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ISRN Metallurgy
VolumeΒ 2012Β (2012), Article IDΒ 431742, 4 pages
http://dx.doi.org/10.5402/2012/431742
Research Article

Defect Studies in bcc and fcc Iron

Department of Physics, Chandernagore College, Chandernagore 712136, India

Received 18 January 2012; Accepted 5 March 2012

Academic Editor: A.Β Dlouhy

Copyright Β© 2012 A. Ghorai and Arjun Das. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Variation of vacancy formation energy (𝐸𝐹1𝑣) with π‘Ÿπ‘ of Ashcroft's empty core model potential (AECMP) model for different exchange and correlation functions (ECFs) show almost independent nature but slight variation with ECF for both bcc 𝛼 iron and fcc 𝛾 iron.

1. Introduction

Recently in some papers, defect structures in 𝛼-Fe were discussed using Monte Carlo (MC) technique [1, 2], ab initio density functional (PP) model [3–5], and molecular dynamics (MD) simulation [6]. Iron exists in different phases, namely 𝛼-Fe (room temperature to 768Β°C, bcc structure, lattice constant π‘Ž=5.48AU, and ferromagnetic), 𝛽-Fe (768Β°C to 910Β°C, bcc structure, π‘Ž=5.48AU, and nonmagnetic), 𝛾-Fe (910Β°C to 1400Β°C, fcc structure, π‘Ž=6.73AU, and paramagnetic), and 𝛿-Fe (1400Β°C to 1535Β°C, bcc structure again). Korhonen et al. [7] predicted that the stability of a self-interstitial in bcc-Fe depends on the range of potential but not on the type, while Osetsky et al. [8] predicted significant larger values of vacancy formation energy than the experimental ones in cases of bcc V and Cr using LDA.

SΓΆderlind et al. [9] incorporated the results of above two groups of researchers using a full-potential, linear muffin-tin-orbital (FP-LMTO) method in conjunction with both the local-density approximation (LDA) and the generalized-gradient approximation (GGA) in bcc metals. A complementary ab initio pseudopotential (PP) method has also been used. They predicted FP-LMTO-LDA and PP-LDA formation energies are nearly identical within or close to experimental error bars for all bcc metals except Cr, and the overall agreement with experiment is better for the 4𝑑 and 5𝑑 metals than the 3𝑑 metals. GGA and LDA formation energies are very similar for the 4𝑑 and 5𝑑 metals but for the 3𝑑 metals, and especially Fe, GGA performs better. The dominant structural effects are an approximate 5% inward relaxation of the first near-neighbor shell for group V metals and a corresponding 1% inward relaxation for group VI metals, with the exception of Mo, for which the second-shell atoms also relax inward by about 1%. Thus it will be interesting to use in this paper the one parameter (π‘Ÿπ‘) Ashcroft’s empty core model potential (called here after AECMP) [10] to study the variation of monovacancy formation energy (𝐸𝐹1𝑣) in iron with the help of nine different types of exchange and correlation functions (called ECF) [11–20].

2. Formulations

The structure-dependent energy of a crystal depends on ion-ion, ion-electron, and electron-electron interactions and is also dependent on the modified lattice wave numbers. The modifications in the lattice wave numbers from their perfect lattice value are necessary to maintain the lattice volume and the number of lattice ions constant. The ion-ion interaction is determined from electrostatic energy and the last two interactions are included in the band structure energy, which is calculated using the second-order perturbation theory incorporating pseudopotential model form [16]. When a vacancy is created the Brillouin zone volume has to be scaled up in order to keep the lattice volume constant and so the lattice wave numbers are modified. Finally, one gets the expressions for vacancy formation energy 𝐸𝐹1𝑣 considering relaxation energy associated with these defect formations as𝐸𝐹1𝑣=ξ…žξ“π‘ž0π‘ž03ξ€·π‘žπœ•π‘ˆ0ξ€Έπœ•π‘ž0+Ξ©2πœ‹2ξ€œβˆ0π‘ˆ(π‘ž)π‘ž2π‘‘π‘ž,(1) where π‘ˆ(π‘ž)=πΏπ‘‘πœ‚β†’βˆ2πœ‹π‘’2𝑧2Ξ©π‘ž2π‘’βˆ’π‘ž2/4πœ‚+[𝑀(π‘ž)]2πœ€(π‘ž)πœ’(π‘ž),(2)𝑀(π‘ž)=βˆ’4πœ‹π‘§π‘’2cosπ‘žπ‘Ÿπ‘Ξ©π‘ž2,(3)πœ€(π‘ž)=1βˆ’8πœ‹π‘’2Ξ©π‘ž2[]1βˆ’π‘“(π‘ž)πœ’(π‘ž),(4)πœ’(π‘ž)=βˆ’3𝑧8𝐸𝐹1+4π‘˜2πΉβˆ’π‘ž24π‘˜πΉπ‘ž||ln2π‘˜πΉ||+π‘ž||2π‘˜πΉ||ξƒ­βˆ’π‘ž.(5)

Here π‘ž0 and π‘ž are the lattice and quasi-continuous wave numbers, respectively, Ξ© is the atomic volume, 𝑒 the electronic charge, 𝑧 the valency, πœ‚ the convergence factor, π‘˜πΉ the Fermi wavenumber, 𝐸𝐹 the Fermi energy, 𝑀(π‘ž) the AECMP with parameter π‘Ÿπ‘, πœ€(π‘ž) the dielectric function or screening factor, πœ’(π‘ž) the perturbation characteristics, and 𝑓(π‘ž) the ECF whose nine different forms have been shown in Table 1.

tab1
Table 1: Different forms of ECF 𝑓(π‘ž) and corresponding fitted value of parameter π‘Ÿπ‘ of AECMP in atomic unit (AU).

3. Discussions

The calculation of (1) needs integration over quasi-continuous wavenumbers by quadrature technique and the discrete sum over lattice wave numbers. The input and output parameters for this purpose for fcc iron (𝛾-Fe) and bcc iron (𝛼-Fe) are shown in Table 2. In the first step the variation of 𝐸𝐹1𝑣 with parameter π‘Ÿπ‘ of AECMP is plotted for nine different ECF from 0 to 5 AU as shown in Figure 1 for fcc iron (𝛾-Fe) and Figure 2 for bcc iron (𝛼-Fe). The two graphs have positive peaks and they look almost similar due to cos2π‘žπ‘Ÿπ‘ term of AECMP for all exchange and correlations but there is a slight variation. It is observed that experimental value of 𝐸𝐹1𝑣 lies near the nodal point corresponding to the condition 𝐸𝐹1𝑣→0 rather than that to the maxima. Fitted value of π‘Ÿπ‘ has been chosen corresponding to the condition π‘Ž0β‰€π‘Ÿπ‘<2πœ‹/π‘˜πΉ, where π‘Ž0 is the Bohr radius.

tab2
Table 2: Input and output parameters for fcc and bcc iron (1 eV = 13.605 Rydberg and 1 atomic unit (AU) = 5.29177 Γ— 10βˆ’2 nm).
431742.fig.001
Figure 1: 𝐸𝐹1𝑣-π‘Ÿπ‘plot for fcc 𝛾-iron (𝛾-Fe).
431742.fig.002
Figure 2: 𝐸𝐹1𝑣-π‘Ÿπ‘plot for bcc 𝛼-iron (𝛼-Fe).

It is observed that the experimental value of 𝐸𝐹1𝑣, obtained from positron annihilation technique, lies within the range of the theoretical value of it obtained from the empirical relation [23–25]:π‘‡π‘š(𝐾)=1200𝐸𝐹1𝑣=660𝑄0=360𝐸coh.(6)

Here π‘‡π‘š is the melting temperature, 𝑄0 the activation energy, and 𝐸coh the cohesive energy of the metal. From the experimental value of 𝐸𝐹1𝑣 we note that the fitted value of π‘Ÿπ‘ is within the first peak value of 2 AU and we note from Table 2 that π‘Ÿπ‘ values lie close to Bohr radius (π‘Ž0=1AU). ECF’s of Kleinman [15], Harrison [16], Vashishta and Singwi [17], and Taylor [18] give reasonably close values of π‘Ÿπ‘ while others give the range over of π‘Ÿπ‘.

In conclusion, it should be noted that the inherent simplicity of AECMP makes it difficult to have a universal π‘Ÿπ‘ parameter for all types of atomic property calculations and we have to use different ECFs of which Taylor, Harrison, Kleinmann, Vashishta and Singwi type of ECF give better results in this case.

References

  1. J. Rottler, D. J. Srolovitz, and R. Car, β€œPoint defect dynamics in bcc metals,” Physical Review B, vol. 71, no. 6, Article ID 064109, 12 pages, 2005. View at Publisher Β· View at Google Scholar Β· View at Scopus
  2. S. M. J. Gordon, S. D. Kenny, and R. Smith, β€œDiffusion dynamics of defects in Fe and Fe-P systems,” Physical Review B, vol. 72, no. 21, Article ID 214104, 10 pages, 2005. View at Publisher Β· View at Google Scholar Β· View at Scopus
  3. C. Domain and C. S. Becquart, β€œDiffusion of phosphorus in α-Fe: An ab initio study,” Physical Review B, vol. 71, no. 21, 13 pages, 2005. View at Publisher Β· View at Google Scholar Β· View at Scopus
  4. C. C. Fu and F. Willaime, β€œAb initio study of helium in α-Fe: dissolution, migration, and clustering with vacancies,” Physical Review B, vol. 72, no. 6, Article ID 064117, 6 pages, 2005. View at Publisher Β· View at Google Scholar
  5. P. Olsson, C. Domain, and J. Wallenius, β€œAb initio study of Cr interactions with point defects in bcc Fe,” Physical Review B, vol. 75, no. 1, Article ID 014110, 12 pages, 2007. View at Publisher Β· View at Google Scholar Β· View at Scopus
  6. C. Bos, J. Sietsma, and B. J. Thijsse, β€œMolecular dynamics simulation of interface dynamics during the fcc-bcc transformation of a martensitic nature,” Physical Review B, vol. 73, no. 10, Article ID 104117, 7 pages, 2006. View at Publisher Β· View at Google Scholar Β· View at Scopus
  7. T. Korhonen, M. J. Puska, and R. M. Nieminen, β€œVacancy-formation energies for fcc and bcc transition metals,” Physical Review B, vol. 51, no. 15, pp. 1926–1927, 1995. View at Publisher Β· View at Google Scholar Β· View at Scopus
  8. Yu. N. Osetsky, M. Victoria, A. Serra, S. I. Golubov, and V. Priego, β€œComputer simulation of vacancy and interstitial clusters in bcc and fcc metals,” in Proceedings of the International Workshop on Defect Production, Accumulation and Materials Performance in an Irradiation Environment, vol. 251, pp. 34–48, Journal of Nuclear Materials, 1997. View at Publisher Β· View at Google Scholar
  9. P. Söderlind, L. H. Yang, J. A. Moriarty, and J. M. Wills, β€œFirst-principles formation energies of monovacancies in bcc transition metals,” Physical Review B, vol. 61, no. 4, pp. 2579–2586, 2000. View at Scopus
  10. N. W. Ashcroft, β€œElectron-ion pseudopotentials in metals,” Physics Letters, vol. 23, no. 1, pp. 48–50, 1966. View at Scopus
  11. W. F. King III and P. H. Cutler, β€œLattice dynamics of beryllium from a first-principles nonlocal pseudopotential approach,” Physical Review B, vol. 2, no. 6, pp. 1733–1742, 1970. View at Publisher Β· View at Google Scholar
  12. W. F. King III and P. H. Cutler, β€œLattice dynamics of magnesium from a first-principles nonlocal pseudopotential approach,” Physical Review B, vol. 3, no. 8, pp. 2485–2496, 1971. View at Publisher Β· View at Google Scholar
  13. L. J. Sham, β€œA Calculation of the phonon frequencies in sodium,” Proceedings of the Royal Society A, vol. 283, no. 1392, pp. 33–49, 1965.
  14. D. J. W. Geldert and S. H. Vosko, β€œThe screening function of an interacting electron gas,” Canadian Journal of Physics, vol. 44, pp. 2137–2171, 1965.
  15. L. Kleinman, β€œExchange and the dielectric screening function,” Physical Review, vol. 172, no. 2, pp. 383–390, 1968. View at Publisher Β· View at Google Scholar Β· View at Scopus
  16. W. A. Harrison, Pseudopotentials in the Theory of Metals, W. A. Benjamin, New York, NY, USA, 1966.
  17. P. Vashishta and K. S. Singwi, β€œElectron correlations at metallic densities. v,” Physical Review B, vol. 6, no. 3, pp. 875–887, 1972. View at Publisher Β· View at Google Scholar Β· View at Scopus
  18. R. Taylor, β€œA simple, useful analytical form of the static electron gas dielectric function,” Journal of Physics F, vol. 8, no. 8, pp. 1699–1702, 1978. View at Publisher Β· View at Google Scholar Β· View at Scopus
  19. H. Hubbard, β€œThe description of collective motions in terms of many-body perturbation Theory. II. The correlation energy of a free-electron gas,” Proceedings of the Royal Society A, vol. 243, no. 1234, pp. 336–352, 1958.
  20. S. D. Mahanti and T. P. Das, β€œTheory of knight shifts and relaxation times in alkali metals-role of exchange core polarization and exchange-enhancement effects,” Physical Review B, vol. 3, no. 5, pp. 1599–1610, 1971. View at Publisher Β· View at Google Scholar Β· View at Scopus
  21. C. Kittel, Introduction to Solid State Physics, Wiley Eastern, New Delhi, India, 5th edition, 1979.
  22. H. Matter, J. Winter, and W. Triftshäuser, β€œPhase transformations and vacancy formation energies of transition metals by positron annihilation,” Applied Physics A, vol. 20, no. 2, pp. 135–140, 1979. View at Publisher Β· View at Google Scholar Β· View at Scopus
  23. A. Ghorai, β€œDependence of mono-vacancy formation energy on the parameter of Ashcroft's potential,” Defect and Diffusion Forum, vol. 278, pp. 25–32, 2008. View at Publisher Β· View at Google Scholar Β· View at Scopus
  24. A. Ghorai, β€œA study of the variation of monovacancy formation energy with the parameter of Ashcroft's potential and different exchange and correlation functions for some group-IIa and group-VIII FCC metals in the active divalent state,” Defect and Diffusion Forum, vol. 293, pp. 11–18, 2009. View at Publisher Β· View at Google Scholar Β· View at Scopus
  25. A. Ghorai, β€œExploration of parameters of Ashcroft's potential using monovacancy formation energy and different exchange and correlation functions for some trivalent FCC metals,” Defect and Diffusion Forum, vol. 294, pp. 113–118, 2009. View at Publisher Β· View at Google Scholar Β· View at Scopus