## 3 Results and Discussion  
The EAM potential [36; 37] used in this work is widely used when simulating the HE of \(\alpha\)-Fe single crystals [28; 34; 51] and is also suitable for elasticity and thermal expansion problems [52]. As shown in Table 1, we calculated that the elastic constants and moduli of \(\alpha\)-Fe single crystals were also very close to the experimental results. After calculating the elastic constants of \(\alpha\)-Fe single crystal at T \(\in\) [0, 500] K using the fluctuation method [44], by fitting Equations (6) and (9), we obtained the following parameters, listed in Table 2:  
Figure 1 was obtained from Equations (9) and (10). \(C_{11}\), \(C_{12}\), and \(C_{44}\) all decreased with the temperature increase and volume expansion, and their changing trend is basically in accordance with previous studies [52; 53]. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Data Source**} & \multicolumn{3}{c}{**Elastic Constants (GPa)**} & \multicolumn{3}{c}{**Elastic Moduli (GPa)**} \\ \cline{2-7}  & **C\({}_{11}\)** & **C\({}_{12}\)** & **C\({}_{44}\)** & **E** & **B** & **G** \\ \hline This work & 247.13 & 146.47 & 116.75 & 231.83 & 180.02 & 90.18 \\ Experiment [53] & 239.55 & 135.75 & 120.75 & 236.50 & 170.35 & 93.21 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The elastic constants and moduli of \(\alpha\)-Fe. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Result** & **Parameter** & **Result** & **Parameter** & **Result** \\ \hline \(y_{1}\) (Å\({}^{3}\)/atom) & \(-\)0.03212 & \(b_{1}\) (Å\({}^{3}\)/atom) & 0.03199 & \(b_{2}\) (K) & \(-\)286.533 \\ \(y_{1}^{11}\) (GPa) & \(-\)121.484 & \(b_{3}^{11}\) (GPa) & 118.488 & \(b_{4}^{11}\) (K) & 1085.129 \\ \(y_{2}^{12}\) (GPa) & \(-\)45.014 & \(b_{3}^{12}\) (GPa) & 45.113 & \(b_{4}^{12}\) (K) & 250.363 \\ \(y_{2}^{44}\) (GPa) & \(-\)44.799 & \(b_{3}^{14}\) (GPa) & 45.431 & \(b_{4}^{14}\) (K) & 253.642 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fitted parameters for the relationships of the elastic constants and the changes of temperatures and volumes. Figure 2 shows the changes of the elastic constants \(C_{ij}\) with \(c_{H}\). \(C_{ij}^{vol}\) and \(C_{ij}^{coh}\) are defined as follows:  
\[ \begin{array}{l}C_{ij}^{vol}(c_{H})=C_{ij}(0)+\Delta C_{ij}^{vol}(\Delta V(c_{ H}))\\ C_{ij}^{coh}(c_{H})=C_{ij}(0)+\Delta C_{ij}^{coh}(\Delta V(c_{H}))\end{array}  \tag{15}\]  
From Figure 2, if the \(c_{H}\) increases, \(\Delta C_{ij}^{vol}\) always decreases, while \(\Delta C_{ij}^{coh}\) continues increasing, and their total effect reflected by \(\langle C_{ij}^{v}\rangle\) is slightly reduced, which is consistent with the first-principle calculation results [27]. Figure 3 shows the effect of the \(c_{H}\) on the elastic moduli. It can be seen from Figure 3a that for E, B, or G, as \(c_{H}\) increases, \(\Delta M^{coh}\) always increases, while \(\Delta M^{vol}\) always decreases. Their common result is a slight decrease, and the reduction of E is more obvious. This indicates that when hydrogen is ideally uniformly distributed in \(\alpha\)-Fe single crystals, the chemical bonding process causes hardening, the volume expansion causes softening, and the overall result is light-softening, which is also consistent with the first-principle calculations [27]. In this work, the \(c_{H}\) was significantly lower than the first-principle calculations but still higher than the general experimental con  
Figure 1: Elastic constants of \(\alpha\)-Fe, influenced by (**a**) the temperatures, and (**b**) the changes of volumes. Figure 2: Elastic constants of \(\alpha\)-Fe influenced by hydrogen concentrations (\(c_{H}\)), in which the \(C_{ij}^{vol}\) decreases and the \(C_{ij}^{coh}\) increases with the increase of \(c_{H}\). softening was still very slight, which may be the reason why the change of E in the experiment was not easily detected [39]. Figure 3b shows that the microscopic distribution of hydrogen caused the elastic moduli deviations and the average results.