## 3 Results  
### \(\beta\)La, CBS and CEL  
The THz spectra of \(\beta\)LA, CBS, and CEL are shown in Figure 2a. Across all spectra, a general trend emerges: the absorption coefficient rises with frequency, transitioning from levels below 10 cm\({}^{-1}\) at 0.3 THz to approximately 200 cm\({}^{-1}\) at 3 THz. The increasing background intensity as a function of frequency is a common trend for organic molecular crystals, and it is due to the increasing vibrational density of states (VDOS) of the amorphous phase in the compounds [7; 30]. Superimposed on this trend, several peaks due to the crystalline phases can be observed, whose frequency and intensity strongly depend on each compound (Table 2). \begin{table}
\begin{tabular}{c c c c c c c} \hline \(\beta\)**LA** & **CBS** & **CEL** & **GLU** & **GAL** & **aLA** & **\(\kappa\)**LM** \\ \hline
1.20 (23.9) & 1.24 (127.7) & 2.15 (103.8) & 1.27 (126.4) & 1.20 (54.1) & 1.19 (145.4) & 0.53 (84.2) \\
1.83 (124.6) & 1.54 (114.0) & 2.73 (136.9) & 1.43 (184.8) & 2.14 (364.6) & 1.86 (278.4) & 1.19 (47.5) \\
2.21 (126.3) & 2.07 (461.3) & 3.00 (199.7) & 2.08 (190.3) & 2.32 (486.8) & 2.20 (179.7) & 1.37 (158.5) \\
2.39 (239.3) & 2.31 (274.1) & 3.19 (218.0) & 2.51 (285.5) & & 2.39 (153.6) & 1.81 (27.9) \\
2.68 (160.3) & 2.66 (240.8) & & 2.68 (277.9) & & 2.68 (218.5) & 2.55 (217.3) \\  & & & 2.92 (293.3) & & & 2.87 (278.3) \\ \end{tabular}
\end{table} 
Table 2: Experimental peak frequencies (\(\mathrm{THz}\)) and the corresponding value of the absorption coefficients (in cm\({}^{-1}\)) in brackets in the 0.3–3.3 THz frequency range. The THz vibrational absorption spectrum of \(\beta\)LA presents five peaks at 0.53, 1.20, 1.37, 1.85, and 2.21 THz in the range 0.3-2.7 THz (Figure 2a). The small intensity peaks at 0.53 and that at 1.37 THz correspond to the spectral fingerprints of \(\alpha\)LM (see Figure 2 and Table 2), which is present in small quantities in \(\beta\)LA [17]. The observed peak at 1.20 THz is assigned to \(\beta\)LA, as already observed in a study focused on the non-destructive evaluation of particle size in lactose powder [16] and on the identification of anomers in lactose [17]. To our knowledge, the other peaks have never been described. As already described, THz-TDS spectroscopy, being able to access the phase of the electromagnetic impulses transmitted through the samples, allows the measurement of their refractive index. Using the procedure described in Section 2.6, it was possible to obtain the refractive index, \(n_{s}\) (Equation (16)), of all the studied compounds. The graphs of \(n_{S}\) as a function of the frequency of the studied saccharides are presented in Figure S11 of Supplementary Materials. The THz vibrational absorption spectrum of CBS presents five peaks at 1.24, 1.54, 2.07, 2.31, and 2.66 THz in the range 0.3-3.2 THz (Table 2). The peak at 1.24 THz was detected in [13], and all peaks were also shown in reference [4]. The THz vibrational spectrum of CEL is characterized by a monotonic increase from 0.3 THz to approximately 1.6 THz, followed by a broad and not very intense peak at 2.15 THz (Table 2). This peak was recently investigated in detail, and it was shown that it is typical of the cellulose I\({}_{\beta}\) polymorph exclusively [32]. In the region around 3 THz, another not intense peak can be appreciated, barely distinguishable from the noise. As reported in [53], the spectroscopic features of CEL around 3 THz depend on the cellulose water content. The CEL, equilibrated at a relative humidity RH = 47.7%, presents well-defined bands at approximately 2.6 and 3 THz. As the RH is decreased, a strong reduction in the intensity and a broadening of the bands are observed. For CEL, equilibrated in a dry environment, as in the present study, only low intensity and broad spectroscopic features, are observed at approximately 2.73, 3.00, and 3.19 THz, hardly detectable from noise.