## 2 Materials and Methods
### Methods  
Initially, the phase changes of REMES under high-temperature and -vacuum conditions were investigated to find the possible obstacles to recovering REF\({}_{3}\) by vacuum distillation. The REMES was compacted as a block with no pretreatment and set at the center of the vertical vacuum distillation furnace. Then, many 10 g REMES blocks were distilled at different temperatures for different times. Based on the distillation results, the possible phase changes and potential obstacles were identified. Then, to resolve the barriers, the fluorination method was proposed, and NH\({}_{4}\)HF\({}_{2}\) was selected as the fluorination reagent. It reacted with REMES in an airtight furnace. After the reaction, the fluorinated REMES was distilled to prove the availability of the fluorination and vacuum distillation. The recovery efficiencies of the REEs and Li were calculated by Equation (1), using the recovery efficiency of the REEs as an example.  
\[\eta_{\rm N}=[1-(\rm{w}_{\rm NC}\times m_{\rm C})/(\rm{w}_{\rm NR}\times m_{ \rm R})]\times 100\% \tag{1}\]  
where \(\eta_{\rm N}\) represents the recovery efficiency of the REEs, %; \(\rm{w}_{\rm NC}\) is the content of the REEs (calculated as oxides) in the distillation residue, %; \(\rm{m}_{\rm C}\) is the mass of the distillation residue, g; \(\rm{w}_{\rm NR}\) represents the content of the REEs (calculated as oxides) in the raw material, %; \(\rm{m}_{\rm R}\) is the mass of the raw material, g.  
Differential scanning calorimetry and thermo-gravimetric analysis were carried out by SDT (TA Q600, TA, New Castle, DE, USA) at a temperature range of 294-1466 K and a constant heating rate of 10 K/min in a N\({}_{2}\) atmosphere. The phase analysis of the sample was conducted by an X-ray diffractometer (D8 Advance, Bruker, Bremen, Germany; ultima IV, Rigaku, Akishima, Japan), which used Cu-Ka radiation with a scanning rate of 10 \({}^{\circ}\)/min, a step size of 0.02\({}^{\circ}\), and a recorded range from 10\({}^{\circ}\) to 90\({}^{\circ}\). An inductively coupled plasma optical emission spectrometer (PE optima 8300 DV, PerkinElmer, Waltham, MA, USA) was employed to analyze the concentration of Li. The XPS analysis was conducted by Thermo SCIENTIFIC ESCALAB 250 Xi (Thermo Fisher Scientific, Waltham, MA, USA), which used an Al K\(\alpha\) source. The XPS data were calibrated for binding energy at C\({}_{18}\) = 248.8 eV. The morphology and element distribution of the slag were studied by a field emission scanning electron microscope (JSM-7800F, JEOL, Akishima, Japan), which was equipped with an Oxford SDD for EDS analysis (Oxford Instruments, High Wycombe, UK). The fluorine contents were determined by DETA titration (GB/T 18114.11-2010), with an analytical error of \(<\)0.4 wt.%. The total amount of rare earth oxides was determined by the oxalate precipitation method (GB/T 14635-2010), with an analytical error of \(<\)0.5 wt.%. The rare earth oxide relative contents were analyzed by ICP-OES (Agilent, Santa Clara, CA, USA) (GB/T 18114.8-2010), with an analytical error of \(<\)0.4 wt.%. The lithium content wasdetected by an inductively coupled plasma optical emission spectrometer (YS/T 739.3-2020), with an analytical error of <0.27 wt.%.