Scattering of light nuclei

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Lawrence Livermore National Laboratory

Scattering of light nuclei

LLNL
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Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA 94551

This work performed under the auspices of the U.S. Department of Energy by

Lawrence Livermore National Laboratory under Contract DE
-
AC52
-
07NA27344

Sofia
Quaglioni

in collaboration with
Petr

Návratil

19
th

International IUPAP Conference

on Few
-
Body Problems in Physics

Bonn, September 4, 2009

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Nuclear reactions


Nuclear physics underlying many key astrophysical processes


Formation of the chemical elements


Solar neutrino problem


Stellar evolution



Tools for studying exotic nuclei


Structure inferred from breakup reactions


Most low
-
lying states are unbound



A formidable challenge to nuclear theory …


Main difficulty:
scattering states

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Disclaimer


As they deserve, nuclear reactions are attracting much attention


There are many interesting new developments …


… forgive me if I miss to mention some of them!




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Microscopic



All nucleons are active



Exact Pauli principle


Few
-
nucleon
techniques using
realistic NN (+ NNN) interactions


Faddeev
, AGS (
Deltuva

et al.
), FY (
Lazauskas

et al.
), HH (
Viviani

et
al.
), LIT (
Bacca

et al.
), RRGM (Hoffman
et al.
), …



Many
-
body
techniques using
realistic NN (+ NNN) interactions


GFMC (
Nollett

et al.
),
NCSM/RGM

(
Navrátil
, SQ), FMD (Neff
et al.
), …



Cluster

techniques using semi
-
realistic NN interactions


RGM, GCM (
Descouvemont

et al
.),
...

Reaction approaches

Cluster few
-
body



N
-
nucleus interactions



(usually) inert core


Techniques using
local/non
-
local


optical potentials


Faddeev
, AGS (
Deltuva

et al
.), …



Techniques using
local optical potentials


CDCC (Moro
et al.
), XCDCC (Summers
et al.
), DWBA,
adiabatic approaches (
Baye

et al.
), …



Halo effective
-
field theories
(
Higa

et al.
), …



PRC 79, 054007 (2009)

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Our goal:

ab

initio

approach to low
-
energy reactions of light nuclei


Start with

the
ab

initio

description of the structure of light nuclei


The
ab

initio

no
-
core shell model (
NCSM
)


A successful
ab

initio

approach to nuclear structure


Capable of employing
chiral

effective field theory (

EFT) NN + NNN potentials for
A
>4


Covers nuclei beyond the
s
-
shell


Incorrect description of wave
-
function asymptotic (
r

>

5

fm), no coupling to continuum



Add

microscopic description of nucleus
-
nucleus scattering


The resonating
-
group method (
RGM
)


A successful microscopic cluster technique (also multi
-
cluster)


Preserves Pauli principle, includes Coulomb force


Describes reactions and clustering in light nuclei (also multichannel, transfer etc.)


Usually simplified NN interactions and internal description of the clusters



Combine
:
NCSM/RGM


ab

initio
bound & scattering

states in light nuclei


NCSM
-

single
-
particle degrees of freedom


RGM
-

clusters and their relative motion

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The
ab initio

no
-
core shell model (NCSM) in brief

The NCSM is a technique for the solution of the
A
-
nucleon bound
-
state problem


Hamiltonian


“realistic” (= reproduce NN data with high precision) NN potentials:


coordinate space: Argonne …


momentum space: CD
-
Bonn,

EFT

N
3
LO
, …


NNN interactions:


Tucson
-
Melbourne TM’,

EFT

N
2
LO


Finite harmonic oscillator (
HO
) basis


A
-
nucleon HO basis states


Jacobi relative
or

Cartesian single
-
particle coordinates


complete
N
max
ħ


model space


translational invariance preserved even with Slater
-
determinant (
SD
) basis


Constructs effective interaction tailored to model
-
space truncation


unitary transformation in a
n
-
body cluster approximation (
n
=2,3)

1
max


N
N
Convergence to exact solution with increasing
N
max

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Resonating
-
group method


Ansatz
:






The many
-
body Schrodinger equation is mapped onto:








Input
: ,



Output
(
e.g
.,
R
-
matrix method on Lagrange mesh): , scattering matrix

Norm
kernel

Hamiltonian

kernel



eigenstates

of
H
(
A
-
a
)
,
H
(
a
)

in the NCSM basis

NCSM/RGM: NCSM microscopic wave functions for the clusters involved,

and realistic (bare or derived NCSM effective) interactions among nucleons.

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Single
-
nucleon projectile: the norm kernel

(A
-
1)

(1)



(A
-
1)


(A
-
1)

(1)

(1,…,A
-
1)

(A)

(1,…,A
-
1)

(A)

“Direct term” treated exactly. “Exchange” term localized


d

expanded in HO radial w.f.

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Single
-
nucleon projectile basis: the Hamiltonian kernel





(A
-
1)





(A
-
1)(A
-
2)




“direct potential”

“exchange potential”

(A
-
1)

(1)

(1,…,A
-
1)

(A)

(1,…,A
-
1)

(A)

+
terms containing NNN potential

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The RGM kernels in the single
-
nucleon projectile basis



(A
-
1)(A
-
2)






(A
-
1)



(A
-
1)

(1)



+

(A
-
1)



“direct

potential”

“exchange

potential”

In the A=5 system

the 1/2
+

(
2
S
1/2
) is a Pauli
-
forbidden state,
therefore
g.s
. in
P

wave

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NCSM/RGM
ab

initio

calculation of
n
-
4
He phase shifts


NCSM/RGM calculation with
n

+ 4He(g.s.)



Low
-
momentum
V
lowk

NN potential:
convergence reached with
bare

interaction




EFT N
3
LO NN potential: convergence
reached with
two
-
body effective

interaction


4
He

n

Is everything else under
control?



need
verification

against
independent
ab

initio

approach!

No fit
.


No free parameters
.

Convergence in
N
max


under control.

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The A=4 system as a test ground for the NCSM/RGM
approach within the single
-
nucleon
-
projectile basis


NCSM/RGM calculation with
n

+
3
H(
g.s.
) and
p

+
3
He(
g.s.
), respectively



EFT N
3
LO NN potential: convergence with
2
-
body effective

interaction


Benchmark:
AGS results (
+
)
,
Deltuva

& Fonseca, PRC
75
, 014005 (2007)


The omission of
A = 3 partial waves with 1/2
<
J

5/2

leads to effects of comparable

magnitude on the AGS results.
Need to include target excited (here breakup) states!

3
H

n

3
He

p

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n
-
4
He phase shifts with

EFT N
3
LO

NN interaction


Very mild effects of
J
p
T

㴠=
+
0 on
2
S
1/2


The negative
-
parity states have larger effects on
P

phases
(coupling to
s
-
wave of relative motion)


0
-
0, 1
-
0 and 1
-
1 affect
2
P
1/2


2
-
0 and 2
-
1 affect
2
P
3/2



NCSM/RGM calculation with
n

+
4
He(g.s.,
ex.
)



EFT N
3
LO NN potential: convergence with
2
-
body effective

interaction

4
He

n

The resonances are sensitive to the inclusion of the first six excited states of
4
He

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Nucleon
-

phase
-
shifts with
EFT N3LO NN interaction


NCSM/RGM calculation with
N
+
4
He(g.s., 0
+
0
0
-
0
1
-
0
1
-
1
2
-
0
2
-
1)



EFT N
3
LO NN potential: convergence with
2
-
body effective

interaction




2
S
1/2
in agreement with Expt. (dominated by
N
-


repulsion
-

Pauli principle
)


Insufficient spin
-
orbit splitting between
2
P
1/2

and
2
P
3/2

(sensitive to interaction)


Fully
ab

initio
, very promising results.

The resonances are sensitive to NNN force.

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n
+
4
He differential cross section and analyzing power


NCSM/RGM calculations with


N

+
4
He(g.s., 0
+
0)


SRG
-
N
3
LO NN potential with
Λ
=2.02 fm
-
1


Differential cross section and analyzing
power @17
MeV

neutron energy


Polarized neutron experiment at Karlsruhe

4
He

n

Good agreement for energies beyond low
-
lying resonances

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NCSM/RGM
ab

initio
calculation of

n
+
7
Li
scattering

7
Li

n


N
max

= 8
NCSM/RGM calculation with
n

+
7
Li(g.s.,1/2
-
, 7/2
-
)


SRG
-
N
3
LO NN potential with
Λ

= 2.02 fm
-
1


Qualitative agreement with experiment:


Calculated broad 1
+

resonance


3
+

resonance not seen when the 7/2
-

state of
7
Li is not included


7
Li

Predicted narrow 0
+

and 2
+

resonances seen at recent
p
+
7
Be experiment at FSU

Expt
:
a
01
=0.87(7) fm


a
02
=
-
3.63(5) fm

Calc:
a
01
=0.73 fm


a
02
=
-
1.42 fm

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11
Be bound states and
n
-
10
Be phase shifts

10
Be

n

NCSM/RGM

NCSM

3.0

2.5

2.0

1.5

1.0

0.5

0.0

-
0.5

-
1.0

E
[MeV]

Expt.

1/2
-

1/2
+

Parity
-
inverted
g.s
. of
11
Be

understood!

11
Be


Exotic nuclei: vanishing of magic numbers,
abnormal spin
-
parity of ground states, …



The
g.s
. of
11
Be one of the best examples


Observed spin
-
parity :

1/2+


p
-
shell expected:
1/2
-



Large
-
scale NCSM calculations,
Forssen

et al
.,
PRC
71
, 044312 (2005)


Several realistic NN potentials


Calculated
g.s
. spin
-
parity
:
1/2
-



NCSM/RGM calculation with CD
-
Bonn


n

+
10
Be(g.s.,2
1
+
,2
2
+
,1
1
+
)


Calculated
g.s
. spin
-
parity :
1/2+

What happens?
Substantial drop of the relative

kinetic energy due to the

rescaling of the relative

wave function when the Whittaker tail i
s recovered

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The deuteron
-
projectile formalism: norm kernel

(A
-
2)

(2)

(1,…,A
-
2)

(A
-
1,A
)

(1,…,A
-
2)

(A
-
1,A
)

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NCSM/RGM
ab

initio

calculation of
d
-
4
He scattering


N
max

= 8 NCSM/RGM calculation with
d
(g.s
.) +
4
He(g.s.)


SRG
-
N
3
LO potential with
Λ

= 2.02 fm
-
1

4
He

d


Calculated two resonances: 2
+
0, 3
+
0


The 1
+
0
g.s
. is still unbound: convergence moves towards bound state

6
Li

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Toward the first
ab

initio
calculation of the

Deuterium
-
Tritium fusion

r




n

r



n

r


3
H


d

r

d

3
H


r




n

r



n

r


3
H


d

r

d

3
H


3
H

d

4
H
e

n





Work in progress on coupling between
d

+
3
H and
n

+
4
He bases

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Conclusions and Outlook


With the NCSM/RGM approach we are extending the
ab

initio
effort to
describe low
-
energy reactions and weakly
-
bound systems



Recent results for nucleon
-
nucleus scattering with NN realistic potentials:


n
-
3
H,
n
-
4
He,
n
-
10
Be and
p
-
3,4
He


S.Q. and P.
Navrátil
,

PRL
101
, 092501 (2008), PRC
79
, 044606 (2009)




New results with SRG
-
N
3
LO:


N
-
4
He,
n
-
7
Li,
(also
N
-
12
C and


N
-
16
O, not presented here)


Initial results for
d
-
4
He scattering


First steps towards
3
H(
d
,
n
)
4
He



To do:


Coupling of
N
+A and
d
+(A
-
1)


Inclusion of NNN force


Heavier projectiles:
3
H,
3
He,
4
He


NCSM with continuum (
NCSMC
)


Three
-
cluster NCSM/RGM and treatment of three
-
body continuum


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Thanks


Petr

Navrátil
, without whom much of this work would not have been
possible


Our collaborators:


R. Roth, GSI, on the Importance
-
truncation NCSM


S.
Bacca
, TRIUMF, on the NCSMC

Thank you for your attention!