Investigation of unbound nuclei
via transfer reaction Tokuro Fukui
Istituto Nazionale di Fisica Nucleare (INFN) - Sezione di Napoli -
In collaboration with
Y. Kikuchi, T. Matsumoto, and K. Ogata
18/October/2016
Motivation 1
Unbound nuclei n  They are measured as a resonance state formed by a dynamical process.
→ Precise reaction analysis is desired.
13F
10O
25O
10N
21C
16B
13Be
8Be
10Li
5He
7He
9He 10He
12Li
15Be 16Be
13Li
26O
Motivation 1
Unbound nuclei n  They are measured as a resonance state formed by a dynamical process.
→ Precise reaction analysis is desired.
13F
10O
25O
26O
10N
21C
16B
13Be
8Be
10Li
5He
7He
9He 10He
12Li
15Be 16Be
13Li
Our study:
Unbound nuclei formed by
transfer reactions (stripping).
p
n
p
n
Motivation 2
Stripping reaction to unbound state: A(d, p)B n  The transition matrix of the post-form representation for the (d, p) reaction
within the distorted-wave Born approximation (DWBA):
D
E
(post)
( )
(+)
TDWBA =
U
n Vpn + VpA
d ↵
The remnant term
D
E
( )
(+)
⇠
VpA U = VpA UpB ⇠ 0
n Vpn
d ↵
Z
Z
⇤( )
= dr d dr n
(r d , r n ) n⇤ (r n )Vpn (r d ) d (r d ) (+)
↵ (r d , r n ).
oscillate
p
attenuate
Vpn: deuteron binding potential
d
rd
rβ
rp
The r n integration does not converge!
rα
n
rn
21
BC
20
Α
C
Motivation 3
Historical summary for the convergence problem n  Some treatments have been suggested under some approximations:
e r , and then take
(1) Introduce “convergence factor”　! 0.
With
R. Huby and J. R. Mines, Rev. Mod. Phys. 37, 406 (1965).
the zero-range
(ZR) approx. (2) Integrate in the complex plane with e r .
C. M. Vincent and H. T. Fortune, Phys. Rev. C 2, 782 (1970).
Vpn (r d )
d (r d )
⇡ D0 (r d ) (3) Divide T-matrix into three parts with an channel radius.
G. Baur and D. Trautmann, Phys. Rep. 25, 293 (1976).
(4) Approximate unbound states as bound states.
n  More precise treatments
(5) Reduce the dimension to surface integration with an channel radius.
V. E. Bunakov, Nucl. Phys. A140, 241 (1970).
(6) Modification of (5) with a continuum-discretized coupled-channels
(CDCC) framework.
A. M. Mukhamedzhanov, Phys. Rev. C 84, 044616 (2011).
Motivation 4
Our approach n  The prior form
D
(prior)
( )
TDWBA =
U↵
n Vn↵ + VpA
D
E
( )
(+)
=
n Vn↵
d ↵
Z
Z
⇤( )
= dr d dr n
(r d , r n )
← If the remnant term VpA
⇤
(+)
n (r n )Vn↵ (r n ) d (r d ) ↵ (r d , r n ).
rd
rβ
These respectively attenuate
for two independent coordinates.
→ The integration does converge.
rp
VnA : interaction for the n-A
rα
n
U↵ = 0
oscillate attenuate attenuate
p
d
(+)
d ↵
E
unbound state
rn
21
BC
20
Α
C
Motivation 5
NOTE 1 (Remnant term)
Generally, with in the standard DWBA the remnant term of the prior form is not
canceled:
VpA U↵ = VpA UdA 6= 0.
Therefore we need to treat the remnant term explicitly,
OR
solve three-body scattering problem, in which there is a p-A interaction only, not
an n-A.
n
No VnA
[H 0 E↵ ] (+)
↵ = 0,
r↵
H 0 = T↵ + U↵ = T↵ + VpA .
rd
We perform it with CDCC.
p
VpA
A
Motivation 6
NOTE 2 (Beyond DWBA)
(1)  In the prior form the distorted wave in the final channel should be exact.
(2)  In a unbound system, resonant-nonresonant couplings may be important.
→ CDCC also in the final channel:
Coupled-channels Born approximation (CCBA).
"i
"f
p + n
n + A
Purpose n  Construct precise model for the stripping reaction to unbound states.
n  Investigate the coupled-channel (CC) effect.
Model
7
Continuum-discretized coupled-channels method (CDCC) n  How to treat breakup channels
(+)
(r, R) =
xb (k0 , r) aA (K0 , R)
+
Truncation & discretization
(+)
X
(r, R) ⇡
Z
1
xb (k, r) aA (K, R)dk
0
Infinite number of continuum states
ii0
i
xb (r) aA (R)
truncate
i
M. Kamimura et al., Prog. Theor. Phys. Suppl. No. 89, 1 (1986).
N. Austern et al., Phys. Rep. 154, 125 (1987) .
M. Yahiro et al., Prog. Theor. Exp. Phys. 2012, 01A209 (2012).
discretize
x + b
n  CDCC equation
h
i
xb |
[h + K + UxA (r, R) + UbA (r, R)
[K + Uii (R)
Uij (R) =
⌦
Ei ]
ii0
aA (R)
=
X
E]
(+)
Uij (R)
(r, R) = 0
ji0
aA (R)
b
R
r
j6=i
i
xb
UxA (r, R) + UbA (r, R) |
j
xb i
A
x
Model
8
How to discretize bin (average) method
discretization
ˆn`m (r) = p
1
kn
Z
kn
`m (k, r)dk
kn
~
"ˆn =
2µr

(kn + kn
4
`m (r)
=
` (r)i
2
1
1)
2
( kn ) 2
+
12
pseudo state
"
Hij
!
Hij =
"n Nij
D
Nij = h
i
i
|
Ĥ
ji
!#
j
E
Cj
!
=0
diagonalization
` (r)
=
iX
max
`
Y`m (r̂),
ci '`i (r),
i
`
'`i (r) = Ni r exp
" ✓
r
⇢i
◆2 #
98.8
2.452
1.40
1.32
0.544
0.783
Model
nds to the main
but inaccurate
he deviation at
Gaussian basis
e !!!k, r" at the
n be solved by
d line) instead.
9.775
1.32
0.783
model space is composed of two k-continua for !=0 and 2.
Since there exists a resonance in !=2, the d-wave
k-continuum is further divided in the Av method into the
resonant part #0%k%0.55$ and the nonresonant part
#0.55%k%2.0$. In the former region the k continuum of
!i,!=2!k, r" varies rapidly with k. The Av method can simulate
this rapid change by taking f i,!=2!k"=1 with bins of an extremely small width. In fact, clear convergence is found for
both the elastic and the breakup S-matrix elements, when the
resonance part is described by 30 bins and the nonresonance
part of the d-wave and the s-wave k-continua by 20 bins.
Another
n Av discretization, which has been widely used as a
convenient prescription [1,5,6,8,10], is also made for comparison,nin which the resonance region is (av)
represented by a
n
single state with the weight factor of Breit-Wigner
type given
n (12).
1 The two sorts of Av discretization are compared
by Eq.
with the real- and complex-range Gaussian PS methods.
With the PS methods, convergence of the S-matrix elements
is found with 21 s-wave breakup channels and 22 d-wave
ones. The level sequences of the resulting discrete eigenstates are shown in Fig. 6 for both the basis functions, which
have the T.
same
properties as in Fig.et3.al.
The parameter sets of
MATSUMOTO
the basis functions, finally taken in the PS methods, are !a1
6 real-range40Gaussian basis and
=1.0, an =30.0, n=30" for the
9
Equivalence of two methods for discretization n  Overlap with true scattering wave
k
V
6
jectile Li has
simplicity, we
s. [1,5,6]. Then
state with "res
energy and the
but at least the
uch by the ne-
onsists of deuach pair of the
Ca scattering at
56 MeV [25],
MeV and r0
the optical po-
k
k̂
k
pseudostate method
average method
D
(k, r) ˆ
(r)
E
n  Observables ( Li + Ca at 156 MeV)
g breakup prond !max =2; the
k
k̂n
D
(k, r)
ˆ(ps) (r)
n
E
PHYSICAL REVI
T. Matsumoto et al., Phys. Rev. C 68, 064607 (2003).
FIG. 8. The same
+ Ca scattering at 156
grazing total angular m
line is the result of th
40
Model
10
Truncation regarding momentum & angular momentum spaces n  Momentum truncation
Z 1
br (r, R) ⌘
xb (k, r)
aA (K, R)dk
0
!
Z
truncate
kmax
xb (k, r) aA (K, R)
x + b
0
n  Angular momentum truncation (Austern-Yahiro-Kawai theorem)
N. Austern et al., Phys. Rev. Lett. 63, 2649 (1989) .
CDCC with ang. mom. truncation
P =
[E
Z
K
dr̂
lm X
X
⇤
Ylm (r̂) Ylm
(r̂) ,
l=0 m
V
PUP]
CDCC
CDCC
= 0.
Distorted-Faddeev equations
[E
K
V
[E
K
Ux
⇣
⌘
ˆ
ˆ
ˆ
PUP] a = V
x+ b ,
⇣
⌘
ˆ
ˆ
Ub ]
P U P ) ˆ a.
x + b = (U
Expected
to be small
can be a good approximation of ˆ d if lm is large enough.
Model
10
Truncation regarding momentum & angular momentum spaces n  Momentum truncation
Z 1
br (r, R) ⌘
xb (k, r)
aA (K, R)dk
0
!
Z
truncate
kmax
xb (k, r) aA (K, R)
x + b
0
Model space should be set so that
n  Angular momentum
truncation (Austern-Yahiro-Kawai
observables
we want to see can be theorem)
N. Austern et al., Phys. Rev. Lett. 63, 2649 (1989) .
described properly.
CDCC with ang. mom. truncation
P =
[E
Z
K
dr̂
lm X
X
⇤
Ylm (r̂) Ylm
(r̂) ,
l=0 m
V
PUP]
CDCC
CDCC
= 0.
Distorted-Faddeev equations
[E
K
V
[E
K
Ux
⇣
⌘
ˆ
ˆ
ˆ
PUP] a = V
x+ b ,
⇣
⌘
ˆ
ˆ
Ub ]
P U P ) ˆ a.
x + b = (U
Expected
to be small
can be a good approximation of ˆ d if lm is large enough.
Model
11
Beyond DWBA n  Coupled-channels Born approximation (CCBA) with CDCC
D
E
(prior)
( )
(+)
TCCBA =
(CDCC) VnA
↵(CDCC)
p
n
Vpn p
VpA
A
VnA
n
VpA
A
n  The CC regarding continuum states (breakup effect) is explicitly included.
n  The CDCC wave functions both in the initial and final channels.
→ Remnant term is canceled out exactly.
→ Rearrangement component is involved implicitly.
A. M. Moro et al., Phys. Rev. C 80, 064606 (2009).
T. Fukui et al., Phys. Rev. C 91, 014604 (2015).
Result 12
Benchmark on 4He(d, p)5He n  The interactions
T. Ohmura et al., Prog. Theor. Phys. 43, 347 (1970).
G. R. Satchler et al., Nucl. Phys. A112, 1 (1968).
H. Kanada et al., Prog. Theor. Phys. 61, 1327 (1979).
Vpn : 1 range Gaussian (Ohmura potential)
VpHe : Real Woods-Saxon for the distorted potential (Satchler pot. for E < 20 MeV)
VnHe : l-dependent multi-range Gaussian (KKNN potential)
→ reproduce p-resonance at 0.8 MeV
(The strength is adjusted because we ignore the nucleon spin.)
n  Restricted model space
d-wave
p-wave
s-wave
CC
p + n
Transiton only
from the s-states
s-wave
CC
d-wave
CC
n + 4He
Very phenomenological and naïve model at the moment.
Result 13
Discretized-continuum states for n-4He n  Pseudostate method (diagonalize the Hamiltonian with Gaussian basis functions)
250
200
n + 4He
p
s
d
0.4
0.2
150
0.1
φl(r) (fm-1/2)
ε (MeV)
0.3
100
0
-0.1
50
-0.2
n-4He
-0.3
resonance (0.797 MeV)
non-resonance (0.345 MeV)
non-resonance (41.982 MeV)
0
-0.4
Zoom
1
0.5
0
0
20
40
60
r (fm)
80
100
Result 14
Angular distribution 10
3
Preliminary
4
He(d,p)5He(3/2-) at 27.3 MeV
Exp.
Prior-CCBA
dσ/dΩ (mb/sr)
102
101
10
10
0
Exp.: H. J. Erramuspe and R. J. Slobodrian, Nucl. Phys. 49, 65 (1963).
-1
0
20
40
60
80
100
θ (deg)
120
140
n  The calculation overestimates the data slightly.
n  The diffraction pattern is shifted to the backward angles.
→ Improper interactions (future work)?
160
180
Result 15
Discretized-energy spectra (prior)i
0
n  The T-matrix TCCBA is
specified by the energy index i0.
1
i0 = 5
i0 = 4
i0 = 3
i0 = 2
i0 = 1
0.5
→ Discretized-energy spectra is obtained.
450
4
Preliminary
400
0
He(d,p)5He at 27.3 MeV
p-wave
n
p-wave
s-wave
350
dσ/dε (mb/MeV)
300
250
200
150
100
50
0
0
1
2
3
ε (MeV)
4
5
0
"inHe
4He
Result 16
Smoothing procedure T. Matsumoto et al., Phys. Rev. C 68, 064607 (2003).
n  Insert the approximate complete set.
n  Originally developed for the breakup reaction.
D
E
( )
(+)
T̃ (k, ✓) =
(k, k ) VnHe ↵(CDCC)
n
D
E
( )
(+)
= 'nHe (k)eik ·r VnHe + U
↵(CDCC)
ED
XD ( )
i0
i0
ik ·r
⇡
'nHe (k) nHe
e
VnHe + U
nHe
i0
⇡
=
XD
( )
'nHe (k)
i0
nHe
i0
XD
i0
( )
'nHe (k)
i0
nHe
ED
E
i0 ( )
(CDCC)
(prior)i0
TCCBA
n  The double differential cross section
2
d2 l
µ↵ µ k X
=
T̃lm (k, ✓) .
2
2
dkd⌦
(2⇡~ ) k↵ m
.
VnHe
p
k
k
(+)
↵(CDCC)
(+)
↵(CDCC)
E
4He
E
Result 17
Smoothed-energy spectra n  Only p-wave spectrum
450
4
400
Preliminary
He(d,p)5He at 27.3 MeV
p-wave
p-wave (smoothing)
without smoothing
s-wave (smoothing)
without smoothing
350
dσ/dε (mb/MeV)
300
250
200
150
100
50
0
0
1
2
3
ε (MeV)
4
5
Summary 18
Stripping reactions to unbound states
n  Prior form + CCBA → Resolve the convergence problem (post form)
4He(d,
p)5He
Angular distribution
103
Energy spectrum
450
4
4
He(d,p)5He(3/2-) at 27.3 MeV
400
Exp.
Prior-CCBA
102
He(d,p)5He at 27.3 MeV
p-wave
p-wave (smoothing)
without smoothing
s-wave (smoothing)
without smoothing
350
dσ/dε (mb/MeV)
dσ/dΩ (mb/sr)
300
101
250
200
150
100
100
50
10-1
0
20
40
60
80
100
θ (deg)
120
140
160
180
0
0
1
2
3
4
ε (MeV)
Future plan
n 
n 
n 
n 
n 
Other interactions (e.g. the JLM interaction) J. -P. Jeukenne et al., Phys. Rev. C 16, 80 (1977).
Large model space
Clarify the reaction mechanism
Other incident energy H. W. Broek and J. L. Yntema, Phys. Rev. 135, B678 (1964).
Other system (e.g. 21C)
5
Backups
ü  They are measured as a resonance state formed by a dynamical process.
“EXPERIMENT”
16Be: A. Spyrou et al., Phys. Rev. Lett. 108, 102501 (2012). [NSCL] {p knockout}
→
reaction
analysis
is108,
desired.
26O: Precise
E. Lunderberg
et al., Phys.
Rev. Lett.
142503 (2012). [NSCL] {p knockout}
21C: S. Mosby et al., Nucl. Phys. A 909, 67 (2013). [NSCL] {p removal}
26O: Z. Kohley et al., Phys. Rev. Lett. 110, 152501 (2013). [NSCL] {p knockout}
13Li: Z. Kohley et al., Phys. Rev. C 87, 011304(R) (2013). [NSCL] {p removal}
13F
10Li, 12Li, 13Li: Yu. Aksyutina et al., Phys. Letter B 666, 430 (2008). [GSI] {p knockout, n knockout}
10He: Z. Kohley et al., Phys. Rev. Lett. 109, 232501 (2012). [NSCL] {p knockout}
10O
25O
10He: S. I. Sidorchuk et al., Phys. Rev. Lett. 108, 202502 (2012). [JINR] {2n transfer}
9He, 10He: H.T. Johansson et al., Nucl. Phys. A 842, 15 (2010). [GSI] {p knockout}
10N
10He: M.S. Golovkov et al., Phys. Letter B 672, 22 (2009). [JINR] {2n transfer}
9He: T. Al Kalanee et al., Phys. Rev. C 88, 034301 (2013). [GANIL] {n transfer}
21C
15Be: J. Snyder et al., Phys. Rev. C 88, 031303(R) (2013). [NSCL] {n transfer}
25O, 26O: C. Caesar et al., Phys. Rev. C 88, 034313 (2013). [GSI] {p knockout}
16B
13Be: Yu. Aksyutina et al., Phys. Rev. C 87, 064316 (2013). [GSI] {p knockout}
13Be
8Be
15Be 16Be
“THEORY”
7He: S. Baroni et al., Phys. Rev. C 87, 034326 (2013). [ULB] {NCSM}
13Li
12
10Li
7He: S. Baroni et al., Phys. Rev. Lett.
110, 022505Li
(2013).
[ULB] {NCSM}
26O: L. V. Grigorenko et al., Phys. Rev. Lett 111, 042501 (2013). [Dubna] {3BHH}
5He
7He
9He 10He
26O: K. Hagino
et al., arXiv:1307.5502.
[Tohoku] {GFFB}
10N, 13F: H. T. Fortune, Phys. Rev. C 88, 024309 (2013). [Pennsylvania] {MED}
11O: H. T. Fortune, Phys. Rev. C 87, 067306 (2013). [Pennsylvania] {MED}
8Be: R.B. Wiringa et al., arXiv:1308.5670. [Argonne] {GFMC}
Systematics (O to Ti): X. Qu et al., arXiv:1309.3987. [Guizhou] {RCHB}
26O
three-body results, the dashed line to CDCC-TR∗ , the dash-dotted
line and the thin solid line to CDCC-BU with lmax = 8 and lmax = 6,
respectively. DESCRIPTION OF DIRECT NUCLEAR . . .
THREE-BODY
Backups
120
dσ/dΩ (mb/sr)
1
Exp. (35.3 MeV)
Faddeev
Faddeev x0.7
10
Faddeev: fixed n- Be
CDCC-TR*
0.1
0
5
10
θc.m. (deg)
15
20
FIG. 8. (Color online) Transfer reaction 1 H(11 Be, 10 Be)d cross
section at ELab /A = 38.4 MeV. The thick solid line corresponds to
the exact three-body result, while the dotted line corresponds to the
same calculation multiplied by 0.7. The thin solid line is the exact
calculation with a partial-wave independent n-10 Be interaction. The
latter is to be compared with the CDCC-TR∗ calculation (dashed line),
as explained in the text. The experimental data are from Ref. [39] at
Ep = 35.3 MeV.
dσ/dE (mb/MeV)
10
respectively.
10
4
2000
exact three-body
wave function
in the surface region, or that
3
10
the choice of the optical potentials appearing in the remnant
1000
term of Eq. (16) is inadequate
for this purpose, which could
2
be connected
10 to the poor description of the 11 Be elastic data.
0 we show
5
15
Finally in Figs. 9 and 10
the10semi-inclusive
11
1
differential10cross section for the breakup Be + p → 10 Be +
p + n, where 10 Be is the detected particle. We present both
the energy distribution
(Fig. 9) and the angular distribution
Faddeev
0
lmax=8
(Fig. 10). 10
For the CDCC-BU:
energy distribution,
two CDCC-BU calCDCC-BU: lmax=6
culations are shown,
one
with
l
!
8
and
one with l ! 6 for
CDCC-TR*
10 10-1
the n- Be motion.
The 50
significant difference
between
these
0
100
150
10 CDCC-BU calculation is
two calculations suggests that
the
θ ( Be) (deg)
dσ/dΩ (mb/sr)
A. Deltuva et al.,PHYSICAL
Phys. Rev. C
76, 064602
REVIEW
C 76,(2007).
064602 (2007)
exact three-body wave function in the surface
Faddeev region, or that
CDCC-BU: lmax=8
the choice
of
the
optical
potentials
appearing
in the remnant
100
CDCC-BU: lmax=6
term of Eq. (16) is inadequate for this purpose,
CDCC-TR* which could
80 to the poor description of the 11 Be elastic data.
be connected
Finally in Figs. 9 and 10 we show the semi-inclusive
60cross section for the breakup 11 Be + p → 10 Be +
differential
p + n, where 10 Be is the detected particle. We present both
40
the energy distribution (Fig. 9) and the angular distribution
(Fig. 10).20For the energy distribution, two CDCC-BU calculations are shown, one with l ! 8 and one with l ! 6 for
the n-10 Be0motion. The significant difference between these
0
2
3
5
6
1
4
two calculations suggestsE that(10Be)
the (MeV)
CDCC-BU calculation is
not converged with respectc.m.to the number of n-10 Be partial
waves.
calculation
with l ! 8 reproduces
reasonably
well
FIG. 9.The
(Color
online) Semi-inclusive
differential
cross section
the
shape
of
the
energy
distribution
predicted
by
the
AGS
for the reaction 1 H(11 Be, 10 Be)pn, at ELab /A = 38.4 MeV, vs 10 Be
calculation,
it underestimates
section
the
center
of mass but
energy.
The thick solid this
line cross
corresponds
to at
exact
∗ could be due to
peak
by
about
20%.
This
underestimation
three-body results, the dashed line to CDCC-TR , the dash-dotted
duelmax
to some
theand
contribution
of higher
n-10 Be partial
= 6,
line
the thin solid
line to CDCC-BU
with waves
lmax = 8orand
c.m.
10
co
se
w
th
bre
th
agr
sc
exc
is
ar
we
th
how
sc
ove
at t
fu
ene
of
con
do
stat
in
cal
of
Fig
th
tha
CD
in
ind
con
sen
w
we
CD
the
in
tha
th
sca
de
is d
nu
are
on
the
11
sca
br
fun
sc
of p
CD
dom
ca
in t
of
of
br
tha
ADWA
not negligible discrepancies are found in the shapes of dσ/d"
independent of the choice of the auxiliary interaction. This is
between FAGS1 and CDCC, accompanied by a significant
5
true for the lowest energy, but a few percent discrepancy starts
0
dependence on the optical potential (FAGS2 and CDCC2),
0
20
40
60
80
to appear as the beam energy increases.
0
which makes the comparison ambiguous.
12
13
In Fig. 6, we show dσ/d" for the C(d, p) C reaction at
θ (degrees)
We connect the present work with the comparative study
8
10
(b)
Ed = 12 and 56 MeV. Just as in the case of Be, the CDCC
the finite-range adiabatic wave approximation
12 [16] between
12
FIG. 6. (Color online) Angular distribution for C(d, p)13 C:
predictions for (d, p) on C at low energy provide a very good
6
(ADWA)
method
and Faddeev-AGS. For that purpose, we
=
12
MeV
and
(b)
E
=
56
MeV.
(a)
E
d
d
approximation to the Faddeev solution (FAGS1). However, the
include
in
Figs.
5–7
the finite-range ADWA predictions
UPADHYAY, A. DELTUVA, AND F. M. NUNES
PHYSICAL REVIEW C 85, 054621 (2012)
disagreement
MeV becomes
(around
N. atJ. 56
Upadhyay
et significant
al., Phys.
Rev.20%).
C 85,4054621 (2012).
10
Whereas in Be there was a strong dependence of the transfer
80
2cross section
on the choice of the energy at which the optical 8
potentials
are
evaluated,(a)for 12 C, no such dependence exists
1 25
(a)
CDCC
0(compare
CDCC
60 FAGS1 and FAGS2 or CDCC
CDCC and CDCC2) and
20
CDCC2
6
disagreement is quantitatively
robust.
CDCC2
3therefore the
CDCC2
(c)
FAGS1
15
In
Fig.
7
we
present
the
angular
distributions
following
FAGS1
FAGS1
40
0.1
49
FAGS2
Ca at 56 MeV. Small but
(d, p) transfer to the ground state of FAGS2
CDCC
FAGS2
10
2not negligible discrepancies are found in the shapes of dσ/d" 4
FAGS ADWA
ADWA
ADWA
20 FAGS1 and CDCC, accompanied
between
by a significant
5
FAGS1
0.01
1dependence on the optical potential (FAGS2 and CDCC2), 2
0
0
which makes
the comparison ambiguous.
We connect the present
0 8 20
40 (b)60
80 100 120
(b) work with the comparative study
0[16] between the finite-range adiabatic wave approximation 0
θ (degrees)
15
0
10
20
30
40
50
60
0
20
40
60
80
6
(ADWA)
method
and
Faddeev-AGS.
For that purpose, we
θ
(degrees)
θ
(degrees)
48
include
in
Figs.
5–7
the
finite-range
ADWA
predictions
IG. 4. (Color online) Elastic cross section for d+ Ca at
dσ / dΩ (mb/sr)
dσ / dΩ (mb/sr)
σ / σR
dσ / dΩ (mb/sr)
dσ / dΩ (mb/sr)
Backups
10
FIG. 5. (Color online) Angular distribution for 10 Be(d, p)11 Be:
Ed = 21.4 MeV, (b) Ed = 840.9 MeV, and (c) Ed = 71 MeV.
2 energy. CDCC calculations should(a)be
5
ntials with beam
4
pendent of the choice of the auxiliary interaction. This is
0
for the lowest energy, but a few percent discrepancy starts
3 energy increases.
ppear as the beam
(c)
n Fig. 6, we show dσ/d" for the 12 C(d, p)13 C reaction at
10
= 12 and 56 MeV.
2 Just as in the case of Be, the CDCC
dictions for (d, p) on 12 C at low energy provide a very good
roximation to the Faddeev solution (FAGS1). However, the
greement at 561MeV becomes significant (around 20%).
FIG. 7. (Color online) Angular distribution for 48 Ca(d, p)49 Ca at
Ed = 56 MeV.
CDCC054621-6
0
20
40
60 CDCC2
80
FAGS1
θ (degrees)
FAGS2
4
FIG. 6. (Color online) Angular distribution ADWA
for 12 C(d, p)13 C:
(a) Ed = 12 MeV and (b) Ed = 56 MeV.
2
0
dσ / dΩ (mb/sr)
= 56 MeV.
6
r)
Whereas in 10 Be there was a strong dependence of the transfer
cross section on the
0
0 choice of the energy at which the optical
12
25
0 (a)
10
20
30
40
50
60
potentials are evaluated,
for 20
C, no such
exists
0
40 dependence
60
80
CDCC
(compare
FAGS1
and
FAGS2
or
CDCC
and
CDCC2)
and
θ
(degrees)
θ
(degrees)
20
CDCC2
therefore the disagreement is quantitatively robust.
FAGS1 for 10 Be(d, p)11 Be: In Fig.
FIG.155. (Color online) Angular distribution
FIG.
7. (Color
online)
for 48following
Ca(d, p)49 Ca at
7 we
present
the Angular
angulardistribution
distributions
49
FAGS2
(a) Ed = 21.4 MeV, (b) Ed = 40.9 MeV, and (c) Ed = 71 MeV.
d = 56 MeV.
(d, p)Etransfer
to the ground state of Ca at 56 MeV. Small but
10
ADWA
not negligible discrepancies are found in the shapes of dσ/d"
054621-6
between FAGS1 and CDCC, accompanied by a significant
5
dependence on the optical potential (FAGS2 and CDCC2),
0
which makes the comparison ambiguous.
We connect the present work with the comparative study
8
(b)
Backups
Description of transfer reactions ü  The transition matrix for the 8B(d,n)9C reaction.
D
E
( )
(+)
TDWBA =
Vpn ↵
d
n
Vxb
tα
8B
p
tγ
=
Uγ
+
Uγ
Uγ
+
Uγ
tβ
Uγ
Uγ
9C
ü  Conventionally the distorted-wave Born approximation (DWBA)
is adopted to describe transfer reactions.
→ Uγ: The optical potential (of d-8B or n-9C) in γ channel
→ The proton transfers with a one-step process.
+…
Backups
CCBA within continuum states Initial channel
a
Final channel
b
x
Vxb
tβ
tα tα
tβ
B
A
+
tα
+
tβ
tα : consists of the x-A and b-A optical potentials (UxA + UbA)
tβ: consists of the b-A optical potential
Backups Rearrangement component Breakup
x
b
Rearrange
x
Nonorthogonal
b
BC
BC
x
b
A
Breakup
x
Rearrange
x
A
Nonorthogonal b
A
BC
BC
x
A
g.s.
g.s.
ü  Transitions from/into the rearrangement channels are implicitly taken
into account by the breakup transfer process.
Backups
Breakup process n  Decomposition of the transition matrix
D
( )
( )
(+)
TCCBA =
+
V
xb
(res)
(br)
↵(el) +
=T
(res),↵(el)
Continuum
+T
(res),↵(br)
+T
(+)
↵(br)
(br),↵(el)
+T
n
n
A
p
d
p
n
Bound
(br),↵(br)
Continuum
p
A
E
B
n
A
Resonance
Back coupling (BC)
Elastic transfer (ET)
Breakup transfer (BT)
Backups
Beyond DWBA (CC on transfer reactions) n  To take into account channel-couplings due to the three-body dynamics,
the coupled-channels Born approximation (CCBA) was proposed.
1.1. Three-body dynamics
7
S. K. Penny and G. R. Satchler, Nucl. Phys. 53, 145 (1964).
hree-body
dynamics
P. J. Iano and N. Austern, Phys. Rev. 151, 853 (1966).
7
(a)
(a)
5-
13C
14N
(b)
-
5
3-
4.49 MeV
40Ca
3.74 MeV
39K
(b)
5- 4.49 MeV
3.74 MeV
0+ 3 0.00 MeV
3/2+
40Ca
0+
K. Low, T. Tamura, and T. Udagawa, Phys. Lett. B67, 5 (1977).
30+
0.00 MeV
39K
0.00 MeV
40Ca
3/2+ 0.00 MeV
39K
Backups
Beyond DWBA (CC on transfer reactions) n  To take into account channel-couplings due to the three-body dynamics,
the coupled-channels Born approximation (CCBA) was proposed.
1.1. Three-body dynamics
7
S. K. Penny and G. R. Satchler, Nucl. Phys. 53, 145 (1964).
hree-body
dynamics
P. J. Iano and N. Austern, Phys. Rev. 151, 853 (1966).
7
(a)
(a)
5-
13C
14N
CCBA were able to (b)
achieve to reproduce experimental data.
3excited
by including the channel-couplings
among
a
few
states.
4.49 MeV
5
3-
40Ca
39K
3.74 MeV
However Continuum(b)
states were not taken into
0+ account
5- 4.49 MeV
for stable nuclei.
- be3.74
→ They are expected
essential
bound system.
MeV
0+ 3to
0.00 MeV
0.00loosely
MeV
3/2+for
40Ca
0+
K. Low, T. Tamura, and T. Udagawa, Phys. Lett. B67, 5 (1977).
39K
0.00 MeV
40Ca
3/2+ 0.00 MeV
39K
Backups
17
Smoothing function n  Only p-wave spectrum
4
450
He(d,p)5He at 27.3 MeV
p-wave
Preliminary
400
400
smoothing
without smoothing
350
dσ/dε (mb/MeV)
300
250
200
6
He(d,p)5He at 27.3 MeV
p-wave
without smoothing
i0=1
i0=2
i0=3
i0=4
i0=5
i0=6
i0=7
i0=8
i0=9
i0=10
350
300
dσ/dε (mb/MeV)
4
250
4
2
0
200
150
150
100
100
50
50
-2
-4
0
0
1
2
3
ε (MeV)
4
5
0
0
1
2
3
ε (MeV)
4
5
-6
smoothing function
450