# Study of Quasi-Static Magnetization with the Random-Field Ising Model
## 4 Results and Discussion
Each pixel/square of the image represents a single spin operator; positive values of the random field (\(h_{\mathrm{i}}>0\)) are marked in red, while negative ones (\(h_{\mathrm{i}}<0\)) are marked in blue. Some clusters are exceptionally highlighted in yellow to show their geometry. A greater magnetic order of the magnetic system and thus a smaller standard deviation of the random field component in each site of the model space affected the exchange energy of the localized spins in the considered site. The ratio of the strength of exchange interactions to the random field \(h_{\rm i}\) was greater, and therefore the energy needed to overcome the magnetic order in each site was also greater. Therefore, the coercive field was the highest when the influence of the random field component in the model space was the lowest. A magnetic system with higher disorder tends to lower magnetic susceptibility and reduces the coercive field as well [19]. Hence, smooth hysteresis loops significantly depend on the symmetrical shape of the \(h_{\rm i}\) distribution rather than the nearest neighbor interactions. Figure 5: Dependence of dynamic changes in magnetization \(\mathrm{d}M/\mathrm{d}H\) of the model on the external field step \(\Delta H\) (\(D=2\), \(L=5000\), \(R=2.00\)). Figure 6: Normalized hysteresis loops \(M(H)\) for given values of disorder \(R\) (\(D=2\), \(L=5000\), \(\Delta H=0.001\)). Figures 7a-d and 8a-d respectively present mapping of the 2D and the 3D metastable states of magnetization due to spin flipping. Single-colored areas (the colors are attributed randomly) refer to regions in which the magnetization processes were avalanche-like. Flip of spins in these areas occurred under conditions of the temporarily determined value of the external field \(H\). Each single spin flip for a given field value was marked with the same color of pixels/cubes on the 2D and 3D maps. Figure 7: (**a–d**) Two-dimensional maps (\(D=2\), \(L=5000\)) illustrating the dynamics of the magnetization process for various values of the external field step \(\Delta H=\{0.1\), \(0.01\), \(0.001\), \(0.0001\)}. Single-colored areas represent spins that were flipped due to avalanche interactions in metastable conditions. Simulations of the quasi-static process were performed in the same way as earlier tests of numerical implementation efficiency of the RFIM model. A single-spin flip method and a sorting algorithm were used to identify spins meeting the conditions described by the relationships 5 and 6. The quasi-static magnetization process, almost independent of the external field step rate, is depicted in Figure 9. Hence, it is possible to achieve a similar course of metastable changes in  
Figure 8: (**a**–**d**) Three-dimensional maps (\(D=3\), \(L=1000\), \(\Delta H=0.001\)) illustrating the dynamics of the magnetization process for various disorders \(R=\{2.5,\,2.0,\,1.5,\,1.0\}\) respectively. Single-colored areas represent spins that were flipped due to avalanche interactions in metastable conditions. magnetization triggered by an external field with a different rate of change. Similar shapes of avalanches presented in Figure 7c,d also may prove the quasi-static magnetization process. The observed damping of the metastable changes in magnetization was generally consistent with experimental data [12] (Vol.II, Ch.3). The small increment of the external field \(H\) had a negligible impact on the occurring avalanche-like changes in magnetization. The avalanche of spin flips caused discrete changes in magnetization called the Barkhausen effect [15; 27]. The greatest impact of the spin-flip avalanche on the change of magnetization was observed in simulations with moderate disorder \(R\)\(\in\)\(<\)1.0, 2.0\(>\) of random field \(h_{1}\).