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Oa n AECL-8392 ATOMIC ENERGY ÄT£™Ä L'ENERGIE ATOMIQUE OF CANADA LIMITED YfiSjr DU CANADA, LIMITEE PODSIP: ACOMPUTER PROGRAM FOR THE EVALUATION OF POINT-DEFECT PROPERTIES IN METALS PODSIP: UN PROGRAMME D'ORDINATEUR POUR L'EVALUATION DES PROPRIETES DES DEFAUTS PONCTUELS DANS LES METAUX C. H. Woo, M. P. Puis Whiteshell Nuclear Research Etablissement de recherches Establishment nucléaires de Whiteshell Pinawa, Manitoba ROE HO November 1985 novembre

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Page 1: ATOMIC ENERGY ÄT£™Ä L'ENERGIE ATOMIQUE OF CANADA … · oa n aecl-8392 atomic energy Ät£™Ä l'energie atomique of canada limited yfisjr du canada, limitee podsip: a computer

Oa n

AECL-8392

ATOMIC ENERGY Ä T £ ™ Ä L'ENERGIE ATOMIQUEOF CANADA LIMITED Y f i S j r DU CANADA, LIMITEE

PODSIP: A COMPUTER PROGRAM FOR THE EVALUATION

OF POINT-DEFECT PROPERTIES IN METALS

PODSIP: UN PROGRAMME D'ORDINATEUR POUR L'EVALUATION

DES PROPRIETES DES DEFAUTS PONCTUELS DANS LES METAUX

C. H. Woo, M. P. Puis

Whiteshell Nuclear Research Etablissement de recherchesEstablishment nucléaires de Whiteshell

Pinawa, Manitoba ROE HONovember 1985 novembre

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Copyright © Atomic Energy of Canada Limited, 1985

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ATOMIC ENERGY OF CANADA LIMITED

PODSIP: A COMPUTER PROGRAM FOR THE EVALUATION

OF POINT-DEFECT PROPERTIES IN METALS

by

C.H. Woo and M.P. Puls

Whlteshell Nuclear Research EstablishmentPinawa, Manitoba ROE 1LO

1985 NovemberAECL-8392

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PODSIP: UN PROGRAMME D'ORDINATEUR POUR L'ÉVALUATION

DES PROPRIÉTÉS DES DÉFAUTS PONCTUELS DANS LES MÉTAUX

par

C.H. Woo et M.P. Puis

RÉSUMÉ

On décrit un programme d'ordinateur, PODSIP, pour effectuer descalculs de défauts ponctuels à l'aide de techniques de simulation informa-tiques. On peut se servir du programme pour évaluer les propriétés desdéfauts dans un matériau cristallin quelconque de forme hexagonale oucubique; on peut les décrire au moyen d'un potentiel de paires à distancecourte. On présente la base physique qu'utilise PODSIP pour évaluer lesénergies et configurations de défauts, tenseurs dipolaires élastiques etmoments dipolaires diaélastiques. On donne un bref compte rendu de lastructure logique globale de PODSIP et de certaines caractéristiques deprogrammation importantes. Une caractéristique importante est la capacitéde PODSIP d'évaluer les propriétés des défauts pour leurs configurationsaux points d'équilibre et de col par un seul passage machine de programme.Pour tester le programme, on donne une définition des propriétés de défautsde la simple lacune et de l'interstitiel en haltère <100> dans le cuivre àl'aide d'un potentiel de paires empirique. Dans cette étude, on seconcentre surtout sur le moment dipolaire diaélastique du défaut au pointde col.

L'Énergie Atomique du Canada, LimitéeÉtablissement de recherches nucléaires de Whiteshell

Pinawa Manitoba ROE 1L01985 novembre

AECL-8392

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PODSIP: A COMPUTER PROGRAM FOR THE EVALUATION

OF POINT-DEFECT PROPERTIES IN METALS

by

C.H. Woo and M.P. Puls

ABSTRACT

A computer program, PODSIP, designed to carry out point-defectcalculations using computer simulation techniques is described. The pro-gram can be used to evaluate defect properties in any hexagonal or cubiccrystalline material that can be described by means of a sensibly short-ranged pair potential. The physical basis used by PODSIP for evaluatingdefect energies and configurations, elastic dipole tensors, and diaelasticpolarizabilif.ies is presented. A brief account is given of the overalllogical structure of PODSIP and some important programming features. Oneimportant feature is PODSIP's ability to evaluate defect properties forboth their equilibrium and saddle-point configurations by means of only oneprogram run. To test the program, a determination of the defect propertiesof the single vacancy and the <100> dumb-bell interstitial in copper, usingan empirical pair potential, is given. Emphasis in this study is on thediaelastic polarizability of the defect in its saddle-point position.

Atomic Energy of Canada LimitedWhiteshell Nuclear Research Establishment

Pinawa, Manitoba ROE 1L01985 November

AECL-8392

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CONTENTS

Page

1. INTRODUCTION 1

2. PHYSICAL MODEL 2

2.1 DEFECT ENERGY 22.2 DIPOLE TENSOR AND DIAELASTIC POLARIZABILITY 5

3. THE COMPUTER MODEL 13

3.1 OVERVIEW 133.2 CRYSTAL LATTICE AND LATTICE INDEXING GENERATION 143.3 ENERGY AND FORCE CALCULATIONS 143.4 INITIAL DEFECT CONFIGURATION 16

3.5 DETERMINATION OF THE FINAL DEFECT STRUCTURE 17

4. RESULTS AND DISCUSSION 19

5. CONCLUSIONS 27

REFERENCES 29

APPENDIX A EXPANSION OF THE ENERG. IN TERMS OF LAGRANGIAN STRAINS A.I

APPENDIX B EXPRESSIONS FOR STRAIN DERIVATIVES OF THE DEFECT ENERGYIN TERMS OF CUBIC STRAIN EIGENTENSORS AND EIGENSTATES B.I

APPENDIX C LOGICAL STRUCTURE AND INPUT FORMAT FOR PODSIP C.I

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1. INTRODUCTION

Numerous investigations during the past two decades have estab-

lished that the macroscopic effects of irradiation damage, such as void

swelling, irradiation creep and growth, are directly related to the continu-

ous production during irradiation of intrinsic point, defects, and thair

movement and subsequent annihilation at crystal defects. Reliable engineer-

ing design equations, on which the successful operation of nuclear reactors

is based, require fundamentally based mechanistic models to quantify such

processes. In this regard, values of certain point-defect properties, such

as their energies of formation and migration, their binding energies with

themselves and impurities, their elastic dipole tensors, and their diaelas-

tic polarizabilities, have to be provided. During the past twenty years,

much effort, both experimental and theoretical, has been expended in an

attempt to provide this information. For zirconium, due to extreme

experimental difficulties, these point-defect properties are largely

unknown. Theoretical calculations represent one of the possible routes to

obtaining estimates of these parameters.

A conventional approach is to calculate these quantities using an

atomistic model based on a largely empirical pair-wise interatomic potent-

ial. However, this method is expected to break down at the defect core,

where the electronic distribution is perturbed and, hence, the pair-wise

potential is likely to be different from that in a perfect lattice environ-

ment. To account properly for this inner region of the defect, we can adopt

a more rigorous approach, as suggested by Melius et al. [1]. In this

so-called "hybrid" approach, the cluster of atoms at the defect core is

treated quantum mechanically and the long-range elastic distortions of the

lattice a-e treated using a conventional pair-wise potential. The defect

configuration can then be determined by minimizing, using an iterative

procedure, the total energy of the system, including both the electronic

energy of the cluster and the relaxation energy. Thus, as required, the

pair-wise potential is used only far away from the defect, where the

pertubation to the electronic distribution due to the defect may be

considered to be screened out by the mobile conduction electrons.

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Two quantum mechanical methods, namely, the multi-scattering-Xa

method [2] and the tight-binding Linear Combination of Atomic Orbitals

(LCAO) method [3], have been investigated as to their applicability in this

scheme- In conjunction with the quantum mechanical cluster calculations, we

have also developed a computer program that carries out the calculation of

the long-range elastic distortion by using a pair-wise potential. In a

previous report [4], a preliminary version, MEDE, of such a program was

described. In the present report, we describe a general program, PODSIP,

evolved from MEPE, which is developed for calculations using the pair-wise

potential. An important and novel feature of PODSIP is its ;bility to carry

out efficient evaluations of the elastic dipole tensor (refeired to in the

following simply as the dipole tensor) and the diaelastic polarizability of

the point defect at either its equilibrium or saddle-point configuration.

The two quantities are important in evaluating the interaction of a point

defect with other crystal defects, for example, a dislocation, and are,

therefore, important in the study of irradiation damage effects.

This report is meant to be supplemented by the earlier report [4].

Emphasis in this report will be on the method employed to implement the

calculation of the dipole tensor and the diaelastic polarizability, the

program PODSIP, and some typical results obtained.

The plan of the report is as follows. In section 2, we discuss

the physical basis of the point-defect model and the equations used to

determine the various defect properties. In section 3, a detailed descrip-

tion of the computer program developed to carry out these calculations is

given. Section 4 contains the results for vacancy and interstitial defects

in copper and section 5, our conclusions.

PHYSICAL MODEL

2.1 DEFECT ENERGY

We assume that the metal consists of an assembly of atoms that

are arranged in a definite crystal structure and whose interaction can be

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described by means of a pair-wise central potential. The validity of this

latter assumption has been the subject of numerous articles [5,6], and we

will not concern ourselves with it in this report.

The total energy E of an assembly of atoms within the pair-wise

potential approximation is*

E = 7 Z *(|ra-r0!) (1)

where the summation extends over all atoms of the crystal, |_r —_r | is the

distance between atoms a and ß located at _r and _r , respectively, and

$( jjr -r_ |) is the corresponding pair-wise potential. The defect energy E

is the difference in the total energy between an assembly of atoms in

mechanical equilibrium, containing the defect, and the corresponding perfect

lattice. Denoting by _r the equilibrium coordinates in the defected lattice

and by R_ the equilibrium coordinates in the perfect lattice, we have that,

following Equation (1),

ED = I Z *< \La~lB0 - i E •( I R V D • (2)a, B a,6

For the defected lattice, summation is over all lattice and interstitial

sites, and vacant lattice sites are excluded. The defect energy so defined

represents, in the case of a vacancy, the energy of the vacancy with the

removed atom taken to infinity and, in the case of an interstitial, the

energy of the interstitial with the inserted atom brought in from infinity.

The formation energy E of an interstitial, or a vacancy, is then defined by

* In this and the following sections, we employ the following notation-Single bars underneath symbols denote vector quantities, and double bars,tensor quantities. Greek superscripted indices range from 1 to 3.Repeated indices in an expression imply summation.

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EF » ED ± EC

where E is the cohesive energy/atom of the crystal (a negative quantity),

and the plus sign applies to the vacancy, and the negative sign, to the

interstitial.

In setting up a scheme for calculating the energy E , we make use

of the fact that the pair-wise potential 4 has a finite range and that, far

from the defect site, the deviations jj_ = £ - It of lattice sites in the

defected crystal, from perfect lattice sites It, can either be expressed by a

harmonic model or be set to zero. A finite crystallite can, therefore, be

set up consisting of two regions: the inner, region I, containing the

defect and a boundary, region II, of sufficient thickness to cover the range

of the pair-wise potential. In calculations of defects in metals, the size

of region I can usually practicably be chosen large enough so that the

displacements in region II are sufficiently small that they can be set to

zero. This means that atom sites in region II are always given by perfect

lattice coordinates It. With this division of regions, the expression for E

given by Equation (2) can be rewritten as

a, Sei

Z *(|Ra-R8|) (4)ael aelBell 0eII

where now all summations are finite and region II consists of a mantle of

perfect lattice sites.

The equilibrium defect coordinates, r_, are determined by the

condition that the total energy is a minimum at equilibrium, i.e.,

dE /dr = 0. This implies an explicit differentiation of the boundary-region

displacements _£ with respect to jr. Writing

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(5)

we see that this implies (3E /3r_) g = 0, provided that (3E /3_Ç) = 0, i.e.,

region II is in equilibrium. This means that atoms must be displaced in

region I until the forces on them are zero, while the region-II atoms are

maintained in their perfect lattice positions. Practically, this is

achieved numerically in PODSIP by using the conjugate gradient method [7].

To study saddle-point, as well as equilibrium, configurations, PODSIP uses a

modified version of the conjugate gradient method, developed by Sinclair and

Fletcher [8]. Given the two adjacent equilibrium configurations, the

Sinclair-Fletcher method makes it possible to find the defect's saddle point

in one optimization step, without the tedium of doing a series of con-

strained minimizations near the top of the energy barrier. This is

described in more detail in section 3.

2.2 DIPOLE TENSOR AND DIAELASTIC POLARIZABILITY

Two important properties of the defect, characteristic of its

long-ranged displacement field, are the defect's dipole tensor P and its

diaelastic polarizability g. These quantities characterize the defect

macroscopically [9] and can be used to evaluate long-range elastic

interactions between the point defect and other crystal defects [9].

A number of formulae have been proposed to evaluate P_. A recent

analysis by Gillan [10,11] shows the limitations of these relations when

approximate methods are used to evaluate them and also some relationships

between them.

A commonly employed expression is the Kanzaki-Hardy formula, which

is written [12]

S ' Z **'** <6>ael

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where the K are fictitious (Kanzaki) forces, originally introduced by

Kanzaki [13]. These forces are chosen to produce the same displacements _£_

in the perfect harmonic crystal as result in the defected real crystal.

Schober and Ingle [12] have shown, however, that this formula is inaccurate

for defects with strong distortion fields, such as interstitials- Provided

region I is sufficiently large that it contains all ehe anharmonic displace-

ments and the host-defect interactions, they propose that a more accurate

relation for P is given by

P = T Ra-Fa (7)~ f T — —ext

where F are the external forces generated in region II as a result of the

relaxation of the defect in region I. Equation (7) can be rearranged into a

more convenient form, which avoids the necessity of specifying neighbours

for region-II atoms, by noting that certain terms cancel and others are zero

because the crystal is in equilibrium in region I [4]. This yields

a'AFa (8)cell

where

AFa = fa(R) " Fa(R) = - -\ [Y, *<|r6-Ra|)-*(|Rß-Ra|)] .~ ~ 3R pel ~ ~ ~ ~

Physically, _f and F_ are partial forces due to interactions between atoms

in region-; II and I. 4F_ is the difference in these forces between the

defected tnd the perfect region I.

The above formulae for the dipole tensor are based on mechanical

models of the moments of the forces set up in the crystal due to the defect.

An alternate approach is in terms of an exact relation between the strain

derivative of the defect formation energy calculated at constant strain.

This give:. [10]

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äAo - - s°

where E is the defect energy given by Equation (4) (i.e., determined using

a rigid boundary model) and e is the infinitesimal strain tensor. However,

the infilitesimal strains do not vanish for arbitrary rotations and are not

the most general strains to use in an expansion of the energy as a function

of strain. The appropriate strains to use are the Lagrangian strains .n

defined by

Here and in the rest of the paper, a dot product g.b between two second-rank

tensors a and b results in a second-rank tensor a.,b, .; a double-dot product

g:b results

being used.

g:b results in a sealer a..b.., and the notation of implied summation is

Expansion of E in terms of j yields

3ED 1 ^ D *ED(n) = ED(0) f- - ^ p j + |-IJ =• «n + ... (11)

33

where we denote the sealer product a. .A. .klb, . of the second-rank tensor

g.b with the fourth-rank tensor 4 ^y §«A.b. As shown in Appendix A, to

second order in _e, this gives (using Equation (9))

, o . i _ e .</* 1 . .<fi= E D(0) - Pfe + i P

fcs |-| - f c-a-e (12)

where we define the diaelastic polarizability

-I ' ( 1 3 )

3e8e/e=O

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Equations (9) and (12) suggest two independent ways of evaluating

the di.pole tensor from a lattice model. In the first approach, E_ is

evaluated at different values of the applied strain. Plotting the

difference, E (e)-E (0), as a function of strain and taking the strain

derivative at e=0 yields

)3e /E-0

- P° . (14)

In the second approach, we develop an expression for P from Equation (9)

that can be evaluated directly. The derivation is as follows. The strain

derivative can be rewritten

- 3R« -

Explicit evaluation of this, using Equation (4) for E yields

aell

ael 3R/ßell } aell 3R J0cl

ael 3£ (ßel ) ael 3£ (ßell )

Combining the first and third, the fifth and sixth, and the second and

fourth terms, respectively, we obtain

= ael ael aell

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where

3R (ßelorll

3r (Pel Pell

and AF_ is defined in Equation (8). Since, at equilibrium, F_ (R) = 0 and

_f_ (_r) = 0, we have that

*'** . (18)T T

aell

Combining Equations (18) and (9), we arrive at an identical relation to

Equation (8), which is the Schober and Ingle [12] formula for evaluating P.

This demonstrates the equivalency of the two ways of calculating P.

Equations (12) and (13) can also be used to evaluate the diaelas-

tic polarizability £ by taking the second strain derivative of E (j). This

requires a determination of E as a function of strain for at least two

non-zero values of the strain. Note that it is not possible to use Equation

(13) with Equation (4) to develop an analytic expression for a, as was done

for the dipole tensor, because near the defect the defected lattice cannot

be strained uniformly, and certain second derivatives, which depend on this

condition, do not vanish, as did the first derivatives in the derivation for

the dipole tensor.

In cubic crystals, it is convenient to write a. ., .. in terras of its

eigenvalues and eigenstates (second-rank eigentensors), in the same way that

the elastic constants can be reduced to their eigenvalues and the corre-

sponding strain eigenstates. The strain derivative of E in Equation (12)

is evaluated in Appendix B. It can be seen that a.... has six eigenvalues,

a to or , where a ' corresponds to the isotropic dilatation mode, a

to o to the two independent (HO)-shear modes, and a ^ to cr to the

three independent (lOO)-shear modes.

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Expressed in terms of the a , we have, from Equation (12),

3e le =0P / P

where now the second derivative of the defect energy E has been reduced to

a simple scalar.

It is worth noting at this point that there is, in fact, another

way of using the strain derivative method to solve for a. This makes use of

the relation

which comes from combining Equations (9) and (13). Expanding P in terms of

the Lagrangian strains yields

3P

P(n) = P(0) + B' J^

In order to deal with scalar quantities throughout, as in the strain

derivative of the energy, we multiply by n and rearrange terms. This gives

fP(n) - P(O)]:n = n'-g 'n . (32)sa

Writing this in terms of e (to second order) yields

[P(|)-£(0)]:| = - [£(|)-P(0)]:^2=-+ e'o-e . (33)

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In terms of the cubic eigenstrains e , this becomes

A?' - 1Dividing by e and taking the derivative with respect to e , we obtain

<»>

or

ij

Thus, in the case of a program such as PODSIP, which calculates both E n and

P, Equation (35) provides an alternate way of evaluating the diaelastic

(p)polarizabilities, o . Given that AP can be evaluated as accurately as

AE , Equation (35) might be expected to give a more accurate value for o

since only the first derivative is required. However, in the results

section we show that, for a given region size, the energy-derivative method

is more accurate. Gillan [10,11] has provided an analysis giving the reason

for this.

Finally, it is of interest to provide expressions that relate the

diaelastic polarizability to the change in elastic constants due to the

defect. Leibfried and Breuer [9] have shown that this is given by (for

convenience we continue to write everything in terms of cubic eigenvalues)

NA C(P) = _ 4 ^ (36)1 defect V

c

where N = total number of atoms in crystal

V = atomic volume (volume of the Wigner-Seitz cell)

and the subscript "1" refers to the case of one defect in the crystal.

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Physically, the changes in elastic constant given above result

from the local distortions induced by the defect and the resulting change in

the defect-lattice atoms interactions. In addition to these changes, in a

real finite crystal, the long-ranged distortion field of the point defect

gives rise to two other changes, A.C^ and 4 C . A C^ is due to the defect'sCO

volume change in an infinite crystal, <$.. v , giving rise to a change in the

lattice parameter [9]; A,C is due to the additional relaxations, 6.v ,

induced by the free surface of the crystal changing both the coupling

constants and the lattice parameter throughout the crystal. These two

contributions can be approximated by [9]

Col C(f>) 6 1 V " V 1 ( C11 + 2 C12 ) 3C(P)

where 3C /3p is the pressure derivative of the elastic constants. The

total change in the elastic constants is, therefore,

NAC1C . = NA.C V 4 h < C ^ ^ + 2 C ^ ' > > ^1 total 1 defect 3 V V 11 12 ap

c c

Previously, Dederichs et al. [14] had assumed that the quantity that is

determined from the atomistic model was not NA C , as given by Equationl detect(p)

(36) above, but a quantity NA ci , where f.s. refers to "fixed surface",j. r . s.

i.e., the rigid boundary relaxation method. This then led them to derive

another expression for the total elastic constant change, viz.,

where

6 v = 6. v + 6 v

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However, the present analysis shows that NA C^P' = NA r P I and that,

therefore, the correct expression for the total elastic constant change is

as given by Equation (38).

3. THE COMPUTER MODEL

3.1 OVERVIEW

The computer code developed to carry out the calculations

described in this report has been given the name PODSIP (POlnt Defect

^Simulation Pjrogram). It is written in FORTRAN and runs on the Chalk River

Nuclear Laboratories CDC computer. PODSIP represents a considerable

improvement over an earlier version, called MEDE [4]. PODSIP is also a

close relation to the programs DIPOS and PDINT, which have been developed at

the Whiteshell Nuclear Research Establishment to carry out defect calci la-

tions in ionic crystals [15,16]. In particular, it makes use of similar

lattice handling and minimization routines.

The plan of the program can be logically divided into four main

tasks:

(1) generation of the crystal lattice and lattice indexing scheme;

(2) calculation of the energy and forces of atoms in region I;

(3) introduction of the defect configuration in region I;

(4) determination of the equilibrium defect configuration and its

properties.

Appendix C gives a brief overview of the input format and logical structure

of PODSIP and a description of the Main calling program. In the following,

we discuss the four tasks carried out by PODSIP, as outlined above, in more

detail.

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3.2 CRYSTAL LATTICE AND LATTICE INDEXING GENERATION

The routines in this part of the program consist of the SSP

package obtained from D,J. Bacon of Liverpool University, U.K. The program

bases for these routines were the lattice handling routines in the program

DEVIL, developed by M.J. Norgett at Harwell, U.K. These routines were also

used in DIPOS and PDINT. A detailed description of the logical structure of

these routines has already been given by Puls [17] and, therefore, need not

be repeated here. Aside from trivial changes in renaming the various

variables and subroutines, the SSP package performs essentially the same

functions. There are, however, two significant differences. One is that

the generation of a particular lattice structure has been greatly simpli-

fied. Â large number of different structures are automatically generated,

as shown in Appendix C, with only two data entries specifying (a) the

general structure: cubic or hexagonal, and (b) the specific lattice

structure desired. The disadvantage of this method is that incorporation of

a lattice type currently not covered by this scheme requires modification of

the code. The second, and important, difference with the lattice handling

routines in PDINT and DIPOS is the distinction between the atom index

designating its type (LTYPE) and the index designating the particular

sublattice the atom is on (LSUBL). In the earlier versions, both indices

were combined into one (LCO), which made it impossible to use the neighbour

list generation scheme in monatoraic crystals requiring two sublattices for

their generation, such as h.c.p. (hexagonal-close-packed) lattices. In SSP

(as in PDINT and DIPOS) the neighbour look-up index ranges over the various

sublattices (bases) needed to generate the crystal. However, the atom types

(LTYPE) on different sublattices can be identical, or different, as deter-

mined by the crystal lattice type chosen. It should be noted that, although

this added feature of the SSP code is significant, it required only a few

small modifications to the original code.

3.3 ENERGY AND FORCE CALCULATIONS

The total energy of all atoms in region I and their individual

forces are calculated in subroutine FUNC. In this subroutine the appropri-

ate neighbour of each atom is found, and the energy and forces are

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calculated by calling SFUNC which, in curn, calls FORCE. SFUNC evaluates

distances between atoms and establishes their types. Forces, or gradients

and energies of interaction, are evaluated in FORCE. There is an option for

choosing either forces (IFOG=O) or gradients (IFOG=1 in order to make use

of either of two conjugate gradient minimization routines described further

on. To improve computing efficiency, the total energy is summed according

to the scheme

E (region I) = /_. ^Li'I^ •i>j 3

region I

With this scheme, the energy/atom of an atom in region I cannot be obtained

by dividing by the total number of region-I atoms. Force evaluations on a

pair of atoms are attributed to each atom in the pair on an equal and

opposite basis, except that no forces are stored for region-II atoms.

Currently, subroutine FORCE contains the option for the following

three types of pair-wise potentials:

(1) Morse:

<Kr) = D{exp[-2a(r-ro)) - 2 exp[-o(r-rQ)]

where D, a and r are constants.

(2) Lennard-Jones:

0(r) = e{(|)12 - 2(^)6}

where e and d are constants.

(3) Spline fit (with Born-Mayer):

•<r) = Ue" r / p -f

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where i=l,2,... labels the splines of the polynomial, a is the

order of the polynomial, and U, p, A and r. are constants.ai i

In Appendix C, we detail the input requirements for each of these

potentials. For the Morse potential, a fifth-order polynomial cut-off

function has been added, which joins smoothly with the Morse and brings the

potential to zero between two specified distances. The distances between

which this function operates must be supplied. The constants for this

spline function are evaluated in subroutine MORSE. For the Born-Mayer

potential used with the spline fit, only its range need be specified. The

constants are determined automatically.

The potentials given above are for monatomic crystals only. If a

crystal structure consisting of more than one atom type is specified, then

appropriate pair potentials need to be generated and added to FORCE, or a

new subroutine written. Information on the atom type (as given by LTYPE)

would have to be carried through to this routine.

3.4 INITIAL DEFECT CONFIGURATION

Defects are introduced into the lattice through subroutine DEFECT.

Substitutional atoms are simply created by entering their perfect lattice

coordinates (referred to the block axes), designating the original atom

basis number on that location and their new type number (zero for

vacancies). The substituted atoms assume the index of the atoms they

replace and can be treated in the same way as lattice atoms in the energy

and force calculation routines.

Interstitials need to be dealt with somewhat differently as the

look-up scheme for creating a neighbour list only applies to lattice atoms.

Moreover, adding interstitials increases the total number of atoms in the

list. Therefore, the first step is to add the interstitials to the top of

the list and to shift the listing of the lattice atoms by the number of

interstitials added. The interstitial coordinates and their atom types are

then read in. A look-up index is then set up in ENTRY INTNBR for each

interstitial. This is used in the establishment of a neighbour list for

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- 17 -

each interstitial read in, which provides for the interaction with lattice

atoms that are within a specified range. Provision is made in this routine

to allow the range for each interstitial to be increased over the range

specified for the lattice atoms. This is sometimes useful, since inter-

stitials are often displaced much greater distances than lattice atoms

during the minimization.

In subroutine FUNG, there is a separate loop that looks after the

interaction of the interstitials among themselves. This is necessary, since

the separate neighbour list for interstitials in DEFECT deals only with the

Interaction of each interstitial with its neighbouring lattice atoms.

Note that if more than one defect configuration (location) is

evaluated in one run, then INTBR must be called to set up a new

interstitial-lattice atoms neighbour list each time new interstitial

coordinates have been read in.

3.5 DETERMINATION OF THE FINAL DEFECT STRUCTURE

This section deals with two topics: the minimization procedure

and the evaluation of the defect structure (dipole tensor) after the

minimization is completed.

Two subroutines are available to minimize the energy and forces in

region I after the introduction of the defect. These are subroutines CONJUG

and COGID. CONJUG is a slightly modified version of the Harwell conjugate

gradient subroutine. It requires values for the total energy of the region-

I configuration and the gradients on each of the atoms (evaluated in FUNC).

COGID is a modified version of this technique developed by Sinclair and

Fletcher [8] to evaluate saddle points. This routine requires only the

values of the forces on each atom*. Because of the ease with which saddle-

* We have nonetheless chosen to evaluate and print out the value of thetotal energy after each iteration step (as in CONJUG), to provide bettercontrol over the validity of the final configuration.

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- 18 -

point configurations can be determined with COGID, generally only COGID is

used. However, in some cases CONJUG may be useful, as it is less likely to

explore invalid minimization paths since the routine monitors both changes

in gradients and energies, so that, if an error is made in differentiating

the potential, or if the interstitial received too large an initial dis-

placement, CONJUG would terminate quickly. However, both CONJUG an3 COGID

contain provisions for limiting the displacement of the atoms by scaling

them with the maximum displacement determined for the region-I atoms.

COGTD's main usefulness lies in its ability to carry out saddle-

point evaluations efficiently. By specifying ISSD=1, PODSIP, through COGID,

carries out a saddle-point minimization. In this mode, COGID requires, as

input, an array of initial search directions s. This array is defined by

eq eq— ^L —2

eq eqwhere x. and _x« are arrays containing the region-I coordinates of two,

immediately adjacent, equilibrium positions for the defect. In addition,

the initial coordinates, x , of the saddle-point configuration are chosen

to be

eq , eqxi + x->sp —1 —2

x = x

In evaluating js_ for two arrays containing interstitials, care must be taken

that this difference is taken between the appropriate atoms. To ensure

this, the following procedure is followed, as illustrated for the dumb-bell

interstitial in copper.

The first equilibrium defect configuration is introduced by

removing two adjacent lattice atoms and reinserting three atoms designated

as "interstitials". One interstitial is put back onto one of the regular

lattice sites where a vacancy had been placed. The other two are placed to

straddle the other original regular lattice site to form the dumb-bell

interstitial. This configuration is then minimized. After minimization,

the coordinates are temporarily stored on disk.

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- 19 -

The second (adjacent) defect configuration is obtained by follow-

ing the same procedure as above, except that the positions of the dumb-bell

and the replacement interstitials are reversed. After minimization, the

previous coordinates are recalled from disk and the arrays s and x calcu-

lated. COGID is then called one more time to carry out the saddle-point

optimization.

It is desirable, in carrying cut the saddle-point optimization, to

have the defect as close to the centre of region I as possible and to have

the search vector s_ derived from two, identical, minimum energy configura-

tions. For this reason, the region-I block size should be chosen to consist

of an even number of lattice planes. Then both equilibrium defect positions

are equally displaced from the centre of the block, whereas the saddle point

is close to the centre. Note that this procedure will not yield the most

accurate value for the energy of the equilibrium defect configuration.

The evaluation of the final defect structure involves calculating

the dipole-force tensor of the defect. This is carried out in subroutine

STRESS. In the case of a saddle-point evaluation, the dipole ^nsor is

determined for both the saddle point and one of the equilibrium configura-

tions (the first one). The method of evaluating the dipole tensor is based

on Equation (8) of section 2. The perfect lattice coordinates required for

this evaluation, stored on a temporary disk space, are read in at this

point. Subroutine STRESS prints out the volume change, the trace of the

dipole tensor, and its normalized components and evaluates the eigenvectors

and eigenvalues of the dipole tensor. Final print-out is the displacement

field of the defect, which gives the amount the atoms have moved from their

perfect lattice positions as a result of the introduction of the defect.

4. RESULTS AND DISCUSSION

In this section, we summarize the results obtained on applying

PODSIP to study two defects in copper: the single vacancy and the <100>

dumb-bell interstitial. The defect configurations are evaluated using the

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- 20 -

MO Morse potential of Schober [18,19], described in section 3, using the

following constants: D = 0.18 eV, r = 0.71346 a and a = 8.38526 a ~ ,

where a is the lattice parameter. The fifth-order polynomial spline was

used to force the potential smoothly to zero between 1.05 a and 1.2 a .

However, ar- shown below, a study was also made to determine the sensitivity

of the results to the range over which the cut-off procedure is invoked.

The tolerance level on the forces (TOL) for the minimization was generally-4 -5 -1

set at either 10 or 10 eV a . The total size of region I used

(number of atoms) is listed in the tables and ranged from 256 to 1372 atoms.

A summary of the defect properties of the vacancy and the <100> dumb-bell

for both equilibrium and saddle-point positions is given in Table 1. Listed

are the formation energy (E_), the migration energy (E ), the trace of ther M

dipole tensor (TrP), its normalized eigenvalues and eigenvectors (p. and x.,

i = 1 to 3, respectively), and the relaxation volume in a finite crystal

divided by the atomic value (<5v/v ). The relaxation volume is obtained

using

5v = (l/3)TrP/K

where K = (C..+2C. _)/3 = bulk modulus. The equilibrium vacancy position is

located at (0,0,0)a , whilst the saddle point consists of an interstitial at

(0.25,0.25,0)a , halfway between two vacancies. When the region-I block

consists of an even number of planes, locating the vacancy at (0,0,0)a

means that it is not exactly at the centre of region I (but the saddle-point

configuration is). A small error is introduced, which is negligible for the

large region-I sizes used. Most of the results presented made use of

region-I sizes consisting of an even number of planes, since the emphasis in

this study was on the saddle-point configuration. The equilibrium dumb-bell

consists of two atoms located at (±O.3O3,O,O)a . The initial position for

the saddle-point configuration has two "interstitials" (replacing one

lattice vacancy located at (0,0,0)a ) with "interstitials" at

(-0.194,-0.022,0)a and (0.402,0.098,0)a ), plus a third "interstitial"

(replacing a lattice vacancy at (0.5,0.5,0)a ) at (0.522,0.694,0)a . The

final configuration has two "interstitials" at (-0.206,-0.047,0)a and

(0.386,0.114,0)a , plus a third at (0.547,0.706,0)a . That is, the true

interstitial position lies at (0.386,0.114,0)a .

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TABLE 1

DEFECT PROPERTIES OF A SINGLE VACANCY AND <1OO> DUMB-BELL INTERSTITIAL IN COPPER*

Defect

Vacancy

(req.1=864)

1 nterstitial

(req.1=864)

Location

equ i 1.

sadd le

point

equi1.

saddle

point

E /eV

1.16613

2.00014

3.80470

3.87576

M

0.

0

/eV

8 3401

07106

«v/

-0.

0.

2.

2.

c

01538

27323

42330

48849

TrP/eV

-0.42749

7.59668

67.37502

69.18745

P,

1

-0

0

0

.0

.153

.965

.855

P2

1.0

0.551

1.017

1.031

P3

1.0

2.602

1.018

1.114

*1

1

00

1

10

100

1-1

0

X

010

1-1

0

01

0

0

01

001

001

001

11

0

Using the modified Morse potential (MO, Schober and Zeller 1191) cut-off between 1.05 and 1.2 a , where

-4 -1a = lattice parameter, reqion I = 864 atoms, forces reduced to 10 eV a E = cohesive energy =o o C

-1.1700564 eV; the dipole tensors have been diaqonaI ized : the diagonal elements can be obtained by multiplying

the values 'isted under p ,...p by the appropriate TrP/3; the eigenvectors of the tensor are listed under

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- 22 -

Comparison with the results of Schober [18] shows good agreement

for the formation energies, but the trace of the dipole tensor shows small

discrepancies. This was puzzling, since the calculations were carried out

with apparently the same potential as used by Schober [18]. In his papers,

however, it is not clearly stated exactly between which distances the spline

cut-off reduces the Morse potential to zero. We have, therefore, investi-

gated how sensitive the results are to variations in the start of the cut-

off range, using a small region-I size. The results are summarized in

Tables 2 and 3, which show values of E and TrP, and the six polarizabili-

ties, respectively, as a function of the cut-off range for the <100> dumb-

bell interstitial. It is evident from this that all the quantities vary

signj(3)

significantly with cut-off range. Two of the most sensitive are a and

, which vary by about a factor of three over the range investigated. We

have also investigated the trend in the variation of E and TrP withF =

region-I size, as summarized in Table h. For this and all other studies,

the Morse cut-off range used was from 1.05 to 1.20. This study shows that

using a region I = 864 atoms introduces an error of only about 0.1 eV in TrP

and about 0.005 eV in E„. This represents an error of approximately 0.2%

in these quantities.

The main focus of this study is an investigation of the polariza-

bility of the vacancy and the <100> dumb-bell interstitial in copper, with

emphasis on the saddle-point configuration. In particular, we wanted to

compare the results obtained using two methods: the second strain deriva-

tive of the energy with the first strain derivative of the dipole tensor.

Gillan [11] has shown that, for the evaluation of P, the energy strain

derivative method should converge at smaller region-I sizes compared with

the direct moment method of Equations (7) or (8) and is, therefore, to be

preferred in cases where use of a large region-I size is not practical (such

as in ionic crystals or in metallic crystals involving the use of more

elaborate potentials, for instance, those based on the tight-binding model).

One might, therefore expect a similar result to hold for the polarizability.

The most difficult case is expected to be the interstitial, since it has the

largest distortion field.

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- 23 -

TABLE 2

EFFECT OF VARYING THE START OF THE SPLINE CUT-OFF RANGE FOR THE MORSE

POTENTIAL ON THE DEFECT PROPERTIES OF ThE <1OO> DUMB-BELL INTERSTITIAL*

Cut-of

1.0

1.03

1.05

1 .08

1.1

f Range/ao

to

to

to

to

to

1

1

1

1

1

.2

.2

.2

.2

.2

TrP/eV

68.65020

68.76190

67.51463

64.22585

60.96954

E /eV

3.931 27

3.88179

3.85246

3.79130

3.72667

* Region I = 256 atoms

TABLE 3

<100> DUMB-BELL INTERSTITIAL

VARIATION OF a TO a 6 ) WITH CUT-OFF RANGE OF MORSE POTENTIAL

Po 1

/(By

arizabil i t y V e V

By Energy \

Dipole Tensor/

( 1 )a

(2)a

(3 )a

(5)a

(6)

1.0-1.2

-272.797

-270.667

-32.809

-34.854

- 17.09 1

-18.870

330.769

328.381

238.193

235. 123

238.193

237.983

Cut-off

1 .05-1 .2

- 1 70.464

-165.495

25.326

36.156

51.349

52.370

360.861

355.571

368.326

369.628

367.826

37 1.352

Range/a

o

1.08- 1 .2

-126.261

- 1 30.349

37.820

36. 1 14

64.330

62.615

362.086

36 !.798

4 16.819

408.885

4 16.319

4 1 1 .083

1. 1

-97

-10 1

49

48

69

68

345

345

349

355

349

352

-1.2

.991

.771

.786

.969

.381

.630

.933

.643

.282

.337

.282

.239

• Equilibrium position, TOL - 10"4 eV a "'. region I * 2'i6 atoms,

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- 24 -

TABLE 4

VARIATION OF E AND P WITH REGION SIZE FOR THE <tOO> DUMB-BELL INTERSTITIALF =

Parameter

E /eV

TrP/eV

256

3.93127

68.6502

Region Size

368

3.90538

68.3621

(Atoms)

864

3.88259

68.0887

1372

3.87541

67.9769

TABLE 5

DIAELASTIC POLAR I ZABILITY FOR A SINGLE VACANCY EVALUATED USING TWO METHODS

Location

Equi1.

Saddle

Point

72

72

62

69

P=l

.8575

.6979

.8822

.025

(fa

P=2

22.

12.

32.

36.

3574

9948

5042

4662

ï)i By Enerqy

IBy Dipole TensorJ

P=3

22.3575

11.8828

37.7271

33.496

44

35

68

70

P=4

.8574

.9894

.0462

.0126

P

44.

35.

68.

70.

=5

8574

9777

0462

0177

44

35

p=6

.8575

.181

112.004

84.0861

-4 -1* Region I = 864 atoms; TOL = 1 0 eV a ; cut-off range for Morse

opotential from 1.05 a to 1.2 a

o o

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- 25 -

All polarizabilities were determined by applying a strain of mag--4

nitude 10 . The results are evaluated in terms of the six eigenvalues, as

given in Appendix B, by applying the appropriate eigenstrains. Table 5

gives a summary of the polarizabilities for the vacancy. Table 6 gives the

same values for the <100> dumb-bell interstitial evaluated at various region

sizes and different levels of force reductions. This table shows that, when

determined by the energy strain derivative method, there are only small dif-

ferences in a in going from a region-I size of 665 atoms to one double

that size, of 1372 atoms. Even the results obtained at a relatively small

region-I size of 256 atoms (Table 3: these were obtained for the equilib-

rium position only) is within 15% of the best answer. On the other hand,

the same values obtained by the change in dipole tensor method become

accurate only at force reduction levels of 10 eV a . At smaller

region-I sizes and TOL = 1 0 eV a , the a P values obtained by means of

the dipole tensor method are quite unreliable, sometimes agreeing closely

and sometimes disagreeing significantly, with the energy method. Particu-

lia

<3>

(2)larly unreliable values are obtained for a and, to a lesser extent, for

anda<

Table 6 shows, as previously found by Dederichs et al. [14], that

in the equilibrium position, the interstitial is quite soft in the

(lOO)-shear modes (e to e^ ' ) . Contrariwise, the (HO)-shears (e^ and(3)e ) have relatively little effect on the distortion of the defect. The

s^ and e -modes apply shears parallel to the dumb-bell oriented along

the x-axis, and must therefore be equal, by symmetry. This is in agreement

with the present results obtained (at the smaller region sizes, there is a

small discrepancy). It is, however, surprising, that there is so little(4)difference between the perpendicular e -mode and the two parallel ones.

This result is in agreement with those of Dederichs et al. [14].

The a ^ results for the saddle-point position are very similar to

those for the equilibrium position. All the a values, although somewhat

different, follow essentially the same trend as before. The largest

contributions come, again, from the (lOO)-shear modes with the e -shear

largest, whilst the e and e -shears are now almost equal and somewhat

reduced.

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- 26 -

TABLE 6

DI AELAST IC POLARIZABILITY» FOR THE <1OO> DUMB-BELL INTERSTITIAL

EVALUATED USING TWO METHODS

Model Size

And Accuracy*

reg.1=665

TOL=10"

reg.1=864

TOL=IO"

reg.1=864

TOL=IO"

reg.1=1372

TOL=1O"

Location

equi 1.

saddle

point

equi1.

sadd le

point

equi1.

sadd le

pofnt

equi1.

sadd le

point

p=l

-173.173

-197.396

-229.925

-226.099

-173.197

-177.906

-231.25

-239.448

-173.199

-172.757

-221.251

-229.539

-173.993

-170.534

-231.265

-231.708

a ( p >

P=2

29.270

87.333

15.724

21.347

29.764

-22.811

15.622

6.344

29.782

31.006

15.623

16.141

30.079

33.336

15.973

17.861

I By Energy(By Dipole Tensor

P=3

58.325

72.072

56.761

61.330

59.153

47.971

57.822

59.563

58.826

59.177

57.821

57.697

59.903

60.670

58.806

59.608

p=4

400.352

380.738

284.251

287.300

404.363

398.578

288.484

297.503

404.363

407.589

288.484

288.277

411.768

413.140

292.943

293.853

VeV

P=5

408.773

410.917

285.255

287.299

412.279

423.866

288.112

295.760

412.276

412.085

288.111

289.432

418.254

416.507

293.315

292.123

P

407409

345548

412421

349369.

412409.

348.

353.

418.

4)6.

354.

354.

=6

.265

.397

.224

.153

.279

,116

.781

853

27627

291337

254

517

975533

• Reglon-1 size in atoms; TOL in eV a ; cut-off range for Morse potential from

1.05 a to 1.2 ao o

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- 27 -

In Table 7, we list the results for the changes in elastic con-

stants, derived using Equation (38) and evaluated using the data given in

Table 8. For brevity, only one set of elastic constant changes is given.

The results compare closely with those of Dederichs et al. [14], evaluated(2)

for the equilibrium position only. The largest discrepancy is for the e(3)and e -shear modes, for which the smallest response to applied strain

occurs. The changes in elastic constants are all negative, except for the

e -mode in the saddle-point position. The result for the e -mode is due

to the interstitial's large volume expansion, negating the large positive

local effects of the interstitial on the change in elastic constant.

It is evident from the calculations summarized in Table 7 that the

equilibrium and the saddle-point configurations both yield similar aniso-

tropies for the <100> dumb-bell in copper, as represented by a Morse

potential.

Finally, in Table 9, we list the orientationally averaged values

of the changes in elastic constants for both the vacancy and the inter-

stitial in their equilibrium positions. These were evaluated by the energy

method.

5. CONCLUSIONS

Application of PODSIP to evaluate the properties of defects

located both in their equilibrium and their saddle-point positions demon-

strates the efficiency of the program in carrying out this calculation in

one program submission step.

Using the MO Morse potential to simulate the metal copper, we find

that the results for the formation energy, trace of the dipole tensor, and

the diaelastic polarizabilities of the <100> dumb-bell interstitial are a

sensitive function of the range over which the Morse potential is smoothly

reduced to zero.

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- 28 -

TABLE 7

CHANGES IN ELASTIC CONSTANTS IN A FINITE CRYSTAL FOR THE SINGLE VACANCY AND THE<1OO> DUMB-BELL INTERSTITIAL EVALUATED USING TWO METHODS

Type

Vacancy*

Interstitial**

Location

equ11.

saddlepoint

equl1.

saddlepoint

P=1

-2.4Î6-2.410-4.072-4.293

-1.120-1.244

0.7411.036

P=2

-4.047-2.300-6.472-7.195

-10.403-il.011-7.834

-6.102

AC(p) i

CC(P) UyP=3

-4.047-2.091-7,584-6.794

-15.971-16.114-15.712-16.037

By EnergyDipole Tensorf

P=4

-2.852-2.259-5.176-5.306

-33.067-33.159-24.968-25.571

P=5

-2.852-2.259-5.175-5.306

-33.501-33.384-24.943-25.455

P=6

-2.852-2.205-8.117-6.249

-33.501-33.385-29.069

-30.412

* Region I = 864 atoms; TOL= 10 eV a

5 °*• Region I = 1372 atoms; TOL = 1 0 eV a

TABLE 8

ELASTIC CONSTANTS OF COPPER AND THEIR PRESSURE DERIVATIVES (FROM REFERENCE [14])

(1)c

38.06

(5C

no 1 0

7.

->3)

N/m 2)

34

C

20.

->6)

48

9c9p

20

(1)

.94

3g"c,p

2.

2->3)

52

3C(

3p

8.

4^6)

22

TABLE 9

ORIENTATIONALLY AVERAGED VALUES OF CHANGES IN ELASTIC CONSTANTS*FOR THE SINGLE VACANCY AND THE <100> DUMB-BELL INTERSTITIAL

IN THEIR EQUILIBRIUM POSITIONS

Defect

Vacancy:EquIIibrlum

Interstitial :Equ11ibrI urn

cK

-2.416

-1.120

Ac1

cC

-4.047

-13.187

A C44

CC,4

-2.852

-33.356

* Based on Equation (38)

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- 29 -

A comparison of two methods of evaluating the diaelastic polariza-

bility, one by means of the second strain derivative of the defect energy,

the other by means of the first strain derivative of the dipole tensor,

shows that the energy method is more accurate for a given region size and

force reduction level.

The defect properties of the single vacancy show large differences

in the energy and in the trace of the dipole tensor, and symmetry between

the equilibrium and saddle-point positions. Except for the dilatation mode,

a , the diaelastic polarizabilities are also strongly changed by factors

of from one and a half to three, a (saddle-point) is almost triple that

of a (equilibrium)). All a values are positive.

The defect properties of the <100> dumb-bell interstitial show

only small differences in the energy and in the trace of the dipole tensor,

and symmetry between the equilibrium and saddle-point positions. The

saddle-point o values are all decreased from the equilibrium a values, the(2)

largest change of a factor of two being for a . Except for a largenegative value for a , all other a values are again positive.

As previously found by Dederichs et al. [14], the <100> dumb-bell

interstitial is very soft in the <100>-shear mode, leading to large

a -values. Contrariwise, the a -values are small and differ little

from the a -values for the vacancy.

REFERENCES

1. C F . Melius, C.L. Bisson and D.W. Wilson, "Quantum-Chemical andLattice-Defect Hybrid Approach to the Calculation of Defects inMetals", Phys. Rev. B _18_, 1647 (1978).

2. A.A. Bahurmuz and C.H. Woo, "The Multi-Scattering-Xa Method forAnalysis of the Electronic Structure of Atomic Clusters", AtomicEnergy of Canada Limited Report, AECL-7798 (1984).

3. R.J. Jerrard, C.H. Woo and J.M. Vail, "A Semi-Empirical Method forPoint-Defect Studies in Transition Metals", Atomic Energy ofCanada Limited Report (in preparation).

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- 30 -

4. C.K. Or.g, J.M. Vail and C.H. Woo, "Computer Simulation of PointDefects in Metals: A First Report", unpublished WhiteshellNuclear Research Establishment Report, WNRE-198 (1982).

5. M.P. Puls and C.H. Woo, "Physical Bases of Dislocation CoreConfiguration Calculations in Metals and Ionic Crystals", inDislocations 1984, P. Veyssière, L. Kubin, J. Castaing, eds.,Editions du CNRS, 1984, p. 93.

6. See for example, Jong K. Lee, ed., Interatomic Potentials andCrystalline Defects, The Metallurgical Society of AIME,Warrendale, Pa., U.S.A., 1981.

7. R. Fletcher and C M . Reeves, "Function Minimization by ConjugateGradients", Comput. J. ]_, 149 (1964).

8. J.E. Sinclair and R. Fletcher, "A New Method of Saddle-PointLocation for the Calculation of Defect Migration Energies", J.Phys. C]_, 864 (1974).

9. G. Leibfried and N. Breuer, Point Defects in Metals, Part I,Springer-Verlag, Berlin, 1978.

10. M.J. Gillan, "The Long-Range Distortion Caused by Point Defects",Phil. Mag. A4^, 903 (1983).

11. M.J. Gillan, "The Elastic Dipole Tensor for Point Defects in IonicCrystals", J. Phys. C17_, 1473 (1984).

12. H.R. Schober and K.W. Ingle, "Calculation of Relaxation Volumes,Dipole Tensors and Kanzaki Forces for Point Defects", J. Phys.Fl£, 575 (1980).

13. H. Kanzaki, "Point Defects in Face-Centred-Cubic Lattice - IDistortion Around Defects", J. Phys. Chem. Solids _2_, 24 (1957).

14. P.H. Dederichs, C. Lehmann and A. Scholz, "Change of ElasticConstants due to Interstitials", Z. Physik B ^ , 155 (1975).

15. C.H. Woo and M.P. Puls, "Atomistic Breathing Shell ModelCalculations of Dislocation Core Configurations in IonicCrystals", Phil. Mag. 21» 7 2 7 (1977).

16. M.P. Puls, C.H. Woo, and M.J. Norgett, "Shell Model Calculationsof Interaction Energies Between Point Defects and Dislocations inIonic Crystals", Phil. Mag. 36^ 1457 (1977).

17. M.P. Puls, "An Atomistic Model to Calculate the Core Structure andEnergy of Dislocations in Ionic Crystals. Part 1: Rigid BoundaryModel", Atomic Energy of Canada Limited Report, AECL-5237 (1975).

18. H.R. Schober, "Single and Multiple Interstitials in fee Metals",J. Phys. F7, 1127 (1977).

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- 31 -

19. H.R. Schober and R. Zeller, "Structure and Dynamics of MultipleInterstitials in fee Metals", J. flucl. Mater. >9_ and 7£, 341(1978).

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- A.I -

APPENDIX A

EXPANSION OF THE ENERGY IN TERMS OF LAGRANGIAN STRAINS

In this appendix, we derive an expression for EL(e), valid to

second order in the infinitesimal strain tensor e, which can be used to

evaluate dipole tensors and diaelastic polarizabilities.

Expansion of the energy, E , to second order in the Lagrangian

strain tensor £ yields

En(n) = E_(0)D -' D 3n =o: JTQ

(A.I)

where

(A.2)

For convenience, we drop the tensor notation in the following. Expansion of

E (n) about e yields

3E,ED(O 3E

3e

Ae (A.3)

Inserting Equation (A.2) into the right-hand side of Equation (A.I) and

using Equation (A.3) for ED(n) yield (to 0(e ))

r.o 1 T,O 2 1 2 , 1 no 2P e - y P e - j ae + - P c (A.4)

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where

- A.2 -

3e 3c e=0

and

a = -3e e=0

Collect ing terms gives

ED(e) = ED(0) -o , 1 _e 2 1 2e + P e ae (A.5)

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- B.I -

APPENDIX B

EXPRESSIONS FOR STRAIN DERIVATIVES OF THE DEFECT ENERGY IN TERMS OF

CUBIC STRAIN EIGENTENSORS AND EIGENSTATES

In this appendix, we derive an expression for En(iE) and its first

and second strain derivatives, in terms of the strain eigentensors and

eigenstates appropriate to a cubic crystal.

let

Let b . , p=l to 6, be the set of orthonormal basis tensors and

'

•13« • vi'H;' • «•»

Given that

ED(|) = ED(0) - P?jEij + I P* . ^ . - \ *tfim*u (B.3)

then, substituting Equations (B.I) and (B.2) into (B.3), yields

where p is a dummy index because of the implied summation.

Using the relation between the change in elastic constants due to

one defect in the crystal and the polarizability, given by

v — <»-5>

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- B . 2 -

where N = number of lattice sites in the crystal

V = volume of the Wigner—Seitz cell of the crystal

Equation (B.4) can also be written

The cubic elastic eigenconstants and eigenstrains are given by

(!) ( 1) ! [ 1 0 °C{1) =C,,+2C ; Sl) = M O 1 0

i Z Ä \0 0

C U ; = C . - C , , ; b ^ ; = - I 0 1 011 U * \O 0 0,

Ll ll Se \ o o 2

l S S ÏHoio

[HI/2 \ l 0 0

0 0 0,

The first derivatives of E with respect to e evaluated at e = 0 yield

M = - p°.

r ( 5 ) _ ? r . v(5)c ~ 2C44 î b

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- B.3 -

or, explicitly,

/3

(pi°rP2°2>3e2/e2=O /2

3 ED

9e3/e3=O

3 e4/e4=0 /2

3e5/e5=O /2

-6/e,=0o

and the second derivatives yield

3 e2 I ij i l ljP /ep=0

or, explicitly,

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- B . 4 -

l - e +p e +p e ) - aL *22 * 3 3 ; 1

'I /ej-0

3e:11 2 2 ' 2

3 / e 3 = 0

3e,23 32 ; " °4

3 2E l

T26

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- C l -

APPENDIX C

LOGICAL STRUCTURE AND INPUT FORMAT FOR PODSIP

A schematic, showing the logical structure of PODSIP in terms of

subroutine calls, is given in Figure C-l. A brief description of the

function of each of the subroutines is given below, but first we give an

overview of the overall calling program MAIN.

MAIN starts the program and is the starting point from which all

other routines (directly or indirectly) are called, as shown in Figure C-l.

Most of the data input/output occurs in this routine. The required input

information is given in Table C-l. After reading in the desired crystal

structure, region sizes, potential and strain state, the program calculates

the perfect lattice energy of region I. The defect is then introduced and

the region-I energy recalculated. The difference between this energy and

the perfect lattice energy is printed. This is the defect energy, E . A

call to COGID or CONJUG then results in minimization of the defected config-

uration. After minimization, a call to STRESS causes an evaluation of the

defect's dipole tensor and the results are printed. If a saddle-point

defect configuration is also desired, new defect coordinates are read in,

and the minimization is repeated two more times, as explained in the main

text. A final call to STRESS evaluates the dipole tensor of the defect in

its saddle-point configuration. The following subroutines are called

through MAIN (in alphabetical order):

CONJUG - Minimizes a function of n atomic position variables (the energy

of region I) with given gradients on the atoms.

COGID - Optimizes a function of n atomic position variables (the energy

of region I) with given forces on the atoms; both minimum and

saddle-point configurations can be found.

CUBSET - Reads in lattice parameters, sizes and ranges of regions I and

II for a crystal with cubic symmetry.

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- C.2 -

MAIN

XSET

—| FL-NC 1 FORCE j

—|IIEFECT/IKTNBR1 I FUNC ] jFORCE

COG ID PRN'T JMC.O2AS

FORCF.

STRESS

KICKS

Figure C-l: The Logical Structure of PODSIP

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TABLE C-1

INPUT PARAMETERS FOR PODS IP

Line No.

(Cal 1 ing

Subroutine)

1

(MAIN)

2

(MAIN)

3

(MAIN)

4

(MAIN)

5

(MAIN)

Input Parameters

TITLE

IPOTEN

RANGS,RCORE,RNN

i) If Morse potential

REQ.D,ALPHA

II) If Lennard-Jones

DD.EE

ili) If SplIne fit

NSPLN

RA(l)

RA(I),RAB(I)

*[Apd,l),Apd,2),Ap(t

[_Ap(l ,4),Apd,5),Apd

*I=2,NSPLN

NTOTE.NINTE

.3,1,6)J

DefIn It ion

Choices of palrwlse potentials

= 1 Morse potential

= 2 Lennard-Jones potential

= 3 Spline fit potential

RANGS = square of cut-off distance for

pairwise potential

RCORE = core radius, defined to harden

or soften the core. If nothing

about core radius needs to be

done, put RCORE«)

RNN = nearest neighbour distance

D!exp(-2ALPHA(R-REQ>)-2exp(-ALPHA(R-REQ))l

EEKDD/r) -2(DD/r)6|

- number of splines

- range of Born-Mayer potential calcu-

lated automatically

6 i-1]T Apd ,j)(r-RAB(l )/a)

j = 1

RAd-l)<r<RA(l)

Total # of atoms, § of interstitial s

(upper bound estimates only, needed)

Unit

Integer

a(lattice)

constant)

a ,eV,a

o o

a eV

o

Integer

ao

O OeV

integer

Format

80H

15

3F20.10

3F20.10

2F20.10

15F20.10

3F20.10

3F20.10

215

I

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TABLE C-l Cont 'd

Line No.(Cal 1InqSubroutine)

6(MAIN)

7(XSET)

8(XSET)

9

(XSET)

10

(XSET)

11

(XSET)

12

(MAIN)

13

(MAIN)

14

(DEFECT)

Input Parameters

SYMME

STRUCT

A(1),A(2),A(3)

Ml 1ler(1),...MIIler(4)Mi 1ler(5),...Ml1ler(8)Mlller(9),...MIller(12)

NSIZE(U,NSIZE(2),NSIZE<3)

RANGE(1),RANGE(2).RANGE(3)

ESR(1,1),ESR(2,1)(ESR(3,1)etc.

ISSD

NSUB.NINT

Definition

Crysta1 symmetry= CU, cubic symmetry= HC, hexagonal symmetry

Crystal structure SC/UB,FC/C,BC/C,DI/AM,SP/HAL,FL/OUR,NA/CL,CS/CI,HC/P,GR/APHITE,WU/RTZ,CO/RUN

Lattice constant (usually use 1) orlattice constants, a, c and length ofa in A

Miller indices for surface planes3 indices for cubic symmetry4 Indices for hexagonal symmetry

Size of region 1 10 of planes In eachdirection) negative for cyclic boundaryconditions

Size of reqlon 11

Strain,tensor,matrix: (3,3)-read In by row

= 0 for energy minimization= 1 for saddle point configuration= -N Initial search direction for saddle

point atom

# of vacancies, 0 of Interstitlals

Unit

character

character

ao ovvA

ao

ao

01 r

1nteger

Integer

Format

A2

A2

3F10.5

41 10411041 10

3110

3110

F11.4JX

15

215

In

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TABLE C-1 Cont'd

Line No.(Cal 1InqSubroutine)

15(DEFECT)

16(MAIN)

17(MAIN)

18(STRESS)

19(MAIN)

Input Parameters

RANAD

X1,Y1,Z1,LB,L2

X1.Y1.Z1.L2

BMOD

IFRZ(J),J=1,N

Definition

Additional range (one tor each) forneighbour list of the Interstitial s

Coordinates and type of vacancy (one setfor each vacancy). LB is type of atomtaken away; L2 type put In 0 for vacancy

Coordinate and type of Interstitial(need another if ISSD>0)

Bulk modulus

N coordinate #'s (usually of intersti-tials) to be frozen during energy mln.

Unit

ao

a ,1nteger

a ,1ntegero

12 210 dyn/cm *

integer

Format

F10.4

3F10.4.2I5

3F10.4.I5

1415

n

I

* 1 dyn/cm = 0.1 Pa

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- C.6 -

DEFECT - Reads in the defect configuration and calculates its neighbour

lists.

EÏGRS - Finds eigenvalues and eigenvectors of a matrix A.

FORCE - Calculates the magnitude of the potential and the force on the

atom at each position.

FUNC - Calculates the total energy of the crystal.

HCF (function) - Calculates the highest common factor z of two numbers x and

y-

HEXSET - Reads in lattice parameters, sizes and ranges of region I and II

for a crystal of hexagonal symmetry.

INTNBR (entry) - Calculates the neighbour list for interstitials.

LATSSP - Calculates the Bravais lattice for each crystal structure.

LATWRT - Writes lattice vectors, basis vectors and block vectors.

LATBLK - Normalizes block edge vectors.

LATMSH - Sets up a cubic mesh aligned with the block edge vectors.

MORSE - Evaluates the coefficients for the Morse spline-fit potential.

MC02AS - Calculates the scalar product of two vectors.

NEIBR3 - Calculates three-body neighbour lists for an atom in region I.

NEIBOR - Calculates two-body neighbour lists for an atom in region I.

PRNT - Prints coordinates of the atoms.

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- C.7 -

PRINT 2 - Prints the two-body neighbour lists.

PRINT 3 - Prints the three-body neighbour lists.

REGDIM - Calculates the total number of atoms in region I and II.

REGPBC, REGSCL - Sets up the supercell, which consists of regions I and II

combined.

REGIND - Sets up a back-conversion table to enable the storage-based

index to be found from the lattice-based index.

REGXLT - Generates all points of the cubic mesh within regions I and II

of the model crystal and checks whether these are proper lattice

sites.

SSP - Main calling program for the lattice generation program.

STRAIN - Displaces the atom coordinates according to a given strain.

STRESS - Calculates dipole-force tensor, its eigenvalues, eigenvectors

and relaxation volume.

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