# Hydrogen Deuteride for Cold Neutron Production: A Model for the Double Differential Cross Section
## 1 Introduction  
Liquid hydrogen and its isotopic forms are among the most renowned cryogenic fluids and, in the specific application to neutron techniques, are the most important low-temperature moderating materials used to realize cold (0.1-10 meV) [1] neutron sources. In recent years, quite an effort has been devoted to deeply refine the ability to predict the neutron scattering properties of these diatomic fluids and build up reliable databases collecting their double differential cross section (DDCS) in as wide kinematic ranges as possible. As far as hydrogen (H\({}_{2}\)) and deuterium (D\({}_{2}\)) are concerned, significant progress was possible some years ago when crucial experimental work was dedicated to accurate measurements of the total scattering cross section \(\sigma_{\mathrm{s}}\) of para-H\({}_{2}\)[2] and, at the same time, quantum calculation methods have been employed, in place of classical computations of the translational dynamics, in the DDCS algorithms for these low-mass molecules undergoing nonnegligible quantum delocalization effects [3, 4]. More recently, the latter approach, which gives the unique opportunity to get rid of adjustable parameters in DDCS evaluations, has been successfully adopted in moderator design and nuclear data processing codes [5, 6] implemented at the European Spallation Source (ESS, Sweden). Recent experiments aimed at a better assessment of the coherent scattering properties of D\({}_{2}\)and providing absolute scale determinations of its DDCS by inelastic neutron scattering [7], will also help further refinement of the scattering libraries for D\({}_{2}\). Therefore, concerning the homonuclear representatives of the hydrogen family, we can be rather satisfied with the present computational and predicting capabilities, now permitting well-grounded and safe applications to forthcoming and existing neutron sources. Nonetheless, another member among the hydrogen liquids, namely hydrogen deuteride (HD), has not received the same attention as its homonuclear partners in recent years, either from a scientific point of view or in applications, although some of its midway properties between H\({}_{2}\) and D\({}_{2}\) might also be conveniently exploited in neutron moderation processes, as explored in previous moderator experimental studies [8; 9]. For instance, at the same (manageable) low temperatures of liquid H\({}_{2}\) and D\({}_{2}\) (\(\sim\)20 K), HD provides stronger scattering than D\({}_{2}\) and lower absorption than H\({}_{2}\), while its molecular mass does not critically decrease the moderation efficiency, as shown by the successful use of an even heavier liquid, like D\({}_{2}\), for cold neutron production. Also, mixtures of H\({}_{2}\) and D\({}_{2}\) (partly combining themselves in the form of HD molecules) could be envisaged. Indeed, it has been suggested that adding a small percentage of H\({}_{2}\) to liquid D\({}_{2}\) might improve the moderation efficiency in the case of small moderator volumes [10; 11]. Nevertheless, when exploring the literature about experimental, theoretical and simulation work on liquid HD and, in particular, its neutron scattering properties, one is compelled to face an unexpected lack of published results, with the only exception being Ref. [9] which, however, requires a rather detailed inquiry about various unreported instrumental parameters needed for an accurate modeling of the experimental conditions. It is also quite surprising that the neutron response of HD, which belongs to an important class of systems anyway, has, for decades, escaped a detailed treatment like those elegantly devised, in the mid 1960s, on homonuclear diatomic molecules by [PERSON] and Koppel (YK) [12] and by [PERSON] [13], or on other linear molecules by [PERSON] [14].