CDCC analysis for momentum
distribution of 22Mg in 23Al+12C
reaction
Y. Aoki, K. Hara, M. Iijima,
T. Motobayashi, and A. Ozawa
0.22
0.20

230.18

22

0.16
0.14

Al+12C, EAl=74A MeV
Mg mom. dist.

PRC76,031601(R)

0.12
0.10
0.08
0.06
0.04
0.02
0.00
-400

-300

-200

-100
0cP|| [MeV]

100200lets write a CDCC program andanalyze the data300

CDCC is a 3 body like approach toscatt. of loosely bound projectile
off the hard nucleus

projectile
r21R
3 targetint.

pot. typestatesV 12

real pot.

bound/ scatt.

V 13

OP pot.

scatt.

V 23

OP pot.

scatt.

Hamiltonian H:
H = Tr + V12(r)+ TR + V13(r13) + V23(r23)wave function φ(r) be defined by{Tr + V12(r) − Ec} φc(r) = 0
scatt. state wf. at large rr
2φc(k, r) →sin(k r − lcπ/2 + δc k )/rπklc
δc kwave numberorb. ang. mom. for ch. cnucl. phase shift

if charge and spin are neglectedtruncation
discretization

k and spin space
k spacehenceContinuum Discretized
Coupled Channels (CDCC)
1φ̂c j = p

Z kj

kj − kj−1 kj−1φc(k, r) dk

and is orthonormalized ashφ̂c j | φ̂c0 j 0 ir = δc c0 δj j 0suffix
c, c’j, j’meaningspins
wave numbers

V13 + V23 :consists of nucl. and Coul. pots.
they are expanded by
Legendre functions PλV13(r13) + V23(r23)X=vλ(r, R) Pλ(r̂ · R̂)λλmultipolarity

λ = 0 equivalent(folded) potential
pre/post acceleration
λ ≥ 1 tidal force
dipole/multipole break up
reorientation

eigen state of H withangular momentum J , M
ΨJ M =1X
R

χJL (R)
cjc, j× [φ̂c j (r) ilc Ylc (r̂) iLc j YLc j (R̂)]J M[ . . . ] for ang. mom. couplingχJL (R) : motion of (1-2) and 3,cj

satisfies the CDCC eq.
(TR − Ec j )χJL (R)cj=−X

h[c j]J |(V13 + V23)|[c0 j 0]J ic0 j 0× χJ
L0c0 j 0(R)

CDCC eq. is solved numericallywith usual boundary cond.
χJLc j (R) →ILc0 δc,c0 δLc j ,Lc0s
Kc 0 J−Sc j, c0 Lc Oc j
0Kc j
where c0 stands for inc. ch.
IL (OL) are usual Coul. wf.

Elastic cross sec
X(geom. factor) ei(σL0 +σL)³´J
2× SL,L−δ
Y(R̂)|L L0LM
0

σel ∝ |fc +triple diff. cross sec.
d3σdΩ1 dΩ2 dE1T ∝∝ (final state dens.) × |T |2Ã

X( geom. factor)ei(δc k +σc k )JMc³i(σL0 +σLc )× e´JSL(k)c L0× [Ylc (r̂) ⊗ YLc (R̂)]J Mk!

numerical aspects
23Al+12C at E

assumed:

Al= 80A MeV
23Al = p + 22Mg

pot. V12 between p and 22Mg
central

Woods-Saxon type

Coul.

uniform. charged spherespin-orbit Thomas typewith common geometric param.
following energies are reproducedlevel
spin E [MeV] widthgsd5/2 −0.125
1st res. s1/2
2nd res. d3/2

0.402

77 [eV]

2.444

76 [keV]intrinsic spins are neglected
to reduce cpu time

φ̂(r),

the 1-2 basereal, even for scatt. statesd-, p-, f -state (bound/ scatt.) wf.
0.45 < k < 0.55 f m−10.6

radial 1-2 wave functions
bound d-statep- d- f- scatt. states
0.45 < k < 0.55/fm

0.4

au

00.2
0 2 4 6 8 101214
0-0.2

050100r [fm]150200dumps at large r, which is alarge merit of binnings, p, d-states have a node r < 4 fm
but not for f -state( No bond states)product of φ̂(bound) × (scatt.) state wf.

1.0E-01c=p-state

φgs(r)*φc(r)c=d-state
1.0E-02c=f-state
1.0E-03

1.0E-04

1.0E-05

051015r [fm]this product is related tobreak up reaction

202530prod. of continuum φ̂’s

φc’(r)*φc(r)1.0E-01

c: d-scatt. c’: p-scatt.
c: d-scatt. c’: f-scatt.

1.0E-02

1.0E-03

1.0E-04
c: bound, c’: p-scatt.
1.0E-05

05

1015r [fm]2025

break up is induced at small r
but at large rcont.-cont. coupling is importantand is called “post acceleration”

30

V13 and V23 :OP pot. are used
V(p−12C)

Watson et al.,V(22Mg−12C) Beunerd et al.,⇑remains ambiguity

if we require φ̂ feels nucleusrmax ≥mMg + mp
mpRmax !

No excitation of 12C nor 22Mg
No coalescence
No Coul. BU includedin the final analysis !

dipole part of potential’multpot’ u 1:2:4
0-5-10-15-20-25-30
-35
-40
-45
-50
-55

vλ(R,r)λ=1100-10-20
-30
-40
-50
-60

0510
R [fm]

1520 0300
250
200150 ]100 [fmr50

C (r, R)|φ̂ (r)ihφ̂c(r)|vλrc0
1.0E-01

Cchannel
1: d bound
2: d scatt.
3: p scatt.
4: f scatt.
0.45 <k<0.55/fm|<c|v λ|c’>|1.0E-02

1.0E-03

1 <-> 4, λ=1
1.0E-04

1 <-> 3, λ=1
1 <-> 2, λ=0

1.0E-05

1 <-> 1, λ=2
1.0E-06

050

100

150
R [fm]

200

off diag. monopole elem. dumps
very rapidly,
due to orthogonality
λ( > 0) elements dump as
R−(λ+1) for large r

250300N (r, R)
Fourier exp. of vλ25
Fourier trans. of vNλ201510

1 <-> 4, λ=1[au]50-5

1 <-> 2, λ=2-10
1 <-> 2, λ=0-15

1 <-> 3, λ=1-20-25-30

00.20.4

0.6

0.8

1
1.2
k [1/fm]channel
1: d bound
2: d scatt.
3: p scatt.
4: f scatt.
0.45 <k< 0.55 /fm
1.4

1.6

λ = 0, 1 and 2 component forc < − > c0 transition

1.82

* CDCC eq. solved numericallyby using 8 point Stömer’s method* safeguard
integrate NOT from the origin* to get indep. sol. vectors
occasional ortho-normalization* S-mat.: cond. num. estimated* Coul. wf. Virmigham approach.
use of continued fraction* 3- 6-j: use of 3 term recur. relation
purge factorial evaluation

J and k dep. of p-wave absorption’SabsP’ u 1:2:3
P wave break up
1.0e+00
1.0e-01
1.0e-021.0e-031.0e-04
1.0e-05
1.0e-06
1.0e-071.0e-08
1.0e-09 0|S| [au]0.5
k [1/fm 1]

1.5

2

just qualitative !

01000
2000 r]3000 [hbaJ
4000
2.5 5000elastic scatt. cross sec.
1.0E+00

23

Al+12C, EAl=80A MeV

dσEL/dσRuth

1.0E-01

p+Mg gs0d state

1.0E-02
1s state
1.0E-03
1.0E-04
1.0E-050510θCM [deg]l=0 to 4 states of 1-2 system
0 < k < 1.5 fm−1
No Coul. break up(π0d state23
gs. of Al
π1s state
No exp. data1520

triple diff. cross sec.
No Coul. BU inluded yets-, d-state for p-22Mg bound state
NOT symmetric about k = 0central supression
dipole break up dominates
Coulomb supression of φ̂
Coul. BU may fill the dip
d-state be preferred for p+22Mg gs

5000
d3σ/dΩpdΩMgdEMg [mb/sr2 MeV]

23

Al+12C, EAl=80A MeV

4000

3000

p-Mg gs: 0d-stateθMg=0 deg
θp=0-15 deg

2000

1000

08100

8200

8300

8400
cP|| [MeV]

8500

8600

8700

8500

8600

8700

5000
d3σ/dΩpdΩMgdEMg [mb/sr2 MeV]

4500
4000
3500
3000

23Al+12C, EAl=80A MeV

p+Mg gs: 1s-stateθMg=0.0 deg
θp=0-15 deg

2500
2000
1500
1000
500
08100

8200

8300

8400
cP|| [MeV]

5000
d3σ/dΩpdΩMgdEMg [mb/sr2 MeV]

23

Al+12C, EAl=80A MeV

4000

3000

p-Mg gs: 0d-stateθMg=0 deg
θp=0-15 deg

2000

1000

08100

8200

8300

8400
cP|| [MeV]

8500

8600

8700

8500

86008700

5000
d3σ/dΩpdΩMgdEMg [mb/sr2 MeV]

4500
4000
3500
3000

23

Al+12C, EAl=80A MeV

p+Mg gs: 1s-stateθMg=0.0 deg
θp=0-15 deg

2500
2000
1500
1000
500
08100

8200

8300

8400
cP|| [MeV]