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R-matrix theory of nuclear reactions. Pierre Descouvemont Université Libre de Bruxelles, Brussels, Belgium. Introduction General collision theory: elastic scattering Phase-shift method R-matrix theory ( theory) Phenomenological R matrix ( experiment) Conclusions. - PowerPoint PPT Presentation
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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 fmradius~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+DA+B C+D+EA+B C*+Detc…
4. radiative capture reactions A+B C +
NOT covered here
covered here
5
Assumptions: elastic scatteringno internal structureno Coulombspins zero
Before collisionAfter collision
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=2E/2
A=amplitude (scattering wave function is not normalized)
f() =scattering amplitude (length)
7
Cross sections
solid angle d
Cross section
• Cross section obtained from the asymptotic part of the wave function
• “Direct” problem: determine from the potential
• “Inverse” problem : determine the potential V from
• Angular distribution: E fixed, variable
• Excitation function: 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')
r must be known
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 systemsspins 0single-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
ll m
u rY
r
When inserted in the Schrödinger equation
)()()()()1(
2 22
22
rEururVrur
ll
dr
dlll
To be solved for any potential (real)
For r∞ V(r) =0
222
2
2 with ,0
1
E
krukrur
llu lll
)()(
)("
=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
For small x
)2/cos(1
)(
)2/sin(1
)(
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 l = phase shift (information about the potential)If V=0 =0
2 1| ( , ) | , with ( , ) (2 1) exp(2 ( )) 1 (cos )
2 l ll
df E f E l i E P
d ik
Cross section:
Provide the cross section
13
factorization of the dependences in E and
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 nexp(2iexp(2in
4. Levinson theorem
• (E=0) - (E=)= N where N is the number of bound states
• Example: p+n, ℓ =0: (E=0) - (E=)= (bound deuteron)
2
0
1| ( , ) | , with ( , ) (2 1) exp(2 ( )) 1 (cos )
2 l ll
df E f E l i E P
d ik
14
Interpretation of the phase shift from the wave function
l<0 V repulsive
l>0 V attractive
u(r)
r
V=0 u(r)sin(kr-l/2) l=0
V≠0 u(r)sin(kr-l/2+l)
)2/sin()(*tan)()( llll lkrkrnkrjru
15
Resonances: =Breit-Wigner approximation
ER=resonance energy=resonance width
3/4
/4
/2
(E)
ER ER+/2ER-/2E
• Narrow resonance: small
• Broad resonance: large
Example: 12C+p
With resonance: ER=0.42 MeV, =32 keV lifetime: t=}/~2x10-20 s
Without resonance: interaction range d~10 fm interaction time t=d/v ~1.1x10-21 s
16
0)()()1( 2
2"
rukru
r
llu lll
3.b Generalization to the Coulomb potential
The asymtotic behaviour(r) exp(ikz) + f()*exp(ikr)/r
Becomes: (r) exp(ikz+ log(kr)) + f()*exp(ikr- log(2kr))/r
With =Z1Z2e2/ℏv =Sommerfeld parameter, v=velocity
0)()2()()1( 2
2"
rukkru
r
llu lll
Bessel equation:
Coulomb equation
Solutions: Fl(,kr): regular, Gl(,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
)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
= Hard-Sphere phase shift
-150
-120
-90
-60
-30
0
30
0 1 2 3 4 5 6
Ecm (MeV)
hard
-sph
ere
l=0
l=2
a+12C
-150
-120
-90
-60
-30
0
30
0 2 4 6
Ecm (MeV)
ha
rd-s
ph
ere
l=0
l=2
p+12C
-150
-120
-90
-60
-30
0
30
0 1 2 3 4 5 6
Ecm (MeV)ha
rd-s
pher
e
l=0
l=2
n+12C
18
example : a+n phase shift l=0
Special case: Neutrons, with l=0:
F0(x)=sin(x), G0(x)=cos(x)
=-ka
hard sphere is a good approximation
19
Elastic cross section with Coulomb:
)(cos)1)2)(exp(12(2
1)( with ,|)(| 2
ll
l Pilik
ffd
d
Still valid, but converges very slowly.
22
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 Cl l l
l ll
C Cl l l l
l
N C
C
f l i Pik
l i i i Pik
f f
f ik
Coulomb: exactNuclear
2|)(| C
C fd
d
= Coulomb cross section: diverges for =0
increases at low energies
20
2
11
42
2
sin~|)(|E
fd
dC
C
• 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 =180o nuclear effect maximum
)()()( NC fff
•fC(): Coulomb part: exact
•fN(): nuclear part: converges rapidly
1
10
100
1000
10000
100000
1000000
0 30 60 90 120 150 180
(degrees)
2
1
4 sin
21
• One channel: phase shift U=exp(2i)• Multichannel: collision matrix Uij, (symmetric , unitary) with i,j=channels
• Good quantum numbers J=total spin =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 (-1) l
Example of quantum numbers
J I l size
1/2+ 1/2 0, 1 1
1/2- 1/2 0, 1 1
3/2+ 1/2 1, 2 1
3/2- 1/2 1, 2 1
X
X
X
X
J I l size
0+ 12
12
1
0- 12
12
1
1+ 12
0, 1, 21, 2, 3
3
1- 12
0, 1, 21, 2, 3
3
a+3He a=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(()(|)(| l
ll Pil
ikff
d
d1212
2
1 with ,2
Cross sections
Multichannel
With: K1,K2=spin orientations in the entrance channel
K’1,K’2=spin orientations in the exit channel
''
''
,,,,
|)(|2121
2121
2
KKKKKKKK
fd
d
One channel:
),(....)( ', '',
'',, '' 0 2121
l
J IllI
JIllIKKKKYUf
Collision matrix
• generalization of Uijijexp(2iij
• 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
l=0a
l=0a
l>0
strong difference between l=0 and l>0
Generalization a=scattering lengthre=effective range
24
Cross section at low E:
In general (neutrons):
Thermal neutrons (T=300K, E=25 meV): only l=0 contributes
example : a+n phase shift l=0
At low E: ~-kawithareplusive
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 ul(0) and ul(h) (Numerov algorithm)
• ul(3h),… ul(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: aa
0+
E=0.09 MeV=6 eV
4+
E~11 MeV~3.5 MeV
2+
E~3 MeV~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)
aa
VB~1 MeV
27
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 »)
3.f Optical model
Phase shift is complex: =R+iI
collision matrix: U=exp(2i)=exp(2iR) where =exp(-2I)<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 spaceNij=<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 E
)()( rucr ii
i
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 functionsSet of N basis functions ui(r)
correct asymptotic behaviour (E>0 and E<0)
Extension to the R-matrix: includes boundary conditions
a. Hamiltonian2
1 2, with Z Z e
H E H Tr
( ) ( ) valid in a limited range
with ( ) ( ) ( )m
i
l
i
l l
i
lI
r c
kr U O kr Y
u r
Principles of the R-matrix theory
31
• Spectroscopy: short distances onlyCollisions : 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+a
Exit channels
12C(2+)+a
15N+p, 15O+n
12C+a
CoulombNuclear+Coulomb:R-matrix parametersCoulomb
32
Here: elastic cross sections
Cross sections
Phase shifts
R matrix
Potentialmodel
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 ,
N
iii rucr
1
)()(int
• At large distances the wave function is Coulomb in the external region:
arfor ,)()()( 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 int
2 2
2 20 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 uLTuuLTu 00 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, 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=2l=0
-3
-2
-1
0
1
-1 0 1 2
l=2
l=0Sl
Pl
E (MeV)
a + na + 3He
S(E)=shift factorP(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 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 aa
• 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
potentialVN(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)
Resonance l=0E=0.42 MeV=32 keV
-20
-15
-10
-5
0
5
10
15
20
0 2 4 6 8
r (fm)
V (
Me
V)
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)
de
lta (
de
g.)
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=7E=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=10E=0.8 MeV
48
Resonance energies
with
• Resonance energy Er defined by R=90o
In general: must be solved numerically
• Resonance width 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,
221
mll YkrW
r
eZZTH
EH
0)()2()()1( 2
2"
rukkru
r
llu lll Positive energies: Coulomb functions Fl, Gl
0)()2()()1( 2
2"
rukkru
r
llu lll Negative energies: Whittaker functions
Asymptotic behaviour:
x
xxW
xxxG
xxxF
)exp()2(
)2log2
cos()(
)2log2
sin()(
2/1,
Starting point of the R matrix:
, 1/2 (2 )W x
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
Iteration ng=6 ng=10
1 -1.500 -2.190
2 -1.498 -1.937
3 -1.498 -1.942
-1.942
Final -1.498 -1.942
Exact -1.942
Left derivative -1.644 -0.405
Right derivative -0.379 -0.406
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 l(r)
56
With = reduced width
Completely equivalent
2 steps: computation of eigen-energies E (=poles) and eigenfunctionscomputation of reduced widths from eigenfunctions
Remarks:
•E=pole energy: different from the resonance energydepend on the channel radius
• proportional to the wave function at a measurement of clustering (depend on a!)large 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
El gl2
-27.9454 2.38E-05
0.112339 0.392282
7.873401 1.094913
26.50339 0.730852
40.54732 0.010541
65.62125 1.121593
107.7871 4.157215
153.8294 1.143461
295.231 0.097637
629.6709 0.019883
-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):
pole El l2
1 -27.95 2.38E-05
2 0.11 3.92E-01
3 7.87 1.09E+00
4 26.50 7.31E-01
5 40.55 1.05E-02
6 65.62 1.12E+00
7 107.79 4.16E+00
8 153.83 1.14E+00
9 295.23 9.76E-02
10 629.67 1.99E-02
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)
02/(E0-E)
61
Solving the Schrödinger equation in a basis with N functions provides:
where E, are calculated from matrix elements between the basis functions
Other approach: consider E as free parameters (no basis)
Phenomenological R matrix
Question: how to relate R-matrix parameters with experimental information?
=1
=3
=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 R=90o
In general: must be solved numerically
pole resonance
-10-8-6-4-202468
10
0 0.2 0.4 0.6 0.8
E (MeV)
R matrix
1/S
pole resonance
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:
• 0, 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)
• obs, Er = “observed” parameters
• model independent, to be compared with experiment
• should not depend on a
• The total width depends on energy through the penetration factor Pfast 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 J
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 channelC. Brune, Phys. Rev. C 66, 044611 (2002) – multi channel
68
Examples: 12C+p and 12C+a
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+a
-120
-60
0
60
120
180
0 1 2 3 4 5
12C+a, 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 0
• l=1, no resonance expectedhard-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
22
|)(cos)12)(exp(12(|4
1 l
ll Pil
kd
d
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
c.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
c.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
c.m. =156.6o (b)
diff
eren
tial c
ross
sec
tion
[mb
/sr]
c.m. energy [MeV]
18Ne+p elastic scattering
Final resultER = 1.066 ± 0.003 MeVp = 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 shiftFrom =101 keV, 2=605 keV, 2=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)
d/d
(m
b/s
r)
10C+p at 180o
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 unboundGoldberg et al. Phys. Rev. C 69 (2004) 031302
74
Data analysis: general procedure for elastic scattering
''
''
,,,,
|)(|2121
2121
2
KKKKKKKK
fd
d
),(....)( ', '',
,'',, '' 0
2121
l
J IllI
JIllIKKKKYUf
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
J l
1/2+ 0 R matrix: one pole (2 parameters)but could be 3 (background)
1/2- 1 No resonance Hard sphere
3/2+ 2 Very narrow resonance Hard sphere
3/2- 1 No resonance Hard sphere
≥5/2 ≥2 Neglected
•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)
J I, l
0+ 1,1 No resonancehard sphere
0- 2,2 No resonance hard sphere
1+ 1,12,12,3
•Res. at 0.63 MeV Rmatrix I=1 OR I=2
•Channel mixing neglected (Uij=0 for i≠j)
1- 1,01,22,2
•Channel mixing neglected
•l=0: scattering length formalism
•l=2: hard sphere
2+ 1,11,32,12,3
•Channel mixing neglected•Hard sphere
2- 1,22,02,22,4
• Channel mixing neglected
• l=0: scattering length formalism
• l=2,4: hard sphere
p+7Be
me
asu
rem
en
t
Finally, 4 parameters: ER and 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: energyreduced 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,a)3He
S-f
acto
r (M
eV
)
Ecm (MeV)
Example of transfer reaction: 6Li(p,a)3He (Nucl. Phys. A639 (1998) 733)
aaaaa
app
p
ppp RRRR
RRR
with ,
Non-resonant reaction: R matrix=constant
Collision matrix
Cross section: ~|Upa|2
matrix R thefrom deduced ,
aaa
a
UU
UUU
p
ppp
78
Radiative capture
E Capture reaction= transition between an initial state at energy E to bound states
Cross section ~ |<f|H|i(E)>|2
Additional pole parameter: gamma width
<f|H|(E)>=<f|H|i(E)>int + <f|H|i(E)>ext
with <f|H|i(E)>int depends on the poles
<)>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 1, 2=internal wave functionsl(,R)=gaussian functionR=generator coordinate (variational parameter)=nucleus-nucleus relative coordinate
• At =a, antisymmetrization is negligible the same R-matrix method is applicable
ninteractionucleon -nucleon
inucleon ofenergy kinetic
with
i
ij
A
jiij
A
ii
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 waysa) To solve the Schrödinger equationb) 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