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Computer simulation on the interaction between vacancy and carbon impurity in α-Fe
Nuclear Instruments and Methods in Physics Research B 267 (2009) 3179–3181
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Computer simulation on the interaction between vacancy and carbon impurity in a-Fe Qiao Jiansheng a,b, Liu yongli b, Huang Yina a, Xiao Xin a, Wan Farong a,c,* a
Department of Materials Physics and Chemistry, University of Science and Technology Beijing, Beijing 100083, China Primary Education College, Xingtai University, Xingtai 054000, China c Beijing Key Lab of Advanced Energy Material and Technology, Beijing 100083, China b
a r t i c l e
i n f o
Article history: Available online 16 June 2009 PACS: 02.70.NS Keywords: Molecular dynamics Pair potential Vacancy Carbon
a b s t r a c t Metals possessing a bcc lattice have been considered as materials with high resistance to irradiation swelling. The aggregation of vacancies is the main reason of the swelling and the interaction of carbon atoms with vacancies affects the mechanical properties of metals. The complex of vacancy-carbon atom in a-Fe is modeled by MD method using a pair inter-atomic potential. The defect formation energy of vacancy-carbon atom complex was obtained by MD method. The stable site of carbon near a vacancy in three dimensions was discussed. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Metals and alloys possessing a bcc lattice have been considered as materials with high resistance to swelling, especially when compared with fcc metals and alloys. Experimental data provided by Gelles on model iron-base alloys showed low swelling levels after being irradiated to doses as high as 200 dpa in the Fast Flux Test Facility (FFTF) [1]. There has been a general perception that the low swelling rate of bcc materials was an intrinsic property of bcc crystal lattice [2]. So that the bcc alloys was considered to be the structural material of the nuclear fusion reactor. Carbon was one of the elements in steel. The properties of iron– carbon (Fe–C) solid solutions have been investigated extensively due to their particular relevance in steel technology. It is known from experiment that the interaction of carbon (C) interstitial atoms with lattice defects significantly affects the mechanical properties of ferritic steels, [3], and properties of point defects. Although it is widely accepted that vacancy (V) diffusivity is significantly decreased due to trapping of C atoms, there has been considerable uncertainty concerning the magnitude and form of the interaction. Vehanen et al. [4] deduced a value of 0.85 eV for the binding energy of a V–C complex, which, when added to a vacancy migration energy value of 0.55 eV in pure Fe, is consis-
* Corresponding author. Address: Department of Materials Physics and Chemistry, University of Science and Technology Beijing, Beijing 100083, China. Tel.: +86 10 62333932; fax: +86 10 62333724. E-mail address: [email protected] (W. Farong). 0168-583X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2009.06.070
tent with an effective vacancy migration energy of 1.35 eV for steels. However, V–C binding energy values of 0.41 and 1.10 eV have also been reported [5,6], the former estimate being close to the result of 0.47 eV obtained in recent ab initio calculations [7]. Vehanen et al. [4] argued that the value of 0.41 eV found by Arndt and Damask [5] should associate with the binding of C atoms to V–C complexes and vacancy agglomerates. In addition, the experiments on V–C interaction were performed by using electrons irradiation to build up a measurable concentration of non-equilibrium vacancies, when self-interstitial atoms were also produced. The role of the self-interstitial atoms (I) on trapping C atoms was unclear. Little and Harries [8] considered the binding energy of 0.41 eV found in [5] to be more probably associated with a C atom–interstitial cluster complex, although the later experiments [4,6] indicated the C–I binding energy to be much smaller, i.e. 0.10 eV, and ab initio calculations actually gave a negative energy in the range 0.20 to 0.40 eV for this pair [7]. Hence, it was desirable to use computer simulation methods to investigate the migration properties and defect interactions of C solute, but an Fe–C inter-atomic potential set was required and the development of such potentials has lagged behind those for pure a-Fe and Fe alloys with substitutional solute, and only a few Fe–C potentials had been proposed to date. The paper is organised as follows. In Section 2, the calculation scheme and the potential sets are described. In Section 3, the MD results on perfect bcc lattice, a vacancy, an interstitial solute and a vacancy–interstitial solute complex in a-Fe are presented. Conclusions are drawn in Section 4.
3180
Q. Jiansheng et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3179–3181
2. Calculation model Two atomic-scale techniques are used here to study static properties of a V, a C atom and V–C complexes in an Fe matrix. MD is used to model a V, an interstitial C atom and V–C complexes using a combination of conjugate gradients potential energy minimization at zero temperature and to calculate formation, binding and migration energies. In these cases the calculations were performed for a simulation box containing 2000 bcc lattice sites with constant volume and periodic boundary conditions. The pair potential that was proposed by Johnson [9] was used for the Fe–Fe interaction and the Fe–C interaction. The pair potential is described below:
VðrÞ Ar 3 þ Br 2 þ Cr þ D;
ð1Þ
where V(r) is the pair potential of the two atoms (Fe–Fe or Fe–C) which is interacted. r is the distance between the two interacting atoms. A, B, C and D are constants which is difference to the different distances of the two atoms. The potential sets allow a correct description of the elastic properties of a-Fe, while retaining almost the same properties of a C atom as the original model of Johnson [9]. This was verified by static simulation, as follows. The octahedral site was found to be the most stable position for a C atom. Both Fe potentials give the insertion energy of a C atom into a-Fe (difference in the crystal energy with and without C atom) of 1.23 eV, which is close to the corresponding energy of 1.34 eV calculated using the pair potential of Johnson [9] for Fe–Fe interactions. The tetrahedral site is the saddle point for a C jump from one octahedral site to another, as with the potential set of Johnson [9] and in the ab initio calculations by Domain and Becquart [7]. In the present work we investigate thermally activated diffusion of C atoms in bcc Fe and V–C interaction by the molecular dynamics (MD) method. Only molecular statics (MS) modeling (i.e. representing 0 K) of the interaction between a C atom and some point defects has been performed with the potentials mentioned above. Since we are not aware of any potential set that accurately represents the Fe–C system, the approach adopted here is to study a ‘model’ system for an octahedral interstitial solute in a bcc metal matrix that has many similarities to the real alloy. We have used for this purpose the pair potential of Johnson [9] for the Fe–C and Fe–Fe interaction. We also use the data obtained, together with results from the ab initio calculations referred to above, to review and interpret the earlier experimental information on C-point defect interactions. We use a molecular dynamics program which is modified for introduction of vacancies &/or interstitials coordinates by user. This program is a greatly modified and extended version of code developed for constant stress molecular dynamics using the parinello-rahman lagrangian. It is designed for use on the cray and exploits vectorisation, the special cray functions cvmgp and cray timing routines. The Johnson Fe potential is used in this version. The original link cell method, or lcm [10] was designed for a large number n of particles, the cpu scales as n because of using lcm. When a neighbour list ‘n-list’ is used in the traditional way, the cpu scales as n at each time-step at which the n-list is consulted, and as n n at each time-step at which it is updated. The vectorised link cell method, or vlcm [11], besides being simpler in its coding than the original lcm, is significantly faster. It has vectorised loops over linked lists of neighbours. It is more than 95% of the time which is spent in the routines kravlc and rhovlc where the cohesive and pair-wise functions are evaluated. A significant speeding up of the code in [11] is made by using the cray compiler
directive to vectorise the density and force array updating loops which end at label 3422 in these two routines. The equations of motion are integrated with a four value gear algorithm (predictor–corrector) in the n-representation, which is clearly described in [12]. 3. Molecular dynamics calculation results In this section we present results of MD modelling of C atom in Fe with the pair potential of Johnson. The simulations were carried out for the temperature at 0 K for up to 50 ns. (High temperatures are required to generate high statistics of C atom jumps in MD simulation, and we note that the F–S potential of Ackland et al. [13] gives the bcc crystal structure at all temperatures. That will be our next work.) The thermal expansion of the lattice was taken into account by a corresponding increase of the lattice parameter, a0, to maintain zero pressure. Time integration was performed using the velocity Verlet algorithm with a variable time step, ts, which was controlled by fixing the maximum displacement of the fastest atom in the system at each step to be 0.01 a0 or 0.02 a0. The mean value of ts was from 0.60 to 0.80 fs. During simulations, the jump direction, displacement and coordinates of the solute atom were recorded every time a jump occurred from one octahedral position to another. 3.1. Properties of perfect bcc lattice of Fe When there is no vacancy and no interstitial atom in a bcc lattice, it can be called a perfect bcc lattice. We calculate the energy of a perfect bcc lattice in two different ways. The energy of a atom is described below:
cohe ¼ 4 Vðr 1 Þ þ 3 Vðr 2 Þ;
ð2Þ
cohe is the energy of a perfect bcc lattice. V(r1), V(r2) are the pair potential described in Eq. (1). r1 is the nearest distance of the two atoms. r2 is the second nearest distance of the two atoms. The energy of a atom in a perfect bcc lattice is 1.54 eV (electron volt) on subroutine cebcc algorithm. The energy of a atom in a perfect bcc lattice is calculated using main program. The whole lattice energy is 3073.22 eV, so the energy of a atom in a perfect bcc lattice is 1.54 eV using the main program. So we can see the energy is same on different calculated algorithm. 3.2. Properties of a vacancy in bcc lattice of Fe There are different defects in the lattice of Fe because of irradiation or thermal motion. We will calculate the energies when a vacancy is in a bcc lattice of Fe. The coordinates of the vacancy is (0, 0, 0). The lattice energy is 3070.15 eV. The defect formation energy Efv is described below:
Efv ¼ lattice energy 1999 echo:
ð3Þ
The defect structural energy Ecv is described below: c
Ev ¼ lattice energy 2000 echo:
ð4Þ
The lattice energy is the energy of bcc lattice of Fe which has a vacancy. The echo is the energy of per Fe atom in perfect bcc lattice. The result of defect formation energy Efv is 1.53 eV. The defect structural energy Ecv is 3.07 eV. 3.3. Properties of a interstitial carbon atom in bcc lattice of Fe When a interstitial carbon atom is in a bcc lattice of Fe, the properties of the bcc lattice of Fe will be changed. This changing will be reflected through the energy. We calculated the lattice en-
Q. Jiansheng et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3179–3181
<001>
5.5
defect formation energy (eV)
5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
3181
ferent directions. A spot of the smallest defect formation energy Efv of a vacancy-carbon atom complex in bcc lattice of Fe can be seen in Fig. 1. Its coordinate is (0.35, 0, 0). That is the most stable site of a carbon atom near a vacancy along h0 0 1i. We also calculated the defect formation energy Efv of a vacancycarbon atom complex on {0 0 1} plane. The vacancy coordinate is (0, 0, 0) too. The carbon atom coordinate changes on 0.02 a0 per step on h1 0 0i and h0 1 0i direction. It maintains 0 spot on the h0 0 1i direction. The result is described in Fig. 2. From the Fig. 2, there are four spots existed on where the defect formation energy Efv of a vacancy-carbon atom complex is smaller than other spots. But the four spots are all not on (0, 0, 0) spot which is the vacancy coordinate. The defect formation energy is 1.52 eV where the carbon coordinate is on (0.35, 0.01, 0.01) and (0.01, 0.35, 0.01). On the other two spots, The defect formation energy is 3.35 eV where the carbon coordinate is on (0.49, 0.97, 0.01) and (0.97, 0.49, 0.01).
X (a0) Fig. 1. The defect formation energy Efv of a vacancy-carbon atom complex in h0 0 1i direction.
defect formation energy (eV)
{110}
4. Conclusions This paper has reported the simulation study to model C in a-Fe, and the simulations have been performed using an empirical pair potential of Johnson et al. for the Fe matrix and the Fe–C interaction. The following conclusions are drawn: (1) The defect formation energy Efv of a vacancy-carbon atom complex in bcc lattice of Fe is same on the three different directions, on h0 0 1i, on h0 1 0i and on h1 0 0i. The spot of the smallest defect formation energy Efv of a vacancy-carbon atom complex in bcc lattice of Fe is (0.35, 0, 0). That is the most stable site of a carbon atom in a vacancy-carbon atom complex in bcc lattice of Fe.
25 20 15 10 5 0 0.0
1.0 0.8
0.2
0.6
0.4
0.4
0.6
X (a
0)
0.2
0.8 1.0
0.0
Y
0)
(a
Fig. 2. The defect formation energy Efv of a vacancy-carbon atom complex on {1 1 0} plane.
ergy, the defect formation energy Efv and the defect structural energy Ecv . The lattice energy is 3072.95 eV. It is higher than that of a vacancy in bcc lattice of Fe. It is same to the defect formation energy Efv and the defect structural energy Ecv is smaller than that of a vacancy in bcc lattice of Fe. The defect formation energy Efv is 0.27 eV. The defect structural energy Ecv is 0.27 eV too. 3.4. Properties of a vacancy-carbon atom complex in bcc lattice of Fe A interstitial carbon atom is inevitable in bcc lattice of Fe. When there is a vacancy in bcc lattice of Fe, what is the site of a interstitial carbon atom near the vacancy. Can it be on the center of the vacancy site? We calculated the defect formation energy Efv of a vacancy-carbon atom complex in bcc lattice of Fe to result the question. The vacancy coordinate is (0, 0, 0). The carbon atom motives in different directions such as h0 0 1i, h0 1 0i and h1 0 0i direction. The step is 0.02 a0. The defect formation energy Efv of a vacancycarbon atom complex in bcc lattice of Fe is same on the three dif-
(2) On {0 0 1} plane, there are four spots existed on where the defect formation energy Efv of a vacancy-carbon atom complex is smaller than other spots. But the four spots are all not on (0, 0, 0) spot which is the vacancy coordinate. Acknowledgements This work is supported by the National Natural Science Foundation of China with Grant No. 50571019 and the National Basic Research and Development Foundation of China with Grant No. 2008cb717802. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
[12] [13]
D.S. Gelles, J. Nucl. Mater. 225 (1995) 163. E.A. Little, D.A. Stow, Ajaccio, Corsica, France, 4–8 June 1979, p. 17. A.H. Cottrell, B.A. Bilby, Proc. Phys. Soc. Lond. A 62 (1949) 49. A. Vehanen, P. Hautojarvi, J. Johansson, J. Yli-Kauppila, P. Moser, Phys. Rev. B 25 (1982) 762. R.A. Arndt, A.C. Damask, Acta Metall. 12 (1964) 341. S. Takaki, J. Fuss, H. Kugler, U. Dedek, H. Schults, Radiat. Eff. 79 (1983) 87. C. Domain, C.S. Becquart, J. Foct, Phys. Rev. B 69 (2004) 144112. E.A. Little, D.R. Harries, Met. Sci. J. 4 (1970) 188. R.A. Johnson, Phys. Rev. 134 (1964) A1329. D. Fincham, D.M. Heyes, in: M.W. Evans (Ed.), Advances in Chemical Physics, Vol. 63, Wiley, New york, 1985, p. 493. D.M. Heyes, W. Smith, Information quarterly for computer simulation of condensed phases, No. 26, September 1987, An informal newsletter associated with collaborative computational Project No. 5 on molecular dynamics, Monte Carlo and lattice simulations of condensed phases, Science and Engineering Research Council, Daresbury Laboratory, Daresbury, Warrington wa4 4ad, England. H.J.C. Berendsen, W.F.V. Gunsteren, in: Proc. Int. School of Physics ‘Enrico Fermi’, 23 July–2 August 1985, North-Holland, 1986. G.J. Ackland, D.J. Bacon, A.F. Calder, T. Harry, Philos. Mag. A 75 (1997) 713.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Computer simulation on the interaction between vacancy and carbon impurity in a-Fe Qiao Jiansheng a,b, Liu yongli b, Huang Yina a, Xiao Xin a, Wan Farong a,c,* a
Department of Materials Physics and Chemistry, University of Science and Technology Beijing, Beijing 100083, China Primary Education College, Xingtai University, Xingtai 054000, China c Beijing Key Lab of Advanced Energy Material and Technology, Beijing 100083, China b
a r t i c l e
i n f o
Article history: Available online 16 June 2009 PACS: 02.70.NS Keywords: Molecular dynamics Pair potential Vacancy Carbon
a b s t r a c t Metals possessing a bcc lattice have been considered as materials with high resistance to irradiation swelling. The aggregation of vacancies is the main reason of the swelling and the interaction of carbon atoms with vacancies affects the mechanical properties of metals. The complex of vacancy-carbon atom in a-Fe is modeled by MD method using a pair inter-atomic potential. The defect formation energy of vacancy-carbon atom complex was obtained by MD method. The stable site of carbon near a vacancy in three dimensions was discussed. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Metals and alloys possessing a bcc lattice have been considered as materials with high resistance to swelling, especially when compared with fcc metals and alloys. Experimental data provided by Gelles on model iron-base alloys showed low swelling levels after being irradiated to doses as high as 200 dpa in the Fast Flux Test Facility (FFTF) [1]. There has been a general perception that the low swelling rate of bcc materials was an intrinsic property of bcc crystal lattice [2]. So that the bcc alloys was considered to be the structural material of the nuclear fusion reactor. Carbon was one of the elements in steel. The properties of iron– carbon (Fe–C) solid solutions have been investigated extensively due to their particular relevance in steel technology. It is known from experiment that the interaction of carbon (C) interstitial atoms with lattice defects significantly affects the mechanical properties of ferritic steels, [3], and properties of point defects. Although it is widely accepted that vacancy (V) diffusivity is significantly decreased due to trapping of C atoms, there has been considerable uncertainty concerning the magnitude and form of the interaction. Vehanen et al. [4] deduced a value of 0.85 eV for the binding energy of a V–C complex, which, when added to a vacancy migration energy value of 0.55 eV in pure Fe, is consis-
* Corresponding author. Address: Department of Materials Physics and Chemistry, University of Science and Technology Beijing, Beijing 100083, China. Tel.: +86 10 62333932; fax: +86 10 62333724. E-mail address: [email protected] (W. Farong). 0168-583X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2009.06.070
tent with an effective vacancy migration energy of 1.35 eV for steels. However, V–C binding energy values of 0.41 and 1.10 eV have also been reported [5,6], the former estimate being close to the result of 0.47 eV obtained in recent ab initio calculations [7]. Vehanen et al. [4] argued that the value of 0.41 eV found by Arndt and Damask [5] should associate with the binding of C atoms to V–C complexes and vacancy agglomerates. In addition, the experiments on V–C interaction were performed by using electrons irradiation to build up a measurable concentration of non-equilibrium vacancies, when self-interstitial atoms were also produced. The role of the self-interstitial atoms (I) on trapping C atoms was unclear. Little and Harries [8] considered the binding energy of 0.41 eV found in [5] to be more probably associated with a C atom–interstitial cluster complex, although the later experiments [4,6] indicated the C–I binding energy to be much smaller, i.e. 0.10 eV, and ab initio calculations actually gave a negative energy in the range 0.20 to 0.40 eV for this pair [7]. Hence, it was desirable to use computer simulation methods to investigate the migration properties and defect interactions of C solute, but an Fe–C inter-atomic potential set was required and the development of such potentials has lagged behind those for pure a-Fe and Fe alloys with substitutional solute, and only a few Fe–C potentials had been proposed to date. The paper is organised as follows. In Section 2, the calculation scheme and the potential sets are described. In Section 3, the MD results on perfect bcc lattice, a vacancy, an interstitial solute and a vacancy–interstitial solute complex in a-Fe are presented. Conclusions are drawn in Section 4.
3180
Q. Jiansheng et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3179–3181
2. Calculation model Two atomic-scale techniques are used here to study static properties of a V, a C atom and V–C complexes in an Fe matrix. MD is used to model a V, an interstitial C atom and V–C complexes using a combination of conjugate gradients potential energy minimization at zero temperature and to calculate formation, binding and migration energies. In these cases the calculations were performed for a simulation box containing 2000 bcc lattice sites with constant volume and periodic boundary conditions. The pair potential that was proposed by Johnson [9] was used for the Fe–Fe interaction and the Fe–C interaction. The pair potential is described below:
VðrÞ Ar 3 þ Br 2 þ Cr þ D;
ð1Þ
where V(r) is the pair potential of the two atoms (Fe–Fe or Fe–C) which is interacted. r is the distance between the two interacting atoms. A, B, C and D are constants which is difference to the different distances of the two atoms. The potential sets allow a correct description of the elastic properties of a-Fe, while retaining almost the same properties of a C atom as the original model of Johnson [9]. This was verified by static simulation, as follows. The octahedral site was found to be the most stable position for a C atom. Both Fe potentials give the insertion energy of a C atom into a-Fe (difference in the crystal energy with and without C atom) of 1.23 eV, which is close to the corresponding energy of 1.34 eV calculated using the pair potential of Johnson [9] for Fe–Fe interactions. The tetrahedral site is the saddle point for a C jump from one octahedral site to another, as with the potential set of Johnson [9] and in the ab initio calculations by Domain and Becquart [7]. In the present work we investigate thermally activated diffusion of C atoms in bcc Fe and V–C interaction by the molecular dynamics (MD) method. Only molecular statics (MS) modeling (i.e. representing 0 K) of the interaction between a C atom and some point defects has been performed with the potentials mentioned above. Since we are not aware of any potential set that accurately represents the Fe–C system, the approach adopted here is to study a ‘model’ system for an octahedral interstitial solute in a bcc metal matrix that has many similarities to the real alloy. We have used for this purpose the pair potential of Johnson [9] for the Fe–C and Fe–Fe interaction. We also use the data obtained, together with results from the ab initio calculations referred to above, to review and interpret the earlier experimental information on C-point defect interactions. We use a molecular dynamics program which is modified for introduction of vacancies &/or interstitials coordinates by user. This program is a greatly modified and extended version of code developed for constant stress molecular dynamics using the parinello-rahman lagrangian. It is designed for use on the cray and exploits vectorisation, the special cray functions cvmgp and cray timing routines. The Johnson Fe potential is used in this version. The original link cell method, or lcm [10] was designed for a large number n of particles, the cpu scales as n because of using lcm. When a neighbour list ‘n-list’ is used in the traditional way, the cpu scales as n at each time-step at which the n-list is consulted, and as n n at each time-step at which it is updated. The vectorised link cell method, or vlcm [11], besides being simpler in its coding than the original lcm, is significantly faster. It has vectorised loops over linked lists of neighbours. It is more than 95% of the time which is spent in the routines kravlc and rhovlc where the cohesive and pair-wise functions are evaluated. A significant speeding up of the code in [11] is made by using the cray compiler
directive to vectorise the density and force array updating loops which end at label 3422 in these two routines. The equations of motion are integrated with a four value gear algorithm (predictor–corrector) in the n-representation, which is clearly described in [12]. 3. Molecular dynamics calculation results In this section we present results of MD modelling of C atom in Fe with the pair potential of Johnson. The simulations were carried out for the temperature at 0 K for up to 50 ns. (High temperatures are required to generate high statistics of C atom jumps in MD simulation, and we note that the F–S potential of Ackland et al. [13] gives the bcc crystal structure at all temperatures. That will be our next work.) The thermal expansion of the lattice was taken into account by a corresponding increase of the lattice parameter, a0, to maintain zero pressure. Time integration was performed using the velocity Verlet algorithm with a variable time step, ts, which was controlled by fixing the maximum displacement of the fastest atom in the system at each step to be 0.01 a0 or 0.02 a0. The mean value of ts was from 0.60 to 0.80 fs. During simulations, the jump direction, displacement and coordinates of the solute atom were recorded every time a jump occurred from one octahedral position to another. 3.1. Properties of perfect bcc lattice of Fe When there is no vacancy and no interstitial atom in a bcc lattice, it can be called a perfect bcc lattice. We calculate the energy of a perfect bcc lattice in two different ways. The energy of a atom is described below:
cohe ¼ 4 Vðr 1 Þ þ 3 Vðr 2 Þ;
ð2Þ
cohe is the energy of a perfect bcc lattice. V(r1), V(r2) are the pair potential described in Eq. (1). r1 is the nearest distance of the two atoms. r2 is the second nearest distance of the two atoms. The energy of a atom in a perfect bcc lattice is 1.54 eV (electron volt) on subroutine cebcc algorithm. The energy of a atom in a perfect bcc lattice is calculated using main program. The whole lattice energy is 3073.22 eV, so the energy of a atom in a perfect bcc lattice is 1.54 eV using the main program. So we can see the energy is same on different calculated algorithm. 3.2. Properties of a vacancy in bcc lattice of Fe There are different defects in the lattice of Fe because of irradiation or thermal motion. We will calculate the energies when a vacancy is in a bcc lattice of Fe. The coordinates of the vacancy is (0, 0, 0). The lattice energy is 3070.15 eV. The defect formation energy Efv is described below:
Efv ¼ lattice energy 1999 echo:
ð3Þ
The defect structural energy Ecv is described below: c
Ev ¼ lattice energy 2000 echo:
ð4Þ
The lattice energy is the energy of bcc lattice of Fe which has a vacancy. The echo is the energy of per Fe atom in perfect bcc lattice. The result of defect formation energy Efv is 1.53 eV. The defect structural energy Ecv is 3.07 eV. 3.3. Properties of a interstitial carbon atom in bcc lattice of Fe When a interstitial carbon atom is in a bcc lattice of Fe, the properties of the bcc lattice of Fe will be changed. This changing will be reflected through the energy. We calculated the lattice en-
Q. Jiansheng et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3179–3181
<001>
5.5
defect formation energy (eV)
5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
3181
ferent directions. A spot of the smallest defect formation energy Efv of a vacancy-carbon atom complex in bcc lattice of Fe can be seen in Fig. 1. Its coordinate is (0.35, 0, 0). That is the most stable site of a carbon atom near a vacancy along h0 0 1i. We also calculated the defect formation energy Efv of a vacancycarbon atom complex on {0 0 1} plane. The vacancy coordinate is (0, 0, 0) too. The carbon atom coordinate changes on 0.02 a0 per step on h1 0 0i and h0 1 0i direction. It maintains 0 spot on the h0 0 1i direction. The result is described in Fig. 2. From the Fig. 2, there are four spots existed on where the defect formation energy Efv of a vacancy-carbon atom complex is smaller than other spots. But the four spots are all not on (0, 0, 0) spot which is the vacancy coordinate. The defect formation energy is 1.52 eV where the carbon coordinate is on (0.35, 0.01, 0.01) and (0.01, 0.35, 0.01). On the other two spots, The defect formation energy is 3.35 eV where the carbon coordinate is on (0.49, 0.97, 0.01) and (0.97, 0.49, 0.01).
X (a0) Fig. 1. The defect formation energy Efv of a vacancy-carbon atom complex in h0 0 1i direction.
defect formation energy (eV)
{110}
4. Conclusions This paper has reported the simulation study to model C in a-Fe, and the simulations have been performed using an empirical pair potential of Johnson et al. for the Fe matrix and the Fe–C interaction. The following conclusions are drawn: (1) The defect formation energy Efv of a vacancy-carbon atom complex in bcc lattice of Fe is same on the three different directions, on h0 0 1i, on h0 1 0i and on h1 0 0i. The spot of the smallest defect formation energy Efv of a vacancy-carbon atom complex in bcc lattice of Fe is (0.35, 0, 0). That is the most stable site of a carbon atom in a vacancy-carbon atom complex in bcc lattice of Fe.
25 20 15 10 5 0 0.0
1.0 0.8
0.2
0.6
0.4
0.4
0.6
X (a
0)
0.2
0.8 1.0
0.0
Y
0)
(a
Fig. 2. The defect formation energy Efv of a vacancy-carbon atom complex on {1 1 0} plane.
ergy, the defect formation energy Efv and the defect structural energy Ecv . The lattice energy is 3072.95 eV. It is higher than that of a vacancy in bcc lattice of Fe. It is same to the defect formation energy Efv and the defect structural energy Ecv is smaller than that of a vacancy in bcc lattice of Fe. The defect formation energy Efv is 0.27 eV. The defect structural energy Ecv is 0.27 eV too. 3.4. Properties of a vacancy-carbon atom complex in bcc lattice of Fe A interstitial carbon atom is inevitable in bcc lattice of Fe. When there is a vacancy in bcc lattice of Fe, what is the site of a interstitial carbon atom near the vacancy. Can it be on the center of the vacancy site? We calculated the defect formation energy Efv of a vacancy-carbon atom complex in bcc lattice of Fe to result the question. The vacancy coordinate is (0, 0, 0). The carbon atom motives in different directions such as h0 0 1i, h0 1 0i and h1 0 0i direction. The step is 0.02 a0. The defect formation energy Efv of a vacancycarbon atom complex in bcc lattice of Fe is same on the three dif-
(2) On {0 0 1} plane, there are four spots existed on where the defect formation energy Efv of a vacancy-carbon atom complex is smaller than other spots. But the four spots are all not on (0, 0, 0) spot which is the vacancy coordinate. Acknowledgements This work is supported by the National Natural Science Foundation of China with Grant No. 50571019 and the National Basic Research and Development Foundation of China with Grant No. 2008cb717802. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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