Investigation of $α$ clustering with knockout reactions
Kazuki Yoshida Kazuyuki Ogata Yoshiko Kanada-En'yo
nucl-th





[]yoshida.kazuki@jaea.go.jp
Advanced Science Research Center, Japan Atomic Energy Agency, 
Tokai, Ibaraki 319-1195, Japan
Research Center for Nuclear Physics (RCNP), Osaka
University, Ibaraki 567-0047, Japan



Research Center for Nuclear Physics (RCNP), Osaka
University, Ibaraki 567-0047, Japan
Department of Physics, Osaka City University, Osaka 558-8585, Japan


Department of Physics, Kyoto University, Kyoto 606-8502, Japan






  Background
Nuclear clustering has been one of the main interests in nuclear physics.
In order to probe the α clustering through reaction observables,
α transfer and α knockout reactions have been studied.
It is very important to probe the α cluster amplitude at nuclear
surface since the α spectroscopic factor is not necessarily a direct
measure of the α clustering.

  Purpose
Our goal is to reveal how the
α cluster amplitude is probed through α
knockout reactions depending on reaction conditions, e.g.,
the incident energy.

  Method
We consider ^20Ne(p, pα)^16O and ^120Sn(p, pα)^116Cd
at 100–400 MeV
within the distorted wave impulse
approximation (DWIA) framework.
We introduce a masking function, which shows how
the reaction amplitude in the nuclear
interior is suppressed and defines the probed region of
the α cluster
wave function.

  Results
It is clearly shown by means of the masking function
that the α knockout reaction probes
the α cluster amplitude in the nuclear surface region,
which is the direct measure of well-developed α cluster states.
The incident energy dependence of the masking effect is
investigated, using a simplified form of the masking function.

  Conclusions
The α knockout reaction can probe the α cluster amplitude
in the nuclear surface region by choosing proper kinematics
owing to the masking effect originated
from absorptions of distorting potentials, and is a suitable method
to investigate how α cluster states are spatially developed.





24.10.Eq, 25.40.-h, 21.60.Gx





Investigation of α clustering with knockout reactions

    Yoshiko Kanada-En'yo
    December 30, 2023
=======================================================






§ INTRODUCTION

Cluster states are one of the main interests in nuclear physics.
For a recent review, see Ref. <cit.>.
When investigating α cluster states,
it should be noted that
a large α spectroscopic factor
does not necessarily indicate
well-developed α cluster states
because of the dual nature of the  cluster and
the shell model <cit.>.
From this point of view,
recently the ^16O(^6Li,d)^20Ne
α transfer reaction has been studied <cit.>
with a three-body reaction model using a macroscopic
cluster wave function,
and the reaction was shown to have high sensitivity
in the nuclear surface region and suitable
to prove the α cluster amplitude there,
i.e., spatially developed α cluster states.


An alternative method to the α transfer reaction is the
proton-induced α knockout reaction, i.e., (p, pα).
In the present study we employ the distorted wave impulse approximation (DWIA)
framework, which has been utilized and well established in α knockout
reaction studies <cit.>
and nucleon knockout reactions <cit.>
as well.
For a recent review on the (p, pN) reactions,
see Ref. <cit.>.
In this paper we examine the
peripherality of ^20Ne(p, pα)^16O and
its incident energy dependence to investigate how the α cluster
amplitude in the nuclear surface region is probed through
the ^20Ne(p, pα)^16O reaction.
The ^120Sn(p, pα)^116Cd reaction is also investigated
in a similar manner.



In Sec. <ref> we describe the DWIA formalism for
(p, pα) reactions and also how the cluster wave function is
constructed in the present study.
The definition of the masking function,
which is the key concept in the present study,
is also given.
In Sec. <ref> we introduce a general feature of
knockout reactions.
Then the absorption effect in α knockout reactions
due to the distorting potential is discussed in terms of the
masking function, which defines the probed region in the (p, pα)
reaction.
The ^120Sn(p, pα)^116Cd reaction is
investigated as a case of strong
absorption and Coulomb effects.
The incident energy dependence of the masking effect
is also discussed with introducing a simplified form of
the masking function.
Finally, a summary is given in Sec. <ref>.



§ THEORETICAL FRAMEWORK






 §.§ DWIA formalism


In the present study we consider the ^20Ne(p, pα)^16O reaction
in normal kinematics.
The incoming and outgoing protons are labeled as particle
0 and 1, respectively.
K_i, Ω_i, E_i, and T_i denote
the momentum (wave number), its solid angle,
total and kinetic energy of particle i (=0, 1, α),
respectively.
All quantities appear below are evaluated in the
center-of-mass (c.m.) frame, except those with
the superscript L which are in the laboratory (L) frame.




In the DWIA framework, the transition amplitude of
A(p, pα)B is given by

    T_ K_i^nlm   =
    χ_1, K_1^(-)χ_α, K_α^(-)
    |t_pα|
    χ_0, K_0^(+)φ_α^nlm,

where χ_i, K_i (i=0, 1, α) are the
distorted waves of p-A, p-B, and α-B systems with
relative momentum (wave number) K_i,
respectively.
The superscripts (+) and (-) are given to specify the outgoing and
the incoming boundary conditions of the scattering waves, respectively.
The α cluster wave function of the α-B system is denoted by
φ_α^nlm with n, l, and m being
the principal quantum number, the angular momentum, and its third component, respectively.
t_pα is the
effective interaction between p and α,
which is the transition interaction in the DWIA framework.










Applying the so-called factorization approximation,
or  the asymptotic momentum approximation, which has
been justified in Ref. <cit.>,
Eq. (<ref>) is reduced to

    T_ K_i^nlm   ≈t̃_pα(κ',κ)
    ∫ d R 
    F_ K_i( R) φ_α^nlm( R),

where κ (κ') is the p-α
relative momentum in the initial (final) state.
t̃_pα and F_ K_i ( R) are defined by

    t̃_pα(κ',κ)
       ≡∫ d s 
    e^-iκ'· s 
    t_pα( s) 
    e^iκ· s, 
    
    F_ K_i( R)
       ≡χ_1, K_1^*(-)( R)
    χ_α, K_α^*(-)( R)
    χ_0, K_0^(+)( R) 
    e^-i K_0· RA_α/A,

where A_α = 4 and A is the mass number of the nucleus A.
By making the on-the-energy-shell (on-shell) approximation
of the final state prescription:

    κ   =
    κ'κ̂,

the matrix element of t_pα in Eq. (<ref>)
can be related with a free p-α differential cross section as

    |
    t̃_pα(κ',κ)
    |^2
    ≈(2πħ ^2)^2/μ_pα^2dσ_pα/dΩ_pα(θ_pα,T_pα),

where the p-α scattering angle θ_pα
is an angle between κ and κ',
and the scattering energy is defined by
T_pα = (ħκ')^2 / (2μ_pα) with
μ_pα being the reduced mass of the p-α system.
The triple differential cross section (TDX) of the
A(p, pα)B reaction is then given by

    d^3σ/dE_1^LdΩ_1^LΩ_2^L   =
    S_αF_kin C_0
    dσ_pα/dΩ_pα(θ_pα,T_pα)
    ∑_m|
    T̅_ K_i^nlm|^2,

where S_α is the so-called α spectroscopic factor, and
F_kin, C_0, and
T̅_ K_i^nlm are defined by

    F_kin   ≡
    J_LK_1 K_α E_1 E_α/(ħ c)^4[
    1+E_α/E_B+E_α/E_B K_1· K_α/K_α^2]^-1, 
    
    C_0
       ≡E_0/(ħ c)^2 K_01/2l+1ħ ^4/(2π)^3 μ_pα^2,

and

    T̅_ K_i^nlm   ≡∫ d R 
    F_ K_i( R) φ_α^nlm( R)

with J_L being the Jacobian from the c.m. frame to the L frame.

The DWIA framework introduced in this section
has been validated in Ref. <cit.> by the benchmark comparison with
the transfer-to-the-continuum model <cit.>
and the Faddeev/Alt-Grassberger-Sandhas method <cit.>.






 §.§ α cluster wave function

The α cluster wave function φ_α^nlm( R)
for the α-^16O system
is defined as the eigenstate of the Schrödinger equation:

    [
    -ħ^2/2μ_α∇_ R^2
    +V_αB(R)
    ]φ_α^nlm( R)
       =
    ε_αφ_α^nlm( R),

where μ_α, V_αB, and
ε_α are
the reduced mass, the binding potential, and the binding energy
of α-B system, respectively.

For the nuclear part of V_αB,
a central Woods-Saxon shape with
the radius parameter r_0 and the diffuseness parameter a_0
is adopted:

    f_WS(R)
       =
    1/1+exp(R-r_0/a_0).

The numerical input for r_0, a_0 and ε_α
are given in Sec. <ref>.
Since V_αB is a central potential,
φ_α^nlm( R) can be simply separated into the radial
and the angular part as

    φ_α^nlm( R)
    =
    ϕ_α^nl(R)Y_lm(Ω),

where ϕ_α^nl(R), Y_lm, and Ω are
the radial part of φ_α^nlm(R),
the spherical harmonics, and
the solid angle of R, respectively.




 §.§ Masking function

Since the square modulus of Eq. (<ref>)
is proportional to the knockout cross section,
it should be worth investigating
the property of the integrand in the right-hand side of Eq. (<ref>)
to reveal a contribution of φ_α^nlm
to the knockout cross section.
Here we define the masking function

    D_lm(R)
    ≡1/√(4π)∫ dΩ 
     F_ K_i( R) Y_lm(Ω)

so that T̅_ K_i^nlm is given by

    T̅_ K_i^nlm   =
    √(4π)∫ dR 
    R^2 D_lm(R) ϕ_α^nl(R).

Therefore the masking function D_lm(R) is a weighting function
which determines
the radial contribution of ϕ_α^nl(R)
to the knockout reaction amplitude.

If the nucleus B can be treated as a spectator,
the total momentum of the p-α system is approximately conserved;

    k_α   ≈q≡ K_1+ K_α
    -(1-A_α/A) K_0,

where q is the so-called missing momentum and
k_α is the momentum of the
α cluster in the nucleus A in the initial state.
It can be shown that k_α is actually the momentum
-K_B^L.
Once all distorting potentials are switched off, i.e.,
the plane wave impulse approximation (PWIA) is adopted,
T̅_ K_i^nlm turns out to be the Fourier transform
of the α cluster wave function,

    T̅_ K_i^nlm   ≈∫ d R 
    e^-i k_α· Rϕ_α^nl( R).

The masking function D_lm(R) is normalized to
unity when an l=0 cluster wave function is considered in PWIA,
as discussed in Sec. <ref>.















§ RESULTS AND DISCUSSION




 §.§ Numerical inputs


In the calculation of ^20Ne(p, pα)^16O, n=4, and l=0 are assumed
for the α cluster orbital.
For the binding potential V_αB(R)=V_0 f_WS(R),
the radius parameter r_0=1.25 × 16^1/3 fm and the
diffuseness parameter a_0 = 0.76 fm are employed.
These parameters are fixed to describe the behavior
of the microscopic cluster model wave function in the tail region <cit.>.
The depth parameter V_0 is determined so as to reproduced the
α separation energy of 4.73 MeV.
In Fig. <ref> we show the obtained
ϕ_α^40(R)
by solving Eq. (<ref>).

The solid, dashed, and dotted lines are
ϕ_α^40(R), Rϕ_α^40(R), and
R^2ϕ_α^40(R), respectively.
The dashed line is equivalent to the PM1 in Fig. 3 of Ref. <cit.>.
The dotted line
in Fig. <ref>
shows a cluster wave function multiplied by R^2, which
will be helpful in the following discussion
because R^2 appears as a weight on ϕ_α^40(R)
in the transition matrix, as shown in Eq. (<ref>).


For the optical potentials of the incoming and outgoing protons,
the global optical potential by Koning and Delaroche <cit.>
is employed, and for the α-^16O potential in the final state,
the parameter set by Nolte, et al. <cit.> is employed.
For calculating the p-α scattering cross section,
the microscopic single folding model <cit.> with a phenomenological
α density and the Melbourne nucleon-nucleon (NN) g-matrix interaction <cit.>
is adopted.




 §.§ ^20Ne(p, pα)^16O reaction


In Fig. <ref> we show the TDX of ^20Ne(p, pα)^16O at 392 MeV
as a function of the recoil momentum p_R defined by

    p_R   =
    ħ K_B^LK_Bz^L/| K_Bz^L|,

where K_Bz^L is the z-component of
K_B^L following the Madison convention.
The kinematics are fixed to be
T_1=352 MeV, θ_1^L = 32.5^∘,
and T_α (θ_α) varies 31–35 MeV (27–108^∘).
The azimuthal angles ϕ_K_1 and ϕ_K_2 are
fixed at 0 and π, respectively,
i.e., the scattered particles are in a coplanar.
The TDX has a peak at p_R∼ 0 MeV/c when l=0, since
p_R corresponds to the momentum of α in the initial state
in quasifree knockout reactions.








In Fig. <ref> we investigate the
masking function D_00(R)
and the radial reaction amplitude I(R) at p_R=0 MeV/c. The latter is defined by

    I(R)
       ≡
    R^2 |D_00(R)| ϕ_α^40(R),

which corresponds to the integrand of R in Eq. (<ref>).

As shown by the solid line in Fig. <ref>(a),
|D_00(R)| is enough small in the nuclear interior region to suppress
contribution of the α amplitude in this region to I(R) because of
the absorption effect of the distorting potentials.
The result without the Coulomb interaction V_c
(dashed line in Fig. <ref>(a))
approaches to unity in the asymptotic region because
all the nuclear potentials are absent there and
F_ K_i( R)=1
when q=0; see Eqs. (<ref>) and (<ref>).
On the other hand, the result of DWIA including V_c (solid line)
never reaches to unity even in the asymptotic region.
This is because F_K_i( R) suffers from
the long-range nature of the Coulomb interaction.
Even if the asymptotic
momenta of the three particles satisfy q=0,
finite values of the Coulomb phase shifts of the scattering waves
make |D_00(R)| deviate from unity when R is finite.

Figure <ref>(b) shows I(R) defined by Eq. (<ref>).
One sees that I(R) of DWIA (solid line) is strongly suppressed in the
interior region, compared with that of PWIA divided by 3 (dotted line).
The result of DWIA without Coulomb interactions (dashed line) agrees well
with that of DWIA (solid line),
since the cluster wave function has an amplitude for R8 fm,
where |D_00(R)| with and without Coulomb interactions
agrees well each other.
Therefore the deviation of the solid line
in Fig. <ref>(a) from unity due to Coulomb interactions
makes no difference in understanding the property of D_00(R).
This allows one to make further simplification of D_00(R)
as discussed in Sec. <ref>.

The masking effect on the cluster wave function due to
nuclear distorting potentials can be
a great advantage of
(p, pα) reactions for probing α cluster states.
A large (p, pα) cross section
corresponds to a large cluster amplitude around the nuclear surface,
which indicates an existence of well-developed α cluster states in a nucleus.
It should be noted that in the
nuclear interior, the NN antisymmetrization plays a significant role,
which makes even the definition of the α cluster unclear. Because of
the masking effect, (p, pα) reactions automatically avoid such an
unclear region and focus the nuclear surface. In consequence of this,
one can conclude that what is determined by (p, pα) is not an
α spectroscopic factor S_α but the surface amplitude of the α
distribution, and the latter is the measure of the α clustering.





 §.§ Coulomb effect in ^120Sn(p, pα)^116Cd reaction

As shown in Fig. <ref> (a),
long-ranged
Coulomb interactions
prohibit the masking function from reaching unity,
even if the kinematics of the knockout process is fixed so as
to satisfy q=0.
This effect becomes significant when the charge of the target nucleus
is large.
In this point of view,
we investigate the peripheral property of the
^120Sn(p, pα)^116Cd reaction at
392 MeV and the masking function as well.
The α cluster wave function is constructed in
the same method as mentioned above but with
r_0=1.25× 116^1/3 fm and
ε_α=-4.81 MeV.
The three-body kinematics are chosen to be
T_1=328 MeV, θ_1^L = 43.2^∘,
T_α=51 MeV, and θ_α=61^∘,
which satisfies q=0.

In Fig. <ref> |D_00(R)| and I(R) of
^120Sn(p, pα)^116Cd are shown in the
same manner as in Fig. <ref>.

The obtained masking function for this system
(the solid line in Fig. <ref> (a))
is only
0.2–0.3 even
in the asymptotic region
by the Coulomb interactions.
On the other hand, D_00(R) without
V_C (the dashed line)
tends to approach unity, as well as
the eikonal masking function D_00^EK(R)
(dot-dashed line).
This peculiar behavior of D_00(R) at larger R, however,
causes no serious problem, because the α cluster wave function
is sufficiently damped in that region.
Consequently, the dashed line
and the dot-dashed line agree quite well with the solid line in
Fig. <ref> (b)
indicating that I(R) can be approximately simulated by analyses of the
masking function without Coulomb interactions and
also the eikonal masking function.
It turns out that only the α amplitude at the nuclear surface can be safely probed by α knockout reactions
even in the case of heavy-mass targets.






 §.§ Incident energy dependence of masking function


It is important to clarify how the aforementioned property of the
masking function depends on the incident energy.
For this purpose, a more simplified functional form of the
masking function is preferable. We thus rely on the eikonal
approximation and use the following form:

    D_00^EK(R)
       ≡1/4π∫ dΩ exp[
    -C_abs∫_-∞^∞ dz
    f_WS( R)
    ]
    e^-i q· R,

which is designated eikonal masking function.
C_abs represents
the total strength of the absorption caused by the distorting potentials
of particles 0–2, and is determined so that the peak height of

    I^EK(R)
       ≡
    R^2
    |
    D_00^EK(R)
    |
    ϕ_α^40(R)

reproduces that of I(R).
Thus, one can characterize the masking function D_00(R) by
just one parameter C_abs.
In a naïve interpretation, C_abs is related to
the mean free path (MFP) λ through
1/λ = 2C_absf_WS(R).



The dot-dashed lines in Fig. <ref>(a) and <ref>(b) represent
|D_00^EK(R)| and I^EK(R),
respectively; C_abs is taken to be 0.69 fm^-1.
One sees in Fig. <ref>(a) that the dot-dashed line
agrees well with the solid line for 4  R 6 fm and slightly
deviates from it for larger R. This asymptotics of the dot-dashed
line can be understood as a result of absence of Coulomb interactions
in Eq. (<ref>).
Nevertheless, as shown in Fig. <ref>(b), the dot-dashed line,
I^EK(R),
agrees well with the solid line for R 4 fm, in which
the integrand I(R) has a meaningful amplitude. This indicates
that Eq. (<ref>) is sufficient
to describe the property of the masking function that is relevant to the
(p,pα) reaction considered. It should be noted that
the small but finite difference between the solid and dot-dashed lines in
Fig. <ref>(a) for R 6 fm appears also in
Fig. <ref>(b); for instance, the difference is about 6% at 8 fm.
We then discuss the incident energy T_0 dependence of C_abs,
which is shown in Fig. <ref>.
At each T_0, the scattering energies and angles of particle 1 and 
α are chosen so as to satisfy p_R = 0 MeV/c.

It is found that the T_0 dependence of C_abs is
weak above 200 MeV, where the quasi-free (impulse) picture
of the knockout reaction is valid.
On the other hand, below 200 MeV, C_abs increases rather
rapidly as T_0 decreases. This behavior can be understood
by the fact that the MFP's of low energy particles are small.
Nevertheless, the general property of the masking function
and the peripherality of the reaction are
found to be robust for T_0 shown in Fig. <ref>.








§ SUMMARY


^20Ne(p, pα)^16O reactions at 100–392 MeV were investigated
within the distorted wave impulse approximation framework.
We have introduced the masking function, which describes the absorption
effect due to distorting potentials of the incident p and also the
emitted p and α and suppresses the α amplitude
contribution in the interior region to the
α knock-out  cross section.
Through the analyses on the masking functions of the reactions,
it is clearly shown that α knockout reactions are peripheral
and suitable for probing the α cluster amplitude in the
nuclear surface, which is regarded as a direct
measure of spatially
developed α cluster states.


As a case of strong Coulomb interactions,
the ^120Sn(p, pα)^116Cd reaction at 392 MeV
has been investigated.
It is shown that
in this case the masking function has a nontrivial behavior
at larger distance
due to Coulomb interactions.
Nevertheless, this
does not cause any problem because the overlap between the
masking function and the cluster amplitude in that region is
negligibly small.



To investigate the incident energy T_0 dependence of the masking
function, we introduce a simplified functional that is
characterized by only one parameter C_abs.
It was found that C_abs depends weakly on T_0
for 200–400 MeV. At lower energies, C_abs increases
rather rapidly, corresponding to stronger absorption.
However, the feature of the masking function turned out to be
robust down to 100 MeV.






§ ACKNOWLEDGMENTS

A part of the computation
was carried out with the computer facilities at the Research
Center for Nuclear Physics, Osaka University.
This work was
supported in part by Grants-in-Aid of the Japan Society for
the Promotion of Science (Grants No. JP16K05352,
JP15J01392, and No. JP26400270).






00
Hor12
H. Horiuchi, K. Ikeda, and K. Kato,
Prog. Theor. Phys. Supplement 192, 1 (2012).

Bay58
B. F. Bayman and A. Bohr, Nucl. Phys. 9, 596 (1958/59).

Fuk16
T. Fukui, Y. Taniguchi, T. Suhara, Y. Kanada-En'yo, and K. Ogata,
Phys. Rev. C 93, 034606 (2016).

Car84
T. A. Carey, P. G. Roos, N. S. Chant, A. Nadasen, and H. L. Chen,
Phys. Rev. C 29, 1273 (1984).

Mab09
J. Mabiala, A. A. Cowley, S. V. Förtsch, E. Z. Buthelezi, R. Neveling, F. D. Smit, G. F. Steyn, and J. J. Van Zyl,
Phys. Rev. C 79, 054612 (2009).

Roo77
P. G. Roos, N. S. Chant, A. A. Cowley, D. A. Goldberg, H. D. Holmgren, and R. Woody, III,
Phys. Rev. C 15, 69 (1977).

Nad80
A. Nadasen, N. S. Chant, P. G. Roos, T. A. Carey, R. Cowen, C. Samanta, and J. Wesick
Phys. Rev. C 22, 1394 (1980).

Nad89
A. Nadasen, P. G. Roos, N. S. Chant, C. C. Chang, G. Ciangaru, H. F. Breuer, J. Wesick, and
E. Norbeck,
Phys. Rev. C 40, 1130 (1989).

Wan85
C. W. Wang, P. G. Roos, N. S. Chant, G. Ciangaru, F. Khazaie, D. J. Mack, A. Nadasen, S. J. Mills, R. E. Warner, E. Norbeck, F. D. Becchetti, J. W. Janecke, and P. M. Lister,
Phys. Rev. C 31, 1662 (1985).

Yos88
T. Yoshimura, A. Okihana, R. E. Warner, N. S. Chant, P. G. Roos, C. Samanta, S. Kakigi, N. Koori,
M. Fujiwara, N. Matsuoka, K. Tamura, E. Kubo, and K. Ushiro,
Nucl. Phys. A 641, 3 (1988).

Sam86
C. Samanta, N. S. Chant, P. G. Roos, A. Nadasen, J. Wesick,
and A. A.  Cowley,
Phys. Rev. C 34, 1610 (1986).

Cha77
N. S. Chant and P. G. Roos,
Phys. Rev. C 15, 57 (1977).

Sam87
C. Samanta, N. S. Chant, P. G. Roos, A. Nadasen, and A. A. Cowley,
Phys. Rev. C 35, 333 (1987).

Jac66
G. Jacob and Th. A. J. Maris,
Rev. Mod. Phys. 38, 121 (1966).

Jac73
G. Jacob and Th. A. J. Maris,
Rev. Mod. Phys. 45, 6 (1973).

Kit85
P. Kitching, W. J. McDonald, Th. A. J. Maris, and C. A. Z. Vasconcellos,
Adv. Phys. Part. Nuclei 15, 43 (1985).

Wak17
T. Wakasa, K. Ogata, and T. Noro,
Progress in Particle and Nuclear Physics 96, 32 (2017).

Yos16
K. Yoshida, K. Minomo, and K. Ogata,
Phys. Rev. C 94 044604 (2016).

Yos18
K. Yoshida, M. Gómez-Ramos, K Ogata, and A. M. Moro,
Phys. Rev. C 97, 024608 (2018).

Mor15
A. M. Moro,
Phys. Rev. C 92, 044605 (2015).

Mar17
M. Gómez-Ramos J. Casal, and A. M. Moro,
Phys. Lett. B 772 115 (2017).

Fad61
L. D. Faddeev,
Zh. Eksp. Teor. Fiz. 39, 1459 (1960)
[Sov. Phys. JETP 12, 1014 (1961)].

Alt67
E. O. Alt, P. Grassberger, and W. Sandhas,
Nucl. Phys. B 2, 167 (1967).

Kon03
A. J. Koning and J. P. Delaroche,
Nucl. Phys. A 713 231 (2003).

Nol87
M. Nolte, H. Machner, and J. Bojowald,
Phys. Rev. C 36, 1312 (1987).

Toy13
M. Toyokawa, K. Minomo, and M. Yahiro,
Phys. Rev. C 88, 054602 (2013).

Amo00
K. Amos, P. J. Dortmans, H. V. von Geramb, S. Karataglidis, and J. Raynal,
Adv. Nucl. Phys. 25, 276 (2000).


