# Significance of the refraction effect on the $p$-$d$ elementary process in
the ($p$,$pd$) reaction

Kazuki Yoshida<EMAIL_ADDRESS>Advanced Science Research Center,
Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan  Yoshiki Chazono
RIKEN Nishina Center for Accelerator-Based Science, 2-1 Hirosawa, Wako
351-0198, Japan  Kazuyuki Ogata  Department of Physics, Kyushu University,
Fukuoka 819-0395, Japan  Research Center for Nuclear Physics, Ibaraki, Osaka
567-0047, Japan

###### Abstract

Background

The proton-induced deuteron knockout reaction, ($p$,$pd$), is one of the
interests in the studies for probing the deuteron-like $p$-$n$ correlation in
nuclei. According to a recent study of the inclusive deuteron-induced
reaction, $(d,d^{\prime}x)$, the refraction effect of the deuteron has a
significant effect on the elementary process, nucleon-deuteron ($N$-$d$)
binary scattering inside a nucleus, of the reaction. In the paper, it is shown
that proper treatment of the local $N$-$d$ relative momentum in the elementary
process is crucial in $(d,d^{\prime}x)$ reactions at $100$ MeV and below.

Purpose

In the present work, we investigate the deuteron refraction effect in the
exclusive ($p$,$pd$) reactions. We also discuss the incident energy dependence
of the refraction effect.

Method

The refraction effect on the $p$-$d$ elementary process is taken into account
by the local semiclassical approximation to the distorted waves. The results
are compared with those obtained with the asymptotic momentum approximation,
which is standardly applied to the distorted wave impulse approximation
framework.

Result

It is shown that the refraction effect drastically changes the energy sharing
distribution of the 16O($p$,$pd$)14N reaction at 101.3 MeV and gives a better
agreement with experimental data. In contrast, it is confirmed that the effect
is negligibly small at 250 MeV.

Conclusion

We have clarified that the deuteron refraction effect is significant in the
16O($p$,$pd$)14N reaction at 101.3 MeV and the experimental data are well
reproduced. The refraction effect plays a significant role in both the shape
and magnitude of the ($p$,$pd$) cross section, while the effect is negligible
at 250 MeV.

###### pacs:

24.10.Eq, 25.40.-h

##  I Introduction

The quasi-free knockout reactions have been used for more than half a century
to probe both the single-particle (s.p.) nature of a nucleon [1, 2, 3, 4, 5,
6, 7, 8, 9] and the $\alpha$ cluster states [10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34] of nuclei.
These reactions have been described by the distorted wave impulse
approximation (DWIA) framework [3, 4] and the experimental data have been well
understood within this framework.

Recently, more exotic clusters, e.g., $d$, $t$, and 3He, have become of
interest [35, 36]. These clusters as well as the traditional $\alpha$ cluster
will be intensively studied by the ONOKORO project [37] using cluster knockout
reactions mainly at HIMAC, RCNP, and RIKEN RIBF in Japan.

The proton-induced deuteron knockout reaction, $(p,pd)$, is expected to be a
probe for the $p$-$n$ spatial and spin-isospin correlations, or, existence of
deuteron, inside a nucleus. One of the difficulties in describing the $(p,pd)$
reaction is the fragileness of the deuteron, which is not explicitly included
in the standard DWIA. In order to incorporate this nature of the deuteron into
a reaction theory, the continuum-discretized coupled-channels (CDCC) wave
function [38, 39, 40] has recently been implemented into the DWIA framework,
and the breakup and the reformation process of the $p$-$n$ pair in the
($p$,$pd$) reaction is studied within a new framework called CDCCIA [41].
Another difficulty is that the refraction of the deuteron by the nuclear
potential is larger compared to that of the nucleon. According to the recent
study on inclusive ($d$,$d^{\prime}x$) reactions [42] using the semiclassical
distorted wave model [43, 44], the refraction effect on the $p$-$d$ elementary
process has a significant impact. The same effect is expected in the exclusive
$(p,pd)$ reactions and it has to be also considered in combination with the
CDCCIA framework.

It should be noted that, in Ref. [25], the refraction effect in the
120Sn$(p,p\alpha)^{116}$Cd was investigated and found to be small. The main
reason of this result is the strong absorption of $\alpha$ in the region where
the refraction is important. In other words, both the real and imaginary
potential for $\alpha$ are strong. Thus, the significant effect of the
deuteron refraction found in Ref. [42] will suggest that the refraction of the
deuteron is much more important than its absorption.

In the present work, we focus on the deuteron refraction effect on the $p$-$d$
elementary process in the ($p$,$pd$) reaction within the DWIA framework and
discuss to what extent the $(p,pd)$ cross section is affected. There is a
limited number of ($p$,$pd$) reaction experiments [45, 46, 47, 48, 49, 50]. In
Ref. [51], the data of 16O($p$,$pd$)14N${}^{*}_{3.95~{}\mathrm{MeV}}$ at 101.3
MeV [47] are reanalyzed using the finite-range DWIA (FR-DWIA) formalism with
two types of $p$-$d$ interactions (“all attractive” and “repulsive core”).

This paper is organized as follows. In Sec. II we recapitulate the DWIA
framework with the local semiclassical momentum approximation (LSCA) [43, 44]
and the asymptotic momentum approximation (AMA) [25]. We discuss that the
finite-range nature of the $p$-$d$ effective interaction is partially taken
into account in the position dependence of its matrix element in the DWIA with
LSCA. In Sec. III the inputs for the DWIA calculations are explained and the
effect of the local momentum on the ($p$,$pd$) cross section is discussed
based on DWIA calculations with LSCA and AMA. The relation between the present
result and the FR-DWIA calculation [51] is also discussed in Sec. III.2.
Finally a summary is given in Sec. IV.

##  II Theoretical framework

In the present work, we basically follow the DWIA framework [3, 4, 7, 52] to
describe the ($p$,$pd$) reaction. Practical calculations are performed using a
newly published code pikoe [52]. The incoming proton, the outgoing proton, and
the outgoing deuteron are denoted as particles 0, 1, and 2, respectively, and
the target (residual) nucleus is by A (B). The total energy and the wave
number vector are represented by $E_{i}$ and $\bm{K}_{i}$
$(i=0,1,2,\mathrm{A,and~{}B})$, respectively. All the quantities with the
superscript L are evaluated in the laboratory (target-rest) frame of the
($p$,$pd$) reaction, while those without the superscript are evaluated in the
center-of-mass frame. As a characteristic ingredient in the present study, we
use the local semiclassical approximation (LSCA) [43, 44] to describe the
propagation of the distorted waves under nuclear potentials, which is nothing
but the refraction of the reaction particle. This method, DWIA with LSCA, has
already been applied to ($p$,$p\alpha$) analyses in Ref. [25].

The DWIA $T$-matrix of the $(p,pd)$ reaction is given by [3, 7, 53, 52].

$\displaystyle T$ $\displaystyle=\int
d\bm{R}\,d\bm{s}\,\chi_{1}^{(-)}(\bm{R}_{1})\chi_{2}^{(-)}(\bm{R}_{2})t_{pd}(\bm{s})\chi_{0}^{(+)}(\bm{R}_{0})\varphi_{d}(\bm{R}_{2}).$
(1)

The coordinates are defined as shown in Fig. 1, and $\chi_{i}(\bm{R}_{i})$ is
the distorted wave describing the scattering state of particle $i$ with
respect to the coordinate $\bm{R}_{i}$.

Figure 1: Definition of the coordinates of the A($p$,$pd$)B reaction.

The distorted waves with the superscripts $(+)$ and $(-)$ satisfy the outgoing
and incoming boundary conditions, respectively. $\varphi_{d}$ is a normalized
bound state wave function of the deuteron and B, and its spectroscopic factor
$S_{d}$ is explicitly considered in Eqs. (11) and (13) below. $t_{pd}$ is the
$p$-$d$ effective interaction which describes the $p$-$d$ elastic scattering.
In this study, it is assumed that $t_{pd}$ does not change the $p$-$n$
relative state and the $p$-$n$ pair remains in the deuteron ground state and
its breakup and reformation are not considered.

LSCA [43, 44] assumes that the short-range propagation from a reference point
$\bm{R}$ to $\bm{R}+\Delta\bm{R}$ is described by a plane wave with a local
momentum $\bm{K}(\bm{R})$ at the reference point $\bm{R}$:

$\displaystyle\chi_{i}(\bm{R}+\Delta\bm{R})\approx\chi_{i}(\bm{R})e^{i\bm{K}_{i}(\bm{R})\cdot\Delta\bm{R}}.$
(2)

This is the first order approximation of a translation by $\Delta\bm{R}$ of
$\chi_{i}(\bm{R})$. The norm of $\bm{K}_{i}(\bm{R})$ is determined by local
energy conservation and the direction of $\bm{K}_{i}(\bm{R})$ is taken to be
parallel with the flux of $\chi_{i}(\bm{R})$. LSCA has been applied to the
scattering of nucleon [43, 44, 54], $\alpha$ particle [25], and deuteron [42].
Applying LSCA to Eq. (1), one obtains the DWIA $T$-matrix with LSCA [25] as

$\displaystyle T^{\mathrm{LSCA}}=\int
d\bm{R}\,F(\bm{R})\varphi_{d}(\bm{R})\tilde{t}_{pd}(\bm{\kappa}(\bm{R}),\bm{\kappa}^{\prime}(\bm{R})),$
(3) $\displaystyle
F(\bm{R})\equiv\chi_{1}^{(-)}(\bm{R})\chi_{2}^{(-)}(\bm{R})\chi_{0}^{(+)}(\bm{R})e^{-i\bm{K}_{0}(\bm{R})\cdot\bm{R}A_{d}/A},$
(4)
$\displaystyle\tilde{t}_{pd}(\bm{\kappa}(\bm{R}),\bm{\kappa}^{\prime}(\bm{R}))\equiv\int
d\bm{s}\,e^{-i\bm{\kappa}^{\prime}(\bm{R})\cdot\bm{s}}t_{pd}(\bm{s})e^{i\bm{\kappa}(\bm{R})\cdot\bm{s}},$
(5)

where $\bm{\kappa}(\bm{R})$ and $\bm{\kappa}^{\prime}(\bm{R})$ are the local
$p$-$d$ relative momentum in the initial and final states, respectively, which
are given by

$\displaystyle\bm{\kappa}(\bm{R})$
$\displaystyle\equiv\frac{A+1}{A}\frac{A_{d}}{A_{d}+1}\bm{K}_{0}(\bm{R})-\frac{1}{A_{d}+1}\bm{K}_{d}(\bm{R}),$
(6) $\displaystyle\bm{\kappa}^{\prime}(\bm{R})$
$\displaystyle\equiv\frac{A_{d}}{A_{d}+1}\bm{K}_{1}(\bm{R})-\frac{1}{A_{d}+1}\bm{K}_{2}(\bm{R})$
(7)

as discussed in Ref. [25]. $A_{d}$ and $A$ are the mass numbers of the
deuteron and the target nucleus, respectively. $\bm{K}_{d}$ is determined to
satisfy the momentum conservation of the $p$-$d$ system:

$\displaystyle\bm{K}_{d}(\bm{R})=\bm{K}_{1}(\bm{R})+\bm{K}_{2}(\bm{R})-\frac{A+1}{A}\bm{K}_{0}(\bm{R}).$
(8)

In the numerical calculation of Eq. (3), in many cases, $\tilde{t}_{pd}$
parameterized in momentum space has been used, rather than calculating the
right-hand-side of Eq. (5). In other words, the integration over $\bm{s}$
seems to be unnecessary in the numerical calculation. Because of this,
sometimes Eq. (3) is referred to as the zero-range approximation. However, it
is obvious from Eq. (5) that the finite-range nature of $t_{pd}(\bm{s})$ is
explicitly taken into account. Furthermore, in DWIA with LSCA, the arguments
of $\tilde{t}_{pd}$ depend on $\bm{R}$, which is not the case with DWIA with
AMA as one sees from Eqs. (13) and (14) below. The $\bm{R}$ dependence of
$\tilde{t}_{pd}$ can be understood that the original 6-dimensional nature is
more respected than in DWIA with AMA. We will return to this point later.

Making the on-the-energy-shell (on-shell) approximation to $\tilde{t}_{pd}$
using the relative momentum and the energy in the final state (final-state
prescription), one obtains

$\displaystyle\left(\frac{\mu_{pd}}{2\pi\hbar^{2}}\right)^{2}\left|\tilde{t}_{pd}(\bm{\kappa}(\bm{R}),\bm{\kappa}^{\prime}(\bm{R}))\right|^{2}\approx\frac{d\sigma_{pd}}{d\Omega_{pd}}\left(\theta_{pd}(\bm{R}),E_{pd}(\bm{R})\right),$
(9)

where $\mu_{pd}$ is the reduced mass of the $p$-$d$ system and
$d\sigma_{pd}/d\Omega_{pd}$ represents the $p$-$d$ elastic cross section. The
reduced $T$-matrix with LSCA is obtained as

$\displaystyle\bar{T}^{\mathrm{LSCA}}$ $\displaystyle=\int
d\bm{R}\,\sqrt{\frac{d\sigma_{pd}}{d\Omega_{pd}}(\theta_{pd}(\bm{R}),E_{pd}(\bm{R}))}F(\bm{R})\varphi_{d}(\bm{R}).$
(10)

The $p$-$d$ scattering angle $\theta_{pd}(\bm{R})$ and the scattering energy
$E_{pd}(\bm{R})$ are determined locally. Using $\bar{T}^{\mathrm{LSCA}}$, the
triple differential cross section (TDX) of the ($p$,$pd$) reaction is given by
[3, 7, 53, 52]

$\displaystyle\frac{d^{3}\sigma^{\mathrm{L}}_{\mathrm{LSCA}}}{dE_{1}^{\mathrm{L}}d\Omega_{1}^{\mathrm{L}}d\Omega_{2}^{\mathrm{L}}}$
$\displaystyle=\frac{(2\pi)^{4}}{\hbar
v}S_{d}F_{\mathrm{kin}}^{\mathrm{L}}\frac{E_{1}E_{2}E_{\mathrm{B}}}{E_{1}^{\mathrm{L}}E_{2}^{\mathrm{L}}E_{\mathrm{B}}^{\mathrm{L}}}\frac{1}{2l+1}$
$\displaystyle\times\left(\frac{2\pi\hbar^{2}}{\mu_{pd}}\right)^{2}\left|\bar{T}^{\mathrm{LSCA}}\right|^{2},$
(11) $\displaystyle
F_{\mathrm{kin}}^{\mathrm{L}}\equiv\frac{E_{1}^{\mathrm{L}}K_{1}^{\mathrm{L}}E_{2}^{\mathrm{L}}K_{2}^{\mathrm{L}}}{(\hbar
c)^{2}}\left[1+\frac{E_{2}^{\mathrm{L}}}{E_{\mathrm{B}}^{\mathrm{L}}}+\frac{E_{2}^{\mathrm{L}}}{E_{\mathrm{B}}^{\mathrm{L}}}\frac{\bm{K}_{\mathrm{X}}^{\mathrm{L}}\cdot\bm{K}_{2}^{\mathrm{L}}}{\left(K_{2}^{\mathrm{L}}\right)^{2}}\right]^{-1},$
(12)

with
$\bm{K}_{\mathrm{X}}^{\mathrm{L}}=\bm{K}_{1}^{\mathrm{L}}-\bm{K}_{0}^{\mathrm{L}}-\bm{K}_{\mathrm{A}}^{\mathrm{L}}$.
Here, $\Omega_{i}$ is the solid angle of $\bm{K}_{i}$ of particle $i$. $v$ is
the relative velocity between particle 0 and the target A, $l$ is the relative
angular momentum between B and the deuteron in A. $S_{d}$ is the
aforementioned deuteron spectroscopic factor. As discussed in Sec. III.1, only
the $l=0$ case is discussed in the present work.

In AMA [7, 25], $\bm{K}_{i}(\bm{R})$ is replaced with its asymptotic momentum
$\bm{K}_{i}$. Thus, $\theta_{pd}$, $E_{pd}$, and $d\sigma_{pd}/d\Omega_{pd}$
are determined only by the asymptotic momenta. In other words,
$d\sigma_{pd}/d\Omega_{pd}$ no longer depends on $\bm{R}$. The factorized form
of the DWIA TDX is given by [3, 7, 53, 52]

$\displaystyle\frac{d^{3}\sigma^{\mathrm{L}}_{\mathrm{AMA}}}{dE_{1}^{\mathrm{L}}d\Omega_{1}^{\mathrm{L}}d\Omega_{2}^{\mathrm{L}}}$
$\displaystyle=\frac{(2\pi)^{4}}{\hbar
v}S_{d}F_{\mathrm{kin}}^{\mathrm{L}}\frac{E_{1}E_{2}E_{\mathrm{B}}}{E_{1}^{\mathrm{L}}E_{2}^{\mathrm{L}}E_{\mathrm{B}}^{\mathrm{L}}}\frac{1}{2l+1}$
$\displaystyle\times\left(\frac{2\pi\hbar^{2}}{\mu_{pd}}\right)^{2}\frac{d\sigma_{pd}}{d\Omega_{pd}}\left|\bar{T}^{\mathrm{AMA}}\right|^{2},$
(13) $\displaystyle\bar{T}^{\mathrm{AMA}}$ $\displaystyle=\int
d\bm{R}\,F(\bm{R})\varphi_{d}(\bm{R}).$ (14)

In this article, we call this the factorized form of the DWIA, not the zero-
range DWIA (ZR-DWIA), because the $p$-$d$ elastic cross section
$d\sigma_{pd}/d\Omega_{pd}$ results from the $p$-$d$ effective interaction
$t_{pd}(\bm{s})$ as seen in Eqs. (5) and (9), which is a finite-range
interaction. It should be noted, however, that as discussed in Sec. III.1, we
directly use experimental data of the $p$-$d$ elastic cross section for
$d\sigma_{pd}/d\Omega_{pd}$ in Eqs. (10) and (13). Therefore, an explicit form
of $t_{pd}(\bm{s})$ is not considered in the present study.

##  III Result and discussion

###  III.1 Numerical input

The differential cross section of the $p$-$d$ elastic scattering is prepared
numerically by fitting the experimental data from $5$ MeV to $800$ MeV with
respect to the scattering energy and the angle as shown in Fig. 2.

Figure 2: Angular distribution of the $p$-$d$ elastic scattering cross
section. The horizontal axis is the center-of-mass scattering angle of the
$p$-$d$ system. The experimental data taken from Refs. [55, 56, 57, 58, 59,
60, 61, 62, 63, 64] are represented by filled circles. The solid lines show
the results of the fitting. See Eq. (15) and Table 1 for details.

The experimental data of the $p$-$d$ elastic scattering are taken from Refs.
[55, 56, 57, 58, 59, 60, 61, 62, 63, 64]. The function used in the fitting is

$\displaystyle
f(\theta_{pd})=\sum_{j=1}^{j_{\textrm{max}}}a_{j}\exp\left[-\left(\frac{\theta_{pd}-c_{j}}{b_{j}}\right)^{2}\right]+d_{0},$
(15)

where $j_{\textrm{max}}$ is $2$ for $5$–$20$ MeV, $3$ for $35$–$250$ MeV, and
$4$ for $425$–$800$ MeV. All the parameters $a_{j}$, $b_{j}$, $c_{j}$, and
$d_{0}$ are listed in Table 1. It should be noted that $c_{1}$
($c_{j_{\textrm{max}}}$) is fixed at $0^{\circ}$ ($180^{\circ}$).

$E_{pd}$ |  | $j$ |  | $a_{j}$ |  | $b_{j}$ |  | $c_{j}$ |  | $d_{0}$
---|---|---|---|---|---|---|---|---|---|---
5 |  | 1 |  | 222.1560 |  | 86.8088 |  | 0.0000 |  | 0.0000
|  | 2 |  | 302.8700 |  | 37.5459 |  | 180.0000 |  |
10 |  | 1 |  | 177.2360 |  | 78.7301 |  | 0.0000 |  | 0.0000
|  | 2 |  | 182.1080 |  | 30.6876 |  | 180.0000 |  |
16 |  | 1 |  | 116.5830 |  | 70.8749 |  | 0.0000 |  | 0.0000
|  | 2 |  | 108.7870 |  | 26.0295 |  | 180.0000 |  |
20 |  | 1 |  | 97.9835 |  | 67.8439 |  | 0.0000 |  | 0.0000
|  | 2 |  | 73.3261 |  | 25.8404 |  | 180.0000 |  |
35 |  | 1 |  | 65.6339 |  | 52.7178 |  | 0.0000 |  | 0.0000
|  | 2 |  | 2.7094 |  | 35.3458 |  | 93.5447 |  |
|  | 3 |  | 28.8682 |  | 24.2590 |  | 180.0000 |  |
46 |  | 1 |  | 52.7869 |  | 44.5933 |  | 0.0000 |  | 0.0000
|  | 2 |  | 2.7551 |  | 48.1972 |  | 80.0000 |  |
|  | 3 |  | 15.7479 |  | 24.9981 |  | 180.0000 |  |
70 |  | 1 |  | 34.6225 |  | 38.2736 |  | 0.0000 |  | 0.7461
|  | 2 |  | 0.5250 |  | 29.8396 |  | 80.0000 |  |
|  | 3 |  | 6.9138 |  | 21.1479 |  | 180.0000 |  |
120 |  | 1 |  | 21.2430 |  | 30.7048 |  | 0.0000 |  | 0.3485
|  | 2 |  | 0.4932 |  | 19.8301 |  | 64.7798 |  |
|  | 3 |  | 1.9542 |  | 18.4435 |  | 180.0000 |  |
135 |  | 1 |  | 17.8934 |  | 31.7476 |  | 0.0000 |  | 0.3064
|  | 2 |  | 0.5316 |  | 20.8098 |  | 61.0116 |  |
|  | 3 |  | 0.9988 |  | 21.4401 |  | 180.0000 |  |
155 |  | 1 |  | 16.6284 |  | 29.5589 |  | 0.0000 |  | 0.2246
|  | 2 |  | 0.4812 |  | 20.5405 |  | 63.6499 |  |
|  | 3 |  | 0.6649 |  | 21.1364 |  | 180.0000 |  |
170 |  | 1 |  | 14.8897 |  | 27.9947 |  | 0.0000 |  | 0.1761
|  | 2 |  | 0.5273 |  | 23.9095 |  | 55.2300 |  |
|  | 3 |  | 0.4100 |  | 26.4076 |  | 180.0000 |  |
190 |  | 1 |  | 13.0640 |  | 28.5562 |  | 0.0000 |  | 0.1305
|  | 2 |  | 0.5110 |  | 26.0810 |  | 50.0000 |  |
|  | 3 |  | 0.3499 |  | 28.3901 |  | 180.0000 |  |
250 |  | 1 |  | 9.6172 |  | 20.2655 |  | 0.0000 |  | 0.0741
|  | 2 |  | 3.1887 |  | 40.5310 |  | 0.0000 |  |
|  | 3 |  | 0.1642 |  | 30.3782 |  | 180.0000 |  |
425 |  | 1 |  | 18.8803 |  | 21.7724 |  | 0.0000 |  | 0.1305
|  | 2 |  | 0.1294 |  | 13.0634 |  | 49.2562 |  |
|  | 3 |  | 0.0613 |  | 38.2106 |  | 80.0000 |  |
|  | 4 |  | 0.1900 |  | 34.5092 |  | 180.0000 |  |
582 |  | 1 |  | 24.7257 |  | 18.9076 |  | 0.0000 |  | 0.1305
|  | 2 |  | 0.1519 |  | 10.1242 |  | 38.0159 |  |
|  | 3 |  | 0.0858 |  | 41.6712 |  | 60.0000 |  |
|  | 4 |  | 0.2437 |  | 33.0826 |  | 180.0000 |  |
800 |  | 1 |  | 43.1368 |  | 15.4699 |  | 0.0000 |  | 0.1305
|  | 2 |  | 0.0780 |  | 33.9306 |  | 40.0000 |  |
|  | 3 |  | 0.0111 |  | 25.3878 |  | 100.0000 |  |
|  | 4 |  | 0.0904 |  | 30.4011 |  | 180.0000 |  |
Table 1: Numerical value of the parameters in Eq. (15). $E_{pd}$ is given in
MeV, $a_{j}$ and $d_{0}$ are in mb/sr, $\theta_{pd}$, $b_{j}$, and $c_{j}$ are
in degrees.

In the present calculation, the fitted $p$-$d$ elastic cross section is
interpolated to the required energy and scattering angle. Compared to the
previous analysis [47] in which only the $p$-$d$ scattering data at $22$,
$35$, and $46$ MeV [56] are used, the wide coverage of the $p$-$d$ scattering
energies improves the reliability of the elementary cross section of the
($p,pd$) reaction at $E_{pd}>46~{}\mathrm{MeV}$.

The proton distorted waves, $\chi_{0}$ and $\chi_{1}$, are obtained as
scattering waves under the phenomenological optical potential by Koning and
Delaroche [65]. For the deuteron distorted wave $\chi_{2}$, the optical
potential proposed by An and Cai [66] is adopted. The nonlocality correction
to the distorted waves is taken into account by the Perey factor [67] with the
range parameter $\beta=0.85$ fm for the proton and $\beta=0.54$ fm for the
deuteron [68].

$\varphi_{d}$ is obtained as a bound state wave function under the Woods-Saxon
shaped potential with the range parameter $R_{0}=1.41\times 14^{1/3}$ fm and
the diffuseness parameter $a_{0}=0.65$ fm [47]. The depth parameter of the
potential is adjusted to reproduce the effective deuteron separation energy of
$24.7$ MeV, which is the sum of the deuteron separation energy of 16O ($20.74$
MeV) and the excitation energy of the 14N residue ($3.95$ MeV). As discussed
in Ref. [47], the 16O($p$,$pd$)14N${}_{1^{+}}$ data show a typical $s$-wave
peak at the recoilless condition, in which the reaction residue is at rest in
the final state. Following Ref. [47], we only consider $l=0$ for the
$d$-14N${}_{1^{+}}$ relative angular momentum.

###  III.2 Refraction effect in ${}^{16}\mathrm{O}(p,pd)^{14}\mathrm{N}$

The TDXs of the 16O($p$,$pd$)14N${}_{1^{+}}$ at $101.3~{}\mathrm{MeV}$
calculated with LSCA and AMA, and the experimental data [47] are shown in Fig.
3 as a function of the proton emission energy $T_{1}^{\mathrm{L}}$.

Figure 3: TDXs of the 16O($p$,$pd$)14N${}_{1^{+}}$ reaction calculated with
LSCA and AMA in comparison with the experimental data [47]. Both are
multiplied by $S_{d}=1.74$.

The proton and deuteron emission angles are
$\theta_{1}^{\mathrm{L}}=40.1^{\circ}$ and
$\theta_{2}^{\mathrm{L}}=40.0^{\circ}$, respectively, and the kinematics are
in coplanar. The TDX with LSCA clearly shows better agreement with the data
than that with AMA in their shape, showing that the refraction effect is
important. The experimental spectroscopic factor $S_{d}=1.74$ is obtained by
fitting the theoretical curve of LSCA to the data. The TDX using AMA is also
multiplied by $S_{d}=1.74$ for comparison. The obtained $S_{d}$ is consistent
with the 1$p$ shell model prediction at early ages [69] $S_{d}=1.75$, which is
referred in the original experimental analysis [47], and is comparable with
1.43 determined in the ZR-DWIA analysis [51].

As discussed in Sec. II, LSCA should partially incorporate the finite-range
nature of the $p$-$d$ effective interaction into the $p$-$d$ cross section
through the $\bm{R}$ dependence of $\bm{\kappa}(\bm{R})$ and
$\bm{\kappa}^{\prime}(\bm{R})$. As seen in the difference between the LSCA and
AMA results in Fig. 3, the shape of the distribution strongly depend on how
$\theta_{pd}$ and $E_{pd}$ are evaluated. The present analysis shows that LSCA
is valid for reproducing both the shape and the height of the experimental
data and the cross section with a reasonable $S_{d}$ as long as
$d\sigma_{pd}/d\Omega_{pd}$ is prepared precisely as in Fig. 2.

The origin of the inconsistency between the present AMA and the ZR-DWIA in
Ref. [51] is not clear. In the present AMA calculation, the shape of the TDX
cannot be reproduced as shown in Fig. 3, while the ZR-DWIA calculation gives
good agreement with the data as shown in Fig. 2 of Ref. [51]. Although the
detail of the ZR-DWIA calculation is not described in Ref. [51], the
inconsistency may arise from the different treatment of the $p$-$d$ elementary
process. In the present study we use the fitted function of the experimental
$p$-$d$ differential cross sections as shown in Fig. 2, while in Ref. [51]
they constructed $t_{pd}(\bm{s})$ (denoted as $t_{L}(E,r)$ in the reference)
based on the $p$-$d$ optical potential [70, 71, 56]. Also the difference
between the present LSCA result and the FR-DWIA in Ref. [51] is not clear. In
two types of the FR-DWIA analysis in Ref. [51], all attractive and repulsive
core, $S_{d}=0.47$ and $0.12$ are obtained, respectively, which are both
smaller than the present value of 1.74, suggesting again that the difference
in the treatment of the $p$-$d$ elementary process.

To investigate the impact of the deuteron refraction at higher energies, we
consider the same reaction at 250 MeV; the results are shown in Fig. 4. The
kinematics conditions are fixed at $\theta_{1}^{\mathrm{L}}=47.1^{\circ}$ and
$\theta_{2}^{\mathrm{L}}=48.8^{\circ}$, which gives the recoilless condition
at around $T_{1}^{\mathrm{L}}=150$ MeV.

Figure 4: Same as Fig. 3 but at 250 MeV.

Since there is no experimental data at higher energies, only the theoretical
curves are shown assuming $S_{d}=1$ in Fig. 4. It is clearly shown that the
result using LSCA is almost equivalent to that using AMA at 250 MeV, in
contrast to the 101.3 MeV case in Fig. 3. This result is consistent with the
previous work on the 120Sn($p$,$p\alpha$)116Cd reaction at 392 MeV [25].

##  IV Summary

The refraction effect on the elementary process of the $(p,pd)$ reaction has
been investigated by using LSCA in the DWIA framework. It is found that the
effect gives a significant change in the shape of the energy sharing
distribution of the 16O($p$,$pd$)14N reaction at 101.3 MeV. The present
analysis using LSCA gives $S_{d}=1.74$, which is consistent with 1$p$ shell
model prediction 1.75 [69] referred in Ref. [47], and comparable with 1.43 by
the previous ZR-DWIA analysis [51]. On the other hand, at 250 MeV, the results
with LSCA and AMA are similar, showing that the local $p$-$d$ kinematics has
little effect on the TDX and the commonly used factorized form of the DWIA
with AMA can be safely applied to the reaction analysis at this energy.

As discussed in Refs. [53, 41], the description of the spatial distribution
and the radial magnitude of $\varphi_{d}(\bm{R})$ based on microscopic
structure theories is still challenging but important for the quantitative
discussion of the deuteron-like $p$-$n$ pair in nuclei. A combination of the
present work and CDCCIA [41], which takes into account the breakup and the
reformation process of the $p$-$n$ pair, will be important for the accurate
description of the $(p,pd)$ reaction. The development of such framework as
well as a numerical code is ongoing.

###### Acknowledgements.

We thank T. Uesaka and S. Ogawa for fruitful discussions. This work is
supported in part by Grants-in-Aid for Scientific Research from the JSPS
(Grants No. JP20K14475, JP21H00125, and JP21H04975).

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