# A Generalized Ising-like Model for Spin Crossover Nanoparticles

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

^{4}to 3d

^{8}embedded in an organic matrix. They are characterized by two different electronic spin configurations, resulting in two stable, magnetic macrostates, termed high spin (HS) and low spin (LS). LS-HS transitions are triggered by an external constraint. Most of the SCO compounds that have been studied are based on the Fe (II) 3d

^{6}configuration. For the 3d

^{8}case (Ni

^{2+}SCO) and following the work of O.A. Qamar et al. [13], the SCO phenomenon is associated with three different lattice distortions, one of them being from a square planar (LS, S = 0) to a tetrahedral geometry (HS, S = 1).

_{h}) as a metal complex, the degenerated orbitals of the cation split into triply degenerate t

_{2g}and doubly degenerate e

_{g}states. The macrostates are related to the micro-configurations permitted by the distribution of the six electrons in the t

_{2g}and e

_{g}states, mediated by the competition between the pairing electron affinity and the strength of the ligand field, which defines the energy gap between the states. The LS state corresponds to the limiting strong field ligand case, which favors the pairing of the six electrons in the t

_{2g}orbitals (${t}_{2g}^{6}{e}_{g}^{0}$ and spin S = 0), and the HS state corresponds to the limiting weak field ligand case, which allows the distribution of the electrons on both the t

_{2g}and e

_{g}states within the frame of Hund’s rule (${t}_{2g}^{4}{e}_{g}^{2}$ and total spin S = 2). The LS state is thus diamagnetic when neglecting the second-order Zeeman effect, and the SCO appears in a deeply colored state, while the HS state is a paramagnetic, fainter-colored state. On top of that, the Fe-ligand distance is greater in the HS state compared to the LS state.

## 2. The Model

## 3. Entropic Sampling

_{i}the probability of obtaining the macrostate $\left(\mathrm{m},{s}_{HH},{s}_{HL},{s}_{LL}\right)$, shown as:

_{i}, (s

_{HH})

_{i}, (s

_{HL})

_{i}, (s

_{LL})

_{i}, to the macrostate (j).

## 4. Numerical Results and Analysis

_{2}(NCS)

_{2}], btr = 4,4′-bis-1,2,4-triazole [16] is used. From the enthalpy H measurements, we derived the value of the energy gap as ∆/k

_{B}= 1300 K. The equilibrium temperature T

_{eq}is deduced from the evolution of ∆H and is fixed at 216.3 K, from which the entropy ∆S = ∆H/T

_{eq}is calculated to be ∆S= 50 J/K/mol. Recall that according to the previous model, the expression of T

_{eq}is given by ${T}_{eq}=\frac{\Delta}{{k}_{B}\mathrm{ln}\left(g\right)}$.

_{up}is the ascending thermal transition temperature, T

_{down}is the descending thermal transition temperature and T

_{eq}= T

_{1/2}is the average temperature between T

_{up}and T

_{down}in which the HS fraction is equal to 1/2. The hysteresis width is defined as $\Delta T={T}_{up}-{T}_{down}$.

- Effect of the Interaction Parameters

`o`- The Case x = J
_{HH}/J_{LL}= 1.0

_{eq}is independent of the $J/{k}_{B}$ values and remains equal to ${T}_{eq}=\Delta /\left({k}_{B}\mathrm{ln}\left(g\right)\right)$ = 216.3 K.

`o`- The Case of x = J
_{HH}/J_{LL}= 0.4

_{eq}(J/k

_{B}) variation gives: $T(J/{k}_{B})=0.24072\times J/{k}_{B}+216.30$.

_{up}) and in the cooling mode (T

_{down}), clearly show that the width of the hysteresis loop ∆T = T

_{up}− T

_{down}increases when the $J/{k}_{B}$ interaction is stronger. Linear regression leads for the Teq Branch to: $T(J/{k}_{B})=0.26019\times J/{k}_{B}+216.30$.

- Effects of the Variation of x = J
_{HH}/J_{LL}

_{eq}toward higher values. The equilibrium temperature, which is equal to 216.3 K when x = 1, reaches the value of 221.85 K when $x=0.2$. The values of the equilibrium temperatures associated with the different $x$ values are reported in the last column of Table 4.

_{eq}gradually moves toward the order-disorder temperature T

_{O.D.}. The hysteresis width decreases and vanishes for weaker $J/{k}_{B}$ values when T

_{eq}becomes greater than T

_{O.D.}.

## 5. Conclusions

_{eq}remains constant. The nature of the transition from LS to HS configuration is governed by the intensity of the interactions and, therefore, the value of the $J/{k}_{B}$ and $G/{k}_{B}$ parameters. Increasing the short-range interaction parameter $J/{k}_{B}$ leads to a hysteretic transition. This behavior is explained by the fact that the Curie (or the order-disorder) temperature designed by T

_{O.D.}increases. When $x$ is set to a value other than 1, a “pseudo-dilution” effect is simulated by gradually reducing $J/{k}_{B}$. Figure 4 and Figure 6 highlight that the equilibrium temperature shows a linear decrease towards $\Delta /\left({k}_{B}\mathrm{ln}\left(g\right)\right)$ = 216.3 K. Moreover, the hysteretic behavior vanishes at a threshold value of ${J}_{c}/{k}_{B}\approx 16K$ and ${J}_{c}/{k}_{B}\approx 15K$ for 6 × 6 and 10 × 10 systems, respectively. This feature is connected to the relative positions of T

_{eq}and T

_{O.D.}and can be explained by the fact that the order-disorder transition T

_{O.D.}decreases faster for small lattices. The condition T

_{eq}> T

_{O.D.}leads to a gradual transition. In the present study, we mainly focused on the effect of varying $x$ when $J/{k}_{B}$ is constant. An important result is that when the difference between ${J}_{HH}/{k}_{B}$ and ${J}_{LL}/{k}_{B}$ is increased, the LS state is stabilized and T

_{eq}is shifted toward higher temperatures. This behavior is more significant for larger lattices. The effects of the interactions between surface molecules and their environment will also be explored in future work.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Coronado, E. Molecular magnetism: From chemical design to spin control in molecules, materials and devices. Nat. Rev. Mater.
**2020**, 5, 87–104. [Google Scholar] [CrossRef] - Gütlich, P.; Hauser, A.; Spiering, H. Thermal and optical switching of Iron(II) Complexes. Angew. Chem.
**1994**, 33, 2024–2054. [Google Scholar] [CrossRef] - Sanvito, S. Molecular Spintronics. Chem. Soc. Rev.
**2011**, 40, 3336–3355. [Google Scholar] [CrossRef] [PubMed] - Decurtins, S.; Gütlich, P.; Köhler, C.P.; Spiering, H.; Hauser, A. Light-induced excited spin state trapping in a transition-metal complex: The hexa-1-propyltetrazole-iron (II) tetrafluoroborate spin-crossover system. Chem. Phys. Lett.
**1984**, 105, 1–4. [Google Scholar] [CrossRef] - Linares, J.; Codjovi, E.; Garcia, Y. Pressure and Temperature Spin Crossover Sensors with Optical Detection. Sensors
**2012**, 12, 4479–4492. [Google Scholar] [CrossRef] - Boukheddaden, K.; Ritti, M.H.; Bouchez, G.; Sy, M.; Dîrtu, M.M.; Parlier, M.; Linares, J.; Garcia, Y. Quantitative contact pressure sensor based on spin crossover mechanism for civil security applications. J. Phys. Chem. C
**2018**, 122, 7597–7604. [Google Scholar] [CrossRef] - Spiering, H. Elastic interaction in spin-crossover compounds. In Spin Crossover in Transition Metal Compounds III Topics in Current Chemistry; Springer: Berlin/Heidelberg, Germany, 2004; Volume 235, pp. 171–195. [Google Scholar] [CrossRef]
- Boukheddaden, K.; Linares, J.; Spiering, H.; Varret, F. One-dimensional Ising-like systems: An analytical investigation of the static and dynamic properties, applied to spin-crossover relaxation. Eur. Phys. J. B
**2000**, 15, 317–326. [Google Scholar] [CrossRef] - Boukheddaden, K.; Linares, J.; Varret, F. Thermodynamic properties of coupled mixed-valence molecules using a cooperative PKS theory: A second-order localized-delocalized transition. Chem. Phys.
**1993**, 172, 239–245. [Google Scholar] [CrossRef] - Constant-Machado, H.; Stancu, A.; Linares, J.; Varret, F. Thermal hysteresis loops in spin-crossover compounds analyzed in terms of classical Preisach model. IEEE Trans. Mag.
**1998**, 34, 2213–2219. [Google Scholar] [CrossRef] - Linares, J.; Jureschi, C.M.; Boukheddaden, K. Surface effects leading to unusual size dependence of the thermal hysteresis behavior in spin-crossover nanoparticles. Magnetochemistry
**2016**, 2, 24. [Google Scholar] [CrossRef] [Green Version] - Rotaru, A.; Linares, J.; Varret, F.; Codjovi, E.; Slimani, A.; Tanasa, R.; Enachescu, C.; Stancu, A.; Haasnoot, J. Pressure effect investigated with first-order reversal-curve method on the spin-transition compounds [FexZn
_{1−x}(btr)_{2}(NCS)_{2}] ·H_{2}O (x = 0.6,1). Phys. Rev. B**2011**, 83, 224107–224114. [Google Scholar] [CrossRef] - Qamar, O.A.; Cong, C.; Ma, H. Solid state mononuclear divalent nickel spin crossover complexes. Dalton Trans.
**2020**, 49, 17106–17114. [Google Scholar] [CrossRef] [PubMed] - Kahn, O. Molecular Magnetism; Wiley-VCH: New York, NY, USA, 1993. [Google Scholar]
- Shepherd, H.J.; Bonnet, S.; Guionneau, P.; Bedoui, S.; Garbarino, G.; Nicolazzi, W.; Bousseksou, A.; Molnar, G. Pressure-induced two-step transition with structural symmetry breaking: X-ray diffraction, magnetic, and Ramman studies. Phys. Rev. B
**2011**, 54, 144107–144115. [Google Scholar] [CrossRef] [Green Version] - Krober, J.; Audière, J.P.; Claude, R.; Codjovi, E.; Kahn, O.; Hassnoot, J.; Grolière, F.; Jay, C.; Bousseksou, A.; Linares, J.; et al. Spin Transitions and Thermal Hysteresis in the Molecular-Based Materials [Fe(Htrz)2(trz)](BF4) and [Fe(Htrz)3](BF4)2.cntdot.H2O (Htrz = 1,2,4-4H-triazole; trz = 1,2,4-triazolato). Chem. Mater.
**1994**, 6, 1404–1412. [Google Scholar] [CrossRef] - Varret, F.; Slimani, A.; Boukheddaden, K.; Chong, C.; Mishra, H.; Collet, E.; Haasnoot, J.; Pillet, S. The Propagation of the Thermal Spin Transition of [Fe(Btr)2(NCS)2]_H2O Single Crystals Observed by Optical Microscopy. New J. Chem.
**2011**, 35, 2333–2340. [Google Scholar] [CrossRef] - Loutete-Dangui, E.D.; Codjovi, E.; Tokoro, H.; Dahoo, P.R.; Ohkoshi, S.; Boukheddaden, K. Spectroscopic ellipsometry investigations of the thermally induced first-order transition of RbMn[Fe(CN)6]. Phys. Rev. B
**2008**, 78, 014303–014312. [Google Scholar] [CrossRef] - Pillet, S.; Hubsch, J.; Lecomte, C. Single crystal diffraction analysis of the thermal spin conversion in [Fe(btr)
_{2}(NCS)_{2}](H_{2}O): Evidence for spin-like domain formation. Eur. Phys. J. B**2004**, 38, 541–552. [Google Scholar] [CrossRef] - Roubeau, O.; Castro, M.; Burriel, R.; Haasnoot, J.G.; Reedijk, J. Calorimetric Investigation of Triazole-Bridged Fe(II) Spin-Crossover One-Dimensional Materials: Measuring the Cooperativity. J. Phys. Chem. B
**2011**, 115, 3003–3012. [Google Scholar] [CrossRef] - Rotaru, A.; Dîrtu, M.M.; Enachescu, C.; Tanasa, R.; Linares, J.; Stancu, A.; Garcia, Y. Calorimetric measurements of diluted spin crossover complexes [Fe
_{x}M_{1−x}(btr)_{2}(NCS)_{2}]·H_{2}O with MII = Zn and Ni. Polyhedron**2009**, 28, 2531–2536. [Google Scholar] [CrossRef] - Rotaru, A.; Varret, F.; Codjovi, E.; Boukheddaden, K.; Linares, J.; Stancu, A.; Guionneau, P.; Létard, J.F. Hydrostatic pressure investigation of the spin crossover compound [Fe(PM-BIA)2(NCS)2] polymorph I using reflectance detection. J. Appl. Phys.
**2009**, 106, 053515–053520. [Google Scholar] [CrossRef] - Rotaru, G.M.; Codjovi, E.; Dahoo, P.R.; Maurin, I.; Linares, J.; Rotaru, A. Monotoring-Spin-Crossover Properties by Diffussed Reflectivity. Symmetry
**2021**, 13, 1148. [Google Scholar] [CrossRef] - Wajnflasz, J.; Pick, R. Transitions “low spin”—“high spin” dans les complexes de Fe
^{2+}. J. Phys. Colloques**1971**, 32, 91–92. [Google Scholar] [CrossRef] - Bousseksou, A.; Nasser, J.; Linares, J.; Boukheddaden, K.; Varret, F. Ising-like model for the two-step spin-crossover. J. Phys. I France
**1992**, 2, 1381–1403. [Google Scholar] [CrossRef] - Linares, J.; Spiering, H.; Varret, F. Analytical solution of 1d ising-like systems modified by weak long-range interaction—Application to spin crossover compounds. Eur. Phys. J. B
**1999**, 10, 271–275. [Google Scholar] [CrossRef] - Chiruta, D.; Jureschi, C.-M.; Linares, J.; Dahoo, P.R.; Garcia, Y.; Rotaru, A. On the origin of multi-step spin transition behaviour in 1d nanoparticles. Eur. Phys. J. B
**2015**, 88, 233. [Google Scholar] [CrossRef] - Ndiaye, M.; Singh, Y.; Fourati, H.; Sy, M.; Lo, B.; Boukheddaden, K. Isomorphism between the electro-elastic modeling of the spin transition and Ising-like model with competing interactions: Elastic generation of self-organized spin states. J. App. Phys.
**2021**, 129, 153901–153921. [Google Scholar] [CrossRef] - Paez-Espejo, M.; Sy, M.; Boukheddaden, K. Elastic frustration causing two-step and multistep transitions in spin crossover solids: Emergence of complex Aantiferroelastic structures. J. Am. Chem. Soc.
**2016**, 138, 3202–3210. [Google Scholar] [CrossRef] - Shteto, I.; Linares, J.; Varret, F. Monte carlo entropic sampling for the study of metastable states and relaxation paths. Phys. Rev. E
**1997**, 56, 5128–5137. [Google Scholar] [CrossRef] - Linares, J.; Cazelles, C.; Dahoo, P.R.; Boukheddaden, K. A first-order phase transition studied by an Ising-like model solved by entropic Sampling Monte Carlo Method. Symmetry
**2021**, 13, 587. [Google Scholar] [CrossRef] - Linares, J.; Enachescu, C.; Boukheddaden, K.; Varret, F. Monte Carlo entropic sampling applied to spin crossover solids: The squareness of the thermal hysteresis loop. Polyhedron
**2003**, 22, 2453–2456. [Google Scholar] [CrossRef] - Constant-Machado, H.; Linares, J.; Varret, F.; Haasnoot, J.G.; Martin, J.P.; Zarembowitch, J.; Dworkin, A.; Bousseksou, A. Dilution effects in a spin crossover system, modelled in terms of direct and indirect intermolecular interactions. J. Phys. I Fr.
**1996**, 6, 1203–1216. [Google Scholar] [CrossRef] [Green Version] - Martin, J.P.; Zarembowitch, J.; Bousseksou, A.; Dworkin, A.; Haasnoot, J.G.; Varret, F. Solid State Effects on Spin Transitions: Magnetic, Calorimetric, and Moessbauer-Effect Properties of [FexCo
_{1−x}(4,4′-bis-1,2,4-triazole)_{2}(NCS)_{2}].cntdot.H_{2}O Mixed-Crystal Compounds. Inorg. Chem.**1994**, 33, 6325–6333. [Google Scholar] [CrossRef]

**Figure 1.**Thermal evolution of the HS fraction in a SCO system with a size of 6 × 6 for different values of the average interaction parameter $J/{k}_{B}$: $J/{k}_{B}=12K$ (red square), $J/{k}_{B}$ = 14 K (blue square), $J/{k}_{B}$ = 16 K (orange square), $J/{k}_{B}$ = 22 K (green square) and $J/{k}_{B}$ = 25 K (black square). The computational parameters are: $\Delta /{k}_{B}$ = 1300 K, $G/{k}_{B}$ = 172.2 K, $x={J}_{HH}/{J}_{LL}=1.0$ and ln(g) = 6.01.

**Figure 2.**Phase diagram T = f (J/k

_{B}) for a 6 × 6 square lattice. The red and blue squares correspond, respectively, to the upper and lower transitions for the heating (T

_{up}) and cooling (T

_{down}) temperatures of the thermal HS fraction. The black squares correspond to the equilibrium temperatures (T

_{eq}). The computational parameters are: $\Delta /{k}_{B}$ = 1300 K, $G/{k}_{B}$ = 172.2 K, $x={J}_{HH}/{J}_{LL}=1.0$ and ln(g) = 6.01.

**Figure 3.**Thermal evolution of the HS fraction in a SCO system with a size of 6 × 6 for different values of the average interaction parameter $J/{k}_{B}$:$J/{k}_{B}=19$ K (magenta square), $J/{k}_{B}$ = 18 K (dark yellow square), $J/{k}_{B}$ = 17 K (red square), $J/{k}_{B}$ = 16 K (purple square), $J/{k}_{B}$ = 15 K (black square), $J/{k}_{B}=13$ K (green square), $J/{k}_{B}=12$ K (orange square) and $J/{k}_{B}=11$ K (blue square). The computational parameters are: $\Delta /{k}_{B}$ = 1300 K, $G/{k}_{B}$ = 172.2 K, $x={J}_{HH}/{J}_{LL}=0.4$ and ln(g) = 6.01.

**Figure 4.**Temperature T versus J/k

_{B}for a 6 × 6 square lattice. The red (T

_{up}) and blue (T

_{down}) squares correspond, respectively, to the transition temperatures of the upper and lower branches of thermal hysteresis. The black squares correspond to the equilibrium temperatures (T

_{eq}) of the gradual transition region. The computational parameters are: $\Delta /{k}_{B}$ = 1300 K, $G/{k}_{B}$ = 172.2 K, $x=({J}_{HH}/{k}_{B})/({J}_{LL}/{k}_{B})=0.4$ and ln(g) = 6.01.

**Figure 5.**Thermal evolution of the HS fraction in a SCO system with a size of 10 × 10 for different values of the average interaction parameter $J/{k}_{B}$:$J/{k}_{B}=19$ K (magenta square), $J/{k}_{B}$ = 18 K (dark yellow square), $J/{k}_{B}$ = 17 K (red square), $J/{k}_{B}$ = 16 K (purple square), $J/{k}_{B}$ = 15 K (black square), $J/{k}_{B}=13$ K (green square), $J/{k}_{B}=12$ K (orange square), $J/{k}_{B}=11$ K (blue square). The computational parameters are: $\Delta /{k}_{B}$ = 1300 K, $G/{k}_{B}$ = 172.2 K, $x={J}_{HH}/{J}_{LL}=0.4$ and ln(g) = 6.01.

**Figure 6.**Phase diagram T = f (J/k

_{B}) for a 10 × 10 square lattice. The red and blue squares correspond, respectively, to the upper and lower transitions for the heating (T

_{up}) and cooling (T

_{down}) temperatures of the thermal HS fraction. The black squares correspond to the equilibrium temperatures (T

_{eq}). The computational parameters are: $\Delta /{k}_{B}$ = 1300 K, $G/{k}_{B}$ = 172.2 K, $x=({J}_{HH}/{J}_{LL})=0.4$ and ln(g) = 6.01.

**Figure 7.**Thermal evolution of the HS fraction in a SCO system with a size of 6 × 6 for different values of the $x$ parameter: $x$ = 1.0 (green square), $x$ = 0.6 (purple square) and $x$ = 0.2 (red square). The computational parameters are: $\Delta /{k}_{B}$ = 1300 K, $J/{k}_{B}$ = 14.8 K, $G/{k}_{B}$ = 172.2 K and ln(g) = 6.01.

**Figure 8.**Thermal evolution of the HS fraction in a SCO system with a size of 10 × 10 for different values of the $x$ parameter: $x$ = 0.2 (red square), $x$ = 0.6 (purple square) and $x$ = 1.0 (green square). The computational parameters are: $\Delta /{k}_{B}$ = 1300 K, $J/{k}_{B}$ = 14.8 K, $G/{k}_{B}$ = 172.2 K and ln(g) = 6.01.

**Table 1.**Values of the thermal hysteresis width $\Delta {T}_{hyst}={T}_{up}-{T}_{down}$ and the equilibrium temperature ${T}_{eq}$ as a function of $J/{k}_{B}$, ${J}_{HH}/{k}_{B}$ and ${J}_{LL}/{k}_{B}$ in a SCO system with a size of 6 × 6. The computational parameters are: $\Delta /{k}_{B}=$ 1300 K, $G/{k}_{B}=$ 172.2 K, $x={J}_{HH}/{J}_{LL}=0.4$ and ln(g) = 6.01.

$\mathit{J}/{\mathit{k}}_{\mathit{B}}\left[\mathbf{K}\right]$ | ${\mathit{J}}_{\mathit{H}\mathit{H}}/{\mathit{k}}_{\mathit{B}}\left[\mathbf{K}\right]$ | ${\mathit{J}}_{\mathit{L}\mathit{L}}/{\mathit{k}}_{\mathit{B}}\left[\mathbf{K}\right]$ | $\left({\mathit{J}}_{\mathit{L}\mathit{L}}-{\mathit{J}}_{\mathit{H}\mathit{H}}\right)/{\mathit{k}}_{\mathit{B}}\left[\mathbf{K}\right]$ | $\Delta {\mathit{T}}_{\mathit{h}\mathit{y}\mathit{s}\mathit{t}}\left[\mathbf{K}\right]$ | ${\mathit{T}}_{\mathit{e}\mathit{q}}\left[\mathbf{K}\right]$ |
---|---|---|---|---|---|

19 | 10.86 | 27.14 | 16.28 | 1.03 | 220.86 |

18 | 10.28 | 25.71 | 15.43 | 0.62 | 220.60 |

17 | 9.72 | 24.29 | 14.57 | 0.33 | 220.37 |

16 | 9.14 | 22.86 | 13.72 | 0.10 | 220.13 |

15 | 8.57 | 21.43 | 12.86 | 0 | 219.90 |

13 | 7.43 | 18.58 | 11.15 | 0 | 219.43 |

12 | 6.85 | 17.14 | 10.29 | 0 | 219.18 |

11 | 6.28 | 15.71 | 9.42 | 0 | 218.93 |

**Table 2.**Evolution of the thermal hysteresis width, $\Delta {T}_{hyst}={T}_{up}-{T}_{down}$, and the equilibrium temperature, ${T}_{eq}$, as a function of $J/{k}_{B}$, ${J}_{HH}/{k}_{B}$ and ${J}_{LL}/{k}_{B}$ in a SCO system with a size of 10 × 10. The computational parameters are: $\Delta /{k}_{B}$ = 1300 K, $G/{k}_{B}$ = 172.2 K, $x={J}_{HH}/{J}_{LL}$ and ln(g) = 6.01.

$\mathit{J}/{\mathit{k}}_{\mathit{B}}\left[\mathbf{K}\right]$ | ${\mathit{J}}_{\mathit{H}\mathit{H}}/{\mathit{k}}_{\mathit{B}}\left[\mathbf{K}\right]$ | ${\mathit{J}}_{\mathit{L}\mathit{L}}/{\mathit{k}}_{\mathit{B}}\left[\mathbf{K}\right]$ | $({\mathit{J}}_{\mathit{L}\mathit{L}}/{\mathit{k}}_{\mathit{B}}-{\mathit{J}}_{\mathit{H}\mathit{H}}/{\mathit{k}}_{\mathit{B}})\left[\mathbf{K}\right]$ | $\Delta {\mathit{T}}_{\mathit{h}\mathit{y}\mathit{s}\mathit{t}}\left[\mathbf{K}\right]10\times 10$ | ${\mathit{T}}_{\mathit{e}\mathit{q}}\left[\mathbf{K}\right]10\times 10$ |
---|---|---|---|---|---|

19 | 10.837 | 27.094 | 16.257 | 1.57 | 221.244 |

18 | 10.267 | 25.669 | 15.402 | 1.13 | 220.941 |

17 | 9.697 | 24.244 | 14.547 | 0.71 | 220.709 |

16 | 9.125 | 22.814 | 13.689 | 0.38 | 220.463 |

15 | 8.555 | 21.389 | 12.834 | 0.09 | 220.160 |

13 | 7.415 | 18.539 | 11.124 | 0 | 219.658 |

12 | 6.845 | 17.114 | 10.269 | 0 | 219.409 |

11 | 6.274 | 15.685 | 9?411 | 0 | 219.149 |

$\mathit{J}/{\mathit{k}}_{\mathit{B}}\left[\mathbf{K}\right]$ | $\Delta {\mathit{T}}_{\mathit{h}\mathit{y}\mathit{s}\mathit{t}}\left[\mathbf{K}\right]6\times 6$ | ${\mathit{T}}_{\mathit{e}\mathit{q}}\left[\mathbf{K}\right]6\times 6$ | $\Delta {\mathit{T}}_{\mathit{h}\mathit{y}\mathit{s}\mathit{t}}\left[\mathbf{K}\right]10\times 10$ | ${\mathit{T}}_{\mathit{e}\mathit{q}}\left[\mathbf{K}\right]10\times 10$ |
---|---|---|---|---|

19 | 1.03 | 220.86 | 1.57 | 221.244 |

18 | 0.62 | 220.60 | 1.13 | 220.941 |

17 | 0.33 | 220.37 | 0.71 | 220.709 |

16 | 0.10 | 220.13 | 0.38 | 220.463 |

15 | 0 | 219.90 | 0.09 | 220.160 |

13 | 0 | 219.43 | 0 | 219.658 |

12 | 0 | 219.18 | 0 | 219.409 |

11 | 0 | 218.93 | 0 | 219.149 |

**Table 4.**Equilibrium temperatures and hysteresis widths calculated for different values of $x={J}_{HH}/{J}_{LL}$ parameter in a 6 × 6 square lattice. The computational parameters are: $\Delta /{k}_{B}$ = 1300 K, $J/{k}_{B}$ = 14.8 K, $G/{k}_{B}$ = 172.2 K and ln(g) = 6.01.

$\mathit{J}/{\mathit{k}}_{\mathit{B}}\left[\mathbf{K}\right]$ | $\mathit{x}$ | ${\mathit{J}}_{\mathit{L}\mathit{L}}/{\mathit{k}}_{\mathit{B}}\left[\mathbf{K}\right]$ | ${\mathit{J}}_{\mathit{H}\mathit{H}}/{\mathit{k}}_{\mathit{B}}\left[\mathbf{K}\right]$ | ${\mathit{J}}_{\mathit{H}\mathit{L}}/{\mathit{k}}_{\mathit{B}}\left[\mathbf{K}\right]$ | $\Delta {\mathit{T}}_{\mathit{h}\mathit{y}\mathit{s}\mathit{t}}\left[\mathbf{K}\right]$ | ${\mathit{T}}_{\mathit{e}\mathit{q}}\left[\mathbf{K}\right]$ |
---|---|---|---|---|---|---|

14.8 | 1.0 | 14.8 | 14.8 | 14.8 | 0.11 | 216.30 |

14.8 | 0.6 | 18.5 | 11.09 | 14.8 | 0 | 218.38 |

14.8 | 0.2 | 24.66 | 4.93 | 14.8 | 0 | 221.85 |

**Table 5.**Equilibrium temperatures and hysteresis widths calculated for different values of the $x=({J}_{HH}/{k}_{B})/({J}_{LL}/{k}_{B})$ parameter in a 10 × 10 square lattice. The computational parameters are: $\Delta /{k}_{B}$ = 1300 K, $J/{k}_{B}$ = 14.8 K, $G/{k}_{B}$ = 172.2 K and ln(g) = 6.01.

$\mathit{J}/{\mathit{k}}_{\mathit{B}}\left[\mathbf{K}\right]$ | $\mathit{x}$ | ${\mathit{J}}_{\mathit{L}\mathit{L}}/{\mathit{k}}_{\mathit{B}}\left[\mathbf{K}\right]$ | ${\mathit{J}}_{\mathit{H}\mathit{H}}/{\mathit{k}}_{\mathit{B}}\left[\mathbf{K}\right]$ | ${\mathit{J}}_{\mathit{H}\mathit{L}}/{\mathit{k}}_{\mathit{B}}\left[\mathbf{K}\right]$ | $\Delta {\mathit{T}}_{\mathit{h}\mathit{y}\mathit{s}\mathit{t}}\left[\mathbf{K}\right]$ | ${\mathit{T}}_{\mathit{e}\mathit{q}}\left[\mathbf{K}\right]$ |
---|---|---|---|---|---|---|

14.8 | 1.0 | 14.8 | 14.8 | 14.8 | 6.10 | 216.30 |

14.8 | 0.6 | 18.48 | 11.087 | 14.78 | 5.16 | 219.88 |

14.8 | 0.2 | 24.59 | 4.92 | 14.75 | 0 | 222.29 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cazelles, C.; Linares, J.; Dahoo, P.-R.; Boukheddaden, K.
A Generalized Ising-like Model for Spin Crossover Nanoparticles. *Magnetochemistry* **2022**, *8*, 49.
https://doi.org/10.3390/magnetochemistry8050049

**AMA Style**

Cazelles C, Linares J, Dahoo P-R, Boukheddaden K.
A Generalized Ising-like Model for Spin Crossover Nanoparticles. *Magnetochemistry*. 2022; 8(5):49.
https://doi.org/10.3390/magnetochemistry8050049

**Chicago/Turabian Style**

Cazelles, Catherine, Jorge Linares, Pierre-Richard Dahoo, and Kamel Boukheddaden.
2022. "A Generalized Ising-like Model for Spin Crossover Nanoparticles" *Magnetochemistry* 8, no. 5: 49.
https://doi.org/10.3390/magnetochemistry8050049