## 1 Introduction  
As the production process of calcium sulfoaluminate (CSA) cement produces fewer CO\({}_{2}\) emissions and consumes less energy compared to Portland cement, CSA cement is treated as a type of alternative and low-carbon binder [1, 2]. Due to characteristics of early strength and tailored expansion, CSA cement has also been widely used as a rapid repairing material since the 1970s and it has recently been promoted for application in numerous domains, such as soft soil stabilization and 3D printing [3, 4, 5]. Ye'elimite is a dominant phase in CSA cement and generally accounts for over 50 wt.% in CSA cikers [6, 7]. Sintering conditions of CSA cikhers are determined by the forming conditions of ye'elimite; meanwhile, the main hydration products of CSA cement result from the reactions between ye'elimite and gypsum [8, 9, 10, 11, 12].  
The chemical formula of stoichiometric ye'elimite is Ca\({}_{4}\)Al\({}_{6}\)SO\({}_{16}\), which can be abbreviated as st-C\({}_{4}\)A\({}_{5}\)S (hereafter, nomenclatures are used--C: CaO, $: SO\({}_{3}\), A: Al\({}_{2}\)O\({}_{3}\), S: SiO\({}_{2}\)). The crystallographic structure of st-C\({}_{4}\)A\({}_{3}\)S has been identified to be orthorhombic at room temperature, with a space group of Pcc2. Meanwhile, the formation and hydration mechanisms of st-C\({}_{4}\)A\({}_{3}\)S have also been intensively investigated by a number of studies [13, 14, 15]. On the other hand, certain types of doping ions (such as iron ions and sodium ions et al.) enter into the lattices of ye'elimite in the manufacturing process of CSA cikhers, which leads to the formation of doped/solid-solution ye'elimite [16, 17, 18]. Specifically, when solid waste is utilized as raw materials, various types of solid-solution ye'elimite tend to be formed more easily [19]. A series of recent studies focused on different types of doped ye'elimite, including strontium-bearing ye'elimite, barium-bearing ye'elimite, and iron-bearing ye'elimite.  
The consensus obtained by studies on doped ye'elimite is that dopants result in a crystallographic system of ye'elimite, partly or totally transforming from an orthorhombic system to a cubic system [16; 17; 20]. Particularly, Ca\({}_{3.8}\)Na\({}_{0.2}\)Al\({}_{5.6}\)Fe\({}_{0.2}\)Si\({}_{0.2}\)SO\({}_{16}\) (abbreviated as ss-C\({}_{4}\)A\({}_{3}\)S), a typical kind of doped solid-solution ye'elimite, has attracted special attention, as the crystallographic system of ss-C\({}_{4}\)A\({}_{3}\)S has been proved to be purely cubic at room temperature [21]. Comparing it to st-C\({}_{4}\)A\({}_{3}\)S, ss-C\({}_{4}\)A\({}_{3}\)S also exhibited obviously distinct hydration characters [22]. Additionally, elemental compositions of ss-C\({}_{4}\)A\({}_{3}\)S approached sintered ye'elimite phases in CSA linkers due to the existence of iron ions and silicon ions in raw materials [16].  
In this study, the formation process of ss-C\({}_{4}\)A\({}_{3}\)S was investigated. Clinkers of ss-C\({}_{4}\)A\({}_{3}\)S were sintered at various temperatures for different holding times. X-ray diffraction (XRD) tests and Rietveld quantitative phase analysis (RQPA) were conducted to determine the phase compositions of the clinkers. According to the results of the RQPA, different models of solid-state reactions were employed to fit the isothermal formation process of ss-C\({}_{4}\)A\({}_{3}\)S. By comparing the obtained correlation coefficients, the formation mechanism of ss-C\({}_{4}\)A\({}_{3}\)S was confirmed. Lastly, the value of activation energy for ss-C\({}_{4}\)A\({}_{3}\)S was acquired by the Arrhenius equation.