T. Gonzalez-Lezana et al. PRL 82 (1999), Diffraction experiments with transmission gratings Carnal & Mlynek, PRL 66 (1991) Hegerfeldt & Kohler, PRL 84, (2000) Three-body recombination in ultra cold atoms The case of He trimer L.H. Thomas, Phys.Rev.47,903(1935)

31. First Observation of Efimov States Letter Nature 440, 315-318 (16 March 2006) | Evidence for Efimov quantum states in an ultracold gas of caesium atoms T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H.-C. Nägerl and R. Grimm

32. Magnetic tuning of the two-body interaction For Cs atoms in their energetically lowest state the s-wave scattering length a varies strongly with the magnetic field. Trap set-ups and preparation of the Cs gases All measurements were performed with trapped thermal samples of caesium atoms at temperatures T ranging from 10 to 250 nK. In set-up A they first produced an essentially pure Bose–Einstein condensate with up to 250,000 atoms in a far-detuned crossed optical dipole trap generated by two 1,060-nm Yb-doped fibre laser beams In set-up B they used an optical surface trap in which they prepared a thermal sample of 10,000 atoms at T 250 nK via forced evaporation at a density of n0 = 1.0 1012 cm-3. The dipole trap was formed by a repulsive evanescent laser wave on top of a horizontal glass prism in combination with a single horizontally confining 1,060-nm laser beam propagating along the vertical direction

33. T. Kraemer et al. Nature 440, 315

34. Observation of an Efimov spectrum in an atomic system. M. Zaccanti et al. Nature Physics 5, 586 (2009) System composed of ultra-cold potassium atoms ( 39 K) with resonantly tunable two-body interaction. Atom-dimer resonance and loss mechanism Large values of a up to 25,000 a o reached. First two states of an Efimov spectrum seen

35. Unlike cold atom experiments we have no control over the scattering lengths. Can we find Efimov Effect in the atomic nucleus?

36. The discovery of 2-neutron halo nuclei, characterized by very low separation energy and large spatial extension are ideally suited for studying Efimov effect in atomic nuclei. Fedorov & Jensen PRL 71 (1993) Fedorov, Jensen, Riisager PRL 73 (1994) P. Descouvement PRC 52 (1995), Phys. Lett. B331 (1994) Conditions for occurrence of Efimov states in 2-n halo nuclei.

37.  n -1 (p)F(p) ≡  (p) and  c -1 (p)G(p) ≡  (p) Where  n -1 (p) =  n -1 – [  r (  r + √p 2 /2a +  3 ) 2 ] -1  c -1 (p) =  c -1 – 2a[ 1+ √2a(p 2 /4c +  3) ] -2 where  n =  2 n /  1 2 and  c =  2 c /2a  1 3 are the dimensionless strength parameters. Variables p and q in the final integral equation are also now dimensionless, p/  1  p & q/  1  q and -mE/  1 3 =  3,  r =  /  1 Factors  n -1 and  c -1 appear on the left hand side of the spectator functions F(p) and G(p) and are quite sensitive. They blow up as p  0 and  3 approaches extremely small value. The basic structure of the equations in terms of the spectator functions F(p) and G(p) remains same. But for the sensitive computational details of the Efimov effect we recast the equations in dimensionless quantities.

38. Mazumdar and Bhasin, PRC 56, R5 First Evidence for low lying s-wave strength in 13 Be Fragmentation of 18 O, virtual state with scattering length < 10 fm Thoennessen, Yokoyama, Hansen Phys. Rev. C 63, 014308

39. Mazumdar, Arora Bhasin Phys. Rev. C 61, 051303(R)

41. The feature observed can be attributed to the singularity in the two body propagator [  C -1 – h c (p)] -1. There is a subtle interplay between the two and three body energies. The effect of this singularity on the behaviour of the scattering amplitude has to be studied.

42. For k  0, the singularity in the two body cut Does not cause any problem. The amplitude has only real part. The off-shell amplitude is computed By inverting the resultant matrix, which in the limit a o (p) p  0  -a, the n- 19 C scattering length. For non-zero incident energies the singularity in the two body propagator is tackled by the CSM. P  p 1 e -i  and q  qe -i  The unitary requirement is the Im(f -1 k ) = -k Balslev & Combes (1971) Matsui (1980) Volkov et al.

43. n- 18 C Energy  3 (0)  3 (1)  3 (2) (keV) (MeV) (keV) (keV) 60 3.00 79.5 66.95 100 3.10 116.6 101.4 140 3.18 152.0 137.5 180 3.25 186.6 ----- 220 3.32 221.0 ----- 240 3.35 238.1 ----- 250 3.37 ----- ----- 300 3.44 ----- ----- Arora, Mazumdar,Bhasin, PRC 69, 061301

44. Ugo Fano (1912 – 2001) Third highest in citation impact of all papers published in the entire Physical Review series.

45. Fitting the Fano profile to the N- 19 C elastic cross section for n- 18 C BE of 250 keV Mazumdar, Rau, Bhasin Phys. Rev. Lett. 97 (2006)  =  o [(q +  ) 2 /(1+  2 )] an ancient pond a frog jumps in a deep resonance

46. The resonance due to the second excited Efimov state for n- 18 C BE 150 keV. The profile is fitted by same value of q as for the 250 keV curve.

47. Comparison between He and 20 C as three body Systems in atoms and nuclei

48. We emphasize the cardinal role of channel coupling. There is also a definite role of mass ratios as observed numerically. However, channel coupling is an elegant and physically plausible scenario. Difference can arrive between zero range and realistic finite range potentials in non-Borromean cases. Note, that for n- 18 C binding energy of 200 keV, the scattering length is about 10 fm while the interaction range is about 1 fm. The extension of zero range to finer details of Efimov states in non-Borromean cases may not be valid. The discrepancy observed in the resonance vs virtual states in 20 C clearly underlines the sensitive structure of the three-body scattering amplitude derived from the binary interactions. Discussion

49. The calculation have been extended to 1)Two hypothetical cases: very heavy core of mass A = 100 (+ 2n) three equal masses m 1 =m 2 =m 3 2) Two realistic cases of 38 Mg & 32 Ne 38 Mg S 2n = 2570 keV n + core ( 37 Mg) 250 keV (bound) 32 Ne S 2n = 1970 keV n + core ( 31 Ne) 330 keV (bound) 38 Mg (S 2n ) Audi & Wapstra (2003) 37 Mg & 31 Ne (Sn) Sakurai et al., PRC 54 (1996), Jurado et al. PLB (2007), Nakamura et al. PRL (2009) We have reproduced the ground state energies and have found at least two Efimov states that vanish into the continuum with increasing n-core interaction. They again show up as asymmetric resonances at around 1.6 keV neutron incident energy in the scattering sector. Mazumdar, Bhasin, Rau Phys. Lett. B (In Press)

50.  o Equal Heavy Core (keV) (keV) (keV) 250 455 4400 300 546 4470 350 637 4550 Ground states for the two cases Mazumdar Few Body Systems, 2009 Equal mass case strikingly different from unequal ( heavy core ) case. Evolution of Efimov states in heavy core of 100 fully consistent with 20 C results.

51. n-Core Energy  2 keV  3 (0) keV  3 (1) keV  3 (2) keV  3 (0) keV  3 (1) keV  3 (2) keV  3 (0) keV  3 (1) keV  3 (2) keV 40 60 80 100 120 140 180 250 300 350 4020 4080 4130 4170 4220 4259 4345 4460 4530 4590 53.6 70.4 86.9 103.1 (119.3) 44.4 61.7 (78.4) 3550 3610 3670 3710 3750 3790 3860 3980 4040 4120 61.3 80.8 99.2 117 134.5 151.6 185.6 49.9 67.1 84.16 101.4 (118.9) 3420 3480 3530 3570 3620 3650 3730 3852 3910 3980 61.5 81.0 99.8 117.5 135 152.5 186.5 50 67.2 84.3 101.5 (118.9) TABLE: Ground and excited states for three cases studied, namely, mass 102 (columns 2, 3, 4), 38 Mg (column 5, 6, 7), and 32 Ne (columns 8, 9, 10) for different two body input parameters. 38 Mg 32 Ne 102 A

52. Mazumdar, Bhasin, Rau Phys. Lett. B (In Press) Heavy Core 38 Mg 32 Ne

53. A possible experimental proposal to search for Efimov State in 2-neutron halo nuclei. Production of 20 C secondary beam with reasonable flux Acceleration and Breakup of 20 C on heavy target Detection of the neutrons and the core in coincidence Measurement of  -rays as well The Arsenal: Neutron detectors array Gamma array Charged particle array Another experimental scenario: 19 C beam on deuteron target: Neutron stripping reaction

54. Summary  A three body model with s-state interactions account for most of the gross features of 11 Li in a reasonable way.  Inclusion of p-state in the n- 9 Li contributes marginally.  A virtual state of a few keV (2 to 4) energy corresponding to scattering length from -50 to -100 fm for the n- 12 Be predicts the ground state and excited states of 14 Be.  19 B, 22 C and 20 C are investigated and it is shown that Borromean type nuclei are much less vulnerable to respond to Efimov effect  20 C is a promising candidate for Efimov states at energies below the n-(nc) breakup threshold.  The bound Efimov states in 20 C move into the continuum and reappear as Resonances with increasing strength of the binary interaction.  Asymmetric resonances in elastic n+ 19 C scattering are attributed to Efimov states and are identified with the Fano profile. The conjunction of Efimov and Fano phenomena my lead to the experimental realization in nuclei.

55. Present Activities: Resonant states above the three body breakup threshold in 20 C. Structure calculations for 12 Be Fano resonances of Efimov states in 16 C, 19 B, 22 C and analytical derivation of the Fano index q. Role of Efimov states in Bose-Einstein condensation. Studying the proton halo ( 17 Ne) nucleus. three charged particle, Belyaev, Shlyk, NPA 790 (2007) Reanalyze profiles of GDR on ground states for its asymmetry. Planning for possible experiments with 20 C beam Epilogue “ the richness of undestanding reveals even greater richness of ignorance” D.H. Wilkinson

56. THANK YOU

57. Kumar & Bhasin, Phys. Rev. C65 (2002) Incorporation of both s & p waves in n- 9 Li potential Ground state energy and 3 excited states above the 3-body breakup threshold were predicted The resulting coupled integral equations for the spectator functions have been computed using the method of rotating the integral contour of the kernels in the complex plane. Dynamical content of the two body input potentials in the three body wave function has also been analyzed through the three-dimensional plots.  -decay to two channels studied: 11 Li to high lying excited state of 11 Be 11 Li to 9 Li + deuteron channel E r E r   x     Data from Gornov et al. PRL81 (1998) 18.3 MeV, bound ( 9 Li+p+n) system Gamow-Teller  -decay strength calculated Branching ratio (1.3X10 -4 ) calculated Mukha et al (1997), Borge et al (1997)

58. Kumar & Bhasin PRC65, (2002)

61. Appearance of Resonance in n- 19 C Scattering The equation for the off-shell scattering amplitude in n- 19 C ( bound state of n- 18 C) can be written as where a k (p) is the off-shell scattering amplitude, normalized such that

62. In the present model, the singularity in the present model appears in the two body propagator