Pierre Descouvemont Universit

1
Pierre DescouvemontUniversité Libre de
Bruxelles, Brussels, Belgium
The R-matrix theory in nuclear physics

1. Introduction
2. General collision theory elastic scattering
3. Phase-shift method
4. Calculable R-matrix (? theory)
5. Phenomenological R-matrix (? experiment)
6. Conclusions

2
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

• Characteristics must be considered
• Energy with respect to the Coulomb barrier
• Level density
• Number of open channels

3
Introduction
Conditions (A1, Z1) (A2, Z2) Energy E
Far above the barrier(Too) many l values? no
partial wave expansionex Eikonal, semi-classic
theories, etc.
V
Nucleus-nucleus potential
Near the barrierLimited l values? partial wave
expansionex CDCC, DWBA, etc.
lgt0
Below the barrierVery few l valuesR matrix, GCM
Z1Z2e2/r
NuclearAstrophysics
r
l0
Barrier energyRB A11/3A21/3VB Z1Z2e2/RB
4
Level density Spectrum of (A1A2) nucleus
V
Nucleus-nucleus potential
High L.D.
lgt0
Low L.D.
Z1Z2e2/r
r
l0
a16O
? different models according to the level density
5

• Stable nuclei
• high level density
• many open channels

• Exotic nuclei
• low level density
• few open channels

d18F
n19Ne
p19F
a16O
20Ne
6

• Low-energy reactions
• Transmission (tunnelling) decreases with l ?
  few partial waves ? semi-classic theories
  not valid
• astrophysics
• thermal neutrons
• Wavelength ??c/E ?(fm)197/E(MeV) example
  1 MeV ? 200 fm radius2-4 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

7

• Assumptions elastic scattering no internal
  structure no Coulomb spins zero

2
q
1
2
1
Before collision
After collision
Center-of-mass system
8
Scattering wave functions
Schrödinger equation V(r)interaction
potential Assumption the potential decreases
faster than 1/r
A large distances V(r)? 0 2 independent
solutions
Incoming plane wave
Outgoing spherical wave
where kwave number k22mE/?2 Aamplitude
(scattering wave function is not
normalized) f(q) scattering amplitude (length)
9

• Cross sections

q
solid angle dW
Cross section

• Cross section obtained from the asymptotic part
  of the wave function
• Direct problem determine s from the potential
• Inverse problem determine the potential V
  from s
• Angular distribution E fixed, q variable
• Excitation function q fixed, E variable

10

• How to solve the Schrödinger equation for Egt0?
• With (z along the beam axis)
• Several methods
• Formal theory Lippman-Schwinger equation ? Born
  approximation
• Eikonal approximation high energies
• Phase shift method
• partial-wave expansion of the wave function
• well adapted to low energies
• common to several theories R-matrix, optical
  model, CDCC, etc.
• Etc

11
Lippman-Schwinger equation
is solution of the Schrödinger equation
With Green function
r is supposed to be large (r gtgt r')

• Y(r) must be known
• Y(r) only needed where V(r)?0 ? approximations
  are possible
• Born approximation
• Eikonal approximation

Valid at high energiesV(r) E
12

1. Definitions, cross sections Simple conditions
   neutral systems spins 0 single-channel
2. Extension to charged systems
3. Extension to multichannel problems
4. Low energy properties
5. General calculation
6. Optical model

13
3.a Definition, cross section

• The wave function is expanded as

When inserted in the Schrödinger equation
To be solved for any potential (real or
complex) For r?8 V(r) 0
Bessel equation ? ul(r) jl(kr), nl(kr)
14
For small x
For large x
Examples
At large distances ul(r) is a linear combination
of jl(kr) and nl(kr)
With dl phase shift (information about the
potential)If V0 ? d0
Cross section
Provide the cross section
15

• ? factorization of the dependences in E and q
• General properties of the phase shifts
• Expansion useful at low energies small number of
  l values
• The phase shift (and all derivatives) are
  continuous functions of E
• The phase shift is known within np
  exp(2id)exp(2i(dnp))
• Levinson theorem
• d(E0) is arbitrary
• d(E0) - d(E?) Np , where N is the number of
  bound states
• Example pn, l 0 d(E0) - d(E?) p (bound
  deuteron)

16
Interpretation of the phase shift from the wave
function
V0? u(r)?sin(kr-lp/2) dl0
u(r)
r
V?0? u(r)?sin(kr-lp/2dl)
dllt0 ? V repulsive
dlgt0 ? V attractive
17
Resonances Breit-Wigner approximation ERreso
nance energyGresonance width

• Narrow resonance G small
• Broad resonance G large

Example 12Cp With resonance ER0.42 MeV, G32
keV ? lifetime t/G2x10-20 s Without
resonance interaction range d10 fm ?interaction
time td/v 1.1x10-21 s
18
3.b Generalization to the Coulomb potential
The asymtotic behaviour Y(r)? exp(ikz)
f(q)exp(ikr)/r Becomes Y(r)? exp(ikzh
log(kr)) f(q)exp(ikr-h log(2kr))/r With
hZ1Z2e2/?v Sommerfeld parameter, vvelocity
Bessel equation
Coulomb equation
Solutions Fl(h,kr) regular, Gl(h,kr)
irregular Ingoing, outgoing functions Il
Gl-iFl, OlGliFl
19

• Example hard sphere

V(r)
V(r) 8 for rlta Z1Z2e2/r for rgta
a
r
Hard-Sphere phase shift
20
Special case Neutrons, with l0 F0(x)sin(x),
G0(x)cos(x) ? d-ka
example an phase shift l0
? hard sphere is a good approximation
21

• Elastic cross section with Coulomb

Still valid, but converges very slowly.
Coulomb exact
Nuclear
Coulomb cross section diverges for
q0 increases at low energies
22

• fC(q) Coulomb part exact
• fN(q) nuclear part converges rapidly

• 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 q180o ? nuclear effect maximum

23
3.c Extension to multichannel problems

• One channel phase shift d ? Uexp(2id)
• Multichannel collision matrix Uij, (symmetric ,
  unitary) with i,jchannels
• Good quantum numbers Jtotal spin ptotal
  parity
• Channel i characterized by II1? I2 channel
  spin JI ? l l angular momentum
• Selection rules I1- I2 I I1 I2 l - I
  J l I pp1p2 (-1) l

Example of quantum numbers
a3He a0, 3He1/2
p7Be 7Be3/2-, p1/2
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
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
X
X
X
X
X
X
X
X
24
Cross sections
One channel
Multichannel With K1,K2spin orientations
in the entrance channel K1,K2spin
orientations in the exit channel

• Collision matrix
• generalization of d Uijhijexp(2idij)
• determines the cross section

25
3.e General calculation
For some potentials analytic solution of the
Schrödinger equation In general no analytic
solution ? numerical approach
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 (Elt0) same idea

26
Example aa
Experimental phase shifts
Experimental spectrum of 8Be
4E11 MeVG3.5 MeV
Potential (Buck) et al. VN(r)-122.3exp(-(r/2.13
)2)
2E3 MeVG1.5 MeV
VB1 MeV
aa
0E0.09 MeVG6 eV
27
3.f Optical model
Goal to simulate absorption channels

• High energies
• many open channels
• strong absorption
• potential model extended to complex potentials
  ( optical )

d18F
n19Ne
p19F
Phase shift is complex ddRidI with
dIgt0collision matrix Uexp(2id)hexp(2idR)
where hexp(-2dI)lt1 Elastic cross
section Reaction cross section
a16O
20Ne
28

• Goals
• To solve the Schrödinger equation (Egt0 or Elt0)
  calculable R-matrix
• Potential model
• 3-body scattering
• Microscopic models
• Many applications in nuclear and atomic physics
• To fit cross sections phenomenological
  R-matrix
• Elastic, inelastic ? spectroscopic information on
  resonances
• Capture, transfer ? astrophysics

29

Principles of the R-matrix theory

• Standard variational calculations
• Hamiltonian
• Set of N basis functions ui(r) with
• ?Calculation of HijltuiHujgt over the full
  space Nijltuiujgt
• (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
• ? 2 regions

30

Principles of the R-matrix theory

• R-matrix divide the space into 2 regions
  (radius a)
• Internal r a Nuclear coulomb
  interactions
• External r gt a Coulomb only

Matching at ra provides collision matrix
U coefficients ci
31

• Use of the R matrix method

Cross sections
measurements
Phase shifts
calculable R-matrix
Phenomenological R-matrix
3 body continuum
32
Formalism of the R-matrix (potential model)

• N basis functions ui(r) are valid in a limited
  range (r a)? in the internal region

(cicoefficients)

• At large distances the wave function is Coulomb
  ? in the external region

• Idea to solve
• with defined in the internal region

• But T is not hermitian over a finite domain,

33
Formalism of the R-matrix

• Bloch-Schrödinger equation
• With L(L0)Bloch operator (surface operator)
  constant L0 arbitrary

• Now we have TL(L0) hermitian

• Since the Bloch operator acts at ra only

• Here L00 (boundary-condition parameter)

34
Formalism of the R-matrix

• Summary

• Using (2) in (1) and the continuity equation

Provides ci(inversion of D)
Dij(E)
Continuity at ra
35
With the R-matrix defined as
If the potential is real R real, U1, d
real

• Procedure (for a given l)
• Compute matrix elements
• Invert matrix D
• Compute R matrix
• Compute the collision matrix U

36
S(E)shift factorP(E)penetration factor
Charged system
Neutral system
a 3He
a n
Pl
Sl
E (MeV)
37

• 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

38
Example aa scattering 1. potential
V(r)-126exp(-(r/2.13)2) (Buck potential)
VN(a)0 ? a gt 6 fm
potential
39
2. Basis functions
Gaussians with different widths aix0a0(i-1)
(geometric progression)typically x00.6 fm,
a01.4
l0
Can be done exactly(incomplete g function)
Matrix elements of H can be calculated
analytically for gaussian potentials Other
potentials numerical integration
40
Elastic phase shifts

• a10 fm too large (needs too many basis
  functions)
• a4 fm too small (nuclear not negligible)
• a7 fm is a good compromise

41
Wave functions at 5 MeV, a 7 fm
42

• Other example sine functions
• Matrix elements very simple
• Derivative ui(a)0

? Not a good basis (no flexibility)
43
Example of resonant reaction 12Cp
Resonance l0E0.42 MeVG32 keV

• potential V-70.5exp(-(r/2.70)2)
• Basis functions ui(r)rlexp(-(r/ai)2)

44
Wave functions
Phase shifts a7 fm
45
Resonance energies
with
Hard-sphere phase shift
R-matrix phase shift

• Resonance energy Er defined by ?dR90o
• In general must be solved numerically
• Resonance width G defined by

46
Application of the R-matrix to bound states
Positive energies Coulomb functions Fl, Gl

Negative energies Whittaker functions
Asymptotic behaviour
47

• R matrix equations

With CANC (Asymptotic Normalization Constant)
important in external processes

• Using (2) in (1) and the continuity equation

Dij(E)

• But L depends on the energy, which is not known
  ? iterative procedure

48
Application to the ground state of 13N12Cp
Resonance l0 Egt0
Negative energy -1.94 MeV
Bound state l1 Elt0

• Potential V-55.3exp(-(r/2.70)2)
• Basis functions ui(r)rlexp(-(r/ai)2) (as
  before)

49

• Calculation with a7 fm
• ng6 (poor results)
• ng10 (good results)

Iteration ng6 ng10
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 -1.942

Left derivative -1.644 -0.405
Right derivative -0.379 -0.406
50
Wave functions (a7 fm)
51
Application to microscopic models

• A-body treatment
• wave function completely antisymmetric (bound
  and scattering states)
• not solvable when Agt3
• Generator Coordinate Method (GCM) basis
  functions with f1, f2internal wave
  functions Gl(r,R)gaussian function Rgenerato
  r coordinate (variational parameter) rnucleus-n
  ucleus relative coordinate
• At ra, antisymmetrization is negligible? the
  same R-matrix method is applicable

52

• 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)
53
Change of basis

• Eigenstates of over the internal region
• expanded over the same basis
• ? standard diagonalization problem

Instead of using ui(r), one uses Yl(r)
54
Completely equivalent
With reduced width
?2 steps computation of eigen-energies El
(poles) and eigenfunctions computation of
reduced widths from eigenfunctions

• Remarks
• Elpole energy different from the resonance
  energy depend on the channel radius
• gl proportional to the wave function at a?
  measurement of clustering (depend on a!) large
  gl ? strong clustering
• dimensionless reduced width Wigner
  limit

55
Example 12Cp

• potential V-70.5exp(-(r/2.70)2)
• Basis functions ui(r)rlexp(-(r/ai)2) with
  aix0a0(i-1)

10 basis functions, a8 fm
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
10 eigenvalues
56

• Link between calculable and phenomenological
  R-matrix
• Calculable R-matrix parameters are calculated
  from basis functions
• Phenomenological R-matrix parameters are fitted
  to data

57
Calculation 10 poles
Fit to data
pole El gl2
1 -27.95 2.38E-05
2 0.11 3.92E-01
3 7.87 1.09E00
4 26.50 7.31E-01
5 40.55 1.05E-02
6 65.62 1.12E00
7 107.79 4.16E00
8 153.83 1.14E00
9 295.23 9.76E-02
10 629.67 1.99E-02
Isolated pole (2 parameters)
Background (high energy)gathered in 1 term E
ltlt El ? R0(E)R0
? In phenomenological approaches (one resonance)
58
12Cp
Approximations R0(E)R0constant
(background) R0(E)0 Breit-Wigner
approximation one term in the R matrix Remark
the R matrix method is NOT limited to resonances
(RR0)
59
Summary
Solving the Schrödinger equation in a basis with
N functions provides where El, gl are
calculated from matrix elements between the basis
functions
l3
Each pole corresponds to a state (bound state or
resonance)Properties energy, reduced width !!
depend on a !! (not physical)
l2
l1
Other approach consider El, gl as free
parameters (no basis) ? Phenomenological R
matrix
Question how to relate R-matrix parameters with
experimental information?
60
Resonance energies !!! Different from pole
energies!
with
61
Resonance energy Er defined by ?dR90o In
general must be solved numerically
Plot of R(E), 1/S(E)
pole
resonance
62
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
63
with

• Remarks
• g0, E0 R-matrix, calculated, formal
  parameters, needed in the R matrix
• depend on the channel radius a
• Defined for resonances and bound states (Erlt0)
• gobs, Er observed parameters
• model independent, to be compared with
  experiment
• should not depend on a
• The total width G depends on energy through the
  penetration factor P ?fast variation with E ?
  low energy narrow resonances (but the reduced
  width can be large)

64
Summary
High-energy states with the same Jp Simulated by
a single pole background
Energies of interest
Isolated resonances Treated individually
Non-resonant calculations are possible only a
background pole
65
Link between calculated and observed
parameters
One pole (N1)
R-matrix parameters(calculated) Observed
parameters(data)
Several poles (Ngt1)
Must be solved numerically
Generalization of the Breit-Wigner formalism
link between observed and formal parameters when
Ngt1 C. Angulo, P.D., Phys. Rev. C 61, 064611
(2000) single channel C. Brune, Phys. Rev. C
66, 044611 (2002) multi channel
66
Examples 12Cp and 12Ca Narrow resonance 12Cp
total
Hard sphere
67
Broad resonance 12Ca
68

• 18Nep elastic scattering C. Angulo et al, Phys.
  Rev. C67 (2003) 014308
• Experiment at Louvain-la-Neuve 18Nep elastic
• ? search for the mirror state of 19O(1/2)

• Phase shifts dl defined in the R-matrix (in
  principle from l0 to 8)
• l0 one pole with 2 parameters energy
  E0 reduced width g0
• l1, no resonance expected?hard-sphere phase
  shift
• l2 (J3/2,5/2) very narrow resonances
  expected ?weak influence
• lgt2 hard sphere
• ? The cross section is fitted with 2 parameters

l4
l0
Experimental region Simple case one isolated
resonance
l2
69
18Nep elastic scattering
Final result ER 1.066 0.003 MeVGp 101 3
keV
differential cross section mb/sr
? Very large Coulomb shift From G101 keV,
g2605 keV, q223 Very large reduced width
c.m. energy MeV
70
Other case 10Cp ?11N analog of 11Be 11Be
neutron rich 11N proton rich
Markenroth et al (Ganil)Phys.Rev. C62, 034308
(2000)
10Cp
Expected 11N (unbound) Energies?Widths?Spins?
71
Other example 14Op ?15F analog of 15C
14Op elastic ground state of 15F
unboundGoldberg et al. Phys. Rev. C 69 (2004)
031302
14Cn
72
Data analysis general procedure for elastic
scattering

• Only unknown quantity
• To be obtained from models

Problem how to determine the collision matrix
U? Consider each partial wave Jp
Simple case 18Nep spins I10 and
I21/2 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
2
2

1/2
1
1

1/2


3/2
(3/2
)
0
0


5/2
(5/2
)
18
Nep
19
19
Na
O
73
More complicated 7Bep spins I13/2- and
I21/2 collision matrix size larger that one
(depends on J)
J I, l
0 1,1 No resonance?hard sphere
0- 2,2 No resonance ? hard sphere
1 1,12,12,3 Res. at 0.63 MeV ? Rmatrix I1 OR I2 Channel mixing neglected (Uij0 for i?j)
1- 1,01,22,2 Channel mixing neglected l0 scattering length formalism l2 hard sphere
2 1,11,32,12,3 Channel mixing neglected Hard sphere
2- 1,22,02,22,4 Channel mixing neglected l0 scattering length formalism l2,4 hard sphere
measurement
p7Be
Finally, 4 parameters ER and G for the 1
resonance, 2 scattering lengths
74
Other processes capture, transfer, inelastic
scattering, etc.
Elastic scattering
Inelastic scattering, transfer
Threshold 2
Poles Elgt0 or Ellt0
Threshold 1
Pole properties energy reduced width in
different channels (? more parameters) gamma
width ? capture reactions
75
Example of transfer reaction 6Li(p,a)3He (Nucl.
Phys. A639 (1998) 733)
Non-resonant reaction R matrixconstant Collisio
n matrix Cross section sUpa2
76
Radiative capture
Capture reaction transition between an initial
state at energy E to bound states Cross section
ltYfHgYi(E)gt2 Additional pole parameter gamma
width ltYfHgYi(E)gtltYfHgYi(E)gtint
ltYfHgYi(E)gtext with ltYfHgYi(E)gtint depends
on the poles ltYfHgYi(E)gtext integral
involving the external w.f.
E
More complicated than elastic scattering! But
many applications in nuclear astrophysics
77

• One R-matrix for each partial wave (limited to
  low energies)
• Consistent description of resonant and
  non-resonant contributions (not limited to
  resonances!)
• The R-matrix method can be applied in two ways
• To solve the Schrödinger equation
• To fit experimental data (low energies, low level
  densities)
• Applications a)
• Tool to get phase shifts and wave functions of
  scattering states
• Application in many fields of nuclear and atomic
  physics
• Stability with respect to the radius is an
  important test
• Applications b)
• Same idea, but the pole properties are used as
  free parameters
• Many applications elastic scattering, transfer,
  capture, beta decay, etc.