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1944-24
Joint ICTP-IAEA Workshop on Nuclear Reaction Data for AdvancedReactor Technologies
Pierre Descouvemont
19 - 30 May 2008
Universite Libre de BruxellesPhysique Nucleaire Thoerique
Cp 2291050 Brussels
BELGIUM
R-matrix theory of nuclear reactions.
1
Pierre Descouvemont
Université Libre de Bruxelles, Brussels, Belgium
R-matrix theory of nuclear reactions
1. Introduction
2. General collision theory: elastic scattering
3. Phase-shift method
4. R-matrix theory ( theory)
5. Phenomenological R matrix ( experiment)
6. Conclusions
2
1.Introduction
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
E
r
V
ℓ=0
ℓ>0
Only a few partial waves
contribute
3
Low-energy reactions
• Transmission (“tunnelling”) decreases with ℓ → few partial waves
→ semi-classic theories not valid
• astrophysics
• thermal neutrons
• Wavelength λ=ℏc/Eλ(fm)~197/E(MeV)
example: 1 MeV: λ ~200 fm
radius~2-4 fm quantum effects important
Some references:
•C. Joachain: Quantum Collision Theory, North Holland 1987
•L. Rodberg, R. Thaler: Introduction to the quantum theory of scattering, Academic Press 1967
•J. Taylor : Scattering Theory: the quantum theory on nonrelativistic collisions, John Wiley 1972
4
Different types of reactions
1. Elastic collision : entrance channel=exit channel
A+B A+B: Q=0
2. Inelastic collision (Q≠0)
A+B A*+B (A*=excited state)
A+B A+B*
A+B A*+B*
etc..
3. Transfer reactions
A+B C+D
A+B C+D+E
A+B C*+D
etc…
4. radiative capture reactions
A+B C + γ
NOT covered here
covered here
5
Assumptions: elastic scattering
no internal structure
no Coulomb
spins zero
Before collisionAfter collision
θ
Center-of-mass system
2. Collision theory: elastic scattering
1
2
1
2
6
Schrödinger equation: V(r)=interaction potential
Assumption: the potential is real and decreases faster than 1/r
Scattering wave functions
A large distances : V(r) 0
2 independent solutions :
Incoming plane wave Outgoing spherical wave
where: k=wave number: k2=2µE/2
A=amplitude (scattering wave function is not normalized)
f(θ) =scattering amplitude (length)
7
Cross sections
θ solid angle dΩ
Cross section
• Cross section obtained from the asymptotic part of the wave function
• “Direct” problem: determine σ from the potential
• “Inverse” problem : determine the potential V from σ
• Angular distribution: E fixed, θ variable
• Excitation function: θ variable, E fixed,
8
How to solve the Schrödinger equation for E>0?
With (z along the beam axis)
Several methods:
• Formal theory: Lippman-Schwinger equation approximations
• Eikonal approximation
• Born approximation
• Phase shift method: well adapted to low energies
• Etc…
9
Lippman-Schwinger equation:
r is supposed to be large (r >> r')
Ψ(r) must be known
Ψ(r) only needed where V(r)≠0 approximations are possible
Born approximation :
Eikonal approximation:
is solution of the Schrödinger equation
With =Green function
Valid at high energies:
V(r) ≪ E
10
3. Phase-shift method
a. Definitions, cross sections
Simple conditions: neutral systems
spins 0
single-channel
b. Extension to charged systems
c. Extension to multichannel problems
d. Low energy properties
e. General calculation
f. Optical model
11
The wave function is expanded as
,
( )(r) ( )ml
l
l m
u rY
rΨ = Ω∑
When inserted in the Schrödinger equation
)()()()()1(
2 22
22
rEururVrur
ll
dr
dlll =+⎟⎟⎠
⎞⎜⎜⎝
⎛ +−−µ
To be solved for any potential (real)
For r∞ V(r) =0
2
22
2
2 with,0
1
Ekrukru
r
llu lll
µ==++− )()()("
=Bessel equation ul(r)= jl(kr), nl(kr)
3.a Definition, cross section
12
x
xxn
x
xxj
)cos()(,
)sin()( 00 −==
1
!)!12()(
!)!12()(
+
−−→
+→
ll
l
l
x
lxn
l
xxj
)(tan)()( krnkrjru llll ∗−→ δ
For small x
)2/cos(1
)(
)2/sin(1
)(
π
π
lxx
xn
lxx
xj
l
l
−−→
−→For large x
Examples:
At large distances: ul(r) is a linear combination of jl(kr) and nl(kr):
With δl = phase shift (information
about the potential)
If V=0 δ=0
[ ]2 1| ( , ) | , with ( , ) (2 1) exp(2 ( )) 1 (cos )
2l l
l
df E f E l i E P
d ik
σ θ θ δ θ= = + −Ω ∑
Cross section:
Provide the cross section
13
factorization of the dependences in E and θ
General properties of the phase shifts
1. Expansion useful at low energies: small number of ℓ values
2. The phase shift (and all derivatives) are continuous functions of E
3. The phase shift is known within nπ: exp(2iδ)=exp(2i(δ+nπ))
4. Levinson theorem
• δl(E=0) is arbitrary
• δ(E=0) - δ(E=∞)= Nπ , where N is the number of bound states
• Example: p+n, ℓ =0: δ(E=0) - δ(E=∞)= π (bound deuteron)
[ ]2
0
1| ( , ) | , with ( , ) (2 1) exp(2 ( )) 1 (cos )
2l l
l
df E f E l i E P
d ik
σ θ θ δ θ∞
=
= = + −Ω ∑
14
Interpretation of the phase shift from the wave function
δl<0 V repulsive
δl>0 V attractive
u(r)
r
V=0 u(r)sin(kr-lπ/2)
δl=0
V≠0 u(r)sin(kr-lπ/2+δl)
15
Resonances: =Breit-Wigner approximation
ER=resonance energy
Γ=resonance width
π
3π/4
π/4
π/2
δ(E)
ER ER+Γ/2ER-Γ/2E
• Narrow resonance: Γ small
• Broad resonance: Γ large
Example: 12C+p
With resonance: ER=0.42 MeV, Γ=32 keV lifetime: t=/Γ~2x10-20 s
Without resonance: interaction range d~10 fm interaction time t=d/v ~1.1x10-21 s
16
0)()()1( 2
2
" =++− rukrur
llu lll
3.b Generalization to the Coulomb potential
The asymtotic behaviour Ψ(r)→ exp(ikz) + f(θ)*exp(ikr)/r
Becomes: Ψ(r)→ exp(ikz+η log(kr)) + f(θ)*exp(ikr-η log(2kr))/r
With η=Z1Z2e2/ℏv =Sommerfeld parameter, v=velocity
0)()2()()1( 2
2
" =−++− rukkrur
llu lll η
Bessel equation:
Coulomb
equation
Solutions: Fl(η,kr): regular, Gl(η,kr): irregular
Ingoing, outgoing functions: Il= Gl-iFl, Ol=Gl+iFl
( ) ( ) tan ( )
( ) ( ),with exp(2 )
( )*cos ( )*sin
l l l l
l l l l l
l l l l
u r F kr G kr
I kr U O kr U i
F kr G kr
δδ
δ δ
→ + ∗→ − ∗ =→ +
)exp()(
)exp()(
ixxO
ixxI
l
l
+→−→
17
Example: hard sphere
V(r)
ra
V(r)= ∞ for r<a
=Z1Z2e2/r for r>a
)(
)(tan0)(
)(tan)()(
kaG
kaFau
krGkrFru
l
lll
llll
−=→=
+=
δ
δ
= Hard-Sphere
phase shift
-150
-120
-90
-60
-30
0
30
0 1 2 3 4 5 6
Ecm (MeV)
ha
rd-s
ph
ere
l=0
l=2
α+12C
-150
-120
-90
-60
-30
0
30
0 2 4 6
Ecm (MeV)
ha
rd-s
ph
ere
l=0
l=2
p+12
C
-150
-120
-90
-60
-30
0
30
0 1 2 3 4 5 6
Ecm (MeV)
ha
rd-s
ph
ere
l=0
l=2
n+12
C
18
example : α+n phase shift l=0
Special case: Neutrons, with l=0:
F0(x)=sin(x), G0(x)=cos(x)
δ=-ka
hard sphere is a good approximation
19
Elastic cross section with Coulomb:
)(cos)1)2)(exp(12(2
1)(with,|)(| 2 θδθθσ
l
l
l Pilik
ffd
d −+==Ω ∑Still valid, but converges very slowly.
2
2
1( ) (2 1)[exp(2 ) 1] (cos )
2
1(2 1)[exp(2 ) exp(2 ) exp(2 ) 1] (cos )
2
( ) ( )
( ) exp( log(sin / 2)) Coulomb amplitude2 sin / 2
N C
l l l
l l
l
C C
l l l l
l
N C
C
f l i Pik
l i i i Pik
f f
f ik
δ δ δ
θ δ θ
δ δ δ θ
θ θηθ η θ
θ
= +
= + −
= + − + −
= +−= − =
∑
∑
Coulomb: exactNuclear
2|)(| θσC
C fd
d =Ω
= Coulomb cross section: diverges for θ=0
increases at low energies
20
2
11
42
2
θθσ
sin
~|)(|E
fd
dC
C =Ω
• The total Coulomb cross section is not defined (diverges)
• Coulomb is dominant at small angles
used to normalize data
• Increases at low energies
• Minimum at θ=180o nuclear effect maximum
)()()( θθθ NC fff +=
•fC(θ): Coulomb part: exact
•fN(θ): nuclear part: converges rapidly
1
10
100
1000
10000
100000
1000000
0 30 60 90 120 150 180
θ (degrees)
2
1
4 θsin
21
• One channel: phase shift δ → U=exp(2iδ)
• Multichannel: collision matrix Uij, (symmetric , unitary) with i,j=channels
• Good quantum numbers J=total spin π=total parity
• Channel i characterized by I=I1⊕ I2 =channel spin
J=I ⊕ l l =angular momentum
• Selection rules: |I1- I2| ≤ I ≤ I1+ I2| l - I| ≤J ≤ l + I
π=π1∗π2∗ (-1) l
Example of quantum numbers
1, 2
1, 2
0, 1
0, 1
l
1
1
1
1
size
1/23/2+
1/23/2-
1/21/2-
1/21/2+
IJ
X
X
X
X
0, 1, 2
1, 2, 3
0, 1, 2
1, 2, 3
1
2
1
2
l
3
3
1
1
size
1
2
1+
1
2
1-
1
2
0-
1
2
0+
IJ
α+3He α=0+, 3He=1/2+ p+7Be 7Be=3/2-, p=1/2+
X
X
X
X
XX
XX
3.c Extension to multichannel problems
22
)(cos)))(exp(()(|)(| θδθθσl
l
l Pilik
ffd
d1212
2
1with,2 −+==
Ω ∑Cross sections
Multichannel
With: K1,K2=spin orientations in the entrance channel
K’1,K’2=spin orientations in the exit channel
∑=Ω ''
''
,,,,
|)(|
2121
2121
2
KKKKKKKK
fd
d θσ
One channel:
),(....)( '
, '',
'',,'' 02121
θθπ
πl
J IllI
J
IllIKKKKYUf ∑ ∑=
Collision matrix
• generalization of δ: Uij=ηijexp(2iδij)
• determines the cross section
23
Then, for very low E (neutral system):
One defines =logarithmic derivative at r=a (a large)
3.d Low-energy properties of the phase shifts
For ℓ=0:
E
δ
l=0, α<0
l=0, α>0
l>0
strong difference
between l=0 and l>0
Generalization α=scattering length
re=effective range
24
Cross section at low E:
In general (neutrons):
Thermal neutrons (T=300K, E=25 meV): only l=0 contributes
example : α+n phase shift l=0
At low E: δ~-kα with α>0 replusive
25
3.e General calculation
with:
Numerical solution : discretization N points, with mesh size h
• ul(0)=0
• ul(h)=1 (or any constant)
• ul(2h) is determined numerically from u
l(0) and u
l(h) (Numerov algorithm)
• ul(3h),… u
l(Nh)
• for large r: matching to the asymptotic behaviour phase shift
Bound states: same idea
For some potentials: analytic solution of the Schrödinger equation
In general: no analytic solution numerical approach
26
-20
-15
-10
-5
0
5
10
0 1 2 3 4 5 6 7 8
r (fm)
V(M
eV
)
nuclear
Coulomb
Total
Example: α+α
0+
E=0.09 MeV
Γ=6 eV
4+
E~11 MeV
Γ~3.5 MeV
2+
E~3 MeV
Γ~1.5 MeV
-50
0
50
100
150
200
0 5 10 15
Ecm (MeV)
de
lta
(de
gre
s)
l=0
l=2
l=4Experimental spectrum of 8Be
Experimental phase shifts
Potential: VN(r)=-122.3*exp(-(r/2.13)2)
α+α
VB~1 MeV
27
Goal: to simulate absorption channels
α+16O
p+19F
d+18F
n+19Ne
20Ne
High energies:
• many open channels
• strong absorption
• potential model extended to complex
potentials (« optical »)
3.f Optical model
Phase shift is complex: δ=δR+iδI
collision matrix: U=exp(2iδ)=ηexp(2iδR)
where η=exp(-2δI)<1
Elastic cross section
Reaction cross section:
28
4. The R-matrix Method
Goals:
1. To solve the Schrödinger equation (E>0 or E<0)
1. Potential model
2. 3-body scattering
3. Microscopic models
4. Many applications in nuclear and atomic physics
2. To fit cross sections
1. Elastic, inelastic spectroscopic information on resonances
2. Capture, transfer astrophysics
References:
• A.M. Lane and R.G. Thomas, Rev. Mod. Phys. 30 (1958) 257
• F.C. Barker, many papers
29
Standard variational calculations
• Hamiltonian
• Set of N basis functions ui(r) with
Calculation of Hij=<ui|H|uj> over the full space
Nij=<ui|uj>
(example: gaussians: ui(r)=exp(-(r/ai)2))
• Eigenvalue problem : → upper bound on the energy
• But: Functions ui(r) tend to zero → not directly adapted to scattering states
H EΨ = Ψ
)()( rucr i
i
i∑=Ψ
0)( =−∑i
iijij cENH
Principles of the R-matrix theory
0
0
( ) ( )
( )
ij i j
ij i j
N u r u r dr
H u T V u dr
∞
∞
=
= +
∫
∫
30
b. Wave functions
Set of N basis functions ui(r)
correct asymptotic behaviour (E>0 and E<0)
Extension to the R-matrix: includes boundary conditions
a. Hamiltonian
2
1 2, w ithZ Z e
H E H Tr
Ψ = Ψ → +
( )
( ) ( ) valid in a limited range
with ( ) ( ) ( )m
i
l
i
l l
i
lI
r c
kr U O kr Y
u rΨ
Ψ ∗
=
→ − Ω
∑
Principles of the R-matrix theory
31
• Spectroscopy: short distances only
Collisions : short and large distances
• R matrix: deals with collisions
• Main idea: to divide the space into 2 regions (radius a)
• Internal: r ≤ a : Nuclear + coulomb interactions
• External: r > a : Coulomb only
Internal region
16O
Entrance channel
12C+α
Exit channels
12C(2+)+α
15N+p, 15O+n
12C+α
CoulombNuclear+Coulomb:
R-matrix parametersCoulomb
32
Here: elastic cross sections
Cross sections
Phase shifts
R matrix
Potential
model
Microscopic
models
Etc,….
measurements
models
33
Plan of the lecture:
• The R-matrix method: two applications
Solving the Schrödinger equation (essentially E>0)
Phenomenological R-matrix (fit of data)
• Applications
• Other reactions: capture, transfer, etc.
34
c. R matrix
• N basis functions ui(r) are valid in a limited range (r ≤ a)
⇒ in the internal region:
arfor, ≤=Ψ ∑=
N
i
ii rucr1
)()(int
• At large distances the wave function is Coulomb
⇒ in the external region:
( ) arfor,)()()( >∗−=Ψ krOUkrIAr lllext
• Idea: to solve
with defined in the internal region
0)( =−∑i
iijij cENH
int
int
|
||
>=<
>=<
jiij
jiij
uuN
uHuH
• But T is not hermitian
over a finite domain,int in t
2 2
2 2
0 0
| | | |
, if a
i j j i
a a
i j j i
u T u u T u
d du u dr u u drdr dr
< > ≠< >
≠ ≠ ∞∫ ∫
(ci=coefficients)
35
• Bloch-Schrödinger equation:
• With L(L0)=Bloch operator (surface operator) – constant L0 arbitrary
intint |)(||)(| >+=<>+< ijji uLTuuLTu00
LL
• Now we have T+L(L0) hermitian:
• Since the Bloch operator acts at r=a only:
• Here L0=0 (boundary-condition parameter)
36
• Summary:
• Using (2) in (1) and the continuity equation:
Dij(E)
Provides ci
(inversion of D)
Continuity at r=a
37
With the R-matrix defined as
If the potential is real: R real,
|U|=1, δ real
Procedure (for a given ℓ):
1. Compute matrix elements
2. Invert matrix D
3. Compute R matrix
4. Compute the collision matrix U
38
Neutral system:Charged system:
1.E-10
1.E-08
1.E-06
1.E-04
1.E-02
1.E+00
1.E+02
-1 0 1 2
l=2l=0
-4
-3
-2
-1
0
-1 0 1 2
l=2
l=0
-1
0
1
2
-1 0 1 2
l=2
l=0
-3
-2
-1
0
1
-1 0 1 2
l=2
l=0
Sl
Pl
E (MeV)
α + nα + 3He
S(E)=shift factor
P(E)=penetration factor
39
Interpretation of the R matrix
From (1):
Then: =inverse of the logarithmic derivative at a
40
Example
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8
r
u(r
)
Can be done exactly
(incomplete γ function)
Matrix elements of H can be calculated analytically for gaussian potentials
Other potentials: numerical integration
l=0
Gaussians with different widths
41
Input data
• Potential
• Set of n basis functions (here gaussians with different widths)
• Channel radius a
Requirements
• a large enough : VN(a)~0
• n large enough (to reproduce the internal wave functions)
• n as small as possible (computer time)
compromise
Tests
• Stability of the phase shift with the channel radius a
• Continuity of the derivative of the wave function
42
Results for α+α
• potential : V(r)=-126*exp(-(r/2.13)2) (Buck potential)
• Basis functions: ui(r)=rl*exp(-(r/ai)2)
with ai=x0*a0(i-1) (geometric progression)
typically x0=0.6 fm, a0=1.4
-20
-15
-10
-5
0
5
10
0 1 2 3 4 5 6 7 8
r (fm)
V(M
eV
)
nuclear
Coulomb
Total
potential
VN(a)=0
a > 6 fm
43
-300
-250
-200
-150
-100
-50
0
0 5 10 15 20 25
del exact
n=20
n=15
n=10
n=9
a=10 fm
-400
-350
-300
-250
-200
-150
-100
-50
0
0 5 10 15 20 25
del exact
n=15
n=10
n=8
n=7
a=7 fm
-300
-250
-200
-150
-100
-50
0
0 5 10 15 20 25
del exact
n=7
n=10
n=6
n=15
a=4 fm
• a=10 fm too large (needs too many basis functions)
• a=4 fm too small (nuclear not negligible)
• a=7 fm is a good compromise
Elastic phase shifts
44
Wave functions at 5 MeV, a= 7 fm
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10 12
exact w f
internal
external
n=15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8
ng=7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8
ng=15
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10 12
exact w f
internal
external
n=7
-400
-350
-300
-250
-200
-150
-100
-50
0
0 5 10 15 20 25
del exact
n=15
n=10
n=8
n=7
a=7 fm
45
Other example : sine functions
• Matrix elements very simple
• Derivative ui’(a)=0
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10 12
a=7 fm
-300
-250
-200
-150
-100
-50
0
0 5 10 15 20
del exact
ng=15
ng=25
ng=35
ng=10
Not a good basis (no flexibility)
46
Example of resonant reaction: 12C+p
• potential : V=-70.5*exp(-(r/2.70)2)
• Basis functions: ui(r)=rl*exp(-(r/ai)2)
Resonancel=0
E=0.42 MeV
Γ=32 keV
-20
-15
-10
-5
0
5
10
15
20
0 2 4 6 8
r (fm)
V(M
eV
)nuclear
Coulomb
Total
V=-70.50*exp(-(r/2.7)2)
47
-20
0
20
40
60
80
100
120
140
160
180
0 0.5 1 1.5 2 2.5 3
E (MeV)
delta
(deg
.)
exact
ng=10
ng=8
ng=7
Phase shifts a=7 fm
Wave functions
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
0 5 10 15
r (fm)
exact w f
internal
external
ng=7
E=0.8 MeV
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
0 5 10 15
r (fm)
exact w f
internal
external
ng=10
E=0.8 MeV
48
Resonance energies
with
• Resonance energy Er defined by δR=90o
In general: must be solved numerically
• Resonance width Γ defined by
Hard-sphere phase shift
R-matrix phase shift
-100
-50
0
50
100
150
200
0 0.5 1 1.5 2 2.5 3
E (MeV)
de
lta
(de
g.)
delta_tot
Hard Sphere
delta_R
49
Application of the R-matrix to bound states
)()2(2/1,
2
21
Ω→Ψ
+→
Ψ=Ψ
+−m
ll YkrW
r
eZZTH
EH
η
0)()2()()1( 2
2
" =−++− rukkrur
llu lll ηPositive energies: Coulomb functions F
l, Gl
0)()2()()1( 2
2
" =−−++− rukkrur
llu lll ηNegative energies: Whittaker functions
Asymptotic behaviour:
ηη
ηπ
ηπ
x
xxW
xxxG
xxxF
)exp()2(
)2log2
cos()(
)2log2
sin()(
2/1,
−→
−−→
−−→
+−
Starting point of the R matrix:
, 1/2 (2 )W xη− +
50
• R matrix equations
• Using (2) in (1) and the continuity equation:
Dij(E)
• But: L depends on the energy, which is not known iterative procedure
With C=ANC (Asymptotic Normalization Constant): important in “external” processes
51
Application to the ground state of 13N=12C+p
Resonance
l=0: E>0
Bound state
l=1: E<0
• Potential : V=-55.3*exp(-(r/2.70)2)
• Basis functions: ui(r)=rl*exp(-(r/ai)2) (as before)
Negative energy: -1.94 MeV
52
-0.406-0.379Right derivative
-0.405-1.644Left derivative
-1.942Exact
-1.942-1.498Final
-1.942
-1.942-1.4983
-1.937-1.4982
-2.190-1.5001
ng=10ng=6Iteration
Calculation with a=7 fm
•ng=6 (poor results)
•ng=10 (good results)
53
Wave functions (a=7 fm)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 2 4 6 8 10 12
r (fm)
exact w f
internal
external
ng=6
0.00001
0.0001
0.001
0.01
0.1
1
0 2 4 6 8 10 12
r (fm)
exact w f
internal
external
ng=6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 2 4 6 8 10 12
r (fm)
exact w f
internal
external
ng=10
0.0001
0.001
0.01
0.1
1
0 2 4 6 8 10 12
r (fm)
exact w f
internal
externalng=10
54
5. Phenomenological R matrix
• Goal: fit of experimental data
• Basis functions
• R matrix equations
with
The choice of the basis functions is arbitrary
BUT must be consistent in R(E) and D(E)
55
Change of basis
• Eigenstates of over the internal region
• expanded over the same basis:
• standard diagonalization problem
Instead of using ui(r), one uses Ψl(r)
56
With = reduced width
Completely equivalent
2 steps: computation of eigen-energies Eλ (=poles) and eigenfunctions
computation of reduced widths from eigenfunctions
Remarks:
•Eλ=pole energy: different from the resonance energy
depend on the channel radius
• γλ proportional to the wave function at a measurement of clustering (depend on a!)
large γλ strong clustering
• dimensionless reduced width = Wigner limit
• http://pntpm.ulb.ac.be/Nacre/Programs/coulomb.htm: web page to compute reduced
widths
57
0
60
120
180
0 1 2 3E (MeV)
de
lta
(de
g.)
Example : 12C+p
• potential : V=-70.5*exp(-(r/2.70)2)
• Basis functions: ui(r)=rl*exp(-(r/ai)2) with ai=x0*a0
(i-1)
10 basis functions, a=8 fm
10 eigenvalues
0.019883629.6709
0.097637295.231
1.143461153.8294
4.157215107.7871
1.12159365.62125
0.01054140.54732
0.73085226.50339
1.0949137.873401
0.3922820.112339
2.38E-05-27.9454
gl2El
-15
-10
-5
0
5
10
15
20
25
30
35
0 0.2 0.4 0.6 0.8
E (MeV)
R matrix
pole energy
58
Link between “standard” and “phenomenological” R-matrix:
• Standard R-matrix: parameters are calculated from basis functions
• Phenomenological R-matrix: parameters are fitted to data
59
Calculation: 10 polesFit to data
Isolated pole (2 parameters)
Background (high energy):
gathered in 1 term
E << El R0(E)~R0
In phenomenological approaches (one resonance):
629.67
295.23
153.83
107.79
65.62
40.55
26.50
7.87
0.11
-27.95
El
1.99E-02
9.76E-02
1.14E+00
4.16E+00
1.12E+00
1.05E-02
7.31E-01
1.09E+00
3.92E-01
2.38E-05
γl2
10
9
8
7
6
5
4
3
2
1
pole
60
12C+p
Approximations: R0(E)=R0=constant (background)
R0(E)=0: Breit-Wigner approximation: one term in the R matrix
Remark: the R matrix method is NOT limited to resonances (R=R0)
-10
-8
-6
-4
-2
0
2
4
6
8
10
0 0.2 0.4 0.6 0.8
E (MeV)
R0(E)
γ02/(E0-E)
61
Solving the Schrödinger equation in a basis with N functions provides:
where Eλ, γλ are calculated from matrix elements between the basis functions
Other approach: consider Eλ, γλ as free parameters (no basis)
Phenomenological R matrix
Question: how to relate R-matrix parameters with experimental information?
λ=1
λ=3
λ=2
Each pole corresponds to a state (bound state
or resonance)
Properties: energy, reduced width
!! depend on a !! (not physical)
Summary
62
Resonance energies
!!! Different from pole energies!
with
-100
-50
0
50
100
150
200
0 0.5 1 1.5 2 2.5 3
E (MeV)
de
lta
(de
g.)
delta_tot
Hard Sphere
delta_R
63
-100
-50
0
50
100
150
200
0 0.5 1 1.5 2 2.5 3
E (MeV)
de
lta
(de
g.)
delta_tot
Hard Sphere
delta_R
Resonance energy Er defined by δR=90o
In general: must be solved numerically
poleresonance
-10
-8
-6
-4
-2
0
2
4
6
8
10
0 0.2 0.4 0.6 0.8
E (MeV)
R matrix
1/S
poleresonance
Plot of R(E), 1/S(E)
64
The Breit-Wigner approximation
Single pole in the R matrix expansion
Phase shift:
Resonance energy Er, defined by: Not solvable
analytically
Thomas approximation:
Then:
Near the resonance energy Er:
65
with
Remarks:
• γ0, E0 =“ R-matrix”, “calculated”, “formal” parameters, needed in the R matrix
•depend on the channel radius a
•Defined for resonances and bound states (Er<0)
• γobs, Er = “observed” parameters
• model independent, to be compared with experiment
• should not depend on a
• The total width Γ depends on energy through the penetration factor P
fast variation with E
low energy: narrow resonances (but the reduced width can be large)
66
Summary
Isolated resonances:
Treated individually
High-energy states with the same Jπ
Simulated by a single pole = background
Energies of interest
Non-resonant calculations are possible: only a background pole
67
Link between “calculated” and “observed” parameters
One pole (N=1)
R-matrix parameters
(calculated)
Observed parameters
(=data)
Several poles (N>1)
Must be solved numerically
Generalization of the Breit-Wigner formalism:
link between observed and formal parameters when N>1
C. Angulo, P.D., Phys. Rev. C 61, 064611 (2000) – single channel
C. Brune, Phys. Rev. C 66, 044611 (2002) – multi channel
68
Examples: 12C+p and 12C+α
Narrow resonance: 12C+p
-60
0
60
120
180
0 0.2 0.4 0.6 0.8 1 1.2
a=4
a=7
12C+p, l = 0
total
Hard sphere
69
Broad resonance: 12C+α
-120
-60
0
60
120
180
0 1 2 3 4 5
12C+α, l = 1 a=5
a=8
a=5
a=8
70
Experimental region
Simple case: one
isolated resonance
18Ne+p elastic scattering: C. Angulo et al, Phys. Rev. C67 (2003) 014308
• Experiment at Louvain-la-Neuve: 18Ne+p elastic
→ search for the mirror state of 19O(1/2+)
19Na
1/2+
3/2+
3/2+
5/2+
5/2+
1/2+
(3/2+)
(5/2+)
19O
18O+n
18Ne+p
Ex (MeV)
0
1
2
3
7/2+
0
1
2
3
Ex (MeV)
0.745
0.120
-0.321
1.472
0.096
• Phase shifts dl defined in the R-
matrix (in principle from l=0 to ∞)
•• l=0: one pole
with 2 parameters: energy E0
reduced width γ0
• l=1, no resonance expected
hard-sphere phase shift
• l=2 (J=3/2+,5/2+): very narrow
resonances expected
weak influence
• l>2: hard sphere
The cross section is fitted with 2
parameters
2
2|)(cos)12)(exp(12(|
4
1 θδσl
l
l Pilkd
d −+=Ω ∑
l=0
l=2
l=4
71
0
100
200
300
400
500
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
θc.m. =120.2o (c)
0
100
200
300
400
500
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
θc.m. =170.2o (a)
0
100
200
300
400
500
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
θc.m. =156.6o (b)
diffe
rentialcro
ss
section [m
b/s
r]
c.m. energy [MeV]
18Ne+p elastic scattering
Final result
ER = 1.066 ± 0.003 MeV
Γp = 101 ± 3 keV
19Na
1/2+
3/2+
5/2+
1/2+
(3/2+)
(5/2+)
19O
18Ne+p
0
1
0
1
0.745
0.120
-0.321
1.472
0.096
Very large Coulomb shift
From Γ=101 keV, γ2=605 keV, θ2=23%
Very large reduced width
72
Other case: 10C+p 11N: analog of 11Be11Be: neutron rich11N: proton rich
10C+p
Expected 11N (unbound)
Energies?
Widths?
Spins?
0.00
200.00
400.00
600.00
800.00
1000.00
1200.00
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Ecm (MeV)
dσ/
dΩ
(mb
/sr)
10C+p at 180
o
Markenroth et al (Ganil)
Phys.Rev. C62, 034308 (2000)
73
Other example: 14O+p 15F: analog of 15C
14C+n
14O+p elastic: ground state of 15F unbound
Goldberg et al. Phys. Rev. C 69 (2004) 031302
74
Data analysis: general procedure for elastic scattering
∑=Ω ''
''
,,,,
|)(|
2121
2121
2
KKKKKKKK
fd
d θσ
),(....)( '
, '',
,
'',,'' 02121
θθπ
πl
J IllI
J
IllIKKKKYUf ∑ ∑=
Problem: how to determine the collision matrix U?
Consider each partial wave Jp
19Na
1/2+
3/2+
5/2+
1/2+
(3/2+)
(5/2+)
19O
18Ne+p
0
1
2
0
1
2
Simple case: 18Ne+p: spins I1=0+ and I2=1/2+
collision matrix 1x1
No resonance Hard sphere13/2-
≥2
2
1
0
l
Neglected
Very narrow resonance Hard
sphere
No resonance Hard sphere
R matrix: one pole (2 parameters)
but could be 3 (background)
3/2+
≥5/2
1/2-
1/2+
J
•Only unknown quantity
•To be obtained from
models
75
More complicated : 7Be+p: spins I1=3/2- and I2=1/2+
collision matrix: size larger that one (depends on J)
• Channel mixing neglected
• l=0: scattering length formalism
• l=2: hard sphere
1,0
1,2
2,2
1-
• Channel mixing neglected
• Hard sphere
1,1
1,3
2,1
2,3
2+
• Channel mixing neglected
• l=0: scattering length formalism
• l=2,4: hard sphere
1,2
2,0
2,2
2,4
2-
•Res. at 0.63 MeV Rmatrix I=1 OR I=2
•Channel mixing neglected (Uij=0 for i≠j)
No resonance hard sphere
No resonancehard sphere
1,1
2,1
2,3
1+
2,20-
1,10+
I, lJ
p+7Be
me
asu
rem
en
t
Finally, 4 parameters: ER and Γ for the 1+ resonance, 2 scattering lengths
76
Other processes: capture, transfer, inelastic scattering, etc.
Poles
El>0 or
El<0
Elastic scattering
Threshold 1
Threshold 2 Inelastic scattering,
transfer
Pole properties: energy
reduced width in different channels ( more parameters)
gamma width capture reactions
77
0
1
2
3
4
5
6
0.01 0.1 1
6Li(p,α)
3He
S-f
acto
r(M
eV
)
Ecm (MeV)
Example of transfer reaction: 6Li(p,a)3He (Nucl. Phys. A639 (1998) 733)
ααααα
αpp
p
pppRR
RR
RRR =⎟⎟⎠
⎞⎜⎜⎝
⎛= with,
Non-resonant reaction: R matrix=constant
Collision matrix
Cross section: σ~|Upa|2
matrixRthefromdeduced,⎟⎟⎠
⎞⎜⎜⎝
⎛=
ααα
α
UU
UUU
p
ppp
78
Radiative capture
E Capture reaction=transition between an initial state at
energy E to bound states
Cross section ~ |<Ψf|Hγ|Ψi(E)>|2
Additional pole parameter: gamma width
<Ψf|Hγ|Ψι(E)>=<Ψf|Hγ|Ψi(E)>int + <Ψf|Hγ|Ψi(E)>ext
with <Ψf|Hγ|Ψi(E)>int depends on the poles
<Ψφ|Ηγ|Ψι(Ε)>ext =integral involving the external w.f.
More complicated than elastic scattering!
But: many applications in nuclear astrophysics
79
Microscopic models
• A-body treatment:
• wave function completely antisymmetric (bound and scattering states)
• not solvable when A>3
• Generator Coordinate Method (GCM)
basis functions
with φ1, φ2=internal wave functions
Γl(ρ,R)=gaussian function
R=generator coordinate (variational parameter)
ρ=nucleus-nucleus relative coordinate
• At ρ=a, antisymmetrization is negligible the same R-matrix method is applicable
ninteractionucleon-nucleon
inucleonofenergykinetic
with
i
=
=
+= ∑∑=<=
ij
A
ji
ij
A
i
i
V
T
VTH11
80
1. One R-matrix for each partial wave (limited to low energies)
2. Consistent description of resonant and non-resonant contributions (not limited
to resonances!)
3. The R-matrix method can be applied in two ways
a) To solve the Schrödinger equation
b) To fit experimental data (low energies, low level densities)
4. Applications a)
• Useful to get phase shifts and wave functions of scattering states
• Application in many fields of nuclear and atomic physics
• 3-body systems
• Stability with respect to the radius is an important test
5. Applications b)
• Same idea, but the pole properties are used as free parameters
• Many applications: elastic scattering, transfer, capture, beta decay, etc.
6. Conclusions
