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