# Hydrogen Deuteride for Cold Neutron Production: A Model for the Double Differential Cross Section
## 2 Materials and Methods  
Before summarizing the basic formalism, it is useful to recall under which hypotheses the adopted modeling of the DDCS holds [12; 18; 19]: (i) molecules are considered to be _free_ vibrating rotors, i.e., the translational center of mass (CM) dynamics is assumed to be completely decoupled from the intramolecular motions; (ii) rotations are treated as independent of vibrations; and (iii) anharmonicity effects are neglected. The first hypothesis corresponds to consider an isotropic interaction among the molecules, with negligible orientational correlations. Assumptions (i) to (iii) are known to be very well satisfied for hydrogen, even in the (low pressure) solid phase [23]. However, since the centers of mass and charge do not coincide in the heteronuclear case, the full validity of hypothesis (i) for HD also can only be verified a posteriori. Moreover, for the case of low temperature liquids, an additional assumption is that all the molecules in the system initially (that is, before interaction with neutrons) lie in their ground vibrational state. Finally, only the case of unpolarized neutrons is considered here. The above general assumptions for the treatment of the neutron cross sections of diatomic low temperature fluids are further accompanied by the important simplifications introduced by the heteronuclear nature of the molecule, which allows us to consider its nuclei as distinguishable particles. The absence of symmetry requirements for the total molecular wave function implies that there is no coupling between the total molecular spin and the rotational state of the molecule. Therefore, one can refer to the so-called \"uncorrelated spin\" case [19], where quantum-statistical averages involving nuclear and neutron spin variables can be carried out separately from those related to position variables. In the mentioned hypotheses and conditions, the basic starting formulas leading to the double differential (per unit solid angle and exchanged frequency interval) cross section are those gathered in Equation (7) of Ref. [19] which, by omitting the superscript _uncorr_, we rewrite as:  
\[\frac{d^{2}\sigma}{d\Omega d\omega}=\frac{k_{1}}{k_{0}}S_{\text{n }}(Q,\omega)=\frac{k_{1}}{k_{0}}\frac{1}{2\pi}\int_{-\infty}^{+\infty}dt\;e^{ -i\omega t}F_{\text{n}}(Q,t)= \tag{1}\] \[=\frac{k_{1}}{k_{0}}\frac{1}{2\pi}\int_{-\infty}^{+\infty}dt\;e^ {-i\omega t}[u(Q)F_{\text{d}}(Q,t)+v(Q,t)F_{\text{s}}(Q,t)],\]  
where \(k_{0}\) and \(k_{1}\) are the incident and scattered neutron wavevectors, and \(S_{\text{n}}(Q,\omega)\) is the total dynamic structure factor provided by neutron scattering, i.e., the time Fourier transform of \(F_{\text{n}}(Q,t)\), which is the neutron weighted combination of the distinct and self intermediate scattering functions \(F_{\text{d}}(Q,t)\) and \(F_{\text{s}}(Q,t)\), respectively. For molecular fluids, the latter are to be identified with the CM functions defined by  
\[F_{\text{d}}(Q,t)=\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1(j\
eq i)}^{ N}\left\langle e^{-i\mathbf{Q}\cdot\mathbf{R}_{i}(0)}e^{i\mathbf{Q}\cdot \mathbf{R}_{j}(t)}\right\rangle \tag{2}\] \[F_{\text{s}}(Q,t)=\frac{1}{N}\sum_{i=1}^{N}\left\langle e^{-i \mathbf{Q}\cdot\mathbf{R}_{i}(0)}e^{i\mathbf{Q}\cdot\mathbf{R}_{i}(t)}\right\rangle \tag{3}\]  
which provide the total CM intermediate scattering function \(F(Q,t)=F_{\text{d}}(Q,t)+F_{\text{s}}(Q,t)\). In the above equations, \(N\) is the total number of molecules, \(\mathbf{R}_{i}(0)\) is the CM position of the \(i\)th molecule in the (arbitrarily chosen) time origin, and \(\mathbf{R}_{j}(t)\) is the CM position of a different molecule at a subsequent time \(t\). The angle brackets denote the quantum canonical ensemble average. The isotropy of the fluid actually makes these functions depend only on the modulus \(Q\) of the exchanged wavevector \(\mathbf{Q}=\mathbf{k}_{0}-\mathbf{k}_{1}\).