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.

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 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 asEF1v=∑q0′q03∂U(q0)∂q0+Ω2π2∫0∝U(q)q2dq,
where
U(q)=Ltη→∝2πe2z2Ωq2e-q2/4η+[w(q)]2ε(q)χ(q),w(q)=-4πze2cosqrcΩq2,ε(q)=1-8πe2Ωq2[1-f(q)]χ(q),χ(q)=-3z8EF[1+4kF2-q24kFqln|2kF+q||2kF-q|].

Here q0 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.

Different forms of ECF f(q) and corresponding fitted value of parameter rc of AECMP in atomic unit (AU).

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.

Input and output parameters for fcc and bcc iron (1 eV = 13.605 Rydberg and 1 atomic unit (AU) = 5.29177 × 10^{−2} nm).

fcc γ-Fe

bcc α-Fe

Valency z

2

2

Lattice constant a AU

6.727

5.424

Volume AU^{3}

76.118

79.765

kF AU^{−1}

0.920

0.905

EF Ryd.

0.846

0.820

Melting temperature Tm(K)

1536

1811

Cohesive energy Ecoh eV

4.28

4.28

Activation energy for self-diffusion Q0 eV

2.943

2.658

Vashishta-Singwi parameters

A

0.929633

0.938470

B

0.309178

0.312208

EF1v(theo) in eV^{1}

1.284–1.619

1.284–1.509

EF1v(expt) in eV^{2}

1.40

1.60

Range of rc fitted to EF1v(expt) in AU

1.0028–1.1282

1.1017–1.2234

Mean rc in AU

1.0492

1.1501

^{1}Using (6) [21]; ^{2}[22].

EF1v-rcplot for fcc γ-iron (γ-Fe).

EF1v-rcplot 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]:Tm(K)=1200EF1v=660Q0=360Ecoh.

Here Tm 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 universal rc 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.

RottlerJ.SrolovitzD. J.CarR.Point defect dynamics in bcc metalsGordonS. M. J.KennyS. D.SmithR.Diffusion dynamics of defects in Fe and Fe-P systemsDomainC.BecquartC. S.Diffusion of phosphorus in α-Fe: An ab initio studyFuC. C.WillaimeF.Ab initio study of helium in α-Fe: dissolution, migration, and clustering with vacanciesOlssonP.DomainC.WalleniusJ.Ab initio study of Cr interactions with point defects in bcc FeBosC.SietsmaJ.ThijsseB. J.Molecular dynamics simulation of interface dynamics during the fcc-bcc transformation of a martensitic natureKorhonenT.PuskaM. J.NieminenR. M.Vacancy-formation energies for fcc and bcc transition metalsOsetskyYu. N.VictoriaM.SerraA.GolubovS. I.PriegoV.Computer simulation of vacancy and interstitial clusters in bcc and fcc metals251Proceedings of the International Workshop on Defect Production, Accumulation and Materials Performance in an Irradiation Environment1997Journal of Nuclear Materials344810.1016/S0022-3115(97)00255-9SöderlindP.YangL. H.MoriartyJ. A.WillsJ. M.First-principles formation energies of monovacancies in bcc transition metalsAshcroftN. W.Electron-ion pseudopotentials in metalsKingW. F.IIICutlerP. H.Lattice dynamics of beryllium from a first-principles nonlocal pseudopotential approachKingW. F.IIICutlerP. H.Lattice dynamics of magnesium from a first-principles nonlocal pseudopotential approachShamL. J.A Calculation of the phonon frequencies in sodiumGeldertD. J. W.VoskoS. H.The screening function of an interacting electron gasKleinmanL.Exchange and the dielectric screening functionHarrisonW. A.VashishtaP.SingwiK. S.Electron correlations at metallic densities. vTaylorR.A simple, useful analytical form of the static electron gas dielectric functionHubbardH.The description of collective motions in terms of many-body perturbation Theory. II. The correlation energy of a free-electron gasMahantiS. D.DasT. P.Theory of knight shifts and relaxation times in alkali metals-role of exchange core polarization and exchange-enhancement effectsKittelC.MatterH.WinterJ.TriftshäuserW.Phase transformations and vacancy formation energies of transition metals by positron annihilationGhoraiA.Dependence of mono-vacancy formation energy on the parameter of Ashcroft's potentialGhoraiA.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 stateGhoraiA.Exploration of parameters of Ashcroft's potential using monovacancy formation energy and different exchange and correlation functions for some trivalent FCC metals