The Effect of Zero-range Pseudo-potential Approximation in the Elastic Scattering Channel Using a Mass-dependent M3Y-type Interaction

Raymond Chivirter Abenga; Gertrude Ashia Bijimi

doi:doi:10.11648/j.wjap.20251002.12

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The Effect of Zero-range Pseudo-potential Approximation in the Elastic Scattering Channel Using a Mass-dependent M3Y-type Interaction

Raymond Chivirter Abenga^*

, Gertrude Ashia Bijimi

Published in World Journal of Applied Physics (Volume 10, Issue 2)

Received: 13 May 2025 Accepted: 11 July 2025 Published: 7 August 2025

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Abstract

The angular distributions of the elastic scattering of deuteron from ¹²C were measured in the double model framework using a mass-dependent M3Y-type interaction. The optical potential of this study was derived using the double folding formalism and subsequently employed in the optical model formalism to determine the reaction cross-sections of d+¹²C at different incident energies. The calculated differential cross-sections were analysed and compared to experimental results. A good fit of the differential cross-section to the experimental data was achieved. The suitability of the fit affirms the present formulation as a suitable tool for the study of nuclear reactions and nuclear structure.

Published in	World Journal of Applied Physics (Volume 10, Issue 2)
DOI	10.11648/j.wjap.20251002.12
Page(s)	41-49
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Angular Distribution, Folding Potential, Optical Potential, Zero-range Pseudo-potential, M3Y-type Interaction, S-Matrix

1. Introduction

The problem of nuclear scattering involves solving the Schrödinger equation for two interacting nuclei. This requires robust theoretical tools and models capable of reproducing fundamental nuclear matter properties such as binding energy, pressure, volume, susceptibility, and incompressibility

[1-3]

. Among these tools, the Michigan-3-Yukawa (M3Y) realistic interaction, derived from G-matrix elements of the harmonic oscillator potential, has been widely used and proven effective in studying nuclear structure and reactions.

In recent years, a variant of the M3Y interaction, known as the mass-dependent M3Y-type interaction, was developed using the lowest-order constrained variational (LOCV) method. This interaction incorporates mass dependence through parameters derived from nuclei with mass numbers 24, 40, and 90

[4]

. Despite this mass dependence, the M3Y-type interaction was recommended for broader applications across different nuclei. However, its effect on reaction calculations remains insufficiently explored-a gap that forms the basis of the present study.

Determining the most appropriate form of nuclear potential to analyse interactions has been a longstanding focus in nuclear physics

[5]

. Both phenomenological and microscopic potentials have been developed and extensively used, particularly in elastic and non-elastic scattering channels

[6]

. Due to the presence of strong absorption effects in heavy-ion collisions, accurate modelling of the nuclear optical potential (OP) is essential. The real parts of the optical potentials are often derived using folding models, while the imaginary parts, accounting for absorption, are typically modelled phenomenologically

[7, 8]

The double-folding model combines realistic nucleon-nucleon (NN) interactions with nuclear density distributions to construct the real part of the OP

[9]

. This approach minimises the ambiguities associated with phenomenological potentials

[10]

. Although folded potentials are inherently real, the imaginary component-representing absorption – can be included phenomenologically or through density-dependent corrections.

This study applies the double-folding model using the mass-dependent M3Y-type interaction to analyse the elastic scattering of deuteron from

{}^{12}C

. Further to the aforementioned goal, this research assesses the impact of the zero-range pseudo-potential and the mass-dependent interactions on scattering predictions and compares the resulting cross-sections with experimental data. Both the real and imaginary parts of the optical potential are evaluated using folding techniques, and the fits to experimental angular distributions are examined to determine the model’s accuracy.

2. M3Y-type Effective Interaction

This study adopts the isoscalar component of the central potential from the mass-dependent M3Y-type interaction, as outlined in prior work

[4]

. The direct and exchange components of this interaction are given by

[11, 12]

V_{00}^{D} (r) = 7419.23 \frac{e^{- 4 r}}{4 r} - 1823.98 \frac{e^{- 2.5 r}}{2.5 r}

(1)

Similarly, the exchange term is determined as:

V_{00}^{Ex} (r) = 4745.02 \frac{e^{- 4 r}}{4 r} - 1984.144 \frac{e^{- 2.5 r}}{2.5 r} - 7.8474 \frac{e^{- 0.7072 r}}{0.7072 r}

(2)

where

V_{00}^{D} (r)

, and

V_{00}^{Ex} (r)

are the direct and exchange parts of the isoscalar central M3Y-type interaction, and

r

is the internucleon distance.

The bare nucleon-nucleon (NN) interactions alone are inadequate in reproducing the saturation properties of nuclear matter. To correct this, a density-dependent factor

f (ρ)

is introduced, ensuring that the interaction achieves nuclear matter saturation

[13]

. This density dependence is expressed as

[14, 15]

V_{00}^{D (Ex)} (ρ, r) = f (ρ) V_{00}^{D (Ex)} (r)

(3)

Concerning the density dependence, an exponential form of the density-dependent term was adopted

[16, 17]

f (ρ) = C (1 + α exp (- βρ))

(4)

Here,

C

and

β

are constants determined to reproduce the correct binding energy and saturation density of symmetric nuclear matter. The values of the constants

C

α

, and

β

are obtained from a prior parameterization and can be found in reference

[18]

The exchange term is analysed using the resonating-group method in a single-channel approximation

[19]

. It includes both single-nucleon and core exchange effects, significant primarily when projectile and target nuclei have similar mass numbers. To localise the otherwise nonlocal exchange term, a zero-range pseudo-potential is added, yielding

[13]

V_{pt} \to V_{pt} (1 - P_{pt})

(5)

where

V_{pt}

are the direct and exchange parts of the effective interaction,

P_{pt}

is the operator that exchanges all the coordinates of the two nucleons by making the exchange part of the effective interaction local. By the zero-range pseudo-potential of equation (5), the interaction

V_{pt}

is transformed in the form

[20, 21]

V_{pt} (1 - P_{EX}) \to V_{pt} (r) + \hat{J} (E) δ (r),

(6)

where the function

\hat{J} (E) = J_{00} g (E)

represents the zero-range pseudo-potential that incorporates the exchange component of the M3Y-type interaction

[22]

. The term,

V_{pt} (r)

is the direct part of the isoscalar interaction defined by Equation (1), but now expressed as the projectile-target interaction. The term

g (E)

represents the energy scaling factor. The magnitude of the exchange amplitude

J_{00}

was evaluated to be about

- 361 MeV

[11, 18]

. The energy scaling factor was approximated to be

[1 - 0.005 (\frac{E}{A})]

, where

E

and

A

are the incident laboratory energy and the mass of the projectile, respectively.

3. The Double-folding Model

To construct the nuclear optical potential, the double-folding model was used. The double-folding model convolves the effective NN interaction with the nuclear density distributions of the projectile and target

[19]

V_{F} (r) = \iint ρ_{p} (r_{p}) ρ_{t} (r_{t}) V_{pt} (r_{tp}) d r_{p} d r_{t}

(7)

where

ρ_{p}

and

ρ_{t}

are the density distributions of the projectile and target nuclei, and

r_{tp} = r_{t} - r_{p}

. The integration of Equation (7) was performed using the default integration parameters of the Nuclear Reaction Video (NRV). The nuclear densities are modelled using a two-parameter Fermi (2pF) distribution

[23-25]

ρ_{n (p)} (r) = ρ_{0 n (p)} {[1 + \exp (\frac{r - R_{p (n)}}{a_{p (n)}})]}^{- 1}

(8)

here,

ρ_{0}

is the central density,

R_{p (n)}

, and

a_{p (n)}

are the half-density radius and the diffuseness parameter, respectively. These parameters are obtained from empirical fits to charge distributions

[26]

The double-folded potential naturally represents the real part of the optical potential. To account for the imaginary part-which captures absorption into non-elastic channels, a similar folding approach is applied but with a different renormalization factor

[27]

U (r) = (N_{r} + i N_{i}) V_{F} (r) + V_{Coul} (r)

(9)

where

N_{r (i)}

are the real and imaginary renormalization factors for the real and imaginary parts of the optical potential, and

V_{Coul} (r)

is the Coulomb potential. These renormalization constants are adjusted to achieve optimal agreement with experimental data. The double-folding model calculations were done using the folding model code of the Nuclear Reaction Video (NRV)

[28]

The optical potential formulation of Equation (9) was modified to include potential geometrical parameters. These geometrical parameters were assumed to have the Woods-Saxon form factor. The nuclear part of the OP of Equation (10) was modified to include the geometrical parameters in the form;

V_{N} = - [V_{0} f (r, R_{v}, a_{v}) + i W_{0} f (r, R_{w}, a_{w})]

(10)

where

V_{0}

and

W_{0}

are the strengths of the real and imaginary folded potentials, respectively, and

f (r, R_{i} {, a}_{i})

is the introduced Woods-Saxon form factor with

i = v, w

. The geometrical form factor introduced in Equation (10) is defined by

[29]

;

f (r, R_{i}, a_{i}) = {(1 + \exp [\frac{r - R_{i}}{a_{i}}])}^{- 1}

(11)

where

R_{i}

and

a_{i}

are the half-value radius and the diffuseness parameters, which describe the decreasing rate of the optical potential. The radius parameters were defined by

[26]

;

R_{i} = [r_{i} (A_{p}^{\frac{1}{3}} + A_{t}^{\frac{1}{3}})]

(12)

while the diffuseness parameter was defined by:

a_{i} = 0.734 - \frac{150}{Z_{t}^{2} + 500}

(13)

where

A_{p (t)}

are the mass numbers of the projectile and target nuclei, and

Z_{t}

is the atomic number of the target.

The introduced geometrical parameters were deduced by fitting the elastic scattering data using the double folding potential by minimising the

χ^{2}

value in the OM code. The best value of

χ^{2}

was defined by

[6, 30]

χ^{2} = \frac{1}{N} \sum_{i = 1}^{N} {[\frac{σ_{cal} (θ_{i}) - σ_{\exp} (θ_{i})}{∆ σ_{\exp} (θ_{i})}]}^{2}

(14)

where

σ_{cal} (θ_{i})

and

σ_{\exp} (θ_{i})

are, respectively, the calculated and experimental values of the elastic scattering differential cross-section at

θ_{i}

and

∆ σ_{\exp} (θ_{i})

is the corresponding experimental error. At the same time,

N

is the number of data points.

4. Results: Folding Model Potential of the Elastic Scattering of d+¹²C

The double-folding model was employed to calculate the real and imaginary parts of the optical potential for the elastic scattering of

d + {}^{12}C

at incident laboratory energies of 28, 110, 120, and 170 MeV. Using these potentials, the reaction

σ_{r}

(mb) and total cross-sections

σ_{tot}

(mb) were computed, and the results are presented in Table 1.

Table 1. Best-fit parameters and cross-sections of elastic scattering of

d + {}^{12}C

Target	$E_{lab}$ (MeV)	$N_{r}$	$N_{i}$	$σ_{r}$ (mb)	$σ_{tot}$ (mb)
${}^{12}C$	28	1.00	0.80	977.95	1671.11
	110	1.60	1.00	752.84	1416.83
	120	1.70	1.00	730.08	1401.19
	170	1.45	0.90	578.19	1119.98

The results indicate good agreement between the calculated reaction cross-sections and existing literature

[29-34]

, confirming the reliability of the double-folding approach. The optical potentials generated in this study exhibited typical Woods-Saxon shapes. The plots of the radial dependence of the strength of the folded potentials are shown in Figures 1-4. The potentials are attractive and short-ranged over small nuclear distances.

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Figure 1. The depth of the real and imaginary folded potential of

d + {}^{12}C

elastic scattering at

E_{lab} = 28 MeV

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Figure 2. The depth of the real and imaginary folded potential of

d + {}^{12}C

elastic scattering at

E_{lab} = 120 MeV .

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Figure 3. The depth of the real and imaginary folded potential of

d + {}^{12}C

elastic scattering at

E_{lab} = 110 MeV

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Figure 4. The depth of the real and imaginary folded potential of

d + {}^{12}C

elastic scattering at

E_{lab} = 170 MeV

5. Analysis of the Differential Cross Section of the Elastic Scattering of d+¹²C

The plots of the angular distributions of the elastic scattering for each incident energy are shown in Figures 5-8. The theoretical differential cross-sections match well with experimental data across all angles. The large

χ²

value at 170 MeV suggests statistical anomalies or increased sensitivity in the data fitting.

Table 2. Derived geometrical parameters of the optical potentials of

d + {}^{12}C

Target	$E_{lab}$ $(MeV)$	$V_{r}$ $(MeV)$	$r_{v}$ $(fm)$	$a_{v}$ $(fm)$	$W_{i}$ $(MeV)$	$r_{w}$ $(fm)$	$a_{w}$ $(fm)$	$χ^{2}$
${}^{12}C$	28	61.00	0.91	0.57	50.00	0.75	1.49	10.66
	110	75.00	0.67	0.90	27.36	0.91	0.63	2.09
	120	76.00	0.67	0.98	25.77	0.96	0.57	4.43
	170	52.00	0.78	0.86	25.00	0.98	0.61	3876440.21

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Figure 5. Angular distribution of

d + {}^{12}C

elastic scattering at

E_{lab} = 28 MeV

The experimental data points were reproduced quite accurately at all energies and angular regions. However, in the data of 170 MeV, a maximum and a minimum were observed between

15 ° \leq θ_{cm} \leq 25 °

and

25 ° \leq θ_{cm} \leq 35 °

. In general, a good fit of the theoretical calculations to experimental data was obtained, and a pronounced improvement in the fit of the differential cross-sections using the present formulation was achieved as compared to those obtained using microscopic and phenomenological potentials

[28, 30, 31]

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Figure 6. Angular distribution of

d + {}^{12}C

elastic scattering at

E_{lab} = 120 MeV

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Figure 7. Angular distribution of

d + {}^{12}C

elastic scattering at

E_{lab} = 110 MeV

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Figure 8. Angular distribution of

d + {}^{12}C

elastic scattering at

E_{lab} = 170 MeV

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Figure 9. Modulus of the elastic S-matrix as a function of total orbital angular momentum of

d + {}^{12}C

E_{lab} = 28 MeV

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Figure 10. Modulus of the elastic S-matrix as a function of total orbital angular momentum of

d + {}^{12}C

E_{lab} = 110 MeV

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Figure 11. Modulus of the elastic S-matrix as a function of total orbital angular momentum of

d + {}^{12}C

E_{lab} = 120

MeV

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Figure 12. Modulus of the elastic S-matrix as a function of total orbital angular momentum of

d + {}^{12}C

E_{lab} = 170 MeV

The plots of the modulus of the S-matrix elements are also shown in Figures 9-12 as a function of total orbital angular momentum

L

for the different incident energies. These plots reveal regions where

|S_{L}| < 1

, indicating strong absorption and the contributions of the nuclear part of the OP. At 28 MeV scattering data, the effect of strong absorption was observed at small impact parameters in the angular momentum range

L \approx 0 - 5

. In the cases of the 110, 120, and 170 MeV scattering data, the contributions of the nuclear optical potentials were observed, but no region of total absorption was observed. At large impact parameters

|S_{L}| \to 1

, the contributing angular momentum values shift to higher L, indicating peripheral scattering contributions. This transition is consistent with the behaviour of the nuclear optical potential at different energies.

6. Conclusion

This study employed the double-folding model in conjunction with the mass-dependent M3Y-type interaction to investigate the elastic scattering of

d + {}^{12}C

over a range of incident laboratory energies (28-170 MeV). The analysis incorporated both the real and imaginary components of the optical potential derived through folding procedures. The imaginary part was incorporated to account for the absorption effects observed in heavy-ion collisions. The calculated reaction and total cross-sections, as well as differential angular distributions, show excellent agreement with experimental data, validating the effectiveness of the M3Y-type interaction in modelling nuclear scattering phenomena. The renormalization factors obtained for the optical potentials reflect the expected energy dependence, and the folded potentials exhibit Woods-Saxon-like shapes consistent with known nuclear interaction behaviours.

Furthermore, the analysis of the elastic S-matrix elements revealed that the nuclear part of the optical potentials contributed most significantly at small impact parameters. The contributions of the optical potentials were gradually suppressed with increasing angular momentum, leaving the Coulomb interaction dominant at large impact parameters. This confirms the realistic absorption effects modelled by the imaginary component of the potential. Overall, the results highlight the success of the mass-dependent M3Y-type interaction and the double-folding model in accurately reproducing scattering observables. It also demonstrated that the M3Y-type interaction is a suitable tool for studying both elastic and potentially non-elastic nuclear reactions. Further studies on similar systems are recommended to take into account this formulation and its application to nonelastic scattering considerations.

Abbreviations

DFM	Double-Folding Model
HI	Heavy-Ion
M3Y	Michigan three Yukawa
NRV	Nuclear Reaction Video
OM	Optical Model

Conflicts of Interest

The authors declare no conflict of interest.

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Abenga, R. C., Bijimi, G. A. (2025). The Effect of Zero-range Pseudo-potential Approximation in the Elastic Scattering Channel Using a Mass-dependent M3Y-type Interaction. World Journal of Applied Physics, 10(2), 41-49. https://doi.org/10.11648/j.wjap.20251002.12

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Abenga, R. C.; Bijimi, G. A. The Effect of Zero-range Pseudo-potential Approximation in the Elastic Scattering Channel Using a Mass-dependent M3Y-type Interaction. World J. Appl. Phys. 2025, 10(2), 41-49. doi: 10.11648/j.wjap.20251002.12

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Abenga RC, Bijimi GA. The Effect of Zero-range Pseudo-potential Approximation in the Elastic Scattering Channel Using a Mass-dependent M3Y-type Interaction. World J Appl Phys. 2025;10(2):41-49. doi: 10.11648/j.wjap.20251002.12

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@article{10.11648/j.wjap.20251002.12,
  author = {Raymond Chivirter Abenga and Gertrude Ashia Bijimi},
  title = {The Effect of Zero-range Pseudo-potential Approximation in the Elastic Scattering Channel Using a Mass-dependent M3Y-type Interaction
},
  journal = {World Journal of Applied Physics},
  volume = {10},
  number = {2},
  pages = {41-49},
  doi = {10.11648/j.wjap.20251002.12},
  url = {https://doi.org/10.11648/j.wjap.20251002.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.wjap.20251002.12},
  abstract = {The angular distributions of the elastic scattering of deuteron from 12C were measured in the double model framework using a mass-dependent M3Y-type interaction. The optical potential of this study was derived using the double folding formalism and subsequently employed in the optical model formalism to determine the reaction cross-sections of d+12C at different incident energies. The calculated differential cross-sections were analysed and compared to experimental results. A good fit of the differential cross-section to the experimental data was achieved. The suitability of the fit affirms the present formulation as a suitable tool for the study of nuclear reactions and nuclear structure.},
 year = {2025}
}

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TY  - JOUR
T1  - The Effect of Zero-range Pseudo-potential Approximation in the Elastic Scattering Channel Using a Mass-dependent M3Y-type Interaction

AU  - Raymond Chivirter Abenga
AU  - Gertrude Ashia Bijimi
Y1  - 2025/08/07
PY  - 2025
N1  - https://doi.org/10.11648/j.wjap.20251002.12
DO  - 10.11648/j.wjap.20251002.12
T2  - World Journal of Applied Physics
JF  - World Journal of Applied Physics
JO  - World Journal of Applied Physics
SP  - 41
EP  - 49
PB  - Science Publishing Group
SN  - 2637-6008
UR  - https://doi.org/10.11648/j.wjap.20251002.12
AB  - The angular distributions of the elastic scattering of deuteron from 12C were measured in the double model framework using a mass-dependent M3Y-type interaction. The optical potential of this study was derived using the double folding formalism and subsequently employed in the optical model formalism to determine the reaction cross-sections of d+12C at different incident energies. The calculated differential cross-sections were analysed and compared to experimental results. A good fit of the differential cross-section to the experimental data was achieved. The suitability of the fit affirms the present formulation as a suitable tool for the study of nuclear reactions and nuclear structure.
VL  - 10
IS  - 2
ER  -

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Author Information

Raymond Chivirter Abenga

Department of Pure and Applied Physics, Veritas University, Abuja, Nigeria

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http://orcid.org/0000-0001-7040-9395
Gertrude Ashia Bijimi

Department of Pure and Applied Physics, Veritas University, Abuja, Nigeria

http://orcid.org/0009-0002-9300-7641

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APA Style

Abenga, R. C., Bijimi, G. A. (2025). The Effect of Zero-range Pseudo-potential Approximation in the Elastic Scattering Channel Using a Mass-dependent M3Y-type Interaction. World Journal of Applied Physics, 10(2), 41-49. https://doi.org/10.11648/j.wjap.20251002.12

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ACS Style

Abenga, R. C.; Bijimi, G. A. The Effect of Zero-range Pseudo-potential Approximation in the Elastic Scattering Channel Using a Mass-dependent M3Y-type Interaction. World J. Appl. Phys. 2025, 10(2), 41-49. doi: 10.11648/j.wjap.20251002.12

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AMA Style

Abenga RC, Bijimi GA. The Effect of Zero-range Pseudo-potential Approximation in the Elastic Scattering Channel Using a Mass-dependent M3Y-type Interaction. World J Appl Phys. 2025;10(2):41-49. doi: 10.11648/j.wjap.20251002.12

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@article{10.11648/j.wjap.20251002.12,
  author = {Raymond Chivirter Abenga and Gertrude Ashia Bijimi},
  title = {The Effect of Zero-range Pseudo-potential Approximation in the Elastic Scattering Channel Using a Mass-dependent M3Y-type Interaction
},
  journal = {World Journal of Applied Physics},
  volume = {10},
  number = {2},
  pages = {41-49},
  doi = {10.11648/j.wjap.20251002.12},
  url = {https://doi.org/10.11648/j.wjap.20251002.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.wjap.20251002.12},
  abstract = {The angular distributions of the elastic scattering of deuteron from 12C were measured in the double model framework using a mass-dependent M3Y-type interaction. The optical potential of this study was derived using the double folding formalism and subsequently employed in the optical model formalism to determine the reaction cross-sections of d+12C at different incident energies. The calculated differential cross-sections were analysed and compared to experimental results. A good fit of the differential cross-section to the experimental data was achieved. The suitability of the fit affirms the present formulation as a suitable tool for the study of nuclear reactions and nuclear structure.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - The Effect of Zero-range Pseudo-potential Approximation in the Elastic Scattering Channel Using a Mass-dependent M3Y-type Interaction

AU  - Raymond Chivirter Abenga
AU  - Gertrude Ashia Bijimi
Y1  - 2025/08/07
PY  - 2025
N1  - https://doi.org/10.11648/j.wjap.20251002.12
DO  - 10.11648/j.wjap.20251002.12
T2  - World Journal of Applied Physics
JF  - World Journal of Applied Physics
JO  - World Journal of Applied Physics
SP  - 41
EP  - 49
PB  - Science Publishing Group
SN  - 2637-6008
UR  - https://doi.org/10.11648/j.wjap.20251002.12
AB  - The angular distributions of the elastic scattering of deuteron from 12C were measured in the double model framework using a mass-dependent M3Y-type interaction. The optical potential of this study was derived using the double folding formalism and subsequently employed in the optical model formalism to determine the reaction cross-sections of d+12C at different incident energies. The calculated differential cross-sections were analysed and compared to experimental results. A good fit of the differential cross-section to the experimental data was achieved. The suitability of the fit affirms the present formulation as a suitable tool for the study of nuclear reactions and nuclear structure.
VL  - 10
IS  - 2
ER  -

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