Applied Mathematics and Physics
Volume 9, 2021 - Issue 1
Website: http://www.sciepub.com/journal/amp

ISSN(Print): 2333-4878
ISSN(Online): 2333-4886

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Review Article

Open Access Peer-reviewed

Chadia Qotni^{ }, Hicham Laribou

Received January 07, 2021; Revised February 08, 2021; Accepted February 22, 2021

We made the numerical simulation of the theoretical results obtained by the resonant means, method in combination with Floquet's theorem of a set of spin particles interacting with an oscillating field. The paces that we obtained by the numerical simulation using Mathematica software of the magnetic susceptibility function theoretically obtained as a function of temperature or as a function of frequency represent a good agreement with the curves of the work published using experimental results. Our objective was to obtain a quality agreement between the experiments and the theoretical model studied.

The study of the magnetic susceptibility of materials has applications in several areas of physics such as electromagnetic, radar application, the optical propagation ^{ 1, 3}, Photonics ^{ 4, 5} and semiconductors ^{ 6}.

The properties of magnetic susceptibility have aroused the interest of many experimental and theoretical researchers ^{ 7}. Experimentally observed many properties of themwere good compared to theoretical predictions based on spin theories of fluctuations .Dependence of temperature and frequency of magnetic susceptibility The theory succeeded in deriving the Curie-Weiss law of the magnetic susceptibility of the nonlinear effect of thermal magnetic amplitudes greater than TC ^{ 14, 19}. It is, however, known that the spontaneous magnetic moment does not vanish within the limit; T → TC ^{ 8, 10} ^{ 20, 21, 22, 23, 24, 25}.

In this article, we will show the dependence of the magnetic susceptibility as a function of the frequency for several values of the temperature that we have chosen arbitrarily to compare our theoretical results ^{ 11} numerically with the results published in the literature, then we have although showed good agreement with the theoretical and experimental results of several authors published on the same subject ^{ 26, 28}.

Consider a quantum system interacting with an oscillating field, Periodical période période and amplitude ω_{1}. The Schrödinger equation governing the evolution of this system is such that

(1) |

(2) |

where H_{0} is the Hamiltonian of the free system , and where H_{1} is the Hamiltonian which reflects the interaction with aperiodic field. Let V (t) the evolution operator on Eq. (1) V (t) satisfies

(3) |

According Floquet 's theorem, there is a decomposition (R, T (t)) such that :

(4) |

where T (t) is a unitary operator, τ Periodical period (or a multiple de τ), and where R is a constant Hermitian operator.

V (t) is the interaction representation

The method of resonant medium (MMR) is due to G. and M. Lochak Thiounn ^{ 12} and is a generalization of the method of Bogolyubov ^{ 13, 28, 29}. It allows the approximate solution of the Schrödinger equation governing the evolution of a quantum system interacting with a periodic field.

The solutions are obtained in the form of a series expansion in powers ω1.The Hamiltonian H_{I} (t) can be written as :

(5) |

where (H_{I})_{k} is a constant set of hermetic operators; Ω_{k} which is a sequence of frequencies, which can extract a suite including the frequency Ω_{k} =0.Suppose that this subsequence contains finite number k of frequencies that can direct Ω_{0 }Ω_{1} Ω_{k}…., And the other Ω_{k} frequencies, Ω_{k+1} are neither harmonics And the other Ω_{k} frequencies, Ω_{k+1} are neither harmonics or harmonic combinations of raw k. Then we can write the interaction Hamiltonian in the form:

(6) |

We define two operators called "middle part" and oscillating part "of (H_{I})_{k} respectively denoted and asking

(7) |

(8) |

Consider a statistical ensemble of N particles with spin 1 without mutual interactions; individual magnetic moment; subjected to a static field and an oscillating field

The resulting Hamiltonian interaction with these fields is given by:

(10) |

From this result we note that the component of the magnetization depends only on the first order of the intensity of the oscillating field.

Knowing that the magnetic susceptibility can be written:

(11) |

we can deduce the value of the magnetic susceptibility at first order improved depending on the density of the oscillating field , we obtained the following results

The term dispersion is given by the equation:

(12) |

The term absorption is written

(13) |

Susceptibility to order a is expressed by the equation :

(14) |

Hence we have deduced the following equation:

(15) |

We note that the magnetic susceptibility to order one is linear, and takes its minimum value if ω tends to zero.

In order three , only the Magnetic susceptibility to this order of approximation is given by

(16) |

We obtained the coefficient of dispersion to order three, it is as

(17) |

with

(18) |

We deduced the absorption coefficient at this order of approximation as

(19) |

or

(20) |

This result highlights the nonlinearity which appeared in the formula of magnetic susceptibility in this order approximation, this translates that has a population of difference in the transitions between the stationary states.

The magnetic susceptibility varies linearly as a function of the frequency, and presents a slope which differs for precise values of the temperature.The simulation of the magnetic susceptibility in absorption therm of the following equation as a function of the temperature shows an interesting decrease as the temperature increases. We compared this curve with the one published in the article ^{ 11} we will find a good agreement.

In conclusion, the linear effect is due to the transition level of the photon. In addition, the increase in relaxation time, may improve this effect ; we have also shown that the nonlinearity depends on the temperature , and other parameters such as the structure of the material. We approached the study of a simple example where it is possible, without too many complications, pushing the calculations in higher orders. We have set thus identified "non-linear" interesting effects: saturation effects and nonlinear susceptibility

[1] | A.H. Zaki, M.A. Hafiez, W.M. El Rouby, S.I. El-Dek, A.A. Farghali, Novel magnetic standpoints in Na2Ti3O7 nanotubes, J. Magn. Magn Mater. 476 (2019) 207-212. | ||

In article | View Article | ||

[2] | R.M. Francisco, J.P. Santos, Magnetic properties of the Ashkin-Teller model on a hexagonal nanotube, Phys. Lett. A 383 (11) (2019) 1092-1098. | ||

In article | View Article | ||

[3] | Z. Zhang, Z. Li, J. Zhang, H. Bian, T. Wang, J. Gao, J. Li, Structural and magnetic properties of porous FexOy nanosheets and nanotubes fabricated by electrospinning, Ceram. Int. 45 (1) (2019) 457-461. | ||

In article | View Article | ||

[4] | P. Robkhob, I.M. Tang, S. Thongmee, Magnetic properties of the dilute magnetic semiconductor Zn 1-x Co x O nanoparticles, J. Supercond. Nov. Magnetism (2019) 1-9. | ||

In article | View Article | ||

[5] | A. Gorczyca-Goraj, T. Domanski, M.M. Maska, Topological superconductivity at finite temperatures in proximitized magnetic nanowires, Phys. Rev. B 99 (23) (2019) 235430. | ||

In article | View Article | ||

[6] | Y. Wang, D. Hu, H. Jia, Q. Wang, Efficient enhancement of light trapping in the double-textured Al doped ZnO films with nanorod and crater structures, Phys. B Condens. Matter 565 (2019) 9-13. | ||

In article | View Article | ||

[7] | N.Y. Schmidt, S. Laureti, F. Radu, H. Ryll, C. Luo, F. d’Acapito, M. Albrecht, Structural and magnetic properties of FePt-Tb alloy thin films, Phys. Rev. B 100 (6) (2019), 064428. | ||

In article | View Article | ||

[8] | R. Das, J.A. Cardarelli, M.H. Phan, H. Srikanth, Magnetically tunable iron oxide nanotubes for multifunctional biomedical applications, J. Alloys Compd. 789 (2019) 323-329. | ||

In article | View Article | ||

[9] | A.L. Elrefai, T. Yoshida, K. Enpuku, Magnetic parameters evaluation of magnetic nanoparticles for use in biomedical applications, J. Magn. Magn Mater. 474 (2019) 522-527. | ||

In article | View Article | ||

[10] | N. Lowa, € J.M. Fabert, D. Gutkelch, H. Paysen, O. Kosch, F. Wiekhorst, 3D-printing of novel magnetic composites based on magnetic nanoparticles and photopolymers, J. Magn. Magn Mater. 469 (2019) 456-460. | ||

In article | View Article | ||

[11] | C. Qotni, A. L. Marrakchi, S. Sayouri, Y. Achkar, Effets non linéaires dans l’interaction d’un ensemble de particules de spin 1 avec un champ oscillant dans l'expression de l'aimantation et de la susceptibilité magnétique, I.J.I.A.S. ISSN 2028-9324 vol.17 No. 3 Aug. 2016, pp.1050-1061. | ||

In article | |||

[12] | G. Lochak. C.R.A.S. Série B, A/272. P.1281.1971. | ||

In article | |||

[13] | A. Erbeia , Résoances magnétiques-Masson, Paris, 1969. | ||

In article | |||

[14] | F. Sedighi, M. Esmaeili-Zare, A. Sobhani-Nasab, M. Behpour, Synthesis and characterization of CuWO 4 nanoparticle and CuWO 4/NiO nanocomposite using co-precipitation method; application in photodegradation of organic dye in water, J. Mater. Sci. Mater. Electron. 29 (16) (2018) 13737-13745. | ||

In article | View Article | ||

[15] | B.Z. Mi, C.J. Feng, J.G. Luo, D.Z. Hu, Magnetic compensation and critical properties of a mixed spin-(2, 3/2) Heisenberg single-walled nanotube superlattice, Superlattice. Microst. 113 (2018) 524-533. | ||

In article | View Article | ||

[16] | M.D. Hossain, R.A. Mayanovic, R. Sakidja, M. Benamara, R. Wirth, Magnetic properties of core-shell nanoparticles possessing a novel Fe (II)-chromia phase: an experimental and theoretical approach, Nanoscale 10 (4) (2018) 2138-2147. | ||

In article | View Article PubMed | ||

[17] | D. Lv, W. Jiang, Y. Ma, Z. Gao, F. Wang, Magnetic and thermodynamic properties of a cylindrical ferrimagnetic Ising nanowire with core/shell structure, Phys. E Low-dimens. Syst. Nanostruct. 106 (2019) 101-113. | ||

In article | View Article | ||

[18] | E. Kantar, Ising-type single-segment ferromagnetic nanowire with core/shell: the dependences of the angle, temperature, and geometry, J. Supercond. Nov. Magnetism 31 (2) (2018) 341-346. | ||

In article | View Article | ||

[19] | H. Magoussi, A. Zaim, M. Kerouad, Theoretical investigations of the phase diagrams and the magnetic properties of a random field spin-1 Ising nanotube with core/shell morphology, J. Magn. Magn Mater. 344 (2013) 109-115. | ||

In article | View Article | ||

[20] | A.S. Tehrani, M.A. Kashi, A. Ramazani, A.H. Montazer, Axially adjustable magnetic properties in arrays of multilayered Ni/Cu nanowires with variable segment sizes, Superlattice. Microst. 95 (2016) 38-47. | ||

In article | View Article | ||

[21] | B. Deviren, M. Keskin, Thermal behavior of dynamic magnetizations, hysteresis loop areas and correlations of a cylindrical Ising nanotube in an oscillating magnetic field within the effective-field theory and the Glauber-type stochastic dynamics approach, Phys. Lett. A 376 (10-11) (2012) 1011-1019. | ||

In article | View Article | ||

[22] | Z. Huang, Z. Chen, S. Li, Q. Feng, F. Zhang, Y. Du, Effects of size and surface anisotropy on thermal magnetization and hysteresis in the magnetic clusters, Eur. Phys. J. B-Condens. Matter Complex Syst. 51 (1) (2006) 65-73. | ||

In article | View Article | ||

[23] | Ü. Akıncı, Crystal field dilution in S-1 Blume Capel model: hysteresis behaviors, Phys. Lett. A 380 (14-15) (2016) 1352-1357. | ||

In article | View Article | ||

[24] | N. Si, J.M. Wang, A.B. Guo, F. Zhang, F.G. Zhang, W. Jiang, Study on magnetic and thermodynamic characteristics of core-shell graphene nanoribbon, Phys. E Lowdimens. Syst. Nanostruct. (2019) 113884. | ||

In article | View Article | ||

[25] | X.W. Quan, N. Si, F. Zhang, J. Meng, H.L. Miao, Y.L. Zhang, W. Jiang, Phase diagrams of kekulene-like nanostructure, Phys. E Low-dimens. Syst. Nanostruct. 114 (2019) 113574. | ||

In article | View Article | ||

[26] | T. Kaneyoshi, Effects of random fields in an antiferromagnetic Ising bilayer film, Phys. E Low-dimens. Syst. Nanostruct. 94 (2017) 184-189. | ||

In article | View Article | ||

[27] | R. Honmura, T. Kaneyoshi, Contribution to the new type of effective-field theory of the Ising model, J. Phys. C Solid State Phys. 12 (19) (1979) 3979. | ||

In article | View Article | ||

[28] | R. Honmura, T. Kaneyoshi, Contribution to the new type of effective-field theory of the Ising model, J. Phys. C Solid State Phys. 12 (19) (1979) 3979. | ||

In article | View Article | ||

[29] | T. Kaneyoshi, J.W. Tucker, M. Jacur, Differential operator technique for higher spin problems, Phys. A Stat. Mech. Appl. 186 (3-4) (1992) 495-512. | ||

In article | View Article | ||

Published with license by Science and Education Publishing, Copyright © 2021 Chadia Qotni and Hicham Laribou

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Chadia Qotni, Hicham Laribou. Numerical Simulation of the Magnetic Susceptibility of a Spin 1 System Interacting with an Oscillating Magnetic Field. *Applied Mathematics and Physics*. Vol. 9, No. 1, 2021, pp 1-4. http://pubs.sciepub.com/amp/9/1/1

Qotni, Chadia, and Hicham Laribou. "Numerical Simulation of the Magnetic Susceptibility of a Spin 1 System Interacting with an Oscillating Magnetic Field." *Applied Mathematics and Physics* 9.1 (2021): 1-4.

Qotni, C. , & Laribou, H. (2021). Numerical Simulation of the Magnetic Susceptibility of a Spin 1 System Interacting with an Oscillating Magnetic Field. *Applied Mathematics and Physics*, *9*(1), 1-4.

Qotni, Chadia, and Hicham Laribou. "Numerical Simulation of the Magnetic Susceptibility of a Spin 1 System Interacting with an Oscillating Magnetic Field." *Applied Mathematics and Physics* 9, no. 1 (2021): 1-4.

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[1] | A.H. Zaki, M.A. Hafiez, W.M. El Rouby, S.I. El-Dek, A.A. Farghali, Novel magnetic standpoints in Na2Ti3O7 nanotubes, J. Magn. Magn Mater. 476 (2019) 207-212. | ||

In article | View Article | ||

[2] | R.M. Francisco, J.P. Santos, Magnetic properties of the Ashkin-Teller model on a hexagonal nanotube, Phys. Lett. A 383 (11) (2019) 1092-1098. | ||

In article | View Article | ||

[3] | Z. Zhang, Z. Li, J. Zhang, H. Bian, T. Wang, J. Gao, J. Li, Structural and magnetic properties of porous FexOy nanosheets and nanotubes fabricated by electrospinning, Ceram. Int. 45 (1) (2019) 457-461. | ||

In article | View Article | ||

[4] | P. Robkhob, I.M. Tang, S. Thongmee, Magnetic properties of the dilute magnetic semiconductor Zn 1-x Co x O nanoparticles, J. Supercond. Nov. Magnetism (2019) 1-9. | ||

In article | View Article | ||

[5] | A. Gorczyca-Goraj, T. Domanski, M.M. Maska, Topological superconductivity at finite temperatures in proximitized magnetic nanowires, Phys. Rev. B 99 (23) (2019) 235430. | ||

In article | View Article | ||

[6] | Y. Wang, D. Hu, H. Jia, Q. Wang, Efficient enhancement of light trapping in the double-textured Al doped ZnO films with nanorod and crater structures, Phys. B Condens. Matter 565 (2019) 9-13. | ||

In article | View Article | ||

[7] | N.Y. Schmidt, S. Laureti, F. Radu, H. Ryll, C. Luo, F. d’Acapito, M. Albrecht, Structural and magnetic properties of FePt-Tb alloy thin films, Phys. Rev. B 100 (6) (2019), 064428. | ||

In article | View Article | ||

[8] | R. Das, J.A. Cardarelli, M.H. Phan, H. Srikanth, Magnetically tunable iron oxide nanotubes for multifunctional biomedical applications, J. Alloys Compd. 789 (2019) 323-329. | ||

In article | View Article | ||

[9] | A.L. Elrefai, T. Yoshida, K. Enpuku, Magnetic parameters evaluation of magnetic nanoparticles for use in biomedical applications, J. Magn. Magn Mater. 474 (2019) 522-527. | ||

In article | View Article | ||

[10] | N. Lowa, € J.M. Fabert, D. Gutkelch, H. Paysen, O. Kosch, F. Wiekhorst, 3D-printing of novel magnetic composites based on magnetic nanoparticles and photopolymers, J. Magn. Magn Mater. 469 (2019) 456-460. | ||

In article | View Article | ||

[11] | C. Qotni, A. L. Marrakchi, S. Sayouri, Y. Achkar, Effets non linéaires dans l’interaction d’un ensemble de particules de spin 1 avec un champ oscillant dans l'expression de l'aimantation et de la susceptibilité magnétique, I.J.I.A.S. ISSN 2028-9324 vol.17 No. 3 Aug. 2016, pp.1050-1061. | ||

In article | |||

[12] | G. Lochak. C.R.A.S. Série B, A/272. P.1281.1971. | ||

In article | |||

[13] | A. Erbeia , Résoances magnétiques-Masson, Paris, 1969. | ||

In article | |||

[14] | F. Sedighi, M. Esmaeili-Zare, A. Sobhani-Nasab, M. Behpour, Synthesis and characterization of CuWO 4 nanoparticle and CuWO 4/NiO nanocomposite using co-precipitation method; application in photodegradation of organic dye in water, J. Mater. Sci. Mater. Electron. 29 (16) (2018) 13737-13745. | ||

In article | View Article | ||

[15] | B.Z. Mi, C.J. Feng, J.G. Luo, D.Z. Hu, Magnetic compensation and critical properties of a mixed spin-(2, 3/2) Heisenberg single-walled nanotube superlattice, Superlattice. Microst. 113 (2018) 524-533. | ||

In article | View Article | ||

[16] | M.D. Hossain, R.A. Mayanovic, R. Sakidja, M. Benamara, R. Wirth, Magnetic properties of core-shell nanoparticles possessing a novel Fe (II)-chromia phase: an experimental and theoretical approach, Nanoscale 10 (4) (2018) 2138-2147. | ||

In article | View Article PubMed | ||

[17] | D. Lv, W. Jiang, Y. Ma, Z. Gao, F. Wang, Magnetic and thermodynamic properties of a cylindrical ferrimagnetic Ising nanowire with core/shell structure, Phys. E Low-dimens. Syst. Nanostruct. 106 (2019) 101-113. | ||

In article | View Article | ||

[18] | E. Kantar, Ising-type single-segment ferromagnetic nanowire with core/shell: the dependences of the angle, temperature, and geometry, J. Supercond. Nov. Magnetism 31 (2) (2018) 341-346. | ||

In article | View Article | ||

[19] | H. Magoussi, A. Zaim, M. Kerouad, Theoretical investigations of the phase diagrams and the magnetic properties of a random field spin-1 Ising nanotube with core/shell morphology, J. Magn. Magn Mater. 344 (2013) 109-115. | ||

In article | View Article | ||

[20] | A.S. Tehrani, M.A. Kashi, A. Ramazani, A.H. Montazer, Axially adjustable magnetic properties in arrays of multilayered Ni/Cu nanowires with variable segment sizes, Superlattice. Microst. 95 (2016) 38-47. | ||

In article | View Article | ||

[21] | B. Deviren, M. Keskin, Thermal behavior of dynamic magnetizations, hysteresis loop areas and correlations of a cylindrical Ising nanotube in an oscillating magnetic field within the effective-field theory and the Glauber-type stochastic dynamics approach, Phys. Lett. A 376 (10-11) (2012) 1011-1019. | ||

In article | View Article | ||

[22] | Z. Huang, Z. Chen, S. Li, Q. Feng, F. Zhang, Y. Du, Effects of size and surface anisotropy on thermal magnetization and hysteresis in the magnetic clusters, Eur. Phys. J. B-Condens. Matter Complex Syst. 51 (1) (2006) 65-73. | ||

In article | View Article | ||

[23] | Ü. Akıncı, Crystal field dilution in S-1 Blume Capel model: hysteresis behaviors, Phys. Lett. A 380 (14-15) (2016) 1352-1357. | ||

In article | View Article | ||

[24] | N. Si, J.M. Wang, A.B. Guo, F. Zhang, F.G. Zhang, W. Jiang, Study on magnetic and thermodynamic characteristics of core-shell graphene nanoribbon, Phys. E Lowdimens. Syst. Nanostruct. (2019) 113884. | ||

In article | View Article | ||

[25] | X.W. Quan, N. Si, F. Zhang, J. Meng, H.L. Miao, Y.L. Zhang, W. Jiang, Phase diagrams of kekulene-like nanostructure, Phys. E Low-dimens. Syst. Nanostruct. 114 (2019) 113574. | ||

In article | View Article | ||

[26] | T. Kaneyoshi, Effects of random fields in an antiferromagnetic Ising bilayer film, Phys. E Low-dimens. Syst. Nanostruct. 94 (2017) 184-189. | ||

In article | View Article | ||

[27] | R. Honmura, T. Kaneyoshi, Contribution to the new type of effective-field theory of the Ising model, J. Phys. C Solid State Phys. 12 (19) (1979) 3979. | ||

In article | View Article | ||

[28] | |||

In article | View Article | ||

[29] | T. Kaneyoshi, J.W. Tucker, M. Jacur, Differential operator technique for higher spin problems, Phys. A Stat. Mech. Appl. 186 (3-4) (1992) 495-512. | ||

In article | View Article | ||