The introduction of high-order terms in N1 and N2 in Eq. (8) could be useful to describe the curved behavior around 25% Co and the moment saturation around 50% as shown in Fig. 4. The cross terms, which contain both N1 and N2, are used to capture the correlated effects of the first and second shells. To determine all the parameters P0, A1, A2, A3, B1, B2, B3, C, D1 and D2, we employed a least square fitting approach in which values of MFe and N1, N1 2 , N1 3 , N2, N2 2 , N2 3 , and N1N2, N1 2 N2, and N1N2 2 of each Fe atom came from our DFT calculations. The DFT data of each Fe atom allows us to construct one equation and a set of about 2200 equations obtained from 20 SQSs and 10 B2-like structures was set up for fitting. The obtained parameters are listed in Table 1. A simpler version of Eq. (8) with fitting only up to the second order in N1, N2 was also performed. As can be observed from Table 1, the fitted parameters A1, A2, A3, B1, B2, and B3 indicates that the impact of the first shell is almost twice as strong as that of the second shell. To check the accuracy and usefulness of this analytical model, it was applied to calculate the magnetic moment of all the Fe atoms in each Fe-Co system based on the respective N1, N2. The predicted atomic moments from the model were then used to calculate the Fe average moments and the magnetization of the structures, where the magnetic moment of each Co atom is considered to be 1.76 µB for all concentrations. The results obtained from this atomic analytical model are plotted in Fig. 9. From Fig. 9(a), the model reproduces very well the magnetization in both the SQSs and the ordered structures with all important features pointed out by DFT and experiments in Fig. 3. The model also correctly predicts magnetic moment variation of Fe with increasing Co content for both the SQSs and the B2-like ordered structures, as shown in Fig. 9(b) and (c), respectively. Then, the model is further verified in calculating moments in additional structures which were not included in the fitting database. We employed the analytical formula (8) to calculate the average magnetic moment of Fe in various well-known ordered structures B2, B32, cI16, and D03 [12]. A comparison between DFT and the model fitted up to the third order in N1, N2 is shown in Fig. 10, and the values are given in Table 2. The results obtained by the model on fitting just up to the second order were also calculated for a comparison. Fig. 10 shows a good agreement between our model and DFT results for all the considered structures. Please note that the visible discrepancy in Co7Fe is because this structure has 87.5% Co and Co atoms fill completely the 1st and 2nd shells of some Fe atom. Such configurations were not represented in the database of fitting that limited to 70% of Co as we explained above.
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