# On the Distribution of Magnetic Moments in a System of Magnetic Nanoparticles

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## Abstract

**:**

## 1. Introduction

## 2. Fittings to the Magnetization Curves

#### 2.1. Samples

#### 2.2. Models

#### 2.3. Fittings

## 3. Approximation to the Actual Magnetic Moment Distribution of an Ideal Sample

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Formula for the Magnetization

**H**. The partition function associated to this Hamiltonian is

## References

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**Figure 1.**Fittings to the magnetization curves of the Mag300-1h, Mag300-2h, Mag500-2h and Mag600-2h samples using the models (M) Modified Langevin function, (L) lognormal, (G) gamma and (W) Weibull. For each sample, the fitted curves are very similar. For this reason we only show the best-fitted curve for the last three samples, i.e., the model which has the lowest Akaike information criterion (AIC) value. Experimental measurements were performed at room temperature (300 K).

**Figure 2.**Representation of the probability density functions used to fit the magnetization curves of the Mag300-1h, Mag300-2h, Mag500-2h and Mag600-2h samples.

**Figure 3.**Representation of the cumulative distribution function F($\mu $) (continuous solid line) associated to the probability density function f($\mu $), defined in Equation (15), and the step function ${F}_{A,n}\left(\mu \right)$ with $A={10}^{5}\phantom{\rule{4pt}{0ex}}{\mu}_{B}$ and n = 500, defined in Equation (11), which approximates to F($\mu $).

**Figure 4.**Representation of the probability density function f($\mu $) (continuous solid line), defined in Equation (15), and its approximation, obtained from taking a numerical derivative of ${F}_{A,n}\left(\mu \right)$ with $A={10}^{5}{\mu}_{B}$ and n = 500.

**Table 1.**Parameters of the models (M) Modified Langevin function, (L) Lognormal, (G) Gamma and (W) Weibull obtained by fitting the magnetization curve of the Mag300-1h, Mag300-2h, Mag500-2h and Mag600-2h samples. The mean magnetic moment (in Bohr magnetons, $\mu $B), the coefficient of determination R and the Akaike information criterion (AIC) value are also shown. Although models (L), (G) and (W) do not have an ${M}_{S}$ parameter, it is convenient to define ${M}_{S}=c\langle \mu \rangle $ in these cases.

Model | $\mathit{\chi}\phantom{\rule{0.277778em}{0ex}}(\times {10}^{-5}\phantom{\rule{4pt}{0ex}}{\mathbf{cm}}^{3}/\mathbf{g})$ | M${}_{\mathit{S}}$ (emu/g) | $\mathit{\lambda}\phantom{\rule{0.277778em}{0ex}}\left({\mathit{\mu}}_{\mathit{B}}\right)$ | $\mathit{\beta}$ | $\mathit{c}\phantom{\rule{0.277778em}{0ex}}(\times {10}^{16}\phantom{\rule{3.33333pt}{0ex}}{\mathbf{g}}^{-1})$ | $\u2329\mathit{\mu}\u232a\phantom{\rule{0.277778em}{0ex}}\left({\mathit{\mu}}_{\mathit{B}}\right)$ | ${\mathit{R}}^{2}$ | AIC |
---|---|---|---|---|---|---|---|---|

Mag300-1h | ||||||||

(M) | 4.9 | 0.47 | 11,042.3 | 0.999133 | −750.955 | |||

(L) | 4.5 | 0.54 | 3133.78 | 1.03 | 1.09 | 5327.46 | 0.999261 | −774.827 |

(G) | 4.5 | 0.24 | 12,531.1 | 0.02 | 24.0 | 106.96 | 0.999264 | −775.482 |

(W) | 4.5 | 0.56 | 1912.33 | 0.57 | 1.96 | 3062.9 | 0.999264 | −775.337 |

Mag300-2h | ||||||||

(M) | 5.0 | 10.2 | 42,216.1 | 0.999860 | −222.430 | |||

(L) | 4.3 | 10.3 | 28,108.2 | 0.69 | 3.12 | 35,634.2 | 0.999898 | −271.347 |

(G) | 4.2 | 10.3 | 17,798.2 | 1.95 | 3.20 | 34,762.0 | 0.999896 | −269.014 |

(W) | 4.2 | 10.3 | 37,355.8 | 1.40 | 3.28 | 34,033.7 | 0.999896 | −268.319 |

Mag500-2h | ||||||||

(M) | 16.7 | 60.5 | 14,924.9 | 0.998594 | 348.987 | |||

(L) | 15.7 | 60.7 | 14,376.2 | 0.19 | 44.7 | 14,640.8 | 0.998595 | 350.913 |

(G) | 15.6 | 60.7 | 580.66 | 25.2 | 44.8 | 14,614.0 | 0.998595 | 350.908 |

(W) | 15.2 | 60.8 | 15,757.3 | 5.16 | 45.2 | 14,494.8 | 0.998596 | 350.880 |

Mag600-2h | ||||||||

(M) | 63.2 | 57.7 | 10,937.0 | 0.999038 | 628.309 | |||

(L) | 45.1 | 61.1 | 6405.28 | 0.68 | 81.6 | 8071.44 | 0.999162 | 608.134 |

(G) | 39.1 | 62.6 | 5723.98 | 1.14 | 103 | 6545.34 | 0.999176 | 605.332 |

(W) | 40.1 | 62.3 | 6898.42 | 1.10 | 101 | 6657.45 | 0.999177 | 605.160 |

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**MDPI and ACS Style**

Rodríguez, M.J.J.; Vieira, D.S.; Nery, R.C.; Dias, G.S.; Santos, I.A.d.; Mendes, R.d.S.; Cotica, L.F.
On the Distribution of Magnetic Moments in a System of Magnetic Nanoparticles. *Magnetochemistry* **2022**, *8*, 129.
https://doi.org/10.3390/magnetochemistry8100129

**AMA Style**

Rodríguez MJJ, Vieira DS, Nery RC, Dias GS, Santos IAd, Mendes RdS, Cotica LF.
On the Distribution of Magnetic Moments in a System of Magnetic Nanoparticles. *Magnetochemistry*. 2022; 8(10):129.
https://doi.org/10.3390/magnetochemistry8100129

**Chicago/Turabian Style**

Rodríguez, Max Javier Jáuregui, Denner Serafim Vieira, Renato Cardoso Nery, Gustavo Sanguino Dias, Ivair Aparecido dos Santos, Renio dos Santos Mendes, and Luiz Fernando Cotica.
2022. "On the Distribution of Magnetic Moments in a System of Magnetic Nanoparticles" *Magnetochemistry* 8, no. 10: 129.
https://doi.org/10.3390/magnetochemistry8100129