# Hydrogen Deuteride for Cold Neutron Production: A Model for the Double Differential Cross Section
## 3 Results
Indeed, the absence of spin correlations in HD does not introduce different scattering cross sections according to the parity of the levels involved in the transitions and the spin statistics of the nuclei in the molecule. Conversely, in para-H\({}_{2}\), if \(E_{0}\) is insufficient to excite the first rotational line \(J_{0}=0\to J_{1}=1\), then only the even-even \(J_{0}=0\to J_{1}=0\) transition contributes, and spin correlations can be found [19] to weight such kinds of transitions only by the coherent cross section. Therefore, at some incident energies (\(E_{0}<14.7\) meV), para-H\({}_{2}\) behaves coherently and the distinct dynamics plays a role. In HD, this is never the case, and the huge incoherent cross section of H dominates the scattering behavior independently of \(E_{0}\). In order to make a comparison with the experimental total cross section possible, the DDCS of HD was calculated at \(T=17\) K as a function of scattering angle \(\theta\) and exchanged energy \(E=\hbar\omega\) at various values of the incident energy \(E_{0}\) ranging between 1 and 85 meV. At such energies, only rotations are excited and the main contribution to the spectra comes from the elastic \(J_{0}=0\to J_{1}=0\) line and the mentioned \(J_{0}=0\to J_{1}=1\) Stokes transition (when excited). Double integration of the DDCS over the exchanged energy and solid angle, according to:  
\[\sigma_{\text{s}}=\int_{\Omega}d\Omega\int_{-\infty}^{\omega_{0}}\,d\omega\frac {d^{2}\sigma}{d\Omega d\omega}, \tag{9}\]  
provides the total scattering cross section \(\sigma_{\text{s}}\) shown in Figure 4, in comparison with [PERSON]'s data [21], from which the known absorption cross section \(\sigma_{\text{a}}(E_{0})\)[41] has been subtracted: in formulas, \(\sigma_{\text{s}}^{expt.}=\sigma^{expt.}-\sigma_{\text{a}}\). The effect of the neglect of the distinct part in the calculation of \(\sigma_{\text{s}}\) is observed to be very small (compare the blue circles with the red stars in Figure 4), and slightly appreciable only at very low \(E_{0}\) values. Unexpectedly, agreement between the data and calculations is not as good as the one found for normal H\({}_{2}\) and D\({}_{2}\)[4, 20]. In Ref. [21], the author reports an average systematic error of \(\pm 8.1\%\) and an average statistical error of \(2.8\%\) for HD. Figure 5 shows the possible effect of shifting [PERSON]'s data upwards by the systematic error (empty circles), while statistical uncertainties are taken into account by the corresponding error bars. By doing so, a good agreement is recovered at all energies (given that the experimental data points are rather scattered), except in the low-energy range \(1<E_{0}<20\) meV, a disagreement which is worth investigating and discussing. Figure 3: Calculated double differential cross section of HD at an incident neutron energy \(E_{0}=80\) meV and at three example \(Q\) values, for \(T=17\) K. In panels (**b**,**c**), the \(J_{0}=0\to J_{1}=1\) rotational line at \(E\simeq 11\) meV starts to be appreciable compared to the elastic one. The progressive broadening and increasing detailed-balance asymmetry of the lines can also be observed as \(Q\) grows. Figure 4: Dependence on the incident energy \(E_{0}\) of the total neutron scattering cross section of HD at 17 K, as obtained by absorption-subtracted [PERSON]’s data [21] (black full squares) and by double integration of Equation (7) following Equation (9) (red stars), using the GA plus Skold approximations for the CM translational dynamics. The blue empty circles represent the self component of \(\sigma_{5}\) alone, as obtained by double integration of the last term in Equation (7). The small effect of the absorption cross section \(\sigma_{\rm a}\) of HD (cyan line) is also shown. Figure 5: Black full squares and red stars with thin line represent [PERSON]’s data and present calculations, respectively, like in Figure 4. Empty circles with error bars take the systematic and statistical uncertainties of the measurements into account, as explained in the text.