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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
Copyright © Atomic Energy of Canada Limited, 1985
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
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
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
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
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.
- 2 -
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
- 3 -
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.
- 4 -
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
- 5 -
(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
- 6 -
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]
- 7 -
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
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
- 9 -
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.
- 10 -
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)
- 11 -
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.
- 12 -
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
- 13 -
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.
- 14 -
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
- 15 -
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
- 16 -
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
- 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.
- 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.
- 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
- 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 .
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
- 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.
- 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,
- 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
- 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.
- 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
- 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.
- 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)
- 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).
- 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).
- 31 -
19. H.R. Schober and R. Zeller, "Structure and Dynamics of MultipleInterstitials in fee Metals", J. flucl. Mater. >9_ and 7£, 341(1978).
- 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)
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)
- 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>
- 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
- 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,
- 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
- 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.
- 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
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
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
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
- 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.
- 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.
ISSN 0067-0367 ISSN 0067-0367
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