## 1 Introduction  
The demand for high performance magnets has been growing due to the increased use of motors in electric/hybrid vehicles and wind turbines [1; 2; 3; 4; 5; 6]. While rare-earth (\(RE\)) permanent magnets, such as Nd\({}_{2}\)Fe\({}_{14}\)B, exhibit excellent magnetic properties, including a high anisotropy field (\(H_{\mathrm{a}}\)), high saturation magnetization (\(M_{\mathrm{s}}\)), and coercivity (\(H_{\mathrm{c}}\)), the supply risk and high price of \(RE\) elements have become significant concerns. Ce, which is a byproduct of the Nd extraction process, has been overstocked, and, so, the price of Ce elements is very low compared to Nd, Pr and Dy [7; 8; 9]. For this reason, Ce-based magnets have emerged as a promising candidate that can replace or reduce the use of Nd in \(RE\) permanent magnets, although the inferior intrinsic magnetic properties still pose a significant challenge for industrial applications. As a result, it is expected that developing Ce-based low-cost permanent magnets, with a performance comparable to Nd-based magnets, can bridge the performance gap between ferrite and Nd-Fe-B magnets [8; 9]. In addition, the fabrication processes for Nd-Fe-B magnets, such as the rapid solidification process (RSP) or melt-spinning, require the use of pure metals for raw materials, complex equipment and the careful consideration of oxidation of metal alloys during the process, which can lead to high production costs. Compared to conventional Nd-Fe-B magnet fabrication processes, the reduction-diffusion (R-D) process, which uses reduction agents of calcium or calcium compounds, exhibits many advantages over the RSP and melt-spinning, such as (1) low production costs due to the use of inexpensive raw materials such as rare-earth oxides and relatively simple equipment, (2) low energy consumption and (3) the direct production of alloy powder form suitable for the subsequent milling process [10; 11; 12; 13; 14; 15; 16; 17; 18]. Prior reports on the fabrication of Nd-Fe-B powders by R-D processes have used precursors (composed of NdFeO\({}_{3}\) and Fe phases) prepared by wet chemical methods such as the spray-drying process [10; 11; 12; 13], the sol-gel process [14; 15] or the co-precipitation method [16; 17; 18]. In this study, 2-14-1 phases, which represent \(RE_{2}\)Fe\({}_{14}\)B phases, were synthesized by a modified reduction-diffusion process, which uses an \(RE\)FeO\({}_{3}\) precursor synthesized by a solid-state reaction using oxide raw materials. The solid-state reaction has been well known for the synthesis of inorganic solid materials, such as ferrites and perovskites, and widely used in mass production processes [19]. To the best of our knowledge, no previous studies have reported the synthesis of the 2-14-1 phase by a reduction-diffusion process using precursors prepared by the solid-state reaction. The solid-state reaction offers many advantages, which include (1) the use of oxide raw materials (which are relatively not costly and easy to store/handle) and (2) easy control of the elemental composition. Additionally, compared to other processes, it requires simple equipment, which can lead to low production costs. Therefore, the modified R-D process proposed in this study, utilizing precursors prepared by the solid-state reaction, offers significant advantages over the conventional R-D process due to the low cost of raw materials and simple processes for production, making it a cost-effective approach for large-scale manufacturing. Moreover, precise control of the elemental composition can be achieved in the process. In this study, not only the Ce\({}_{2}\)Fe\({}_{14}\)B phase (Ce 2-14-1) but also (Ce\({}_{0.5}\)Nd\({}_{0.5}\))\({}_{2}\)Fe\({}_{14}\)B (Ce\({}_{0.5}\)Nd\({}_{0.5}\) 2-14-1) and Nd\({}_{2}\)Fe\({}_{14}\)B (Nd 2-14-1) phases were synthesized using the modified R-D process. To obtain single-phase Ce-based 2-14-1 compounds, the elemental composition and heat-treatment condition were optimized.