## 3 Results
### aLM, aLA, bLM  
The THz absorption spectra of aLA, \(\beta\)LA, and aLM are shown in Figure 4a. Furthermore, for these spectra, as a function of frequency, an increasing background intensity due to the amorphous phase in the compounds is observed [7; 30]. Superimposed on this trend, several peaks due to the crystalline phases can be observed whose frequency and intensity depend on each compound (Table 2). The experimental absorption spectra of the two anhydrous isomeric forms of lactose, aLA and \(\beta\)LA, are qualitatively and quantitatively very similar, showing five peaks of similar intensity in the range 0.3-2.9 THz centered at 1.20, 1.83, 2.21, 2.39, and at approximately 2.68 THz for \(\beta\)LA, and at 1.19, 1.86, 2.20, 2.39, and at approximately 2.68 THz for aLA (Table 2). Moreover, \(\beta\)LA shows two low-intensity peaks at 0.53 and 1.37 THz.  
The spectra of these anhydrous forms are partially different from that of the lactose hydrous form aLM, which shows peaks with smaller widths in the range 0.3-3 THz at 0.53, 1.19, 1.37, 1.81, 2.55, and 2.87 THz. The most evident peaks are at 0.53, 1.37, 2.55, and 2.87 THz. The observed peaks are at frequencies compatible with those found in the previous studies for monohydrate lactose [4; 6; 15; 16] and consistent with frequencies observed at 300 K in a temperature-dependent experiment [23].  
Interestingly, the small intensity peaks found in the anhydrous forms at 0.53 and 1.37 THz were shown to belong to a small amount of \(\alpha\)LM [17]. Two spectral regions at approximately 1.19 THz and at 1.83 THz show peaks that appear in all spectra of the anhydrous or hydrous lactose forms, even if their spectral width is lower in aLM. Interestingly, as was recently observed, the peaks of the anhydrous forms of lactose are at frequencies that differ from those found for aLM submitted to heating at 377 K to remove water molecules [6]. This suggests that the phase transformations of lactose are complex, as was also stated by the same authors of Ref. [6].  
Figure 3: (**a**) THz absorption coefficient of GLU, GAL, and aLA. All the spectra show absorption peaks superimposed to a background intensity increasing as a function of frequency. (**b**) Simulated low-frequency THz vibrational modes (color vertical lines)The frequencies of the simulated peaks of \(\alpha\)LM and \(\alpha\)LA in the 0.3-3.3 THz range are shown in Figure 4b and in Table 3 (for \(\beta\)LM see Section 3.1). The most evident peaks of \(\alpha\)LA at 1.19, 1.86, and approximately 2.68 THz were assigned to the calculated modes at 1.04, 1.93, and 2.84 THz, which correspond to mode numbers 2, 4, and 7, respectively. The smaller peaks at 2.20 and 2.38 THz were assigned to the modes at 2.25 and 2.38 THz, respectively, which correspond to mode numbers 5 and 6. The most intense peaks of \(\alpha\)LM at 0.53, 1.37, 2.55, and 2.87 THz were assigned to the calculated modes at 1.12, 1.40, 2.51, and 2.82 THz, which correspond to mode numbers 1, 2, 6, and 7 (Figure 4b), respectively. The discrepancy between the experimental peak position at 0.53 THz and the calculated frequency for mode 1 is 0.67 THz, which corresponds to \(\approx\)2.8 meV. As was recently demonstrated, the frequencies of the calculated THz peaks of \(\alpha\)LM strongly depend on the position of the added water molecule in the \(\alpha\)-lactose molecule. This can explain the discrepancy between the experimental and the calculated frequency of the first peak, since in our simulations, we have not changed the position of the water molecule but we have used the coordinates obtained from X-ray diffraction measurements from PubChem ([[https://pubchem.ncbi.nlm.nih.gov/compound/alpha-lactose](https://pubchem.ncbi.nlm.nih.gov/compound/alpha-lactose)]([https://pubchem.ncbi.nlm.nih.gov/compound/alpha-lactose](https://pubchem.ncbi.nlm.nih.gov/compound/alpha-lactose)), accessed on 24 February 2020).