# Hydrogen Deuteride for Cold Neutron Production: A Model for the Double Differential Cross Section
## 2 Materials and Methods
Concerning the self properties, it is seen instead that the dependence of \(v(Q,t)\) on time prevents one from expressing the second term as directly proportional to the self dynamic structure factor \(S_{\rm s}(Q,\omega)\). However, Equation (A14) shows that time enters \(v(Q,t)\) only in exponential form; therefore, it is possible to write  
\[\int_{-\infty}^{+\infty}dt\;e^{-i\omega t}v(Q,t)F_{\rm s}(Q,t)=\] \[=\sum_{b_{0}\uparrow_{1}v_{0}v_{1}}\int_{-\infty}^{+\infty}dt\;e^ {-i(\omega-\omega_{b_{1}}-\omega_{b_{0}}v_{1})t}\mathcal{F}(Q;b_{0}\uparrow_{ 1}v_{0}v_{1})F_{\rm s}(Q,t) \tag{6}\]where the explicit expression for \(\mathcal{F}(Q;J_{0}J_{1}v_{0}v_{1})\) in the low temperature case is given in Equation (A13) of Appendix A. In Equation (6), the rotational \((\omega_{J_{0}J_{1}})\) and vibrational \((\omega_{v_{0}v_{1}})\) transition frequencies, involving the initial (subscript 0) and final (subscript 1) quantum numbers, are seen to determine the time dependence of \(v(Q,t)\). In the special case of a low temperature sample, all molecules are assumed to lie initially in the vibrational ground state (\(v_{0}=0\)); therefore, it is possible to write \(\omega_{v_{0}v_{1}}=v_{1}\omega_{v}\), with \(\omega_{v}\) indicating the frequency of the harmonic oscillator of reduced mass \(\mu=\frac{m_{1}m_{2}}{M_{12}}\), with \(m_{1}\) and \(m_{2}\) the mass of the two (different) atoms in the molecule and \(M_{12}=m_{1}+m_{2}\). Consequently, the DDCS finally reads  
\[=\frac{k_{1}}{k_{0}}\frac{1}{2\pi}\int_{-\infty}^{+\infty}dt\;e^ {-i\omega t}[u(Q)F_{\text{d}}(Q,t)+v(Q,t)F_{\text{s}}(Q,t)],\]0 \[=\frac{k_{1}}{k_{0}}\frac{1}{2\pi}\int_{-\infty}^{+\infty}dt\;e^ {-i\omega t}[u(Q)F_{\text{d}}(Q,t)+v(Q,t)F_{\text{s}}(Q,t)],\]1  
Equation (7) shows that the single-molecule contribution to the DDCS corresponds to a comb of lines centered at the frequencies of the possible rotovibrational transitions. These spectral components are therefore either central or shifted replicas of the lineshape describing the CM \(S_{\mathrm{s}}(Q,\omega)\), with amplitudes ruled by the involved quantum numbers, the initial state probabilities and the nuclear scattering lengths. Calculations of the DDCS, of course, require a modeling of both \(S(Q,\omega)\) and \(S_{\mathrm{s}}(Q,\omega)\). It is worth recalling that a comparison with the experiment is only possible if the model lineshapes obey the detailed balance principle. Therefore, if classical (i.e., symmetric) models are used for the dynamic structure factors, these must be duly asymmetrical via multiplication by the factor \(\frac{h\omega}{k_{\mathrm{B}}T}[n(\omega)+1]=\frac{h\omega}{k_{\mathrm{B}}T}[1 -\exp(-\frac{h\omega}{k_{\mathrm{B}}T})]^{-1}\) prior to their inclusion in Equation (7), where \(n(\omega)\) is the Bose factor [24], with \(k_{\mathrm{B}}\) and \(\hbar\) indicating the Boltzmann and Planck constants, respectively. Moreover, the finite energy resolution of the spectroscopic data needs to be taken into account by performing comparisons with calculations only after these have been properly broadened by the experimental resolution function. As mentioned, the present work focuses on HD. It is well known that the low mass (between 2 and 6 a.m.u.) and relatively low temperatures (e.g., around 20 K) of molecular hydrogen and its isotopes in the liquid phase make the de Broglie thermal wavelength [25]\(\Lambda=h/\sqrt{2\pi Mk_{\mathrm{B}}T}\) reach values of the order of the molecular size, while remaining inferior to the average intermolecular distance [26]. Therefore, the quantum delocalization of individual particles affects the static and dynamic CM properties of these systems with respect to classical behavior, while indistinguishability can still be assumed to play a negligible role, thus justifying the use of Boltzmann statistics. These overall assumptions are commonly summarized by saying that hydrogens are _moderate_ quantum fluids, if compared to the paradigmatic case of helium.