Indranil Mazumdar Dept. of Nuclear & Atomic Physics, Tata Institute of Fundamental Research, Mumbai 400 005 Bose Institute, Kolkata 26 th August, 2011.

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

Halo Nuclei:Exotic Structures and exotic effects

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

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)

terra incognita

Stable Nuclei

Known nuclei

R = ROA1/3

R = R0 A1/3

Shell Structure

Magic Numbers

11Li

208Pb

Advent of Radioactive Ion BeamsInteraction cross section measurements

et

I = [RI(P) + RI(T)]2

R = R0 A1/3

Shell Structure

Magic Numbers

11Li

208Pb

“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

Exotic Structure of 2-n Halo Nuclei

11LiZ=3N=8

Radius ~3.2 fm

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-

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)

Neutron skin

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)

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)

NN Interaction & Nuclear Many-Body ProblemNov. 17 – 26, 2010, TIFR

AN Mitra

LD Faddeev

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

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

<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

Dasgupta, Mazumdar, Bhasin, PRC 50, R550 Data:Ieki et al,PRL 70 ,1993

2-particle correlations:Hagino et al , PRC, 2009

11LiVs6He

11Li 6He

11Li is different from 6He

•Strong influence of virtual s-state n-core interaction in 11Li

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

Belyaev, Faddeev, EfimovsNov. 2010, TIFR

Efimov, 1990Ferlaino & Grimm 2010

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)

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

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

T. Kraemer et al. Nature 440, 315

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

Unlike cold atom experiments we have no control over the scattering lengths.

Can we find Efimov Effect in the atomic nucleus?

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.

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.

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

Mazumdar, Arora BhasinPhys. Rev. C 61, 051303(R)

• 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.

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.

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

Ugo Fano(1912 – 2001)

Third highest in citation impact of all papers published in the entire Physical Review series.

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

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.

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

•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

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)

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.

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

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

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

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.

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

THANK YOU

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)

Kumar & BhasinPRC65, (2002)

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

• 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