Toward a fundamental understanding
of nuclear reactions and exotic nuclei
Advances in Radioactive Isotope Science
Tokyo, June 1-6, 2014
S. Quaglioni
Contributors:
G. Hupin (LLNL)
P. Navrátil, C. Romero-Redondo,
J. Dohet-Eraly, F. Raimondi (TRIUMF)
R. Roth, J. Langhammer, A. Calci (TU
Darmstadt)
LLNL-PRES-655514
This work was performed under the auspices of the U.S. Department
of Energy by Lawrence Livermore National Laboratory under contract
DE-AC52-07NA27344. Lawrence Livermore National Security, LLC
Light exotic nuclei offer an exciting opportunity to test
our understanding of the interactions among nucleons

But this is not an easy goal …

Challenging for experiment

•
Short half lives or unbound
•
Minute production cross sections
Challenging for theory
•
Low (multi-)particle emission
thresholds or unbound
•
Bound, resonant and scattering
states may be strongly coupled
Need advanced experimental techniques and
ab initio nuclear theory including the continuum
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To develop such an ab initio nuclear theory we:
1) Start with accurate nuclear forces (and currents)
NN force

Two- plus three-nucleon (NN+3N)
forces from chiral effective field
theory (EFT)
•
NN potential at N3LO (by
Entem & Machleidt)
•
3N force at N2LO (in the
local form by Navratil)
•
Guided by quantum
chormodynamics (QCD)
•
Entirely constrained in the
2- and 3-nucleon systems
NNN force
NNNN force
Q0
LO
Q2
NLO
Q3
N2LO
Q4
N3LO
+ ...
+ ...
+ ...
Worked out by Van Kolck, Keiser,
Meissner, Epelbaum, Machleidt, ...
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2) ‘Soften’ the interactions using unitary transformations
 Similarity Renormalization Group (SRG) method
Phys. Rev. Lett. 103,
082501 (2009)
H l = Ul H l=¥Ul+ Þ
Unitary
transformations
dH l é
= ëh ( l ), H l ùû
dl
Flow parameter
SRG
NN+3N
Bare
NN+3N
Decouples low and
high momenta
l = 20
fm-1
Induces 3-body (&
higher-body) forces
l = 2 fm-1
For the lightest nuclei SRG-evolved NN+3N forces allow to obtain
unitarily equivalent results in much smaller model spaces
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3) Solve the many-body problem using a unified
approach to nuclear bound and continuum states



Ab initio no-core shell model (NCSM) …
•
Bound states, narrow resonances
•
Clusters’ structure, short range
N  N max  1
… with resonating-group method (RGM)
•
Bound & scattering states, reactions
•
Dynamics between clusters, long range
r
Most efficient: ab initio no-core shell model with continuum (NCSMC)
NCSM/RGM
channel states
NCSM
eigenstates
Y
( A)
= å cl
l
, l + å ò dr g v (r ) Aˆn
n
r
( A - a)
( a) , n
Unknowns
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Discrete and continuous variational amplitudes are
determined by solving the coupled NCSMC equations
( A)
ElNCSM dll¢
æ H
ç NCSM
ç
h
è
r'
Ù
H An
h
H RGM
Ù
Ù
An ¢ H An
r
( A)
dll¢
öæ
æ 1
ö
÷ç c ÷ = E ç NCSM
÷ç g ÷
ç g
è
ø
ø
è
r
r'
r
Ù
An
g
N RGM
Ù
Ù
An ¢ An
öæ
ö
c
֍
÷÷
ç
÷ g
ø
øè
r
 Scattering matrix (and observables) from matching solutions to
known asymptotic with microscopic R-matrix on Lagrange mesh
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7He:
An ideal system to showcase new achievements
made possible by the ab initio NCSM with continuum
is unbound – cannot be
reasonably described with NCSM

7He

6He
•

n+6He Ground State Resonance
core polarization important
n+6He NCSM/RGM calc. with
more than few 6He states difficult!
NCSMC calculation:
•
SRG-N3LO NN with l = 2.02 fm-1
•
Nmax= 12, ħW = 16 MeV
•
Up to 3 6He and 4 7He states
Convergence with number of 6He eigenstates
7He
states compensate for
missing higher 6He excitations
Lawrence Livermore National Laboratory
S. Baroni, P. Navratil, and S. Quaglioni, Phys. Rev. Lett.
110, 022505 (2013); Phys. Rev. C 87, 034326 (2013)
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7He:
An ideal system to showcase new achievements
made possible by the ab initio NCSM with continuum

NCSMC yields 3/2- g.s. and 5/2resonances close to experiment

Is there a low-lying 1/2- state?
n+6He Low-Lying Continuum
?
Expt.
Results obtained with NCSMC approach
Low-lying 1/2- state not
supported by our calculation
Lawrence Livermore National Laboratory
S. Baroni, P. Navratil, and S. Quaglioni, Phys. Rev. Lett.
110, 022505 (2013); Phys. Rev. C 87, 034326 (2013)
LLNL-PRES-655514
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picture?The unbound 5He nucleus is an excellent testing
ground

5He

SRG-evolved chiral NN+3N with
l = 2.0 fm-1
resonances sensitive to
strength of spin-orbit force
•

n+4He Scattering Phase Shifts
NCSMC
Both Induced and initial 3N
forces are included
NCSMC calculation:
+
•
Nmax = 13 model space
•
Up to 14 5He and 7 4He states
•
Excellent convergence
Convergence with number of 4He eigenstates
G. Hupin, S. Quaglioni, and P. Navratil, in preparation
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picture?The unbound 5He nucleus is an excellent testing
ground

Are the 5He states really
needed to accurately describe
the n+4He continuum?
n+4He Scattering Phase Shifts
NCSM/RGM

NCSM/RGM calculation:
•
Nmax = 13 model space
•
Up to first 7 states of 4He
•
Not sufficient!
4He
core polarization is non
negligible. 5He states essential
to describe resonances
Lawrence Livermore National Laboratory
Convergence with number of 4He eigenstates
G. Hupin, J. Langhammer, P. Navratil, S. Quaglioni, A.
Calci, And R. Roth, Phys. Rev. C 88, 054622 (2013)
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Are 3N forces necessary to achieve a complete
picture?The unbound 5He nucleus is an excellent testing
ground
 3N force enhances 1/2-  3/2- splitting; essential at low energies!
Elastic scattering of neutrons on 4He
NN+3N
NN (with ind. terms)
NN+NNN
NCSMC
x
Expt.
G. Hupin, S. Quaglioni, and P. Navratil, in preparation
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9Be:
Does the 3N force deteriorate the description of the
low-lying spectrum? No, need to include the continuum!

9Be
vs. 9Be+n-8Be(0+,2+) calculations: preliminary Nmax=10 results
Expt.
E
E
Expt.
n+8Be
n+8Be
J. Langhammer, P. Navratil, R. Roth et al., in progress
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What about scattering with composite projectiles?
angular differential cross section [mb/sr]
d+4Hed+4He with & without inclusion of: 6Li states / 3N forces
1000
Expt.
4
d+ He
4
6
(d+ He) + Li
preliminary
100
preliminary
NN-only
l = 2.0 fm-1
Ed = 2.935 MeV
10
0
30
60
90
120
150
(d+4He) + 6Li
180
center-of-mass angle [deg]
 (d,N) transfer and deuterium scattering on p-shell nuclei underway
G. Hupin, S. Quaglioni, and P. Navratil, in progress
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We want to describe also systems for which the lowest
threshold for particle decay is of the 3-body nature
 Borromean halo nuclei
11Li
• 6He (= 4He + n + n )
• 6Be (= a + p + p )
• 11Li (= 9Li + n + n )
• 14Be (= 12Be + n + n )
• …
n
4He
S. Quaglioni, C. Romero-Redondo, and P. Navratil,
Phys. Rev. C 88, 034320 (2013)
Lawrence Livermore National Laboratory
separation (fm)
n
Probability
density
of 6He g.s.
Probability
density
of 6He g.s.
4He-neutrons
 Constituents do not bind
in pairs!
208Pb
neutron’s separation (fm)
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Experimental picture for the excited states of 6He
Recent expt. @SPIRAL, GANIL: PLB 718 (2012) 441
8He(p,3H)
Soft dipole
mode?
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Can we gain insight from an ab initio calculation?
Need to treat 3-cluster continuum: 4He(g.s.)+n+n
C. Romero-Redondo, S. Quaglioni, and P. Navratil, arXiv:1404.1960
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Can we gain insight from an ab initio calculation?
Need to treat 3-cluster continuum: 4He(g.s.)+n+n
Scattering phase shifts
Energy spectrum of states
new
Present results do not support a
three-body soft-dipole resonance
C. Romero-Redondo, S. Quaglioni,
and P. Navratil, arXiv:1404.1960
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The ab initio description of light dripline nuclei with
QCD-guided NN+3N forces is now becoming possible
 The NCSMC is an efficient ab initio theory including the continuum
• NCSM eigenstates  short- to medium-range A-body structure
• NCSM/RGM cluster states  scattering physics of the system

New developments
• 3N force in nucleon- and
deuterium-nucleus scattering
• Three clusters in the continuum
• Many I have not presented
 see J. Dohet-Eraly, PS2-B010
 Even more exciting work ahead!
Lawrence Livermore National Laboratory
G. Hupin, S. Quaglioni, and P. Navratil, in preparation
LLNL-PRES-655514
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Extras
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✓
◆✓Is◆this
◆ ◆
✓ ✓basis
◆✓ ◆
◆problem?
✓ NCSM
✓
✓
NCSM
calculations.
a
HO
size
Is Is
¯
HN
h
calculations.
Is
this
a
HO
basis
size
c c
1 g
¯
c c ◆problem?
¯
CSM
HN CSM
h
1
g
¯
= E= E
−
−
thish¯this
anh¯interaction
dependent
problem?
χ
¯
g
¯problem?
1
¯ χ
−
−
N
HN
an interaction
dependent
χ
¯
g
¯ 1 χ
¯
N HN
Breakup threshold influences
s-wave
continuum
Breakup
thresholds
impact
S-waves
Breakup thresholds
impact
S-waves
Need
to switch
to NCSMC!
Need
to switch
to NCSMC!
1
2
1
2
1
2
1
2
Continuum
important
for for
other
Continuum
important
other
waves
as well
waves
as well
8
7
+7
6
5
4
3
2
1
3/2
6
5+
5/2
4
1/23
2
1
+
0
0
-1
-1
-2
-3
3/2
9
Be
-1
SRG L =SRG
2.0 fm
L = 2.0 fm
+
5/2
1/2
+
-hW = 20 -hMeV
W = 20 MeV
-W
11hW 11h
+
8
8
n+ Be n+ Be
preliminary
preliminary
-2
-3
9 N3LO NN+3NF(400)
3
Be N LO NN+3NF(400)
-1
NCSM NCSM
Expt
Expt
NCSMC NCSMC
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180
180
150
150
120
120
90
2
NCSMC
6
D5/2 D5/2n-8Be(08+,2NCSMC
D
+) ++ 9+Be 9
D9/2
9/2
n- Be(0 ,2 ) + Be
2
2
60
S1/2
60
30
30
0
0
2
4
S1/2
2
2
D3/2 D3/2
-30
-60
preliminary
-60preliminary
0
-90
01
12
4
S3/2 S3/2
-30
-90
6
90
d [deg]
8
9/2
+
d [deg]
9
10
+
9/2 9
Energy (MeV)
Energy (MeV)
10
23
6
6
S5/2 S5/2
34
45
56
67
Ekin [MeV]
Ekin [MeV]
78
89
910 10
33
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20
Microscopic three-cluster problem
 Starts from:
( a2 )
3-body
channels
h23
Y ( A) = å òò dx dy Gn (x, y) Aˆn Fn xy
n
h1,23
( A - a23 )
( a3 )
ya(A-a )ya(a )ya(a )d ( x - h23 ) d ( y - h1,23 )
23
1
2
2
3
3
 Projects (H - E)Y ( A) = 0 onto the channel basis:
å òò dx dy éëH
n
v¢v
( x¢, y¢, x, y) - E N v¢v ( x¢, y¢, x, y)ùû Gv (x, y) = 0
Fn ¢x¢y¢ Aˆn ¢ HAˆn Fn xy
Fn ¢x¢y¢ Aˆn ¢ Aˆn Fn xy
Hamiltonian kernel
Norm or Overlap kernel
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This can be turned into a set of coupled-channels
Schrödinger equations for the hyperradial motion
y
 Hyperspherical Harmonic (HH)
functions form a natural basis:
Fn xy = åf K x y (a ) Fn K r
* ,
r
a
µ¡
K

ìï x = r cosa
í
ïî y = r sin a
K
L
x y
x,
y,
x
y
x
d (r - rh )
(ah , hˆ1,23, hˆ23 ) 5/2 5/2
r rh
¢x , ¢y
Then, with orthogonalization and projection over fK ¢ (a ¢) :
uKv (r )
uK ¢v¢ (r¢)
5 é -1/2
-1/2 ù
¢
d
r
r
N
H
N
(
r
,
r
)
=
E
åò
ë
ûn ¢n
5/2
¢5/2
r
nK
r
é N 1/2Gù (x, y) = r -5/2 åun K (r ) fKx y (a )
ë
ûn
K
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These equations can be solved using R-matrix theory
Internal region
(r  a)
External region
(r > a)
r
a
0
Expansion on a basis
Bound state asymptotic behavior
uKv (r ) = CKn k r K K+2 (k r )
uKv (r ) = å cnKn fn (r )
n
Scattering state asymptotic behavior
uKv (r ) = AKn éëH-K (k r ) dnn ¢dKK¢ - SvK,v¢K¢H+K (k r )ùû
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Results for 6He ground state
6-body diagonalization vs 4He(g.s)+n+n calculation
NCSM/RGM 4He(g.s.)+n+n
NCSM 6-body diagonalization
Ntot = N0 + Nmax
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
Differences between NCSM 6-body
and NCSM/RGM 4He(g.s.)+n+n
results due to core polarization

Contrary to NCSM, NCSM/RGM
wave function has appropriate
asymptotic behavior
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Other convergence tests
 HH expansion
c n (x, y) =
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1
r 5/2
K max
å un
K
(r ) fKx y (a )
K
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Other convergence tests
 Extended-size HO
expansion
(
VA-1A 1- PˆA-1,A
N ext
) µ åR
ny
y
( y¢)Rny y (y)
ny
• Sizable effects only when
neutrons are in 1S0 partial
wave (strong attraction)
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Started with NCSM/RGM approach & gradually built up
capability to describe fusion reactions with NN force

Astrophysical S-factors
Fusion reactions important to
solar astrophysics, Big Bang
nucleosynthesis, fusion research,
atomic physics

Good convergence with harmonic
oscillator (HO) basis size (Nmax)

Slower convergence with number
of clusters’ eigenstates
Still required for a fundamental
description are also:
• 3N forces
• three-body dynamics
Lawrence Livermore National Laboratory
PLB 704, 379 (2011)
3H(d,n)4He
astrophysical S-factor
PRL 108, 042503 (2012)
Electron
screening
LLNL-PRES-655514
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We started with nucleon-nucleus collisions …
PRL 107, 122502 (2011)
The elastic n-3H cross section for
14 MeV neutrons, important for
understanding how the fuel is
assembled in an implosion at NIF,
was not known precisely enough
n
n (14 MeV)
3H
3H
d
3H
4H
n+3Hn+3H cross section
Corrected
for target
breakup
En=14 MeV
e
We delivered evaluated data
for fusion diagnostic at NIF
with required 5% uncertainty.
Lawrence Livermore National Laboratory
Ab initio theory reduces uncertainty due to
conflicting data (,,, ,)
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Similarity Renormalization Group evolution of operators*
Oˆ l = Uˆ l Oˆ l=¥Uˆ *l
Uˆ l = å ya (l ) ya (l = ¥)
a
Root-mean-square radius and total dipole strength
4He
3H
*M.D. Schuster, S. Quaglioni, C.W. Johnson, E.D. Jurgenson, and P. Navratil, in preparation
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Latest Publications
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