Reduced-activation ferritic/martensitic (RAFM) steels are one of the most important candidates for the first-wall structural materials in fusion reactors,^[1] partly due to their low swelling rates.^[2] However, in a fusion reactor the energetic He atoms enter the first-wall structural material, while at the same time large amounts of He are produced from transmutation reactions. Due to the strong self-trapping,^[3] and strong binding to vacancies,^[4] hydrogen,^[5] grain boundaries,^[6–8] and dislocations,^[9] high concentration He will form He bubble and induce void swelling.^[10,11] Thus these steels still suffer He-induced embrittlement.^[12]

Great effort has been made to suppress He bubble formation to improve the mechanical properties of the materials, and some alloying additions appear to work this way. For example, an experiment has demonstrated that the segregated Cr at the He bubble surface in a Fe-9Cr% steel slows down the Brownian motion of a small He bubble.^[13] Hao et al., using first-principles density functional theory (DFT) calculations, revealed that the segregated Cr increases the charge density at the bubble/Fe interface which prevents further He atoms passing through, and thus slows down the growth of He bubbles.^[14] DFT calculations also demonstrated that Au^[15] and Ce^[16] have a more pronounced effect than Cr. C is one of the elementary chemical elements in ferritic steels, whose effect on the He migration and agglomeration in bcc Fe have also been investigated.^[17] It is found that C inhibits the formation of higher order clusters composed of He and vacancies, i.e., He_nV_m, and thus increases the amount of He at substitutional positions at room temperature.

Due to its low solubility in ferritic steels, the excess carbon will precipitate in the form of carbides in quenched steels.^[18,19] In the very initial stage, the C atoms occupied the octahedral sites in bcc Fe.^[20] When the C content exceeds the equilibrium solubility, the cementite, θ-Fe₃C, is formed. A further increase of the C content, the η-Fe₂C and χ-Fe₅C₂ will also appear. Recently, the structure and stabilization of the iron carbides, especially, θ-Fe₃C, η-Fe₂C, and χ-Fe₅C₂, have been investigated by first-principles studies.^[21–26] Faraoun et al.^[23] using DFT calculation pointed out that the energy needed for the formation of the χ-Fe₅C₂ is three times larger than that to form η-Fe₂C. Denisov et al. studied the influence of heat treatment on the interaction between hydrogen and RUSFER-EK-181 Russian ferritic–martensitic steel, using the method of thermodesorption mass spectrometry, and they found that after soaking of martensitic steels at low-temperature tempering, clusters of carbon atoms are observed and metastable carbides (Fe₂C) appear.^[27] Despite this, η-Fe₂C is not stable at any temperature and has been scarcely investigated experimentally in the modified martensitic steel.^[28] However, previous investigations have showed that it may be stabilized by impurity elements^[22,29] and may be produced by the irradiation in extreme conditions in connection with realization of international projects ITER.^[30] Furthermore, Mazumder et al.^[31] examined He implantation of an F82H tempered martensitic steel at different temperatures, and suggested that the carbide precipitates prevent He bubbles formation by trapping the diffusing He at low temperature. Our previous studies have declared that θ-Fe₃C in ferritic steels might mitigate He bubble growth^[32] just as dispersed oxide particles^[33,34] and smaller grains do.^[35] Thus, η-Fe₂C is one of the most frequently observed carbide-precipitate phases, its effect on He bubble growth in bcc iron is in urgent need to understand. Due to the lattice mismatch at the interface leaves abundant free volumes (low electron density region) which are strong traps for He, the carbide/Fe interface will be as crucial as, if not more than, the inside of the carbide particles. As a first step to tackle the η-Fe₂C effect on He accumulation, in this work, we focus on the properties of He inside the grain of a η-Fe₂C particle in micro-meter scale, which can be well represented by a bulk system. Our calculations demonstrate that He is more soluble in η-Fe₂C than in bcc Fe and the He self-trapping in η-Fe₂C is less powerful than that in bcc Fe, indicating that η-Fe₂C particles in ferritic steels will trap more He atoms and might slow down He bubble growth at the initial stage, which may make the steel more swelling resistant.

Due to its closed 1s shell, He can only interact repulsively with other elements under compression, thus He almost always prefers larger free volume, i.e., low electron density space. In the unit cell of η-Fe₂C, two octahedral (O1 and O2) and three tetrahedral interstices (T1, T2, and T3) have been investigated (Fig. 1). The O1 site is embraced by six Fe atoms “A”, “B”, “C”, “D”, “E”, and “F” and the O2 site is also embraced by six Fe atoms “D”, “H”, “I”, “J”, “K”, and “L”. The two octahedral interstices have the same volume (9.5 ų), 26% larger than that in bcc Fe. The T1 and T2 sites are inside the O1 and O2 octahedrons, respectively, while the T3 site is surrounded by four Fe atoms “D”, “E”, “F”, and “G”.

The formation energies of the He atoms at different interstices were calculated according to Eq. (1) and listed in Table 1. Interstitial He at T1 and T2 sites are not stable, and they both slide to the neighboring octahedral interstices after full optimization. The formation energy of the He atom at T3 site is 4.22 eV. Although the two octahedral interstices have the same volume, the formation energies of the He atoms at the two sites are very different. The octahedral interstice O2 is the most favorable site for He with the formation energy of 3.76 eV, while the structure with the He atom at O1 site has a much larger formation energy of 4.25 eV. In order to find the underlying reason for the discrepancy, we provided the initial and optimized structures with He at O1 and O2 sites and the charge densities of the octahedral planes in Fig. 2. The relevant structural parameters are given in Table 2. From Table 2 and Fig. 2, we can see that, for the initial structure of He at O1 site, the distances between the He atom and its surrounding Fe atoms are between 1.91 Å and 1.95 Å, while for the optimized structure, the corresponding distances are increased to 2.09 Å and 2.18 Å, respectively. For He at O2 site, the distances of He–D and He–J are increased to 1.78 Å from 1.29 Å, and the other four bond lengths are increased to 2.53 Å from 2.40 Å. The volume of the optimized O1 interstice with He is 12.7 ų, while that of O2 is 14.4 ų. Thus, much larger volume change for O2 interstice (4.9 ų) than O1 interstice (3.2 ų) may be responsible for the lower formation energy of He at O2 than at O1 site. Due to the great difference between the bond lengths of He and its surrounding Fe atoms in interstice O2, we provided the charge densities of two octahedral planes in Figs. 2(f) and 2(g). The charge density of the plane composed by Fe atoms “A”, “C”, “D”, and “F” in O1 is shown in Fig. 2(e). It is found that the charge densities near the He atom at O2 site are much lower than those at O1 site, which is in accordance with the smaller formation energy for He at O2 than O1 site.

Table 1.

Table 1.

Table 1. The formation energies (in eV) of He at interstitial and substitutional sites in η-Fe2C, as well as the formation energies of He at the tetrahedral interstice (T) and the substitutional site (Fe) in bcc Fe. For an explanation of He positions in η-Fe2C, please see Fig. 1. . Interstitial He at Substitutional He for bcc Fe η-Fe2C bcc Fe η-Fe2C T O1 O2 T3 Fe Fe C 4.61 4.25 3.76 4.22 4.38 3.49 5.09

Table 1. The formation energies (in eV) of He at interstitial and substitutional sites in η-Fe2C, as well as the formation energies of He at the tetrahedral interstice (T) and the substitutional site (Fe) in bcc Fe. For an explanation of He positions in η-Fe2C, please see Fig. 1. .

We have also calculated the spin magnetic moments of the systems with He at O1 or O2 sites, as shown in Table 2. The induced magnetic moments on the He atoms are less than 0.03μ_B, and the magnetic moments on the C atoms have no change compared with the pure η-Fe₂C. By contrast, because He behaves as a bonding insulator leading to the d-orbital hybridization reduction of its neighboring Fe atoms from one another, all the spin magnetic moments on Fe atoms surrounding the He atom are increased.^[32,44]

The formation energies for He substitution of a Fe and a C atoms were calculated according to Eq. (2), and the corresponding formation energies are 3.49 eV and 5.09 eV, respectively. The He atom at the substitutional Fe site is 0.27 eV more stable than that at the interstitial site and it is much smaller than that in bcc Fe (4.38 eV). The formation energy of substitutional He can be decomposed into two events, forming a vacancy and trapping an interstitial He atom into a vacancy. The Fe and C vacancy formation energies are 1.80 eV and 0.94 eV, respectively, and the attraction between an interstitial He atom and the Fe vacancy is 2.07 eV, which is a little smaller than that in bcc iron, 2.49 eV. Thus, the smaller formation energy of substitutional He in η-Fe₂C is due to the smaller formation energies of Fe vacancy and interstitial He.

Fig. 2.

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	Fig. 2. The local configurations for He at O1 site before (a) and after (b) optimization, and at O2 site before (c) and after (d) optimization. Panel (e) is the calculated total valence charge densities for the octahedral planes of O1, and panels (f) and (g) are for O2. Lines start from 0.01 e/a.u.3, and increase successively by a factor of 101/5.

Table 2.

Table 2.

Table 2. Spin magnetic moments (M) of Fe atoms neighboring the He atom implanted into η-Fe2C and their distances (dO1−X and dO2−X) away from the He atom, which is located at interstitial site O1 or O2, before and after relaxation. For an explanation of lattice sites A–L, and the interstitial sites O1 and O2, see Fig. 1. . Site X dO1−X/Å M/μB Site X dO2−X/Å M/μB No He He@O1 He@O1 No He He@O2 He@O2 A 1.95 2.18 1.82 D 1.29 1.78 1.84 B 1.91 2.09 1.71 H 2.40 2.53 1.76 C 1.91 2.09 1.71 I 2.40 2.53 1.76 D 1.95 2.18 1.82 J 1.29 1.78 1.84 E 1.91 2.09 1.71 K 2.40 2.53 1.76 F 1.91 2.09 1.71 L 2.40 2.53 1.76

Table 2. Spin magnetic moments (M) of Fe atoms neighboring the He atom implanted into η-Fe2C and their distances (dO1−X and dO2−X) away from the He atom, which is located at interstitial site O1 or O2, before and after relaxation. For an explanation of lattice sites A–L, and the interstitial sites O1 and O2, see Fig. 1. .

Fig. 3.

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	Fig. 3. The optimized structures of He pairs with two interstitial He atoms (a), one He atom at the substitutional site and the other at the interstitial site (b) and two He atoms at the substitutional sites (c).

The formation of a He pair is the initial stage of He bubble growth. Our objective in the present work is to elucidate the accumulation of He at the very initial state, i.e., pairing of He, and compare its easiness with that in the bcc Fe matrix. The binding energies between the two He atoms of a He pair can be considered as the reduction of the total energy when the two favorable interstitial (or substitutional), as well as one substitutional and one interstitial He atoms come nearby. The calculated binding energies are listed in Table 3, according to Eqs. (3)–(5). We have investigated four int–int, five sub–sub and three sub–int He pairs. The most stable int–int He pair is shown in Fig. 3(a). The two He atoms are at the neighboring O2 sites with the binding energy of 0.31 eV, as large as that in bcc Fe. The calculated diffusion barrier of interstitial He is 0.35 eV, much larger than that in bcc Fe (0.06 eV), indicating that it is more difficult for two interstitial He atoms to form a He pair in η-Fe₂C than in bcc Fe. The most stable sub–sub He pair is formed by two He atoms replacing two Fe atoms “D” and “J” with a binding energy of 0.76 eV, which is much smaller than that in bcc Fe (1.18 eV), shown in Fig. 3(c). The formation energies for the three investigated sub–int He pairs are the same, 1.28 eV. All of them are much smaller than the binding energy of the sub–int He pair in bcc Fe (1.86 eV). Figure 3(b) is the optimized structure of the sub–int He pair with one He atom replacing Fe atom “D” and one He atom at O2 site. From Fig. 3, we can see that the distances between the two He atoms in the int–int, sub–sub, and sub–int He pairs are 1.76 Å, 1.67 Å, and 1.59 Å, respectively, indicating stronger binding corresponding to smaller distance between the two He atoms in a He pair.

Table 3.

Table 3.

Table 3. The binding energies (in eV) of int–int, sub–int, and sub–sub He pairs in η-Fe2C. The values in the parentheses are the binding energies of the most stable int–int, sub–int, and sub–sub He pairs in bcc Fe. For an explanation of He positions, please see Fig. 1. . Int-int Sub-int Sub-sub He@ Eint−int He@ Esub−int He@ Esub−sub O2 and O2 0.31(0.31) D and O2 1.28(1.86) D and B 0.03 O2 and O1 −0.07 D and O1 1.28 D and H 0.1 O2 and T2 −0.64 D and T3 1.28 D and J 0.76(1.18) O2 and T3 −0.11 B and C −0.02 B and H −0.06

Table 3. The binding energies (in eV) of int–int, sub–int, and sub–sub He pairs in η-Fe2C. The values in the parentheses are the binding energies of the most stable int–int, sub–int, and sub–sub He pairs in bcc Fe. For an explanation of He positions, please see Fig. 1. .

From the numerical results above, we know that the formation energies of substitutional He are smaller than those of the interstitial He in both η-Fe₂C and bcc Fe. The occupation probabilities of He at the interstitial and substitutional sites in η-Fe₂C and bcc Fe could be estimated by McLean’s equation as

From the calculations of the He pairs, we know that the binding energies between the two He atoms of a sub–sub He pair (0.76 eV) and a sub–int He pair (1.28 eV) are much smaller than those in bcc Fe (1.18 eV and 1.86 eV). In order to find the reason for this, we provided the charge densities of He at the substitutional site, a sub–int He pair and a sub–sub He pair in η-Fe₂C in Fig. 5. From Fig. 5(a), we can see that there is a high charge density area near He. This is because in η-Fe₂C, one C atom bonded with two Fe atoms and when one Fe atom substituted by the He atom, one C–Fe bond broke, which will create higher electron density area near the C atom. This may explain why the attraction between an interstitial He atom and the Fe vacancy in η-Fe₂C (2.07 eV) is much smaller (0.42 eV) than that in bcc Fe (2.49 eV). From Figs. 5(b) and 5(c), we can see that there are one and two C atoms near the sub–int He pair and the sub–sub He pair, respectively, due to one C–Fe bond and two C–Fe bonds breaking. Furthermore, the charge densities for the transition state of He diffusion in Fig. 5(d) shows that the densities of the area near the C–Fe bonds are much higher than those of the other areas encircling the inserted He atoms. Thus, we attribute the less powerful self-trapping of He in η-Fe₂C than in bcc Fe to the repulsive interaction between He and C. Our results are in good agreement with the previous research, in which, it is suggested that C inhibits the formation of higher order clusters composed of He and vacancies, i.e., He_nV_m, and thus increases the number of He at substitutional positions at room temperature.^[17]

Fig. 4.

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	Fig. 4. The calculated occupation probability of He as a function of the energy difference at different temperatures and bulk concentrations based on the McLean’s equation.

Fig. 5.

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	Fig. 5. The calculated total valence charge densities for (a) the (001) plane with He at the substitutional site, (b) the plane with a sub–int He pair, (c) the plane with a sub–sub He pair and (d) the (001) plane of the transition state of He diffusion. Lines start from 0.01e/a.u.3, and increase successively by a factor of 101/5.

In summary, we have systematically investigated the stability of He inside the η-Fe₂C bulk by first-principles calculations based on the density functional theory. The formation energies of the He atoms at the substitutional and interstitial sites in η-Fe₂C are 3.49 eV and 3.76 eV, respectively, which is much smaller than those in bcc Fe. The binding energies between two interstitial He atoms is as large as that in bcc Fe, 0.31 eV. However, the diffusion barrier of the interstitial He is 0.35 eV, which is much larger than that in bcc Fe. Furthermore, the binding potencies between two substitutional He atoms and one substitutional and one interstitial He atoms (0.76 eV and 1.28 eV) are much weaker than those in bcc Fe. The larger solubility of a He atom and weaker binding potency between two He atoms in η-Fe₂C than in bcc Fe might may make the small and dense η-Fe₂C particles in ferritic steels serve as scattered trapping centers for He and mitigate He bubble growth at the initial stage. We hope that this study will promote further experimental and theoretical efforts to design more swelling resistant steels taking advantage of carbides.