# L2CS-Net: Fine-Grained Gaze Estimation in Unconstrained Environments

###### Abstract

Human gaze is a crucial cue used in various applications such as human-robot
interaction and virtual reality. Recently, convolution neural network (CNN)
approaches have made notable progress in predicting gaze direction. However,
estimating gaze in-the-wild is still a challenging problem due to the
uniqueness of eye appearance, lightning conditions, and the diversity of head
pose and gaze directions. In this paper, we propose a robust CNN-based model
for predicting gaze in unconstrained settings. We propose to regress each gaze
angle separately to improve the per-angel prediction accuracy, which will
enhance the overall gaze performance. In addition, we use two identical
losses, one for each angle, to improve network learning and increase its
generalization. We evaluate our model with two popular datasets collected with
unconstrained settings. Our proposed model achieves state-of-the-art accuracy
of 3.92°and 10.41°on MPIIGaze and Gaze360 datasets, respectively. We make our
code open source at https://github.com/Ahmednull/L2CS-Net.

Index Terms—  Appearance-based gaze estimation, Gaze Analysis, Gaze Tracking,
Convolutional Neural Network.

##  1 Introduction

Eye gaze is one of the essential cues used in a large variety of applications.
It indicates the user’s level of engagement in human-robot interaction [1, 2],
and open dialogue systems [3]. Furthermore, it is used in augmented reality
[4] to predict the users’ attention, which improves the device’s awareness and
reduces power consumption. Therefore, researchers developed multiple methods
and techniques for accurately estimating the human gaze. These methods are
divided into two categories: model-based and appearance-based approaches.
Model-based methods generally require dedicated hardware that makes them
difficult to use in an unconstrained environment. On the other hand,
appearance-based methods regress the human gaze directly from the images
captured by inexpensive off-the-shelf cameras, making them easy to generate in
different locations with unconstrained settings.

Recently, CNN-based appearance-based methods are the most commonly used gaze
estimation methods as they provide better gaze performance [5, 6, 7, 8]. Most
of the related work [9, 10, 5, 7, 6] focussed on developing novel CNN-based
networks which mainly consist of popular backbones (e.g. VGG [11], ResNet-18
[9], ResNet-50 [12]) to extract gaze features and finally outputs the gaze
direction. The input to these networks can be a single stream [12, 9](e.g.,
face or eye images) or multiple streams [5] (e.g., face and eye images). The
most common loss function used for the gaze estimation task is the mean square
loss or $\ell_{2}$ loss. However, Petr et al. [9] proposed a novel pinball
loss that estimates the gaze direction and error bounds together, which
improves the accuracy of gaze estimation, especially in unconstrained
settings. Although CNN-based methods achieve improved gaze accuracy, they lack
robustness and generalization, especially in unconstrained environments.

In this paper, we introduce a new method to estimate 3D gaze angles from RGB
images using a multi-loss approach. We propose to regress each gaze angle
(yaw, pitch) independently using two fully-connected layers to enhance the
prediction accuracy of each angle. Furthermore, we use two separate loss
functions for each gaze angle. Each loss consists of gaze binary
classification and regression components. Finally, the two losses are
backpropagated through the network, which accurately fine-tunes the network
weights and increases network generalization. We perform gaze bin
classification by utilizing a softmax layer along with cross-entropy loss so
that the network estimates the neighborhood of the gaze angle in a robust
manner. Based on the proposed loss function and the softmax layer ($\ell_{2}$
loss+ cross-entropy loss+ softmax layer), we present a new network (L2CS-Net)
to predict 3D gaze vector in unconstrained settings. Finally, we evaluate the
robustness of our network on two popular datasets, MPIIGaze [10] and Gaze360
[9]. The proposed L2CS-Net achieved state-of-the-art performance on MPIIGaze
and Gaze360 datasets.

##  2 Related Work

According to the literature, appearance-based gaze estimation can be divided
into conventional and CNN-based methods. Conventional gaze estimation methods
use a regression function to create a person-specific mapping function to the
human gaze, e.g., adaptive linear regression [13] and gaussian process
regression [14]. These methods show reasonable accuracy in constrained setup
(e.g., subject-specific and fixed head pose and illumination), however they
significantly decrease when tested on unconstrained settings.

Recently, researchers have gained more interest in CNN-based gaze estimation
methods, as they can model a highly nonlinear mapping function between images
and gaze. Zhang et al. [10] first proposed a simple VGG CNN-based architecture
to predict gaze using a single eye image. Also, they designed a spatial
weights CNN in [15] to give more weight to those regions of the face that
related to the gaze in appearance. Krafka et al. [16] proposed a multichannel
network that takes eye images, full-face images, and face grid information as
inputs.

Combining statistical models with deep learning is a good solution for gaze
estimation as in [17], which they introduced a mixed effect model that
integrates information from statistics within CNN architecture based on eye
images. Chen et al. [7] adopted dilated convolutions to make use of the high-
level features extracted from images without decreasing spatial resolution. In
addition, they expanded their work by proposing GEDDNet [18], which utilizes
both gaze decomposition with dilated convolutions and reported better
performance than using dilated convolutions only. Fischer et al. [11] append
the head pose vector along with features extracted using a VGG CNN with eye
crops to predict gaze angels. Additionally, they used an ensemble scheme to
improve gaze accuracy.

Motivated by the two-eye asymmetry property, Cheng et al. [19] proposed FAR-
Net that estimates 3D gaze angels for both eyes with an asymmetric approach.
They give asymmetric weights to each loss of the two eyes and finally sum
these losses. The proposed model presented a top performance in multiple
public datasets. Wang et al. [20] integrated adversarial learning with the
Bayesian approach in one framework, which demonstrates an increased gaze
generalization performance. Cheng et al. [6] proposed a coarse-to-fine
adaptive network (CA-Net) that first uses face image to predict primary gaze
angels and adapt it with the residual estimated from eye crops. Then, they
proposed a bi-gram model to bridge the primary gaze with the eye residual.
Kellnhofer et al. [9] used a temporal model (LSTM) with a sequence of 7 frames
to predict gaze angels. In addition, they adopt pinball loss to jointly
regress the gaze direction and error bounds together to improve gaze accuracy.

The most recent work with gaze estimation is AGE-Net [5] which they propose
two parallel networks for each eye image, one is used to generate a feature
vector using CNN, and the other is used to generate a weight feature vector
using an attention-based network. The output of the two parallel networks is
multiplied and then refined with the output of VGG CNN of the face images.

Fig. 1: L2CS-Net with combined classification and regression losses.

##  3 METHOD

###  3.1 Proposed loss function

Most CNN-based gaze estimation models predict 3D gaze as the gaze direction
angles (yaw, pitch) in spherical coordinates. Further, they adopt the mean-
squared error ($\ell_{2}$ loss) for penalizing their networks. We propose to
use two identical losses for each gaze angle. Each loss contains a combined
cross-entropy loss and mean-squared error. Instead of directly predicting
continuous gaze angels, we used a softmax layer with cross-entropy to predict
binned gaze classification. Then, we estimate the expectation of the gaze
binned output to fine-grain the predictions. Finally, we add a mean-squared
error to the output to improve the gaze predictions. Using $\ell_{2}$ together
with Softmax can tune the nonlinear softmax layer with immense flexibility.

The cross entropy loss is defined as:

$\displaystyle H(y,p)=-\sum_{i}y_{i}\log p_{i}$

And the mean-squared error is defined as:

$\displaystyle MSE(y,p)=\frac{1}{N}\sum_{0}^{N}\left(y-p\right)^{2}$

Our proposed loss for each gaze angle is a linear combination of the mean-
squared error and cross-entropy losses, which is defined as:

$\displaystyle CLS(y,p)=H(y,p)+\beta\cdot MSE(y,p)$

Where $CLS$ is the combined loss, $p$ is the predicted values, $y$ is the
ground-truth values and $\beta$ is the regression coefficient. We change the
weight of the mean-squared loss during the experiments in Section 4 to obtain
the best gaze performance.

To the best of our knowledge, all related works which estimated gaze using
CNN-based methods do not consider the combined classification and regression
loss in their techniques.

###  3.2 L2CS-Net Architecture

We propose a simple network architecture (L2CS-Net) based upon the proposed
classification and regression losses. It takes face images as input and feeds
them to ResNet-50 as a backbone to extract spatial gaze features from images.
In contrast to most previous work that regresses the two gaze angles together
in one fully-connected layer, we predict each angle separately using two
fully-connected layers. These two fully-connected layers share the same
convolution layers in the backbone. Also, we use two loss functions, one for
each gaze angle (yaw, pitch). Using this approach will improve network
learning, as it has two signals that backpropagate through the network.

For each output from the fully-connected layer, we first use a softmax layer
to convert the network output logits into a probability distribution. Then, a
cross-entropy loss is applied to calculate the bin classification loss between
output probabilities and target bin labels. Next, we calculate the expectation
of the probability distribution to get fine-grained gaze predictions. Finally,
we calculate the mean square error for this prediction and add it to the
classification loss. The detailed architecture of L2CS-Net is shown in Figure
1.

###  3.3 Datasets

With the development of appearance-based gaze estimation methods, large-scale
datasets have been proposed to improve gaze performance. These datasets were
collected with different procedures, varying from laboratory-constrained
settings to unconstrained indoor and outdoor environments. In order to get a
valuable evaluation of our network, we train and evaluate our model using two
popular datasets collected with unconstrained settings: Gaze360 and MPIIGaze.

Gaze360 [9] provides the widest range of 3D gaze annotations with a range of
360 degrees. It contains 238 subjects of different ages, genders, and
ethnicity. Its images are captured using a Ladybug multi-camera system in
different indoor and outdoor environmental settings like lighting conditions
and backgrounds.

MPIIGaze [15] provides 213.659 images from 15 subjects captured during their
daily routine over several months. Consequently, it contains images with
diverse backgrounds, time, and lighting that make it suitable for
unconstrained gaze estimation. It was collected using software that asks the
participants to look at randomly moving dots on their laptops.

##  4 Experiments

###  4.1 Data preprocessing

We follow the same procedures in [15] to normalize images in the two datasets.
In summary, this process applies rotation and translation to the virtual
camera to remove the head’s roll angle and keep the same distance between the
virtual camera and a reference point (the center of the face). Furthermore, we
split up the continuous gaze target in each dataset (pitch and yaw angles)
into bins with binary labels for classification based on the range of the gaze
annotations. As a result, both datasets have two different target annotations:
continuous and binned labels make them suitable for our combined regression
and classification losses. Furthermore, we change the regression coefficient
in the combined loss function during the experiments to obtain the best gaze
performance.

###  4.2 Training and results

We use an ImageNet-pretrained ResNet-50 as the backbone network. Our proposed
network (L2CS-Net) was trained in PyTorch framework using Adam optimizer with
a learning rate of 0.00001. We train our proposed network for 50 epochs using
a batch size of 16. We evaluate our proposed network on MPIIGaze and Gaze360
datasets. We change the regression coefficient during the experiments and
compare the output performance with the state-of-the-art gaze estimation
methods. We utilize gaze angular error $(\degree)$ as the evaluation metric
following most gaze estimation methods. Assuming the ground-truth gaze
direction is $\mathrm{g}$ $\epsilon$ $\mathbb{R}^{3}$ and the predicted gaze
vector is $\mathrm{\hat{g}}$ $\epsilon$ $\mathbb{R}^{3}$ , the gaze angular
error $(\degree)$ can be computed as:

$\displaystyle\mathcal{L}_{angular}=\frac{\mathrm{g}\cdot\mathrm{\hat{g}}}{\left\|\mathrm{g}\right\|\left\|\mathrm{\hat{g}}\right\|}$

We adapt leave-one-subject-out cross-validation on MPIIGaze dataset as used in
the related works [15, 6, 5]. In order to obtain the best performance, we
train L2CS-Net on the MPIIGaze dataset with different regression coefficients
($\beta$) of 1 and 2. Table. 1 shows the comparison of mean angular error
between our proposed model and state-of-the-art methods on MPIIGaze dataset.
Our proposed L2CS-Net ($\beta=1$) achieved state-of-the-art gaze performance
with 3.92°mean angular error. Furthermore, we present the gaze accuracy of
each subject of the MPIIGaze dataset and compare it with FARE-Net [19] as they
presented the subject-wise gaze accuracy. Out of 15 subjects, our proposed
method achieves better gaze accuracy for 11 subjects, as shown in Fig 2.

Methods |  | MPIIFaceGaze
---
iTracker (AlexNet) [16] | 5.6°
MeNets [17] | 4.9°
FullFace (Spatial weights CNN)[15] | 4.8°
Dilated-Net [7] | 4.8°
RT-Gene(1 model) [11] | 4.8°
GEDDNet [18] | 4.5°
RT-Gene(4 ensemble) [11] | 4.3°
Bayesian Approach [20] | 4.3°
FAR-Net [19] | 4.3°
CA-Net [6] | 4.1°
AGE-Net [5] | 4.09°
|  L2CS-Net $(\beta=1)$
---
L2CS-Net $(\beta=2)$
| 3.96°
---
3.92°

Table 1: Comparison of mean angular error between our proposed model and SOTA
methods on MPIIGaze dataset

Methods | Front 180° | Front Facing
---|---|---
FullFace | 14.99° | N/A
Dilated-Net | 13.73° | N/A
RT-Gene (4 ensemble) | 12.26° | N/A
CA-Net | 12.26° | N/A
Gaze360 (LSTM) [9] | 11.4° | 11.1°
|  L2CS-Net $(\beta=2)$
---
L2CS-Net $(\beta=1)$
| 10.54°
---
10.41°
| 9.13°
---
9.02°

Table 2: Comparison of mean angular error between our proposed model and SOTA
methods on Gaze360 dataset

Kellnhofer et al. [9] divided the Gaze360 dataset into train-val-test sets and
presented three evaluation scopes based on the range of gaze angles: all 360°,
front 180°, and front-facing (within 20°). We follow the same evaluation
criteria in [9], but only with the front 180°and front-facing for a fair
comparison with all related methods that are trained and evaluated on datasets
within 180°range. We trained L2CS-Net on Gaze360 dataset with different
regression coefficients of 1 and 2. Table. 2 shows the comparison of mean
angular error between our proposed model and State-of-the-art methods on
Gaze360 dataset. We used the results from [21] as they implement typical gaze
estimation methods on the Gaze360 dataset. Our proposed L2CS-Net ($\beta=1$)
achieves state-of-the-art gaze performance with 10.41°mean angular error on
front 180°and 9.02°on front facing.

p00p01p02p03p04p05p06p07p08p09p10p11p12p13p14Avg$2$$3$$4$$5$$6$$7$Mean Angular
Error (degrees)L2CS-NetFARE-Net Fig. 2: Comparison of subject gaze accuracy
between our proposed model and FARE-Net [19] on MPIIGaze dataset.

##  5 Conclusion

In this paper, we present a robust CNN-based model (L2CS-Net) for predicting
3D gaze directions in unconstrained environments. We propose to predict each
gaze angle individually with two fully-connected layers and a ResNet-50
backbone. In order to improve the network learning, we used two separate loss
functions for each gaze angle, each of them is a linear combination of
regression and classification losses. Further, we use a reliable softmax layer
to predict gaze bins. Furthermore, we changed the regression coefficient
during the experiments to obtain the best gaze performance. To show the
robustness of our model, we validate our network using two of the most
unconstrained gaze datasets: MPIIGaze and Gaze360, and we followed the same
evaluation criteria used in each dataset. Our model achieved state-of-the-art
gaze accuracy with the lowest angular error in both datasets.

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# BiFold: A Python code for the calculation of double folded (bifold)
potentials with density-in/dependent nucleon-nucleon interactions

Mesut Karakoç Department of Physics, Faculty of Science,
Akdeniz University, TR 07070, Antalya, Turkey

###### Abstract

BiFold calculates the density-dependent (DDM3Y$n$, BDM3Y$n$, CDM3Y$n$) or
independent double folded potentials between two colliding spherical nuclei.
It is written in a Python package form to give the ability to use the
potentials directly in a nuclear reaction/structure code. In addition to using
Woods-Saxon/Fermi or Gaussian functions, the code also allows for the
definition of nuclear matter densities using pre-calculated densities in a
data file. The manuscript provides an overview of the double folding model and
the use of the code.

###### keywords:

Nuclear interaction; Double folded potentials; Density-dependent NN
interactions; M3Y-interaction.

††journal: Computer Physics Communications

PROGRAM SUMMARY

Program Title: BiFold
CPC Library link to program files: (to be added by Technical Editor)
Developer’s repository link: https://github.com/mkarakoc/BiFold
Code Ocean capsule: (to be added by Technical Editor)
Licensing provisions: GPLv3
Programming language: Python 3.x
Nature of problem: BiFold calculates the real part of the nuclear potential
between two colliding spherical nuclei by integrating a density-
independent/dependent nucleon-nucleon (NN) interaction [1, 2, 3] over the
nuclear matter densities of the two nuclei. The code based on M3Y Reid/Paris
NN interactions [1, 2, 3] by default, but it is possible to define custom NN
interactions when necessary.
Solution method: The code uses the Fourier transform method in spherical
coordinates to calculate the potential. The method simplifies the sixfold
integration [1] and makes the calculation a lot faster. The integration is
done by default using Simpson’s integration method, but Filon’s integration
method is also available.

## References

  * [1] G.R. Satchler and W.G. Love, Phys. Rep. 55 (1979) 183.
  * [2] A. Kobos, B. Brown, P. Hodgson, G. Satchler abd A. Budzanowski, NPA 384 (1) (1982) 65–87.
  * [3] D. T. Khoa, W. von Oertzen, H. G. Bohlen, S. Ohkubo,J. Phys. G: Nucl. Part. Phys 34 (3) (2007) R111–R164.

##  1 Introduction

The folded potential model [1] is a well-known model for describing the mean-
field nuclear interaction between two colliding nuclei. It has been widely
used in the literature [1, 2, 3, 4] since accepted as a more realistic
approach than a phenomenological potential such as the well-known Woods-Saxon.
The latter usually needs three free parameters to fit data, and this may
create possible known ambiguities [5]. On the contrary, a folded potential
usually has only one free parameter called the re-normalization factor to
compensate for higher-order effects in a nuclear reaction. Folded potentials
have two categories single folded (SF) potentials and double folded (DF)
potentials.

The SF potential describes the interaction between two nuclei using one of the
two nuclei’s densities and a phenomenological nucleon-nucleus interaction
potential. The shortcomings of this treatment are density dependence, surface
features, or couplings of a nuclear system are not well defined, as pointed
out in the report [1] of Satchler and Love. These shortcomings make the depth
of the SF potentials unrealistically deep [6, 2] to explain the rainbow
scattering.

The usual DF potential between two spherical nuclei is constructed by
integrating over an effective nucleon-nucleon (NN) interaction with nuclear
matter densities representing nucleons of both nuclei. Although, DF potentials
would overcome the shortcomings of SF potentials. Some cases for DF potentials
can still overestimate the depth of the nuclear potential. Additionally, it is
necessity to add the antisymmetrization-exchange effects and the density-
dependent saturation effects in the effective NN interaction [2] for a more
realistic nuclear potential.

The main reason for density dependence (DD) at NN interaction is the Pauli
principle effects in the nuclear medium of both colliding nuclei. The DD of NN
interaction has several treatments in the literature, but the code BiFold is
built on the treatments of Satchler and Love [1], Kobos et al. [2], and Khoa
et al. [3]. All these three treatments have the frozen density approximation
while the case of Kobos et al. [2] has energy dependence, but the other two
have no energy dependence.

Many studies _(see the references in the present work)_ have used these DD
treatments of NN interactions, but there are very few published codes [7, 8]
to be able to reproduce the results of these works. In addition, the codes
have limitations, and not all are updated regularly. For example, DFPOT [7]
cannot calculate the potentials with the DDM3Y, BDM3Y$n$ ($n=1,2,3$), and
CDM3Y$n$ ($n=1\ldots 6$) type density-dependent interactions [3]. While DFMSPH
[8] can calculate many of those, it does not support BDM3Y2, BDM3Y3 [9], and
the first version of DDM3Y [2]. And it is a well-known fact results of a study
must be reproducible in science. Therefore, the code BiFold will help the
community in these manners.

##  2 The model

###  2.1 The usual double folded potential

It can be claimed that Coulomb potential between two spherical charges is the
inspiration for the double folded potentials. And it is formulated as [1, 4]

$U_{DF}(\vec{R})=\int\int
d\vec{r}_{p}d\vec{r}_{t}~{}\rho_{p}(\vec{r}_{p})\rho_{t}(\vec{r}_{t})v({\vec{s}}),$
(1)

where $\rho_{p}$ and $\rho_{t}$ are charge densities, and $v(\vec{s})=1/s$
interaction between charges in Coulomb potential case, while $\rho_{p}$ and
$\rho_{t}$ are nuclear matter densities, and $v(\vec{s})$ is the effective NN
interaction between point-like nucleons in nuclear potential case. Its
schematic representation and definitions of the vectors are given in Fig. 1.

Figure 1: It is the schematic representation of DF potential in Eq. (1) where
$\vec{R}$ is the vector between the centers of the projectile (p) and the
target (t) nuclei. $\vec{r}_{p}$ and $\vec{r}_{t}$ are the locations of the
interacting parts of the nucleon distributions of both nuclei. And
$\vec{s}=\vec{R}+\vec{r}_{t}-\vec{r}_{p}$ is the separation between them.

The effective NN interaction $v(\vec{s})$ is density-independent in this usual
definition of the DF potentials. The medium effects are included in the
calculations by changing $v(\vec{s})$ with
$v(\rho,\vec{s})=F(\rho)v(\vec{s})$, where $F(\rho)$ describes the density
dependence of the NN interaction.

###  2.2 Effective density-independent NN interactions

Although BiFold can use a wide range of density-independent NN interactions in
the DF potential calculations, M3Y type interactions [10, 9, 11] are defined
by default in the code since they are perhaps the most widely used ones.

These interactions are called M3Y-Reid [12] and M3Y-Paris [13] effective
interactions. The former is derived from the solution of the Bethe-Goldstone
equation with Reid [12] soft-core interaction on a harmonic oscillator basis
to obtain G-matrix. The latter is derived from a more fundamental Paris NN
potential [13] to generate all components of the effective interaction [14].

Both versions of the NN interactions have direct ($v_{d}$) and exchange
($v_{ex}$) parts,

M3Y-Reid: $\displaystyle\quad v_{d}(\vec{s})$
$\displaystyle=\left[7999\frac{\mathrm{e}^{-4s}}{4s}-2134\frac{\mathrm{e}^{-2.5s}}{2.5s}\right]~{}\mathrm{MeV},$
(2) $\displaystyle\text{ZR:}\quad v_{ex}(\vec{s})$
$\displaystyle=J_{ex}(E)\delta(\vec{s})\quad\text{where}\quad
J_{ex}(E)\approx-276[1-0.005\varepsilon]~{}\mathrm{MeV}\mathrm{fm^{3}},$ (3)
$\displaystyle\text{FR:}\quad v_{ex}(\vec{s})$
$\displaystyle=\left[4631\frac{\mathrm{e}^{-4s}}{4s}-1787\frac{\mathrm{e}^{-2.5s}}{2.5s}-7.847\frac{\mathrm{e}^{-0.7072s}}{0.7072s}\right]~{}\mathrm{MeV},$
(4) M3Y-Paris: $\displaystyle\quad v_{d}(\vec{s})$
$\displaystyle=\left[11062\frac{\mathrm{e}^{-4s}}{4s}-2538\frac{\mathrm{e}^{-2.5s}}{2.5s}\right]~{}\mathrm{MeV},$
(5) $\displaystyle\text{ZR:}\quad v_{ex}(\vec{s})$
$\displaystyle=J_{ex}(E)\delta(\vec{s})\quad\text{where}\quad
J_{ex}(E)\approx-590[1-0.002\varepsilon]~{}\mathrm{MeV}\mathrm{fm}^{3},$ (6)
$\displaystyle\text{FR:}\quad v_{ex}(\vec{s})$
$\displaystyle=\left[-1524\frac{\mathrm{e}^{-4s}}{4s}-518.8\frac{\mathrm{e}^{-2.5s}}{2.5s}-7.847\frac{\mathrm{e}^{-0.7072s}}{0.7072s}\right]~{}\mathrm{MeV},$
(7)

while the direct one defines the usual nuclear interaction between nucleons.
The exchange part describes the interchange (knock-on exchange) of the
nucleons of the colliding nuclei [9, 11]. The DF potential will have two parts
since the effective NN interactions have two parts; then Eq. (1) will have a
new form

$\displaystyle U_{DF}(\vec{R})$
$\displaystyle=U_{D}(\vec{R})+U_{EX}(\vec{R})=$ $\displaystyle\int\int
d\vec{r}_{p}d\vec{r}_{t}~{}\rho_{p}(\vec{r}_{p})\rho_{t}(\vec{r}_{t})v_{d}({\vec{s}})+\int\int
d\vec{r}_{p}d\vec{r}_{t}~{}\rho_{p}(\vec{r}_{p})\rho_{t}(\vec{r}_{t})v_{ex}({\vec{s}}),$
(8)

where $U_{D}$ and $U_{EX}$ direct and exchange folding potentials,
respectively.

There are two approaches for the exchange parts of the interactions; these are
zero-range (ZR) [15, 16] or finite-range (FR) [17] knock-on exchange
interactions. The ZR approaches given in Eqs. (3) and (6), where
$\varepsilon=E/a_{p}$ is the projectile’s incident energy per nucleon in the
laboratory frame, are widely used in the literature due to their simplicity in
calculations. One needs only to put the ZR interactions in 3 or Eq. (6) to the
exchange part of Eq. (8) to obtain the exchange part of the DF potential
($U_{EX}$). The depths of the ZR interactions are defined by $J_{ex}(E)$.
Determination of $J_{ex}(E)$ is empirically done; the detailed information can
be found in Refs. [15, 16, 11].

As is pointed out by Khoa [17], the exchange interaction, in general, must be
nonlocal [18]. Thus, the exact numerical calculation of exchange interaction
can become too complicated. A plain wave [19] approximation for the relative
motion of nucleons can overcome this complication and lead to an equivalent
local potential. The plain wave given in Ref. [3] is

$\chi(\vec{R}+\vec{s})\approx\exp\left(\frac{i\vec{K}(\vec{R})\cdot\vec{s}}{M}\right)\chi(\vec{R}),$
(9)

where $M=a_{p}a_{t}/(a_{p}+a_{t})$ is the recoil factor (or reduced mass),
while $a_{p}$ and $a_{t}$ are the mass numbers of the projectile and target
nuclei, respectively. And $\vec{K}(\vec{R})$ is the local momentum of the
relative motion given by [18]

$K^{2}(\vec{R})=\frac{2mM}{\hbar^{2}}\left[E_{\text{c.m.
}}-U_{D}(\vec{R})-U_{EX}(\vec{R})-U_{C}(\vec{R})\right],$ (10)

where $E_{\text{c.m. }}$ is relative energy in the center-of-mass system, $m$
is the nucleon mass, and $U_{C}$ is the Coulomb potential. Then, local
exchange potential will take the form of [17, 18, 20, 21, 22, 23]:

$\displaystyle U_{EX}(\vec{R})=\int\int
d\vec{r}_{p}d\vec{r}_{t}~{}\rho_{p}(\vec{r}_{p},\vec{r}_{p}+\vec{s})\rho_{t}(\vec{r}_{t},\vec{r}_{t}-\vec{s})v_{ex}({\vec{s}})\exp\left(\frac{i\vec{K}(\vec{R})\cdot\vec{s}}{M}\right),$
(11)

where $\rho_{p}(\vec{r}_{p},\vec{r}_{p}+\vec{s})$ and
$\rho_{t}(\vec{r}_{t},\vec{r}_{t}-\vec{s})$ are one-body density matrices [22,
23, 24] of the projectile and target nucleons. This local potential becomes an
FR exchange potential when $v_{ex}(\vec{s})$ is chosen as one of the FR
interactions in Eqs. (4) or (7) (M3Y-Reid/Paris-FR). One should realize that
the exchange part of Eq. (8) needs to be replaced by Eq. (11) for the FR
exchange potential.

The next step in the exchange potential (Eq. (8)) is the calculation of
density matrices. Although the matrices can be obtained from single-particle
wave functions [18], Khoa [17] has chosen a realistic local approximation from
Ref. [25]:

$\rho_{p,t}(\vec{r},\vec{r}\pm\vec{s})\approx\rho_{p,t}\left(\vec{r}\pm\frac{\vec{s}}{2}\right)~{}\hat{j}_{1}\left(k_{\text{F}_{p,t}}(\vec{r}\pm\frac{\vec{s}}{2})s\right),$
(12)

where $\hat{j_{1}}(x)=3(\sin x-x\cos x)/x^{3}$. $k_{F}$ is the average local
Fermi momentum from Refs. [25, 26, 27, 28]:

$k_{F}(\vec{r})=\left\\{\left[\frac{3}{2}\pi^{2}\rho(\vec{r})\right]^{2/3}+C_{S}\frac{5}{3}\frac{[\nabla\rho(\vec{r})]^{2}}{\rho^{2}(\vec{r})}+\frac{5}{36}\frac{\nabla^{2}\rho(\vec{r})}{\rho(\vec{r})}\right\\}^{1/2},$
(13)

where rho is the nuclear matter densities of the projectile or the target and
$C_{S}$ is the strength of the Weizsäcker term, representing the surface
contribution to the kinetic energy density[27]. The strength term is usually
$C_{S}\approx\frac{1}{36}$ in the literature, but Khoa _et al._ [27] have
taken it as $C_{S}\approx\frac{1}{4}$ for the given reasons in their work. The
default value is $C_{S}\approx\frac{1}{36}$ in BiFold, but the user has the
option to change the value.

After this point, the FR exchange potential will have the following form [17]:

$U_{EX}(\vec{R})=4\pi\int_{0}^{\infty}v_{EX}(s)s^{2}ds\int
f_{p}(\vec{r},\vec{s})f_{t}(\vec{r}-\vec{R},\vec{s})j_{0}(K(\vec{R})s/M)d\vec{r}$
(14)

where
$f_{p,t}(\vec{r},\vec{s})=\rho_{p,t}(\vec{r})~{}\hat{j}_{1}(k_{\text{F}_{p,t}}(\vec{r})s)$
and $j_{0}(x)=\sin x/x$. Now, the exchange potential with Fourier transforms
in spherical coordinates will take the form [1, 17]:

$U_{EX}(R)=4\pi\int_{0}^{\infty}G(R,s)j_{0}(K(R)s/M)v_{EX}(s)s^{2}ds,$ (15)

where

$G(R,s)=\frac{1}{2\pi^{2}}\int_{0}^{\infty}f_{p}(q,s)f_{t}(q,s)j_{0}(qR)q^{2}dq,$
(16)

$f_{p,t}(q,s)=4\pi\int_{0}^{\infty}f_{p,t}(r,s)j_{0}(qr)r^{2}dr.$ (17)

Finally, it is necessary to solve a self-consistency problem to obtain the
exchange part (Eq. (11)) of the double folded potential (Eq. (8)) at each
radial point. Since the exchange potential (Eq. (11)) contains the local
momentum of the relative motion (Eq. (10)) and $\vec{K}(\vec{R})$ depends on
the total double folded potential, this problem can be solved exactly by an
iterative method given in Refs. [27, 29, 22, 23].

###  2.3 Effective density-dependent NN interactions

The effective density-dependent NN interaction is proposed [2, 27, 28, 30] in
the following form for both direct ($v_{d}$) and exchange ($v_{ex}$) parts;

$v_{d,ex}(\rho,E,\vec{s})=g(E)F_{d,ex}(\rho)v_{d,ex}(\vec{s})$ (18)

where $\rho$ is the overlapping density of the nuclear medium of both nuclei.
$g(E)$ is the weak intrinsic energy dependence proposed by Khoa et al. [31].
The density-dependent folding potential with an FR exchange part can be
calculated by replacing this new form with $v_{d}(\vec{s})$ in Eq. (8) and
with $v_{ex}(\vec{s})$ in Eq. (11). In the case of a ZR exchange part [32], it
can be calculated by using the new form for both $v_{d}(\vec{s})$ and
$v_{ex}(\vec{s})$ in Eq. (8).

The overlapping density for the direct part and the ZR exchange part of the
folded potential has been approximated in most of the folding potential
calculations [2, 33, 34, 27, 9, 35, 29, 32] as

$\rho=\rho_{p}(\vec{r}_{p})+\rho_{t}(\vec{r}_{t}),$ (19)

since it permits the separation of variables in the integrals of Eqs. (8) and
(11). The overlapping density for the FR exchange part of the folded potential
has been assumed [2, 33, 34, 27, 9, 35, 29, 32] as

$\rho=\rho_{p}(\vec{r}_{p}+\frac{\vec{s}}{2})+\rho_{t}(\vec{r}_{t}-\frac{\vec{s}}{2}).$
(20)

Table 1: BiFold can calculate double folding potentials for the interactions
marked with “$\checkmark$” where ZR and FR stand for zero-range and finite-
range exchange interactions, respectively. The $\times$ stands for not
supported or non existing interactions. The nuclear incompressibility values
($K[\mathrm{MeV}]$) are only exist for finite range versions of the
interactions [35, 29].

Interaction names ZR FR $C$ $\alpha$ $\beta~{}[\mathrm{fm}^{3}]$
$\gamma[\mathrm{fm}^{3n}]$ n $K[\mathrm{MeV}]$ Refs. DDM3Y Reid $\checkmark$
$\times$ $C(E)$ $\alpha(E)$ $\beta(E)$ $0.0$ 0 [2, 34] DDM3Y1 Reid
$\checkmark$ $\checkmark$ 0.2845 3.6391 2.9605 $0.0$ 0 171 [31, 27, 9, 3]
DDM3Y1 Paris $\checkmark$ $\checkmark$ 0.2963 3.7231 3.7384 $0.0$ 0 176 [9,
35, 3] BDM3Y0 Reid $\times$ $\times$ 1.3827 0.0 0.0 $1.1135$ 2/3 232 [31]
BDM3Y1 Reid $\checkmark$ $\checkmark$ 1.2253 0.0 0.0 $1.5124$ 1 232 BDM3Y2
$\checkmark$ $\checkmark$ 1.0678 0.0 0.0 $5.1069$ 2 354 [31, 27, 9, 3] BDM3Y3
$\checkmark$ $\checkmark$ 1.0153 0.0 0.0 $21.073$ 3 475 BDM3Y1 Paris
$\checkmark$ $\checkmark$ 1.2521 0.0 0.0 $1.7452$ 1 270 BDM3Y2 $\checkmark$
$\checkmark$ 1.0664 0.0 0.0 $6.0296$ 2 418 [9, 35, 3] BDM3Y3 $\checkmark$
$\checkmark$ 1.0045 0.0 0.0 $25.115$ 3 566 CDM3Y1 Paris $\checkmark$
$\checkmark$ 0.3429 3.0232 3.5512 $0.5$ 1 188 [29, 3] CDM3Y2 $\checkmark$
$\checkmark$ 0.3346 3.0357 3.0685 $1.0$ 1 204 CDM3Y3 $\checkmark$ $\checkmark$
0.2985 3.4528 2.6388 $1.5$ 1 217 CDM3Y4 $\checkmark$ $\checkmark$ 0.3052
3.2998 2.3180 $2.0$ 1 228 CDM3Y5 $\checkmark$ $\checkmark$ 0.2728 3.7367
1.8294 $3.0$ 1 241 CDM3Y6 $\checkmark$ $\checkmark$ 0.2658 3.8033 1.4099 $4.0$
1 252 M3Y Reid $\checkmark$ $\checkmark$ usual density-independent M3Y [1, 17,
32] M3Y Paris $\checkmark$ $\checkmark$

The code can calculate double folding potentials for the interactions marked
with a “$\checkmark$” in Table 1. The density dependence of these interactions
is defined by $F(\rho)$ in Eq. (18). Different versions of 18 are proposed in
the Refs. [2, 31, 9, 29]. In a more recent study by Khoa et al. [3], these
different versions of $F(\rho)$ merged into one formula;

$F(\rho)=C[1+\alpha\exp(-\beta\rho)-\gamma\rho^{n}].$ (21)

The parameters of this formula are given in Table 1 other than the original
DDM3Y [2] since its parameters are energy-dependent. The values of these
energy-dependent parameters can be obtained from the Refs. [2, 34].

The final part of the density-dependent NN interaction (Eq. (18)) is $g(E)$.
It is $g(E)=1$ for the original DDM3Y [2] since $g(E)$ does not exist for this
interaction. For the remaining density-dependent interactions in Table 1,
$g(E)\approx 1-\kappa\varepsilon$, where $\kappa=0.002$ and $\kappa=0.003$ for
the M3Y-Reid and M3Y-Paris types of interactions, respectively, and
$\varepsilon=E/a_{p}$ is energy (in MeV) per nucleon.

##  3 The code

The file structure of BiFold code given is in Fig. 2 on the left. The bold
ones are directories, and the rest are Python files in the given file
structure. The two sub-directories in the bifold directory (simpson and filon)
have the same structures other than the integration methods. _‘bifold.py’_
file calls _‘simpson.py’_ file since the Simpson integration method is the
default integration method of BiFold. Both _‘simpson.py’_ and _‘filon.py’_
have the functions with the names starts-with _‘u_’_ since the symbols of the
double folding potentials are $U_{DF}$ or $U_{D}$ or $U_{EX}$ in Eq. (8).
Therefore, _‘simpson.py’_ (_‘filon.py’_) is the main part of the code where
the calculation of the double folding integrals takes place through
_‘integrals.py’_. Table 2 shows the names of these functions that correspond
to the calculation of double folding potential types. Table 2 also lists the
Python functions of the _‘interactions.py’_ file for the computation of the
effective NN interactions in Eqs. (2) through (7). The _‘functions.py’_ file
contains Python functions for nuclear matter densities and effective NN
interactions. Table 3 shows the mathematical representations of the densities
and interactions versus Python representations. The _‘matematik.py’_ contains
the spherical Bessel functions, numerical derivation functions, and other
mathematical tools. The _‘constants.py’_ is for all the necessary physical and
mathematical constants. The other two files (_‘graph_tools.py’_ and
_‘print_tools.py’_) are for drawing and printing the results.

.1 bifold. .2 bifold.py. .2 interactions.py. .2 functions.py. .2 constants.py.
.2 matematik.py. .2 graph_tools.py. .2 print_tools.py. .2 simpson. .3
simpson.py. .3 integrals.py. .2 filon. .3 filon.py. .3 integrals.py.

⬇

1 from bifold import *

2

3 r = mesh(zero, 10, 0.05) # fm

4 q = mesh(zero, 3, 0.05) # fm^-1

5

6 e_lab = 141.7 # MeV

7 a_proj = 4

8

9 rho_p = f_2prm_gaussian(r, 0.4229, (1/0.7024)**.5)

10 rho_t = f_2prm_fermi(r, 0.169, 3.60, 0.523)

11

12 u = u_m3y_reid_zr(e_lab, a_proj, rho_p, rho_t, r, q)

13

14 title = ”a + 40Ca @ Elab = 141.7 MeV using M3Y-Reid/ZR”

15 print_all(u, r, q, title=title)

16 plot_potentials(u, r, part=”all”)

Figure 2: The file tree structure of BiFold code is on the left. The bold ones
are directories, and the rest are Python files. And an example Python file to
calculate a double folding potential for an $\alpha+^{40}$Ca elastic
scattering on the right. Table 2:  The presented names are the Python
functions in BiFold to calculate the effective NN interactions and the
doubling potentials. The $\times$ stands for not supported or non existing
potentials/interactions.

Interaction names _‘simpson.py’_ / _‘filon.py’_ _‘interactions.py’_ $U_{DF}$
with ZR $U_{DF}$ with FR $v_{d}(\vec{s})$ ZR: $v_{ex}(\vec{s})$ FR:
$v_{ex}(\vec{s})$ DDM3Y Reid u_ddm3y_reid_zr $\times$ v_m3y_reid_d
v_m3y_reid_ex_zr $\times$ DDM3Y1 Reid u_xdm3yn_zr u_xdm3yn_fr v_m3y_reid_d
v_m3y_reid_ex_zr v_m3y_reid_ex_fr BDM3Y$n$; $n=1,2,3$ DDM3Y1 Paris u_xdm3yn_zr
u_xdm3yn_fr v_m3y_paris_d v_m3y_paris_ex_zr v_m3y_paris_ex_fr BDM3Y$n$;
$n=1,2,3$ CDM3Y$n$; $n=1\ldots 6$ M3Y Reid u_m3y_reid_zr u_xdm3yn_fr
v_m3y_reid_d v_m3y_reid_ex_zr v_m3y_reid_ex_fr M3Y Paris u_m3y_paris_zr
v_m3y_paris_d v_m3y_paris_ex_zr v_m3y_paris_ex_fr

[b]  Python function Math formula Python function Math formula f_exp_decay(r,
V0, a) $V_{0}~{}\textrm{e}^{-ar}$ f_sog(r, Ris, Qis, RP, Ze)
$\sum\limits_{i}A_{i}\left\\{\textrm{e}^{-\left[(r-R_{i})/\gamma\right]^{2}}+\textrm{e}^{-\left[(r+R_{i})/\gamma\right]^{2}}\right\\}$
[36] f_yukawa(r, V0, a, b) $V_{0}~{}\textrm{e}^{-ar}/br$ f_dirac_delta(r, V0)
$V_{0}~{}\delta(r)$ f_2prm_fermi(r, V0, R, a)
$V_{0}~{}\frac{1}{1+\textrm{e}^{\frac{r-R}{a}}}$ f_external(r, external_data)1
reads from a local file f_3prm_fermi(r, V0, w, R, a)
$V_{0}~{}\frac{1+wr^{2}}{1+\textrm{e}^{\frac{r-R}{a}}}$ f_internet(r, url)1
reads a file from a given url f_2prm_gaussian(r, V0, a)
$V_{0}~{}\textrm{e}^{-(r/a)^{2}}$ f_ripl(r, Z, A)1 HFB14 densities from RIPL
[37, 38, 39, 40] f_3prm_gaussian(r, V0, w, a)
$V_{0}~{}(1+wr^{2})~{}\textrm{e}^{-(r/a)^{2}}$ https://www-nds.iaea.org/RIPL 1
These functions create Python arrays using data from local or remote files.
These arrays contain interpolated values to match the given $r$ array defined
by the $mesh(r_{min},r_{max},dr)$ function.

Table 3:  The presented names are the Python functions in BiFold to calculate
the nuclear matter densities or, if necessary, to write a new type of
interaction or to calculate phenomenological potentials.

An example Python file for calculating a density-independent double potential
using M3Y-Reid NN interaction with the ZR exchange part (see Eqs. (2) and (3))
for an $\alpha+^{40}$Ca elastic scattering is on the right side of Fig. 2. And
the results of the calculation are shown in Figs. 3 and 4. One may agree that
the code is easy to understand. The first line of the Python code imports
BiFold since it is a Python package. On the third and fourth lines, the _mesh_
functions defines the numerical integration grids for $r=zero$ fm to $r=10$ fm
with $0.05$ fm steps and $q=zero$ fm-1 to $r=3$ fm-1 with $0.05$ fm-1 steps
where $zero=1\times 10^{-10}$.The sixth and seventh lines are the laboratory
energy (_e_lab_) and the atomic mass number (_a_proj_) of the projectile
($\alpha~{}-~{}\textrm{particle}$), respectively. The ninth and tenth lines
are the nuclear matter densities of the projectile (_rho_p_) and the target
(_rho_t_) nuclei, respectively. The twelfth line is the double folding
calculation in Eq. (8) for M3Y-Reid NN interaction with the ZR exchange part
(see Eqs. (2) and (3)). The rest of the file is optional if one needs to print
(Fig. 3) the calculation information and draw (Fig. 4) the potentials versus
radial distance between the two nuclei. This code is a simple example of how
to use BiFold to calculate a double-folded potential. The related GitHub
(https://github.com/mkarakoc/BiFold) page has the BiFold code, a more detailed
user guide, and more examples.

═══════════════════════════════╣a + 40Ca @ Elab = 141.7 MeV using
M3Y-Reid/ZR╠═══════════════════════════════density/interaction L norm renorm
vol2 vol4
msr─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────total
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - -u_R : u_m3y_reid_zr 0 None 1.000 -59536.370 -982867.456
16.509───────direct - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - - - -u_R : u_direct 0 None 1.000 -23281.699
-486747.684 20.907rho_p : f_2prm_gaussian 0 None 1.000 4.000 8.543 2.136rho_t
: f_2prm_fermi 0 None 1.000 39.908 461.090 11.554vnn : f_yukawa 0 None 1.000
1570.558 588.970 0.375vnn : f_yukawa 0 None 1.000 -1716.459 -1647.805
0.960───────exchange - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - - - -u_R : u_exchange_zr 0 None 1.000 -36254.670
-496119.772 13.684rho_p : f_2prm_gaussian 0 None 1.000 4.000 8.543 2.136rho_t
: f_2prm_fermi 0 None 1.000 39.908 461.090 11.554vnn : f_dirac_delta 0 None
1.000 -227.114 0.000 0.000 R u_R  R u_R  R u_R  R u_R ─────── ───────────────
─────── ─────────────── ─────── ─────────────── ─────── ─────────────── 0.000
-224.563778558 2.500 -157.142066726 5.000 -33.591208578 7.500 -1.294809340
0.050 -224.538322266 2.550 -154.527402817 5.050 -32.017487683 7.550
-1.195282003 0.100 -224.461928239 2.600 -151.883273803 5.100 -30.494524998
7.600 -1.102914652 0.150 -224.334521330 2.650 -149.211853484 5.150
-29.022094646 7.650 -1.017241399 ... ... ... 2.300 -167.264509196 4.800
-40.395194787 7.300 -1.774804137 9.800 -0.024215666 2.350 -164.788335884 4.850
-38.617900517 7.350 -1.641422081 9.850 -0.022007393 2.400 -162.274567943 4.900
-36.891404224 7.400 -1.517343866 9.900 -0.019983036 2.450 -159.725143549 4.950
-35.215829117 7.450 -1.401989622 9.950 -0.018129460

Figure 3: It is the output of the double folding potential calculation for the
$\alpha+^{40}$Ca elastic scattering. Figure 4: It is the double folding
potential [dash-dot] with direct [solid] and exchange [dash] parts for the
$\alpha+^{40}$Ca elastic scattering.

The output of the BiFold calculation shown in Fig. 3 gives individual
information about the potentials and the functions used in the calculations.
This individual information from left to right contains a Python dictionary
key, name of the function/potential, multi-polarity (L), normalization (norm)
and re-normalization (renorm), volume integrals (vol2 and vol4), and mean
square radii (msr). The output also contains printout of the calculated
potential $U_{R}$ versus radial distance $R$ between two nuclei.

The dictionary keys _func_i_ , _func_r_ and _func_q_ store the information
about the function, numeric values of the function at every point defined by
$mesh(r_{min},r_{max},dr)$, and numeric values of the function’s Fourier
transform at every point defined by $mesh(q_{min},q_{max},dq)$, respectively.
The output does not list these keys. The keys _total_ , _direct_ and
_exchange_ are to reach total, direct and exchange individual parts of the
calculations under the previous keys. And _u_R_ , _rho_p_ , _rho_t_ , and
_vnn_ keys store information about every individual part under the previous
keys. Finally, one can reach the information tabulated in the output by using
_L_ , _norm_ , _renorm_ , _vol2_ , _vol4_ and _msr_ keys. For example, using
_func_i_ , _direct_ and _rho_p_ keys in the given order will produce the
following output about the density of the projectile nucleus in the example
code in Fig. 2:

⬇

1input : u[’func_i’][’direct’][’rho_p’]

2output: [{’name’: ’f_2prm_gaussian’, ’L’: 0, ’norm’: None, ’renorm’: 1.0,

3 ’vol2’: 4.000237160, ’vol4’: 8.542647693, ’msr’: 2.135535308}]

The following formula defines the volume integrals for all radial dependent
functions and potentials,

$vol(n+2)=renorm~{}4\pi\int f(r)r^{n+2}dr.$ (22)

If the value of _norm_ is _None_ , then _renorm = 1_ , otherwise if _norm_ is
a real number, then _renorm_ is,

$renorm=norm\left[4\pi\int f(r)r^{n+2}dr\right]^{-1}.$ (23)

The mean square radii (msr) of all radial dependent functions and potentials
are $\langle r^{2}\rangle~{}=~{}vol4/vol2$.

##  4 Test cases

This section compares BiFold computations to three examples to demonstrate the
code’s reliability. The first one is an analytical calculation, and the second
one is a numerical calculation using DFPOT [7]. And the work of Khoa et al.
[35] is the last one. One can assume that the examples are reasonably accurate
since the first example is an exact solution to the double folding integrals,
the second example is a published code used and tested many times in the
literature, and the final one is a reliable published work. Therefore, these
examples are reference calculations to use in the following formula,

$\xi^{2}=\frac{1}{N}\sum\limits_{i}^{N}\left(\frac{U_{A}(R_{i})-U_{B}(R_{i})}{U_{A}(R_{i})+U_{B}(R_{i})}\right)^{2}.$
(24)

This formula defines a mean relative error (mre) [8] for comparing BiFold
calculations with reference calculations where $U_{A}$ and $U_{B}$ are the
reference and the present double folding potentials with the radial distance
$R_{i}$, respectively. It is better if $\xi^{2}$ is getting closer to zero, as
it will mean that the results of both computations are getting more
consistent. The results of the mre calculations are in Table 4, and the
details of the test cases are in the following sections.

Table 4:  The mean relative errors, defined by Eq. (24), are given for the
three test cases. The cases are compared with BiFold’s calculations using both
Simpson’s and Filon’s integration.

Integration $\xi^{2}$ $\overbrace{\rule{120.0pt}{0.0pt}}^{\textrm{\normalsize
Analytical}}$ DFPOT Khoa et al. [35] method of BiFold
$q_{max}=3~{}\textrm{fm}^{-1}$ $q_{max}=10~{}\textrm{fm}^{-1}$ total
$2.16\times 10^{-3}$ $6.66\times 10^{-8}$ $1.31\times 10^{-7}$ $6.58\times
10^{-5}$ Simpson direct $1.84\times 10^{2}$ $1.19\times 10^{-7}$ $8.10\times
10^{-5}$ $6.65\times 10^{-3}$ exchange $1.22\times 10^{0}$ $5.87\times
10^{-6}$ $1.40\times 10^{-6}$ $2.48\times 10^{-5}$ total $2.16\times 10^{-3}$
$6.64\times 10^{-8}$ $1.31\times 10^{-7}$ $6.57\times 10^{-5}$ Filon direct
$1.84\times 10^{2}$ $1.16\times 10^{-7}$ $8.13\times 10^{-5}$ $6.65\times
10^{-3}$ exchange $1.22\times 10^{0}$ $5.87\times 10^{-6}$ $1.40\times
10^{-6}$ $2.48\times 10^{-5}$

###  4.1 BiFold vs. analytical calculation

In this case, $\alpha+\alpha$ scattering with projectile energy 50 MeV in the
laboratory system is in consideration. Satchler and Love [1] suggested a
Gaussian shaped nuclear matter distribution for an
$\alpha~{}-~{}\textrm{particle}$ is

$\rho(r)=0.4229~{}\textrm{e}^{-0.7024r^{2}}~{}\textrm{fm}^{-3},$ (25)

with a mean square radius $\langle r^{2}\rangle=2.1355~{}\textrm{fm}^{2}$. A
new nuclear matter distribution for the $\alpha~{}-~{}\textrm{particle}$ is

$\rho(r)=2.12~{}\textrm{e}^{-2.3705r}~{}\textrm{fm}^{-3},$ (26)

proposed using the msr value since the previous one does not allow to obtain
an analytical solution. Therefore, the reason for choosing this distribution
is to obtain an exact analytical double-folded potential since the density-
independent NN effective interaction (M3Y-Reid, Eq. (2) also has a similar
mathematical form. Then this is easily achieved by using the Fourier transform
techniques as usual [1] for the double folding integral given in Eq. (8), but
this time with an analytical integration. Thus, the analytical double-folded
potential with both direct and ZR exchange NN interactions included is

$\displaystyle U(R)=~{}$
$\displaystyle\frac{1}{R}~{}2747\left(\textrm{e}^{-4R}-31323.3246~{}\textrm{e}^{-2.5R}-10.729866914~{}\textrm{e}^{-2.3705R}~{}\times\right.$
$\displaystyle\left.[R^{3}-24.345R^{2}+377.89705R-2919.17177]\right).$ (27)

Both computations agree very well, as supported by the $\xi^{2}$ values in
Table 4 and shown in Fig. 5(a). The $\xi^{2}$ values are almost the same for
both integration methods. There is a caveat to be careful of about this case.
The effective NN interaction and the new nuclear matter distribution slowly go
to zero when $r$ goes to infinity, contrary to the Gaussian-shaped density in
Eq. (25). This leads to the problem shown in Fig. 5(b), where
$q_{max}~{}=~{}3~{}\textrm{fm}^{-1}$ is not enough to obtain a numerically
accurate solution. Therefore it was necessary to raise $q_{max}$ to
$10~{}\textrm{fm}^{-1}$ in this case while this value of $q_{max}$ is usually
enough most of the time, as mentioned in Ref. [7]. This problem illustrates it
is better to make a consistency check by raising the value of $q_{max}$ till
the calculation reaches a saturation point where the potential does not change
anymore.

(a) $q_{max}~{}=~{}10~{}\textrm{fm}^{-1}$

(b) $q_{max}~{}=~{}3~{}\textrm{fm}^{-1}$

Figure 5: The full circles, up triangles, and down triangles are the
calculations of the total, direct and exchange parts of the analytically
calculated double folding potentials, respectively. The solid lines are the
computations using BiFold.

###  4.2 BiFold vs. DFPOT

This test case compares the two codes for an $\alpha+^{40}\textrm{Ca}$ elastic
scattering system where the energy of the $\alpha$ projectile is 141.7 MeV in
the laboratory system. The effective NN interaction is for both codes is the
density-independent M3Y-Paris with the ZR exchange part given in Eqs. (5) and
(6). BiFold can perform the calculation for this case by changing
_‘u_m3y_reid_zr’_ to _‘u_m3y_paris_zr’_ in the twelfth line of Python code
shown in Fig. 2.

The nuclear matter density for $\alpha~{}-~{}\textrm{particle}$ is in Eq.
(25), and the density [34] of ${}^{40}\textrm{Ca}$ is

$\rho(r)=0.169~{}\left[1+\exp{\left(\frac{r-3.60}{0.523}\right)}\right]^{-1}~{}\textrm{fm}^{-3},$
(28)

with a msr value of $~{}\langle r^{2}\rangle=11.553~{}\textrm{fm}^{2}$. As
seen in Fig. 6, the computations of both codes are in very well agreement with
each other, and the $\xi^{2}$ values in Table 4 also support the claim.

Figure 6: The full circles, up triangles, and down triangles are the
calculations of the total, direct and exchange parts of the double folding
potentials using DFPOT [7], respectively. The solid lines are the computations
using BiFold.

###  4.3 BiFold vs. Khoa et al. ’s calculation

The ${}^{16}\textrm{O}+^{16}\textrm{O}$ elastic scattering system at several
incident energies was studied by Khoa et al. [35] using various density-
dependent M3Y-Paris-based NN interactions. The case chosen for the comparison
is the one at 250 MeV incident energy. And the effective NN interaction used
for the double folding potential is the BDM3Y1-Paris NN interaction with the
FR exchange part. The calculations of Khoa et al. [35] were obtained by
digitizing the related figure in the reference with the help of the programs
Inkscape [41] and Engauge Digitizer [42] .

The nuclear matter density of the ${}^{16}\textrm{O}$ nuclei with the msr
value $~{}\langle r^{2}\rangle=6.625~{}\textrm{fm}^{2}$ [34, 35] is

$\rho(r)=0.181~{}\left[1+\exp{\left(\frac{r-2.525}{0.45}\right)}\right]^{-1}~{}\textrm{fm}^{-3}.$
(29)

Since BDM3Y1 is a density-dependent NN interaction, the double folding
integral in Eq. (11) has to be solved to obtain the potentials in Fig. 7. This
integral contains the Coulomb potential between the two nuclei through the
local momentum $\vec{K}(\vec{R})$ of the relative motion. The Coulomb
potential modeled for uniformly charged spherical nuclei is

$U_{C}(R)=z_{p}z_{t}\frac{e^{2}}{4\pi\varepsilon_{0}}\begin{cases}\frac{1}{R}&\left(R\geq
R_{C}\right)\\\
\frac{1}{2R_{c}}\left[3-\left(\frac{R}{R_{c}}\right)^{2}\right]&\left(R\leq
R_{C}\right)\end{cases}$ (30)

used in the calculations where $z_{p}$, $z_{t}$ are proton numbers of the
projectile and the target, respectively. The Coulomb radius is
$R_{c}=1.405\left(a_{p}^{1/3}+a_{t}^{1/3}\right)~{}\textrm{fm}$.

Figure 7: The full circles, up triangles, and down triangles are the
calculations of the total, direct and exchange parts of the double folding
potentials from Khoa et al. [35], respectively. The solid lines are the
computations using BiFold.

As can be seen from the $\xi^{2}$ values in Table 4 and the potentials in Fig.
7, BiFold is also consistent with the final reference work. It is important to
note here that the comparison is made till 5.6 fm in this case since the
resolution of the digitized figure [35] was not enough to recover the data
with enough precision.

## Declaration of competing interest

The author declares that he has no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgements

The author would like to thank Prof. Dr. O. Bayrak for stimulating discussions
and useful comments on the manuscript.

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