Atomic-scale Effects of Hydrogen in Iron toward Hydrogen Embrittlement: Ab-initio Study

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1 , pp Atomic-scale Effects of Hydrogen in Iron toward Hydrogen Embrittlement: Ab-initio Study Yoshitaka TATEYAMA and Takahisa OHNO Computational Materials Science Center, National Institute for Materials Science, Sengen, Tsukuba, Ibaraki Japan. (Received on August 31, 2002; accepted in final form on December 18, 2002 ) We have investigated atomic-scale effects of hydrogen atoms in a-fe by means of ab-initio calculations and examined microscopic mechanism of hydrogen embrittlement in Fe-based materials. Our calculations indicate that accumulation of interstitial hydrogen in a tensile-stress-concentrated region ahead of a crack tip is estimated to be hundredfold increase of the concentration at most, which is much lower than the assumption of the representative models for hydrogen embrittlement. On the contrary, we quantitatively demonstrate that hydrogen significantly facilitates vacancy formation in a-fe and the resultant vacancies trapping hydrogen incline to the anisotropic clusterization. These results suggest considerable contribution of vacancy-related processes to hydrogen embrittlement in those materials, rather than support interstitial hydrogen effects on which the existent models mainly based. KEY WORDS: ab-initio calculation; hydrogen; lattice defect; iron; hydrogen embrittlement. 1. Introduction Hydrogen-induced catastrophic degradation of mechanical properties, which is known as hydrogen embrittlement, in Fe-based structural materials (e.g. steels) has been a technologically important issue and extensively studied over several decades. 1,2) Nevertheless, the fundamental mechanism as well as efficient techniques to solve the problem has not been established yet. It is mainly because even atomic-scale effects of hydrogen in a-fe, which is a most fundamental knowledge of hydrogen embrittlement, have been still open questions. Irrespective of recent developments in resolution of electron microscopy approach, 3) direct detection of solute hydrogen behaviors in a-fe is still very difficult due to the extremely low solubility. What effects of solute hydrogen need to be elucidated, then? As universal mechanisms of hydrogen embrittlement, several models such as lattice decohesion 4,5) and hydrogenenhanced localized plasticity (HELP) 6 8) were already proposed. These models are essentially based on the effects of interstitial hydrogen atoms. In particular, they postulate extreme accumulation of interstitial hydrogen in a stress-concentrated region ahead of a crack tip for the crack propagation. Thus interstitial hydrogen behavior in a-fe is a key issue. Oriani et al. 5) presented a numerical model for the lattice decohesion that increase of interstitial hydrogen concentration reduces the maximum lattice cohesive force. According to this model, they suggested that fold increase of interstitial hydrogen concentration is needed to reduce the cohesive force sufficiently. In the HELP theory, elastic shielding, 6 8) namely elastic relaxation of dislocation dislocation repulsion by the presence of interstitial hydrogen, is regarded as the main mechanism to enhance dislocation mobility leading to fracture through material thinning. The estimation with a finite element method 8) indicated that, at least, 10 2 of interstitial hydrogen concentration seems necessary for sufficient reduction of the repulsion force. Since interstitial hydrogen concentration in a-fe at ambient condition of hydrogen gas pressure (fugacity) is in the order of 10 7 by Sievert s law, 9) this value corresponds to fold increase of the concentration. Because of universality of interstitial hydrogen, 2) both mechanisms have been widely applied to various materials without any examination of the hydrogen concentration. However, their application to materials highly endothermic for hydrogen is still a matter of discussion. In fact, there has been no direct evidence of such high concentration of interstitial hydrogen in Fe-based materials even in stress-concentrated regions due to experimental difficulty in the insitu observations, to our knowledge. Quantitative estimation on the accumulation behavior of interstitial hydrogen ahead of a crack tip is thus an essential point to be clarified. On the contrary, contributions of point defects such as vacancies to hydrogen embrittlement in steels were also suggested from experimental studies ) It might be promising, since it is well known that a certain amount of vacancies can actually exist in materials and largely affect their mechanical properties. 13) On the other hand, there has been still ambiguity in hydrogen states trapped by vacancies and vacancy behaviors affected by hydrogen in a-fe. The thermal desorption data in the experiments 10,11) indicated formation of some strong traps for hydrogen in steels by loading, which affected susceptibility for hydrogen embrittlement. However, characterization of trapped hydrogen states as well as configurations of point defects has been ISIJ

2 still unsettled because of difficulty in determination of the trapping energies of hydrogen. 10) The nature of vacancy hydrogen complexes (V m H n ) in a-fe was extensively examined in terms of plasma wall interaction in fusion reactors, 14,15) not hydrogen embrittlement. Implantation-annealing experiments revealed some hydrogen trapping energies of vacancies. 14,15) Besides, a theory on multiple trapping of hydrogen in a monovacancy was proposed in conjunction with results of the effective medium theory (EMT) calculations, 15,16) which has been referred as a settled theory to date. 9,17,18) However, we should point out that the theory itself still involves several inconsistencies. In general, there are six sites possible to trap hydrogen in monovacancy of bcc metal, which are located near octahedral sites adjacent to the vacancy, 19) as shown in Fig. 1. An experiment implies two characteristic states of hydrogen in a monovacancy of a-fe. Their energies for trapping hydrogen from interstitial sites were estimated to be 60.8 and 41.5 kj/mol-h by use of a transport model. 15) The EMT calculations, on the other hand, gave the result that the hydrogen trapping energy for VH and VH 2 formation is about 77 kj/mol-h, while those of VH 3 VH 6 are estimated between kj/mol-h. 15,16) From these results, it was concluded that the experimental 60.8 kj/mol-h state corresponds to hydrogen trapping for VH and VH 2, while the 41.5 kj/mol- H state for VH 3 VH 6. 15,16) Since the heat of solution for interstitial hydrogen in a-fe is 25 kj/mol-h, 9) this conclusion implies that all of the six sites in a monovacancy are exothermic for hydrogen 9,16 18) and thus VH 6 is the most major complex at ambient condition of hydrogen gas pressure. However, one can see that there is a distinct difference between the experiment and calculations in the absolute values of hydrogen trapping energies. Besides, the theory implies significant low formation energy of monovacancy in a-fe by trapping six hydrogen atoms, because experimental monovacancy formation energy in a-fe is about kj/mol-vacancy 13) and the sum of heats of solution for VH VH 6 formations is estimated to be 138 kj/mol according to the above assignments, which corresponds to decrease of the formation energy. Taking entropy effects into account, 17) this energy decrease leads to serious increase of vacancy concentration even at ambient pressure of hydrogen gas and room temperature, whereas such extreme vacancy formation has not been reported at the condition as yet. More precise examinations should be thus indispensable in order to elucidate the nature of vacancy hydrogen complexes in a-fe. In spite of such necessity, experimental approaches to solute hydrogen in a-fe are quite limited as described above. In that situation, accurate calculations would be a most promising tool to resolve the problem. In this paper, we apply an ab-initio calculation method with high accuracy, which is based on the density functional theory (DFT), to investigation of atomic-scale properties of solute hydrogen in a-fe. Our calculations give more reasonable results that can explain various experimental results, compared to the previous calculations, as shown later. In order to estimate change of hydrogen concentrations under typical stress field ahead of a crack tip, we calculate Fig. 1. Schematic view of monovacancy in bcc Fe. Gray spheres show Fe atoms, while bold cubic at the center expresses monovacancy. Six octahedral sites around the monovacancy are also shown by open circles. pressure dependence of the energy of interstitial hydrogen in a-fe, which is a key for the relevance of existent models to hydrogen embrittlement in Fe-based materials. Also, our findings of energetics of vacancy hydrogen complexes in a-fe, which will be reported in detail elsewhere, 20,21) are briefly introduced. Then we present a new aspect of hydrogen effects on monovacancies and vacancy clusters in a-fe, different from the existent theory believed to date. 9,15,16) Based on these results of interstitial and trapped hydrogen in a-fe, we finally discuss the microscopic mechanism of hydrogen embrittlement in Fe-based materials. This paper is organized as follows: In Sec. 2, we briefly explain the method and conditions of our ab-initio calculations and demonstrate the accuracy. The pressure dependence of stability of interstitial hydrogen in a-fe is presented in Sec. 3. In Sec. 4, the calculation results of properties of vacancy hydrogen complexes in a-fe are briefly introduced with some new aspects. Finally, microscopic mechanism of hydrogen embrittlement in Fe-based materials is discussed in Sec Calculation Method Our calculations are based on DFT. The theory was developed by Hohenberg and Kohn in 1965, 22,23) and is widely used in ab-initio calculations of electronic states of condensed matter systems nowadays. Compared to EMT 24) used in the previous calculations, DFT has an advantage that it can deal with various chemical states such as covalent, ionic as well as metallic bonds equivalently, which is important in study on defect properties with bond breaking. There are several types of calculation methods based on DFT, in practice. Among them, we use a method employing plane wave basis with pseudopotential, because it can most easily obtain relaxed atomic configurations as well as optimized electronic states with accuracy. 25) The calculation conditions are as follows: Here we use an ultra-soft pseudopotential, 26) with the cutoff energies for the wave functions and the augmented charge, respectively, are 25 and 289 Ry. (1 Ry ev) A spin-polarized generalized gradient approximation (GGA) 27) is used for the exchange-correlation energy. GGA is necessary to obtain the bcc ferromagnetic phase as the ground state of iron at zero pressure. 28) Treatment of spin-polarization is crucial for the defect stability, since it was reported that calculated energy of vacancy formation is affected by spin configuration. 29) 2003 ISIJ 574

3 We use a supercell scheme 30) with a 54-atoms cell and 27 k points for integration of the full Brillouin zone (FBZ), which corresponds to about k points in FBZ of the bcc primitive cell. These conditions lead to 2.85 Å of lattice constant, 152 GPa of bulk modulus and 2.24 m B /atom of magnetic moment of pure a-fe. As shown in Table 1, these are in good accordance with experiments 31,32) as other ab-initio calculations with plane wave basis. 33) Furthermore, our calculations excellently reproduce the monovacancy formation energy and volume compared to other calculations using Green function or linear-muffin-tin-orbital methods, 29,34) also shown in Table 1. The zero-point motion energy of hydrogen is important for the heat of solution. Here we have calculated the energy by solving the Schrödinger equation with the adiabatic potential surface for hydrogen, which is evaluated by ab-initio calculations of several hydrogen positions in an optimized Fe lattice. We have then obtained 19.3 kj/mol-h of the zeropoint motion energy for interstitial hydrogen in a-fe. 35) Taking the potential energy into account, this leads to 32.8 kj/mol-h of heat of solution for interstitial hydrogen in a- Fe as shown in Table 1. This is in rather good agreement with the experimental value of 25 kj/mol-h. 9) We have also calculated the zero-point motion energies of hydrogen in VH n complexes and found them to be about kj/mol- H. As described above, our study utilizes most effective calculation techniques and conditions for the investigation of hydrogen and defects in a-fe. Besides, the calculations well reproduce properties of pure lattice, vacancy as well as interstitial hydrogen in a-fe as shown in Table 1. These results demonstrate the accuracy of our calculations for the present target, namely atomic-scale effects of solute hydrogen in a-fe. Other characteristic of our calculations will be explained in detail elsewhere. 20,35) Table 1. Fundamental properties of pure a-fe as well as vacancy and hydrogen in a-fe, obtained in our ab-initio calculations with the results of other calculations and experiments. The symbol 0 denotes the atomic volume of pure a-fe without vacancy. 3. Pressure Dependence of Interstitial Hydrogen Concentration in a-fe Accumulation of interstitial (diffusible) hydrogen in a stress-concentrated region ahead of a crack tip is a key assumption of the existent models for hydrogen embrittlement. Here we simply estimate the change of thermodynamic concentration of interstitial hydrogen in a-fe based on the energy dependence on pressure (volume) of lattice. Within the volumes examined here, change in the zeropoint motion energy of hydrogen is estimated to be sufficient small by preliminary calculations, so that the contribution of this energy is neglected in the pressure dependence. We calculate the total energy of a-fe with an interstitial hydrogen atom for each lattice volume. Using 152 GPa of bulk modulus in our calculations, we link each volume with a pressure to obtain the pressure dependence of the energy, which is shown in Fig. 2. The results show that the reduction of interstitial hydrogen energy at 2 and 4 GPa of the hydrostatic tensile pressure are 4.8 and 9.6 kj/mol-h, respectively. These decreases correspond to 5- and 30-fold increases of hydrogen concentration according to Boltzmann statistics at room temperature. Fig. 2. Pressure (volume) dependence of heat of solution for interstitial hydrogen in a-fe obtained in our ab-initio calculations. The energy calculations are performed for each volume. Using bulk modulus of a-fe in our calculations (152 GPa), the hydrostatic tensile pressure corresponding to each volume is obtained. In practice, typical tensile stress applied to structural materials could be about 500 MPa. Finite elements calculations suggest that the stress concentration ahead of a crack tip is about 4 10 times as large as the externally applied stress. 36,37) Thus realistic pressure in a stress-concentrated region ahead of a crack tip is estimated to be 2 5 GPa. Considering these tensile pressures and the calculated energy change in a-fe, increase of interstitial hydrogen concentration ahead of a crack tip in Fe-based materials will be about 100-fold at most under practical conditions. This increase is much lower than the assumptions on which both representative models for hydrogen embrittlement strongly depend. For establishment of accumulation properties of interstitial hydrogen, shear stress effects, interactions between accumulated hydrogen atoms and diffusion behaviors of hydrogen will need to be clarified ISIJ

4 However, the present results significantly indicate that one should be careful to apply these existent models of hydrogen embrittlement to Fe-based materials. 4. Vacancy Hydrogen Complexes in a-fe 4.1. Energetics of Monovacancy Hydrogen Complexes The hydrogen trapping energies in our calculations 20,21) are shown in Fig. 3 together with the EMT and experimental results. Here we define the trapping energy for VH n, E trap (n), as energy gain by trapping an interstitial hydrogen atom into VH n 1. Our energies are obtained by electronic states optimization and structure relaxation as well as estimation of zero-point motion energy of hydrogen. The result is that the VH 6 formation has almost zero in hydrogen trapping energy. On the contrary, our energies for VH and VH 2 formations are about 57.9 kj/mol-h, which correspond to the experimental 60.8 kj/mol-h state. The experimental 41.5 kj/mol-h state is also regarded as the VH 3 formation, since the trapping energy is 38.6 kj/mol-h in our calculations. Agreement between our calculations and experimental results is quantitatively excellent. The heat of solution for VH n, the energy from H 2 molecule in vacuum at ambient condition, indicates that the VH and VH 2 formations are completely exothermic for hydrogen and VH 3 is almost zero. On the contrary, the VH 4 and VH 5 formations are slightly endothermic, and VH 6 is in the same situation as interstitial hydrogen. These suggest that VH 2 is the major complex at ambient condition, and VH 3 may increase the population when the chemical potential of hydrogen slightly increases. This conclusion is in good agreement with the experimental observation that the 60.8 kj/mol-h state was seen even at ambient condition, but the 41.5 kj/mol-h state appeared with increase of implanted hydrogen isotopes. 15) It is also consistent with the observation of 44.4 kj/mol-h of hydrogen trapping energy after quenching from high hydrogen gas pressure. 17) Thus our results well explain the experiments quantitatively as well as qualitatively, in contrast to the EMT calculations. In consequence, it is demonstrated that the major complex in a-fe at ambient condition is VH 2, not VH 6 as referred to date. 9,16 18) The formation energy of VH n is defined as the monovacancy formation energy minus the sum of heats of solution for VH 1 VH n. At ambient condition of hydrogen gas pressure, the VH 2 formation energy is kj/mol-vacancy in our calculations. 20,21) This does not lead to an unrealistic large amount of vacancy formation resulting from the existent theory as described above. On the contrary, we should point out that the VH 2 formation energy is lower than that without hydrogen by about 50 kj/mol-vacancy at ambient condition. According to Boltzmann statistics at 300 K, this decrease of formation energy leads to 10 7 times increase of vacancy density by two hydrogen trapping. This result is a clear evidence of enhancement of vacancy formation by hydrogen trapping suggested by Fukai et al. 17,18,38) In order to understand the stability of VH n, we have examined the electronic states of VH n. As shown in Fig. 4(a), density of states in the VH 2 complex indicates that a new state appear below the 4s band of pure a-fe by trapping two hydrogen atoms. Partial electron density of these new states clearly shows Fe 3d H 1s hybridization as shown in Fig. 4(b). This causes the large hydrogen trapping energies for VH and VH 2 formations through the termination of broken Fe bonds. The new state has a bonding character mainly consisting of H 1s orbital and is doubly occupied by electrons. This indicates electron transfer to the region around hydrogen atoms from the neighbor Fe occurs. The resultant negatively charged hydrogen atoms repel each other. The repulsive interaction will become more dominant with increase of the number of trapped hydrogen. This is likely to cause the abrupt decrease of hydrogen trapping energies for VH n (n 2). Consequently, the competition between hybridization and Coulomb repulsion makes VH 2 the major complex at ambient condition. Note that these results demonstrate that the EMT method is not adequate to the present system, because it mainly deals with hydrogen interaction with delocalized states (e.g. Fe 4s) of electrons. 24) Fig. 3. Calculated hydrogen trapping energies in a-fe in our calculations (filled circles). The open circles and arrows denote the results of EMT calculations and experiments, respectively. Hydrogen energies at the interstitial site and H 2 molecule in vacuum at ambient condition are shown by horizontal broken and solid lines, respectively. The difference between the energy in H 2 molecule and the VH n one corresponds to the heat of solution at hydrogen trapping from vacuum into VH n 1 forming VH n Clusterization of Monovacancy Hydrogen Complexes We next investigate the binding energy (energy gain at binding) of two VH n complexes. The results are shown in Table 2. Here we focus on two types of divacancies with 100 and 111 alignments, because these were suggested to be most stable in the study using classical Johnson s potential. 39) Our calculations have clarified that binding of two VH 6 is energetically unfavorable. Following the existent theory based on the VH 6 dominance, one might have concluded that the vacancy binding is unfavorable in the presence of hydrogen. On the contrary, our theory indicating VH 2 dominance leads to the opposite conclusion. The V 2 H formation by binding two VH 2 complexes is calculated to be 17.4 kj/mol, and thus energetically favorable as shown in Table 2. This indicates that the binding is preferred in the 2003 ISIJ 576

5 Fig. 5. Schematic view of probable vacancy cluster configurations in the presence of hydrogen. The light gray cubes mean monovacancies and the dark gray planes denote hydrogen occupation near them. Anisotropic vacancy clusters on the {100} and {110} planes are expected to be favorable. Fig. 4. Table 2. (a) Density of states for up and down spins in pure a-fe and the VH 2 complex system. (1 ev J) New states clearly appearing in the VH 2 system are pointed by black arrows. The energy origin is set to the Fermi energy. (b) Partial electron density of the new states, which is expressed by the gray 3D isosurface and 2D contours on the (001) plane. The dark-gray and lightgray spheres indicate H and Fe atoms, respectively. This clearly shows Fe 3d H 1s hybridization. The diamonds means unoccupied octahedral sites adjacent to the vacancy. Binding energies of two monovacancy hydrogen complexes (VH n ) in a-fe in our ab-initio calculations. Binding direction in divacancy is also shown by using 100 or 111. Positive binding energy means that the binding is energetically favorable. presence of hydrogen at an ambient condition. The binding energy without hydrogen is 14.5 kj/mol, so that the presence of hydrogen slightly enhances (or does not suppress) the vacancy binding. These results imply that formation of multi-vacancy beyond divacancy will be energetically allowed in the case of VH 2 as well as vacancy without hydrogen. In the VH 2 clusterization, furthermore, it should be noticed that the energy gain by Fe H hybridization is about 50 kj/mol-h 2 in our calculations. If a hydrogen atom in a VH 2 complex loses its site at the binding, the loss of 25 kj/mol-h is expected in the binding energy. Since this energy is larger than 17.4 kj/mol of the intrinsic binding energy, the loss of hydrogen sites leads to the negative binding energy. Thus the VH 2 complexes are likely to favor tabular vacancy clusters such as {100} or {110} ones as shown in Fig. 5 as well as line clusters along 111 so as to keep the hydrogen sites. 20,21) 4.3. Roles of Vacancy-related Processes in Hydrogen Embrittlement Our findings on the vacancy hydrogen complexes will present not only a fundamental progress in defect physics in a-fe but also a break-through to elucidate microscopic mechanism of hydrogen embrittlement in Fe-based materials. Our hydrogen trapping energies may be related with strong trap sites observed in the experiment of steel, 10) though more quantitative comparison is necessary. The decrease of vacancy formation energy by trapping hydrogen causes vacancy stabilization leading to their survival over the temperature of recovery stage (stage III), which is about 200 K in a-fe. This is consistent with some experiments. 11,15) Also, the energy decrease can explain the experimentally observed enhancement of dislocation mobility. 40,41) Dislocation motions such as cutting or climb-up motion of jog are accompanied with vacancy formation, and especially the former behavior is of great importance in a region with a high dislocation density. Thus the decrease in the vacancy formation energy could be regarded as the reduction of activation energies of these dislocation motions toward total increase of the mobility ISIJ

6 The anisotropic vacancy clusterization enhanced by hydrogen could be also linked with anisotropy observed in fracture phenomena. The vacancy arrays along the 111 directions, which are slip ones in bcc metal, can be connected with the dislocation cutting or jog motions. The {100} tabular clusters can directly lead to the void formation or crack nucleation on these cleavage planes of a-fe. Furthermore, the {110} cluster is the first reasonable evidence of the enhancement of fracture along these slip planes experimentally observed. 41,42) These vacancy-related processes have an advantage that they do not require improbable accumulation of interstitial hydrogen. Thus our present results quantitatively indicate that hydrogen-enhanced fracture through vacancy processes, which has been hardly discussed as yet, is expected to be more relevant to the microscopic mechanism of hydrogen embrittlement in Fe-based materials. Although further study should be necessary, this new point of view may be of great importance for full understanding of whole hydrogen embrittlement mechanisms in those materials. 5. Concluding Remarks In this paper, we have presented several quantitative evidences leading to hydrogen-enhanced fracture through vacancy processes based on the investigations of atomic-scale effects of hydrogen and vacancy in a-fe by means of abinitio calculations with sufficient accuracy. We have found the following results in our ab-initio calculations so far. (1) The accumulation of interstitial (diffusible) hydrogen under typical stress ahead of crack tip is estimated to be hundredfold increase of the concentration at most. (2) VH 2 is the major monovacancy hydrogen complex in a-fe at ambient condition, which revises the theory believed to date. (3) Trapping of two hydrogen cause 50 kj/mol-vacancy decrease of vacancy formation energy, which leads to fold increase of vacancy concentration in the presence of hydrogen. (4) The resultant vacancies trapping hydrogen incline to the anisotropic clusterization such as 111 line as well as {100} or {110} tabular clusters. The amount of interstitial hydrogen accumulated in a stress-concentrated region around a crack tip is expected to be much lower than the assumptions in the existent models for hydrogen embrittlement. Thus it should be careful to apply these models to Fe-based materials. On the contrary, the results related to vacancy properties can explain the experimental observations of strong trap sites of hydrogen, vacancy stabilization and enhancement of dislocation mobility in the presence of hydrogen. Furthermore, anisotropic vacancy clusterization seems a most reasonable explanation of crack nucleation on the characteristic planes experimentally observed. Acknowledgements We acknowledge Prof. M. Nagumo for his helpful discussion and suggestion. We also thank to Dr. K. Tsuzaki for his fruitful discussion. The content in this paper is based on the presentation in the first international conference on advanced structural steels at Tsukuba on May 2002 (ICASS2002). The calculations in this work were performed on Numerical Materials Simulator in National Institute for Materials Science. REFERENCES 1) R. A. Oriani, J. P. Hirth and M. Smialowski: Hydrogen Degradation of Ferrous Alloys, Noyes Publications, NJ, (1985). 2) S. M. Myers, M. I. Baskes, H. K. Birnbaum, J. W. Corbett, G. G. Deleo, S. K. Estreicher, E. E. Haller, P. Jena, N. M. Johnson, R. Kirchheim, S. J. Pearton and M. J. Stavola: Rev. Mod. Phys., 64 (1992), ) T. Tsuchida, T. Hara and K. Tsuzaki: Tetsu-to-Hagané, 88 (2002), ) A. R. Triano: Trans. Am. Soc. Met., 52 (1960), 54. 5) R. A. Oriani and R. H. Josephic: Acta Metall., 22 (1974), ) H. K. Birnbaum and P. Sofronis: Mater. Sci. Eng., A176 (1994), ) E. Sirois and H. K. Birnbaum: Acta Metall., 40 (1992), ) P. Sofronis and H. K. Birnbaum: J. Mech. Phys. Solids, 43 (1995), 49. 9) Y. Fukai: The Metal Hydrogen System, Springer-Verlag, Berlin, (1993), A2; 5. 10) M. Nagumo, M. Nakamura and K. Takai: Metall. Mater. Trans., 32A (2001), ) M. Nagumo, K. Ohta and H. Saitoh: Scr. Mater., 40 (1999), ) M. Nagumo: ISIJ Int., 41 (2001), ) P. Ehrhart, P. Jung, H. Schultz and H. Ullmaier: Atomic Defects in Metals, ed. by P. H. Ullmaier, Landolt-Boernstein, New Series, Group III, Vol. 25, Springer-Verlag, Berlin, (1991). 14) S. M. Myers, D. M. Follstaedt, F. Besenbacher and J. Boettiger: J. Appl. Phys., 53 (1982), ) F. Besenbacher, S. M. Myers, P. Nordlander and J. K. Noerskov: J. Appl. Phys., 61 (1987), ) P. Nordlander, J. K. Noerskov, F. Besenbacher and S. M. Myers: Phys. Rev. B, 40 (1989), ) M. Iwamoto and Y. Fukai: Mater. Trans. JIM, 40 (1999), ) Y. Fukai, Y. Kurokawa and H. Hiraoka: J. Jpn. Inst. Met., 61 (1997), ) S. M. Myers, P. M. Richards, W. R. Wampler and F. Besenbacher: J. Nucl. Mater., 165 (1989), 9. 20) Y. Tateyama and T. Ohno: Stability and clusterization of vacancy hydrogen complexes in a-fe: Ab-initio study, submitted to Phys. Rev. B. 21) Y. Tateyama and T. Ohno: First-principles study on the stability of vacancy hydrogen complexes in a-iron, Proc. of Int. Conf. on Hydrogen Effects on Material Behavior and Corrosion Deformation Interaction, ed. by A. W. Thompson and N. R. Moody, TMS, Warrendale, PA, (2003), in press. 22) P. Hohenberg and W. Kohn: Phys. Rev., 136 (1964), B ) W. Kohn and S. J. Sham: Phys. Rev., 140 (1965), A ) J. K. Noerskov: Phys. Rev. B, 26 (1982), ) M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias and J. D. Joannopoulos: Rev. Mod. Phys., 64 (1992), ) D. Vanderbilt: Phys. Rev. B, 41 (1990), ) J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh and C. Fiolhais: Phys. Rev. B, 46 (1992), ) P. Bagno, O. Jepsen and O. Gunnarsson: Phys. Rev. B, 40 (1989), ) P. A. Korzhavyi, I. A. Abrikosov, B. Johansson, A. V. Ruban and H. L. Skriver: Phys. Rev. B, 59 (1999), ) M. J. Gillan: J. Phys.: Condens. Matter., 1 (1989), ) Y. Liu and D. J. Singh: Phys. Rev. B, 47 (1993), ) D. J. Singh: Phys. Rev. B, 45 (1992), ) E. G. Moroni, G. Kresse, J. Hafner and J. Fuerthmuller: Phys. Rev. B, 56 (1997), 15629, and reference therein. 34) P. Soederlind, L. H. Yang, J. A. Moriarty and J. M. Wills: Phys. Rev. B, 61 (2000), ) Y. Tateyama, T. Miyazaki and T. Ohno: in preparation. 36) A. Needleman and V. Tvergaard: ASTM STP, 803 (1983), I ) T. Inoue: Private communication. 38) Y. Fukai and N. Okuma: Jpn. J. Appl. Phys., 32 (1993), L ) J. R. Beeler, Jr. and R. A. Johnson: Phys. Rev., 156 (1967), ) T. Tabata and H. K. Birnbaum: Scr. Metall., 17 (1983), ) T. Tabata and H. K. Birnbaum: Scr. Metall., 18 (1984), ) A. Kimura and H. Kimura: Mater. Sci. Eng., 77 (1986), ISIJ 578