Ab Initio Description of Nucleon and Deuteron
Scattering for Systems with up to A=6 Nucleons
«Understanding nuclear structure and reactions microscopically, including the continuum»
FUSTIPEN topical workshop, GANIL Caen, March17th 2014.
Guillaume Hupin
Collaborators:
S. Quaglioni (LLNL)
P. Navrátil (TRIUMF)
R. Roth (TU Darmstadt)
J. Langhammer (TU Darmstadt)
C. Romero-Redondo (TRIUMF)
F. Raimondi (TRIUMF)
J. Dohet-Eraly (TRIUMF)
Quest for
the nuclear
interaction
Fusion based
energy generation
LLNL-PRES-XXXXXX
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
Nuclear astrophysics
Ab initio NCSM/RGM: formalism for binary clusters
S. Quaglioni and P. Navrátil, PRL101 (2008); PRC79 (2009)
Ex: n-4He scattering

Starts from:
A)
Y (RGM
= å ò d r gv (r ) Aˆn F(vrA-a,a)
n
Relative wave
function (unknown) Antisymmetrizer

Channel
basis
ya(A-a )ya(a )d( r - rA-a,a )
1
2
Cluster expansion
technique
Entrance
channel
Exit
channel
Schrödinger equation on channel basis:
( A)
HY (A)
=
EY
RGM
RGM
å ò dr
n
éH v¢v (r¢, r ) - E N v¢v (r¢, r )ù gv (r ) = 0
ë
û
 RGM accounts for: 1) interaction (Hamiltonian kernel), 2) Pauli principle (Norm
kernel) between clusters.
 NCSM accounts for: internal structure of clusters.
 Together with the same microscopic nuclear interaction.
Lawrence Livermore National Laboratory
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Going around the hard core problem
E. Jurgenson, Navrátil, R. J. Furnstahl PRL103 (2009)
In configuration interaction methods
we need to soften interaction to
address the hard core
We use the Similarity-RenormalizationGroup (SRG) method
H l = Ul H Ul+
Unitary
transformations
ì dH
l
ï
= éh l , H ù
ï dl ë ( ) l û
í
ï h ( l ) = dUl U +
l
ïî
dl
Flow parameter
Bare potential
Evolution with flow
parameter l
Evolved potential
Preserves the physics
Decouples high and low
momentum
k(fm-1)
Induces many-body
forces
k’(fm-1)
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k(fm-1)
k’(fm-1)
3
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Demonstrated capability to describe binary-cluster reactions
starting from NN interactions
 Nucleon-nucleus collisions
n-He4 phase-shift NN only
 n-3H, p-3He, N-4He, n-10Be
scattering with N3LO NN (mod.
Lee-Suzuki eff. Int.)
 Nucleon scattering on 3H,
3,4He,7Li,7Be,12C,16O with SRGN3LO
 7Be(p,γ)8B radiative capture with
SRG-N3LO
Deuterium-nucleus collisions
 d-4He scattering and 6Li structure
with SRG-N3LO
Lee-Suzuki
l
dependenteffective interaction
Good reproduction, but still incomplete
(d,N) transfer reactions
 3H(d,n)4He and 3He(d,p)4He
reactions with SRG-N3LO
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n-4He scattering
4
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Including the NNN force into the NCSM/RGM approach
nucleon-nucleus formalism
A-1
( A -1)
æ
ö
V NNN ç1- å PˆiA ÷
è i=1 ø ( a =1) r
( a¢ =1)
( A -1)
FnJ ¢rT¢ Aˆv¢V NNN Aˆv FnJr T =
p
p
r¢
Direct potential:
µ
SD
+ +
( A-1)
ya( A-1)
a
a
a
a
y
¢
i j l k
a
1
1
SD
Exchange potential:
µ
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SD
ya( A-1)
ah+ ai+ a+j am al ak ya( A-1)
¢
1
1
SD
5
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n-4He scattering: NN versus NNN interactions
G. Hupin, J. Langhammer et al. PRC88 (2013)
n-4He scattering
Three scenarii of nuclear Hamiltonians
More spin-orbit
NNN vs
NN “bare”
(NN+NNN-ind)
splitting
• The NNN interactions influence
mostly the P waves.
• The largest splitting between P
waves
is
obtained
with
NN+NNN.
• The agreement of the P3/2
phase-shifts between NN-only
and
NN+NNN
forces
is
accidental.
Comparison between NN+NNN
-ind and NN+NNN at Nmax=13
with six 4He states.
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n-4He scattering: study of the RGM convergence in the NNN case
G. Hupin, J. Langhammer et al. PRC88 (2013)
Convergence with respect to the
target polarization
Target polarization
(g.s., low-lying states)
Exp.
Convergence of the phase-shifts as a function
of the 4He excited states.
• We have included the first 6 low-lying
states of 4He.
• Convergence is difficult to assess.
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4He(d,d)4He
with NN+NNN interaction
G. Hupin, S. Quaglioni and P. Navrátil, work in progress
d-4He
scattering
Comparison of the d-α phase-shifts with
different interactions
• 6Li loses binding energy with NNN
interactions.
d-4He(g.s.) scattering phase-shifts for NN, NN+NNNinduced and NN+NNN potential with l=2.0 fm-1.
Lawrence Livermore National Laboratory
• The largest splitting between 3D3 and
3D partial waves is obtained with
2
NN+NNN.
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Study of the influence of d* continuum
G. Hupin, S. Quaglioni and P. Navrátil, work in progress
Comparison of the d-α phase-shifts with
different interactions
d pseudo states
3S -3D
1
1
channel, 3D2 channel…
7d*+4d’*+7d`*
7d*
d (g.s.)
d-4He(g.s.) scattering phase-shifts
interaction and l=2.0 fm-1 at Nmax=7.
Lawrence Livermore National Laboratory
with
NN+NNN
Pseudo-states in each
deuteron channels
d (g.s.,3S1-3D1, 3D2, 3D3-3G3)
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To overcome the difficulty: couple NCSM and NCSM/RGM (NCSMC)
S. Baroni, P. Navrátil and S. Quaglioni PRL110 (2013)
•
Methods develop in this presentation to solve the many body problem
A)
Y (NCSM
= å cl Al J p T
l
Mixing
coefficients(unknown)
A-body harmonic
oscillator states
A)
Y (RGM
= å ò d r gv (r ) Aˆn F(vrA-a,a)
n
Relative wave
function (unknown) Antisymmetrizer
•
Channel
basis
Al J p T
SD
A
j 00 ( Rc.m.
)
Second quantization
ya(A-a )ya(a )d( r - rA-a,a )
1
2
Cluster expansion
technique
Can address bound
and low-lying
resonances (short
range correlations)
Design to account
for the continuum of
scattering state
(long range
correlations)
The many body quantum problem best describe by superposition of both
A)
Y (NCSMC
= å cl Al J p T + å ò d r gv (r ) Aˆn F(vrA-a,a)
l
NCSMC
n
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Including the NNN force into the NCSMC approach
nucleon-nucleus formalism
( A -1)
r¢( a¢ =1)
V NNN
A-body compound system
…
Target and projectile in
relative motion
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n-4He scattering with NCSMC
G. Hupin, S. Quaglioni and P. Navrátil, work in progress
n-4He scattering
Study of the convergence with respect to
the # of 4He low-lying states
Experimental low-lying states of the A=5
nucleon systems.
n-4He scattering phase-shifts for NN+NNN potential with
l=2.0 fm-1 and 8 low-lying state of 5He.
Lawrence Livermore National Laboratory
• The convergence pattern looks good.
• The experimental phase-shifts are
well reproduced.
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How n-4He elastic cross-sections compare ?
Comparison of the elastic cross-section
between NN and NN+3N with 4He (g.s.)
∂σ
(θc. m . )[mb/ sr]
∂Ω
G. Hupin, S. Quaglioni and P. Navrátil, work in progress
5
NN+NNN
N N + N N N -i nd
ex pt .
2
102
n- 4 He (g.s.)
0
(NN+3N-induced)
45
90
135
180
θc. m . [deg]
Differential cross-section at Eneutron =0.84
MeV between NN+3N-ind and NN+3N.
n-4He elastic cross-section for NN+3N-induced, NN+3N
potentials compared to expt. and ENDF evaluation .
Lawrence Livermore National Laboratory
• A better agreement with experiment
is obtained with NN+NNN.
• The NNN force is essential to get the
resonance right.
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How n-4He elastic cross-sections compare ?
G. Hupin, S. Quaglioni and P. Navrátil, work in progress
103
9
8
∂σ
(θc. m . )[mb/ sr]
∂Ω
Comparison of the elastic cross-section
between NN and NN+3N with 4He (g.s.)
7
NN+NNN
N N + N N N -i nd
ex pt .
6
5
4
3
n- 4 He (g.s.)
2
0
(NN+3N-induced)
45
90
135
180
θc. m . [deg]
Differential cross-section at Eneutron =1.79
MeV between NN+3N-ind and NN+3N.
n-4He elastic cross-section for NN+3N-induced, NN+3N
potentials compared to expt. and ENDF evaluation .
Lawrence Livermore National Laboratory
• A better agreement with experiment
is obtained with NN+NNN.
• The NNN force is essential to get the
resonance right.
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p-4He scattering: NCSM/RGM and NCSMC
G. Hupin, S. Quaglioni and P. Navrátil, work in progress
p-4He scattering
NN+NNN NCSM/RGM and NCSMC
• Preliminary calculation, the
NCSMC results are obtained
with only the 4He ground state.
• We can see an improvement in
the
reproduction
of
the
experimental phase-shifts.
Comparison between NCSM/RGM
and NCSMC at Nmax=13 and
l=2.0 with NN+NNN.
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Analyzing power and differential cross section
G. Hupin, S. Quaglioni and P. Navrátil, work in progress
p-4He reaction observables compared to experiment
∂σ
(θc. m . )[mb/ sr]
∂Ω
104
p- 4 He (g.s.,0+,0- ,2- ,2- ,1- ,1- )
λ =2.0 fm - 1 , N m ax =13
NCSM C
7.89 M eV
11.99 M eV
NCSM / RGM
7.89 M eV
11.99 M eV
NCSM C
5.95 M eV
9.89 M eV
NCSM / RGM
5.95 M eV
9.89 M eV
103
102
(b)
(a)
10
1
A y (θc. m . )
0.8
0.4
expt . Schwandt et al.
7.89 M eV
11.99 M eV
expt . Schwandt et al.
5.95 M eV
9.89 M eV
0
-0.4
-0.8
(d)
(c)
-1.2
0
45
90
135
θc. m . [deg]
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180
45
90
135
180
θc. m . [deg]
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4He(d,d)4He
with NCSMC
d-4He
scattering
G. Hupin, S. Quaglioni and P. Navrátil, work in progress
Effects of the short-range correlation of the compound nuclei on the d-a
differential cross-section
∂σ
(θc. m . )[mb/ sr]
∂Ω
104
N CSM C
2.935 M eV
8.971 M eV
103
RG M
2.935
8.971
ex pt .
2.935
8.971
N CSM C
6.965 M eV
12 M eV
M eV
M eV
M eV
M eV
RG M
6.965 M eV
12 M eV
ex pt .
6.965 M eV
12 M eV
102
10
d- 4 He (g.s.)
1
0
45
NN-only
90
135
θc. m . [deg]
0
45
90
135
180
θc. m . [deg]
d-4He(g.s.) differential cross-section for NN-only potential using NCSM/RGM andNCSMC.
• Preliminary results in a small model space
(Nmax=9).
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• The coupling to the compound nuclei
addresses some missing correlation.
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4He(d,d)4He
with NN+NNN interaction
G. Hupin, S. Quaglioni and P. Navrátil, work in progress
Comparison of the d-a phase-shifts wrt the
number of d* pseudo-states
before
d-4He(g.s.) scattering phase-shifts for NN-only with
different numbers of deuteron pseudo-states.
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• The
NCMSC
weakens
the
dependence on the d* pseudo-states.
• Residual dependence could be
attributed to the missing breakup
channel.
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4He(d,d)4He
with NN+NNN interaction
G. Hupin, S. Quaglioni and P. Navrátil, work in progress
Comparison of the d-a phase-shifts with
different interactions
-1.43
d+4He(g.s.) scattering phase-shifts for NN-only,
NN+NNN-induced, NN+NNN potential with l=2 fm-1.
Lawrence Livermore National Laboratory
-1.15
-1.73
• Preliminary results in a small model
space (Nmax=9).
• The 3D3 resonance is not quite
reproduced but the 3N force is
helping to get the correct position.
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First step towards Ab initio many-body calculations of fusion
P. Navrátil, S. Quaglioni, PRL108 042503 (2012)
3He(d,p)4He
astrophysical S-factor
3H(d,n)4He
astrophysical S-factor
e- lab
screening
l =1.5 fm-1
Pseudo excited
states
Complete picture:
includes break-up
Calculated S-factors converge with the inclusion of the
virtual breakup of the deuterium, obtained by means of
excited 3S1-3D1 (d* ) and 3D2 (d’* ) pseudo-states.
Lawrence Livermore National Laboratory
Evidence of
incomplete model
(nuclear force)
NCSM/RGM results for the 3He(d,n)4He astrophysical Sfactor compared to beam-target measurements.
Incomplete nuclear interaction: requires NNN force
(SRG-induced + “real”)
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First step towards Ab initio many-body calculations of fusion
J. Dohet-Eraly, S. Quaglioni, P. Navrátil et al. work in progress
p-4He bremsstrahlung (preliminary)
• Perspective to diagnose plasmas in
fusion experiments from t(d,ng)4He
radiative transfer reaction
• Provides another way to probe
cluster wave functions
10
∂σ
[µb/ M eV ]
∂Eγ
8
N N +3N - full, N m ax =7
E γ =1MeV
E γ =5MeV
6
4
 NCSM/RGM exchange terms
of a non-scalar operator most
difficult part
2
p- 4 He
0
0
4
8
12
16
E p [MeV ]
p+4He(g.s.,0+) bremsstrahlung for NN+NNN potential
with l=2 fm-1.
Lawrence Livermore National Laboratory
 Transitions of electric dipole
operator between scattering
states
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Conclusions and Outlook
 We are extending the ab initio
NCSM/RGM
approach
to
describe low-energy reactions
with two- and three-nucleon
interactions.
 We are able to describe:
• Nucleon-nucleus collisions with
NN+NNN interaction
• Deuterium-nucleus collisions with
NN+NNN interaction
• NCSMC for single- and two-
nucleon projectile
 Work in progress
• Fusion reactions with our best
complete ab initio approach
• The present NNN force is
Evolution of stars, birth, main sequence, death
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"incomplete”, need to go to N3LO
22
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