Interplay between the Edge Dislocation and Hydrogen in Tungsten at Electronically Excited States

Under the continuous irradiation of high‐density and high‐energy photons from the plasma core of a fusion reactor, the plasma‐facing materials (PFMs) of tungsten (W) are in electronically excited states. How hydrogen (H) interacts with defective PFMs in an electronically excited state is an open question. The authors report the developed W‐H tight‐binding (TB) potential model and employ this model to systemically investigate the interaction between an edge dislocation in tungsten with H at different electronically excited states. With the enhancement of electronic excitation, the strong attraction of the dislocation core to H slightly fluctuates, while the attraction to H is significantly enhanced in the region outside the dislocation core. When the electronic excitation energy is ≈0.86 eV, the region of tensile stress can trap H without additional energy. Additionally, the electronic excitation simultaneously makes H migration easy. It is revealed that the transfer of partial energy of the excited electrons to the lattice leads to the nonthermal expansion of the system and affects the interaction between the edge dislocation and H. These results not only show the nonthermal effect of tuning the interaction between hydrogen and the edge dislocation in tungsten but also uncover the nature underlying these phenomena.


Introduction
Tungsten (W) is one of the most promising plasma-facing materials (PFMs) in a fusion reactor. [1] Although W-PFM has the advantages of good thermodynamic properties and a low physical sputtering rate, its safe service in the extreme environment of the fusion reactor is still a challenge. In a fusion reactor, the W-PFM DOI: 10.1002/apxr.202200115 is exposed to irradiation from the plasma, including hydrogen (H) isotope irradiation, which creates defects such as vacancies, dislocations, grain boundaries, and H bubbles, [2][3][4][5][6][7] etc. Among these defects, the formation and evolution of H bubbles have received much attention. To explore the formation mechanism of H bubbles, the interaction of vacancies, dislocations, or grain boundaries with H has been extensively studied. [8][9][10][11][12][13] For the dislocation-mediated W system, it is realized that for both screw dislocation and edge dislocation, H atoms are attracted to the dislocation core and prefer to arrange themselves as clusters, and the clusters can then trap more H atoms to eventually form H blisters. [14][15][16][17] Further studies [18,19] proposed a two-shell model to describe the rearrangement of H atoms around an edge dislocation at a finite lattice temperature. Deeply, the energy contribution and differential charge density associated with the interaction between H and dislocations in tungsten were examined at the level of density functional theory (DFT). [20] These studies are beneficial for gaining deep insight into the performance of W-PFMs working in a fusion reactor.
However, it is worth noting that almost all of the studies on the interaction between H and defects in W-PFMs were carried out on the system in the electronic ground state. In fact, the material will be in electronically excited states during the operation in the fusion reactor due to the following two factors. First, the plasma temperature that maintains the fusion reaction is as high as 100 million K in a real fusion reactor. According to the Wien displacement law of black body radiation, the plasma core with such high temperature will continuously emit high-energy photons such as X-rays, so the irradiation of high-density and high-energy photons on PFM is continuous, and some electrons in PFMs are excited constantly although part of the excited electrons will be deexcited. Therefore, statistically, there will be a certain number of excited electrons located in the energy levels of some excited states in PFM. Second, when exposed to irradiation, the W-PFM is bombarded by high-energy swift heavy ions, which cause partial electrons from atoms in materials to be excited through Coulomb interactions. In physics, changes in electronic structure will affect the strength of the interatomic bonds in the material, thereby modifying the structure and properties of the material. [21,22] Hence, for W-PFMs working in the extreme environment of a fusion reactor, it is emergent to study the interaction between H and defects under strongly excited states. Currently, several works have focused on systems with strong excited states induced by laser pulses or high-energy irradiation from fusion reactors. Irradiated with ultrashort laser pulses, some unique phenomena, such as ultrafast melting and ultrafast phase transitions occurring on the scale of hundreds of femtoseconds, have been observed, and the movement of atoms in the process can be directly observed experimentally by using ultrafast X-ray diffraction. [23][24][25][26][27][28][29] Theoretically, the excited electrons can reach their equilibrium states under the irradiation of high-energy photons by electron-electron interactions on a time scale of fs. Therefore, there will be a nonequilibrium state in which hot electrons and cold ions coexist. Since the time scale of the electronphonon interaction is generally on the order of ps, it is speculated that the excited electrons transfer their fractional energy to the ionic system through nonthermal effects, which induces various novel phenomena on the fs scale. Several researchers have theoretically discussed the novel phenomena exhibited in the structure and properties of materials. By calculating the phonon spectra, [30][31][32] it is found that different materials respond dissimilarly to intense electronic excitation; for example, the melting temperature of gold increases sharply, while the lattice of silicon becomes unstable due to bond weakening. Furthermore, the process of nonthermal phase transition and nonthermal melting can be simulated by modeling the dynamic process of systems under femtosecond electronic excitation. [33][34][35][36] In particular, the melting and phase transition of W under intense electronic excitation were studied using two-temperature molecular dynamics (2T-MD) simulations based on empirical potentials. [37,38] They proposed that electronic excitation would lead to volume expansion, a decrease in melting point, and solid-solid phase transformation. However, to the best of our knowledge, studies on W systems with complex defects in electronically excited states have mainly focused on the evolution of defects, while there is a lack of understanding from a quantum theory perspective. The tight-binding (TB) potential model has the advantage of dealing with large systems within the framework of quantum mechanics, which is suitable for investigating systems with complicated defects under electronic excitation. In this paper, we report our developed TB potential for the interaction between W and H based on the previously obtained W-TB potential [39] and H-TB potential. [40] We applied our TB potential to study the solution and migration of H under different excited states in the W system containing an <001> edge dislocation. It is found that as the degree of electronic excitation increases, the dislocation core remains a strong attraction to H except for weak fluctuations, while the ability of the region beyond the dislocation core to trap H is enhanced significantly, and even H can be captured without an external energy supply in the region of tensile stress when the electronic excitation energy is ≈0.86 eV (corresponding to an electronic temperature of 10 000 K). Furthermore, the mobility of H is enhanced with the increase in the degree of electronic excitation. We proposed that these phenomena are essentially attributed to the changes in local structure tuned by the electronic excitations in the dislocation region.

Interaction Between H and an Edge Dislocation at Low Electronic Temperature
As a PFM, W has a typical operating temperature of ≈800 K in a fusion reactor. [41] This means that the lattice temperature of W-PFM is ≈800 K, at which the interaction of H with defects in the system has already been studied. It is noted that such an operating temperature, together with the irradiation of high-energy photons and high-energy particles from the plasma core, causes some electrons in the atoms to be excited, so the electronic temperature in such a system is higher than 800 K. As mentioned in the Introduction, the excited electrons should contribute a nonthermal effect to the system. To obtain the net effect of the nonthermal effect, the effect of the lattice temperature is ignored in this paper. It is noted that the electronic temperature of 800 K corresponds to excitation energy of ≈0.07 eV for the electrons. This electronic excitation energy is rather small, and thus, the WH interaction in this situation approaches that in the ground state. In the present work, the results evaluated at an electronic temperature of 800 K are used for comparison of the results achieved at high electronic excited states to show the nonthermal effects from the high electronic excited states.

The Region Beyond the Dislocation Core
Since the <001> edge dislocation was observed when W was exposed to deuterium plasma, [42] we take this edge dislocation as an example to investigate the behavior of H in dislocationcontaining systems when the system is in electronically excited states. The structural model containing the edge dislocation is composed of 2480 atoms, with the dislocation line along the [100] direction and the Burgers vector along the [001] direction. The detail in generating such a structural model was mentioned in previous work. [43] In general, the behavior of an H atom in a dislocation-containing system tightly correlates with the existing hydrostatic stress of the site in the system. [44,45] Therefore, before introducing H atoms into a dislocation-containing system, we examined the distribution of hydrostatic stress in this system, which is shown in Figure 1a. According to the calculated hydrostatic stresses distributed in the system, one can find two regions separated by the glide plane (010): one is the tensile-stress region (notated as region T), which possesses positive hydrostatic stress, and the other one is the compressive-stress region (notated as region C), which has negative hydrostatic stress. Clearly, the closer the atom is to the glide plane, the greater the distortion of the structure and the stress amplitude of the atom. In particular, the degree of structural distortion at the dislocation core is the largest in the whole system, making the interaction between W and H atoms complex, so we will discuss this region later, which is marked with a circle in Figure 1a.
We place H atoms at the sites guided by the stress distribution calculated above. In detail, a single H atom was placed in each of the different TISs at varying distances from the dislocation line, where the considered TISs were distributed along the [001], [010], [01-1], and [011] directions. However, the TISs in the dislocation system have different degrees of distortion, so we adopted the method reported in reference [46] to identify the stable sites for the location of H. For the strained sites mentioned above, the relative stability of sites can be identified using the solution energy F sol (H) of H at the site in the system: where F dis and F dis + H are the free energies of the dislocationmediated system and dislocation-mediated system with H, respectively. (H 2 ) represents the chemical potential of H 2 . The smaller value of solution energy corresponds to the stronger attraction of the site to H. The solution energy obtained is shown in Figure 1b. Notably, the solution energies of H drop sharply near the dislocation core, indicating a strong attraction for H by the dislocation core. For the sites in region T, the solution energies at most of the considered sites are lower than those at the TIS in the pristine W system, which means that this region attracts H favorably with respect to the pristine W system. In addition, in region T, H can be effectively trapped in the spatial regime within ≈25 Å from the dislocation line. In contrast, in region C, the solution energies are all larger than that case at a TIS in pristine W, implying that this region is not conducive to the trapping of H. Furthermore, by comparing the energy curves in the cases assigned to the different directions mentioned above, it is clearly observed that the interaction of the H atom with the edge dislocation shows an obvious anisotropy: the difference in the solution energies of the H arranged in the [010] direction is the largest between region T and region C, and as the locations of H in both regions gradually approach the [001] direction, the difference in the solution energy slowly decreases. The ref. [44] reported the interaction between the edge dislocation 1/2<111> in W material and H by using the empirical potential. It was found that H is most easily captured in the dislocation core, and the compressive stress region repels H while the tensile stress region attracts H, which is in agreement with our results. In addition, the connection region between the tensile stress region and compressive stress region of 1/2<111> dislocation is easier to capture H than the TIS of the perfect W system. However, our calculated results show that the capturing capability of H in this region is close to that of the perfect W system, which is different from the reference. [44] This difference is probably caused by the different types of dislocations and corresponding glide planes, leading to the influence of shear stress on H being different. The different effects of different dislocations on H have also been discussed in the relevant studies of BCC-Fe. [45] Overall, the main features from our calculations are qualitatively consistent with previous studies considered in the ground state. [44,45] This suggests that when the electronic excitation of the system is not strong, the disturbance of such an excited state to the property in the ground state is very weak. In other words, although PFM is always in the electron excited state during the operation of the fusion reactor, as long as the electronic temperature is not high, the research results using the ground state theory are reliable. However, this situation does not remain in the case of high electronic temperatures, which will be discussed later.
We then analyze the discrepancy in the solution energy of H in region T and region C. To obtain representative results, we choose the cases in which the H atom locates along the [010] direction and its solution energy is close to the average solution energy of region T or region C in this direction. Here, the difference in solution energy of H between the two regions in this direction is the largest. We first examined the local structure around the H atom for each case and found that in the region T, the W-W distance increases under tensile stress, and the averaged distance of the W-H bonds is up to 1.919 Å, which is slightly larger than the value (1.908 Å) of the W-H bond length when H is in a TIS in the pristine W system. These enlarged interatomic distances weaken the repulsion between W atoms and H atoms in region T, and thus, an H atom can be more easily trapped in region T. However, the local structural feature in region C is opposite to that in region T. Therefore, the solution energy of H in region C is systematically higher than that in region T.
Second, we studied the electronic structure of W atoms in region T and region C, since the distortion of the atomic structure arising from the edge dislocation should change the electronic structure of the system to some extent. Figure 2 shows that the number of occupied d states in the energy window (−2 to 0 eV) near the chemical potential (0 eV) for W atoms in region T increases relative to that in pristine W, while the number of occupied d states in this energy window for W atoms in region C decreases. These results indicate that the contribution of the band structure energy to the total energy of the W atoms in region T increases, and the stability of the W atoms becomes weaker than that of the pristine W, which is beneficial to trapping H. The situation in region C is the opposite. Therefore, it is easier for an H atom to react with W in the TIS in region T than in both region C and the pristine system.

The Region in the Dislocation Core
As mentioned above, serious structural distortions exist at the core of our considered dislocation. How about the interaction of H with such a highly distorted structure? To answer this question, we calculated the solution energies of H at TISs and at OISs in the core of the dislocation (Figure 3). The OIS is discussed because the H atom at the OIS is energetically more favorable than the H atom at the TIS when large tensile stress is applied to the bcc crystal. [47] In Figure 3, the spheres numbered 1 to 5 represent the five positions where the solution energy of H varies from small to large. As the sites of H deposition are distributed from the core to the T region, the value of the solution energy of H gradually changes from negative to positive. In particular, as sites are distributed from the core to the C region, the corresponding solution energies abruptly change from negative to large positive values. This means that the strongest repulsion of H by the system occurs in the C region near the core, while the strongest attraction to H occurs in the core. Furthermore, after checking the local structures around H, which correspond to the five sites numbered 1 to 5, we found that positions marked with 1, 2, 4, or 5 in Figure 3 are all OISs.
To gain insight into the strong attraction of the dislocation core to H, we calculate the DOS of the H atom at position 1 and its first nearest neighbor (1NN) W. As shown in Figure 4a, near -10 eV below the chemical potential, the electronic density of states of the H atom highly overlaps with that of the associated W atoms, which is very similar to the case where H is located at the TIS in region T (Figure 4b). However, the electronic density of states of W around position 1 is broader and overlaps more with that of H in the dislocation core than the case in region T. This means that the attraction between H and W around site 1 in the dislocation core is stronger than that of H at the TIS in region T.

Interaction Between H and an Edge Dislocation at High Electronic Temperature
During the operation of the fusion reactor, W-PFM is inevitably irradiated by high-energy particles and high-energy photons. Thus, some electrons in the system are not only heated by the operation temperature of the W-PFM but excited by the highenergy particles and high-energy photons. It is necessary to explore the interaction between H and an edge dislocation in highly  electronic states. Usually, the degree of electronic excitation in W-PFM is uncertain. Therefore, different electronic temperatures are chosen in our treatment to discuss how the concerned solution energy evolves with electronic temperature in tungsten mediated by dislocation. In this section, we study the H solution energy in the region beyond the dislocation core and in the dislocation core at high electronic temperatures.

The Region Beyond the Dislocation Core
We first treat the case in which the H atom is located in the region beyond the dislocation core. As shown in Figure 5a, the solution energy in region T is always smaller than that in re-gion C at a given electronic temperature, which means that at each given electronic temperature, region T is more attractive to H than region C. Meanwhile, as the electronic temperature increases, the solution energies in both regions T and C decrease, and the attraction to H increases. When the electronic temperature is greater than ≈5000 K, the whole system, including the dislocation core, region T, and region C, is easier to capture H than the perfect W at an electronic temperature of 800 K. That is, H prefers to stay in the dislocation system with high electron excitation than in the perfect system with no or low electron excitation.
On the other hand, due to the existence of edge dislocations, the whole system exhibits different structural distortions around each atom. Therefore, for either region T or region C, the solution energies of H located at different sites in the [010] direction are different even at a given electronic temperature. To outline the trend of solution energy with increasing electronic temperature in either region T or region C, we extracted the average solution energies in region T and region C, respectively ( Figure 5b). As shown in Figure 5b, several aspects as addressed below are exhibited.
1) The averaged solution energies in region T and region C show a decreasing trend, and the averaged solution energies in region C are always larger than those in region T, similar to those shown in Figure 5a.
The above phenomena correlate to the changes in the volumes of the system as the electronic temperature increases. To uncover this, we extracted the averaged Wigner-Seitz-cell volume of W atoms near the TISs in region T and region C at different electronic temperatures, where the TISs have solution energies close to the average solution energies in region T and region C, respectively. As shown in Figure 6, with increasing electronic temperature, the average volumes in both region T and region C increase, and the volumes in region T are always larger than that in region C. As we know, the larger the volume of a W atom is, the greater the distance between a W atom and its neighbors is, and accordingly, the easier it is to trap H at the site within a certain distance range. Furthermore, the phenomena observed above essentially originate from changes in the electronic structure as the electronic temperature rises. Thus, we calculated the DOS of W atoms near the TISs in region T and region C at different electronic temperatures. In our calculations, the integrations of W d-DOS weighted by twice the Fermi-Dirac distribution function f( l ,T e ) are extracted within an energy window of ( −2 eV, ) at different electronic temperatures. For a given electronic temperature, the value obtained above is equal to the electron charges in the W d states associated with the given energy window. As shown in Figure 7, as the electronic temperature increases, the integral value increases. This indicates that as the electronic temperature increases, more electrons occupy higher energy states, and thus W atoms in both region T and region C become less sta- ble. Usually, the weaker the W stability is, the easier it is to attract H. Therefore, as the electronic temperature increases, the two regions are more likely to trap H atoms. In addition, the number of W electrons near the chemical potential in region T is always larger than that in region C, which suggests that the sites in region T are more capable of trapping H than sites in region C over the entire electronic temperature range.
2) The difference in solution energies between region T and region C decreases as the electronic temperature increases. Similarly, we understand this phenomenon in terms of both atomic structure and electronic structure: as the electronic temperature increases, the average volumes of W atoms in region T and region C tend to be close to each other (seen in Figure 6), and the integrations of W d-DOS in region C are clearly close to that in region T (seen in Figure 7). These indicate that the difference in solution energies for H between region T and region C is reduced with increasing the electronic temperature.
3) Surprisingly, when the electronic temperature is higher than ≈10 000 K, the average solution energy in region T is negative, showing exothermic absorption of H there. This result strongly suggests that proper electronic excitation induced by strong irradiation can turn the tungsten material that originally does not like to adsorb H into one that can exothermically adsorb H.

The Region in the Dislocation Core
As displayed in Figure 5a, the spatial distribution of the solution energy of H has a shape, where the lowest solution energy is located at the dislocation core. The dislocation core is thus a potential well that traps H at the considered electronic temperatures. As the temperature of the electrons rises, the depth of the abovementioned potential well becomes shallower, and the width of the potential well becomes wider. These two changes are respectively caused by the enhanced attraction for H in region T and region C and the expansion of the system as electronic temperature increases as mentioned before. However, the evolution of  interaction between the dislocation core and H with electronic temperature is unclear. To solve this, three sites that most easily capture H are selected to examine the change in solution energy with increasing electronic temperature. The resulting solution energies are listed in Table 1. It can be seen from Table 1 that at a given electronic temperature, the relative magnitudes of the solution energies at different sites remain, among which the solution energies of sites 1 and 2 are basically the same because they have distributed symmetrically around the [010] direction. For a given site, the solution energy of H oscillates slightly in the range of 800 K to 12 000 K. This implies that the attraction of the dislocation core to H varies nonlinearly with electron temperature. We speculate that the above nonlinear change in the attraction of the dislocation core to the H atom at each concerned site is correlated with the nonlinear change in the repulsive interaction, which is directly correlated with the local structure around the site. To examine this, we take site 1 as an example to investigate the repulsive effect arising from electronic excitation on the ability of sites to trap H in the dislocation core. First, we extract the Wigner-Seitz-cell volumes of the W atoms labeled A-F in Figure 8a around site 1 at different electronic temperatures. It is clear that as the electronic temperature rises, the volumes of the six W atoms increase except for a slight decrease of atom B and atom D in the range of 800 K to 5000 K. This means that the local structure around each site expands with increasing electronic temperature. However, the change in volume of atoms A, C, E, or F is obviously larger than that of either atom B or atom D, which leads to apparent structural distortion. To clearly reveal the effect of structural distortion on the WH interaction with increasing electronic temperature, we divided the solution energy into repulsive energy ΔE rep , band structure energy ΔE band , energy arising from the electron entropy T e ΔS e , and chemical potential of H 2 , (H 2 ), namely, Both ΔE band and T e ΔS e above are tightly related to the electronic structure for a given atomic structure of the system, while the repulsive energy ΔE rep is directly related to the contribution of the local atomic structure. Figure 8b shows the evolution of both ΔE rep and solution energy with increasing electronic temperature, where the trend of the repulsive energy is similar to that of the solution energy. Therefore, the nonlinear trend in the solution energy is also reflected in the trend of the repulsive energy. On the one hand, the solution energy presents the ability of the site to trap H; on the other hand, the change in the repulsive energy is directly correlated with the distortion of the local structure induced by the nonthermal effect. Therefore, we conclude that the distortion of the local structure induced by the nonthermal effect plays a crucial role in the nonlinear trends of the ability of the site in the dislocation core to trap H with increasing electronic temperature.

Migration of H in the W System with an Edge Dislocation
Since the sites in the different regions, such as region T, region C, and the dislocation core, have different abilities to attract H at a given electronic temperature, we are naturally concerned about whether the adsorbed H atoms at these sites migrate easily. Generally, the degree of difficulty in migrating an atom in a host material is measured using the energy barriers of migration. Therefore, we calculated the migration energy barriers of an H atom located in region T, region C, and the dislocation core at an electronic temperature of 800 K. As shown in Figure 9a, when H at the low-energy site migrates to the high-energy site, the energy barriers in region T and region C are almost the same and equal to 0.23 eV, but when H migrates from the high-energy site to the low-energy site, the energy barriers in the two regions are different and slightly less than 0.2 eV, the energy barrier between two neighboring TISs in the pristine W system. This result shows that the energy barriers in region C and region T have a similar magnitude to that in pristine W. Since an H atom prefers to migrate from high-energy sites to low-energy sites, more H atoms are finally trapped in the core of dislocation. For the dislocation core, we examine the energy barriers of H migrating between site 1 and its neighbors. As shown in Figure 9b, the migration energy barriers from site 1 to sites denoted a to e are all greater than 1.0 eV, even for moving from site 1 to its equivalent site 1' along the dislocation line. In contrast, H at some sites only needs to overcome an energy barrier of ≈0.2 eV (sites c and d) or even zero (site e) to reach site 1. Overall, the H atom migrates easily from the surrounding sites of site 1 to site 1, but it is difficult to migrate out once the H atom is captured by site 1. Therefore, we reasoned that H in the dislocation core would accumulate at sites similar to site 1 along the dislocation line after a long evolution time.
By extension, the arrangement of multiple H atoms in a dislocation-mediated W system is simulated using a classical kinetic Monte Carlo method, due to the TB-MD calculations being expensive for multiple H atoms interacting with the edge dislocation. In our simulations, the energies of trapping a single H at different positions in the dislocation core are used to infer the energy barriers for this H atom to migrate between adjacent sites. Our simulations show that multiple H atoms first occupy those sites along the direction parallel to the dislocation line with the lowest solution energy for a single H atom, and then other H atoms arrange in the same direction and around the core, where the sites have a strong attraction to a single H atom, such as the sites in the core and region T. The arrangement of multiple H atoms is in agreement with the conclusions obtained in other literature. [15] Before finishing this article, the effect of electronic temperature on H migration in the W system is also considered. For convenience, we calculated the energy barriers for TIS-TIS in pristine W at different electronic temperatures (Figure 9c). It can be seen from the figure that the energy barrier gradually decreases with increasing electron temperature. From this, we speculate that as the electronic temperature increases, the migration of H atoms in the dislocation-mediated W system becomes easier, but the migration of H atoms near the core is still difficult. Combining all the above analyses, we can draw the following conclusions: when the electron temperature increases, the range of H trapping in the dislocation region becomes wider, and the ability to trap H is stronger. At this time, the dislocation-mediated W system attracts more H atoms, and more H atoms accumulate in the core region of the dislocation.

Conclusions
In a fusion reactor subjected to continuous irradiation with highdensity and high-energy photons from the plasma core, statistically, there are a certain number of excited electrons in the energy levels of some excited states in PFM. In order to investigate the interaction of plasma-induced defects in W-PFM under electronically excited states, we, in this paper, developed a tight-binding potential model for the interaction between W and H and employed the tight-binding potential model to investigate the interaction between the <001> edge dislocation and H under different electronic excited states. Our main results are listed below: (1) In the low electronic excited state, the dislocation core has the strongest attraction to H, and the tensile-stress region attracts H while the compressive-stress region repels H compared to the pristine W system. With the increase in electronic excitation, the ability of the tensile-stress region and compressive-stress region to capture H is enhanced, and even around the electronic excitation energy of 0.86 eV, the capture of H in the tensile-stress region becomes an exothermic process. In addition, as the electronic temperature increases, the ability of the two regions to capture H gradually approaches, but the tensile-stress region always more easily captures H than the compressive-stress region. At the same time, the dislocation core always maintains a strong attraction to H but slightly fluctuates.
(2) The nature underlying the findings above was uncovered. By analyzing the electronic structure and related atomic structures, we proposed that in an electronically excited state, the electronic system transfers partial electronic excitation energy to the lattice system, resulting in a nonuniform expansion of the system. The expansion of the system can lead to a decrease in the stability of W atoms, thereby enhancing the attraction to H; the nonuniform expansion corresponds to the distortion of the local structure, which has a weak modulation of the WH interaction.
(3) In the low electronic excited state, the migrations of H in the tensile-stress region and compressive-stress region are similar to that in the pristine W system, while in the dislocation core, the H atom is easily trapped but difficult to move out. Therefore, H atoms prefer to stay in the dislocation core. Furthermore, the migration of H in the concerned system becomes easier as the degree of electronic excitation increases. Combined with the enhanced attraction of the system to H, it can be inferred that more H atoms are trapped in the dislocation-mediated system and mainly accumulate in the dislocation core.

Tight-Binding Potential for W-H
Within the framework of quantum solid-state theory, a system can be described with a Hamilton operatorĤ. Generally, when this system is in a steady state, we havê Both { n } and {| n 〉} are the eigenenergies and the corresponding eigenvectors, respectively. When the atoms in the system possess localized atomic orbitals, the overlap between the atomic orbitals belonging to nearest-neighboring atoms is quite weak. In this scenario, the eigenvectors above can be expressed well using the basis set of the Bloch function composed of the orbitals of the atoms in the system. Under this treatment, Equation (1) is transformed into where H is the Hamiltonian matrix, S is the overlap matrix, and c n is the corresponding coefficient matrix. Similar to references, [39,48] the scheme of the orthogonal basis set is employed in the present work so that S in Equation (2) becomes a unit matrix. Based on the two-center approximation proposed by Slater and Koster, [49] the hopping terms in the Hamiltonian matrix H can be expressed as a linear combination of related bond integrals V ll′m (l and l' = s, p, d…; m = , , …). For such binary systems, there are bonding integrals associated with the interactions between W atoms, between H atoms, and between a W atom and an H atom, which are notated as V WW ′ m , V HH ′ m , and V WH ′ m . To reduce computation, each bonding integral is expressed as a parameterized empirical function, which is commonly used in tight-binding potential models. Previous efforts have developed tight-binding potential models for pure W systems [39] and pure hydrogen systems, [40] in which V WW ′ m and V HH ′ m have been reported. In the present work, the bond integrals V WH ′ m between atom i and atom j with a bond length of r ij are written as where 1 , 2 , 3 , and 4 are parameters and is a cutoff function. In this cutoff function, w is the spatial range in which the cutoff function works. In the case of W-H, w = 0.1 Å. In addition, r cut is the cutoff distance, which takes the value of 3.0 Å for W-H. Note that for our concerned WH binary system, the atomic orbitals of 5d6s6p for W and 1s for H are selected. Therefore, we have ll′m =ss , sp , and sd with l standing for the orbital of the H atom and l′ for the orbital of the W atom. By taking the sum of the single electron energies { n } of the occupied states, we have a band structure energy where f n is the Fermi-Dirac distribution function If the system is electronically excited, the electronic temperature T e > 0 K. For this case, we defined the electronic excitation energy E e = k B T e to express the degree of electronic excitation. k B is the Boltzmann constant. The chemical potential satisfies the condition ∑ n N n f ( n ) = N e , where N e is the number of valence electrons of the system in the ground state and N n is the number of electrons occupied in the nth state.
The parameters in the bond integrals obtained in our TB model are listed in Table 2. It is noted that the TB potential models for H and W were developed separately, where the onsite energies of W and H are independent. If we use these two TB models to deal with a W/H system, the relative energies of these onsite terms correlate with each other. In the present WH TB potential, the energy of the H 1s orbital shifts 5.1518380 eV so that the Fermi level of a W/H system predicted from TB calculations matches that from DFT calculations.  Furthermore, the repulsion energy of the system is required in the tight-binding total energy calculations. For a W/H binary system, the repulsion energy,E rep , is actually contributed by the interaction between atoms, which is expressed as In Equation (9), both f 1 (x) and f 2 (x) correspond to the pure W system and the pure H system, respectively; ϕ WW (r ij ) and ϕ HH (r ij ) are the pairwise potentials for pure W and pure H systems respectively. They are reported in the literature. [39,40] ϕ WH (r ij ) is the pairwise potential between a W atom and an H atom, which, for simplicity, has the same equation as the bond integralsV ll′m . The parameters of pairwise potential for W-H are listed in Table 3. In addition, the dimensionless parameters are d WH = 0.138052 and d HW = 0.568446.
Once both E band and E rep are achieved for a W/H system, the total energy of the system can be expressed as It is necessary to verify the reliability of the TB potential for WH above. For this purpose, the solution energies of H at the tetrahedral interstitial site (TIS) and Octahedral interstitial site (OIS) in pristine W and the migration energy barrier of a single H atom between the neighboring TISs at a temperature of 0.0 K were calculated according to the CI-NEB method, [50] which are summarized in Table 4. Clearly, the data from our TB calculations agree well with the data from the DFT calculations. In addition, the trapping energy E trap of each H atom in a W monovacancy was calculated. The expression of E trap is  [12] 1.26 [12] 0.20 [53]  where E va + nH and E va + (n − 1)H represent the total energy of a W system with n and (n−1) H atoms in a monovacancy, respectively. E p represents the total energy of the pristine W system, and E TIS represents the total energy of a W system with an H atom in the TIS. Taking the solution energy of an H atom at TIS in pristine W as a reference, our TB potential predicts that a monovacancy in the W system can accommodate 14 H atoms at 0.0 K (Figure 10), which is slightly larger than those (10 or 12 H atoms) predicted by DFT calculations. [8,51,52] Overall, our WH-TB potential model is reliable for treating the behavior of H atoms in the W system.

The Treatment of Nonthermal Effects
In the present work, ultrafast excitation in the W/H system is considered. When the system is irradiated with high-energy photons and high-energy particles, partial electrons of the system are excited. This electronic system can reach an equilibrium state in 10 to 100 fs due to the interaction between the excited electrons. Thus, the Fermi-Dirac distribution function can be used to describe the distribution of electrons in the landscape of energy under such a high electronic excitation. As mentioned before, the electronic temperature T e or electronic excitation energy E e = k B T e in the Fermi-Dirac distribution function is used to express the degree of electronic excitation.
Usually, excited electrons transfer their fractional energy by nonthermal effects and electron-phonon coupling. However, the www.advancedsciencenews.com www.advphysicsres.com timescales required for the former are on the order of fs and for the latter are on the order of ps. Since we restrict our concerns to ultrafast irradiation in the present work, the nonthermal effects are treated in our study only. According to the literature, [54] the event of ultrafast irradiation can be regarded as a sudden perturbation to the ground state of the system. In this case, the system exposed to such irradiation has electronic entropy S e , which is expressed as which contributes to the free energy By the partial derivative of Equation (14) with respect to the coordinate r i , we have the force f i on atom i We should stress that the first term of the right-hand side (band structural force) in Equation (15) is completely correlated with the quantum state of the system, in which the nonthermal effect is included. According to Equation (15), changing the electronic state alters the forces acting on ions in the system, which can cause several intriguing nonthermal effects.