# Defect Studies in bcc and fcc Iron

Authors: A. Ghorai; Arjun Das
Journal: ISRN Metallurgy
 (2012)

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## Abstract

Variation of vacancy formation energy (EF1v) with rc 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.


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## Body

## 1. Introduction

Recently in some papers, defect structures inα-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 768°C, bcc structure, lattice constant a=5.48AU, and ferromagnetic), β-Fe (768°C to 910°C, bcc structure, a=5.48AU, and nonmagnetic), γ-Fe (910°C to 1400°C, fcc structure, a=6.73AU, and paramagnetic), and δ-Fe (1400°C to 1535°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.Söderlind 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 (rc) 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 as(1)EF1v=∑q0′q03∂U(q0)∂q0+Ω2π2∫0∝U(q)q2dq,
where
(2)U(q)=Ltη→∝2πe2z2Ωq2e-q2/4η+[w(q)]2ε(q)χ(q),(3)w(q)=-4πze2cos⁡⁡qrcΩq2,(4)ε(q)=1-8πe2Ωq2[1-f(q)]χ(q),(5)χ(q)=-3z8EF[1+4kF2-q24kFqln⁡|2kF+q||2kF-q|].Hereq0 and q are the lattice and quasi-continuous wave numbers, respectively, Ω is the atomic volume, e the electronic charge, z the valency, η the convergence factor, kF the Fermi wavenumber, EF the Fermi energy, w(q) the AECMP with parameter rc, ε(q) the dielectric function or screening factor, χ(q) the perturbation characteristics, and f(q) the ECF whose nine different forms have been shown in Table 1.Table 1
Different forms of ECFf(q) and corresponding fitted value of parameter rc of AECMP in atomic unit (AU).
Different forms off(q)bccα-Fefccγ-FeKing and Cutler [11, 12]q22(q2+2kF2)1.10171.0028Sham [13]q22(q2+kF2+2kF/πa0)1.11761.0204Geldert and Vosko [14]q22[q2+2kF2/(1+(0.026m′/m)(3Ω/4πza03)1/3)]1.10691.0083Kleinman [15]14[q2kF2+2kF/πa0+q2q2+kF2+2kF/πa0]1.15361.0496Harrison [16]q22(q2+kF2+kF2/3)1.14321.0478Vashishta and Singwi [17]A(1-e-Bq2/kF2)1.15391.0501Taylor [18]q24kF2(1+0.1534πkF)1.16871.0566Hubbard [19]q22(q2+kF2)1.17231.0789Mahanti and Das [20]q2kF2[1+(2/(1+(0.026m′/m)(3Ω/4πza03)1/3))2]+q2q2+kF2[1+(2/(1+(0.026m′/m)(3Ω/4πza03)1/3))2]1.23341.1282


## 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 rc 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 cos2qrc 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 rc has been chosen corresponding to the condition a0≤rc<2π/kF, where a0 is the Bohr radius.Table 2
Input and output parameters for fcc and bcc iron (1 eV = 13.605 Rydberg and 1 atomic unit (AU) = 5.29177 × 10−2 nm).
fccγ-Febccα-FeValencyz22Lattice constanta  AU6.7275.424Volume AU376.11879.765kF AU−10.9200.905EF Ryd.0.8460.820Melting temperatureTm(K)15361811Cohesive energyEcoh eV4.284.28Activation energy for self-diffusionQ0 eV2.9432.658Vashishta-Singwi parametersA0.9296330.938470B0.3091780.312208EF1v(theo) in eV11.284–1.6191.284–1.509EF1v(expt) in eV21.401.60Range ofrc fitted to EF1v(expt) in AU1.0028–1.12821.1017–1.2234Meanrc in AU1.04921.15011Using (6) [21]; 2[22].Figure 1
EF1v-rcplot for fcc γ-iron (γ-Fe).Figure 2
EF1v-rcplot for bcc α-iron (α-Fe).It is observed that the experimental value ofEF1v, obtained from positron annihilation technique, lies within the range of the theoretical value of it obtained from the empirical relation [23–25]:(6)Tm(K)=1200EF1v=660Q0=360Ecoh.HereTm 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 rc is within the first peak value of 2 AU and we note from Table 2 that rc values lie close to Bohr radius (a0=1AU). ECF’s of Kleinman [15], Harrison [16], Vashishta and Singwi [17], and Taylor [18] give reasonably close values of rc while others give the range over of rc.In conclusion, it should be noted that the inherent simplicity of AECMP makes it difficult to have a universalrc 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.



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*Source: 431742-2012-03-26.xml*