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![Page 1: Indranil Mazumdar Dept. of Nuclear & Atomic Physics, Tata Institute of Fundamental Research, Mumbai 400 005 Bose Institute, Kolkata 26 th August, 2011.](https://reader035.fdocuments.net/reader035/viewer/2022062423/56649e445503460f94b388e2/html5/thumbnails/1.jpg)
Indranil Mazumdar
Dept. of Nuclear & Atomic Physics,Tata Institute of Fundamental Research,
Mumbai 400 005
Bose Institute, Kolkata26th August, 2011
Halo World:The story according to Faddeev,Efimov and Fano
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Halo Nuclei:Exotic Structures and exotic effects
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Plan of the talk
Introduction to Nuclear Halos Three-body model of 2-n Halo nucleus
probing the structural properties of 11Li
Efimov effect in 2-n halo nucleiFano resonances of Efimov statesProbing few other candidates: The Experimental Angle
Summary and future scope
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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. RichardPhys. 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)
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terra incognita
Stable Nuclei
Known nuclei
R = ROA1/3
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R = R0 A1/3
Shell Structure
Magic Numbers
11Li
208Pb
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Advent of Radioactive Ion BeamsInteraction cross section measurements
et
I = [RI(P) + RI(T)]2
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R = R0 A1/3
Shell Structure
Magic Numbers
11Li
208Pb
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“The neutron halo of extremely neutron rich nuclei”Europhys.Lett. 4, 409 (1987)
P.G.Hansen, B.Jonson
Pygmy ResonancePDR in 68Ni
O. Wieland et. al. PRL 102 , 2009
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Exotic Structure of 2-n Halo Nuclei
11LiZ=3N=8
Radius ~3.2 fm
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Typical experimental momentumdistribution of halo nuclei from fragmentation reaction
S2n = 369.15 (0.65) keV
S2n = 12.2 MeV for 18O
Kobayashi et al., PRL 60, 2599 (1988)
Accepted lifetime:8.80 (0.14) ms J = 3/2-
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RIBF, RIKEN JAEA, Tokai HIMAC, Chiba CYRIC, Tohoku RCNP, Osaka HIRF, LanzhouCIAE, Beijing
Vecc, Kolkata
FRIB, MSUATLAS, ANLHRIBF, Oak Ridge
TRIUMF, Canada
GSI, DarmstadtSPIRAL, Ganil
FLNR-JINR, Dubna
Production Mechanisms
•ISOL
•In-Flight projectile fragmentation
Courtesy: V. Oberacker, Vanderbilt Univ. H. Sakurai, NIM-B (2008)
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Neutron skin
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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, 6,7Li, 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) 11Be Sn = 820 keV Sn = 504 keV Audi & Wapstra (2003)
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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)
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NN Interaction & Nuclear Many-Body ProblemNov. 17 – 26, 2010, TIFR
AN Mitra
LD Faddeev
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Dasgupta, Mazumdar, Bhasin, Phys. Rev C50,550
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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 11Li
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<r2>matter = Ac/A<r2>core + 1/A<2>2 = r2
nn + r2nc
Fedorov et al (1993)Garrido et al (2002) (3.2 fm)
The rms radius rmatter calculated is ~ 3.6 fmDasgupta, Mazumdar, BhasinPRC50, R550
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Dasgupta, Mazumdar, Bhasin, PRC 50, R550 Data:Ieki et al,PRL 70 ,1993
2-particle correlations:Hagino et al , PRC, 2009
11LiVs6He
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11Li 6He
11Li is different from 6He
•Strong influence of virtual s-state n-core interaction in 11Li
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Efimov effect:
“ From questionable to pathological to exotic to a hot topic …”
Nature Physics 5, 533 (2009)
Vitaly EfimovUniv. of Washington, Seattle
ToEfimov Physics
2010: The 40th year of a remarkable discovery
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Belyaev, Faddeev, EfimovsNov. 2010, TIFR
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Efimov, 1990Ferlaino & Grimm 2010
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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 & FonsecaPhys. 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 ofHe trimer
L.H. Thomas,Phys.Rev.47,903(1935)
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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
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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
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T. Kraemer et al. Nature 440, 315
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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 (39K) with resonantly tunable two-body interaction.
• Atom-dimer resonance and loss mechanism
• Large values of a up to 25,000 ao reached. • First two states of an Efimov spectrum seen
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Unlike cold atom experiments we have no control over the scattering lengths.
Can we find Efimov Effect in the atomic nucleus?
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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 & JensenPRL 71 (1993)
Fedorov, Jensen, RiisagerPRL 73 (1994)
P. DescouvementPRC 52 (1995), Phys. Lett. B331 (1994)
Conditions for occurrence of Efimov states in 2-n halo nuclei.
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n-1(p)F(p) ≡ (p) and c
-1(p)G(p) ≡ (p)Where
n-1(p) = n
-1 – [ r (r + √p2/2a + 3)2 ]-1
c-1(p) = c
-1 – 2a[ 1+ √2a(p2/4c + 3) ]-2
where n = 2n/12 and c = 2c/2a1
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 theequations in terms of thespectator functions F(p) and G(p) remains same. But for the sensitive computationaldetails of the Efimov effect werecast the equations in dimensionless quantities.
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Mazumdar and Bhasin, PRC 56, R5
First Evidence for low lying s-wave strength in 13BeFragmentation of 18O, virtual state with scattering length < 10 fmThoennessen, Yokoyama, HansenPhys. Rev. C 63, 014308
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Mazumdar, Arora BhasinPhys. Rev. C 61, 051303(R)
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• The feature observed can be attributed to the singularity in the two body propagator [C
-1 – hc(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.
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For k 0, the singularity in the two body cutDoes 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 ao(p)p0 -a, the n-19C scattering length.
For non-zero incident energies the singularity in the two body propagator is tackled by the CSM.
P p1e-i and q qe-i
The unitary requirement is the Im(f-1k) = -k
Balslev & Combes (1971)Matsui (1980)Volkov et al.
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n-18C Energy 3(0) 3(1) 3(2)(keV) (MeV) (keV) (keV)
60 3.00 79.5 66.95100 3.10 116.6 101.4 140 3.18 152.0 137.5180 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
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Ugo Fano(1912 – 2001)
Third highest in citation impact of all papers published in the entire Physical Review series.
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Fitting the Fano profile to the N-19C elastic cross section forn-18C BE of 250 keV
Mazumdar, Rau, BhasinPhys. Rev. Lett. 97 (2006)
= o[(q + )2/(1+2)]
an ancient ponda frog jumps ina deep resonance
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The resonance due to the second excited Efimov state forn-18C BE 150 keV. The profile is fitted by same value of q as for the250 keV curve.
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Comparison between He and 20C as three body Systems in atoms and nuclei
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•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-18C 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 20C clearly underlines the sensitive structure of the three-body scattering amplitude derived from the binary interactions.
Discussion
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The calculation have been extended to
1) Two hypothetical cases: very heavy core of mass A = 100 (+ 2n)
three equal masses m1=m2=m3
2) Two realistic cases of 38Mg & 32Ne
38Mg S2n = 2570 keV n + core (37Mg) 250 keV (bound)32Ne S2n = 1970 keV n + core (31Ne) 330 keV (bound)
38Mg (S2n) Audi & Wapstra (2003)
37Mg & 31Ne (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 foundat 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)
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o Equal Heavy Core
(keV) (keV) (keV) 250 455 4400
300 546 4470
350 637 4550
Ground states for the two cases
MazumdarFew 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 20C results.
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n-Core Energy 2
keV3(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), 38Mg (column 5, 6, 7), and 32Ne (columns 8, 9, 10) for different two body input parameters.
38Mg32Ne102A
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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00
500
1000
1500
2000
2500
3000
3500
4000
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00
500
1000
1500
2000
2500
3000
3500
4000
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00
500
1000
1500
2000
2500
3000
3500
4000
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00
500
1000
1500
2000
2500
3000
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00
1000
2000
3000
4000
5000
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00
500
1000
1500
2000
2500
3000
3500
4000
2 = 250 keV
Core Mass = 100
2 = 250 keV
Core Mass = 36
2 = 250 keV
Core Mass = 30
2 = 150 keV
Core Mass = 100
el(b
)
2 = 150 keV
Core Mass = 36
Ei (keV)
2 = 150 keV
Core Mass = 30
Mazumdar, Bhasin, RauPhys. Lett. B (In Press)
Heavy Core 38Mg 32Ne
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A possible experimental proposal to search for Efimov Statein 2-neutron halo nuclei.
•Production of 20C secondary beam with reasonable flux
•Acceleration and Breakup of 20C 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:
19C beam on deuteron target:Neutron stripping reaction
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Summary
A three body model with s-state interactions account for most of the gross features of 11Li in a reasonable way.
Inclusion of p-state in the n-9Li 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-12Be predicts the ground state and excited states of 14Be.
19B, 22C and 20C are investigated and it is shown that Borromean type nuclei are much less vulnerable to respond to Efimov effect
20C is a promising candidate for Efimov states at energies below the n-(nc) breakup threshold.
The bound Efimov states in 20C move into the continuum and reappear as Resonances with increasing strength of the binary interaction.
Asymmetric resonances in elastic n+19C 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.
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Present Activities:
Resonant states above the three body breakup threshold in 20C.
Structure calculations for 12Be
Fano resonances of Efimov states in 16C, 19B, 22C and analytical derivation of the Fano index q.
Role of Efimov states in Bose-Einstein condensation.
Studying the proton halo (17Ne) nucleus. three charged particle, Belyaev, Shlyk, NPA 790 (2007)
Reanalyze profiles of GDR on ground states for its asymmetry.
Planning for possible experiments with 20C beamEpilogue
“ the richness of undestanding reveals even greater richness of ignorance” D.H. Wilkinson
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THANK YOU
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Kumar & Bhasin,Phys. Rev. C65 (2002)
Incorporation of both s & p waves in n-9Li 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 thethree body wave function has also been analyzed through
the three-dimensional plots.
-decay to two channels studied:
11Li to high lying excited state of 11Be 11Li to 9Li + deuteron channel
Er Er (T) (x)(T)
()()()
Data fromGornov et al. PRL81 (1998)
18.3 MeV, bound (9Li+p+n) systemGamow-Teller -decay strength calculated
Branching ratio (1.3X10-4) calculated
Mukha et al (1997), Borge et al (1997)
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Kumar & BhasinPRC65, (2002)
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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
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• In the present model, the singularity in the present model appears in the two body propagator
12
322
222
23
2
11
);,()/2(
)(
phaddp
phcc