Overlap integrals in threebody systems

1
Overlap integrals in three-body systems

• N.K. Timofeyuk1)
• in collaboration with I.J. Thompson2) and J.A.
  Tostevin1)
• 1)University of Surrey, UK
• 2)Livermore National Laboratory, USA

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• Non-standard behaviour of overlap integrals in
  three-body systems
• Energy dependence of spectroscopic factors in
  three-body systems
• Spectroscopic factors and asymptotic
  normalization coefficients in mirror three-body
  systems

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A class of the 2Ncore cluster systems
(A-2) 2N
A-2
A
(A-1) N
N
A-1
Nucleus A
N
Examples
4.522 MeV
16O2p
17Fp
3.673 MeV
10Be2n
3.922 MeV
11Ben
3.169 MeV
1.437 MeV
7Be2p
8Bp
1.300 MeV
18Ne
9C
12Be
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When one nucleon moves outside the range of
nuclear radius, the A-1 core tends to restore its
shape as it would be without this nucleon. If the
A-1 core is loosely bound, the probability to
find the valence nucleon outside the range of the
coreN interaction is large.
r
Core
N
c.m.
N
As the result, althought the distance r can be
large, the interaction between two valence
nucleons can cause abnormal behaviour of the
overlap integral.
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Three-body calculations for overlap integrals
?coreN coreNN ? 3-body
Schrodinger equation for coreNN system with
V(r) Vcore-N(rc - rN1) Vcore_N(rc -
rN2) VNN(rN1 - rN2) Antisymmetrization is
taken into account using Pauli projections
technique. Wave functions is expanded onto
hypersherical basis.
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Pseudo 12Be 10Benn No deformation and
excitation of the 10Be core n10Be interaction is
not zero only in s-wave Simple gaussian nn
interaction with scattering length of -20
fm. Hyperspherical expansion to Kmax 120
E(11Be) (MeV) 0.102 0.496 E(12Be) (MeV)
0.656 1.75 Sn(12Be) (MeV) 0.56
1.25 ?r2?½ (fm) 7.650
5.807 ?r2?½standard (fm) 6.983 5.423
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Effective local potential as a sum of two
Woods-Saxon potentials
E(11Be) E(12Be) Sn(12Be) V0 r0
a0 V1 r1 a1 MeV
MeV MeV MeV fm
fm MeV fm fm 0.102 0.656
0.56 53.5 1.20 0.70 4.8
0.989 1.85 0.496 1.75 1.25
55.9 1.07 0.82 1.5 1.124 3.1
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Realistic 12Be 10Benn model Deformation
and excitation of the 10Be core as in F.M.Nunes,
I.J.Thompson and J.A.Tostevin, Nucl.Phys. A703,
593 (2002) n10Be interaction in all partial
waves, 11Be(1/2) and 11Be(1/2-) energies are
reproduced GPT potential for NN interaction,
hyperspherical expansion to Kmax 34
SF ?r2?½ ?r2?½standard
½ 0.325 4.413 4.298 ½- 0.110
3.588 3.473 5/2 0.322 3.738
3.244 E(12Be) 3.641 MeV Eexp(12Be) 3.673
0.015 MeV
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12Be wave functions
n
y
10Be
x
n
Effective local n11Be potential
V0 r0 a0 V1
r1 a1 MeV
fm fm MeV fm fm
½ 50.0 1.33 0.75 6.9
1.17 1.40 1/2- 50.0 0.99 0.79
9.8 1.12 1.55
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One-nucleon knockout cross sections from 12Be at
90 MeV/Nucleon on the 9Be target leading to
11Be(1/2), calculated for different two-body
n11Be Woods-Saxon potentials.
2 WS potentials
standard WS
11
Knockout cross sections for 12Be projectile at 90
MeV/Nucleon on 9Be target leading to 11Be(1/2-)
calculated for different two-body n11Be
Woods-Saxon potentials.
3-body model
standard WS potential
12
One-nucleon knockout cross sections from 12Be at
90 MeV/Nucleon on 9Be target leading to
11Be(5/2) calculated for different two-body
n11Be Woods-Saxon potentials.
3-body model
standard WS potential
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12Be ? 11Be(1/2), l 0
14
12Be ? 11Be(1/2-), l 1
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• Parallel momentum distribution for one-nucleon
  knockout from 12Be to 11Be(1/2) calculated for
  rms radius of 4.57 fm for two cases
• WS potential with r0 1.35 fm and a 0.8 fm
• A sum of two WS potentials with
• r0 1.33 a 0.75 fm and r0 1.17 fm a 1.4 fm

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11Be(n,?)12Be
17
18Ne 16O p p Standard geometry for the 16O
p potential GPT Coulomb potential for pp
interaction Hyperspherical expansion to Kmax 50
0 0.94
0 0.07
16O2p
2 -0.90
0 -0.94
0 -1.14
4 -1.14
2 -1.23
4 -1.76
2 -2.63
2 -2.79
0 -3.58
0 -4.52
theory
experiment
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18Ne(0) wave functions
p
16O
y
x
p
first excited state
ground state
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18Ne(01)

18Ne(02)
17F(5/2) 17F(1/2) 17F(5/2)
17F(1/2) ?r2?½
(fm) 3.602 4.068
3.511 4.486
?r2?½standard (fm) 3.397
3.602 3.426 3.947
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Spectroscopic factors and Asymptotic
Normalization Coefficients in Mirror three-body
systems
A-2
A-2
A
A
p
n
A-1
A-1
p
n
?core n core n n ?
?core p core p p?
At large distances
Asymptotic Normalization Coefficient (ANC)
Whittaker function
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A simple analytical formula for the ratio of
mirror ANCs (N.K. Timofeyuk, R.C. Johnson and
A.M. Mukhamedzhanov, PRL 91, 0232501 (2003))
where Fl is the regular Coulomb wave function,
jl is the Bessel function
?p (2µep)1/2, ?n (2µen)1/2, RN
1.2(A-1)1/3
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Important assumption made to derive this
approximation is that internal wave functions in
mirror nuclei are the same. If we assume that
in internal region ?p/?n Sp/Sn R (Cp/Cn)2
Sp/Sn R0 or R/R0 Sp / Sn (this is
expected to be valid if Veff(p)Veff(n) )
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I(r2) ? dr1 ?2(r1) ?3(r1,r2) ?3(r1,r2)
?eff(r1)?eff(r2) where ?eff(r) is a solution of
the Shrödinger equation with E E3-E2 S (?
dr ?2(r) ?eff(r))2
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small SF
small SF
large SF
large SF
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small SF
large SF
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large SF
small SF
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Conclusions Due to the NN correlations, the
overlap ?core N core 2N ? can noticeably
differ from the two-body model wave functions
obtained with standard Woods-Saxon potentials
that are widely used to model these
overlaps. The three-body dynamics lead to
larger radii of the overlap integrals as compared
to the mean field values. Using bigger overlap
radii in knockout calculations would lead to
smaller spectroscopic factors extracted from
experimental data. Knowledge of three-body
dynamics could be important to predict energy
dependence of the (n,?) cross sections from
weakly bound nuclei at stellar energies.
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• Threshold effects are seen in spectroscopic
  factors behaviour. They are caused by the strong
  NN correlations and the recoil of the core, which
  lead to mismatch between the effective
  single-particle wave functions in the A and the
  A-1 wave function.
• Where several channels are possible, the split of
  the spectroscopic strength between these channels
  depends strongly on the binding energy of the
  two-body subsystem. Coupled-channels effects can
  strongly modify near-threshold behaviour of
  spectroscopic factors.
• Spectroscopic factors in mirror nuclei can
  differ.
• Ratio of mirror ANCs can differ from simple
  analytical predictions and is generally
  correlated with symmetry breaking in mirror
  spectroscopic factors but can also be caused by
  differences in effective local potentials in
  mirror nuclei.

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• Challenges for theory
• Three-body dynamics should be treated explicitly.
• Even when internal structure of the core is
  neglected, the model space, needed to describe
  the nucleon motion at large distances in coupled
  channel calculations, becomes huge.
• Antisymmetrization should be taken explicitly.
• To reproduce large distance behaviour within
  the models of the no-core shell model type, at
  least 30 additional majour shells are needed
  which do not influence much the total binding
  energy.
• A microscopic cluster model is available that
  combined three-body dynamics with
  antisymmetrization but it can use only simplest
  structure of the core wave function and requires
  huge model space to be extended for larger
  distances.

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• Challenges for experiment
• One-nucleon knockout experiments.
• Proper size of the overlap integral should be
  used in reaction theories used to determine
  spectroscopic factors. Parallel momentum
  distributions should be measured with good
  precision to be able to determine the size of the
  overlap integrals.
• 2) Transfer reactions
• These reactions could be used to get
  complementary information on the shape and size
  of the overlap integrals.
• For example, 12C(12Be,11Be)13C.
• However, more efforts should be aimed both from
  experimental and theoretical sides to make this
  information precise enough.

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Feynman diagramm aproach to asymptotics of the
overlap integrals L.D. Blokhintsev, Bull. Rus.
Acad. Sci. 65, 77 (2001)
This diagram generates normal asymptotic
behaviour I(r) ? C exp(-?r)/r for the ?AB?C?
overlap
A
B
C
This diagram generates the following contribution
to the asymptotic behaviour I1(r) ? C1 ?1 /(?12 -
?2) exp(-?1r)/rs2
A
D
E
F
C
B
37
Pre-asymptotic behaviour of some overlap
integrals N.K. Timofeyuk, L.D. Blokhintsev and
J.A. Tostevin, PRC 68, 021601(R) (2003)
This diagram generates normal asymptotic
behaviour I(r) ? C exp(-?r)/r , ? 0.372 fm-1
12Be
11Beg.s.
n
This diagram generates abnormal asymptotic
behaviour I1(r) ? C1 ?1 /(?12 - ?2)
exp(-?1r)/r7/2 , ?1 0.766 fm-1
12Be
n
10Be
n
n
n
11Beg.s.
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Effective local two-body potential Veff(r) for
asymptotic part of the overlap Ias(r)I0(r)I1(r)
?1 - ? 0.394 Effeclive local potential for
n11Be(g.s.) for different values of
c10C1/C??13/2
Standard Woods-Saxon potential
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Modelling abnormal asymptotic behaviour
V(r) Vcore(r) Vhalo(r)
Model wave function of n11Be(g.s.) for different
values of C1/C??13/2
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Feynman diagramm aproach to asymptotics of the
overlap integrals L.D. Blokhintsev, Bull. Rus.
Acad. Sci. 65, 77 (2001)
This diagram generates normal asymptotic
behaviour I(r) ? C exp(-?r)/r for the ?AB?C?
overlap
A
B
C
This diagram generates the following contribution
to the asymptotic behaviour I1(r) ? C1 ?1 /(?12 -
?2) exp(-?1r)/rs2
A
D
E
F
C
B
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Longitudinal momentum distributions from
one-neutron knockout from 12Be populating
11Be(g.s.) residues.
Partial cross sections contributions from
one-neutron knockout from 12Be populating
11Be(g.s.) residues.
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Pseudo 9C 7Be p p No deformation and
excitation of the 7Be core, J?(7Be)0 GPT
Coulomb potential for pp interaction Hyperspherica
l expansion to Kmax 50 E(8B) 0.149 MeV,
E(9C) 4.675 MeV
?r2?½ 3.175 fm ?r2?½standard 3.016 fm
?8B9C?
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Effective local p8B potential
V0 r0 a0 V1 r1
a1 Vso rso aso MeV
fm fm MeV fm
fm MeV fm fm 54.24
1.10 0.81 2.7 1.20 1.5 8.91
1.10 0.81
9C wave functions
p
y
7Be
x
p
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8B(p,?)9C from the d(8Li,9Li)p reaction B.Guo et
al, Nucl. Phys. A761, 162 (2005)
Timofeyuk 1993