Halo World:
The story according to Faddeev,Efimov and Fano
Indranil Mazumdar
Dept. of Nuclear & Atomic Physics,
Tata Institute of Fundamental Research,
Mumbai 400 005
Bose Institute, Kolkata
26th August, 2011
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 nuclei
Fano resonances of Efimov states
Probing 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. 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)
Known nuclei
terra incognita
R = ROA1/3
Stable Nuclei
R = R0 A1/3
Shell Structure
Magic Numbers
208Pb
11Li
Advent of Radioactive Ion Beams
Interaction cross section measurements
I /IO = e-srt
sI = p[RI(P) + RI(T)]2
R = R0 A1/3
Shell Structure
Magic Numbers
208Pb
11Li
Pygmy Resonance
PDR in 68Ni
O. Wieland et. al.
PRL 102 , 2009
“The neutron halo of extremely neutron rich nuclei”
Europhys.Lett. 4, 409 (1987)
P.G.Hansen, B.Jonson
Exotic Structure of 2-n Halo Nuclei
11Li
Z=3
N=8
Radius ~3.2 fm
S2n = 369.15 (0.65) keV
Accepted lifetime:
Jp = 3/28.80 (0.14) ms
S2n = 12.2 MeV for 18O
Typical experimental momentum
distribution of halo nuclei from
fragmentation reaction
Kobayashi et al., PRL 60, 2599 (1988)
TRIUMF, Canada
RIBF, RIKEN
JAEA, Tokai
HIMAC, Chiba
CYRIC, Tohoku
RCNP, Osaka
GSI, Darmstadt
SPIRAL, Ganil
HIRF, Lanzhou
CIAE, Beijing
FRIB, MSU
ATLAS, ANL
HRIBF, Oak Ridge
FLNR-JINR, Dubna
Vecc, Kolkata
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)
Talmi & Unna, PRL 4, 496 (1960) 11Be
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
Pieper, Wiringa
•
Cluster model
• Three-body model ( for 2n halo nuclei )
• RMF model
• EFT Braaten & Hammer, Phys. Rep. 428 (2006)
He,
Sn = 820 keV
Sn = 504 keV
Audi & Wapstra (2003)
6,7Li, 8,9,10 Be
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)
Ludwig Dmitrievich Faddeev
A.N. Mitra, Nucl. Phys. 32, 529 (1962)
LD Faddeev
AN Mitra
NN Interaction & Nuclear Many-Body Problem
Nov. 17 – 26, 2010, TIFR
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
b-decay of 11Li
The rms radius rmatter calculated is ~ 3.6 fm
<r2>matter = Ac/A<r2>core + 1/A<r2>
r2 = r2nn + r2nc
Dasgupta, Mazumdar, Bhasin
PRC50, R550
Fedorov et al (1993)
Garrido et al (2002) (3.2 fm)
Dasgupta, Mazumdar, Bhasin, PRC 50, R550 Data:
Ieki et al,
PRL 70 ,1993
2-particle correlations:
Hagino et al , PRC, 2009
11Li
Vs
6He
11Li
6He
11Li
is different from 6He
•Strong influence of virtual s-state n-core interaction in 11Li
Efimov effect:
To
Efimov Physics
“ From questionable to
pathological to exotic to
a hot topic …”
Nature Physics 5, 533 (2009)
Vitaly Efimov
Univ. of Washington, Seattle
2010: The 40th year of a remarkable discovery
Belyaev, Faddeev, Efimovs
Nov. 2010, TIFR
Efimov, 1990
Ferlaino & Grimm 2010
V. Efimov:
Theoretical searches in Atomic Systems
Sov. J. Nucl. Phys 12, 589 (1971)
Phys. Lett. 33B (1970)
Nucl. Phys A 210 (1973)
Comments Nucl. Part. Phys.19 (1990)
T.K. Lim et al. PRL38 (1977)
Amado & Noble:
The case of
He trimer
Cornelius & Glockle, J. Chem Phys. 85 (1986)
T. Gonzalez-Lezana et al. PRL 82 (1999),
Phys. Lett. 33B (1971)
Phys. Rev. D5 (1972)
Fonseca et al.
Nucl. PhysA320, (1979)
Adhikari & Fonseca
Diffraction experiments with transmission gratings
Carnal & Mlynek, PRL 66 (1991)
Hegerfeldt & Kohler, PRL 84, (2000)
Phys. Rev D24 (1981)
Three-body recombination in ultra cold atoms
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
Can we find Efimov Effect in the atomic nucleus?
Unlike cold atom experiments we have no control over the
scattering lengths.
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)
Conditions for occurrence of Efimov states
in 2-n halo nuclei.
Fedorov, Jensen, Riisager
PRL 73 (1994)
P. Descouvement
PRC 52 (1995), Phys. Lett. B331 (1994)
tn-1(p)F(p) ≡ f(p) and tc-1(p)G(p) ≡ c(p)
Where
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.
tn-1(p) = mn-1 – [ br (br + √p2/2a + e3)2 ]-1
tc-1(p) = mc-1 – 2a[ 1+ √2a(p2/4c + e3) ]-2
where mn = p2ln/b12 and mc = p2lc/2ab13
are the dimensionless strength parameters.
Variables p and q in the final integral equation
are also now dimensionless,
p/b1  p & q/b1  q
and
-mE/b13 = e3, br = b/b1
Factors tn-1 and tc-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 e3 approaches extremely small
value.
Mazumdar and Bhasin, PRC 56, R5
First Evidence for low lying s-wave strength in 13Be
Fragmentation of 18O, virtual state with scattering length < 10 fm
Thoennessen, Yokoyama, Hansen
Phys. Rev. C 63, 014308
Mazumdar, Arora Bhasin
Phys. Rev. C 61, 051303(R)
•
The feature observed can be attributed to the singularity in the
two body propagator [LC-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 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 ao(p)p0  -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-if and q  qe-if
The unitary requirement is the Im(f-1k) = -k
Balslev & Combes (1971)
Matsui (1980)
Volkov et al.
n-18C Energy e3(0)
(keV)
(MeV)
e3(1)
(keV)
60
100
140
180
220
240
250
300
79.5 66.95
116.6 101.4
152.0 137.5
186.6 ----221.0 ----238.1 ---------------------
3.00
3.10
3.18
3.25
3.32
3.35
3.37
3.44
e3(2)
(keV)
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 for
n-18C BE of 250 keV
s = so[(q + e)2/(1+e2)]
Mazumdar, Rau, Bhasin
Phys. Rev. Lett. 97 (2006)
an ancient pond
a frog jumps in
a deep resonance
The resonance due to the
second excited Efimov state for
n-18C BE 150 keV. The profile is
fitted by same value of q as for the
250 keV curve.
Comparison between He and 20C as three body
Systems in atoms and nuclei
Discussion
•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.
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
38Mg
& 32Ne
S2n = 2570 keV n + core (37Mg) 250 keV (bound)
S2n = 1970 keV n + core (31Ne) 330 keV (bound)
(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 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)
eo
Equal
Heavy Core
(keV)
250
(keV)
455
(keV)
4400
300
546
4470
350
637
4550
Ground states for the two cases
•Equal mass case strikingly different from
unequal ( heavy core ) case.
•Evolution of Efimov states in heavy core
of 100 fully consistent with 20C results.
Mazumdar
Few Body Systems, 2009
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.
n-Core Energy e2
keV
e3(0)
keV
40
4020
60
e3(1)
keV
32Ne
38Mg
102A
e3(2)
keV
e3(0)
keV
e3(1)
keV
e3(2)
keV
e3(0)
keV
e3(1)
keV
e3(2)
keV
53.6
44.4
3550
61.3
49.9
3420
61.5
50
4080
70.4
61.7
3610
80.8
67.1
3480
81.0
67.2
80
4130
86.9
(78.4)
3670
99.2
84.16
3530
99.8
84.3
100
4170
103.1
3710
117
101.4
3570
117.5
101.5
120
4220
(119.3)
3750
134.5
(118.9)
3620
135
(118.9)
140
4259
3790
151.6
3650
152.5
180
4345
3860
185.6
3730
186.5
250
4460
3980
3852
300
4530
4040
3910
350
4590
4120
3980
38Mg
Heavy Core
4000
4000
sel(b)
Core Mass
= 100
e 2 = 250 keV
e 2 = 250 keV
e 2 = 250 keV
3500
32Ne
4000
3500
Core Mass = 36
3500
3000
3000
3000
2500
2500
2500
2000
2000
2000
1500
1500
1500
1000
1000
1000
500
500
500
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
3000
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
5000
e 2 = 150 keV
e 2 = 150 keV
Core Mass = 36
Core Mass = 100
Core Mass = 30
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
4000
e 2 = 150 keV
3500
Core Mass = 30
2500
4000
3000
2000
2500
3000
1500
2000
2000
1500
1000
1000
1000
500
500
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Mazumdar, Bhasin, Rau
Phys. Lett. B (In Press)
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Ei (keV)
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
A possible experimental proposal to search for Efimov State
in 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 g-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 beam
Epilogue
“ 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
Er
(T)
•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.
Er
(Ex)
G
(T)
0.038 0.03(0.04) 0.056
1.064 1.02(0.07) 0.050
2.042 2.07(0.12) 0.500
Data from
Gornov et al. PRL81 (1998)
b-decay to two channels studied:
11Li
to high lying excited state of 11Be
11Li
to 9Li + deuteron channel
18.3 MeV, bound (9Li+p+n) system
Gamow-Teller b-decay strength calculated
Branching ratio (1.3X10-4) calculated
Mukha et al (1997), Borge et al (1997)
Kumar & Bhasin
PRC65, (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
L - h ( p ) 
= ( p  2 d 
-1
-1 c
c
2
2
3
- d  / a ) h( p , ; )
2
2
2
2
2
2
3

-1