## 2 Materials and Methods  
The Tb-doped LPS, YPS, GPS, and LaPS samples are carefully synthesized by following orders. First, the 99.99% purity powder of raw materials in including SiO\({}_{2}\), Tb\({}_{4}\)O\({}_{7}\), Lu\({}_{2}\)O\({}_{3}\) (for LPS), Y\({}_{2}\)O\({}_{3}\) (for YPS), Gd\({}_{2}\)O\({}_{3}\) (for GPS), and Lu\({}_{2}\)O\({}_{3}\) (for LaPS) has been weighted and mixed by an agate mortar. Tb concentration in every sample is fixed to the 1.0 mol\(\%\) with respect to each rare-earth site (Lu, Y, Gd, and La). The well-mixed powder has been shaped to the rod by applying isostatic pressure. Each obtained rod is sintering in the furnace for 10 h at 1200 \({}^{\circ}\)C for LPS, YPS, and GPS, and 1400 \({}^{\circ}\)C for LaPS. By these sintered ceramic rods, single crystal growth is done by desktop floating-zone furnace (FZD0192, Canon machinery). The furnace uses two halogen lamps as a heat source. The crystal growth rate is 4 mm/h. The obtained single crystal rods are cut to 1 mm in thickness and polished on both sides of the surface. The leftover crystals pieces from the cutting process have been ground for phase confirmation by powder X-ray diffraction (MiniFlex600, Rigaku Corporation, Tokyo, Japan). The polished samples have been used for diffused transmittance measurement by spectrophotometer (SolidSpec-3700, Shimadzu Corporation, Kyoto, Japan) at a wavelength of 200-800 nm. The PL emission contour graph and PL quantum yield (\(Q\gamma\)) are achieved by using Quantaurus-QY (C11347-01, Hamamatsu Photonics K.K., Shizuoka, Japan), with excitation wavelength from 250-400 nm and observation wavelength from 300-700 nm. On the other hand, The PL decay time analysis is performed by Quantaurus-\(\tau\) (C11367, Hamamatsu Photonics K.K., Shizuoka, Japan). The excitation light is controlled by a halogen lamp with the applied 340-390 nm bandpass filter. The observation wavelength is 550 nm which is generated from Tb\({}^{3+}\) 4f-4f emission.  
In the scintillation properties, X-ray induced scintillation spectra are performed by our original setup [56]. The irradiated X-ray dose used for spectrum observation is 1 Gy. X-ray induced scintillation decay time and the afterglow analysis are also performed by using our original system equipped with the pulsed X-rays source and the afterglow characterization system [57]. The observation wavelength is from 160-650 nm. The calculation of the afterglow level is consistent with our previous works as the following equation [58; 59], \(Af_{20}=(I_{20}\)-\(I_{\rm bg})/I_{\rm max}\) when the \(Af_{20}\) is the afterglow level at 20 ms after X-ray irradiation, \(I_{\rm bg}\) is the signal intensity for X-ray irradiation or so call background intensity, and \(I_{\rm max}\) is the maximum value of the signal intensity during the X-ray irradiation. The pulse height spectra and absolute scintillation light yield of the Tb-doped pyrosilicate samples are measured by the recently developed ms decay time pulse height setup [55]. Strictly speaking, the spectrum observed by this setup is not a pulse height but a pulse area. However, the physical meaning is the same, and to avoid complexity, the common term of pulse height is used in this work. In this measurement, the voltage of \(-\)550 V has been applied to the PMT (R7600U-200, Hamamatsu Photonics), thus the signal from PMT is sent to the digitizer (Model 1819-16, Clear Pulse). \({}^{137}\)Cs has been selected for a \(\gamma\)-ray source (662 keV). In addition, a Bi\({}_{4}\)Ge\({}_{3}\)O\({}_{12}\) (BGO) scintillator with a scintillation light yield of 8200 ph/MeV has been selected as a reference sample for this measurement [60].