In this paper, an approximate effective nucleon-nucleon interaction for nuclear matter and finite studies has been derived using the lowest order constrained variational (LOCV) approach. The LOCV method, a functional minimization procedure, uses a normalization constraint to keep higher-order terms as small as possible. As a first step, two-body matrix elements based on the Reid93 nucleon-nucleon potential were calculated for the nuclear system A = 16 in a harmonic oscillator basis, with the oscillator size parameter ћω = 14.0 MeV, and separated into the central, spin-orbit and tensor channels in conformity with the potentials for Inelastic scattering. Following this, a least squares fitting of the matrix elements to a sum of Yukawa functions was performed to determine the strengths of the effective interaction in the singlet-even, singlet-odd, triplet-even and triplet-odd (Central); tensor-even and tensor-odd (Tensor); spin-orbit-even and spin-orbit-odd (Spin-orbit) channels. Of all the matrix elements, only the triplet-even and tensor-even components, being attractive, are affected by the tensor correlations (a = 0.05); and are shown to exhibit the same trend of variation in conformity with past work, in terms of magnitude, as one goes from the lower-node quantum numbers (n’, n) = (0, 0) to higher ones (n’, n) = (2, 2). When compared with the G-matrix results of previous researchers, the results obtained herein have been found to be in good agreement. This, therefore, gives hope that the new effective interaction promises to be a reliable tool for nuclear matter and nuclear structure studies.
Published in | International Journal of Applied Mathematics and Theoretical Physics (Volume 10, Issue 1) |
DOI | 10.11648/j.ijamtp.20241001.12 |
Page(s) | 21-27 |
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), 2024. Published by Science Publishing Group |
Matrix Elements, Effective Interaction, Nuclear Matter, Finite Nuclei
SE | TE | ||||||
---|---|---|---|---|---|---|---|
(S/S) | n = 0 | 1 | 2 | (S/S) | n = 0 | 1 | 2 |
n’ = 0 | -7.69 | -6.75 | -5.49 | n’ = 0 | -10.50 | -8.59 | -6.49 |
(-6.64) | (-5.41) | (-3.80) | (-9.93) | (-8.84) | (-6.81) | ||
1 | -6.63 | -5.60 | 1 | -8.32 | -6.76 | ||
(-4.73) | (-3.30) | (-7.84) | -5.71) | ||||
2 | -4.96 | 2 | -6.03 | ||||
(-2.23) | (-3.76) | ||||||
SO | TO | ||||||
(P/P) | n = 0 | 1 | 2 | (P/P) | n = 0 | 1 | 2 |
n’ = 0 | 2.54 | 2.66 | 2.61 | n’ = 0 | 0.52 | 0.62 | 0.72 |
(2.52) | (2.37) | (2.11) | (-0.08) | (-0.23) | (-0.09) | ||
1 | 3.56 | 3.74 | 1 | 0.94 | 1.14 | ||
(3.03) | (3.00) | (-0.24) | (-0.05) | ||||
2 | 4.30 | 2 | 1.45 | ||||
(-2.23) | (0.04) | ||||||
TNE | TNO | ||||||
(S/D) | n = 0 | 1 | 2 | (P/P) | n = 0 | 1 | 2 |
n’ = 0 | -5.07 | -7.06 | -8.22 | n’ = 0 | 0.71 | 0.70 | 0.61 |
(-5.76) | -8.12) | (-9.71) | (0.78) | (0.75) | (0.63) | ||
1 | -2.50 | -4.84 | -6.61 | 1 | 0.85 | 0.81 | |
(-2.81) | (-5.52) | (0.92) | (0.87) | ||||
2 | -1.28 | -2.77 | -4.45 | 2 | 0.87 | ||
(-1.52) | (-3.23) | (-5.30) | (0.95) | ||||
LSE | LSO | ||||||
(D/D) | n = 0 | 1 | 2 | (P/P) | n = 0 | 1 | 2 |
n’ = 0 | -0.26 | 0.20 | 0.92 | n’ = 0 | -0.43 | -0.74 | -0.98 |
(-0.14) | (-0.13) | (-0.12) | (-0.60) | (-0.89) | (-1.06) | ||
1 | 0.30 | 0.74 | 1 | -1.13 | -1.45 | ||
(-0.18) | (-0.21) | (-1.26) | (-1.52) | ||||
2 | -0.96 | 2 | -1.81 | ||||
(-0.26) | (-1.84) |
S/N | Channel | R_{1} = 0.25 fm | R _{2} = 0.40 fm | R _{3} = 0.70 fm | R_{4} = 1.414 fm |
---|---|---|---|---|---|
1 | SE | 15228.20 | -5025.36 | -10.463 | |
(12455) | -3835 | (-10.463) | |||
2 | TE | 18258.78 | -5800.38 | -10.463 | |
(21227) | (-6622) | (-10.463) | |||
3 | SO | -17.92 | 1465.22 | 31.389 | |
(-1418) | (950) | (31.389) | |||
4 | TO | 10066.15 | -1048.60 | 3.488 | |
(11345) | (-1900) | (3.488) | |||
5 | TNE | -1062.2 | -26.71 | ||
(-1369) | (-10.69) | ||||
6 | TNO | 232.79 | 13.73 | ||
(-19.71) | (27.06) | ||||
7 | LSE | -111.27 | 3085.20 | ||
(-5101) | (-337) | ||||
8 | LSO | -5911.92 | 93.19 | ||
(-2918) | (-483) |
NN | Nucleon Nucleon |
LOCV | Lowest Order Constrained Variational |
RFM | Relativistic Mean Field |
EFT | Effective Field Theory |
QCD | Quantum Chromodynamics |
HF | Hartree-Fock |
M3Y | Michigan-3-Yukawa |
SE | Singlet Even |
SO | Singlet Odd |
TE | Triplet Even |
TNE | Tensor Even |
TNO | Tensor Odd |
LSE | Spin-Orbit Even |
LSO | Spin-Orbit Odd |
OPEP | One-Pion Exchange Potential |
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APA Style
Ochala, I., Fiase, J., Adeyemi, J., Abubakar, S. (2024). Approximate Effective Interaction for Nuclear Matter and Finite Nuclei. International Journal of Applied Mathematics and Theoretical Physics, 10(1), 21-27. https://doi.org/10.11648/j.ijamtp.20241001.12
ACS Style
Ochala, I.; Fiase, J.; Adeyemi, J.; Abubakar, S. Approximate Effective Interaction for Nuclear Matter and Finite Nuclei. Int. J. Appl. Math. Theor. Phys. 2024, 10(1), 21-27. doi: 10.11648/j.ijamtp.20241001.12
AMA Style
Ochala I, Fiase J, Adeyemi J, Abubakar S. Approximate Effective Interaction for Nuclear Matter and Finite Nuclei. Int J Appl Math Theor Phys. 2024;10(1):21-27. doi: 10.11648/j.ijamtp.20241001.12
@article{10.11648/j.ijamtp.20241001.12, author = {Isaiah Ochala and Joseph Fiase and John Adeyemi and Shuaibu Abubakar}, title = {Approximate Effective Interaction for Nuclear Matter and Finite Nuclei }, journal = {International Journal of Applied Mathematics and Theoretical Physics}, volume = {10}, number = {1}, pages = {21-27}, doi = {10.11648/j.ijamtp.20241001.12}, url = {https://doi.org/10.11648/j.ijamtp.20241001.12}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20241001.12}, abstract = {In this paper, an approximate effective nucleon-nucleon interaction for nuclear matter and finite studies has been derived using the lowest order constrained variational (LOCV) approach. The LOCV method, a functional minimization procedure, uses a normalization constraint to keep higher-order terms as small as possible. As a first step, two-body matrix elements based on the Reid93 nucleon-nucleon potential were calculated for the nuclear system A = 16 in a harmonic oscillator basis, with the oscillator size parameter ћω = 14.0 MeV, and separated into the central, spin-orbit and tensor channels in conformity with the potentials for Inelastic scattering. Following this, a least squares fitting of the matrix elements to a sum of Yukawa functions was performed to determine the strengths of the effective interaction in the singlet-even, singlet-odd, triplet-even and triplet-odd (Central); tensor-even and tensor-odd (Tensor); spin-orbit-even and spin-orbit-odd (Spin-orbit) channels. Of all the matrix elements, only the triplet-even and tensor-even components, being attractive, are affected by the tensor correlations (a = 0.05); and are shown to exhibit the same trend of variation in conformity with past work, in terms of magnitude, as one goes from the lower-node quantum numbers (n’, n) = (0, 0) to higher ones (n’, n) = (2, 2). When compared with the G-matrix results of previous researchers, the results obtained herein have been found to be in good agreement. This, therefore, gives hope that the new effective interaction promises to be a reliable tool for nuclear matter and nuclear structure studies. }, year = {2024} }
TY - JOUR T1 - Approximate Effective Interaction for Nuclear Matter and Finite Nuclei AU - Isaiah Ochala AU - Joseph Fiase AU - John Adeyemi AU - Shuaibu Abubakar Y1 - 2024/08/30 PY - 2024 N1 - https://doi.org/10.11648/j.ijamtp.20241001.12 DO - 10.11648/j.ijamtp.20241001.12 T2 - International Journal of Applied Mathematics and Theoretical Physics JF - International Journal of Applied Mathematics and Theoretical Physics JO - International Journal of Applied Mathematics and Theoretical Physics SP - 21 EP - 27 PB - Science Publishing Group SN - 2575-5927 UR - https://doi.org/10.11648/j.ijamtp.20241001.12 AB - In this paper, an approximate effective nucleon-nucleon interaction for nuclear matter and finite studies has been derived using the lowest order constrained variational (LOCV) approach. The LOCV method, a functional minimization procedure, uses a normalization constraint to keep higher-order terms as small as possible. As a first step, two-body matrix elements based on the Reid93 nucleon-nucleon potential were calculated for the nuclear system A = 16 in a harmonic oscillator basis, with the oscillator size parameter ћω = 14.0 MeV, and separated into the central, spin-orbit and tensor channels in conformity with the potentials for Inelastic scattering. Following this, a least squares fitting of the matrix elements to a sum of Yukawa functions was performed to determine the strengths of the effective interaction in the singlet-even, singlet-odd, triplet-even and triplet-odd (Central); tensor-even and tensor-odd (Tensor); spin-orbit-even and spin-orbit-odd (Spin-orbit) channels. Of all the matrix elements, only the triplet-even and tensor-even components, being attractive, are affected by the tensor correlations (a = 0.05); and are shown to exhibit the same trend of variation in conformity with past work, in terms of magnitude, as one goes from the lower-node quantum numbers (n’, n) = (0, 0) to higher ones (n’, n) = (2, 2). When compared with the G-matrix results of previous researchers, the results obtained herein have been found to be in good agreement. This, therefore, gives hope that the new effective interaction promises to be a reliable tool for nuclear matter and nuclear structure studies. VL - 10 IS - 1 ER -