AB INITIO STRUCTURE AND
REACTIONS OF LIGHT
NUCLEI
Collaborators:
A. Calci (TRIUMF)
J. Dohet-Eraly (TRIUMF)
J. Langhammer (Private sector)
P. Navrátil (TRIUMF)
F. Raimondi (U. of Surrey)
C. Romero-Redondo (LLNL)
R. Roth (TU Darmstadt)
S. Quaglioni (LLNL)
www.cea.fr
Guillaume Hupin
ECT*, Trento, June 6th 2016.
INTRODUCTION
Ab initio structure and reactions
Structure of exotic nuclei
Nuclear astrophysics
Fusion based
energy generation
Materials science
2
FUSION PROCESSES PLAY AN IMPORTANT ROLE IN
DETERMINING THE EVOLUTION OF OUR UNIVERSE, E.G.
NUCLEOSYNTHESIS, STELLAR EVOLUTION …
• What powers stars ?
• How long does a star live ?
• …
Nuclear astrophysics community relies on
accurate fusion reaction observables among
others.
Challenging for both experiment and theory:
Low rates: due to Coulomb repulsion
between target/projectile and low-energy
reactions (quantum tunneling effects).
S-factor for 3He(3He,2p)4He
Projectile and target are not fully ionized in a
lab. This leads to laboratory electron
screening.
Fundamental theory is still missing.
3
E. G. Adelberger et al., RMP83 (2011)
CHIRAL EFFECTIVE FIELD THEORY: TOWARDS PRECISION
NUCLEAR PHYSICS
Importance
Chiral effective field theory is a lowenergy expansion of the interaction
between nucleons with an estimation
of the model uncertainties.
B. D. Carlsson et. al.
PRX6 (2016)
n-p
• Yet uncompleted…
NLON4LO
Constrained to provide an accurate description of the
A=2 and A=3 systems.
Predictions for nuclear structure and dynamic (A>3).
E. Epelbaum, et al.
PRL115 (2015)
4
EFFECTIVE INTERACTION AND THE HARD CORE
PROBLEM
E. D. Jurgenson, P. Navrátil, R. J. Furnstahl PRL103 (2009); PRC83 (2011)…
In configuration interaction
methods we need to soften
interaction to address the hard
core. We use the SimilarityRenormalization-Group (SRG)
method
𝐻𝜆 = 𝑈𝜆 𝐻𝑈𝜆†
Unitary
transformation
𝑑𝐻𝜆
= 𝜂 𝜆 , 𝐻𝜆
𝑑𝜆
𝑑𝑈𝜆 †
𝜂 𝜆 =
𝑈𝜆
𝑑𝜆
Flow parameter
Bare potential
Evolution with flow
parameter l
Evolved potential
Preserves the physics
Decouples high and
low momentum
k(fm-1)
k’(fm-1)
Induces many-body
forces
k(fm-1)
k’(fm-1)
5
EQUAL TREATMENT OF BOUND AND RESONANT STATES:
COUPLE NCSM AND NCSM/RGM (NCSMC)
• Methods develop in this presentation to solve the many
body problem
(𝐴)
Ψ𝑁𝐶𝑆𝑀 = 𝐴𝜆𝐽𝜋 𝑇 =
𝑐𝛼
𝛼
Mixing
coefficients(unknown)
𝐴𝛼𝑗𝑧 𝜋 𝑡𝑧
A-body
harmonic
oscillator states
𝐴𝜆𝐽𝜋 𝑇
𝐴
𝑅𝑐.𝑚.
Second quantization
HO basis
Nmax
𝑆𝐷 𝜙00
Can address bound
and low-lying
resonances (short
range correlations)
Natural Orbital basis
ℏΩ
≈ 𝑒 −𝛼𝑟
No-Core Shell Model
(𝐴)
𝑐𝜆 𝐴𝜆𝐽𝜋 𝑇
Ψ𝑁𝐶𝑆𝑀𝐶 =
𝜆
Ch. Constantinou, et al. arXiv:1605.04976
6
EQUAL TREATMENT OF BOUND AND RESONANT STATES:
COUPLE NCSM AND NCSM/RGM (NCSMC)
• Methods develop in this presentation to solve the many
body problem
(𝐴)
Ψ𝑁𝐶𝑆𝑀 = 𝐴𝜆𝐽𝜋 𝑇 =
𝑐𝛼
𝛼
Mixing
coefficients(unknown)
𝐴𝛼𝑗𝑧 𝜋 𝑡𝑧
A-body
harmonic
oscillator states
𝐴𝜆𝐽𝜋 𝑇
𝑆𝐷 𝜙00
𝐴
𝑅𝑐.𝑚.
Second quantization
Can address bound
and low-lying
resonances (short
range correlations)
𝑟𝐴−𝑎,𝑎
(𝐴)
(𝐴−𝑎,𝑎)
Ψ𝑅𝐺𝑀 =
𝑑𝑟 𝑔𝑣 𝑟 𝐴𝑣 Φ𝑣𝑟
(𝐴−𝑎)
𝜓𝛼1
𝑣
Relative wave
function
(unknown)
(𝐴 − 𝑎)
Channel
basis
Antisymmetrizer
𝑎
(𝑎)
𝜓𝛼2 𝛿(𝑟 − 𝑟𝐴−𝑎,𝑎 )
Cluster expansion
technique
NCSM/RGM
Cluster formalism for
elastic/inelastic
(𝐴)
(𝐴−𝑎,𝑎)
𝑐𝜆 𝐴𝜆𝐽𝜋 𝑇 +
Ψ𝑁𝐶𝑆𝑀𝐶 =
𝜆
𝑑 𝑟 𝑔𝑣 𝑟 𝐴𝑣 Φ𝑣𝑟
𝑣
7
EQUAL TREATMENT OF BOUND AND RESONANT STATES:
COUPLE NCSM AND NCSM/RGM (NCSMC)
S. Baroni, P. Navrátil and S. Quaglioni PRL110 (2013); PRC93 (2013)
• Methods develop in this presentation to solve the many
body problem
(𝐴)
Ψ𝑁𝐶𝑆𝑀 = 𝐴𝜆𝐽𝜋 𝑇 =
𝑐𝛼
𝛼
Mixing
coefficients(unknown)
𝐴𝛼𝑗𝑧 𝜋 𝑡𝑧
A-body
harmonic
oscillator states
𝐴𝜆𝐽𝜋 𝑇
𝑆𝐷 𝜙00
𝐴
𝑅𝑐.𝑚.
Second quantization
Can address bound
and low-lying
resonances (short
range correlations)
𝑟𝐴−𝑎,𝑎
(𝐴)
(𝐴−𝑎,𝑎)
Ψ𝑅𝐺𝑀 =
𝑑𝑟 𝑔𝑣 𝑟 𝐴𝑣 Φ𝑣𝑟
(𝐴−𝑎)
𝜓𝛼1
𝑣
Relative wave
function
(unknown)
(𝐴 − 𝑎)
Channel
basis
Antisymmetrizer
𝑎
(𝑎)
𝜓𝛼2 𝛿(𝑟 − 𝑟𝐴−𝑎,𝑎 )
Cluster expansion
technique
Design to account
for scattering states
( best for long range
correlations)
• The many body quantum problem is best described by
the superposition of both type of wave functions
(𝐴)
(𝐴−𝑎,𝑎)
𝑐𝜆 𝐴𝜆𝐽𝜋 𝑇 +
Ψ𝑁𝐶𝑆𝑀𝐶 =
𝜆
𝑑 𝑟 𝑔𝑣 𝑟 𝐴𝑣 Φ𝑣𝑟
𝑣
NCSMC
8
EQUAL TREATMENT OF BOUND AND RESONANT STATES:
COUPLE NCSM AND NCSM/RGM (NCSMC)
S. Quaglioni and C. Romero-Redondo et al. PRC88 (2013); PRL113 (2014)
• Methods develop in this presentation to solve the many
body problem
(𝐴)
Ψ𝑁𝐶𝑆𝑀 = 𝐴𝜆𝐽𝜋 𝑇 =
𝑐𝛼
𝛼
Mixing
coefficients(unknown)
𝐴𝛼𝑗𝑧 𝜋 𝑡𝑧
A-body
harmonic
oscillator states
𝐴𝜆𝐽𝜋 𝑇
𝑆𝐷 𝜙00
𝐴
𝑅𝑐.𝑚.
Second quantization
Can address bound
and low-lying
resonances (short
range correlations)
𝑟𝐴−𝑎,𝑎
(𝐴)
(𝐴−𝑎,𝑎)
Ψ𝑅𝐺𝑀 =
𝑑𝑟 𝑔𝑣 𝑟 𝐴𝑣 Φ𝑣𝑟
(𝐴−𝑎)
𝜓𝛼1
𝑣
Relative wave
function
(unknown)
(𝐴 − 𝑎)
Antisymmetrizer
𝑎
(𝑎)
𝜓𝛼2 𝛿(𝑟 − 𝑟𝐴−𝑎,𝑎 )
Cluster expansion
technique
Channel
basis
𝑟𝐴−𝑎12,𝑎12
Adding three-cluster degrees of freedom:
(𝐴−𝑎1 −𝑎2 ,𝑎1 ,𝑎2 )
𝑑𝑥𝑑𝑦𝑥 2 𝑦 2 𝐺𝑣 𝑥, 𝑦 𝐴𝑣 Φ𝑣𝑥𝑦
+
𝑣
…
6He
Design to account
for scattering states
(best for long range
correlations)
properties: C. Romero-Redondo, et al. arXiv:1606.00066
(𝐴−𝑎1 −𝑎2 )
(𝑎 )
𝑟𝑎1,𝑎2
𝑎2
𝐴 − 𝑎1 − 𝑎2
𝜓𝛼1
𝑎1
(𝑎 )
𝜓𝛼21 𝜓𝛼32 𝛿(𝑟 − 𝑟𝑎1,𝑎2 ) ×
𝛿(𝑟 − 𝑟𝐴−𝑎12,𝑎12 )
Cluster expansion
technique 9
COUPLED NCSMC EQUATIONS
S. Baroni, P. Navrátil and S. Quaglioni PRL110 (2013); PRC93 (2013)
(𝐴 − 1)
𝐻𝒜
(𝐴 − 1)
𝒜
𝑟
(𝑎 = 1)
𝑟
(𝑎 = 1)
𝐸𝜆 𝛿𝜆𝜆′
𝑐
1𝑁𝐶𝑆𝑀
𝑔
=
𝐸
𝛾
𝑔
𝑁𝑅𝐺𝑀
𝐻𝑁𝐶𝑆𝑀
ℎ
ℎ
𝐻𝑅𝐺𝑀
(𝐴 − 1)
𝑟′
(𝐴 − 1)
′
(𝑎 = 1)
𝒜𝐻𝒜
(𝑎 = 1)
𝑟
Scattering matrix (and observables) from matching solutions to
known asymptotic with microscopic R-matrix on Lagrange mesh
10
MATRIX ELEMENTS OF TRANSLATIONALLY INVARIANT
OPERATORS
Translational invariance is preserved also with SD cluster basis
(𝐴−𝑎′ ,𝑎 ′ )
𝑆𝐷 Φ𝑓𝑆𝐷
(𝐴−𝑎,𝑎)
Ô𝑡.𝑖. Φ𝑖𝑆𝐷
𝑆𝐷
(𝐴−𝑎′ ,𝑎′ )
𝑀𝑖𝑆𝐷 𝑓𝑆𝐷,𝑖𝑟𝑓𝑟 Φ𝑓𝑟
=
(𝐴−𝑎,𝑎)
Ô𝑡.𝑖. Φ𝑖𝑟
𝑖𝑟 𝑓𝑟
Observables calculated in the
translationally invariant basis
What we calculate in the
“SD” channel basis
(𝐴 − 𝑎)
(𝐴 − 𝑎)
𝑎
(𝐴−𝑎)
𝑅𝑐.𝑚.
(𝐴−𝑎)
𝜓𝛼1
•
Matrix inversion
(𝑎)
𝑅𝑐.𝑚.
(𝑎)
𝑆𝐷
𝑟𝐴−𝑎,𝑎
𝑎
𝐴
𝑅𝑐.𝑚.
(𝑎)
(𝐴−𝑎)
𝜓𝛼2 𝜑𝑛𝑙 𝑅𝑐.𝑚.
𝜓𝛼1
(𝑎)
𝜓𝛼2 𝜑𝑛𝑙 𝑟𝐴−𝑎,𝑎
Advantage: can use powerful second quantization techniques
(𝐴−𝑎′ ,𝑎′ )
𝑆𝐷
Φ𝑣 ′ 𝜂 ′
𝐴−𝑎′ ,𝑎′
(𝐴−𝑎,𝑎)
Ô𝑡.𝑖. Φ𝑣𝜂
𝑆𝐷
∝ 𝑆𝐷 Φ𝑣 ′ 𝜂′
𝐴−𝑎,𝑎
𝑎† 𝑎 Φ𝑣𝜂
𝐴−𝑎′ ,𝑎′
𝑆𝐷 , 𝑆𝐷 Φ𝑣 ′ 𝜂 ′
𝐴−𝑎,𝑎
𝑎† 𝑎† 𝑎𝑎 Φ𝑣𝜂
𝑆𝐷
…
11
INCLUDING THE 3N FORCE INTO THE NCSM/RGM
APPROACH
nucleon-nucleus formalism
(𝐴 − 1)
(𝐴 − 1)
𝐽𝜋 𝑇
Φ𝑣 ′ 𝑟 ′
𝐴𝑣 ′ 𝑉
𝑁𝑁𝑁
𝐽𝜋 𝑇
𝐴𝑣 Φ 𝑣 𝑟
𝒱𝑣𝑁𝑁𝑁
𝑟, 𝑟 ′ =
′𝑣
=
𝑟′
ℛ𝑛′ 𝑙′ 𝑟 ′ ℛ𝑛𝑙 𝑟
𝑉 𝑁𝑁𝑁 1 −
(𝑎′ = 1)
𝐴−1
𝑖=1 𝑃𝑖𝐴
𝑟
(𝑎 = 1)
(𝐴 − 1)(𝐴 − 2) 𝐽𝜋𝑇
𝐽 𝜋𝑇
Φ𝑣 ′ 𝑛′ 𝑉𝐴−2𝐴−1𝐴 1 − 2𝑃𝐴−1𝐴 Φ𝑣𝑛
2
Direct potential:
𝐴−1
∝ 𝑆𝐷 𝜓𝛼′
1
−
𝐴−1
𝑎† 𝑎† 𝑎𝑎 𝜓𝛼1
𝑆𝐷
~1Go
(𝐴 − 1)(𝐴 − 2)(𝐴 − 3) 𝐽𝜋𝑇
𝐽 𝜋𝑇
Φ𝑣 ′ 𝑛′ 𝑃𝐴−1𝐴 𝑉𝐴−3𝐴−2𝐴−1 Φ𝑣𝑛
2
Exchange potential:
𝐴−1
∝ 𝑆𝐷 𝜓𝛼′
1
many two-body states
𝐴−1
𝑎† 𝑎† 𝑎† 𝑎𝑎𝑎 𝜓𝛼1
𝑆𝐷
12
n-4He SCATTERING: NN VERSUS 3N
INTERACTIONS
G. Hupin, J. Langhammer et al. PRC88 (2013); P. Navrátil et al. Phys. Script.91 (2016)
n-4He scattering
Two scenarii of nuclear Hamiltonians
More spin-orbit
splitting
• The 3N interactions
influence mostly the P
waves.
3N vs
NN “bare”
(NN+3N-ind)
• The largest splitting
between P waves is
obtained with NN+3N.
NN+3N
NN+3N-induced
Comparison between NN+3N-ind and NN+3N
Nmax=13 with six 4He states and 14 5He states.
at
13
AB INITIO n-4He SCATTERING
G. Hupin, J. Langhammer et al. PRC88 (2013); P. Navrátil et al. Phys. Script.91 (2016)
Study of the convergence with respect to the # of
4He low-lying states
4He
lowlying states
Target polarization
n-4He scattering phase-shifts for NN+3N potential with
l=2.0 fm-1 and first 14th low-lying state of 5He.
• The convergence
pattern looks good.
• The experimental
phase-shifts are well
reproduced.
14
n-4He SCATTERING: STUDY OF THE RGM ONLY
CONVERGENCE
G. Hupin, J. Langhammer et al. PRC88 (2013)
Convergence with respect to the target
polarization
Experimental low-lying states
of the A=5 nucleon systems.
• Convergence is rather
slow.
Convergence of the phase shifts as a function of the 4He
excited states.
15
PREDICTION FOR RUTHERFORD
BACKSCATTERING (RBS)
G. Hupin, S. Quaglioni and P. Navrátil, PRC90 (2014)
p-4He scattering
p-4He differential cross-section compared to
experiment
For the non-destructive
physical, electrical and
chemical characterization of
materials, nuclear physics is
routinely used for energies
above the Rutherford
scattering.
Impurities, e.g. 4He
p-4He differential cross-section for NN+3N.
16
4He(d,d)4He
COMPARISON OF
INTERACTION
G. Hupin, S. Quaglioni and P. Navrátil, PRL114 (2015)
d-4He
scattering
Comparison of the d-a phase-shifts with
different interactions (Nmax=11)
~1.5M CPU/h
3N-full
3N-ind
d-4He(g.s.) scattering phase-shifts for NN-only,
NN+3N-induced, NN+3N-full potential with
l=2.0 fm-1.
• Best results in a decent model
space (Nmax=11).
• The 3D3 resonance is reproduced
but the 3D2 and 3D1 resonance
positions are underestimated.
• The 3N force corrects the D-wave
resonance positions by increasing
the spin-orbit splitting.
• There is room for improvements.
17
6Li
SPECTRUM, NCSMC VS
NCSM/RGM
G. Hupin, S. Quaglioni and P. Navrátil, PRL114 (2015)
4He(d,4He)d
Comparison between NCSMC vs NCSM
differential
cross section at φ=30⁰
• The 3N force is essential to get the
correct 6Li g.s. energy and splitting
between the 3+ and 2+ states.
• The 6Li g.s. is well reproduced.
• There is room for improvements, in
particular regarding the 3+ state.
18
FIRST APPLICATION TOWARDS HEAVIER
SYSTEMS: n-8Be COLLISION
J. Langhammer, P. Navrátil et al., PRC91 (2015)
n-8Be scattering
9Be
vs. 9Be+n-8Be(0+,2+) with chiral NN+3N(400) and l = 2.0 fm-1
Negative parity phase-shifts
3N-full
3N-induced
NCSM
NCSMC
19
THE COMBINATION OF THE TWO METHODS
IS ESSENTIAL
J. Langhammer, P. Navrátil et al., PRC91 (2015)
9Be
vs. 9Be+n-8Be(0+,2+) with chiral NN+3N(400) and l = 2.0 fm-1
Positive parity phase-shifts
NCSM
NCSMC
20
NEUTRON-RICH HALO NUCLEUS 11BE
In the shell model picture g.s. expected to be
Jπ=1/2-
Z=4
N=7
(Z=6, N=7) 13C and (Z=8, N=7) 15O have Jπ=1/2- g.s.
1s1/2
0p1/2
0p3/2
In reality, 11Be g.s. is Jπ=1/2+ -- parity inversion
Very weakly bound: Eth=-0.5 MeV Halo state -dominated by 10Be-n in the S-wave
The
1/2-
state also bound -- only by 180 keV
0s1/2
Single particle interpretation
using nuclear shell model
Can we describe 11Be in ab initio calculations?
Continuum must be included
Does the 3N interaction play a role in the
parity inversion?
21
11Be
WITHIN NCSMC: DISCRIMINATION AMONG CHIRAL
NUCLEAR FORCES
A. Calci, P. Navratil, G. Hupin, S. Quaglioni, R. Roth et al. with IRIS collaboration, in preparation
preliminary
Parity inversion
22
FIRST STEPS TOWARDS AB INITIO
CALCULATIONS OF FUSION
G. Hupin, S. Quaglioni, P. Navrátil work in progress
d-t fusion
n-4He phaseshifts with NCSMC and the
chiral two- and three-nucleon force
Nmax=9
• Perspective to provide accurate
t(d,n)4He fusion cross-section for
the effort toward earth-based
fusion energy generation.
• The d-t fusion is known to be
very sensitive to the spin-orbit
and isospin part of the nuclear
interaction.
d-t channel
n+4He(g.s.) phase shifts with NN+3N potential,
l=2.0 fm-1, with eigenstates of 5He at Nmax =9.
23
FIRST STEPS TOWARDS AB INITIO
CALCULATIONS OF FUSION
G. Hupin, S. Quaglioni, P. Navrátil work in progress
Towards d-t fusion with NCSMC:
comparison between effective interactions
NCSM convergence of compound and
cluster states.
n+4He(g.s.) phase shifts with NN+3N potential,
with eigenstates of 5He.
24
FIRST STEPS TOWARDS AB INITIO
CALCULATIONS OF FUSION
G. Hupin, S. Quaglioni, P. Navrátil work in progress
Towards d-t fusion with NCSMC:
comparison between effective interactions
Relative error (%) with respect to
converged value
n+4He(g.s.) phase shifts with NN+3N potential,
with eigenstates of 5He.
A smaller frequency allows us to
capture the dilute nature of the 3/2+
resonance.
25
FIRST STEPS TOWARDS AB INITIO
CALCULATIONS OF FUSION
G. Hupin, S. Quaglioni, P. Navrátil work in progress
Towards d-t fusion with NCSMC:
comparison between effective interactions
The eigenvalue changes a huge
amount but the eigenvector only
changes a little.
n+4He(g.s.) phase shifts with NN+3N potential,
with eigenstates of 5He.
C. Johnson, “Spectral distribution theory and the
evolution of forces under the similarity
renormalization group” at workshop “Progress in Ab
Initio Techniques in Nuclear Physics” TRIUMF,
Canada
26
FIRST STEPS TOWARDS AB INITIO
CALCULATIONS OF FUSION
G. Hupin, S. Quaglioni, P. Navrátil work in progress
Ab initio d-t fusion cross-section
Preliminary..
Polarized cross-section
will be a valued input to
the design of energy
based fusion technology.
27
FIRST STEPS TOWARDS AB INITIO
CALCULATIONS OF FUSION
G. Hupin, S. Quaglioni, P. Navrátil work in progress
Ab initio d-t fusion cross-section
Cluster sector only
28
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+3N
interaction
Deuterium-nucleus collisions with
NN+3N interaction as the n-n
NCSMC for single- and two-nucleon
projectile
Work in progress:
Fusion reactions with our best
complete ab initio approach
The present NNN force is "incomplete”,
need to go to N3LO
Evolution of stars, birth, main sequence, death
29