## 4 Discussion
The absorption spectrum of \(\alpha\)LA shows five peaks at 1.19, 1.86, 2.20, 2.38, and at approximately 2.68 THz (see also Section 3.3). The most evident peaks were at 1.19, 1.86, and approximately 2.68 THz. These insights underscore the inadequacy of a single-molecule approach for modeling spectra within the THz range. This is due to the presence of vibrational modes that extend across multiple molecules, in contrast to the mid-infrared spectroscopic range where vibrations tend to be localized around individual or a few molecular bonds [6]. A relevant contribution to the THz spectra comes from the interactions between the molecular units. This properly explains the complexity of the low-energy dynamics of these compounds. Conversely, it can serve as an experimental tool to investigate and differentiate the crystalline structure of a molecular crystal specimen. In order to obtain quantitative data, the peaks at 0.53 THz and 1.37 THz in \(\alpha\)LM were subjected to Lorentzian curve fitting, yielding full-width-at-half-maximum (FWHM) values of (0.021 \(\pm\) 0.002) THz and (0.047 \(\pm\) 0.002) THz, respectively. This analysis draws upon both classical and quantum physics, where a Lorentzian line shape signifies homogeneous damping, indicating that the molecular oscillators accountable for absorption experience equivalent scattering or damping mechanisms, regardless of their individual characteristics or states [15]. Mechanistically, this dynamic is often governed by an exponential decay over time, characterized by a time constant \(\tau\) linked to the Lorentzian linewidth via \(\tau=(\pi FWHM)^{-1}\). Consequently, for the vibrations responsible for the 0.53 THz band, \(\tau\) amounts to 15.1 ps, while for the 1.37 THz band, it stands at 6.8 ps. These \(\tau\) values notably exceed the room-temperature damping times associated with molecular vibrations, including collective ones (such as phonons) in solids [15]. This suggests that the spectral features at 0.53 THz and 1.37 THz stem from a collective vibrational transition that evades the typically rapid scattering mechanisms. One plausible explanation lies in the significant reduction of phonon-phonon scattering, attributed to a relatively subdued lactose crystal intermolecular potential energy. This diminished potential energy likely arises from the presence of water molecules within the \(\alpha\)LM crystal structure [6]. The comparison of the spectra of \(\alpha\)LM, \(\alpha\)LA, and \(\beta\)LA shows the importance of water molecules in affecting the THz vibrational spectra by modifying the intermolecular interactions mediated by H-bonds. The qualitative analyses of the THz spectra show the complexity of the observed features and the need for computational simulations in order to assign the experimental peaks to optically active vibrational modes and to obtain information on their vibrational characters. In the comparison between the experimental and calculated spectra, it must be considered that the theoretical method is able to determine the normal mode frequencies of crystalline samples in the harmonic oscillator approximation. This methodology forms the foundation for associating THz absorption peaks with crystalline samples [4]. Nevertheless, even with a grasp of the normal modes within a molecular crystalline system, attributing these modes remains intricate due to the presence of anharmonic effects. These effects induce changes in phonon energies and concurrently curtail their lifetimes. As a consequence, the observed THz peaks generally exhibit frequencies that deviate from those predicted by the harmonic model, accompanied by significantly broader line shapes. Hence, anticipating a flawless alignment between a THz spectrum and a normal mode simulation grounded in the harmonic approximation would be unrealistic, even when employing the most precise modeling methodologies [4].