## 3 Results and Discussion
The rise of \(c_{H}\) caused \(\gamma_{\text{us}}\) of the {112}\(<\)111\(>\) slip system to rise and that of the {110}\(<\)111\(>\) slip system to fall [28], that is, to inhibit the slips of the {112} planes and promote the slips of the {110} planes. Therefore, at most NNs, \(\rho_{[110]}>\rho_{[112]}\) is clearly beneficial to the occurrence of shears and slips when \(c_{H}\) rises. This is the reason why the uniformly distributed hydrogen generally induces softening in \(\alpha\)-Fe. Figure 6: Clustering and averaged partial PCF analysis for the hydrogen distribution effects on \(\Delta B^{dis}\). (**a**) Mean-shift clustering for \(\Delta B^{dis}\), with an x axis of \(\Delta M^{dis}_{EG}\), transformed using Equation (16). (**b**) The averaged partial PCF \(\overline{g}(r)\) for structures in clustered groups. \(\Delta B^{dis}\) decreased with the hydrogen atoms distributed in the 7 NN. At the same time, \(\rho_{[112]}\) was the greatest at 10 NN and was significantly larger than \(\rho_{[110]}\); therefore, an increment of occupied hydrogen atoms at this position will inhibit shears and slips, which can increase E and G, and cause hardening. Figure 7b shows that \(\rho_{[112]}\) at 7 NN is second only to 10 NN but is still smaller than \(\rho_{[110]}\). This indicates that the increase of hydrogen on 7 NN will affect the elastic properties, but without inducing hardening. At 13 NN, 14 NN, and 27 NN, \(\rho_{[112]}\) and \(\rho_{[110]}\) are very close, and their effects on shears and slips may cancel each other out. It can be seen from Figure 5 that the hydrogen occupancy of 5 NN also significantly promoted E and G, and Figure 6 shows that hydrogen occupancy of 7 NN significantly inhibited B; thus, we analyzed the planes that 5 NN and other nearby neighbors jointly affect, and the effect of s7 NN planes. The positions of 5 NN and 6 NN (denoted as s\({}^{5\text{NN-6 NN}}\)) jointly determined the \(\{112\}\) planes, while considering the positions of 5 NN and 10 NN (denoted as s\({}^{5\text{NN-10 NN}}\)) increased \(\rho_{[110]}\). And the geometric structure of the densest planes (DPs) of s\({}^{7\text{NN}}\) may be the reason why it affects B. Figure 8a,c,e shows the positions of s\({}^{5\text{NN-6 NN}}\), s\({}^{5\text{NN-10 NN}}\), and s\({}^{7\text{NN}}\) in the lattice of \(\alpha\)-Fe, and the DPs determined by them. The DPs determined by s\({}^{5\text{NN-6 NN}}\) belong to the \(\{210\}\) and \(\{531\}\) planes, which are colored by green and pink, respectively, in Figure 8a, and they are located on both sides of Z. There are only two DPs of s\({}^{5\text{NN-10 NN}}\), belonging to the \(\{100\}\) family, and they are perpendicular to each other, and are colored by green and pink respectively in Figure 8c; The DPs of s\({}^{7\text{NN}}\) contain two different sets of \(\{120\}\) (blue and pink respectively) and one set of \(\{100\}\) faces (green). The details of these DPs are listed in Table 3.  
Figure 7: Statistics of the densities of the sites on nth nearest neighbors, which can be occupied by hydrogen atoms on different slip planes in \(\alpha\)-Fe. (**a**) The sketch map of the sites can be occupied by hydrogen atoms on the section of (hkl) planes and the spherical shells can be determined by the positions at the same distance from the reference site Z. The positions are distributed on the concentric spherical shells determined by the distances from Z, and those positions with the same distance from Z are distributed on the same spherical shell. Using one of the \(\{\text{hkl}\}\) planes to intersect a certain shell, the hydrogen atoms shared by the shell and the plane determine the local \(c_{H}\) on this plane. A set of parallel planes intersects this spherical shell, and the effect on the slip depends on the averaged concentration of this set of crystal planes. (**b**) The Tet-site densities \(\rho_{[110]}\) and \(\rho_{[112]}\) on the \(\{110\}\) and \(\{112\}\) planes at the nth NN. They are greater than 0 at many NN, and generally \(\rho_{[110]}>\rho_{[112]}\).