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1
R-matrix theory of nuclear reactions
1. General definitions: types of reactions, cross sections, etc.
2. Reaction models (basics)
1. Single-channel potential/optical model (as simple as possible)
2. Phase-shift method
3. Generalizations (Coulomb, numerical calculation, spins, multichannel,
absorption)
3. Reaction models (advanced)
1. The CDCC method (Continuum Discretized Coupled Channel)
2. Technical aspects: R-matrix, Lagrange meshes
3. The Eikonal method
Reaction theories – Structure models for light nuclei
P. Descouvemont
Physique Nucléaire Théorique et Physique Mathématique, CP229,
Université Libre de Bruxelles, B1050 Bruxelles - Belgium
4. Structure models for light nuclei
1. Clustering in nuclei
2. Non-microscopic models
3. Microscopic cluster models
5. Recent applications
1. CDCC (11Be+64Zn, 9Be+208Pb, 7Li+208Pb)
2. Eikonal (three-body breakup, microscopic eikonal)
6. Nuclear astrophysics: brief overview
7. Conclusion
3
General context:
• Two-body systems
• Low energies (E ≲ Coulomb barrier), few open channels (one)
• Low masses (A ≲ 15-20)
• Low level densities (≲ a few levels/MeV)
• Reactions with neutrons AND charged particles
1. Introduction
Main questions to be addressed: determine the choice of the method
A. Type of reactions
• Elastic, inelastic, transfer, etc.
B. Energy range
• Partial wave expansion
• Number of open channels influences the absorption
C. Level densities
4
A. Different types of reactions
1. Elastic collision : entrance channel=exit channel
A+B A+B: Q=0
2. Inelastic collision Q≠
A+B A*+B (A*=excited state)
A+B A+B*
etc..
3. Transfer reactions
A+B C+D
A+B C+D+E
et …
4. Radiative capture reactions
A+B C + g
5. Breakup: main tool to investigate exotic nuclei
1. Introduction
5
B. Energy
Low energies (~ Coulomb barrier)
E
r
V
Partial-wave expansion
• Only a few partial waves contribute
ℓ >
ℓ =
a+16O
p+19F
d+18F
n+19Ne
20Ne
Single-channel calculation (no absorption)
1. Introduction
6
No partial-wave expansion (high E)
• Many partial waves
E
r
V
High energies
a+16O
p+19F
d+18F
n+19Ne
20Ne
Many open channels absorption important
1. Introduction
7
C. Level density
11C+p: Low level density (typical of exotic nuclei)
p+19F
19F+p: High level density (typical of stable nuclei)
different models
1. Introduction
8
Various models
r
V
ℓ=0
ℓ>0
Z1Z2e2/r
Barrier energy
RB~A11/3+A2
1/3
VB~Z1Z2e2/RB
Below the barrier
Very few ℓ values
R matrix, GCM
Far above the barrier
(Too) many ℓ values
no partial wave expansion
ex: Eikonal, semi-classic
Near the barrier
Limited ℓ values partial wave expansion
ex: Optical model, CDCC, DWBA, etc.
Nucleus-nucleus potential
Nuclear
Astrophysics
1. Introduction
9
2. Single-channel potential/optical model
10
Scheme of the collision (elastic scattering)
Before collision
After collision
q
Center-of-mass system
1
2
1
2
2. Single-channel optical model
11
A. Definitions
Schrödinger equation: � , � , …�� = � , � , …�� with > : scattering states
• A-body equation (microscopic models) = + , � − �
• Optical model: internal structure of the nuclei is neglected
the particles interact by a potential
absorption simulated by the imaginary part = optical potential� = − ℏ + � � = �• Additional assumptions: elastic scattering
no Coulomb interaction
spins zero
2. Single-channel optical model
r
r
12
Two contributions to the optical potential: nuclear and Coulomb
Typical nuclear potential: (short range, attractive)
• examples: Gaussian = − exp − /Woods-Saxon: = − +ex �−��
• parameters are fitted to experiment
• no analytical solution of the Schrödinger equation
-60
-50
-40
-30
-20
-10
0
0 2 4 6 8 10
V (
Me
V)
r (fm)
Woods-Saxon potential
=range (~sum of the radii)�= diffuseness (~0.5 fm)
Figure: =50 MeV, =5 fm, � = . fm
2. Single-channel optical model
13
Coulomb potential: long range, repulsive
• « point-point » potential : =• « point-sphere » potential : (radius )= f�r= − f�r
Total potential : = + : presents a maxium at the Coulomb barrier
• radius =• height =
2. Single-channel optical model
14
B. General solution � = − ℏ + � � = �with � = Φ + (Φ � corresponds to =0)
At large distances : � → ⋅� + � � �(with z along the beam axis)
where: �=wave number: � = � /ℏ=amplitude (scattering wave function is not normalized)� =scattering amplitude (length)
2. Single-channel optical model
Incoming plane wave
Outgoing spherical wave
15
C. Cross sections
q solid angle dW
Cross section:� = � , � = ∫ �
• Cross section obtained from the asymptotic part of the wave function
General problem for scattering states: the wave function must be known up to large
distances
• Direct p o le : dete i e s from the potential
• Inverse p o le : dete i e the pote tial V f o s• Angular distribution: E fixed, q variable
• Excitation function: q variable, E fixed,
2. Single-channel optical model
16
Main issue: determining the scattering amplitude � (and wave function � )
• Method 1: partial wave expansion: � = ,�o Must be determined for each partial wave phase-shit method
o At low energies, few partial waves
o � determined by the long-range part of �• Method 2: Formal theory- Lippman-Schwinger equation � = − ℏ ∫ exp − � ′ c�s �′ �′ �′
• equivalent to the Schrödinger equation
• � has a short range � is not necessary at large distances
• approximations: valid if is small or is large
o Born approximation : � = exp ⋅ �o Eikonal approximation � = exp ⋅ � �
2. Single-channel optical model
17
3. Phase-shift method: potential model
18
3. Phase-shift method: potential model
• Goal: solving the Schrodinger equation− ℏ + � � = �with a partial-wave expansion� = ℓ, ℓ ℓ ℓ
• Simplifying assumtions
• neutral systems (no Coulomb interaction)
• spins zero
• single-channel calculations elastic scattering
• Generalizations briefly illustrated in the next section
19
• The wave function is expanded as� = ℓ, ℓ ℓ ℓ• This provides the Schrödinger equation for each partial wave ( = → � = )
− ℏ − ℓ ℓ + ℓ + ℓ = ℓ• Large distances : → ∞, →
ℓ′′ − ℓ ℓ + ℓ + � ℓ = Bessel equati�Ω → ℓ = ℓ � , ℓ �• Remarks
– must be solved for all values
– at low energies: few partial waves in the expansion
– at small : ℓ → ℓ+
3. Phase-shift method: Definition, cross section
20
For small x: � → + ‼� → − − ‼+For large x: � → siΩ � − /� → − c�s � − /
Examples: � = si, � = − c s
3. Phase-shift method: Definition, cross section
At large distances: ℓ is a linear combination of ℓ � aΩd ℓ �ℓ → ℓ � − taΩ ℓ × ℓ �
With ℓ = phase shift (information about the potential):
If V=0 ℓ =
-1.5
-1
-0.5
0
0.5
1
1.5
0 5 10 15 20
j0(x)
n0(x)
21
Derivation of the elastic cross section
• Identify the asymptotic behaviours� → ⋅� + � � �� → ℓ ℓ ℓ � − taΩ ℓ × ℓ � ℓ ℓ+
• Provides coefficients ℓ and scattering amplitude �� , = ℓ=∞ ℓ + exp ℓ − ℓ c�s� , = � ,• Integrated cross section (neutral systems only)� = � ℓ=∞ ℓ + siΩ ℓ• In practice, the summation over ℓ is limited to some ℓ
3. Phase-shift method: Definition, cross section
22
factorization of the dependences in and qlow energies: small number of ℓ values ( ℓ → when ℓ increases)
high energies: large number ( alternative methods)
General properties of the phase shifts
1. The phase shift (and all derivatives) are continuous functions of
2. The phase shift is known within np: exp = exp +3. Levinson theorem• ℓ = is arbitrary
• ℓ − ℓ ∞ = N , where N is the number of bound states in partial wave ℓ• Example: p+n, ℓ = : − ∞ = (bound deuteron)ℓ = : − ∞ = (no bound state for ℓ = )
3. Phase-shift method: Definition, cross section� , = � , with � , = � ℓ=∞ ℓ + exp ℓ − ℓ c�s
23
• Example: hard sphere (radius a)
• continuity at = � ℓ �� − taΩ ℓ × ℓ �� = taΩ ℓ = ℓℓ = −��
At low energies: ℓ → − ℓ+ℓ+ ‼ − ‼ , in general: ℓ ∼ � ℓ+ Strong difference between ℓ = (no barrier) et ℓ ≠ (centrifugal barrier)
3. Phase-shift method: example
V(r)
ra
24
example : a+n phase shift ℓ =consistent with the hard sphere (� ∼ . fm)
3. Phase-shift method: example
25
Resonances: ≈ ataΩ Γ�− = Breit-Wigner approximation
ER=resonance energyG=resonance width
p3p/4
p/4
p/2
d(E)
ER ER+G/2ER-G/2E
• Narrow resonance: G small
• Broad resonance: G large
• Bound states: G =0 ( < )
3. Phase-shift method: resonances
26
Cross section� = ℓ ℓ + exp ℓ − maximum for =Near the resonance: � ≈ ℓ + Γ /�− +Γ / , where ℓ =resonant partial wave
ER
s G
E
In practice:
• Peak not symmetric (G depends on E)
• « Background » neglected (other ℓ values)
• Differences with respect to Breit-Wigner
3. Phase-shift method: resonances
27
Example: n+12C
Ecm=1.92 MeVG= 6 keV
Comparison of 2 typical times:
a. Lifetime of the resonance: � = ℏ/ ≈ . × . − ≈ . × −b. Interaction time without resonance: � = / ≈ . × − tNR<< tR
3. Phase-shift method: resonances
28
3. Phase-shift method: resonances
Narrow resonances
• Small particle width
• long lifetime
• can be approximetly treated by neglecting the asymptotic behaviour of the wave
function
proton width=32 keV
can be described in a bound-state
approximation
29
Broad resonances
• Large particle width
• Short lifetime
• asymptotic behaviour of the wave function is important
rigorous scattering theory
bound-state model complemented by other tools (complex scaling, etc.)
3. Phase-shift method: resonances
ground state unstable: G=120 keV
very broad resonance: G=1990 keV
30
4. Generalizations
• Extension to charged systems
• Numerical calculation
• Optical model (high energies absorption)
• Extension to multichannel problems
31
Generalization 1: charged systems
Scattering energy E
: weak coulomb effects ( negligible with respect to )< : strong coulomb effects (ex: nuclear astrophysics)
4. Generalizations
32
A. Asymptotic behaviour
Neutral systems
− ℏ + − � =� → exp ⋅ � + � exp �Charged systems
− ℏ + + − � =�→ exp ⋅ � + � lΩ ⋅ � − �+ � exp � − � lΩ �
� = ℏ• Sommerfeld parameter
• « measurement » of coulomb effects
• Increases at low energies
• Decreases at high energies
4. Generalizations
33
B. Phase shifts with the coulomb potential
Neutral system: − ℓ ℓ+ + � ℓ =Bessel equation : solutions ℓ � , ℓ �
Charged system: − ℓ ℓ+ − + � ℓ = :
Coulomb equation: solutions ℓ �, � , ℓ �, �
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 5 10 15 20 25 30eta=0
eta=1
eta=5
eta=10
�0(�,�)
-4
0
4
8
12
0 5 10 15 20 25 30
eta=0
eta=1
eta=5
eta=10
0 (�,�)
�
4. Generalizations
34
• Incoming and outgoing functions (complex)ℓ �, � = ℓ �, � − ℓ �, � → − −ℓ�− l +�ℓ : incoming waveℓ �, � = ℓ �, � + ℓ �, � → −ℓ�− l +�ℓ : outgoing wave
• Phase-shift definition
o neutral systems ∶ ℓ → ℓ � − taΩ ℓ ℓ �o charged systems: ℓ → ℓ �, � + taΩ ℓ ℓ �, �→ c�s ℓ ℓ �, � + siΩ ℓ ℓ �, �→ ℓ �, � − ℓ ℓ �, �
3 equivalent definitions (amplitude is different)
Collision matrix (=scattering matrix)ℓ = ℓ : module | ℓ| =
4. Generalizations
35
Example: hard-sphere potential
= � > � � < �
phase shift: taΩ ℓ = − ℓ ,ℓ ,r
a
V
-540
-360
-180
0
0 0.5 1 1.5 2 2.5
ℓ=ℓ=ℓ=�=2
�=0
� (��− )
�=4 fm
4. Generalizations
36
C. Rutherford cross section
For a Coulomb potential ( = ):
• scattering amplitude : � = − si / � − l si /• Coulomb phase shift for ℓ = : � = arg + �We get the Rutherford cross section: � = � = siΩ /• Increases at low energies
• Diverges at = no integrated cross section
4. Generalizations
37
D. Cross sections with nuclear and Coulomb potentials
• The general defintions� = � ℓ=∞ ℓ + exp ℓ − ℓ c�s� = �are still valid
• r�bleΨ ∶ very sl�w c�ΩvergeΩce with ℓ separation of the nuclear and coulomb phase shiftsℓ = ℓ + �ℓ�ℓ = arg + ℓ + �
• Scattering amplitude � written as � = � + �• � = ℓ=∞ ℓ + exp �ℓ − ℓ c�s = − si / � − l si /
analytical
• � = ℓ=∞ ℓ + exp �ℓ exp ℓ − ℓ c�s converges rapidly
4. Generalizations
38
Total cross section: � = � = � + �
• Nuclear term dominant at
• Coulomb term coulombien dominant at small angles used to normalize experiments
• Coulomb amplitude strongly depends on the angle � Ω �� Ω• Integrated cross section ∫ � d is not defined
System 6Li+58Ni
• =• Coulomb barrier∼ . ∼ MeV• Below the barrier: � ∼ �• Above : � is different from �
4. Generalizations
39
with:
Numerical solution : discretization N points, with mesh size h
• =• ℎ = (or any constant)
• ℎ is determined numerically from and ℎ (Numerov algorithm)
• ℎ ,… ℎ• for large r: matching to the asymptotic behaviour phase shift
Bound states: same idea (but energy is unknown)
Generalization 2: numerical calculation
For some potentials: analytic solution of the Schrödinger equation
In general: no analytical solution numerical approach
4. Generalizations
40
-20
-15
-10
-5
0
5
10
0 1 2 3 4 5 6 7 8
r (fm)
V (
Me
V)
nuclear
Coulomb
Total
Example: a+a
0+
E=0.09 MeVG=6 eV
4+
E~11 MeVG~3.5 MeV
2+
E~3 MeVG~1.5 MeV
-50
0
50
100
150
200
0 5 10 15
Ecm (MeV)
de
lta
(d
eg
res)
l=0
l=2
l=4
Experimental spectrum of 8Be
Experimental phase shifts
Potential: VN(r)=-122.3*exp(-(r/2.13)2)
a+a
VB~1 MeV
4. Generalizations
41
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 10 20 30 40 50 60 70
r (fm)
fon
cti
on
d'o
nd
e
-1.5
-1
-0.5
0
0.5
1
1.5
0 10 20 30 40 50 60 70
r (fm)
fon
cti
on
d'o
nd
ea+a wave function for ℓ =
E=0.2 MeV
• <• Small amplitude for r small
E=1 MeV
• ≈41
4. Generalizations
42
Generalization 3: complex potentials
Goal: to simulate absorption channels
a+16O
p+19F
d+18F
n+19Ne
20Ne
High energies:
• many open channels
• strong absorption
• potential model extended to complex
potentials (« optical »)
Phase shift is complex: = +collision matrix: = exp = � exp
where � = exp − <Elastic cross section
Reaction cross section:
4. Generalizations
43
In astrophysics, optical potentials are used to compute fusion cross sections
Fusion cross section: includes many channels
Example: 12C+12C: Essentially 20Ne+a, 23Na+p, 23Mg+n channels
absorption simulated by a complex potential = +
many
states
E
12C+12C
experimental cross section
Satkowiak et al. PRC 26 (1982) 2027
4. Generalizations
44
Origin of the complex potential
• Time-dependent Schrödinger equation: ℏ = = +is equivalent to (real potential): + div = ,with = � constant current J
for a complex potential: = +�� + div = ℏVI < simulates absorption (inelastic, transfer, etc) not explicitly included
Simple interpretation
• Let us assume a constant potential = − wave function=plane wave ∼ exp � ∼ exp +ℏ =
• For a complex potential = − − ( small)
wave function ∼ exp � exp −� : ∼ exp − � incoming particles « disappear » (=absorption)
4. Generalizations
45
Generalization 4 : system with spins (multichannel)
• Allows to deal with inelatic, transfer, etc..
• Phase shift (single-channel) collision (scattering) matrix
A. Quantum numbers
• Good quantum numbers: total angular momentum and parity
• Additional indices
• Channel � defined by 2 nuclei with spins , et parités ,• Channel spin = +• Relative angular momentum ℓ
with = + ℓ= − ℓExamples:
1) a+n : =0, =1/2 =1/2, ℓ = | − | or + : channel number =1
2) p+n : = =1/2 = or 1: channel number depends on
4. Generalizations
46
3) Reaction �i + → �e + �• channel 1: �i + , spin(6Li) =1+, spin(p) =1/2+
• channel 2: �e + �, spin (3He)=1/2+, spin(�)=0+
Size of the collision matrix is: 3x3 or 4x4
channel � = channel � = � ℓ/ + = / , ℓ == / , ℓ = = / , ℓ = 3 values
/ − = / , ℓ == / , ℓ = = / , ℓ = 3 values
/ + = / , ℓ == / , ℓ = , = / , ℓ = 4 values
/ − = / , ℓ == / , ℓ = , = / , ℓ = 4 values
4. Generalizations
47
B. Coupled-channel wave functions
1. Internal wave functions of nuclei 1 and 2 : Φ and Φ2. Coupling of the projectile+target spins: = ⊕Φ = < | >Φ Φ = Φ ⊗Φ3. Channel function is defined by ( = ℓ⊕ )ℓ = Φ ⊗ ℓ4. Total wave function for given and := ℓ ℓ ℓ
Φ Φ� = ,4. Generalizations
48
4. Total wave function for given and : = ℓ ℓ ℓ5. Radial functions ℓ are obtained from a set of coupled equations
• Single-channel: − ℏ − ℓ ℓ+ ℓ + ℓ = ℓWith → ℓ �, � − ℓ ℓ �, �ℓ = exp ℓ = « matrix » 1x1
• Multichannel
− ℏ − ℓ ℓ+ ℓ + ′ ′ℓ′ ℓ, ′ ′ℓ′ ′ ′ℓ′ = ℓwith ℓ → ℓ − ℓ, ′ ′ℓ′ ℓ′Collision matrix provides cross sections (several J values are necessary)
4. Generalizations
49
Comments on the multichannel system
− ℏ − ℓ ℓ+ ℓ + ′ℓ′ ℓ, ′ ′ℓ′ ′ ′ℓ′ = ℓo Standard form of many scattering theories (CDCC, folding, microscopic, 3-body, etc.)
o Theories differ by the calculation of the potentials
o Diagonal and non-diagonal potentials ℓ, ′ ′ℓ′non-diagonal ℓ, ′ ′ℓ′ → for large r
diagonal ℓ, → � �for large r
o Main problems
Sometimes: more than 100 channels are included
Long range of the potential numerical difficulties
o Numerical resolution can be time consuming
2 main methods: Numerov (+ improvements)
R-matrix method
4. Generalizations
50
C. Cross sections in a multichannel formalism
One channel:� = �� = ℓ=∞ ℓ + exp ℓ − ℓ c�s
Multi channel:� � → �′ = ′ ′ � , ′ ′
� , ′ ′ = , ′ ′… ℓ, ′ ′ℓ′ ′ ,
With: =spin orientations in the entrance channel′ ′=spin orientations in the exit channel
Collision matrix
• generalization of d: Uij=hijexp(2idij)
• determines the cross section
4. Generalizations
