- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Volume 2012 (2012), Article ID 431742, 4 pages
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. Dlouhy
Copyright © 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.
Variation of vacancy formation energy () with 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 iron and fcc iron.
Recently in some papers, defect structures in -Fe were discussed using Monte Carlo (MC) technique [1, 2], ab initio density functional (PP) model [3–5], and molecular dynamics (MD) simulation . Iron exists in different phases, namely -Fe (room temperature to 768°C, bcc structure, lattice constant AU, and ferromagnetic), -Fe (768°C to 910°C, bcc structure, AU, and nonmagnetic), -Fe (910°C to 1400°C, fcc structure, AU, and paramagnetic), and -Fe (1400°C to 1535°C, bcc structure again). Korhonen et al.  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.  predicted significant larger values of vacancy formation energy than the experimental ones in cases of bcc V and Cr using LDA.
Söderlind et al.  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 and metals than the metals. GGA and LDA formation energies are very similar for the and metals but for the 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 () Ashcroft’s empty core model potential (called here after AECMP)  to study the variation of monovacancy formation energy () in iron with the help of nine different types of exchange and correlation functions (called ECF) [11–20].
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 . 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 considering relaxation energy associated with these defect formations as where
Here and are the lattice and quasi-continuous wave numbers, respectively, is the atomic volume, the electronic charge, the valency, the convergence factor, the Fermi wavenumber, the Fermi energy, the AECMP with parameter , the dielectric function or screening factor, the perturbation characteristics, and the ECF whose nine different forms have been shown in Table 1.
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 (-Fe) and bcc iron (-Fe) are shown in Table 2. In the first step the variation of with parameter of AECMP is plotted for nine different ECF from 0 to 5 AU as shown in Figure 1 for fcc iron (-Fe) and Figure 2 for bcc iron (-Fe). The two graphs have positive peaks and they look almost similar due to term of AECMP for all exchange and correlations but there is a slight variation. It is observed that experimental value of lies near the nodal point corresponding to the condition rather than that to the maxima. Fitted value of has been chosen corresponding to the condition , where is the Bohr radius.
Here is the melting temperature, the activation energy, and the cohesive energy of the metal. From the experimental value of we note that the fitted value of is within the first peak value of 2 AU and we note from Table 2 that values lie close to Bohr radius . ECF’s of Kleinman , Harrison , Vashishta and Singwi , and Taylor  give reasonably close values of while others give the range over of .
In conclusion, it should be noted that the inherent simplicity of AECMP makes it difficult to have a universal 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.
- J. Rottler, D. J. Srolovitz, and R. Car, “Point defect dynamics in bcc metals,” Physical Review B, vol. 71, no. 6, Article ID 064109, 12 pages, 2005.
- S. M. J. Gordon, S. D. Kenny, and R. Smith, “Diffusion dynamics of defects in Fe and Fe-P systems,” Physical Review B, vol. 72, no. 21, Article ID 214104, 10 pages, 2005.
- C. Domain and C. S. Becquart, “Diffusion of phosphorus in α-Fe: An ab initio study,” Physical Review B, vol. 71, no. 21, 13 pages, 2005.
- C. C. Fu and F. Willaime, “Ab 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.
- P. Olsson, C. Domain, and J. Wallenius, “Ab initio study of Cr interactions with point defects in bcc Fe,” Physical Review B, vol. 75, no. 1, Article ID 014110, 12 pages, 2007.
- C. Bos, J. Sietsma, and B. J. Thijsse, “Molecular 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.
- T. Korhonen, M. J. Puska, and R. M. Nieminen, “Vacancy-formation energies for fcc and bcc transition metals,” Physical Review B, vol. 51, no. 15, pp. 1926–1927, 1995.
- Yu. N. Osetsky, M. Victoria, A. Serra, S. I. Golubov, and V. Priego, “Computer 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.
- P. Söderlind, L. H. Yang, J. A. Moriarty, and J. M. Wills, “First-principles formation energies of monovacancies in bcc transition metals,” Physical Review B, vol. 61, no. 4, pp. 2579–2586, 2000.
- N. W. Ashcroft, “Electron-ion pseudopotentials in metals,” Physics Letters, vol. 23, no. 1, pp. 48–50, 1966.
- W. F. King III and P. H. Cutler, “Lattice dynamics of beryllium from a first-principles nonlocal pseudopotential approach,” Physical Review B, vol. 2, no. 6, pp. 1733–1742, 1970.
- W. F. King III and P. H. Cutler, “Lattice dynamics of magnesium from a first-principles nonlocal pseudopotential approach,” Physical Review B, vol. 3, no. 8, pp. 2485–2496, 1971.
- L. J. Sham, “A Calculation of the phonon frequencies in sodium,” Proceedings of the Royal Society A, vol. 283, no. 1392, pp. 33–49, 1965.
- D. J. W. Geldert and S. H. Vosko, “The screening function of an interacting electron gas,” Canadian Journal of Physics, vol. 44, pp. 2137–2171, 1965.
- L. Kleinman, “Exchange and the dielectric screening function,” Physical Review, vol. 172, no. 2, pp. 383–390, 1968.
- W. A. Harrison, Pseudopotentials in the Theory of Metals, W. A. Benjamin, New York, NY, USA, 1966.
- P. Vashishta and K. S. Singwi, “Electron correlations at metallic densities. v,” Physical Review B, vol. 6, no. 3, pp. 875–887, 1972.
- R. Taylor, “A simple, useful analytical form of the static electron gas dielectric function,” Journal of Physics F, vol. 8, no. 8, pp. 1699–1702, 1978.
- H. Hubbard, “The 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.
- S. D. Mahanti and T. P. Das, “Theory 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.
- C. Kittel, Introduction to Solid State Physics, Wiley Eastern, New Delhi, India, 5th edition, 1979.
- H. Matter, J. Winter, and W. Triftshäuser, “Phase transformations and vacancy formation energies of transition metals by positron annihilation,” Applied Physics A, vol. 20, no. 2, pp. 135–140, 1979.
- A. Ghorai, “Dependence of mono-vacancy formation energy on the parameter of Ashcroft's potential,” Defect and Diffusion Forum, vol. 278, pp. 25–32, 2008.
- A. Ghorai, “A 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.
- A. Ghorai, “Exploration 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.