Comprehensive nuclear materials 1 01 fundamental properties of defects in metals

Comprehensive nuclear materials 1 01 fundamental properties of defects in metals Comprehensive nuclear materials 1 01 fundamental properties of defects in metals Comprehensive nuclear materials 1 01 fundamental properties of defects in metals Comprehensive nuclear materials 1 01 fundamental properties of defects in metals Comprehensive nuclear materials 1 01 fundamental properties of defects in metals Comprehensive nuclear materials 1 01 fundamental properties of defects in metals Comprehensive nuclear materials 1 01 fundamental properties of defects in metals

1.01.6 Anisotropic Diffusion in Strained Crystals of Cubic Symmetry 21

1.01.7.4.3 Surface stresses and bulk stresses for spherical cavities 30

1.01.8.2.2 Dislocation bias with size and modulus interactions 35

Appendix A Elasticity Models: Defects at the Center of a Spherical Body 38

Appendix B Representation of Defects by Atomic Forces and by Multipole Tensors 41

1

bcc Body-centered cubic

CD Center of dilatation model

dpa Displacements per atom

fcc Face-centered cubic

hcp Hexagonal closed packed

INC Inclusion model

IHG Inhomogeneity model

SIA Self-interstitial atom

1.01.1 Introduction

Several fundamental attributes and properties of

crystal defects in metals play a crucial role in radiation

effects and lead to continuous macroscopic changes of

metals with radiation exposure These attributes and

properties will be the focus of this chapter However,

there are other fundamental properties of defects that

are useful for diagnostic purposes to quantify their

concentrations, characteristics, and interactions with

each other For example, crystal defects contribute to

the electrical resistivity of metals, but electrical

resis-tivity and its changes are of little interest in the design

and operation of conventional nuclear reactors What

determines the selection of relevant properties can

best be explained by following the fate of the two

most important crystal defects created during the

primary event of radiation damage, namely vacancies

and self-interstitials

The primary event begins with an energetic

parti-cle, a neutron, a high-energy photon, or an energetic

ion, colliding with a nucleus of a metal atom When

sufficient kinetic energy is transferred to this nucleus

or metal atom, it is displaced from its crystal lattice

site, leaving behind a vacant site or a vacancy The

recoiling metal atom may have acquired sufficient

energy to displace other metal atoms, and they in

turn can repeat such events, leading to a collision

cascade Every displaced metal atom leaves behind a

vacancy, and every displaced atom will eventually

dis-sipate its kinetic energy and come to rest within the

crystal lattice as a self-interstitial defect It is

immedi-ately obvious that the number of self-interstitials is

exactly equal to the number of vacancies produced,

and they form Frenkel pairs The number of Frenkel

pairs created is also referred to as the number of

displacements, and their accumulated density is

expressed as the number of displacements per atom(dpa) When this number becomes one, then on aver-age, each atom has been displaced once

At the elevated temperatures that exist in nuclearreactors, vacancies and self-interstitials diffusethrough the crystal As a result, they will encountereach other, either annihilating each other or formingvacancy and interstitial clusters These events occuralready in their nascent collision cascade, but if defectsescape their collision cascade, they may encounter thedefects created in other cascades In addition, migrat-ing vacancy defects and interstitial defects may also becaptured at other extended defects, such as disloca-tions, cavities, grain boundaries and interface bound-aries of precipitates and nonmetallic inclusions, such

as oxide and carbide particles The capture events atthese defect sinks may be permanent, and the migrat-ing defects are incorporated into the extendeddefects, or they may also be released again

However, regardless of the complex fate of eachindividual defect, one would expect that eventuallythe numbers of interstitials and vacancies that arrive

at each sink would become equal, as they are duced in equal numbers as Frenkel pairs Therefore,apart from statistical fluctuations of the sizes andpositions of the extended defects, or the sinks, themicrostructure of sinks should approach a steadystate, and continuous irradiation should change theproperties of metals no further

pro-It came as a big surprise when radiation-inducedvoid swelling was discovered with no indication of asaturation In the meantime, it has become clear thatthe microstructure evolution of extended defects andthe associated changes in macroscopic properties ofmetals in general is a continuing process with dis-placement damage

The fundamental reason is that the migration

of defects, in particular that of self-interstitials andtheir clusters, is not entirely a random walk but is

in subtle ways guided by the internal stress fields

of extended defects, leading to a partial segregation ofself-interstitials and vacancies to different types of sinks.Guided then by this fate of radiation-producedatomic defects in metals, the following topics arepresented in this chapter:

1 The displacement energy required to create aFrenkel pair

2 The energy stored within a Frenkel pair that sists of the formation enthalpies of the self-interstitial and the vacancy

con-3 The dimensional changes that a solid suffers whenself-interstitial and vacancy defects are created,

and how these changes manifest themselves either

externally or internally as changes in lattice

parameter These changes then define the

forma-tion and relaxaforma-tion volumes of these defects and

their dipole tensors

4 The regions occupied by the atomic defects within

the crystal lattice possess a distorted, if not totally

different, arrangement of atoms As a result, these

regions are endowed with different elastic

proper-ties, thereby changing the overall elastic constants

of the defect-containing solid This leads to the

concept of elastic polarizability parameters for

the atomic defects

5 Both the dipole tensors and elastic polarizabilities

determine the strengths of interactions with both

internal and external stress fields as well as their

mutual interactions

6 When the stress fields vary, the gradients of the

interactions impose drift forces on the diffusion

migration of the atomic defects that influences their

reaction rates with each other and with the sinks

7 At these sinks, vacancies can also be generated by

thermal fluctuations and be released via diffusion

to the crystal lattice Each sink therefore possesses

a vacancy chemical potential, and this potential

determines both the nucleation of vacancy defect

clusters and their subsequent growth to become

another defect sink and part of the changing

microstructure of extended defects

The last two topics, 6 and 7, as well as topic 1, will be

further elaborated in other chapters

1.01.2 The Displacement Energy

Scattering of energetic particles from external sources,

be they neutrons, electrons, ions, or photons, or

emis-sion of such particles from an atomic nucleus, imparts a

recoil energy When this recoil energy exceeds a critical

value, called the threshold displacement energy, Td,

Frenkel pairs can be formed To measure this

displace-ment energy, an electron beam is employed to produce

the radiation damage in a thin film of the material, and

its rise in electrical resistivity due to the Frenkel pairs is

monitored By reducing the energy of the electron

beam, the resistivity rise is also reduced, and a

thresh-old electron energy, Emin, can be found below which no

Frenkel pairs are produced The corresponding recoil

energy is given by relativistic kinematics as

The approximation on the right

is adequate because the electron mass, m, is muchsmaller than the mass, M, of the recoiling atom.Changing the direction of the electron beam inrelation to the orientation of single crystal film speci-mens, one finds that the threshold energy variessignificantly However, for polycrystalline samples,values averaged over all orientations are obtained,and these values are shown inFigure 1for differentmetals as a function of their melting temperatures.1First, we notice a trend that Tdincreases with themelting temperature, reflecting the fact that largerenergies of cohesion or of bond strengths betweenatoms also lead to higher melting temperatures

We also display values of the formation energy

of a Frenkel pair Each value is the sum of thecorresponding formation energies of a self-interstitialand a vacancy for a given metal These energies arepresented and further discussed below The importantpoint to be made here is that the displacement energyrequired to create a Frenkel pair is invariably larger thanits formation energy Clearly, an energy barrier exists forthe recoil process, indicating that atoms adjacent to theone that is being displaced also receive some additionalkinetic energy that is, however, below the displacementenergy Tdand is subsequently dissipated as heat.The displacement energies listed in Table 1andshown inFigure 1are averaged not only over crystalorientation but also over temperature for those metals

0 10 20 30 40 50

bcc Frenkel pair (eV)

Figure 1 Energies of displacement and energies of Frenkel pairs for elemental metals as a function of their melting temperatures.

where the displacement energy has been measured as

a function of irradiation temperature For some

mate-rials, such as Cu, a significant decrease of the

dis-placement energy with temperature has been found

However, a definitive explanation is still lacking

Close to the minimum electron energy for Frenkel

pair production, the separation distance between

the self-interstitial and its vacancy is small Therefore,

their mutual interaction will lead to their

recombina-tion With increasing irradiation temperature, however,

the self-interstitial may escape, and this would

mani-fest itself as an apparent reduction in the displacement

energy with increasing temperature On the other

hand, Jung2has argued that the energy barrier involved

in the creation of Frenkel pairs is directly dependent

on the temperature in the following way This energy

barrier increases with the stiffness of the repulsive part

of the interatomic potential; a measure for this stiffness

is the bulk modulus Indeed, asFigure 2demonstrates,

the displacement energy increases with the bulk

mod-ulus Since the bulk modulus decreases with

tempera-ture, so will the displacement energy

The correlation of the displacement energy with

the bulk modulus appears to be a somewhat better

empirical relationship than the correlation with themelt temperature However, one should not readtoo much into this, as the bulk modulus B, atomic

Table 1 Displacement and Frenkel pair energies of elemental metals

Bulk modulus (GPa)

Figure 2 Displacement energies for elemental metals as a function of their bulk modulus.

volume O, and melt temperature of elemental metals

approximately satisfy the rule

BO  100kBTm

discovered by Leibfried3and shown inFigure 3

1.01.3 Properties of Vacancies

1.01.3.1 Vacancy Formation

The thermal vibration of atoms next to free surfaces,

to grain boundaries, to the cores of dislocations, etc.,

make it possible for vacancies to be created and then

diffuse into the crystal interior and establish an

equi-librium thermal vacancy concentration of

V is the vacancy formationentropy The thermal vacancy concentration can be

measured by several techniques as discussed in

Dam-ask and Dienes,4 Seeger and Mehrer,5 and Siegel,6

and values for EVf have been reviewed and tabulated

by Ehrhart and Schultz;7 they are listed inTable 2

When these values for the metallic elements are

plotted versus the melt temperature inFigure 4, an

approximate correlation is obtained, namely

EfV Tm=1067 ½3

Using the Leibfried rule, a new approximate relation emerges for the vacancy formation enthalpythat has become known as the cBO model8; the con-stant c is assumed to be independent of temperatureand pressure As seen from Figure 5, however, theexperimental values for Ef

cor-V correlate no better with

BO than with the melting temperature

It is tempting to assume that a vacancy is just avoid and its energy is simply equal to the surfacearea 4pR2times the specific surface energy g0 Takingthe atomic volume as the vacancy volume, that is,

O ¼ 4pR3

/3, we show in Figure 6 the measuredvacancy formation enthalpies as a function of4pR2g0, using for g0 the values9 at half the meltingtemperatures It is seen that Ef

Vis significantly less, byabout a factor of two, compared to the surface energy

of the vacancy void so obtained Evidently, this ple approach does not take into account the fact thatthe atoms surrounding the vacancy void relax intonew positions so as to reduce the vacancy volume Vf

The difference between the observed vacancy mation enthalpy and the value from the simplisticsurface model has recently been resolved It will beshown in Section 1.01.7 that the specific surfaceenergy is in fact a function of the elastic strain tan-gential to the surface, and when this surface strainrelaxes, the surface energy is thereby reduced At thesame time, however, the surface relaxation creates astress field in the surrounding crystal, and hence astrain energy As a result, the energy of a void afterrelaxation is given by

for-FC½eðRÞ; e ¼ 4pR2g½eðRÞ; e þ 8pR3me2ðRÞ ½5The first term is the surface free energy of a void withradius R, and it depends now on a specific surfaceenergy that itself is a function of the surface straine(R) and the intrinsic residual surface strain e* for asurface that is not relaxed The second term is thestrain energy of the surrounding crystal that depends

on its shear modulus m The strain dependence of thespecific surface energy is given by

g½e; e ¼ g0þ 2ðmSþ lSÞð2eþ eÞe ½6Here, g0 is the specific surface energy on a surfacewith no strains in the underlying bulk material

Figure 3 Leibfried’s empirical rule between melting

temperature and the product of bulk modulus and atomic

volume.

However, such a surface possesses an intrinsic,

resid-ual surface strain e*, because the interatomic bonding

between surface atoms differs from that in the bulk,

and for metals, the surface bond length would be

shorter if the underlying bulk material would allow

the surface to relax Partial relaxation is possible for

small voids as well as for nanosized objects In addition

to the different bond length at the surface, the elastic

constants, mS and lS, are also different from the

corresponding bulk elastic constants However, they

can be related by a surface layer thickness, h, to bulk

elastic constants such that

mSþ lS ¼ ðm þ lÞh ¼ mh=ð1  2nÞ ½7

where l is the Lame’s constant and n is Poisson’s

ratio for the bulk solid Computer simulations on

freestanding thin films have shown10that the surfacelayer is effectively a monolayer, and h can be approxi-mated by the Burgers vector b For planar crystal sur-faces, the residual surface strain parameter e* is found to

be between 3 and 5%, depending on the surface tation relative to the crystal lattice On surfaces withhigh curvature, however, e* is expected to be larger.The relaxation of the void surface can now beobtained as follows We seek the minimum of thevoid energy as defined by eqn [5] by solving

orien-@FC=@e ¼ 0 The result iseðRÞ ¼  ðmSþ lSÞe

mR þ ðmSþ lSÞ¼ 

h eð1  2nÞR þ h ½8and this relaxation strain changes the initially unre-laxed void volume

Table 2 Crystal and vacancy properties

2

½12This equation is evaluated for Ni and the results areshown inFigure 7as a function of the vacancy relaxa-tion volume Vrel

V =O

  It is seen that relaxation volumes

of 0.2 to 0.3 predict a vacancy formation energycomparable to the experimental value of 1.8 eV

Bulk modulus * atomic volume (eV)

Figure 5 Vacancy formation energy versus the product of

bulk modulus and atomic volume.

0 0.5 1 1.5 2 2.5 3 3.5

Exp value Computed values Bulk strain energy

Vacancy relaxation volume Ni

Figure 7 Vacancy formation energy and its dependence

on the relaxation volume.

melting temperature.

0 0.5 1 1.5 2 2.5 3 3.5 4

fcc Hf/v, eV bcc Hf/v, eV hcp Hf/v, eV

Surface energy of a vacancy (eV) Figure 6 Correlation between the surface energy of a vacancy void and the vacancy formation energy.

Few experimentally determined values are

avail-able for the vacancy relaxation volume, and their

accuracy is often in doubt In contrast, vacancy

for-mation energies are better known Therefore, we

volumes from experimentally determined vacancy

formation energies The values so obtained are listed

possible, they are compared with the values reported

from experiments Computed values for the vacancy

relaxation volumes are between 0.2O and 0.3O

for both fcc and bcc metals The low experimental

values for Al, Fe, and Mo then appear suspect

The surface energy model employed here to

deriveeqn [12] is based on several approximations:

isotropic, linear elasticity, a surface energy

parame-ter, g0, that represents an average over different

crys-tal orientations, and extrapolation of the energy of

large voids to the energy of a vacancy

Nevertheless, this approximate model provides

satisfactory results and captures an important

con-nection between the vacancy relaxation volume and

the vacancy formation energy that has also been

noted in atomistic calculations

Finally, a few remarks about the vacancy

forma-tion entropy, SVf, are in order It originates from the

change in the vibrational frequencies of atoms

sur-rounding the vacancy Theoretical estimates based on

empirical potentials provide values that range from 0.4k

to about 3.0k, where k is the Boltzmann constant As a

result, the effect of the vacancy formation entropy on

the magnitude of the thermal equilibrium vacancy

con-centration, CVeq, is of the same magnitude as the

statisti-cal uncertainty in the vacancy formation enthalpy

1.01.3.2 Vacancy MigrationThe atomistic process of vacancy migration consists

of one atom next to the vacant site jumping into thissite and leaving behind another vacant site The jump

is thermally activated, and transition state theorypredicts a diffusion coefficient for vacancy migration

in cubic crystals of the form

d0is the nearest neighbor distance between atoms, Sm

V

is the vacancy migration entropy, and Em

V is the energyfor vacancy migration It is in fact the energy of anactivation barrier that the jumping atom must over-come, and when it temporarily occupies a position atthe height of this barrier, the atomic configuration isreferred to as the saddle point of the vacancy It will

be considered in greater detail momentarily

Values obtained for EVm from experimental surements are shown inFigure 8as a function of themelting point While we notice again a trend similar

mea-to that for the vacancy formation energy, we find that

EVmfor fcc and bcc metals apparently follow differentcorrelations However, the correlation for bcc metals

is rather poor, and it indicates that Em

of the jumping atom These nearest neighbor atoms

Table 3 Vacancy relaxation volumes for metals

lie at the corners of a rectangular plane as shown in

Figure 9 As the jumping atom crosses this plane,

they are displaced such as to open the channel This

coordinated motion can be viewed as a particular

strain fluctuation and described in terms of phonon

excitations In this manner, Flynn11 has derived the

following formula for the energy of vacancy

migra-tion in cubic crystals

EVm¼ 15C11C44ðC11 C12Þa3w

2½C11ðC11 C12Þ þ C44ð5C11 3C12Þ ½14

Here, a is the lattice parameter, C11, C12, and C44are

elastic moduli, and w is an empirical parameter that

characterizes the shape of the activation barrier and

can be determined by comparing experimental

vacancy migration energies with values predicted by

for fcc metals and w¼ 0.020 for bcc metals

In the derivation of Flynn,11only the four nearestneighbor atoms are supposed to move, while all otheratoms are assumed to remain in their normal latticepositions On the other hand, Kornblit et al.13treat theexpansion of the diffusion channel as a quasistaticelastic deformation of the entire surrounding mate-rial The extent of the expansion is such that theopened channel is equal to the cross-section of thejumping atom, and a linear anisotropic elasticity cal-culation is carried out by a variational method todetermine the energy involved in the channel expan-sion A vacancy migration energy is obtained for fccmetals of

EmV ¼ 0:01727a3C11

p0p1 p2 2

p2þ2

9p0p2þ1

9p2 ½15and the parameters piwill be defined momentarily.For bcc metals,14 the activation barrier consist oftwo peaks of equal height Emax

V with a shallow valley

in between with an elevation of EVmin, where

EVmax¼ 0:003905a3C11

q0q1 q2 2

q22

11q0p2þ3

88q2 ½16and

EminV ¼ 0:002403a3C11 s0s1 s2

s2 0:29232s0s2þ 0:0413s2 ½17The parameters pi,qi,and siare linear functions of theelastic moduli with coefficients listed inTable 4 Forexample,

q1¼ 3:45C11 0:75C12þ 4:35C44

If the depth of the valley is greater than the thermalenergy of the jumping atom, that is, greater than3

2kT,then it will be trapped and requires an additionalactivation to overcome the remaining barrier of

melting temperature.

Figure 9 Second nearest neighbor atom (blue) jumping

through the ring of four next-nearest atoms (green) into

adjacent vacancy in a fcc structure.

Table 4 Coefficients for the Kornblit energy expressions

EmaxV  Emin

V

As a result, Kornblit14assumes that the

vacancy migration energy for bcc metals is given by

EVm¼ EVmax; if EmaxV  Emin

V 3kT2Emax

Using the formulae of Flynn and Kornblit, we compute

the vacancy migration energies and compare them

with experimental values inFigure 10

With a few exceptions, both the Flynn and the

Kornblit values are in good agreement with the

exper-imental results

The self-diffusion coefficient determines the

transport of atoms through the crystal under

condi-tions near the thermodynamic equilibrium, and it is

The most accurate measurements of diffusion

coef-ficients are done with a radioactive tracer isotope of

the metal under investigation, and in this case one

obtains values for the tracer self-diffusion

coeffi-cientDSDT ¼ fDSDthat involves the correlation factor

f For pure elemental metals of cubic structure, f is

a constant and can be determined exactly by

com-putation.15For fcc crystals, f¼0.78145, and for bcc

V and forthe attempt frequency nLV Based on theoretical esti-mates, Seeger and Mehrer5 recommend a value of2.5 k for the former The atomic vibration of nearestneighbor atoms to the vacancy is treated within asinusoidal potential energy profile that has a maxi-mum height of Em

V For small-amplitude vibrations,the attempt frequency is then given by

nLV¼1a

ffiffiffiffiffiffi

Em V

M

rfor fcc and by

nLV¼1a

ffiffiffiffiffiffiffiffi2Em V

3M

r

crystals where M is the atomic mass

In contrast, Flynn11assumes that the atomic tions can be derived from the Debye model for whichthe average vibration frequency is

vibra-nLV¼

ffiffiffi35

In fact, the computed values change little from onemetal to another, and the Flynn model predicts valuesabout twice as large as the model by Seeger andMehrer Either model can therefore be used to provide

a reasonable estimate of the preexponential factorwhere no experimental value is available

1.01.3.3 Activation Volume forSelf-Diffusion

When the crystal lattice is under pressure p, the diffusion coefficient changes and is then given by

self-DSDðT;pÞ ¼ D0

SDexpðQSD=kTÞexpðpVSD=kTÞ ½24The activation volume VSDcan be obtained experi-mentally by measuring the self-diffusion coefficient

as a function of an externally applied pressure Suchmeasurements have been carried out only for a few

Experimental vacancy migration energy (eV)

Figure 10 Comparison of computed vacancy migration

energies according to models by Flynn and Kornblit with

measured values.

metals, and it has been found that the activation

volumes have positive values Therefore, self-diffusion

decreases with applied pressure However, it has been

noticed that the self-diffusion coefficient at melting

appears to be constant, and this can be explained

by the fact that the melting temperature increases

in general with pressure It follows then from the

dTmdp

Brown and Ashby16 have used this relationship

to evaluate the activation volumes for self-diffusion

for a variety of metals Using more recent valuesfor the pressure derivative of the melting tempera-ture by Wallace17 and Wang et al.,18 one obtainsactivation volumes as shown in Figure 12 Theyare in reasonably good agreement with the experi-mental values where they exist With the exception

of Pt, the predicted values are also similar, giving

an activation volume of about 0.85O for fcc metals,0.65O for hcp metals, and around 0.4O for bccmetals

10 –6

10 –5

Seeger Mehrer Flynn

2 s

–1 )

Experimental preexponential Do (m 2 s –1 )

Figure 11 Comparison of preexponential factors for

tracer self-diffusion as computed with two models and as

measured.

Table 5 Preexponentials for tracer self-diffusion

V (eV) Experimental value S & M Flynn (m 2 s1)

0

Ag Al Cu Fe Ni Pb Pt Cd Mg Tl Zn Cs K Li Na Rb

Fe

Experiment Brown and Ashby Wallace

The equilibrium vacancy concentration in a solid

under pressure p is given by

where VVf is the vacancy formation volume Since

the self-diffusion coefficient is the product of the

thermal vacancy concentration and the vacancy

migra-tion coefficient, the activamigra-tion volume for self-diffusion

is the sum of two contributions, namely

If one takes the average of the predicted activation

volumes shown inFigure 12, and the vacancy

relax-ation volumes fromTable 3, one obtains values for

VVmlisted inTable 6and also shown inFigure 12

1.01.4 Properties of Self-Interstitials

1.01.4.1 Atomic Structure of

Self-Interstitials

The accommodation of an additional atom within a

perfect crystal lattice remained a topic of lively

debates at international conferences on radiation

effects for many decades The leading question was

the configuration of this interstitial atom and its

surrounding atoms This scientific question has now

been resolved, and there is general agreement that

this additional atom, a self-interstitial, forms a pair

with one atom from the perfect lattice in the form of a

dumbbell The configuration of these dumbbells can

be illustrated well with hard spheres, that is, atoms

that repel each other like marbles

Let us first consider the case of an fcc metal In the

perfect crystal, each atom is surrounded by 12 nearest

neighbors that form a cage around it as shown on the

left ofFigure 13 When an extra atom is inserted in

this cage, the two atoms in the center form a pair

whose axis is aligned in a [001] direction This [001]dumbbell constitutes the self-interstitial in the fcclattice The centers of the 12 nearest neighbor atomsare the apexes of a cubo-octahedron that encloses thesingle central atom in the perfect lattice, and it can beshown19that the cubo-octahedron encloses a volume

of VO¼ 10O/3 However, around a self-interstitialdumbbell, this cubo-octahedron expands and distorts,and now it encloses a larger volume of V001¼ 4.435O.The volume expansion is the difference

DV ¼ V001 V0¼ 1:10164 O ½28which happens to be larger than one atomic volume

We shall see shortly that the volume expansion of theentire crystal is even larger due to the elastic strainfield created by the self-interstitial that extendsthrough the entire solid

We consider next the self-interstitial defect in abcc metal Here, each atom is surrounded in theperfect crystal by eight nearest neighbors as shown onthe left ofFigure 14 When an extra atom is inserted, itagain forms a dumbbell configuration with anotheratom, and the dumbbell axis is now aligned in the[011] direction, as shown on the right ofFigure 14.The cage formed by the eight nearest neighbor atomsbecomes severely distorted It is surprising, however,that the volume change of the cage is only

it by inserting an extra atom only creates disorder andlower packing density

As already mentioned, the large inclusionvolume DV of self-interstitials leads to a strain field

Table 6 Activation volume for vacancy migration

throughout the surrounding crystal that causes

changes in lattice parameter and that is the major

source of the formation energy for self-interstitials

In order to determine this strain field, we treat in

of a spherical solid with isotropic elastic properties

Although this represents a rather simplified model

for self-interstitials, for vacancies, and for complex

clusters of such defects, it is a very instructive model

that captures many essential features

1.01.4.2 Formation Energy of

Self-Interstitials

In contrast to the formation energy of vacancies,

there exists no direct measurement for the formation

energy of self-interstitials We have mentioned

required to create a Frenkel pair is much larger

than the combined formation energies of the vacancy

and the self-interstitial As pointed out, there exist a

large energy barrier to create the Frenkel pair,

namely the displacement energy Td, and this barrier

is mainly associated with the insertion of the

intersti-tial into the crystal lattice However, although this

barrier should be part of the energy to form a

self-interstitial, it is by convention not included Rather,

the formation energy of a self-interstitial is

consid-ered to be the increase of the internal energy of a

crystal with this defect in comparison to the energy

of the perfect crystal In contrast, since vacancies

can be created by thermal fluctuations at surfaces,

grain boundaries, and dislocation cores by accepting

an atom from an adjacent lattice site and leaving

it vacant, no similar barrier exists The activation

energy for this process is simply the sum of the actual

formation energy Ef

V and the migration energy Em

V,that is, the energy for self-diffusion, Q

When Frenkel pairs are created by irradiation

at cryogenic temperatures, self-interstitials and cancies can be retained in the irradiated sample.Subsequent annealing of the sample and measuringthe heat released as the defects migrate and thendisappear provide an indirect method to measurethe energies of Frenkel pairs Subtracting from thesecalorimetric values, the vacancy formation energyshould give the formation energy of self-interstitials.The values so obtained for Cu7 vary from 2.8 to4.2 eV, demonstrating just how inaccurate calorimet-ric measurements are Besides, measurements haveonly been attempted on two other metals, Al and

va-Pt, with similar doubtful results As a result, cal calculations or atomistic simulations provide per-haps more trustworthy results

theoreti-For a theoretical evaluation of the formationenergy, we can consider the self-interstitial as aninclusion (INC) as described inAppendix A Accord-ingly, a volume O of one atom is enlarged by theamount DV, or in other words, is subject to the trans-formation strain

eij ¼ dij where 3 ¼ DV =O ½30The energy associated with the formation of thisinclusion is given inTable A2, and it can be written as

U0¼ 2K mO3K þ 4m

DVO

2

½31This expression for the so-called dilatational strainenergy provides a rough approximation to the forma-tion energy of a self-interstitial in fcc metals when theabove volume expansion results are used

However, as the nearest neighbor cells depicted inFigures 13 and 14 show, their distorted shapes can-not be adequately described with a radial expansion

of the original cell in the ideal crystal as implied

indicates, the [001] dumbbell interstitial in the fcclattice does not change the cell dimension in the[001] direction In fact, it shortens it slightly, imply-ing that e33¼ 0.01005 The volume change is there-fore due to the nearest neighbor atoms moving onaverage away from the dumbbell axis This can berepresented by the transformation strain components

e11¼ e22being determined by

2e11þ e33¼DV

O ¼ 1:10164The transformation strain tensor for the [001] self-interstitials in fcc crystals is then

Figure 14 On the left is the unit cell of the bcc crystal

structure The central atom shown darker is surrounded by

eight nearest neighbors On the right is the arrangement

when a self-interstitial occupies the center of the cell.

1 C A

and it can be divided into an dilatational part,d ij,

and a shear part,~e ij, as shown

To find the transformation strain tensor for the

[011] self-interstitial in bcc crystal, it is convenient to

use a new coordinate system with the x3-axis as the

dumbbell axis and the x1- and x2-axes emanating from

the midpoint of the dumbbell axis and pointing toward

the corner atoms While the distance between these

corner atoms and the central atom in the original bcc

unit cell is the interatomic distance r0, in the distorted

cell containing the self-interstitial, their distance is

2  1  0:134the remaining strain component is then determined by

@

1 C

@

1 C A

The strain energy associated with the shear part can beshown (see Mura20) to be

U1¼2ð9K þ 8mÞmO5ð3K þ 4mÞ I2

where I2¼ ~e11~e22~e22~e33~e33~e11 ½34and is equal to Ifcc

2 ¼ 0:1068 and Ibcc

2 ¼ 0:3633 for interstitials in fcc and bcc metals, respectively

self-We consider now the total strain energy as areasonable approximation for the formation energy

of self-interstitials, namely

EIf  U ¼ U0þ U1 ½35The dilatational and the shear strain energies forsome elements are listed in Table 7 in the fourthand fifth column, respectively For the fcc elements,the ratio of U1/U0is about 0.2 In contrast, for the bccelements (in italics) this ratio is about two

Table 7 Strain energies and relaxation volumes of self-interstitials

1.01.4.3 Relaxation Volume of

Self-Interstitials

If the elastic distortions associated with

self-interstitials could be adequately treated with linear

elasticity theory, and if the repulsive interactions

between the dumbbell atoms with their nearest

neigh-bors were like that between hard spheres, then the

volume change of a solid upon insertion of an

intersti-tial atom would be equal to the volume change DV as

derived above This follows from the analysis of the

inclusion in the center of a sphere given inAppendix A

From the results listed in Table A2, under column

INC, we see that the volume change of the solid with

a concentration S of inclusions is simply given by

DV

V ¼ 3Swhere 3 is the volume dilatation per inclusion as if it

were not confined by the surrounding matrix This

remarkable result has been proven by Eshelby21to be

valid for any shape of the solid and any location of the

inclusion within it, provided the inclusion and the solid

can be treated as one linear elastic material In other

words, the elastic strains within the inclusion and

within the matrix must be small

However, this is not the case for the elastic

strains produced by self-interstitials Here, the

elas-tic strains are quite large For example, the volume of

the confined inclusion, also listed inTable A2under

column INC, is given by

strained volume for a Poisson’s ratio of n¼ 0.3

This amounts to an elastic compression of 42% of

the ‘volume’ of the self-interstitial in fcc materials,

and 25% for the self-interstitial in bcc materials

Clearly, nonlinear elastic effects must be taken into

account

Zener22has found an elegant way to include the

effects of nonlinear elasticity on volume changes

pro-duced by crystal defects such as self-interstitials and

dislocations If U represents the elastic strain energy of

such defects evaluated within linear elasticity theory,

and if one then considers the elastic constants in the

formula for U to be in fact dependent on the pressure,

then the additional volume change dV produced by

the defects can be derived from the simple expression

dV ¼@U

@p 

U

found by Schoeck.23Its application to the strain energy

of self-interstitials leads to the following result

m0

m 1K

U1 ½37Here, m0 and K0 are the pressure derivatives of theshear and bulk modulus, respectively The first termarising from the dilatational part of the strain energywas derived and evaluated earlier by Wolfer.19 It isthe dominant term for the additional volume changefor self-interstitials in fcc metals Here, we evaluateboth terms using the compilation of Guinan andSteinberg24 for the pressure derivatives of the elasticconstants, and as listed inTable 7

The calculated relaxation volumes for interstitials,

self-VIrel¼ DV þ dV ½38are given in the eighth column ofTable 7, and theycan be compared with the available experimentalvalues also listed

We shall see that the relaxation volume of interstitials is of fundamental importance to explainand quantify void swelling in metals exposed to fastneutron and charged particle irradiations

self-1.01.4.4 Self-Interstitial MigrationThe dumbbell configuration of a self-interstitial gives

it a certain orientation, namely the dumbbell axis,and upon migration this axis orientation may change.This is indeed the case for self-interstitials in fccmetals, as illustrated inFigure 15

Suppose that the initial location of the interstitial is as shown on the left, and its axis isalong [001] A migration jump occurs by one atom

self-of the dumbbell (here the purple one) pairing upwith one nearest neighbor, while its former partner

Figure 15 Migration step of the self-interstitial in fcc metals.

(the blue atom) occupies the available lattice site.

Computer simulations of this migration process have

shown25that the orientation of the self-interstitial has

rotated to a [010] orientation, and that this combined

migration and rotation requires the least amount of

thermal activation

Similar analysis for the migration of

self-interstitials in bcc metals has revealed that a rotation

may or may not accompany the migration, and these

two diffusion mechanisms are depicted inFigure 16

Which of these two possesses the lower activation

energy depends on the metal, or on the interatomic

potential employed for determining it

In general, however, the activation energies for

self-interstitial migration are very low compared to

the vacancy migration energy, and they can rarely be

measured with any accuracy Instead, in most cases

only the Stage I annealing temperatures have been

measured In the associated experiments, specimens

for a given metal are irradiated at such low

tempera-tures that the Frenkel pairs are retained Their

con-centration is correlated with the increase of the

electrical resistivity Subsequent annealing in stages

then reveals when the resistivity declines again

upon reaching a certain annealing temperature The

first annealing, Stage I, occurs when self-interstitials

become mobile and in the process recombine with

vacancies, form clusters of self-interstitials, or aretrapped at impurities Table 8 lists the Stage Itemperature,7 Tm

I , for pure metals as well as twoalloys that represent ferritic and austenitic steels.For a few cases, an associated activation energy Em

I

is known, and in even fewer cases, a preexponentialfactor, DI

0, has been estimated

1.01.5 Interaction of Point Defects with Other Strain Fields

1.01.5.1 The Misfit or Size InteractionMany different sources of strain fields may exist inreal solids, and they can be superimposed linearly

if they satisfy linear elasticity theory If this is thecase, we need to consider here only the interactionbetween one particular defect located at rd and anextraneous displacement field u0(r) that originatesfrom some other source than the defect itself Inparticular, it may be the field associated with externalforces or deformations applied to the solid, or it may

be the field generated by another defect in the solid

To find the interaction energy, we assume that thedefect under consideration is modeled by applying aset of Kanzaki26 forces f(a)(R(a)), a¼ 1, 2, , z, at

z atomic positions R(a)as described in greater detail

Figure 16 Two migration steps are favored by self-interstitials in bcc metals; the left is accompanied by a rotation, while the right maintains the dumbbell orientation.

in Appendix B We imagine that once applied, the

extraneous source is switched on, whereupon the

atoms are displaced by the field u0(r), and work is

done by the Kanzaki forces of the amount

W ¼ Xz

¼1

fðÞ u0 rdþ RðÞ

½39

If the displacement field varies slowly from one

atom position to the next, we may employ the Taylor

expansion for each displacement component,

W1 Pij e0ijðrdÞ ½42where e0

ijðrdÞ is the extraneous strain field at thelocation of the defect

Point defects can also be modeled as inclusions,

as we have seen, and they can be characterized by atransformation strain tensor, ekl InAppendix Bit isshown, witheqn [B25], that the dipole tensor is thengiven by

Pij ¼ OCijklekl ½43and the interaction energy can be written as

W ¼ OCijklekle0ij ¼ O ekls0kl ½44Finally, for the simplest model of a point defect as amisfitting spherical inclusion, as treated inAppendix A,

ekl ¼13

DV

O dkland

or defect size gave this energy the name of misfit or sizeinteraction

1.01.5.2 The Diaelastic or ModulusInteraction

The definition of the dipole tensor and of multipoletensors with Kanzaki forces assumes that they areapplied to atoms in a perfect crystal and selectedsuch that they produce a strain field that is identical

to the actual strain field in a crystal with the defectpresent In particular, the dipole tensor reproducesthe long-range part of this real strain field, and it can

be determined from the Huang scattering ments of crystals containing the particular defects.The actual specification of the exact Kanzaki forces

measure-is therefore not necessary However, if the crystal measure-is

Table 8 Annealing temperatures for Stage I and

migra-tion properties estimated from them for self-interstitials

Metal Stage I T m

I (K) E m

0 (106m 2 s1)

Source: Ehrhart, P.; Schultz, H In Landolt-Bo¨rnstein;

Springer-Verlag: Berlin, 1991; Vol III/25.

under the influence of external loads, the Kanzaki

forces may be different in the deformed reference

crystal Consider for example the case of a crystal

with a vacancy and under external pressure In the

absence of pressure, the vacancy relaxation volume

has a certain value However, under pressure, the

volume of the vacancy may change by a different

amount than the average volume per atom, and

therefore, the Kanzaki forces necessary to reproduce

this additional change will have to change from their

values in the crystal under no pressure The change

of the Kanzaki forces induced by the extraneous

strain field may then also be viewed as a change in

the dipole tensor by dPij Assuming that this change

is to first-order linear in the strains,

dPij ¼ aijkle0kl ½46

The tensor aijkl has been named diaelastic

polariz-ability by Kro¨ner27 based on the analogy with

dia-magnetic materials

When the change of the dipole tensor is included

in the derivation of the interaction energy

per-formed in the previous section, an additional

contri-bution arises, namely

The factor of 1/2 appears here because when the

extraneous strain field is switched on for the purpose

of computing the work, the induced Kanzaki forces

are also switched on This additional contribution W2,

the diaelastic interaction energy, is quadratic in the

strains in contrast to the size interaction, eqn [44],

which is linear in the strain field

A crystalline sample that contains an atomic

frac-tion n of well-separated defects and is subject to

external deformation will have an enthalpy of

per unit volume

It follows from this formula that the presence of

defects changes the effective elastic constants of the

sample by

DCijkl ¼ n

Such changes have been measured in single crystal

specimens of only a few metals that were irradiated at

cryogenic temperatures by thermal neutrons or

elec-trons Significant reductions of the shear moduli C44

and C0¼ (C11–C12)/2 are observed from which the

corresponding diaelastic shear polarizabilities listed

Frenkel pair, and hence each one is the sum of theshear polarizabilities of a self-interstitial and avacancy By annealing these samples and observingthe recovery of the elastic constants to their originalvalues, one can conclude that the shear polarizabil-ities of vacancies are small and that the overwhelmingcontribution to the values listed in Table 9comesfrom isolated, single self-interstitials

The softening of the elastic region around the interstitial to shear deformation is not intuitivelyobvious However, the theoretical investigations byDederichs and associates29on the vibrational proper-ties of point defects have provided a rather convinc-ing series of results, both analytical and by computersimulations According to these results, the self-interstitial dumbbell axis is highly compressed, up

self-to 0.6 of the normal interaself-tomic distance betweenneighboring atoms Therefore, the dumbbell axiscan be easily tilted by shear of the surrounding latticeand thereby release some of this axial compression.The weak restoring forces associated with this tiltintroduce low-frequency vibrational modes that arealso responsible for the low migration energy of self-interstitials in pure metals

Computer simulations carried out by Dederichs

et al.30with a Morse potential for Cu gave the resultspresented inTable 10

While the shear polarizabilities compare ably with the experimental results for Cu listed inTable 9, the bulk polarizability in the last column of

sign for the self-interstitial The experimental resultsfor Cu indicate that the bulk polarizability for theFrenkel pair is close to zero Atomistic simulations

Table 9 Diaelastic shear polarizabilities per Frenkel pair

a44 (eV) (a11a12)/2

(eV)

(a11þ2a12)/3 (eV)

have also been reported by Ackland31using an

effec-tive many-body potential The predicted diaelastic

polarizabilies all turn out to be of the opposite sign

than those reported by Dederichs and those obtained

from the experimental measurements Furthermore,

Ackland also reports that the simulation results are

dependent on the size of the simulation cell, that is,

on the number of atoms Evidently, the predictions

depend very sensitively on the type and the

particu-lar features of the interatomic potential as well as on

the boundary conditions imposed by the periodicity

of the simulation cell

The model of the inhomogeneous inclusion

pio-neered by Eshelby21 may be instructive to explain

the diaelastic polarizabilities of vacancies and

self-interstitials A defect is viewed as a region with elastic

constants different from the surrounding elastic

con-tinuum We suppose that this region occupies a

spherical volume of NO, has isotropic elastic

con-stants K*and G*, and is embedded in a medium with

elastic constants K and G Here, O is the volume per

atom, N the number of atoms in the defect region,

and K and G the bulk and shear modulus, respectively

As Eshelby21 has shown, when external loads are

applied to this medium and they produce a strain

field e0ij in the absence of the spherical

inhomogene-ity, then an interaction is induced upon forming it

For an isotropic crystal,eqn [47]assumes the same

form aseqn [50], and the diaelastic polarization

ten-sor has then only two components These can now be

identified with the two coefficients ineqn [50]to give

the bulk polarizability

Let us first apply the formulae [52] to [55] to avacancy It seems plausible to select N ¼ 1 and assumethat K * ¼ G * ¼ 0 Then

Next, we consider the bulk polarizability of interstitials The two atoms that form the dumbbellare under compression, and the local bulk modulusthat controls their separation distance may be esti-mated as follows:

Du0¼ Du  DV ¼ ð1=gE 1ÞDV ½59where DV is the relaxation volume of the self-interstitial as evaluated for the linear elastic medium

As we have seen in Section 1.01.4, DV ¼ 1.10164Ofor fcc and DV ¼ 0.6418O for bcc crystals With therelations[58] and [59]we obtain

O þ gE ½60and with it the bulk polarizability of self-interstitials as

Numerical values for it are listed in the thirdcolumn of Table 11 We note that these valuesare negative, meaning that self-interstitials increasethe effective bulk modulus of irradiated samples, andthis is in contrast to the results from atomistic simu-lations by Dederichs et al.29 obtained with a Morsepotential for Cu

To rationalize the shear polarizability of interstitials, we recall that the crystal lattice that

self-surrounds the dumbbell becomes significantly dilated

due to nonlinear elastic effects This additional

dila-tation, dV/O, has to be added to DV/O to obtain

relaxation volumes that agree with experimental

values We repeat the values for dV/O in the last

column of Table 11 as a reminder As a result of

this additional dilatation, the atomic structure

adja-cent to the dumbbell is more like that in the liquid

phase, as it lost its rigidity with regard to shear For

this dilated region, consisting of NG atoms that

include the two dumbbell atoms, we assume that its

shear modulus G*¼ 0 Then

BI ¼ BV ¼15ð1  nÞ

7 5nand

aG

I ¼ NGOGBI ½62

If the dilated region extends out to the first,

sec-ond, or third nearest neighbors, then NG¼ 14, 20, or

44, respectively, for fcc crystals, and NG¼ 10, 16,

or 28 for bcc crystals From these numbers we shall

select those that enable us to predict a value for aG

I

that comes closest to the experimental value

Matching it for fcc Cu indicates that the dilated

region reaches out to third nearest neighbors, and

hence NG¼ 44 However, the best match for bcc

Mo is obtained with NG¼ 10, a region that only

includes the dumbbell and its first nearest neighbors

These respective values for NGare also adopted for

the other fcc and bcc elements inTable 11, and the

shear polarizabilities so obtained are listed in the fifth

column

To compare these estimates with experimental

results, the approximation given ineqn [55]is used

with the data inTable 9for the shear polarizabilities

of Frenkel pairs These isotropic averages are listed

in the sixth column of Table 11, and they are to

be compared with aGþ aG

It is seen that the

inhomogeneity model is quite successful in ing the experimental results, in spite of its simplicityand lack of atomistic details

explain-1.01.5.3 The Image InteractionThis interaction arises not from the strain field ofother defects or from applied loads but is caused bythe changing strain field of the point defect itself as itapproaches an interface or a free surface of the finitesolid We have shown inSection 1.01.4that the strainenergy associated with a point defect is given by

U0¼ 2K mO3K þ 4m

Vrel

O

2

¼2mð1 þ nÞ9ð1  nÞ

ðVrelÞ2

when the defect is in the center of a spherical bodywith isotropic elastic properties or when the defect issufficiently far removed from the external surfaces of

a finite solid This strain energy has been obtained byintegrating the strain energy density of the defectover the entire volume of the solid, and since thisdensity diminishes as r6, where r is the distance fromthe defect center, it is concentrated around the defect.Nevertheless, close to a free surface, the strain field

of the defect changes, and with it the strain energy.This change is referred to as the image interactionenergy Uim, and the actual strain energy of the defectbecomes

U ðhÞ ¼ U0þ UimðhÞ ½64Here, h is the shortest distance to the free surface.The strain energy of the defect, U(h), changes with

h for two reasons First, as the defect approaches thesurface, the integration volume over regions of highstrain energy density diminishes, and second, thestrain field around the defect becomes nonsphericaland also smaller The evaluation of both of thesecontributions requires advanced techniques for solv-ing elasticity problems

Table 11 Diaelastic polarizabilities in electron volts for vacancies and self-interstitials estimated with an Isotropic Inhomogeneity Model

Eshelby21 has shown that the strain energy of a

defect, modeled as a misfitting inclusion of radius r0,

in an elastically isotropic half-space, is given by

where h is the distance from the center of the defect

to the surface Equation [65] clearly demonstrates

that the strain energy of the defect decreases as it

approaches the surface The minimum distance h0is

obviously that for which U(h) ¼ 0, and it is given by

h0¼ ð1 þ nÞ

4

Another case for the image interaction has been

solved by Moon and Pao,32 namely when a point

defect approaches either a spherical void of radius R

or, when inside a solid sphere of radius R, approaches

its outer surface

For a defect in a sphere, its strain energy changes

with its distance r from the center of the sphere

2n%

½67

while the strain energy of a defect at a distance r from

the void center is given by

2nþ2%

½68

Again, at a distance of closest approach to the

void, h0(R), the strain energy of the defect vanishes

The numerical solutions of US(R þ h0)¼ 0 and of

UV(Rh0) ¼ 0 gives the results for h0/r0 shown in

Figure 17 There is a modest dependence on the radius

of curvature of the surface Approximately, however,

the defect strain energy becomes zero about halfway

between the top and first subsurface atomic layer,

assuming that r0is equal to the atomic radius

1.01.6 Anisotropic Diffusion in

Strained Crystals of Cubic Symmetry

The diffusion of the point defects created by the

irradiation and their subsequent absorption at

dislocations and interfaces in the material is themost essential process that restores the material toits almost normal state The adjective of ‘almost nor-mal’ is anything but a casual remark here, but it hints

at some subtle effects arising in connection with thelong-range diffusion that constitute the root cause forthe gradual changes that take place in crystallinematerials exposed to continuous irradiation at ele-vated temperatures If these effects were absent, then

a steady state would be reached in the material ject to continuous irradiation at a constant rate andtemperature in which the rate of defect generationwould be balanced by their absorption at sinks, mean-ing the above-mentioned dislocations and interfaces

sub-As vacancies and self-interstitials are created asFrenkel pairs in equal numbers, they would also beabsorbed in equal numbers at these sinks At thispoint, the microstructure of these sinks would also

be in a steady, unchanging state While this steadystate would be different from the initial microstruc-ture or the one reached at the same temperature but

in the absence of irradiation, it would correspond tomaterial properties that reached constant values.The subtle effects alluded to in the above remarksarise from the interactions of the point defectswith strain fields created both internally by thesinks and externally by applied loads and pressures

on the materials that constitute the reactor nents The internal strain fields from sinks give rise tolong-range forces that render the diffusion migrationnonrandom, while the external strains induce aniso-tropic diffusion throughout the entire material In the

compo-0.55 0.6 0.65 0.7 0.75 0.8 0.85

1

Sphere Halfspace Void

Surface radius/atomic radius

Ni Poisson's ratio = 0.287

Figure 17 Distance to surface where the defect strain energy disappears.

next section, we derive the diffusion equations for

cubic materials to clearly expose these two

funda-mental effects

1.01.6.1 Transition from Atomic to

Continuum Diffusion

During the migration of a point defect through the

crystal lattice, it traverses an energy landscape that

is schematically shown in Figure 18 The energy

minima are the stable configurations where the

defect energy is equal to Ef(r), the formation

energy, but modified by the interactions with

inter-nal and exterinter-nal strain fields, which in general vary

with the defect locationr In order to move to the

adjacent energy minimum, the defect has to be

thermally activated over the saddle point that has

an energy

ESðrÞ ¼ EfðrÞ þ EmðrÞ ½69

where EmðrÞ is the migration energy As the

prop-erties of the point defect, such as its dipole tensor

and its diaelastic polarizability, are not necessarily

the same in the saddle point configurations as in

the stable configuration, the interactions with the

strain fields are different, and the envelope of the

saddle point energies follows a different curve than

the envelope of the stable configuration energies,

as indicated inFigure 18 For a self-interstitial, we

must also consider the different orientations that itmay have in its stable configuration Accordingly,let Cmðr; tÞ be the concentration of point defects atthe location r and at time t with an orientation m.For instance, the point defect could be the self-interstitial in an fcc crystal, in which case, there arethree possible orientations for the dumbbell axisand m may assume the three values 1, 2, or 3 if theaxis is aligned in the x1, x2, or x3direction, respec-tively The elementary process of diffusion consistsnow of a single jump to one adjacent site atr þ R,where R is one of the possible jump vectors.The rate of change with time of the concentration

To circumvent this complication, one considers anensemble of identical systems, all having identicalmicrostructures, and identical internal and externalstress fields The ensemble average of the defectconcentration at each site, denoted simply as C(r,t)without a subscript, is now assumed to be the ther-modynamic average such that

Substituting this into eqn [70] on both sidesconstitutes another assumption To see this, supposethat the defect concentrations Cnðr  R; tÞ on allneighbor sites happen, at the particular instance t,

to be aligned in one direction Since their new

Potential profile

Envelope for formation energies

Envelope for saddle points

r r + R/2 r + R

Figure 18 Schematic of the potential energy profile for a

migrating defect.