Dynamic investigations on hydrogen–helium interaction around the vacancy in BCC iron from ab-initio calculations

Coexistence of hydrogen (H) and helium (He) under vacancy (V) supersaturation in the fusion environment alters the dynamic evolution of cavities and ultimately influences the swelling of structural materials. Herein, we investigate H–He interaction around a V as one prototype trapping site for H and He in body-centered cubic (BCC) iron (Fe) utilizing ab initio calculations from the thermal dynamics. First, we demonstrate the significantly stronger He–V interaction than H–V interaction by comparing the dynamic trapping and de-trapping of H with those of He. Furthermore, we confirm the repulsive H–He interaction around the V by examining their hopping around H–He–V complexes. The prior He in the V imposes weak influence on the dynamic trapping of H while enhances H de-trapping. Due to the prior He, more H atoms can be accommodated in the V resulting from larger H–H distances to attenuate repulsive H–H interaction. The dynamic trapping of He by the V is weakly influenced, even though the V is densely coated by the prior H. There exists a critical density of the prior H in the V, below which the prior H enhances He de-trapping. Above this critical density, He de-trapping is inhibited by the prior H. This work provides significant dynamic insights at the atomic scale toward a better understanding of the cavity nucleation and H isotopes/He retention in structural materials in the fusion environment.


Introduction
Fusion energy is an efficient and environmentally friendly energy source that provides a crucial pathway to tackle the energy crisis. The development of nuclear structural materials that can withstand extremely hostile service conditions is recognized as one of the primary challenges in developing advanced nuclear energy systems [1][2][3]. Under the intense flux of irradiation particles, lattice atoms in nuclear structural materials are displaced from their equilibrium sites, generating high excess concentrations of vacancies and interstitials and resulting in displacement damage. Long-time evolution of these displacement defects leads to complex microstructural variations of nuclear structural materials that degrade material properties, involving swelling, embrittlement, hardening, and creep [4]. Among these irradiation damage effects, one pivotal concern is the swelling of structural materials, which stems from the generated cavities, severely threatening the integrity of nuclear structural materials and the operational safety of nuclear reactors. Herein, the term cavity refers to a small volume consisting of vacancies and (or) hydrogen (H) and (or) helium (He) gas atoms as a generic term for void and bubble [5]. Cavity swelling rate has been utilized as one of the pivotal criteria for material designing and screening in the reactor design [6,7].
In the blanket structural materials of fusion reactors, neutron irradiation generates a substantial amount of H and He atoms via nuclear transmutation (n, p) and (n, α) reactions due to the hard energy spectra [8,9]. It was shown experimentally that the coexistence of H and He significantly alters the cavity swelling of nuclear structural materials, although these experimental results are not always consistent [10][11][12][13][14][15][16][17][18][19][20]. Experiments on materials irradiated by fusion neutron at low dose revealed that the coupling of H with He significantly enhances the cavity swelling of nickel and copper [10,11]. Due to the lack of available high-flux fusion-neutron dedicated sources, triple-ion beams (heavy ion + H + He) are considered as one promising method for investigating materials behaviors under fusion-neutron irradiation. Substantial experimental studies utilizing the triple-ion beams indicate that the coexistence of H and He can enhance the cavity swelling of materials in the fusion environment [12][13][14][15][16][17]. On the other hand, there are some experimental results that show that the cavity swelling is not invariably improved under the triple-ion beams [18][19][20]. To better understand the H-He synergetic behaviors in altering the cavity swelling of nuclear structural materials in the fusion environment, investigations on H-He interaction around the vacancy (V) defect are necessary.
There are few direct experimental studies on H-He interaction in the blanket structural materials because of the lack of high-flux fusion-neutron sources, while plasma-facing materials (PFMs) will encounter similar simultaneous damages from fusion neutrons, H isotopes, and He. Meanwhile, extensive experimental efforts have been devoted toward understanding H-He interaction in PFMs, which facilitates the current understanding of H-He interaction in the blanket structural materials. After He pre-irradiation, the trapping of subsequently implanted H isotope within the He-rich zone at the near surface is enhanced in body-centered cubic (BCC) W, and the Hisotope diffusion into the bulk region is hindered, significantly reducing the H-isotope retention in the bulk region [21][22][23][24][25][26]. A similar phenomenon is also found for deuterium (D) retention in the bulk region of reduced activation ferritic/martensitic steel under He ion pre-injection [27]. He atoms added into the D plasma are also found to reduce the D retention in the bulk region of BCC W and Be [28,29]. Under the simultaneous H and He exposures in BCC W, blistering through the subsurface region is suppressed [24,30]. These experimental results indicate that the He-rich regions can serve as effective traps for H. In contrast, the prior H atoms impose a weak influence of He retention [22,23]. It is noteworthy that He irradiation does not invariably decrease H-isotope retention in the bulk region [31,32]. For instance, Ialovega et al have observed that in BCC W pre-irradiated by He, D retention at the near surface alters little compared with the pristine W after annealing at relatively low temperature (below 900 K), indicating a weak influence of He on H-isotope retention [31].
Computational tools can unravel atomic details on H-He interaction in materials. Iron-based alloys are proposed as promising candidates for the blanket structural materials of future fusion reactors [4,[33][34][35][36]. Therefore, computational studies on H-He interaction in BCC iron (Fe) are crucial in developing advanced blanket structural materials. Ortiz et al have shown the repulsive H-He interaction in the perfect lattice of BCC Fe [37]. The He-V complexes are demonstrated to trap H in BCC Fe [37][38][39]. However, the binding energies of H are reduced compared with those of H to corresponding pure V defects [37,39], implying the repulsive H-He interaction in the V defects. A similar phenomenon has been found for H trapping at BCC Fe-He interfaces [40] and at BCC W-He interfaces [41]. The H atoms in the V defects reduce the binding energies of subsequent He atoms [37,39]. Note that H-He interaction may be metal-dependent. For instance, in the region of perfect BCC W lattice, He is shown to attract H (a binding energy of ∼0.25 eV) by repelling the surrounding electron gas [42][43][44], while the case is not for BCC Fe lattice [37]. Additionally, Zhou et al have shown that He can hinder the formation of H 2 molecules in the V in BCC W, while Chen et al have demonstrated that the formation of H 2 molecules is enhanced by increasing He pressure in the V defects in BCC Fe [45]. Most calculations, by calculating the binding energies of H and He to the H-He-V complexes, have exhibited the considerable H-He interaction around the V defects from a static viewpoint. Nonetheless, in the fusion environment, the evolution of H-He-V complexes is essentially a dynamic process involving the processes of formation and dissolution of H-He-V complexes, where the dynamic trapping and de-trapping of H and He play a crucial role. Investigations on these dynamic processes of H and He can provide crucial insights into better understanding of the evolution of H-He-V complexes at longer timescale.
The present work aimed at investigating the H-He interaction around the V as one prototype trapping site for H and He in BCC Fe using ab initio calculations from the dynamic viewpoint. We first compare H-V interaction and He-V interaction from the dynamic trapping and de-trapping of H and He. Furthermore, the influences of He in the V on the dynamic trapping and de-trapping of H and the influences of H in the V on the dynamic trapping and de-trapping of He are studied. This work provides important dynamic insights at the atomic scale into a better understanding of the cavity nucleation and H isotopes/He retention in structural materials in the fusion environment.

Computational methods
Energy calculations in this work are performed using the density functional theory (DFT) [46,47], as implemented in the VASP code [48]. The projector augmented-wave method [49] is applied to describe the interactions between the valence electrons and ions, and the Perdew-Burke-Ernzerhof functional within the framework of generalized gradient approximation [50] is applied. Plane waves [48] are included to a cutoff of 500 eV, and the first-order Methfessel-Paxton smearing scheme is applied with a smearing of 0.2 eV. As done in previous studies [20,42], we utilize the supercells of size 4 × 4 × 4 to simulate the behaviors of H and He around a V. For the Brillouin zone sampling, a Γ-centered 3 × 3 × 3 Monkhorst-Pack k-mesh [51] is applied. Energy calculations are ended when the total energy is converged to 10 −5 eV and the forces exerted on the ions are converged to 0.01 eV Å −1 . Spin polarization is considered in all ab initio calculations. The zero-point energy (ZPE) of H is considered by summing over its normal vibrational modes.
The binding energy of H/He to a defect X is defined to be [52] where E f (H/He), E f (X), and E f (HX/HeX) denote the formation energy of H/He at the tetrahedral interstitial site (TIS), the formation energy of the defect X, and the formation energy of the complex HX/HeX, respectively. The present work primarily investigates the dynamic trapping of H/He by a V and the dynamic de-trapping of H/He from the V, as shown in figure 1, focusing on the minimum energy pathway (MEP) and the hopping barrier, as shown in figure 1, and the hopping rate at finite temperature. Herein, the dynamic evolution of the state of a system can be interpreted as the alteration of the coordinate of the hopping atom [53]. The dynamic trapping of H/He refers to the process where H/He hops from one site to an adjacent site with a hopping rate at finite temperature to gradually approach the V and finally be trapped. The dynamic de-trapping of H/He refers to the process where the trapped H/He hops to one site beyond the interaction radius between H/He and the V at a hopping rate at finite temperature.
The energy barrier calculations are performed utilizing the climbing-image nudged elastic band (CI-NEB) method [54].
To calculate the hopping rates of H and He, we adopt a dynamic model called single-atom statistical model (SASM) based on the canonical ensemble theory within the framework of statistical mechanics [55][56][57][58]. Details on how the SASM works and validations of ab initio calculations and the SASM can be found in the supplementary materials.

Results and discussion
3.1. Dynamic hopping of H and He: from the perfect lattice to a V We first study the dynamic hopping of H and He from the perfect lattice to a V (denoted as V 1 ) as one prototype trapping site. Table 1 shows the discrete grid of possible binding sites (local energy minima) for H and He around V 1 that are constructed according to their distances from V 1 and the symmetrical consideration. Figure 2(a) schematically illustrates partial sites among the binding sites listed in table 1. The binding energies of H and He at the feasible binding sites are given in table 1. Except for the nearest binding site around V 1 , the binding energies of H and He at these binding sites are low (less than 0.10 eV). The nearest binding sites before being trapped by V 1 for H and He are the T 2 and T 3 sites, respectively, indicating a larger trapping radius of He by the V than that of H.
The trapped H preferentially occupies the site with a slight offset from the O 1 site toward V 1 by 0.24 Å, where the electron gas density is 0.12 e Å −3 . In contrast, the trapped He occupies the center of V 1 with the lowest electron gas density (0.04 e Å −3 ). The distinct trapping sites of H and He in the V are attributed to their electronic structures and may account for the 'shell structure' of the H-He-V complexes observed experimentally [59,60] and computationally [39,45]. Additionally, the solution energies of H and He in the homogeneous electron gas also indicate the distinct trapping sites of H and He in the V [61]. Herein, the binding energies of H and He to V 1 are calculated as 0.71 and 2.44 eV, respectively, in close agreement with previous computational [62][63][64] and experimental studies [65,66]. He strongly repulses the electron gas in metals due to its closed-shell structure [67]. Therefore, it is expected that the binding energy of He to the V is appreciably higher than that of H to the V, since the strong repulsive interaction between He and the electron gas can be significantly attenuated in the V. The computationally stronger He-V interaction than H-V interaction in metals is consistent with experimental observations, such as the Doppler broadening spectroscopy for Steel of Institute of Modern Physics steels irradiated by H and He ions [68] and the higher peak temperature of He desorption spectra than that of H in Be [69].
To illustrate the motion of H and He around the trapping site from the dynamic viewpoint, the hopping rates of H and He around V 1 are calculated. When determining the feasible hopping pathways of H and He at a given TIS around V 1 , we use the hopping pathways of H and He in the perfect lattice as reference, i.e. hopping to the first nn TIS and 2nn-TIS (see figure S2(a)), and ultimately determine the feasible pathways based on CI-NEB calculations. Figure 2(b) shows the hopping barriers of H at the T 5 , T 4,1 , T 3 , and T 2 sites along all feasible directions. Schematic illustrations of these hopping processes can be found in figure S3. The hopping barriers when  The discrete grid of possible binding sites of H and He around V 1 , the equivalent number (Neq) of these sites, and the binding energies of H and He at these sites. Op, Lp, and Tp,q denote the pth nearest-neighbor (nn) octahedral interstitial site of V 1 , the p-nn lattice site of V 1 , and the p-nn TIS of kind q. For the coordinates of these sites, the center of V 1 is taken as the original point. For each kind of TIS, x, y, and z can be exchanged due to the symmetry. The symbol '×' indicates that H/He initially assigned at this site will be spontaneously trapped by V 1 after relaxation, or this site is a maximum energy site.

Notation
Neq H hops toward V 1 are slightly lower than those in the perfect lattice (for details on the comparison, see tables S2 and S3 in the supplementary materials), indicating the attractive interaction between H and V 1 . Figure 2(d) shows the hopping rates at these TISs at 500 K-1000 K. At most sites, the hopping rates slightly exceed those in the perfect lattice due to the lower hopping barriers when H hops toward V 1 . The slightly lower hopping rate at the T 2 site than that in the perfect lattice is attributed to the increased hopping barriers when H hops away from V 1 . Nevertheless, the hopping rates at these TISs alter slightly compared with hopping in the perfect lattice. This slight alteration is attributed to the appreciably low hopping barriers of H in the perfect lattice (about 0.03-0.04 eV) despite the attractive interaction between H and V 1 . Figure 2(c) shows the hopping barriers of He at the T 3 and T 4,1 sites, and figure 2(e) shows the hopping rates at 500 K-1000 K. Schematic illustrations of these hopping pathways can be found in figure S4. Akin to H, the hopping rates of He at these sites slightly exceed the hopping rate in the perfect lattice. Before being trapped, both H and He hop quite rapidly around V 1 (∼10 13 Hz). The hopping rates decay with the reciprocal temperature following the Arrhenius relation, while these rates are not appreciably sensitive to temperature due to low hopping barriers. Compared with hopping around V 1 before being trapped, the hopping rates of the trapped H and He alter dramatically, as indicated by the binding energies in table 1. The de-trapping barrier provides an appropriate depiction of the required energy for the trapped H and He to migrate away from the trapping site. Due to being trapped at the inner surface of V 1 , the trapped H can vibrate around V 1 (hopping to another O 1 site), or de-trap from V 1 , while the trapped He can only move along the de-trapping pathway. The de-trapping pathway is essentially determined by the initial site and the final site. Herein, we construct the possible de-trapping pathways according to the feasible binding sites in table 1 and determine the feasible de-trapping pathways using CI-NEB calculations. Figure 3  He, and figures 3(b) and (c) show the energy profiles and the de-trapping rates, respectively. Owing to the relatively small binding energy of H to V 1 compared with He, the trapped H will de-trap from V 1 at a frequency of about 10 7 Hz at 500 K, while the trapped He can hardly de-trap even at 1000 K. From figures 3(c) and 2(d), (e), the de-trapping rates of H and He from V 1 alter with varying temperature more remarkedly in comparison to the hopping rates of H and He around V 1 before being trapped. Additionally, we investigate the trapping of multiple H and He atoms in V 1 . The maximum number of H atoms that can be trapped by V 1 is calculated as six, while the binding energy of He to V 1 remains larger than 1.5 eV even if V 1 has already trapped six He atoms (see figure S5), which further confirms the stronger He-V interaction than H-V interaction.

Influence of prior He in the V on H trapping and de-trapping
The H-He interactions in the vacuum environment and the perfect lattice are repulsive, as shown in figure S6. To investigate the H-He interaction around the V from the dynamic viewpoint, the impact of the prior He in V 1 on subsequent H trapping is examined. We first investigate the dynamic process of H trapping by the He 1 V 1 complex. The symmetry of the He 1 V 1 complex is preserved as that of V 1 , since the trapped He exactly occupies the center of V 1 . Therefore, table 1 can be directly applied to depict the binding sites of H around the He 1 V 1 complex. Figure 4(a) shows the binding energies of H at TISs around the He 1 V 1 complex. The binding energies at most sites are appreciably low, indicating that the hopping barriers of H before being trapped by the He 1 V 1 complex will not evidently deviate from those in the perfect lattice. The binding energy of H at the T 2 site of the He 1 V 1 complex is 0.20 eV, exceeding that of H at the T 2 site of V 1 (0.07 eV). When trapped by the He 1 V 1 complex, the binding energy of H is 0.55 eV, lower than that of H to V 1 (0.71 eV). The lower binding energy of H to the He 1 V 1 complex than that of H to V 1 indicates the repulsive H-He interaction in the V. The number of H atoms trapped in He bubbles in steels is experimentally shown to increase with decreasing He pressure, also indicating the repulsive H-He interaction [70]. Similar repulsive H-He interaction has been reported in BCC W, as indicated by the experimental shift of the D desorption peak to lower temperature in He bubbles-enriched W compared with non-damaged W [31].
The reduced binding energy of H to the V resulting from the prior He can be interpreted from the electronic state. As shown in figure 5(a)   increases in the lower energy region (−2.1 to −0.7 eV). This indicates that the prior He in the V can stabilize the electrons of the neighboring Fe matrix atom. Since the trapped H in the V exhibits attractive interaction with the neighboring Fe matrix atom [71], the prior He in the V can weaken the attraction of H with the neighboring Fe matrix atom and then reduce the binding energy of subsequently trapped H to the V. Figures 5(b) and (c) show the overlap between the s-orbital PDOS of H and the d-orbital PDOS of the neighboring Fe matrix atom when H is trapped in V 1 and He 1 V 1 , respectively. Compared with being trapped in V 1 , this overlap becomes weaker when H is trapped in the He 1 V 1 complex. This indicates that the prior He in the V can attenuate the hybridization between the trapped H and the neighboring Fe matrix atom, thus attenuating the binding of H to the V. Figure 4(b) shows the hopping rates of H at the T 3 and T 4,1 sites of the He 1 V 1 complex. These hopping rates alter slightly compared with those of H at the T 3 and T 4,1 sites of V 1 . Therefore, the prior He in the V will not evidently inhibit the subsequent H trapping. After being trapped by the He 1 V 1 complex, H resides near the O 1 site of V 1 , and He deviates from the center of V 1 by 0.43 Å. Figure 4(c) illustrates the feasible de-trapping pathways of H from the He 1 V 1 complex by CI-NEB calculations. For the de-trapping pathway from the O 1 site to the T 2 site, the energy at the saddle point is lower than that at the T 2 site after ZPE correction. Therefore, the H atom needs to migrate further to one TIS (the T 3 site or the T 4,1 site) to de-trap from the He 1 V 1 complex. Figure 4(d) shows the corresponding energy profiles for these de-trapping pathways. The potential energy profiles of H de-trapping from the He 1 V 1 complex become smoother, with the lower barrier heights and the increased barrier widths, compared with H detrapping from V 1 , as shown in figure 3(b). Figure 4(e) shows the de-trapping rates of H from the He 1 V 1 complex and V 1 at 500 K-1000 K. The residence time of H in the He 1 V 1 complex is evidently reduced compared with that in V 1 despite the capability of the He 1 V 1 complex to trap H. For instance, the prior He in V 1 increases the detrapping rate of H by over one order of magnitude at 500 K. The increased de-trapping rate of H from the He 1 V 1 complex is associated with the collapse of the area of low electron density caused by He [42]. The ratio of the de-trapping rate of H from the He 1 V 1 complex to that of H from V 1 becomes progressively larger with descending temperature, indicating that the influence of the prior He on H de-trapping is expected to be more significantly at lower temperatures. The number of H atoms in the trapping site is determined both by the trapping rate and de-trapping rate of H. The hopping rate of H before being trapped by the He 1 V 1 complex is not sensitive to temperature, as shown in figure 4(b). Therefore, the prior He in the V influences the number of trapped H atoms more significantly at lower temperatures. The reduced de-trapping barrier of H from the He 1 V 1 complex compared with that of H from V 1 indicates the repulsive H-He interaction in the V. With increasing number of prior He atoms in the V, the repulsive H-He interaction will become more evident, and the binding energy of H to the V is reduced more remarkably [37,39]. Consequently, the de-trapping rate of the trapped H is expected to increase more significantly.
Although experimental results indicate that He bubbles can trap H atoms [59], this is associated with areas of low electron density around the He bubbles, rather than the direct attractive H-He interaction. Displacement defects (interstitial and V) can be annihilated by annealing [72], while He-V complexes cannot be readily removed by annealing since He will hinder the recombination between displacement defects [73]. Therefore, at elevated temperatures where pure V defects are relatively lacking, the He-V complexes can serve as strong trapping sites for H and dominate over the effect of displacement damage on H-isotope retention, as also observed in experiment [31,74].
To further investigate the effect of the prior He in V 1 on subsequent H binding, we calculate the binding energies of H + H i-1 He 1 V 1 = H i He 1 V 1 (i = 1−12). Herein, we apply the Wigner-Seitz (W-S) primitive cell method [64] to characterize the configurations of the H i He 1 V 1 complexes. Figure 6(a) shows the W-S primitive cell corresponding to the He 1 V 1 complex. There exist six W-S squares within the {100} planes, and on each W-S square, the center is an O 1 site, and four corners are the T 1 sites. Figure 6(b) shows the unrelaxed configurations of the H i He 1 V 1 complexes for structure optimization. The He atom is assigned at the center of V 1 , H atoms are assigned at the centers of the W-S squares with i ⩽ 6, and partial H atoms are assigned at the corners of the W-S squares with i > 6 since there only exist six O 1 sites around V 1 . The introduction of H atoms into the He 1 V 1 complex leads to an evident deviation of He from the center of V 1 after relaxation. With the consecutive introduction of H atoms, partial H atoms move toward the corners of W-S squares even with i ⩽ 6. Unless otherwise stated, we consider the most stable H i He 1 V 1 complexes in the following. Figure 6(c) shows the cumulative distributions of H-He instances in the H i He 1 V 1 complexes for i ⩽ 6 and i > 6. With i ⩽ 6, the H-He distances are primarily distributed around 1.7 Å, while with increasing number of H atoms the H-He distances evolve into a bimodal distribution, which centers at about 1.8 and 2.1 Å. The increasing H-He distances with increasing number of H atoms can attenuate the repulsive H-He interaction and reduce the total energy of the whole system. Figure 6(d) shows the binding energies of H to the H i He 1 V 1 complexes, where those of H to the H i V 1 complexes are given for comparison. The binding energies of H are reduced by the prior He in V 1 when the number of H atoms is relatively small (i < 6). In contrast, the binding energies of H to the H i He 1 V 1 complexes exceed those to the H i V 1 complex with i > 6. The He 1 V 1 complex can trap up to 11 H atoms, exceeding the number of H atoms that V 1 can trap. Therefore, the influence of the prior He in the V on H binding is two-sided, which weakens H binding to the V while improves the maximum number of H that can be trapped. Similar results have been found in BCC W [42], Cr [20], and Mo [75]. From the dynamic viewpoint, the number of H atoms in the V defect is determined by both the trapping site of H and the de-trapping site of H. Since H atoms randomly diffuse into the V defect, high production rate of H indicates that more H atoms can be trapped in the V defect. Given that the prior He in the V defect can increase the de-trapping rate of H, high production of H can compensate the effect of the prior He in the V defect on H de-trapping. Under the low production rate of H, this compensation effect is weaker, and consequently, the influence of H on the V evolution will also become weaker due to the prior He.
The V defect can only trap a finite number of H atoms due to the repulsive H-H interaction [64]. Figure 6( Generally, the cavity is required to reach the critical size to be nucleated, a prerequisite for the subsequent growth of the cavity [20,76]. H atoms can be trapped on the surface of the cavity and facilitate the nucleation of the cavity [12,77]. Figure 6(f ) shows the distances between the trapped H atoms and the center of V 1 for the H i He 1 V 1 complexes and the H i V 1 complexes. The H-V distances are increased by the prior He in V 1 with i ⩽ 6; meanwhile, the maximum number of H atoms that can be trapped also increases. This indicates a larger size of the H i He 1 V 1 complex than the H i V 1 complex. Therefore, the prior He atoms in the cavity can facilitate the cavity nucleation by making it easier for the cavity to reach up the critical size. This may account for the larger cavity density under the triple-ion beam irradiation than that under the heavy ion + H irradiation [20,77]. Additionally, He atoms can facilitate the cavity nucleation by improving the binding energies of vacancies to V clusters to stabilize the V clusters [63,73].

Influence of prior H in the V on He trapping and de-trapping
To probe how the prior H in the V influences subsequent He trapping and de-trapping, we first examine the dynamic trapping of He by the H 1 V 1 complex and the dynamic de-trapping of He from the H 1 V 1 complex. The prior H in V 1 changes the original isotropic symmetry of V 1 , and the number of various TISs around the H 1 V 1 complex increases compared with V 1 . Herein, the concept of 'boundary trial' [78] is applied. The results show the repulsive H-He interactions in the perfect lattice and the V of BCC Fe. To study the binding energy of He around the H 1 V 1 complex, two limiting TISs are considered in DFT calculation for each kind of TISs in table 1. These two limiting TISs are separated from the prior H atom by the longest and shortest distances, distinguished by the symbols l and s, respectively. It can be reasonably expected that the behavior of He at the given kind of TISs shall be ranged between these two limiting TISs. Figure 7(a) shows the binding energies of He at the TISs around the H 1 V 1 complex. For these two limiting TISs, the binding energy of He at the TIS symbolized by l slightly exceeds that at the TIS symbolized by s. This indicates the repulsive H-He interaction around V 1 . The He atom initially assigned at the T 3 -s site repels one adjacent Fe atom to the original V 1 , spontaneously trapped by the new V 1 after relaxation. The He atom initially assigned at the T 3 -l site can be trapped by the original V 1 with a significantly low barrier (less than 0.05 eV). Thus, the prior H in V 1 hardly hinders the subsequent He trapping.
The dynamic hopping of He at the T 4,1 -l and T 4,1 -s sites is also examined. Figure 7(b) shows the hopping barriers along all feasible directions at these two sites. Schematics of the corresponding MEPs can be found in figure S7. The barriers for the He atom at the T 4,1 -l and T 4,1 -s sites to be directly trapped by the H 1 V 1 complex are both calculated as 0.20 eV, equal to the barrier when He hops from one TIS to its 2nn-TIS in the perfect lattice. This indicates that the prior H in V 1 will not evidently hinder the subsequent He trapping. Figure 7(c) shows the hopping rates of H at the T 4,1 -l and T 4,1 -s sites. The hopping rates of He at the T 4,1 -l and T 4,1 -s sites are slightly influenced by the prior H compared with the hopping rate in the perfect lattice. It is reasonably expected that when the V is surrounded by H atoms of low density, the subsequent He trapping by the V will not be significantly hindered. The influence of the prior H on subsequent He trapping can even be weakened by the interstitial atoms that can knock out the H around the V, such as in Pd [79].
After being trapped by the H 1 V 1 complex, He can de-trap to the T 3 site or the T 4,1 site, with a significantly low rate since the binding energy of He to the H 1 V 1 complex still reaches up to 2.30 eV. The barriers of the de-trapping pathway of He to these two limiting T 4,1 sites differ little (2.52 vs 2.49 eV). The de-trapping of He to the T 3 -s site is not feasible since He initially assigned at the T 3 -s site will be trapped after relaxation. We calculate the upper limit of the de-trapping rate of He from the H 1 V 1 complex by assuming that He can de-trap to all T 3 sites and T 4,1 sites. Figure 7(d) shows the de-trapping rate of He from the H 1 V 1 complex, where the de-trapping rate of He from V 1 is also given for comparison. The de-trapping rate of He from the H 1 V 1 complex remains appreciably low and solely reaches up to 60 Hz at 1000 K, further demonstrating the strong He-V interaction even with the prior H in the V.
To examine the density of prior H atoms in the V on subsequent He trapping and de-trapping, we calculate the binding energies of He to the H i V 1 complexes (i = 2−12), as shown in figure 8(a). According to figure 6(d), the maximum number of H atoms that can be trapped by V 1 is six. From figure 8(a), there exists a critical density of the prior H in the V below which the prior H enhances He de-trapping. Specifically, when the number of H atoms that aggregate around V 1 does not exceed the maximum number of accommodated H atoms in V 1 (i < 6), the binding energies of He to the H i V 1 complexes decrease with increasing number of prior H atoms, while the de-trapping rates of He from the H i V 1 complexes remain low since the binding energy of He to H 5 V 1 complex still reaches up to 1.66 eV. The prior H atoms in the V can lead to the collapse of low-electron-density isosurface (see figure  S5(a)). This collapse will improve the density of electron gas around the trapped He. Since He strongly repulses the electron gas [67], the prior H atoms consequently reduce the binding energy of He to the V. When the number of H atoms that aggregate around V 1 exceeds the maximum number of accommodated H atoms (i > 6), the binding energies increase with the accumulation of prior H atoms, indicating the improved de-trapping barrier of He, and H de-trapping is inhibited. However, this does not indicate the attractive H-He interaction. The increased binding energy of He is associated with the larger H-H distances caused by He to attenuate the repulsive H-H interaction, lowering the total energy of the whole system. It has been also computationally reported that H atoms of high density slow down the release of He atoms from He bubbles in Pd [80].
For the most stable H i He 1 V 1 complexes (i = 2−6) obtained herein, we calculate the de-trapping barriers of He along two limiting pathways, i.e. to the T 3 -l and T 3 -s sites. The He atom will bypass the prior H atoms as much as possible during detrapping to the T 3 -l site, while it will encounter the prior H atoms as much as possible during de-trapping to the T 3 -s site. Taking the H 6 V 1 complex as an example, figure 8(b) illustrates these two limiting de-trapping pathways. Figure 8(c) shows the energy profiles along these limiting de-trapping pathways of He from the H i V 1 complexes, where two results should be noticed. First, the de-trapping barriers decrease with increasing number of prior H atoms in V 1 with i < 6. Second, for He de-trapping from each H i V 1 complex, the barriers along these two limiting de-trapping pathways differ little (with a difference lower than 0.05 eV). Additionally, we find during relaxation that when migrating to the T 3 -s site, the He atom can squeeze out the prior H atoms owing to the strong repulsive H-He interaction.
The de-trapping rates of He from the H i V 1 complexes (i = 2−6) can be estimated using the energy profiles for He de-trapping to the T 3 -l site. This is a reasonable approximation since the barriers of these limiting de-trapping pathways for each H i V 1 complex are almost the same. Figure 8(d) shows the de-trapping rates of He from the H i V 1 complexes at 500 K-1000 K. The largest de-trapping rates of He are found for the H 5 V 1 and H 6 V 1 complexes. The de-trapping rates of He from the H 5 V 1 and H 6 V 1 complexes at 500 K are extremely low (less than 10 −3 Hz), calculated as 8 × 10 4 and 4 × 10 4 Hz at 1000 K, respectively. This indicates that it is difficult for the trapped He to de-trap from the V even with prior H atoms in the V. Thus, the influence of He on the long-time evolution of the cavity is weakly altered by H atoms around the cavity, especially when the temperature is not high. The weak influence of H isotope on the He retention observed experimentally [22,23] also indicates the weak effect of the prior H on the behaviors of He around lattice defects. In the CI-NEB calculations, we find that during He de-trapping from the H i V 1 complexes (i = 2−6), the He atom can squeeze out partial H atoms when the He atom approaches these H atoms. Therefore, even if the prior H atoms densely coat the V, it is expected that the prior H atoms will not evidently inhibit subsequent He trapping since the He atom can squeeze out the prior H atoms in the V.
Taking a V as one prototype trapping site, the present work examines the trapping and de-trapping of H and He and confirms the repulsive H-He interaction. The trapping barriers of H and He, as shown in figure 1(b), are not evidently influenced by the repulsive H-He interaction, approximately equal to the hopping barriers in the perfect lattice. In contrast, the detrapping barriers of H and He from the V are both decreased due to the repulsive H-He interaction. The de-trapping rate of H increases with the accumulation of He in the V due to the H-He repulsive interaction, and it is expected to increase with the accumulation of H due to the repulsive H-H interaction (see figure S5(c) and [64]). The de-trapping rate of He increases with the accumulation of H in the V before the H density reaches a critical value, while the density of H above the critical value in the V inhibits He de-trapping. The binding energy of He generally decreases with increasing number of He in the V (see figure S5(d) and [63]), also leading to the increasing de-trapping rate of He with the accumulation of He.
The obtained results based on ab initio calculations in this work can also be applied in upper-scale simulations. This work investigates three hopping scenarios of H and He at finite temperature: hopping in the perfect lattice, trapping process by the V, and de-trapping from the V. The obtained diffusion coefficients of H and He, the hopping rates of H and He during trapping, and the de-trapping rates of H and He can be integrated in upper-scale simulations, such as the kinetic Monte Carlo simulation and the cluster dynamics simulation. This can contribute to the feasible comparison with experiments on cavity evolution, which will be performed in the future.
The repulsive H-He interaction in the V cluster shall hinge on its size. The V cluster can extend its size by trapping vacancies and consequently providing a broader area with electron gas of low density (the central area of the V cluster) for He trapping and meanwhile more binding sites on its surface for H trapping. The larger V cluster can promote H-He distance and attenuate the repulsive H-He interaction. Consequently, the capability of He (H) in the V cluster to reduce the de-trapping barrier of H (He) is attenuated. Meanwhile, the above critical H density in the V cluster determining whether the prior H in the V cluster enhances or inhibits He de-trapping shall alter. Additionally, the trapped H and He have shown to stabilize the V cluster [73,81]. Future efforts shall be devoted to quantitatively evaluate the H-He interaction in the V cluster and the coupling effect of H and He on the evolution of the V cluster, aiming at elucidating the H-He synergetic behaviors in altering the nucleation and growth of the cavity in the fusion environment.

Conclusions
In summary, we investigated the H-He interaction around a V as one prototype trapping site for H and He in BCC Fe based on ab initio calculations from the thermal dynamic viewpoint. The significantly stronger He-V interaction than H-V interaction is demonstrated by comparing the dynamic trapping and de-trapping of H with those of He. The trapped He can hardly de-trap from the V across the temperature from 500 K to 1000 K, while the trapped H can de-trap with a frequency of about 10 7 Hz at 500 K. Furthermore, the H-He interaction around the V was studied by calculating their hopping rates around the H-He-V complexes. The prior He in the V will not evidently inhibit the dynamic trapping of H. In contrast, the prior He in the V reduces the binding energy of H to the V and enhances H de-trapping, indicating the repulsive H-He interaction. On the other hand, the maximum number of accommodated H atoms increases due to larger H-H distances to attenuate repulsive H-H interaction. During He trapping, He can squeeze out the prior H atoms in the V when approaching the prior H atoms owing to the repulsive H-He interaction. Consequently, the dynamic trapping of He by the V is slightly influenced even if the V is densely coated by prior H atoms. Whether the prior H in the V enhances or inhibits He de-trapping hinges on the H density in the V. Below the critical density, the prior H in the V enhances He de-trapping, while the de-trapping rate of He remains low due to the strong He-V interaction. Above the critical density, the prior H inhibits He de-trapping. These results contribute to a better understanding of the cavity nucleation and H isotopes/He retention in nuclear structural materials in the fusion environment at the atomic scale.