Development of BOP unsrc and BOP scr

As in the nonmagnetic BOPunscr, the unscreened bond integrals βτ have the GSP functional form described by eq. (3.2) [69], supplemented by the fifth-order polynomial as the cut-off function (see Section 3.1 for details). We use the same parameters as in Fe BOP developed by Mrovec et al. [35] with minor modification of the cut-off function to ensure that no unphysical bumps occur. This is achieved by increasing BI

cut

R from the original value of 3.7 Å used by Mrovec et al. to 4.0 Å with all the other parameters remaining the same. Table 5-1 summarizes the parameters used in the unscreened bond integrals, and Figure 5-1 shows the R-dependence of the bond integrals.

96

Table 5-1. The parameters of unscreened bond integrals (eq. (3.2)) used in BOPunscr for

bcc Fe. The parameters are taken from [35], with only BI cut

R changed to a larger value in order to avoid any unphysical bumps in the tail function. The unit of βτ

( )

R0 is eV; R0,

c

R , R1BI and RcutBI are in Å; na and nc are dimensionless.

( )

R0 τ β R₀ R_{c R1}BI BI cut R na nc ddσ – 0.6450 2.4930 4.2336 3.0 4.0 3.5805 6.7179 ddπ 0.3380 2.4930 4.2336 3.0 4.0 5.0703 5.3437 ddδ – 0.0450 2.4930 4.2336 3.0 4.0 8.5802 0.0272

97

As in the nonmagnetic BOPs, screened bond integrals are analytically represented by the functional form ij

( )(

1 ij

)

ij

R S

τ τ τ

β =β − (eq. (3.3)), where βτ

( )

Rij takes the

simplified GSP function eq. (3.4) and _Sij

τ is defined in eq. (3.5). Again βτ

( )

Rij is cutoff

by a fifth-order polynomial (see Section 3.1 for details). The parameters in βτ

( )

R_ij and

ij

Sτ are used to fit the OTB-sd data obtained from the method of Urban et al. [32] for extracting bond integrals from DFT calculations using the projection formalism. The calculation of the OTB-sd bond integrals was done by our collaborator Dr. Mrovec. The parameters of screened bond integrals used in Fe BOPscr are summarized in Table 5-2.

The R-dependence of the OTB-sd bond integrals, deduced from DFT calculations, the screened bond integrals ij

τ

β and the simplified GSP function βτ

( )

Rij are shown in Figure

5-2. The general features of the OTB-sd bond integrals are similar to those found in group 5 and 6 non-magnetic transition metals. In OTB-sd data, the discontinuity of bond integrals in the region between the first and second nearest neighbor of the bcc lattice is again apparent in ddπ and ddδ bond integrals and less pronounced in ddσ bond integral. This is well reproduced by the screened bond integrals represented by

ij

( )(

_{1 ij}

)

ij

R S

τ τ τ

β =β − . Although the reproduction of the discontinuity in ddδ bond

integrals is less satisfactory than for ddπ integrals, these bond integrals are approaching zero at the second nearest neighbors and, presumably, play only a minor role in the overall performance of the resulting BOPsrc.

98

Table 5-2. Parameters entering equations (3.3), (3.4) and (3.5) for screened bond integrals used in BOPscr for bcc Fe. Units of βτ

( )

R₀ and O Rτ

( )

0 are eV; R0, Rc, R1BI and RcutBI in

Å.

( )

R0 τ β O Rτ

( )

₀ R0 Rc R1BI RcutBI sdσ 0.845 –0.045 2.4825 0.71 2.6 4.0 ddσ – 0.620 0.040 2.4825 0.71 2.6 4.0 ddπ 0.410 – 0.030 2.4825 0.47 2.6 4.0 ddδ – 0.062 0.020 2.4825 0.31 2.6 4.0

Figure 5-2. Screened bond integrals used in BOPscr for bcc Fe. Dots: OTB-sd data. Solid

line: analytic screened bond integrals. Dashed line: simplified GSP function (eq. (3.4)). The red and blue color correspond to R from the regions of the first and second nearest neighbors, respectively.

99

The repulsive part of the cohesive energy is again composed of two contributions, a pairwise interaction (_Epair_{) and an environment dependent contribution (E}env_{). The}

functional forms of _Epair _{and E}env _{are the same as in the BOPs for non-magnetic metals}

(see Chapter 3). The parameters in _Epair _{and E}env _{are adjusted so that the resulting BOPs}

reproduce the fundamental physical properties of the equilibrium ferromagnetic bcc Fe, the cohesive energy, three elastic constants (C11, C12 and C44) and the equilibrium lattice

constant. The experimental values of these properties to which the BOPs are fitted are summarized in Table 5-3. Moreover, in order to assure the BOP behaves reasonably when atoms are significantly closer to each other than the first nearest neighbor spacing in the equilibrium bcc lattice, the parameter R0 in Epair is chosen to be slightly smaller than the first nearest neighbor spacing. The parameter B₀ in _Epair _{is then adjusted to}

reproduce the energy dependence on the lattice parameter when the bcc lattice is hydrostatically compressed, calculated using a DFT method (VASP) with spin polarization included. Figure 5-3 shows the comparison for this energy dependence calculated using DFT and BOPs, respectively. The results obtained using BOPunscr and

BOPscr are both in a very good agreement with the DFT calculation. The parameters used

in _Eenv _{and E}pair _{for both BOP}

unscr and BOPscr are summarized in Tables 5-4 and 5-5.

The R-dependence of the pair potential Φ in _Epair _{is shown in Figure 5-4. As in the BOPs}

for non-magnetic metals, a slight attractive part exists in the vicinity of the separation of the second nearest neighbors in both BOPunscr and BOPscr, which presumably arises from

100

Table 5-3. Experimental values [81, 112] to which the BOPs for Fe are fitted. Cohesive energy (_Ecoh_{) is in eV, elastic moduli in 10}11 _{Pa, and lattice parameter (a) in Å.}

Ecoh _C

11 C12 C44 a

4.28 2.431 1.381 1.219 2.8665

Figure 5-3. Dependence of the energy difference between the hydrostatically compressed bcc lattice of Fe and its equilibrium bcc lattice, ∆E, on acom/a, where acom is the lattice

parameter of the compressed bcc lattice. Lines: BOPs. Dots: DFT.

Table 5-4. Parameters used in the environment dependent repulsion (Eenv_{), given by}

equations (3.7), (3.8) and (3.9) in Chapter 3, for BOPunscr and BOPscr for bcc Fe. Rij is in

units of the lattice parameter a. Rs = 0.52 (in units of a). µ, g and ν are dimensionless.

Fe A [eV] µ g ν _R1rep _[Å] rep

cut

R [Å]

BOPunscr 5.498×101 7.0 20.0 6.0 3.0 4.0

101

Table 5-5. Parameters used in the pair potential Φ, given by eq. (3.10) in Chapter 3, for BOPunscr and BOPscr for bcc Fe. Rk is in Å and Bk in eV Å-3, when Rij is in Å. In all cases,

R0 is chosen to be 0.9 of the first nearest neighbor spacing of the equilibrium bcc lattice.

Fe BOPunscr BOPscr

k R_k Bk Rk Bk 0 2.23 35.0 2.23 30.0 1 2.75 − 0.486 028 26 2.75 − 0.972 681 58 2 2.90 2.428 666 99 2.90 2.516 912 64 3 3.50 1.198 115 96 3.50 0.693 543 45 4 3.70 − 0.780 182 45 3.80 − 0.412 698 08

Figure 5-4. The R-dependence of the pair potential Φ used in BOPunscr and BOPscr for Fe.

The first, second and third neighbor spacings in the equilibrium bcc lattice are marked as vertical lines denoted as 1st_{, 2}nd _{and 3}rd_{, respectively.}

102

In document Development of Bond-Order Potentials for Body-Centered-Cubic Transition Metals and Their Application in atomistic Studies of Plastic Properties Mediated by Dislocations (Page 122-129)