# Hydrogen Deuteride for Cold Neutron Production: A Model for the Double Differential Cross Section
## 2 Materials and Methods
Furthermore, in the last member of Equation (1) the functions \(u(Q)\) and \(v(Q,t)\) play the role of inter- and intramolecular form factors weighting, respectively, the distinct and self CM dynamics. In analogy with the monatomic case, purely coherent scattering characterizes \(u(Q)\), that is, only the so-called coherent part of the neutron cross section of the scattering unit probes the interparticle translational dynamics. As a consequence, \(u(Q)\) exclusively contains the coherent scattering lengths of the various nuclei present in the molecule, and is independent of time. Differently, the intramolecular form factor is a function of time, and generally depends on both the coherent and incoherent nuclear scattering lengths. Explicit expressions for the form factors of a heteronuclear diatomic molecule at a low temperature are derived in Appendix A for direct implementation in computing code. There, the various parameters relevant to HD are also given. The neutron DDCS is then obtained by inserting Equations (A14) and (A17) into Equation (1). Before discussing reasonable modelings of the CM translational dynamics, we wish to point out a few basic facts that are significant for the comparison of calculations with the (possibly available) neutron experimental spectra. The first point regards what is actually accessed by experiments. It is well known that conventional neutron spectroscopy provides the Fourier transforms of space and time correlation functions. In Equations (2) and (3), we introduced the time autocorrelation of the space Fourier transform of the microscopic density, i.e., the total intermediate scattering function \(F(Q,t)\), separated into its _distinct_ and _self_ parts. The latter were used in particular to describe the neutron version of the molecular scattering function as \(F_{\rm n}(Q,t)=u(Q)F_{\rm d}(Q,t)+v(Q,t)F_{\rm s}(Q,t)\). Such a separation of the total \(F_{\rm n}(Q,t)\), inherited from the formalism used to describe neutron scattering from monatomic fluids, i.e., \(b_{\rm coh}^{2}F_{\rm d}(Q,t)+(b_{\rm coh}^{2}+b_{\rm inc}^{2})F_{\rm s}(Q,t)\), has the merit of highlighting the relationship between the coherent scattering and the distinct dynamics. This also holds true for molecular liquids, since we saw that \(u(Q)\) only contains the coherent scattering lengths of the constituent atoms. However, despite its conceptual significance, the mentioned separation has no feedback from reality, since neutrons provide a different combination and can only be the probe of \"true\" correlation functions, such as \(F(Q,t)\) and \(F_{\rm s}(Q,t)\), differently from \(F_{\rm d}(Q,t)\). This means that the self and distinct contributions to the total dynamics cannot be disentangled in a neutron measurement on a totally coherent sample. Conversely, it is incoherent scattering that provides an (as remarkable as exclusive) pathway to the self dynamics. In other words, the output of a neutron experiment on a monatomic sample with nonzero coherent and incoherent scattering lengths is actually \(b_{\rm coh}^{2}F(Q,t)+b_{\rm inc}^{2}F_{\rm s}(Q,t)\), the molecular version of which is  
\[F_{\rm n}(Q,t)=u(Q)F(Q,t)+[v(Q,t)-u(Q)]F_{\rm s}(Q,t). \tag{4}\]  
The second thing worth recalling concerns the spectral properties and the general features of the resulting DDCS. By completing the switching to Fourier \((Q,\omega)\) space, Equation (4) becomes  
\[S_{\rm n}(Q,\omega)=\frac{1}{2\pi}\Bigg{\{}u(Q)\int_{-\infty}^{+ \infty}dt\;e^{-i\omega t}F(Q,t)+\] \[+\int_{-\infty}^{+\infty}dt\;e^{-i\omega t}[v(Q,t)-u(Q)]F_{\rm s} (Q,t)\Bigg{\}}= \tag{5}\] \[=u(Q)\big{[}S(Q,\omega)-S_{\rm s}(Q,\omega)\big{]}+\Bigg{[}\frac{ 1}{2\pi}\int_{-\infty}^{+\infty}dt\;e^{-i\omega t}v(Q,t)F_{\rm s}(Q,t)\Bigg{]}.\]  
The first term is, therefore, related to distinct CM dynamic structure factor we are interested in when studying the translational collective dynamics of the system.