philosophical m a,2001,vol. 81,no. 11, 2591± 2612physics.gmu.edu/~ymishin/entropy.pdf · atomistic...

PHILOSOPHICAL MAGAZINE A 2001 VOL 81 NO 11 2591plusmn2612

Calculation of point-defect entropy in metals

Yuri MishinySchool of Computational Sciences MSN 5C3 George Mason University

Fairfax Virginia 22030 USA

Mads R Soslashrensenz and Arthur F Voter

Theoretical Division Los Alamos National Laboratory Los AlamosNew Mexico 87545 USA

[Received 27 June 2000 and accepted in revised form 19 January 2001]

AbstractWe investigate previously used methods and propose a new method for

atomistic calculations of point-defect entropies in metals within the harmonicapproximation to lattice vibrations The key problem is to predict accuratelythe defect formation entropy in a macroscopic crystal from atomisticcalculations performed on a small system containing relatively few atoms Theresults of atomistic calculations may depend signiregcantly on the system sizegeometry and boundary conditions Two previously used methods which wecall the supercell and embedded-cluster methods are analysed in two waysregrstly within a linear elasticity model of a point defect and secondly byatomistic calculations for a vacancy and an interstitial in copper using anembedded-atom potential The results of atomistic calculations conregrm thelinear elasticity analysis and show that the supercell method is much moreaccurate than the embedded-cluster method However the latter is useful forcomputing the defect core entropy which turns out to be a well-deregnedphysical quantity We propose a new method of defect entropy calculationsthat combines the embedded-cluster method with a quasicontinuumapproximation outside the cluster This method which we call an elasticallycorrected embedded-cluster method has an accuracy comparable with that ofthe supercell method and extends defect entropy calculations towards largersystem sizes

1 IntroductionPoint defects mediate atomic di usion creep and other processes in solids at

elevated temperatures (Flynn 1972) If the external pressure is zero (p ˆ 0) theequilibrium atomic concentration of point defects of a certain type is given by

C ˆ exp iexcl E iexcl TS

kBT

sup3 acute hellip1dagger

Philosophical Magazine A ISSN 0141plusmn8610 printISSN 1460-6992 online 2001 Taylor amp Francis Ltd

httpwwwtandfcoukjournals

DOI 10108001418610110043145

E-mail ymishingmueduz Present address Novo Nordisk AS Scientiregc Computing B2131 Novo Nordisk Park

DK-2670 Maaloev Denmark

where T is the temperature E and S are the defect formation energy and entropyrespectively and kB is the Boltzmann constant Given a suitable model of atomicinteractions E and S can be determined by atomistic calculations in the zero-tem-perature limit using molecular statics and the harmonic approximation to atomicvibrations

The physical quantities of interest are the defect energy and entropy in a macro-scopic crystal under constant-pressure (usually zero-pressure) conditions Howeveratomistic calculations of these quantities are limited to relatively small systems withapproximately 103 atoms because the diagonalization of large dynamic matrices iscomputationally demanding and associated with numerical problems For systems ofthis size results of atomistic calculations depend on the system size geometry andboundary conditions How should such calculations be performed in order to obtainaccurate estimates of the defect properties in a macroscopic crystal This question isthe main topic of the present paper

Di erent problems associated with defect entropy calculations have beenreviewed by Catlow (1997) Foiles (1994) and Harding (1990) Most of the previouswork was done on ionic solids in which the electric charges of defects have a sig-niregcant e ect on entropy calculations In this paper we shall be concerned withmetallic systems in which point defects bear no electric charge This will makeour analysis less complex and will allow us to correlate our atomistic simulationresults with simple calculations based on linear elasticity theory Previous atomisticcalculations of vacancy formation entropies in metals were undertaken by severalworkers (for example Hatcher et al (1979) Fernandez and Monti (1993) Debiaggiet al (1996) Fernandez et al (1996 2000) and Frank et al (1996)) There are twodi erent approaches to such calculations which will be referred to in this paper asthe `supercell methodrsquo and the `embedded-cluster methodrsquo In the supercell methodall free atoms (ie atoms allowed to move during the energy minimization) are at thesame time dynamic atoms (ie their vibrations are taken into account in the defectentropy calculations)y In the embedded-cluster method only some of the freeatoms namely those forming a cluster centred a the defect core are considered asdynamic while all other free atoms are treated as static

In this paper we investigate the supercell and embedded-cluster methods andpropose an alternative approach to defect entropy calculations In sect 2 we explainhow entropy calculations have previously been conducted within the supercell andembedded-cluster methods In sect 3 we undertake a simple linear-elasticity analysis ofthe energy and entropy associated with the elastic strain regeld around a point defectWe examine how these quantities depend on the overall system size cluster size andthe boundary conditions imposed on the system In sect 4 these dependences are com-pared with atomistic simulation results performed for vacancies and interstitials incopper These simulations are also used for testing the accuracy of the entropycalculation methods considered in this paper In sect 5 we present our new methodfor defect entropy calculations a so-called elastically corrected embedded-clustermethod This method can be viewed as an extension of the embedded-cluster methodin which the entropy of the lattice regions outside the cluster is included in a quasi-

2592 Y Mishin et al

Often especially in regrst-principles calculations the term `supercellrsquo refers to a periodicblock In this paper we use this term whenever all relaxed atoms are included in the entropycalculation irrespective of the boundary conditions

continuum approximation In sect 6 we summarize and outline possible applications ofour analysis

2 Methods of point-defect calculations

21 Defect formation energyIn a typical atomistic simulation scheme a point defect is created in the centre of

a simulation block and the total energy of the block is minimized with respect toatomic displacements subject to chosen boundary conditions If the defect is a singlevacancy its formation energy can be estimated as

~EEN ˆ EdhellipN iexcl 1dagger iexcl E0hellipNdagger hellipN iexcl 1daggerN

hellip2dagger

EdhellipN iexcl 1dagger being the relaxed energy of the defected block containing N iexcl 1 atomsand E0hellipNdagger the energy of the initial perfect-lattice block containing N atoms In thisand all other equations a tilde indicates quantities evaluated in a regnite-size systemEquation (2) can be modireged for other defects for example for an interstitial N iexcl 1should be replaced by N Dagger 1 If N is large enough equation (2) provides an accurateestimate of the true energy E that would be found in an inregnitely large system atp ˆ 0 Even if the block size is not very large E can be estimated by computing ~EEN

for several di erent N values and making a linear extrapolation of the plot ~EEN versus1=N to 1=N 0 Indeed in metals the point-defect energy can be divided into thedefect core energy Ec and the energy Eel of the elastic strain of the surrounding latticeinduced by the defect

E ˆ Ec Dagger Eel hellip3daggerThe strain-energy density decreases with increasing distance r from the defect as 1=r6

(for example Hatcher et al (1979)) (see also sect 3 of this paper) Thus the total strainenergy stored within a sphere of radius R converges to Eel with R 1 as 1=R3 or as1=N assuming that N is proportional to R3 As will be shown below this extrapola-tion converges to the exact point-defect energy E regardless of whether the volume ofthe block remains regxed or is allowed to relax during the energy minimization tomimic a zero external pressure

22 Defect entropy in the supercell methodWhile the point-defect energy can be calculated almost routinely the vibrational

entropy calculation is a more delicate matter We shall regrst consider the supercellmethod of entropy calculations This method uses a cubic or rectangular latticeblock usually (but not always) with periodic boundary conditions in all directionsThe defect formation entropy is determined from the total entropy change of theblock Thus for a vacancy in a periodic block we have

~SSN ˆ SdhellipN iexcl 1dagger iexcl S0hellipNdagger hellipN iexcl 2daggerhellipN iexcl 1dagger hellip4dagger

where SdhellipN iexcl 1dagger is the entropy of the hellipN iexcl 1dagger-atom defected block after its staticrelaxation and S0hellipNdagger is the entropy of the initial N-atom perfect-lattice block In theharmonic approximation the vibrational entropy of an N-atom periodic block isgiven by

Calculation of point-defect entropy in metals 2593

S ˆ iexclkB

X3Niexcl3

nˆ1

lnhcediln

kBT

sup3 acuteDagger hellip3N iexcl 3daggerkB hellip5dagger

cediln being frequencies of normal vibrations and h the Planck constant Equations (4)and (5) take into account that three eigenvalues of the dynamic matrix out of 3Nare zero owing to the translational invariance of the total energy By combiningequations (4) and (5) we have

~SSN ˆ iexclkB lnY3Niexcl6

nˆ1

cediln

Y3Niexcl3

nˆ1

cedil0n

Aacute hellipNiexcl2dagger=hellipNiexcl1dagger 3

5

2

4 hellip6dagger

cedil0n being normal vibrations of the perfect-lattice block In calculations with regxed

boundary conditions all eigenvalues of the dynamic matrix are positive so the right-hand sides of equations (5) and (6) should be modireged by increasing N by 1 Forinterstitials or other defects these equations must also be modireged accordingly Notethat these equations are only valid at temperatures above the Debye temperatureWe thus assume that the harmonic approximation remains valid at such tempera-tures

In the so-called full harmonic approximation (FHA) the normal vibrationalfrequencies are calculated exactly by diagonalizing the dynamic matrix of the system

Dinotjreg ˆ 1

mnotmreg

iexcl cent1=2

q2E

qRinot qRjreg

hellip7dagger

where the vectors Rinot and Rjreg determine the equilibrium positions of atoms not and regand mnot and mreg are the respective atomic masses Then the normal frequencies aregiven by cediln ˆ para1=2

n where the index n numbers positive eigenvalues paran of the dynamicmatrix To simplify entropy calculations some workers (for example LeSar et al(1989)) proposed to use the local harmonic (Einsteinrsquos) approximation (LHA) inwhich the atoms are treated as independently vibrating oscillators In this approx-imation the vibrational frequencies are calculated by diagonalizing the localdynamic 3 pound 3 matrices Dinotjnot There are even further simpliregcations of the LHAin which only diagonal elements Dinotinot are taken into account for example thesecond-moment approximation of the vibrational spectrum (Sutton 1989 1992Foiles 1994)

As in energy calculations the regnite-size e ects can be eliminated by calculating~SSN for several di erent block sizes and making a linear extrapolation of ~SSN versus1=N to 1=N 0 The convergence of this extrapolation will be shown later in thispaper It will also be shown that similarly to the defect energy the defect entropycan be divided into the core entropy Sc and the entropy Sel associated with elasticdistortions of the surrounding lattice

S ˆ Sc Dagger Sel hellip8dagger

In contrast with the defect energy however the defect entropy converges to di erentvalues denoted as Sp and SV depending on whether the volume V of the block isallowed to vary during the energy minimization (zero-pressure condition p ˆ 0) orremains regxed (constant-volume condition V ˆ constant) For practical purposes weare more interested in the zero-pressure condition The di erence between Sp and SV

2594 Y Mishin et al

originates mainly from the elastic term Sel and can be described by the relation(Flynn 1972 Hatcher et al 1979 Catlow et al 1981)

Sp iexcl SV ˆ shy BVrel hellip9daggershy being the thermal expansion factor at p ˆ 0 deregned as Viexcl1 qV=qThellip daggerpˆ0 B theisothermal bulk modulus and Vrel the relaxation volume of the defect The lattercan be determined directly as the volume change of the simulation block due to therelaxation although other methods are also available (Finnis and Sachdev 1976Gillan 1981) In practice supercell calculations of defect entropy are limited torelatively small blocks (N ordm 103) because the diagonalization of larger dynamicmatrices becomes computationally demanding and is associated with numerical pro-blems This prevents the method from being applied to complex defects defectplusmndefect interactions and other situations in which a large size of the simulationblock is essential

23 Defect entropy in the embedded-cluster methodThe embedded-cluster method takes a di erent approach namely the block size

is chosen as large as appropriate for the inclusion of the distorted lattice regionsaround the defect(s) under study typically with N ordm 103plusmn105 As before the equili-brium conregguration of the block is obtained by its static relaxation For entropycalculations however a cluster of atoms enclosed by a sphere centred at a chosendefect is selected The atoms included in the cluster are considered as dynamic whileall other atoms are considered as static The defect formation entropy is then calcu-lated by applying the previous equations to the dynamic cluster For example for thevacancy formation entropy equation (6) takes the form

~SSNcurrenˆ iexclkB ln

Y3Ncurreniexcl3

nˆ1

cediln

Y3Ncurren

nˆ1

cedil0n

Aacute hellipNcurreniexcl1dagger=hellipNcurrendagger 3

5

2

4 hellip10dagger

where cedil0n and cediln are normal vibrational frequencies of the cluster before and after the

introduction of the vacancy and N is the initial number of atoms in the cluster Foran interstitial defect N iexcl 1 in equation (10) should be replaced by N Dagger 1 while3N iexcl 3 should be replaced by 3N Dagger 3 The cluster size must be much smaller thanthe overall block size and must be suitable for normal frequency calculations Theobtained entropy will obviously depend on the cluster size which gives rise to twoissues what extrapolation must be applied for obtaining a size-independent entropyand what is the physical meaning of that entropy The scheme previously proposed(Hatcher et al 1979) and used for vacancy calculations in metals (Fernandez andMonti 1993 Debiaggi et al 1996 Fernandez et al 1996) suggests that the plot ~SSNcurrenversus 1=N must be linearly extrapolated to 1=N 0 and that the obtainedentropy has the meaning of the defect formation entropy under constant-volumeconditions (SV ) provided that the simulation block was relaxed under a regxedvolume Our analysis described below does not conregrm this scheme Firstly evenif the total volume of the block remains regxed the volume of the cluster generallychanges during the relaxation Moreover this volume change is comparable with thedefect relaxation volume Vrel Secondly both the linear-elasticity analysis ( 3)and atomistic simulations performed by Fernandez et al (2000) and in this papershow that ~SSNcurren

does not converge with increasing cluster size at least not in metalsMore speciregcally it will be shown in this paper that the elastic part of the entropy is


approximately proportional to N=N N being the total number of atoms in theblock Thus the entropy diverges if formally extrapolated to 1=N 0 and con-verges to the defect core entropy in the N 0 limit A more adequate procedure ofentropy calculations within the cluster approach should therefore be developed

3 Linear-elasticity analysis

31 Elastic model of a point defectFor the evaluation of the elastic energy and entropy associated with a point

defect the latter will be represented as a dilatation centre located in the centre ofa large sphere of radius R (reggure 1) This sphere serves as an analogue of a simula-tion block used in atomistic simulations A small spherical region of radius rc aroundthe defect will represent the defect core while the region outside the core r gt rc willrepresent the elastically strained lattice around the defect For simplicity the latticewill be treated as an isotropic elastic medium with Youngrsquos modulus E and Poissonrsquosratio cedil The volume e ect associated with the defect formation will be modelled by asmall expansion or contraction of the core resulting in a change in its radius rc by asmall amount uc The elastic strain regeld induced by this dilatation is described by thedisplacement function (Eshelby 1956 Landau and Lifshitz 1986)

uhelliprdagger ˆ ar Daggerb

r2 hellip11dagger

with uhelliprcdagger ˆ uc The coe cients a and b depend on uc and the boundary conditionat the surface of the sphere The regrst term in equation (11) represents the regnite-size e ect and vanishes when R 1 Expressions for a and b under p ˆ 0 andV ˆ constant conditions at r ˆ R are given in Appendix A

From equation (11) the principal components rr sup3sup3 and iquestiquest (sup3 and iquest beingpolar angles) of the strain tensor are

rr ˆ a iexcl 2b

r3 sup3sup3 ˆ a Dagger b

r3 hellip12dagger

and iquestiquest ˆ sup3sup3 while the local dilatation equals

D ˆ rr Dagger 2sup3sup3 ˆ 3a hellip13dagger

2596 Y Mishin et al

Figure 1 Schematic spherical model of a point defect for elasticity theory calculations

Note that this dilatation is homogeneous and arises exclusively owing to imageforces

32 Relaxation volume of a point defectThe volume change of the sphere which represents the relaxation volume of the

defect equals

~VVrelhellipRdagger ˆ 4pR2uhellipRdagger ˆ 4phellipaR3 Dagger bdagger hellip14daggerUnder zero-pressure conditions using expressions for a and b from appendix A andassuming that R frac34 rc we regnd that

~VVrelhellipRdagger ˆ Vrel 1 iexcl 2hellip1 iexcl 2cedildaggerr3c

hellip1 Dagger cedildaggerR3

Aacute

hellip15dagger

where

Vrel ˆ 12phellip1 iexcl cedildaggerucr2c

1 Dagger cedilhellip16dagger

is the relaxation volume of the defect in an inregnitely large system (Eshelby 1956Flynn 1972) It is reasonable to assume that r3

c=R3 ordm Nc=N Nc being the number ofatoms in the defect core Then equation (15) suggests that the absolute value of therelaxation volume evaluated on a regnite system is slightly lower than its true absolutevalue and tends to that value as 1=N N 1

33 Defect formation energyWe shall now examine the defect energy The strain energy density around the

defect is given by the general expression (Landau and Lifshitz 1986)

Whelliprdagger ˆ hellip1 iexcl cedildaggerED2

2hellip1 Dagger cedildaggerhellip1 iexcl 2cedildagger iexclE 2

sup3sup3 Dagger 2rrsup3sup3

iexcl cent

1 Dagger cedil hellip17dagger

Using equations (12) we obtain

Whelliprdagger ˆ 3Ea2

2hellip1 iexcl 2cedildagger Dagger 3Eb2

hellip1 Dagger cedildaggerr6 hellip18dagger

The elastic strain energy stored in the sphere equals

~EEelhellipRdagger ˆ 4phellipR

rc

W helliprdaggerr2 dr ˆ 4pE R3 iexcl r3c

iexcl cent a2

2hellip1 iexcl 2cedildagger Dagger b2

hellip1 Dagger cedildaggerr3cR3

Aacute

hellip19dagger

In the limit of an inregnitely large system we have a 0 and b ucr2c (see appendix

A) and equation (18) gives Whelliprdagger 1=r6 (Hatcher et al 1979) In this limiting caseequation (19) reduces to the well-known expression (for example Eshelby (1956))

Eel ˆ hellip1 Dagger cedildaggerEV2rel

27hellip1 iexcl cedildagger2V0

hellip20dagger

where V0 ˆ 4pr3c=3 is the defect core volume In a regnite system ~EEelhellipRdagger will depend on

the boundary condition at r ˆ R Using the expressions for a and b from appendix Aand limiting our calculations to linear terms in helliprc=Rdagger3 we regnd that


~EEelphellipRdagger ˆ Eel 1 iexcl 3hellip1 iexcl cedildaggerr3c


Aacute hellip21dagger

for p ˆ 0 and

~EEelVhellipRdagger ˆ Eel 1 Dagger3hellip1 iexcl cedildaggerr3

c

2hellip1 iexcl 2cedildaggerR3

Aacute

hellip22dagger

for V ˆ constant These equations conregrm that as the block size increases thedefect formation energy tends to the same value Ec Dagger Eel regardless of the boundarycondition The asymptotic behaviour is always 1=N but the slope d ~EE=dhellip1=Ndagger andeven its sign are di erent for p ˆ 0 and V ˆ constant Relaxation under a regxedvolume slightly overestimates the defect energy while relaxation under zero pressureslightly underestimates the defect energy In other words we always have

~EEp 4 E 4 ~EEV hellip23daggerwhere ~EEp and ~EEV are estimates of the true defect energy E obtained on the samesimulation block under p ˆ 0 and V ˆ constant conditions respectively Thisrelation does not depend on the sign of Vrel and is valid for both vacancies(Vrel lt 0) and interstitials (Vrel gt 0) Note that the ratio of the slopes d ~EEp=dhellip1=Ndaggerand d ~EEV =dhellip1=Ndagger depends on the elastic properties of the lattice only

d ~EEp

d ~EEV

ˆ iexcl2hellip1 iexcl 2cedildagger


This relation assumes that the defect core energy does not depend on the boundaryconditions

34 Defect formation entropy

341 General expressionsTurning now to the defect entropy the general expression for the density of

entropy associated with the elastic strain is (Landau and Lifshitz 1986)

S ˆ shy BD hellip25daggerIn the context of the cluster method we are interested in the total elastic entropy of asphere with an intermediate radius r

such that rc lt r

4 R (reggure 1) For this

`clusterrsquo entropy we have

~SSelhellipr Rdagger ˆ 4phelliprcurren

rc

Sr2 dr ˆ 4pshy Bahellipr3 iexcl r3

cdagger hellip26dagger

where we used equation (13) for D Note that the R dependence enters equation (26)through the coe cient a Using the expressions for a (appendix A) we regnd in the regrstorder in helliprc=Rdagger3

~SSelphellipr Rdagger ˆ Selp

r3 iexcl r3

c

R31 iexcl

2hellip1 iexcl 2cedildaggerr3c


Aacute

hellip27dagger

for p ˆ 0 and

2598 Y Mishin et al

~SSelVhellipr Rdagger ˆ SelV

r3 iexcl r3

c

R31 Dagger r3

c

R3


for V ˆ constant In these equations

Selp ˆ 2hellip1 iexcl 2cedildagger3hellip1 iexcl cedildagger shy BVrel hellip29dagger

and

SelV ˆ iexcl 1 Dagger cedil

3hellip1 iexcl cedildagger shy BVrel hellip30dagger

are the asymptotic values of ~SSelhellipr Rdagger when r

ˆ R 1 Obviously they have themeaning of the elastic parts of defect formation entropy in an inregnitely large systemunder p ˆ 0 and V ˆ constant conditions respectively Note that they satisfy therelation

Selp iexcl SelV ˆ shy BVrel hellip31dagger

which is identical with equation (9) provided that the core entropy is independent ofthe boundary condition Thus our elastic model provides more speciregc expressionsfor Sp and SV than equation (9) namely

Sp ˆ Sc Dagger 2hellip1 iexcl 2cedildagger3hellip1 iexcl cedildagger shy BVrel hellip32dagger

and

SV ˆ Sc iexcl 1 Dagger cedil


342 Supercell methodThe case of r

ˆ R frac34 rc corresponds to the supercell method of atomistic simu-

lations In this case when R 1 the entropy tends to either Sp or SV depending onthe boundary condition More precisely to regrst order in the small parameter helliprc=Rdagger3equations (27) and (28) give

~SSelphellipR Rdagger ˆ Selp 1 iexcl 3hellip1 iexcl cedildaggerr3c



for p ˆ 0 and

~SSelVhellipR Rdagger ˆ SelV hellip35dagger

for V ˆ constant In the latter case regnite-size corrections are predicted to be of theorder of helliprc=Rdagger6 or higher These equations suggest that the defect entropy deter-mined by the supercell method must converge very rapidly that is at least as fast asthe defect energy Equation (34) also suggests that under zero-pressure conditionsthe obtained entropy can be reregned by a linear extrapolation of the plot ~SS versus1=N to 1=N 0 In practice this turns out to be also valid for the constant-volumecondition (sect 43)


343 Embedded-cluster methodThe embedded-cluster scheme of atomistic simulations corresponds to

rc frac12 r frac12 R In this case equations (27) and (28) become

~SSelphellipr Rdagger ˆrR

sup3 acute3

Selp hellip36dagger

and

~SSelVhellipr Rdagger ˆ

r

R

sup3 acute3

SelV hellip37dagger

respectively These equations show that the elastic part of the entropy determined bythe embedded-cluster method is approximately proportional to N=N In the limit ofa relatively small cluster or a relatively large supercell (N

=N 0) this methodconverges to the defect core entropy In the limit of 1=N 0 (Hatcher et al1979) the entropy diverges which casts doubts on the extrapolation scheme usedpreviously (Hatcher et al 1979 Fernandez and Monti 1993 Debiaggi et al 1996Fernandez et al 1996) Instead the true elastic entropy of the defect can be obtainedas the slope

d ~SS

dhellipN=Ndagger hellip38dagger

Knowing also Sc from the extrapolation to N=N 0 we can determine the full

entropy S from equation (8) This scheme is essentially equivalent to a linear extra-polation of the plot ~SS versus N=N to N=N 1 that is extrapolation of theembedded-cluster scheme to its supercell limit This extrapolation has recentlybeen discussed by Fernandez et al (2000)

4 Atomistic simulations

41 Simulation proceduresAtomistic simulations were performed for a vacancy and an interstitial in copper

using an embedded-atom potential constructed by Voter (1993 1998) The potentialgeneration procedure was described elsewhere (Voter 1993 1994) The potential isregtted to reproduce accurately experimental values of the equilibrium lattice periodcohesive energy elastic constants vacancy formation energy and other properties ofcopper The elastic constants cij and bulk modulus B ˆ hellipc11 Dagger 2c12dagger=3 predicted bythe potential are given in table 1

For comparison with the isotropic elasticity calculations (sect 3) we need to evaluatePoissonrsquos ratio for copper Because the elastic properties of copper are strongly

2600 Y Mishin et al

Table 1 Elastic constants and moduli and thermal expansion of copper predicted by theembedded atom potential used in this work (Voter 1993)

shy pound 10iexcl5

c11 c12 c44 B middot

(eV AEcirc iexcl3) (eV AEcirc iexcl3) (eV AEcirc iexcl3) (eV AEcirc iexcl3) (eV AEcirc iexcl3) cedil LHA FHA

11192 07690 05057 08857 03404 03466 5621 5479

anisotropic (anisotropy ratio A ˆ 2c44=hellipc11 iexcl c12dagger ordm 289) the shear modulus middot andPoissonrsquos ratio cedil signiregcantly depend on the crystallographic direction In particular middotvaries between middot ˆ hellipc11 iexcl c12dagger=2 ˆ 01751 eV AEcirc iexcl3 and middot ˆ c44 ˆ 05057 eV AEcirc iexcl3 Weestimated middot as an average between these values that is middot ˆ 03404eV AEcirc iexcl3 Using thisvalue of middot we obtained cedil as cedil ˆ c12=2hellipmiddot Dagger c12dagger ordm 03466

The thermal expansion factor shy at T 0 was determined from the thermody-namic relation (Landau and Lifshitz 1980)

qS

qV

sup3 acute

T

ˆ shy B hellip39dagger

The entropy S of a periodic perfect-lattice block was calculated from equation (5)under various volume expansions ranging from 0 to 05 and the derivative inequation (39) was evaluated by a linear approximation of the plot of S versus V The calculation were performed in both the FHA and the LHA approximations ineach case for various block sizes from N ˆ 4 to N ˆ 864 The shy values obtained onblocks with N 5 32 were found to be almost identical so the results for N ˆ 864were taken as converged values and used in all further calculations (table 1)

The point defects were simulated by molecular statics in a cubic block (supercell)with periodic boundary conditions The interstitial was considered in a split [100]dumbbell conregguration which was found to be the lowest-energy conregguration Thedefect formation energies were calculated as discussed in sect 1 The defect entropieswere calculated by both the supercell and the embedded-cluster methods in eachcase in both the LHA and the FHA approximations The results are presentedbelow

42 Point-defect energies and relaxation volumesThe point-defect energies were calculated for blocks containing from N ˆ 32 to

N ˆ 5324 atoms The obtained energies plotted against 1=N show excellent linearityfor N gt 200 (reggure 2) Moreover as 1=N 0 the energies obtained under zero-pressure and constant-volume conditions tend to the same value which can beidentireged as the defect formation energy in an inregnite crystal This behaviour is inagreement with equations (21) and (22) derived from the elastic model There is evena reasonable quantitative agreement between the derivatives d ~EEp=d ~EEV determinedfrom the plots and calculated from equation (24) (table 2) This agreement conregrmsthat the size dependence of calculated defect formation energies in metals arisesprimarily from elastic interactions of the defects with the block boundaries andthat this dependence can be e ectively eliminated by a linear extrapolation to1=N 0 Figure 2 also shows that the volume relaxation of the block does notguarantee signiregcantly more accurate energies One can save computational e ortsby carrying out a set of constant-volume calculations followed by the linear extra-polation to 1=N 0

The relaxation volumes were computed as functions of N (with N up to 4000)both directly and from the equation Vr=O0 ˆ iexclP=B where P is the sum of hydro-static stresses on all atoms after the relaxation at V ˆ constant and O0 is the atomicvolume (The stresses were calculated from the virial expression at T ˆ 0) ForN gt 200 both calculation methods gave very similar volumes which behaved line-arly with 1=N as predicted by equation (15) The relaxation volumes extrapolated to1=N 0 are given in table 2


2602 Y Mishin et al

Figure 2 Calculated formation energies of (a) a vacancy and (b) an interstitial in copper asfunctions of the inverse number 1N of atoms in the simulation block Two types ofboundary condition were considered p ˆ 0 and V ˆ constant The energies relating toa macroscopic crystal are obtained by a linear extrapolation to 1=N 0

Table 2 Calculated formation energies E relaxation volumes Vrel and

derivatives d ~EEp=d ~EEV for a vacancy and an interstitial in copper O0

is the atomic volume

d ~EEp=d ~EEV

EDefect (eV) Vrel=O0 Simulation Equation (24)

Vacancy 12580 iexcl02575 iexcl03987 iexcl04557Interstitial 32372 20608 iexcl04659 iexcl04557

43 Defect entropies from the supercell methodSupercell calculations of defect entropies were carried out for N ˆ 32 108 256

500 and 864 The LHA calculations additionally included blocks with N ˆ 1372 and2048 The obtained entropies show a good linearity when plotted versus 1=N forboth types of boundary condition (reggure 3) The entropies obtained by a linearextrapolation to 1=N 0 are listed in table 3 (the point for N ˆ 32 was not includedin the regt) This table also compares the obtained di erences Sp iexcl SV with the pro-ducts shy BVrel appearing in equation (9) This comparison shows that equation (9)works very well for both vacancies and interstitials Overall the supercell methodappears to be very accurate and reliable Already a block with a few hundred atomsis su cient for obtaining a good estimate of the defect entropy plus this estimate canbe further reregned by applying the linear extrapolation scheme Figure 3 demon-strates that the LHA provides only a poor approximation of defect entropies

Although the observed 1=N convergence of entropy is in qualitative agreementwith the elasticity analysis (sect 3) quantitative agreement with equations (34) and (35)regarding the slopes of the lines was not observed and in fact was not even expectedin view of the approximate character of those equations As an additional complica-tion in FHA calculations even the perfect-lattice entropy per atom slightly dependson the block size In fact similarly to the defect entropy the perfect-lattice entropywas found to behave fairly linearly when plotted against 1=N Although the reasonsfor this linear behaviour are di erent from those for the defect entropy the variationin the perfect-lattice entropy with the block size could have a ected the slopes of theplots in reggure 3 Despite this discrepancy with linear-elasticity equations regardingthe slopes of the lines the supercell method provides a high accuracy and fastconvergence of calculations In the following the defect entropies obtained by thismethod will be used as reference values for the evaluation of other methods

44 Defect entropies from the embedded-cluster methodEmbedded-cluster calculations of the defect entropies were performed for a large

set of block and cluster sizes typically with N ˆ 2048 to N ˆ 16 384 and N

ˆ 249to N

ˆ 1289 In reggure 4 we plot the whole set of vacancy and interstitial entropiescalculated in the FHA as functions of N=N The observed linearity conregrms equa-tions (36) and (37) Because the elastic entropies Selp and SelV are proportional toVrel (see equations (29) and (30)) the slopes of the plots must have di erent signs fora vacancy and an interstitial under the same boundary condition which is indeedobserved in reggure 4 Importantly despite the scatter of the points the entropiesobtained under di erent boundary conditions converge as 1=N 0 to the sameentropy Sc within sect0005kB This excellent agreement demonstrates that the defectcore entropy is a well-deregned physical quantity that can be determined very accu-rately by the embedded-cluster method Similar calculations were performed in theLHA The linearity of the plots and convergence to the same core entropy regardlessof the boundary conditions were followed with even better accuracy

Using the slopes of the plots in reggure 4 the defect entropies Sp and SV werecalculated by the scheme described in sect 343 Defect entropies were also calculateddirectly from equations (32) and (33) using the obtained Sc values The results arelisted in table 4 in comparison with defect entropies obtained by the supercellmethod Note that the entropies determined from the slopes equation (38) dofollow the right trends but are not very accurate Surprisingly even equations (32)and (33) seem to work better despite the underlying assumptions and the uncertainty


2604 Y Mishin et al

(a)

(b)

Figure 3 Calculated (a) vacancy and (b) interstitial formation entropies in copper as func-tions of the inverse number 1N of atoms in the simulation block Calculations wereperformed by the supercell method in the FHA and LHA Two types of boundarycondition were considered p ˆ 0 and V ˆ constant The entropies relating to a macro-scopic crystal are obtained by a linear extrapolation to 1=N 0


Table 3 Vacancy and interstitial formation entropies in copper calculated by the

supercell method in the FHA The di erences Sp iexcl SV and the productsshy BVrel are also listed for comparison (see equation (9))

Sp SV Sp iexcl SV shy BVrel

Defect (units of kB) (units of kB) (units of kB) (units of kB)

Vacancy 1231 2969 iexcl1738 iexcl1713Interstitial 11096 iexcl2791 13888 13707

Figure 4 Formation entropies of (a) a vacancy and (b) an interstitial in copper calculated bythe embedded-cluster method in the FHA The entropies are plotted as functionsof the ratio N=N (N is the number of atoms in the cluster and N the numberof atoms in the block) Two types of boundary condition were considered p ˆ 0and V ˆ constant In the limit N=N 0 the entropy converges to the defect coreentropy regardless of the boundary condition

in Poissonrsquos ratio These discrepancies indicate that the slopes dS=dhellipN=Ndagger area ected by factors lying beyond our elastic model such as the cubic shape of theblock and the elastic anisotropy of the lattice Overall the embedded-cluster methodis less accurate than the supercell method even for a single defect

5 Elastically corrected embedded-cluster methodThe main problem with the original embedded-cluster method is that a signireg-

cant part of the defect formation entropy is stored in lattice regions outside thecluster In order to recover that part one has to repeat the calculations for di erentcluster and block sizes and apply a linear extrapolation to N

=N 1 Because thisextrapolation relies on the spherical symmetry and other approximations themethod is not only computationally expensive but rather inaccurate too

We propose an alternative version of the embedded-cluster method that cures itsmain problem by including the outside part of the entropy in a quasicontinuumapproximation Indeed returning for a moment to the spherical model of sect 3 thefull entropy of defect formation equals

S ˆ Sc Dagger 4phelliprcurren

rc

Sr2 dr Dagger 4phellipR

rcurren

Sr2 dr hellip40dagger

The regrst two terms represent the entropy change of an embedded cluster and inatomistic calculations are approximated by the harmonic entropy ~SSNcurren

given byequation (10) The last term represents the elastic entropy associated with distortedlattice regions around the cluster If the lattice distortions are small the associatedentropy density S is proportional to the hydrostatic strain D (see equation (25))Assuming that the lattice regions follow linear elasticity their entropy densitybecomes

S ˆ shy p hellip41daggerwhere p ˆ BD is the local hydrostatic stress We emphasize that this relation does notpresume that the lattice is elastically isotropic

The idea of our method is to apply the elastic continuum expression (41) to thediscrete lattice regions surrounding the cluster Then the third term in equation (40)can be approximated by the sum

Sq ˆ shyX

not

pnotOnot hellip42dagger

over all lattice sites not lying outside the cluster In this sum pnot are local hydrostaticstresses and Onot local atomic volumes assigned to individual sites Although a deregni-

2606 Y Mishin et al

Table 4 Vacancy and interstitial formation entropies in copper calculated by the cluster

method in the FHA The entropies determined by the supercell method and calculatedfrom equations (32) and (33) are listed for comparison

Sp (units of kB) SV (units of kB)

Sc Equation EquationDefect (units of kB) Cluster (32) Supercell Cluster (33) Supercell

Vacancy 1729 0985 1193 1231 2718 2906 2969Interstitial 5752 14145 10042 11096 0178 iexcl3665 iexcl2791

tion of local atomic volume is not unique (Sutton and Vitek 1982 1983) we assumethat these volumes are chosen in such a way that the overall volume of a latticeregion equals the sum of individual atomic volumes Importantly despite the uncer-tainty in local atomic volumes within the embedded-atom method each productpnotOnot is determined in a unique manner (for example Foiles and Adams (1989))For a monatomic metal the quasicontinuum expression (42) can be calculated as

Sq ˆ shy

3

X

not

X

not 6ˆreg

permil12iquest 0hellipRnotregdagger Dagger F 0hellipraquonotdaggerraquo0hellipRnotregdaggerŠRnotreg hellip43dagger

where the second sum runs over all atoms reg other than atom not In this expression iquestand raquo are the pair interaction energy and electron density respectively as functions ofthe interatomic distance Rnotreg Fhellipraquodagger is the embedding function and

raquonot ˆX

reg 6ˆnot

raquohellipRnotregdagger hellip44dagger

is the total electron density on atom not Equation (43) also holds for a pair potentialby omitting the embedding-energy term

Thus in our method the total point-defect entropy is calculated as the sum

S ˆ ~SSNcurrenDagger Sq hellip45dagger

with ~SSNcurrenand Sq given by equations (10) and (43) respectively We emphasize that

although this method was inspired by the spherical-model expression (40) the regnalequation (45) does not require a spherical symmetry of the strain regeld Not only theblock but also the cluster can have any other shape (eg a cubic shape) Howeverthis method does assume that the strained lattice outside the cluster complies withthe linear-elasticity approximation and the linear relation between the vibrationalentropy and hydrostatic stress It also assumes that gradients of the hydrostaticstress outside the cluster are small enough to justify the continuum approximationIn contrast with the supercell method this new method can be readily applied tolarge simulation blocks because the computation of Sq is a fast linear-N procedureAn advantage over the original embedded-cluster method is that the total defectentropy is obtained from one computational run and does not have to be derivedby extrapolation of cluster entropies computed by multiple runs

The new method was tested by conducting many entropy calculations for avacancy and an interstitial in copper with various block and cluster sizes and theresults were very encouraging For example table 5 summarizes our results for a


Table 5 Vacancy and interstitial formation entropies in copper calculated in the FHA by the

elastically corrected embedded-cluster method The calculations were performed on a5324-atom cubic supercell with periodic boundary conditions containing a spherical

cluster with N

ˆ 459 627 and 767 atoms The quasicontinuum entropy Sq (see

equation (43)) was computed with the thermal expansion factor shy ˆ 5479 pound 10iexcl5 The entropies determined by the supercell method are listed for comparison

Sp (units of kB) SV (units of kBdaggerDefect N

ˆ 459 627 767 Supercell N

ˆ 459 627 767 Supercell

Vacancy 1177 1195 1193 1231 2916 2935 2934 2969Interstitial 10563 10642 10722 11096 iexcl3170 iexcl3097 iexcl3017 iexcl2791

5324-atom cubic supercell with three di erent sizes of a spherical cluster If thecluster size is large enough (Ncurren ˆ 500plusmn800) the point-defect entropy almost doesnot change with N The obtained entropy values are close to those found in sect 43 bythe supercell method (cf table 3) A more detailed cluster-size dependence of thedefect entropy obtained by this `elastically correctedrsquo embedded-cluster method isillustrated in reggure 5 For N

gt 100 the vacancy entropy follows a linear depen-dence on 1=N

and extrapolates to 1212kB in excellent agreement with the supercell

result 1231kB The same agreement was found for an interstitial in copper Theagreement improves still further as we increase the block size

2608 Y Mishin et al

Figure 5 Summary of vacancy formation entropies in copper at p ˆ 0 as functions of theinverse number 1N of dynamic atoms in the simulation block Calculations wereperformed in the FHA by the supercell method () original embedded-clustermethod (~) and elastically corrected embedded-cluster method (amp) The embedded-cluster calculations were carried out on a 5324-atom cubic supercell with periodicboundary conditions The entropies calculated in the LHA () are shown for com-parison

6 Discussion and conclusionsIn this paper problems associated with entropy calculations were analysed for

individual point defects in metals However those problems are quite general andhave signiregcant importance in many other situations For example in calculations ofjump frequencies of point defects from the harmonic transition state theory(Vineyard 1957) the jump attempt frequency involves the so-called migrationentropy of a defect The latter is calculated by basically the same scheme as thedefect formation entropy and is therefore also vulnerable to regnite-size e ects andboundary conditions In this case however the e ect of boundary conditions can beless signiregcant than for defect formation because the number of atoms in the systemis conserved and the volume e ect is smaller As another example the stability ofpoint-defect clusters arising under irradiation or upon quenching depends on theirentropy which in this case includes contributions associated with defectplusmndefectinteractions

In principle defect formation entropies can be calculated at regnite temperaturesusing molecular dynamics or Monte Carlo methods (De Lorenzi and Jacucci 1986Foiles 1994) However because such calculations are complex and computationallyexpensive routine calculations are performed by combining molecular statics withthe harmonic approximation to atomic vibrations Within this scheme there are twodi erent approaches to entropy calculations called the supercell and the embedded-cluster methods

The supercell method is the most straightforward and by far the most accurate Itis ideal for a single defect which can be conveniently simulated with a block contain-ing around 103 atoms In more complex situations that require signiregcantly largerblocks the application of this method is hampered by numerical problems associatedwith handling large dynamic matrices For such situations the original embedded-cluster method previously o ered the only reasonable alternative Our elasticityanalysis shows that the defect entropy determined by the embedded-cluster methodequals approximately

S ˆ Sc DaggerN

N

Sel hellip46dagger

Sc being the defect core entropy Sel the elastic part of the entropy and N thenumber of atoms in the cluster This expression includes only a part of the totalelastic entropy Sel in proportion to N

=N Therefore the entropies delivered by themethod are essentially incomplete and must be subject to further processingNamely the entropies must be computed for a set of di erent block andorcluster sizes and the results must be plotted against N

=N Then a linear extra-polation to N

=N 0 gives the core entropy while a linear extrapolation toN

=N 1 allows us to recover the full entropy Sc Dagger Sel In practice the accuracyof the method is not high because equation (46) relies on strong approximationssuch as spherical symmetry of the elastic regeld around the defect and the simula-tion block Under real conditions these assumptions are not fulreglled even for asingle defect On the other hand the method provides a very accurate estimate ofthe core entropy of an individual defect which is a useful quantity by itselfUnlike the full defect formation entropy the defect core entropy does not dependsensitively on the external conditions or the system size and characterizes thedegree of local atomic distortions in the defect core region Note that the defect


core energy cannot be calculated as easily because of the singularity of the elasticstrain-energy density at r 0

As an alternative to the supercell and the embedded-cluster methods we haveproposed a new elastically corrected embedded-cluster method that seems to alle-viate e ectively the problems inherent in both previous methods That is as in thesupercell method the defect entropy is determined from the entropy change of thewhole set of free atoms However this entropy change is not calculated exactly but israther split into two parts (equation (45)) that are calculated in di erent ways Theregrst part is the entropy change ~SSNcurren

of an N-atom cluster containing the defect This

part can be determined from equation (10) as in the original embedded-clustermethod but it can also be calculated by any other atomistic simulation methodbeyond the harmonic approximation for example by the Monte Carlo method(Foiles 1994) The second part Sq is the entropy change of the lattice regionssurrounding the cluster This part is calculated from equation (42) which is basedon a quasicontinuum approximation of the strained-lattice entropy This calculationassumes that the strained lattice around the cluster follows the linear-elasticityapproximation and the linear relation (41) between the entropy and local hydrostaticstress Since the calculation of Sq is a fast linear-N procedure the computation timeof the method is dominated by the calculation of ~SSNcurren

In other words the entropyassociated with the lattice regions around the defect is included at almost no cost Incontrast with the original embedded-cluster method the total defect entropy isobtained by making just one computational run The obtained entropy can befurther reregned by a linear extrapolation to 1=N

0 with a regxed N as illustrated

in reggure 5 These computational advantages make the elastically correctedembedded-cluster method applicable to much larger simulation blocks which inturn may allow us to perform entropy calculations for more complex systemsincluding those involving multiple length scales For example the new method canbe useful in modelling point defects in or near grain boundaries dislocations andother extended defects

Figure 5 summarizes the vacancy formation entropies in copper calculated bydi erent methods as functions of the inverse number of dynamic atoms in the simu-lation block This reggure illustrates that for a given number of dynamic atomsN

gt 500 (and thus a given amount of computational e orts) the elastically cor-rected embedded-cluster method provides an accuracy comparable to that of thesupercell method but can be applied to much larger simulation blocks The entropiesobtained by the original embedded-cluster method show a larger scatter of datapoints and do not approach the defect formation entropy at least not within therange of accessible cluster sizes As was shown by Fernandez et al (2000) andconregrmed in this paper the entropy predicted by this method diverges (tends toDagger1 or iexcl1) when extrapolated to 1=N 0 Figure 5 also demonstrates that theLHA although very fast provides a very poor approximation of the defect entropy

In the elastically corrected embedded-cluster method the cluster must encom-pass all atoms around the defect core that experience signiregcant (nonlinear) dis-placements from their perfect-lattice positions Indeed the idea behind the basicrelations (25) and (41) assumed in this method is that the entropy density S of auniformly strained crystal being a scalar quantity can depend only on geometricinvariants of the strain tensor If the strain is small S must be proportional to theregrst-order invariant that is to the volume deformation D or assuming linear elas-ticity to the hydrostatic stress p For higher strain levels equation (25) would need

2610 Y Mishin et al

to be modireged by including second- or higher-order invariants while equation (41)might need to include nonlinear elasticity corrections Moreover even for su cientlysmall strains it should be remembered that the quasicontinuum approximation (42)neglects strain gradients which makes it similar in spirit to the local-density approx-imation in electronic theory (Kohn and Shan 1965) Our quasicontinuum approx-imation can in principle be improved by including strain-gradient corrections Suchhigher-order quasicontinuum approximations involving nonlinear e ects and gradi-ent corrections lie beyond the scope of this paper

ACKNOWLEDGEMENTS

Y M wishes to thank the US Department of Energy (DOE) O ce of BasicEnergy Sciences (BES) for the regnancial support of this work (contract DE-FG02-99ER45769) A F V is also grateful to DOE BES for regnancial support Thework at Los Alamos National Laboratory was supported by DOE under contractW-7405-ENG-36 (M R S and A F V)

APPENDIX A

EL A STI C STR A IN FI EL D A RO U N D A P O IN T D E F EC T

The elastic displacement regeld around a point defect located in the centre of asphere of radius R (reggure 1) is given by (Eshelby 1956 Landau and Lifshitz 1986)


r2 hellipA 1dagger

with the boundary condition uhelliprcdagger ˆ uc The regrst term in this equation appearsowing to image forces which also make a and b functions of R These functionsdepend on the boundary condition at r ˆ R

If the external pressure is zero (p ˆ 0) the boundary condition at r ˆ R is zeroradial stress

frac14rrhellipRdagger ˆ 0 hellipA 2daggerUsing the appropriate expression for frac14rr we regnd that

a ˆ 2hellip1 iexcl 2cedildaggerucr2c

hellip1 Dagger cedildaggerR31 Dagger



Aacute iexcl1

hellipA 3dagger

b ˆ ucr2c 1 Dagger 2hellip1 iexcl 2cedildaggerr3

c


Aacute iexcl1

hellipA 4dagger

If the volume of the sphere is regxed (V ˆ constant) the boundary condition is

uhellipRdagger ˆ 0 hellipA 5daggerand we obtain

a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger


b ˆ ucr2c 1 iexcl r3

c

R3

Aacute iexcl1

hellipA 7dagger

Considering helliprc=Rdagger3 as a small parameter and limiting our calculations to linearterms in that parameter the above equations become


hellip1 Dagger cedildaggerR31 iexcl 2hellip1 iexcl 2cedildaggerr3

c


Aacute

hellipA 8dagger

b ˆ ucr2c 1 iexcl 2hellip1 iexcl 2cedildaggerr3

c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger

for V ˆ constant We note that for an inregnitely large system (R 1) these equa-tions give a ˆ 0 and b ˆ ucr

2c

ReferencesCatlow C R A 1997 New Trends in Materials Chemistry edited by C R A Catlow and

A Cheetham (Dordrecht Kluwer) p 141Catlow C R A Corish J Jacobs P W M and Lidiard A B 1981 J Phys C 14

L121Debiaggi S B Decorte P M and Monti A M 1996 Phys Stat sol (b) 195 37De Lorenzi G and Jacucci G 1986 Phys Rev B 33 1993Eshelby J D 1956 Solid St Phys 3 79Fernandez J R and Monti A M 1993 Phys Stat sol (b) 179 337Fernandez J R Monti A M and Pasianot R C 1996 J nucl Mater 229 1 2000

Phys status solidi (b) 219 245Finnis M W and Sachdev M 1976 J Phys F 6 965Flynn C P 1972 Point Defects and Di usion (Oxford Clarendon)Foiles S M 1994 Phys Rev B 49 14 930Foiles S M and Adams J B 1989 Phys Rev B 40 5909Frank W Breier U ElsaEgrave sser C and FaEgrave hnle M 1996 Phsy Rev Lett 77 518Gillan M J 1981 Phil Mag A 43 301Harding J H 1990 Rep Prog Phys 53 1403Hatcher R D Zeller R and Dederichs P H 1979 Phys Rev B 19 5083Kohn W and Sham L J 1965 Phys Rev 140 A1133Landau L D and Lifshitz E M 1980 Statistical Physics Part 1 third edition (Oxford

Pergamon) 1986 Theory of Elasticity third edition (Oxford Pergamon)LeSar R Najafabadi R and Srolovitz D J 1989 Phys Rev Lett 63 624Sutton A P 1989 Phil Mag A 60 147 1992 Phil Trans R Soc A 341 233Sutton A P and Vitek V 1982 Acta metall 30 2011 1983 Phil Trans R Soc A 309

1Vineyard G H 1957 J Phys Chem Solids 3 121Voter A F 1993 Unclassireged Technical Report LA-UR 93-3901 Los Alamos National

Laboratory 1994 Intermetallic Compounds Vol 1 (New York Wiley) chapter 4 p77 1998 Phys Rev B 57 13 985

2612 Calculation of point-defect entropy in metals

where T is the temperature E and S are the defect formation energy and entropyrespectively and kB is the Boltzmann constant Given a suitable model of atomicinteractions E and S can be determined by atomistic calculations in the zero-tem-perature limit using molecular statics and the harmonic approximation to atomicvibrations

The physical quantities of interest are the defect energy and entropy in a macro-scopic crystal under constant-pressure (usually zero-pressure) conditions Howeveratomistic calculations of these quantities are limited to relatively small systems withapproximately 103 atoms because the diagonalization of large dynamic matrices iscomputationally demanding and associated with numerical problems For systems ofthis size results of atomistic calculations depend on the system size geometry andboundary conditions How should such calculations be performed in order to obtainaccurate estimates of the defect properties in a macroscopic crystal This question isthe main topic of the present paper

Di erent problems associated with defect entropy calculations have beenreviewed by Catlow (1997) Foiles (1994) and Harding (1990) Most of the previouswork was done on ionic solids in which the electric charges of defects have a sig-niregcant e ect on entropy calculations In this paper we shall be concerned withmetallic systems in which point defects bear no electric charge This will makeour analysis less complex and will allow us to correlate our atomistic simulationresults with simple calculations based on linear elasticity theory Previous atomisticcalculations of vacancy formation entropies in metals were undertaken by severalworkers (for example Hatcher et al (1979) Fernandez and Monti (1993) Debiaggiet al (1996) Fernandez et al (1996 2000) and Frank et al (1996)) There are twodi erent approaches to such calculations which will be referred to in this paper asthe `supercell methodrsquo and the `embedded-cluster methodrsquo In the supercell methodall free atoms (ie atoms allowed to move during the energy minimization) are at thesame time dynamic atoms (ie their vibrations are taken into account in the defectentropy calculations)y In the embedded-cluster method only some of the freeatoms namely those forming a cluster centred a the defect core are considered asdynamic while all other free atoms are treated as static

In this paper we investigate the supercell and embedded-cluster methods andpropose an alternative approach to defect entropy calculations In sect 2 we explainhow entropy calculations have previously been conducted within the supercell andembedded-cluster methods In sect 3 we undertake a simple linear-elasticity analysis ofthe energy and entropy associated with the elastic strain regeld around a point defectWe examine how these quantities depend on the overall system size cluster size andthe boundary conditions imposed on the system In sect 4 these dependences are com-pared with atomistic simulation results performed for vacancies and interstitials incopper These simulations are also used for testing the accuracy of the entropycalculation methods considered in this paper In sect 5 we present our new methodfor defect entropy calculations a so-called elastically corrected embedded-clustermethod This method can be viewed as an extension of the embedded-cluster methodin which the entropy of the lattice regions outside the cluster is included in a quasi-

2592 Y Mishin et al

Often especially in regrst-principles calculations the term `supercellrsquo refers to a periodicblock In this paper we use this term whenever all relaxed atoms are included in the entropycalculation irrespective of the boundary conditions






hellip2dagger










S ˆ iexclkB

X3Niexcl3

nˆ1

lnhcediln

kBT




nˆ1

cediln

Y3Niexcl3

nˆ1

cedil0n


5

2

4 hellip6dagger




Dinotjreg ˆ 1

mnotmreg

iexcl cent1=2

q2E

qRinot qRjreg

hellip7dagger






2594 Y Mishin et al






Y3Ncurreniexcl3

nˆ1

cediln

Y3Ncurren

nˆ1

cedil0n


5

2

4 hellip10dagger










r2 hellip11dagger



rr ˆ a iexcl 2b


r3 hellip12dagger



2596 Y Mishin et al




defect equals




Aacute

hellip15dagger

where










iexcl cent








rc


iexcl cent a2



Aacute

hellip19dagger





hellip20dagger







for p ˆ 0 and


c


Aacute

hellip22dagger



d ~EEp

d ~EEV








such that rc lt r




rc





r3 iexcl r3

c

R31 iexcl



Aacute

hellip27dagger

for p ˆ 0 and

2598 Y Mishin et al


r3 iexcl r3

c

R31 Dagger r3

c

R3




and








and









for p ˆ 0 and







sup3 acute3

Selp hellip36dagger

and


r

R

sup3 acute3

SelV hellip37dagger



d ~SS








2600 Y Mishin et al


shy pound 10iexcl5



11192 07690 05057 08857 03404 03466 5621 5479



qS

qV

sup3 acute

T








2602 Y Mishin et al





d ~EEp=d ~EEV








ˆ 249to N




2604 Y Mishin et al

(a)

(b)















rc


rcurren






Sq ˆ shyX

not



2606 Y Mishin et al







Sq ˆ shy

3

X

not

X

not 6ˆreg



raquonot ˆX

reg 6ˆnot











cluster with N











2608 Y Mishin et al






S ˆ Sc DaggerN

N

Sel hellip46dagger

















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c














hellip2dagger










S ˆ iexclkB

X3Niexcl3

nˆ1

lnhcediln

kBT




nˆ1

cediln

Y3Niexcl3

nˆ1

cedil0n


5

2

4 hellip6dagger




Dinotjreg ˆ 1

mnotmreg

iexcl cent1=2

q2E

qRinot qRjreg

hellip7dagger






2594 Y Mishin et al






Y3Ncurreniexcl3

nˆ1

cediln

Y3Ncurren

nˆ1

cedil0n


5

2

4 hellip10dagger










r2 hellip11dagger



rr ˆ a iexcl 2b


r3 hellip12dagger



2596 Y Mishin et al




defect equals




Aacute

hellip15dagger

where










iexcl cent








rc


iexcl cent a2



Aacute

hellip19dagger





hellip20dagger







for p ˆ 0 and


c


Aacute

hellip22dagger



d ~EEp

d ~EEV








such that rc lt r




rc





r3 iexcl r3

c

R31 iexcl



Aacute

hellip27dagger

for p ˆ 0 and

2598 Y Mishin et al


r3 iexcl r3

c

R31 Dagger r3

c

R3




and








and









for p ˆ 0 and







sup3 acute3

Selp hellip36dagger

and


r

R

sup3 acute3

SelV hellip37dagger



d ~SS








2600 Y Mishin et al


shy pound 10iexcl5



11192 07690 05057 08857 03404 03466 5621 5479



qS

qV

sup3 acute

T








2602 Y Mishin et al





d ~EEp=d ~EEV








ˆ 249to N




2604 Y Mishin et al

(a)

(b)















rc


rcurren






Sq ˆ shyX

not



2606 Y Mishin et al







Sq ˆ shy

3

X

not

X

not 6ˆreg



raquonot ˆX

reg 6ˆnot











cluster with N











2608 Y Mishin et al






S ˆ Sc DaggerN

N

Sel hellip46dagger

















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c









S ˆ iexclkB

X3Niexcl3

nˆ1

lnhcediln

kBT




nˆ1

cediln

Y3Niexcl3

nˆ1

cedil0n


5

2

4 hellip6dagger




Dinotjreg ˆ 1

mnotmreg

iexcl cent1=2

q2E

qRinot qRjreg

hellip7dagger






2594 Y Mishin et al






Y3Ncurreniexcl3

nˆ1

cediln

Y3Ncurren

nˆ1

cedil0n


5

2

4 hellip10dagger










r2 hellip11dagger



rr ˆ a iexcl 2b


r3 hellip12dagger



2596 Y Mishin et al




defect equals




Aacute

hellip15dagger

where










iexcl cent








rc


iexcl cent a2



Aacute

hellip19dagger





hellip20dagger







for p ˆ 0 and


c


Aacute

hellip22dagger



d ~EEp

d ~EEV








such that rc lt r




rc





r3 iexcl r3

c

R31 iexcl



Aacute

hellip27dagger

for p ˆ 0 and

2598 Y Mishin et al


r3 iexcl r3

c

R31 Dagger r3

c

R3




and








and









for p ˆ 0 and







sup3 acute3

Selp hellip36dagger

and


r

R

sup3 acute3

SelV hellip37dagger



d ~SS








2600 Y Mishin et al


shy pound 10iexcl5



11192 07690 05057 08857 03404 03466 5621 5479



qS

qV

sup3 acute

T








2602 Y Mishin et al





d ~EEp=d ~EEV








ˆ 249to N




2604 Y Mishin et al

(a)

(b)















rc


rcurren






Sq ˆ shyX

not



2606 Y Mishin et al







Sq ˆ shy

3

X

not

X

not 6ˆreg



raquonot ˆX

reg 6ˆnot











cluster with N











2608 Y Mishin et al






S ˆ Sc DaggerN

N

Sel hellip46dagger

















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c














Y3Ncurreniexcl3

nˆ1

cediln

Y3Ncurren

nˆ1

cedil0n


5

2

4 hellip10dagger










r2 hellip11dagger



rr ˆ a iexcl 2b


r3 hellip12dagger



2596 Y Mishin et al




defect equals




Aacute

hellip15dagger

where










iexcl cent








rc


iexcl cent a2



Aacute

hellip19dagger





hellip20dagger







for p ˆ 0 and


c


Aacute

hellip22dagger



d ~EEp

d ~EEV








such that rc lt r




rc





r3 iexcl r3

c

R31 iexcl



Aacute

hellip27dagger

for p ˆ 0 and

2598 Y Mishin et al


r3 iexcl r3

c

R31 Dagger r3

c

R3




and








and









for p ˆ 0 and







sup3 acute3

Selp hellip36dagger

and


r

R

sup3 acute3

SelV hellip37dagger



d ~SS








2600 Y Mishin et al


shy pound 10iexcl5



11192 07690 05057 08857 03404 03466 5621 5479



qS

qV

sup3 acute

T








2602 Y Mishin et al





d ~EEp=d ~EEV








ˆ 249to N




2604 Y Mishin et al

(a)

(b)















rc


rcurren






Sq ˆ shyX

not



2606 Y Mishin et al







Sq ˆ shy

3

X

not

X

not 6ˆreg



raquonot ˆX

reg 6ˆnot











cluster with N











2608 Y Mishin et al






S ˆ Sc DaggerN

N

Sel hellip46dagger

















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c














r2 hellip11dagger



rr ˆ a iexcl 2b


r3 hellip12dagger



2596 Y Mishin et al




defect equals




Aacute

hellip15dagger

where










iexcl cent








rc


iexcl cent a2



Aacute

hellip19dagger





hellip20dagger







for p ˆ 0 and


c


Aacute

hellip22dagger



d ~EEp

d ~EEV








such that rc lt r




rc





r3 iexcl r3

c

R31 iexcl



Aacute

hellip27dagger

for p ˆ 0 and

2598 Y Mishin et al


r3 iexcl r3

c

R31 Dagger r3

c

R3




and








and









for p ˆ 0 and







sup3 acute3

Selp hellip36dagger

and


r

R

sup3 acute3

SelV hellip37dagger



d ~SS








2600 Y Mishin et al


shy pound 10iexcl5



11192 07690 05057 08857 03404 03466 5621 5479



qS

qV

sup3 acute

T








2602 Y Mishin et al





d ~EEp=d ~EEV








ˆ 249to N




2604 Y Mishin et al

(a)

(b)















rc


rcurren






Sq ˆ shyX

not



2606 Y Mishin et al







Sq ˆ shy

3

X

not

X

not 6ˆreg



raquonot ˆX

reg 6ˆnot











cluster with N











2608 Y Mishin et al






S ˆ Sc DaggerN

N

Sel hellip46dagger

















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c











defect equals




Aacute

hellip15dagger

where










iexcl cent








rc


iexcl cent a2



Aacute

hellip19dagger





hellip20dagger







for p ˆ 0 and


c


Aacute

hellip22dagger



d ~EEp

d ~EEV








such that rc lt r




rc





r3 iexcl r3

c

R31 iexcl



Aacute

hellip27dagger

for p ˆ 0 and

2598 Y Mishin et al


r3 iexcl r3

c

R31 Dagger r3

c

R3




and








and









for p ˆ 0 and







sup3 acute3

Selp hellip36dagger

and


r

R

sup3 acute3

SelV hellip37dagger



d ~SS








2600 Y Mishin et al


shy pound 10iexcl5



11192 07690 05057 08857 03404 03466 5621 5479



qS

qV

sup3 acute

T








2602 Y Mishin et al





d ~EEp=d ~EEV








ˆ 249to N




2604 Y Mishin et al

(a)

(b)















rc


rcurren






Sq ˆ shyX

not



2606 Y Mishin et al







Sq ˆ shy

3

X

not

X

not 6ˆreg



raquonot ˆX

reg 6ˆnot











cluster with N











2608 Y Mishin et al






S ˆ Sc DaggerN

N

Sel hellip46dagger

















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c












for p ˆ 0 and


c


Aacute

hellip22dagger



d ~EEp

d ~EEV








such that rc lt r




rc





r3 iexcl r3

c

R31 iexcl



Aacute

hellip27dagger

for p ˆ 0 and

2598 Y Mishin et al


r3 iexcl r3

c

R31 Dagger r3

c

R3




and








and









for p ˆ 0 and







sup3 acute3

Selp hellip36dagger

and


r

R

sup3 acute3

SelV hellip37dagger



d ~SS








2600 Y Mishin et al


shy pound 10iexcl5



11192 07690 05057 08857 03404 03466 5621 5479



qS

qV

sup3 acute

T








2602 Y Mishin et al





d ~EEp=d ~EEV








ˆ 249to N




2604 Y Mishin et al

(a)

(b)















rc


rcurren






Sq ˆ shyX

not



2606 Y Mishin et al







Sq ˆ shy

3

X

not

X

not 6ˆreg



raquonot ˆX

reg 6ˆnot











cluster with N











2608 Y Mishin et al






S ˆ Sc DaggerN

N

Sel hellip46dagger

















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c










r3 iexcl r3

c

R31 Dagger r3

c

R3




and








and









for p ˆ 0 and







sup3 acute3

Selp hellip36dagger

and


r

R

sup3 acute3

SelV hellip37dagger



d ~SS








2600 Y Mishin et al


shy pound 10iexcl5



11192 07690 05057 08857 03404 03466 5621 5479



qS

qV

sup3 acute

T








2602 Y Mishin et al





d ~EEp=d ~EEV








ˆ 249to N




2604 Y Mishin et al

(a)

(b)















rc


rcurren






Sq ˆ shyX

not



2606 Y Mishin et al







Sq ˆ shy

3

X

not

X

not 6ˆreg



raquonot ˆX

reg 6ˆnot











cluster with N











2608 Y Mishin et al






S ˆ Sc DaggerN

N

Sel hellip46dagger

















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c












sup3 acute3

Selp hellip36dagger

and


r

R

sup3 acute3

SelV hellip37dagger



d ~SS








2600 Y Mishin et al


shy pound 10iexcl5



11192 07690 05057 08857 03404 03466 5621 5479



qS

qV

sup3 acute

T








2602 Y Mishin et al





d ~EEp=d ~EEV








ˆ 249to N




2604 Y Mishin et al

(a)

(b)















rc


rcurren






Sq ˆ shyX

not



2606 Y Mishin et al







Sq ˆ shy

3

X

not

X

not 6ˆreg



raquonot ˆX

reg 6ˆnot











cluster with N











2608 Y Mishin et al






S ˆ Sc DaggerN

N

Sel hellip46dagger

















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c











qS

qV

sup3 acute

T








2602 Y Mishin et al





d ~EEp=d ~EEV








ˆ 249to N




2604 Y Mishin et al

(a)

(b)















rc


rcurren






Sq ˆ shyX

not



2606 Y Mishin et al







Sq ˆ shy

3

X

not

X

not 6ˆreg



raquonot ˆX

reg 6ˆnot











cluster with N











2608 Y Mishin et al






S ˆ Sc DaggerN

N

Sel hellip46dagger

















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c









2602 Y Mishin et al





d ~EEp=d ~EEV








ˆ 249to N




2604 Y Mishin et al

(a)

(b)















rc


rcurren






Sq ˆ shyX

not



2606 Y Mishin et al







Sq ˆ shy

3

X

not

X

not 6ˆreg



raquonot ˆX

reg 6ˆnot











cluster with N











2608 Y Mishin et al






S ˆ Sc DaggerN

N

Sel hellip46dagger

















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c














ˆ 249to N




2604 Y Mishin et al

(a)

(b)















rc


rcurren






Sq ˆ shyX

not



2606 Y Mishin et al







Sq ˆ shy

3

X

not

X

not 6ˆreg



raquonot ˆX

reg 6ˆnot











cluster with N











2608 Y Mishin et al






S ˆ Sc DaggerN

N

Sel hellip46dagger

















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c









2604 Y Mishin et al

(a)

(b)















rc


rcurren






Sq ˆ shyX

not



2606 Y Mishin et al







Sq ˆ shy

3

X

not

X

not 6ˆreg



raquonot ˆX

reg 6ˆnot











cluster with N











2608 Y Mishin et al






S ˆ Sc DaggerN

N

Sel hellip46dagger

















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c






















rc


rcurren






Sq ˆ shyX

not



2606 Y Mishin et al







Sq ˆ shy

3

X

not

X

not 6ˆreg



raquonot ˆX

reg 6ˆnot











cluster with N











2608 Y Mishin et al






S ˆ Sc DaggerN

N

Sel hellip46dagger

















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c










Sq ˆ shy

3

X

not

X

not 6ˆreg



raquonot ˆX

reg 6ˆnot











cluster with N











2608 Y Mishin et al






S ˆ Sc DaggerN

N

Sel hellip46dagger

















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c













2608 Y Mishin et al






S ˆ Sc DaggerN

N

Sel hellip46dagger

















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c













S ˆ Sc DaggerN

N

Sel hellip46dagger

















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c



















2610 Y Mishin et al


ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c










ACKNOWLEDGEMENTS


APPENDIX A




r2 hellipA 1dagger








Aacute iexcl1

hellipA 3dagger


c


Aacute iexcl1

hellipA 4dagger



a ˆ iexcl ucr2c

R31 iexcl r3

c

R3

Aacute iexcl1

hellipA 6dagger



c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c










c

R3

Aacute iexcl1

hellipA 7dagger




c


Aacute

hellipA 8dagger


c


Aacute

hellipA 9dagger

for p ˆ 0 and

a ˆ iexcl ucr2c

R31 Dagger r3

c

R3

Aacute

hellipA 10dagger

b ˆ ucr2c 1 Dagger

r3c

R3

Aacute

hellipA 11dagger


2c









philosophical m a,2001,vol. 81,no. 11, 2591± 2612physics.gmu.edu/~ymishin/entropy.pdf · atomistic...

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