Data assimilation method for experimental and first-principles data: Finite-temperature magnetization of ${(\mathrm{Nd},\mathrm{Pr},\mathrm{La},\mathrm{Ce})}_2{(\mathrm{Fe},\mathrm{Co},\mathrm{Ni})}_{14}\mathrm{B}$

We propose a data assimilation method for evaluating the finite-temperature magnetization of a permanent magnet over a high-dimensional composition space. Based on a general framework for constructing a predictor from two data sets including missing values, a practical scheme for magnetic materials is formulated in which a small number of experimental data in limited composition space are integrated with a larger number of first-principles calculation data. We apply the scheme to ${\left({\mathrm{Nd}}_{1-\alpha -\beta -\gamma }{\mathrm{Pr}}_{\alpha }{\mathrm{La}}_{\beta }{\mathrm{Ce}}_{\gamma }\right)}_2{\left({\mathrm{Fe}}_{1-\delta -\zeta }{\mathrm{Co}}_{\delta }{\mathrm{Ni}}_{\zeta }\right)}_{14}\mathrm{B}$. The magnetization in the whole $(\alpha ,\beta ,\gamma ,\delta ,\zeta )$ space at arbitrary temperature is obtained. It is shown that the Co doping does not enhance the magnetization at low temperatures, whereas the magnetization increases with increasing $\delta$ above 320 K.

Figure 1

An example for the data assimilation. The true models are given by hand as (i) (blue solid curve) and (ii) (violet solid curve). Sample data generated from the true models are shown as blue squares and violet points. The prediction from the data assimilation is shown as red for the model (i) and cyan for the model (ii). The prediction from the OLS fit is also shown as a dotted-dashed line for comparison.

Figure 2

Flowchart of data assimilation method for finite-temperature magnetization.

Figure 3

The magnetization of ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ as a function of temperature. The violet points denote the experimental observation. The red dashed curve is a regression curve derived from a least-square fitting to Eq. (21). The obtained values for ${\mu }_{0}M(T=0\mathrm{K})$ and $T_{\mathrm{C}}$ are also shown.

Figure 4

Color map of the magnetization ${\mu }_{0}M(\alpha =0,\beta ,\gamma ,\delta ,\zeta =0,T)$ at $T=0$, 300, and 400 K. Experimental values, estimated by the data assimilation method, on $(\beta ,\gamma ,\delta =0)$, $(\beta ,\gamma =0,\delta )$, and $(\beta =0,\gamma ,\delta )$ planes are shown. The values of the magnetization are described in Tesla. Blue squares are sampling points of the first-principles calculation data and violet points are those of the experimental data. Contour lines are also shown with interval of 0.1 T. Note that the upper triangle region on the $(\beta ,\gamma )$ plane is irrelevant because $\beta +\gamma$ cannot exceed 1.

Figure 5

Prediction (red solid lines) for the magnetization ${\mu }_{0}M$ at $T=300$ K is shown along $(0,\beta ,0,0,0)$, $(0,0,\gamma ,0,0)$, and $(0,0,0,\delta ,0)$. Corresponding result for the OLS fit is shown as dark red dashed lines. Violet points indicate the experimental sample data observed at 300 K.

Figure 6

Co concentration dependence of the Curie temperature of ${\mathrm{Nd}}_2{\left({\mathrm{Fe}}_{1-\delta }{\mathrm{Co}}_{\delta }\right)}_{14}\mathrm{B}$. Blue squares denote the first-principles data and violet points denote the experimental data.

Figure 7

Scatter plots of the magnetization at (a) 300 K and (b) 400 K as a function of Nd and Pr concentration ($1-\beta -\gamma$). The dotted straight line connects the magnetizations of two systems: ${\mathrm{La}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ and ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$. (c) Cobalt concentration associated with chemical composition giving maximum magnetization. The horizontal axis is temperature in Kelvin. The inset is an example of the magnetization at 400 K of ${\mathrm{Nd}}_2{\left({\mathrm{Fe}}_{1-\delta }{\mathrm{Co}}_{\delta }\right)}_{14}\mathrm{B}$. A green point denotes the ${\delta }_{\max }$ at the temperature. (d) Magnetization at 400 K as a function of $\gamma /(\beta +\gamma )$. Other variables are fixed in each line and sampled within the range of $0.0\le \alpha \le 0.2$, $0.8\le \beta +\gamma \le 1.0$, $0.0\le \delta \le 0.3$, and $0.0\le \zeta \le 0.1$.

Figure 8

The magnetization of ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ predicted by using the two-sublattice model, Eqs. (B1) and (B2).