Comprehensive nuclear materials 1 01 fundamental properties of defects in metals

1.01

Fundamental Properties of Defects in Metals

W. G. Wolfer
Ktech Corporation, Albuquerque, NM, USA; Sandia National Laboratories, Livermore, CA, USA

Published by Elsevier Ltd.

1.01.1
1.01.2
1.01.3
1.01.3.1
1.01.3.2
1.01.3.3
1.01.4
1.01.4.1
1.01.4.2
1.01.4.3
1.01.4.4
1.01.5
1.01.5.1
1.01.5.2
1.01.5.3
1.01.6
1.01.6.1
1.01.6.2
1.01.6.3
1.01.7
1.01.7.1
1.01.7.2

1.01.7.3
1.01.7.4
1.01.7.4.1
1.01.7.4.2
1.01.7.4.3
1.01.7.4.4
1.01.8
1.01.8.1
1.01.8.2
1.01.8.2.1
1.01.8.2.2
1.01.8.3
1.01.9
Appendix A
A1
A2
Appendix B
B1
B2
B3
B4
B5
References

Introduction
The Displacement Energy
Properties of Vacancies
Vacancy Formation
Vacancy Migration
Activation Volume for Self-Diffusion

Properties of Self-Interstitials
Atomic Structure of Self-Interstitials
Formation Energy of Self-Interstitials
Relaxation Volume of Self-Interstitials
Self-Interstitial Migration
Interaction of Point Defects with Other Strain Fields
The Misfit or Size Interaction
The Diaelastic or Modulus Interaction
The Image Interaction
Anisotropic Diffusion in Strained Crystals of Cubic Symmetry
Transition from Atomic to Continuum Diffusion
Stress-Induced Anisotropic Diffusion in fcc Metals
Diffusion in Nonuniform Stress Fields
Local Thermodynamic Equilibrium at Sinks
Introduction
Edge Dislocations
Dislocation Loops
Voids and Bubbles
Capillary approximation
The mechanical concept of surface stress
Surface stresses and bulk stresses for spherical cavities
Chemical potential of vacancies at cavities
Sink Strengths and Biases
Effective Medium Approach
Dislocation Sink Strength and Bias
The solution of Ham
Dislocation bias with size and modulus interactions
Bias of Voids and Bubbles
Conclusions and Outlook
Elasticity Models: Defects at the Center of a Spherical Body

An Effective Medium Approximation
The Isotropic, Elastic Sphere with a Defect at Its Center
Representation of Defects by Atomic Forces and by Multipole Tensors
Kanzaki Forces
Volume Change from Kanzaki Forces
Connection of Kanzaki Forces with Transformation Strains
Multipole Tensors for a Spherical Inclusion
Multipole Tensors for a Plate-Like Inclusion

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1


2

Fundamental Properties of Defects in Metals

Abbreviations
bcc
CA
CD
dpa
EAM
fcc
hcp
INC
IHG
SIA
Vac

Body-centered cubic
Cavity model
Center of dilatation model
Displacements per atom
Embedded atom method
Face-centered cubic
Hexagonal closed packed
Inclusion model
Inhomogeneity model
Self-interstitial atom
Vacancy

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 resistivity 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 particle, 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 dissipate its kinetic energy and come to rest within the
crystal lattice as a self-interstitial defect. It is immediately 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 average, each atom has been displaced once.
At the elevated temperatures that exist in nuclear
reactors, vacancies and self-interstitials diffuse
through the crystal. As a result, they will encounter
each other, either annihilating each other or forming
vacancy and interstitial clusters. These events occur
already in their nascent collision cascade, but if defects
escape their collision cascade, they may encounter the
defects created in other cascades. In addition, migrating vacancy defects and interstitial defects may also be
captured at other extended defects, such as dislocations, cavities, grain boundaries and interface boundaries of precipitates and nonmetallic inclusions, such
as oxide and carbide particles. The capture events at
these defect sinks may be permanent, and the migrating defects are incorporated into the extended
defects, or they may also be released again.
However, regardless of the complex fate of each
individual defect, one would expect that eventually
the numbers of interstitials and vacancies that arrive
at each sink would become equal, as they are produced in equal numbers as Frenkel pairs. Therefore,
apart from statistical fluctuations of the sizes and
positions of the extended defects, or the sinks, the
microstructure of sinks should approach a steady
state, and continuous irradiation should change the
properties of metals no further.
It came as a big surprise when radiation-induced
void swelling was discovered with no indication of a
saturation. In the meantime, it has become clear that
the microstructure evolution of extended defects and
the associated changes in macroscopic properties of
metals in general is a continuing process with displacement damage.
The fundamental reason is that the migration

of defects, in particular that of self-interstitials and
their 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 of
self-interstitials and vacancies to different types of sinks.
Guided then by this fate of radiation-produced
atomic defects in metals, the following topics are
presented in this chapter:
1. The displacement energy required to create a
Frenkel pair.
2. The energy stored within a Frenkel pair that consists of the formation enthalpies of the selfinterstitial and the vacancy.
3. The dimensional changes that a solid suffers when
self-interstitial and vacancy defects are created,


Fundamental Properties of Defects in Metals

4.

5.

6.

7.

and how these changes manifest themselves either
externally or internally as changes in lattice
parameter. These changes then define the formation and relaxation volumes of these defects and
their dipole tensors.
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 properties, thereby changing the overall elastic constants
of the defect-containing solid. This leads to the
concept of elastic polarizability parameters for
the atomic defects.
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.
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.
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.

3

where mc2 is the rest energy of an electron and
L ¼ 4 mM/(m þ M)2. The approximation on the right

is adequate because the electron mass, m, is much
smaller than the mass, M, of the recoiling atom.
Changing the direction of the electron beam in
relation to the orientation of single crystal film specimens, one finds that the threshold energy varies
significantly. However, for polycrystalline samples,
values averaged over all orientations are obtained,
and these values are shown in Figure 1 for different
metals as a function of their melting temperatures.1
First, we notice a trend that Td increases with the
melting temperature, reflecting the fact that larger
energies of cohesion or of bond strengths between
atoms also lead to higher melting temperatures.
We also display values of the formation energy
of a Frenkel pair. Each value is the sum of the
corresponding formation energies of a self-interstitial
and a vacancy for a given metal. These energies are
presented and further discussed below. The important
point to be made here is that the displacement energy
required to create a Frenkel pair is invariably larger than
its formation energy. Clearly, an energy barrier exists for
the recoil process, indicating that atoms adjacent to the
one that is being displaced also receive some additional
kinetic energy that is, however, below the displacement
energy Td and is subsequently dissipated as heat.
The displacement energies listed in Table 1 and
shown in Figure 1 are averaged not only over crystal
orientation but also over temperature for those metals

50
Displacement energy Td(eV)

fcc Frenkel pair (eV)
bcc Frenkel pair (eV)

Scattering of energetic particles from external sources,
be they neutrons, electrons, ions, or photons, or emission 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 displacement 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 threshold electron energy, Emin, can be found below which no
Frenkel pairs are produced. The corresponding recoil
energy is given by relativistic kinematics as


2mc 2 þ Emin
m
Emin
½1Š
E
%
4
1
þ
Td ¼ LEmin
min
2mc 2 þ LEmin
2mc 2
M

Displacement and Frenkel pair energies (eV)

1.01.2 The Displacement Energy
40

30

20

10

0

0

500

1000 1500 2000 2500 3000 3500
Melting temperature (K)

4000

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


4

Fundamental Properties of Defects in Metals

Table 1
Element

Displacement and Frenkel pair energies of elemental metals
Symbol

Z

Melt temp. ( K)

M

Td (eV)

Frenkel pair
energy (eV)
fcc

Silver
Aluminum
Gold
Cadmium
Cobalt
Chromium
Copper
Iron
Indium
Iridium

Magnesium
Molybdinum
Niobium
Neodymium
Nickel
Lead
Palladium
Platinum
Rhenium
Tantalum
Titanium
Vanadium
Tungsten
Zinc
Zirconium

Ag
Al
Au
Cd
Co
Cr
Cu
Fe
In
Ir
Mg
Mo
Nb
Nd

Ni
Pb
Pd
Pt
Re
Ta
Ti
V
W
Zn
Zr

47
13
79
48
27
24
29
26
49
77
12
42
41
60
28
82
46
78

75
73
22
23
74
30
40

107.9
26.98
197.0
112.4
58.94
52.01
63.54
55.85
114.8
192.2
24.32
95.95
92.91
144.3
58.71
207.2
106.4
195.1
186.2
181.0
47.90
50.95

183.9
65.38
91.22

fcc
fcc
fcc
hcp
hcp
bcc
fcc
bcc
tetragonal
fcc
hcp
bcc
bcc
hcp
fcc
fcc
fcc
fcc
hcp
bcc
hcp
bcc
bcc
hcp
hcp

1235
933.5
1337
594.2
1768
2180
1358
1811
429.8
2719
923.2
2896
2750
1289
1728
600.6
1828
2041
3458
3290
1941
2183
3695
692.7
2128

26.0
15.3
34.0
19.0

23.0
28.0
18.3
17.4
10.5
46.0
10.0
32.4
28.2
9.30
22.0
11.8
34.0
34.0
44.0
26.7
20.8
28.0
44.0
12.0
22.5

bcc

6.52
4.96
5.98

4.40
6.66

4.03

6.69
4.01
10.5
3.24
6.51
12.7
5.77
3.26
8.90

Source: Displacement energies from Jung, P. In Landolt-Bo¨rnstein; Springer-Verlag: Berlin, 1991; Vol. III/25, pp 8–11.

50
Td fcc
Td bcc
Td hcp and others
Td, eV

40
Displacement energy (eV)

where the displacement energy has been measured as
a function of irradiation temperature. For some materials, such as Cu, a significant decrease of the displacement 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 recombination. With increasing irradiation temperature, however,

the self-interstitial may escape, and this would manifest itself as an apparent reduction in the displacement
energy with increasing temperature. On the other
hand, Jung2 has 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, as Figure 2 demonstrates,
the displacement energy increases with the bulk modulus. Since the bulk modulus decreases with temperature, so will the displacement energy.
The correlation of the displacement energy with
the bulk modulus appears to be a somewhat better

30

20

10

0
0

50

100

150

200

250

300

350

400

Bulk modulus (GPa)
Figure 2 Displacement energies for elemental metals as a
function of their bulk modulus.

empirical relationship than the correlation with the
melt temperature. However, one should not read
too much into this, as the bulk modulus B, atomic


Fundamental Properties of Defects in Metals

volume O, and melt temperature of elemental metals
approximately satisfy the rule
BO % 100kB Tm
discovered by Leibfried3 and shown in Figure 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 equilibrium thermal vacancy concentration of
 f

EV À TSVf
eq
½2Š
CV ðT Þ ¼ exp À
kB T
given in atomic fractions. Here, EVf is the vacancy
formation enthalpy, and SVf is the vacancy formation
entropy. The thermal vacancy concentration can be
measured by several techniques as discussed in Damask 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 in Table 2.
When these values for the metallic elements are
plotted versus the melt temperature in Figure 4, an
approximate correlation is obtained, namely
EVf % Tm =1067

½3Š

4000

Bulk mod.*atom. vol./(100 k)

3500
3000
2500

1500
1000
500
0

Using the Leibfried rule, a new approximate correlation emerges for the vacancy formation enthalpy
that has become known as the cBO model8; the constant c is assumed to be independent of temperature
and pressure. As seen from Figure 5, however, the
experimental values for EVf correlate no better with
BO than with the melting temperature.
It is tempting to assume that a vacancy is just a
void and its energy is simply equal to the surface
area 4pR2 times the specific surface energy g0. Taking
the atomic volume as the vacancy volume, that is,
O ¼ 4pR3/3, we show in Figure 6 the measured
vacancy formation enthalpies as a function of
4pR2g0, using for g0 the values9 at half the melting
temperatures. It is seen that EVf is significantly less, by
about a factor of two, compared to the surface energy
of the vacancy void so obtained. Evidently, this simple approach does not take into account the fact that
the atoms surrounding the vacancy void relax into
new positions so as to reduce the vacancy volume VVf
to something less than O. The difference
VVrel ¼ VVf À O

0

500

1000 1500 2000 2500 3000 3500 4000

Melt temperature (K)

Figure 3 Leibfried’s empirical rule between melting
temperature and the product of bulk modulus and atomic
volume.

½4Š

is referred to as the vacancy relaxation volume. The
experimental value7 for the vacancy relaxation of Cu
is À0.25O, which reduces the surface area of the
vacancy void by a factor of only 0.825, but not by a
factor of two.
The difference between the observed vacancy formation enthalpy and the value from the simplistic
surface model has recently been resolved. It will be
shown in Section 1.01.7 that the specific surface
energy is in fact a function of the elastic strain tangential to the surface, and when this surface strain
relaxes, the surface energy is thereby reduced. At the
same time, however, the surface relaxation creates a
stress field in the surrounding crystal, and hence a
strain energy. As a result, the energy of a void after
relaxation is given by
FC ½eðRÞ; eà Š ¼ 4pR2 g½eðRÞ; eà Š þ 8pR3 me2 ðRÞ

2000

5

½5Š

The first term is the surface free energy of a void with
radius R, and it depends now on a specific surface
energy that itself is a function of the surface strain
e(R) and the intrinsic residual surface strain e* for a
surface that is not relaxed. The second term is the
strain energy of the surrounding crystal that depends
on its shear modulus m. The strain dependence of the
specific surface energy is given by
g½e; eà Š ¼ g0 þ 2ðmS þ lS Þð2eà þ eÞe

½6Š

Here, g0 is the specific surface energy on a surface
with no strains in the underlying bulk material.


6

Fundamental Properties of Defects in Metals

Table 2

Crystal and vacancy properties

Metal

Crystal

Melt temp. (K)

KO (eV)

Debye temp. (K)

EfV (eV)

Em
v (eV)

g0 (J mÀ2)

Ag
Al
Au
Be
Co
Cr
Cs
Cu
Fe
Hf
Ir
K
Li
Mg
Mn
Mo
Na
Nb
Nd

Ni
Os
Pb
Pd
Pt
Re
Rh
Ru
Sb
Sr
Ta
Ti
Tl
U
V
W
Zn
Zr

fcc
fcc
fcc
hcp
hcp
bcc
bcc
fcc
bcc
hcp
fcc

bcc
bcc
hcp
bcc
bcc
bcc
bcc
hcp
fcc
hcp
fcc
fcc
fcc
hcp
fcc
hcp
rbh*
fcc
bcc
hcp
hcp
bco**
bcc
bcc
hcp
hcp

1235
933.5
1337

1560
1768
2180
301.6
1358
1811
2506
2719
336.7
453.7
923
1519
2896
371
2750
1289
1728
3306
600.6
1828
2041
3458
2237
2607
904
1050
3290
1941
577.2
1408

2183
3695
693
2128

10.9
7.89
18.1
6.57
13.1
12.1
41.2
10.1
12.3
15.3
31.3
1.55
1.63
5.13
9.15
25.4
1.70
19.3
6.8
12.5
36.7
8.46
17.7
26.7
34

32.6
18.6
7.9

229.2
430.6
162.7

1.11
0.67
0.93
0.8

588.4

2.1

1.09
1.02
1.33
1.30
2.22
2.01

349.6
483.3

1.28
1.90

0.66
0.61
0.71
0.87
0.72
0.95
0.084
0.70
0.55

25.3
11.8
7.7
13
13.5
30.8
6.49
13.8

92.7
369.5

0.48
0.80

473.4
157.1
254.6

3.10

0.34
2.70

481.4

1.79

0.038
0.50
1.30
1.35
0.03
0.55
0.81
1.04

106.6
277.9

0.58
1.85
1.35
3.10
2.50

0.43
1.03
1.43
2.20
1.50

264.7

3.10

0.70

399.4
384.3

2.10
3.60
0.54

0.50
1.70
0.42
0.58

1.57
2.12
1.92
2.65
0.129
0.472
0.688
2.51
0.234
2.31
2.38

2.95
0.54
1.74
2.20
3.13
2.33
2.65
0.461
0.358
2.49
1.75
0.55
1.78
2.30
2.77
0.896
1.69

*

rbh: rhombohedral
bco: body-centered orthorhombic

**

However, such a surface possesses an intrinsic, residual 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
mS þ lS ¼ ðm þ lÞh ¼ mh=ð1 À 2nÞ

½7Š

where l is the Lame’s constant and n is Poisson’s
ratio for the bulk solid. Computer simulations on

freestanding thin films have shown10 that the surface
layer is effectively a monolayer, and h can be approximated by the Burgers vector b. For planar crystal surfaces, the residual surface strain parameter e* is found to
be between 3 and 5%, depending on the surface orientation relative to the crystal lattice. On surfaces with
high curvature, however, e* is expected to be larger.
The relaxation of the void surface can now be
obtained as follows. We seek the minimum of the
void energy as defined by eqn [5] by solving
@FC =@e ¼ 0. The result is
eðRÞ ¼ À

ðmS þ lS ÞeÃ
h eÃ
¼À
mR þ ðmS þ lS Þ
ð1 À 2nÞR þ h

½8Š

and this relaxation strain changes the initially unrelaxed void volume


Fundamental Properties of Defects in Metals

4

4
bcc Hf/v, eV
fcc Hf/v, eV
hcp Hf/v, eV

3
2.5
2
1.5
1

3
2.5
2
1.5
1
0.5

0.5

0

0

1
2
3
4
5
Surface energy of a vacancy (eV)

6

Figure 6 Correlation between the surface energy of a
vacancy void and the vacancy formation energy.
3.5
Vacancy formation energy and its bulk contribution (eV )

4
fcc Hf/v, eV
bcc Hf/v, eV
hcp Hf/v, eV

3.5
3
2.5
2
1.5
1
0.5
0

0

500 1000 1500 2000 2500 3000 3500 4000
Melting temperature (K)

Figure 4 Vacancy formation energies as a function of
melting temperature.

Vacancy formation enthalpy (eV)

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

3.5
Vacancy formation enthalpy (eV)

Vacancy formation energy (eV)

3.5

0

7

0

5

10

15

20

25

30

35

40

Exp. value
Computed values
Bulk strain energy

3

2.5

Ni

2

1.5

1

0.5

0
–0.35

–0.3

Bulk modulus * atomic volume (eV)

4p 3
½9Š
R
3
consisting of n aggregated vacancies, by the amount
nO ¼

½10Š

Applying these equations to a vacancy, for which
n ¼ 1 and R % b, we obtain
VVrel
O

¼À

Ã

3e
2ð1 À nÞ

and for the vacancy formation energy

–0.2

–0.15

–0.1

–0.05

0

Vacancy relaxation volume

Figure 5 Vacancy formation energy versus the product of
bulk modulus and atomic volume.

V rel ðRÞ ¼ 3nOeðRÞ

–0.25

½11Š

Figure 7 Vacancy formation energy and its dependence
on the relaxation volume.

(
EVf ¼ 4pR2


2 )

2ð3 À 4nÞ VVrel
g0 À mb
9ð1 À 2nÞ O

 rel 2
2
V
þ mO V
O
3

½12Š

This equation is evaluated for Ni and the results are
shown in Figure
7 as a function of the vacancy relaxa
tion volume VVrel =O. It is seen that relaxation volumes
of À0.2 to À0.3 predict a vacancy formation energy
comparable to the experimental value of 1.8 eV.


8

Fundamental Properties of Defects in Metals

Few experimentally determined values are available for the vacancy relaxation volume, and their
accuracy is often in doubt. In contrast, vacancy formation energies are better known. Therefore, we
use eqn [12] to determine the vacancy relaxation
volumes from experimentally determined vacancy
formation energies. The values so obtained are listed

in Table 3, and for the few cases7 where this is
possible, they are compared with the values reported
from experiments. Computed values for the vacancy
relaxation volumes are between À0.2O and À0.3O
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
derive eqn [12] is based on several approximations:
isotropic, linear elasticity, a surface energy parameter, g0, that represents an average over different crystal 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 connection 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 formation entropy, SVf , are in order. It originates from the
change in the vibrational frequencies of atoms surrounding 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 coneq
centration, CV , is of the same magnitude as the statistical uncertainty in the vacancy formation enthalpy.
Table 3

1.01.3.2

Vacancy Migration

The atomistic process of vacancy migration consists
of one atom next to the vacant site jumping into this
site and leaving behind another vacant site. The jump

is thermally activated, and transition state theory
predicts a diffusion coefficient for vacancy migration
in cubic crystals of the form
DV ¼ nLV d02 expðSVm ÞexpðÀEVm =kB T Þ
¼ DV0 expðÀEVm =kB T Þ

½13Š

Here, nLV is an average frequency for lattice vibrations,
d0 is the nearest neighbor distance between atoms, SVm
is the vacancy migration entropy, and EVm is the energy
for vacancy migration. It is in fact the energy of an
activation barrier that the jumping atom must overcome, and when it temporarily occupies a position at
the height of this barrier, the atomic configuration is
referred to as the saddle point of the vacancy. It will
be considered in greater detail momentarily.
Values obtained for EVm from experimental measurements are shown in Figure 8 as a function of the
melting point. While we notice again a trend similar
to that for the vacancy formation energy, we find that
EVm for fcc and bcc metals apparently follow different
correlations. However, the correlation for bcc metals
is rather poor, and it indicates that EVm may be related
to fundamental properties of the metals other than
the melting point.
The saddle point configuration of the vacancy
involves not just the displacement of the jumping
atom but also the coordinated motion of other
atoms that are nearest neighbors of the vacancy and
of the jumping atom. These nearest neighbor atoms

Vacancy relaxation volumes for metals

Metal

g 0 (J mÀ2)

m (GPa)

n

HfV (eV)

V rel
V =V (model)

Ag
Al
Au
Cu
Ni
Pb
Pd
Pt
Cr
a-Fe
Mo
Nb
Ta
V
W

1.19
1.1
1.45
1.71
2.28
0.57
1.91
2.40
2.23
2.31
2.77
2.54
2.76
2.51
3.09

33.38
26.18
31.18
54.7
94.6
10.38
53.02
65.1
117.0
90.4
125.8
39.6
89.9

47.9
160.2

0.354
0.347
0.412
0.324
0.276
0.387
0.374
0.393
0.209
0.278
0.293
0.397
0.324
0.361
0.280

1.11 Æ 0.05
0.67 Æ 0.03
0.93 Æ 0.04
1.28 Æ 0.05
1.79 Æ 0.05
0.58 Æ 0.04
1.7, 1.85
1.35 Æ 0.05
2.0 Æ 0.3
1.4, 1.89
3.2 Æ 0.09

2.6, 3.07
2.2, 3.1
2.2 Æ 0.4
3.1, 4.1

À0.247 Æ 0.005
À0.311 Æ 0.003
À0.262 Æ 0.003
À0.259 Æ 0.005
À0.236 Æ 0.004
À0.282 Æ 0.005
À0.239, À0.225
À0.260 Æ 0.003
À0.218 Æ 0.02
À0.278, À0.245
À0.191 Æ 0.004
À0.284, À0.258
À0.264, À0.228
À0.298 Æ 0.028
À0.201, À0.161

V rel
V =V (experiment)
À0.05, À0.38
À0.15 to À0.5
À0.25
À0.2

À0.24, À0.42
À0.05

À0.1


Fundamental Properties of Defects in 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 migration in cubic crystals.
EVm ¼

15C11 C44 ðC11 À C12 Þa3 w
2½C11 ðC11 À C12 Þ þ C44 ð5C11 À 3C12 ފ

½14Š

Here, a is the lattice parameter, C11, C12, and C44 are
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
eqn [14]. Ehrhart et al.7,12 recommend that w ¼ 0.022
for fcc metals and w ¼ 0.020 for bcc metals.

Vacancy migration energy (eV)

2.5

In the derivation of Flynn,11 only the four nearest
neighbor atoms are supposed to move, while all other
atoms are assumed to remain in their normal lattice
positions. On the other hand, Kornblit et al.13 treat the
expansion of the diffusion channel as a quasistatic
elastic deformation of the entire surrounding material. The extent of the expansion is such that the
opened channel is equal to the cross-section of the
jumping atom, and a linear anisotropic elasticity calculation is carried out by a variational method to
determine the energy involved in the channel expansion. A vacancy migration energy is obtained for fcc
metals of
EVm ¼ 0:01727a3 C11

p02

p0 p1 À p22
þ 29 p0 p2 þ 19 p22

½15Š

and the parameters pi will be defined momentarily.
For bcc metals,14 the activation barrier consist of
two peaks of equal height EVmax with a shallow valley
in between with an elevation of EVmin , where

fcc Hm/v, eV
hcp Hm/v, eV
bcc Hm/v, eV

2

9

q0 q1 À q22
q02 À 112 q0 p2 þ 883 q22

½16Š

s0 s1 À s22
2
s0 À 0:29232s0 s2 þ 0:0413s22

½17Š

EVmax ¼ 0:003905a 3 C11
and

1.5

EVmin ¼ 0:002403a 3 C11
1

The parameters pi, qi, and si are linear functions of the
elastic moduli with coefficients listed in Table 4. For
example,

0.5

q1 ¼ 3:45C11 À 0:75C12 þ 4:35C44
0

0

500 1000 1500 2000 2500 3000 3500 4000
Melt temperature (K)

Figure 8 Vacancy migration energy as a function of
melting temperature.

If the depth of the valley is greater than the thermal
energy of the jumping atom, that is, greater than 32 kT ,
then it will be trapped and requires an additional
activation to overcome the remaining barrier of

Table 4
expressions

Figure 9 Second nearest neighbor atom (blue) jumping
through the ring of four next-nearest atoms (green) into
adjacent vacancy in a fcc structure.

Coefficients

for

the

Kornblit

energy

Function

C11

C12

C44

p0
p1
p2
q0
q1
q2
s0
s1
s2

5.29833
0.86667
1.41903
6.36429
3.45
3.32143
3.62621
1.57190
1.46564

À4.76499

À0.3333
À0.88570
À3.66429
À0.75
À0.62143
À2.88241
À0.82810
À0.72184

9.35238
1.9111
1.64444
12.92142
4.35
3.70714
11.30366
4.21984
3.68855


10

Fundamental Properties of Defects in Metals

À max
Á
EV À EVmin . As a result, Kornblit14 assumes that the

vacancy migration energy for bcc metals is given by
&

EVm ¼

EVmax ;
max
2EV À EVmin ;

if EVmax À EVmin 32 kT
if EVmax À EVmin > 32 kT

½18Š

Using the formulae of Flynn and Kornblit, we compute
the vacancy migration energies and compare them
with experimental values in Figure 10.
With a few exceptions, both the Flynn and the
Kornblit values are in good agreement with the experimental results.
The self-diffusion coefficient determines the
transport of atoms through the crystal under conditions near the thermodynamic equilibrium, and it is
defined as

ÀÀ
Á Á
m eq
DSD ¼ DV
CV ¼ nLV a 2 exp SVf þ SVm =k expðÀQSD =kT Þ
½19Š

0
expðÀQSD =kT Þ

¼ DSD

where the activation energy for self-diffusion is
Q SD ¼ EVf þ EVm

½20Š

The most accurate measurements of diffusion coefficients are done with a radioactive tracer isotope of
the metal under investigation, and in this case one
obtains values for the tracer self-diffusion coeffiT ¼ fD
cient DSD
SD that involves the correlation factor
f. For pure elemental metals of cubic structure, f is
a constant and can be determined exactly by computation.15 For fcc crystals, f ¼ 0.78145, and for bcc
crystals, f ¼ 0.72149.

Theoretical vacancy migration energy (eV)

2
Evm, Flynn, fcc
Evm, Kornblit, fcc
Evm, Flynn, bcc
Evm, Kornblit, bcc

1.5

1

To determine the preexponential factor for selfdiffusion
ÀÀ

Á Á
0
¼ nLV a 2 exp SVf þ SVm =k
½21Š
DSD
requires the values for the entropy SVf þ SVm and for
the attempt frequency nLV. Based on theoretical estimates, Seeger and Mehrer5 recommend a value of
2.5 k for the former. The atomic vibration of nearest
neighbor atoms to the vacancy is treated within a
sinusoidal potential energy profile that has a maximum height of EVm . For small-amplitude vibrations,
the attempt frequency is then given by
rffiffiffiffiffiffi
1 EVm
for fcc and by
nLV ¼
a M
rffiffiffiffiffiffiffiffi
1 2EVm
½22Š
nLV ¼
for bcc
a 3M
crystals where M is the atomic mass.
In contrast, Flynn11 assumes that the atomic vibrations can be derived from the Debye model for which
the average vibration frequency is
rffiffiffi
3 kYD
½23Š
nLV ¼
5 h

where YD is the Debye temperature and h is Planck’s
constant.
The calculated preexponential factors for some
fcc metals according to the models by Seeger and
Mehrer5 and by Flynn are listed in Table 5 together
with experimental values. They are also shown in
Figure 11. While the computed values are of the
right order of magnitude, there exists no clear correlation between the experimental and theoretical values.
In fact, the computed values change little from one
metal to another, and the Flynn model predicts values
about twice as large as the model by Seeger and
Mehrer. Either model can therefore be used to provide
a reasonable estimate of the preexponential factor
where no experimental value is available.
1.01.3.3 Activation Volume for
Self-Diffusion

0.5

When the crystal lattice is under pressure p, the selfdiffusion coefficient changes and is then given by
0

0

0.5

1

1.5

Experimental vacancy migration energy (eV)
Figure 10 Comparison of computed vacancy migration
energies according to models by Flynn and Kornblit with
measured values.

2

0
DSD ðT ; pÞ ¼ DSD
expðÀQSD =kT ÞexpðÀpVSD =kT Þ

½24Š

The activation volume VSD can be obtained experimentally by measuring the self-diffusion coefficient
as a function of an externally applied pressure. Such
measurements have been carried out only for a few


Fundamental Properties of Defects in Metals

11

Preexponentials for tracer self-diffusion

Metal

M

a (nm)

QD (K)

Em
V (eV)

Experimental value

S&M

Flynn (m2 sÀ1)

Ag
Al
Au
Cu
Ni
Pb
Pd
Pt

107.9
26.98
197
63.54
58.71
207.2
106.4
195.1

0.409

0.405
0.408
0.361
0.352
0.495
0.389
0.392

229
430.6
162.7
349.6
481.4
106.6
278
240

0.66
0.61
0.71
0.70
1.04
0.43
1.0
1.4

4.5eÀ6
4.7eÀ6
3.5eÀ6
1.6eÀ5

9.2eÀ5
6.65eÀ5
2.1eÀ5
6.0eÀ6

3.00eÀ6
5.69eÀ6
2.29eÀ6
3.55eÀ6
4.39eÀ6
2.11eÀ6
3.53eÀ6
3.11eÀ6

5.90eÀ6
1.08eÀ6
4.16eÀ6
7.02eÀ6
9.18eÀ6
4.01eÀ6
6.46eÀ6
5.66eÀ6

bcc Fe

Seeger Mehrer
Flynn

Experiment
Brown and Ashby

Wallace
Wang et al.

Rb
Na

10–5

Li
K
Cs
Zn
Tl
Element

Theoretical preexponential (m2 s–1)

Table 5

10–6
10–6

Mg
Cd
Pt

10–5

0.0001

Experimental preexponential Do (m2 s–1)

Pb
Ni

Figure 11 Comparison of preexponential factors for
tracer self-diffusion as computed with two models and as
measured.

fcc Fe
Cu
Al
Ag

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
condition
d½ln DSD ðp; Tm ðpÞފ=dpjp¼0 ¼ 0

that


Q dTm 
VSD ¼ 0
Tm dp p¼0

½25Š

where Tm0 is the melting temperature under ambient
conditions.
Brown and Ashby16 have used this relationship
to evaluate the activation volumes for self-diffusion

0

0.2
0.4
0.6
0.8
1
Activation volume/atomic volume

1.2

Figure 12 Activation volumes of elements divided by their
atomic volumes from experiments and from relationship
[25], using the change of the melting temperature with
pressure from different sources.

for a variety of metals. Using more recent values
for the pressure derivative of the melting temperature by Wallace17 and Wang et al.,18 one obtains
activation volumes as shown in Figure 12. They
are in reasonably good agreement with the experimental 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 bcc
metals.


12

Fundamental Properties of Defects in Metals

Table 6

Activation volume for vacancy migration

Metal

VSD =O

VfV =V

Vm
V =V

Ag
Al
Cu
Ni
Pb
bcc Fe

0.872
0.835

0.895
0.841
0.791
0.655

0.753
0.689
0.741
0.764
0.718
0.722

0.119
0.146
0.154
0.077
0.073
À0.067
Figure 13 An atom with its 12 nearest neighbors in the
perfect fcc lattice, on the left, and a [001] self-interstitial
dumbbell with the same nearest neighbors, on the right.

The equilibrium vacancy concentration in a solid
under pressure p is given by
 f
 

E À TSVf
pV f
eq

exp À V
½26Š
CV ðT Þ ¼ exp À V
kB T
kB T
where VVf is the vacancy formation volume. Since
the self-diffusion coefficient is the product of the
thermal vacancy concentration and the vacancy migration coefficient, the activation volume for self-diffusion
is the sum of two contributions, namely
VSD ¼ VVf þ VVm ¼ O þ VVrel þ VVm

½27Š

VVm

with
being the activation volume for vacancy
migration.
If one takes the average of the predicted activation
volumes shown in Figure 12, and the vacancy relaxation volumes from Table 3, one obtains values for
VVm listed in Table 6 and also shown in Figure 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 of Figure 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 fcc
lattice. The centers of the 12 nearest neighbor atoms
are the apexes of a cubo-octahedron that encloses the
single central atom in the perfect lattice, and it can be
shown19 that the cubo-octahedron encloses a volume
of VO ¼ 10O/3. However, around a self-interstitial
dumbbell, this cubo-octahedron expands and distorts,
and now it encloses a larger volume of V001 ¼ 4.435O.
The volume expansion is the difference
DV ¼ V001 À V0 ¼ 1:10164 O

½28Š

which happens to be larger than one atomic volume.
We shall see shortly that the volume expansion of the
entire crystal is even larger due to the elastic strain

field created by the self-interstitial that extends
through the entire solid.
We consider next the self-interstitial defect in a
bcc metal. Here, each atom is surrounded in the
perfect crystal by eight nearest neighbors as shown on
the left of Figure 14. When an extra atom is inserted, it
again forms a dumbbell configuration with another
atom, and the dumbbell axis is now aligned in the
[011] direction, as shown on the right of Figure 14.
The cage formed by the eight nearest neighbor atoms
becomes severely distorted. It is surprising, however,
that the volume change of the cage is only
DV ¼ 0:6418 O

½29Š

less than the volume of the inserted atom to create the
self-interstitial in the bcc structure.
The reason for this is that the bcc structure does
not produce the most densely packed arrangement of
atoms, and some of the empty space can accommodate
the self-interstitial. In contrast, the fcc structure has in
fact the densest arrangement of atoms, and disturbing
it by inserting an extra atom only creates disorder and
lower packing density.
As already mentioned, the large inclusion
volume DV of self-interstitials leads to a strain field


Fundamental Properties of Defects in Metals

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.

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
Appendix A the case of spherical defects in the center
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
in Section 1.01.2 that the displacement energy
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 interstitial into the crystal lattice. However, although this
barrier should be part of the energy to form a selfinterstitial, it is by convention not included. Rather,
the formation energy of a self-interstitial is considered 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 EVf and the migration energy EVm,
that is, the energy for self-diffusion, QSD.

13

When Frenkel pairs are created by irradiation
at cryogenic temperatures, self-interstitials and vacancies can be retained in the irradiated sample.
Subsequent annealing of the sample and measuring
the heat released as the defects migrate and then
disappear provide an indirect method to measure
the energies of Frenkel pairs. Subtracting from these
calorimetric values, the vacancy formation energy
should give the formation energy of self-interstitials.
The values so obtained for Cu7 vary from 2.8 to
4.2 eV, demonstrating just how inaccurate calorimetric measurements are. Besides, measurements have
only been attempted on two other metals, Al and
Pt, with similar doubtful results. As a result, theoretical calculations or atomistic simulations provide perhaps more trustworthy results.
For a theoretical evaluation of the formation
energy, we can consider the self-interstitial as an
inclusion (INC) as described in Appendix A. Accordingly, a volume O of one atom is enlarged by the
amount DV, or in other words, is subject to the transformation strain
eij ¼ dij

where 3 ¼ DV =O

½30Š

The energy associated with the formation of this
inclusion is given in Table A2, and it can be written as
 
2K mO DV 2
½31Š
U0 ¼
3K þ 4m O
This expression for the so-called dilatational strain
energy provides a rough approximation to the formation energy of a self-interstitial in fcc metals when the
above volume expansion results are used.
However, as the nearest neighbor cells depicted in
Figures 13 and 14 show, their distorted shapes cannot be adequately described with a radial expansion
of the original cell in the ideal crystal as implied
by eqn [31]. As the detailed analysis by Wolfer19
indicates, the [001] dumbbell interstitial in the fcc
lattice does not change the cell dimension in the
[001] direction. In fact, it shortens it slightly, implying that e33 ¼ À0.01005. The volume change is therefore due to the nearest neighbor atoms moving on
average away from the dumbbell axis. This can be
represented by the transformation strain components
e11 ¼ e22 being determined by
2e11 þ e33 ¼

DV
¼ 1:10164
O

The transformation strain tensor for the [001] selfinterstitials in fcc crystals is then


14

Fundamental Properties of Defects in Metals
0

1
0:556
0
0
B
C
eij ¼ B
0:556
0 C
@ 0
A
0
0
À0:010
0
1 0
0:189
0
0
0:367
0
0

B
C B
B
0:189
0
¼B
0:367
0 C
@ 0
Aþ@ 0
0
0
À0:377
0
0
0:367
¼ dij þ~eij

The transformation strain tensor can again be separated into a dilatational and a shear part as
1
C
C
A
½32Š

and it can be divided into an dilatational part, dij,
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
pffiffiffi the self-interstitial, their distance is
reduced to 3r0 =2 as shown by Wolfer.19 As a result,
pffiffiffi
3
À 1 % À0:134
e11 ¼ e22 ¼
2
the remaining strain component is then determined by
2e11 þ e33 ¼

DV
¼ 0:6418
O

0

B
eij ¼ B
@ 0
0

U1 ¼

0:214

0

1

0

C B
B
0 C
Aþ@
0:214

À0:348

0

0

À0:348

0

0

30.7
26.2
27.3
49.8
161.4
91.3

8.08
48.5
64.7
116.9
87.6
0.914
3.95
125.6
1.99
38.1
0.434
89.9
47.9
160.2

1.89
1.51
1.93
2.00
7.17
3.03
1.00
2.70
3.78
1.23
1.04
0.086
0.102
2.05
0.097

0.91
0.055
1.81
0.81
2.58

0.38
0.31
0.35
0.42
1.60
0.69
0.18
0.52
0.72
3.19
2.42
0.170
0.209
4.63
0.194
1.72
0.102
3.89
1.63
5.97

C
0 C
A

0:696

2ð9K þ 8mÞmO
I2
5ð3K þ 4mÞ

EIf % U ¼ U0 þ U1

½34Š

½35Š

The dilatational and the shear strain energies for
some elements are listed in Table 7 in the fourth
and fifth column, respectively. For the fcc elements,
the ratio of U1/U0 is about 0.2. In contrast, for the bcc
elements (in italics) this ratio is about two.

dK/dP

dG/dP

Vrel/O
Theory

102.3
76.1
170.7
137.7
354.7

183.7
44.7
192.7
283.0
161.9
167.7
3.30
12.1
261.7
6.90
172.3
2.20
225.0
155.7
311.0

1

and is equal to I2fcc ¼ 0:1068 and I2bcc ¼ 0:3633 for selfinterstitials in fcc and bcc metals, respectively.
We consider now the total strain energy as a
reasonable approximation for the formation energy
of self-interstitials, namely

K (GPa)

Ag
Al
Au
Cu
Ir

Ni
Pb
Pd
Pt
Cr
Fe
K
Li
Mo
Na
Nb
Rb
Ta
V
W

0

½33Š

where I2 ¼ À~e11~e22 À~e22~e33 À~e33~e11

Metal

U1 (eV)

0

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

Strain energies and relaxation volumes of self-interstitials
U0 (eV)

0

¼ dij þ~eij

Table 7

G (GPa)

0:214

6.12
4.42
6.29
5.48
4.83
6.20
5.53
5.35
5.18
4.89
5.29
3.96
3.53
4.40
4.69
6.91

3.63
3.15
3.50
3.95

Elements in italics are bcc, all others are fcc. Experimental values from Ehrhart and Schultz.7

1.40
1.80
1.05
1.35
3.40
1.40
1.10
0.54
1.60
1.40
1.80
0.79
0.42
1.50
0.80
0.53
0.72
1.10
0.94
2.30

1.94
2.03

1.80
1.87
2.67
1.98
1.73
1.44
2.05
1.21
1.43
0.97
0.76
1.26
0.99
0.91
0.97
1.05
1.03
1.58

Experiment
1.9 Æ 0.4
1.55 Æ 0.3
1.8

1.86 Æ 0.3
1.1

1.1 Æ 0.2
1.1

Fundamental Properties of Defects in Metals

1.01.4.3 Relaxation Volume of
Self-Interstitials
If the elastic distortions associated with selfinterstitials could be adequately treated with linear
elasticity theory, and if the repulsive interactions
between the dumbbell atoms with their nearest neighbors were like that between hard spheres, then the
volume change of a solid upon insertion of an interstitial 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 in Appendix 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
¼ 3S
V
where 3 is the volume dilatation per inclusion as if it
were not confined by the surrounding matrix. This
remarkable result has been proven by Eshelby21 to 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 elastic strains are quite large. For example, the volume of
the confined inclusion, also listed in Table A2 under
column INC, is given by
Du

3
3
1þn
¼
¼
¼ 3
u
1 þ o gE
3ð1 À nÞ
and so it is reduced to about 62% of the unconstrained volume for a Poisson’s ratio of n ¼ 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.
Zener22 has found an elegant way to include the
effects of nonlinear elasticity on volume changes produced 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 U
À
@p K

½36Š

15

found by Schoeck.23 Its application to the strain energy
of self-interstitials leads to the following result.
&
'
3K m0 =m þ 4mK 0 =K 1
À
U0
dV ¼
3K þ 4m
K
&
'
12ðK 0 m À K m0 Þ
m0 1
þ À
U1 ½37Š
þ
ð3K þ 4mÞð9K þ 8mÞ m K
Here, m0 and K 0 are the pressure derivatives of the
shear and bulk modulus, respectively. The first term
arising from the dilatational part of the strain energy
was derived and evaluated earlier by Wolfer.19 It is
the dominant term for the additional volume change
for self-interstitials in fcc metals. Here, we evaluate
both terms using the compilation of Guinan and
Steinberg24 for the pressure derivatives of the elastic
constants, and as listed in Table 7.

The calculated relaxation volumes for selfinterstitials,
VIrel ¼ DV þ dV

½38Š

are given in the eighth column of Table 7, and they
can be compared with the available experimental
values also listed.
We shall see that the relaxation volume of selfinterstitials is of fundamental importance to explain
and quantify void swelling in metals exposed to fast
neutron and charged particle irradiations.
1.01.4.4

Self-Interstitial Migration

The 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 fcc
metals, as illustrated in Figure 15.
Suppose that the initial location of the selfinterstitial is as shown on the left, and its axis is
along [001]. A migration jump occurs by one atom
of the dumbbell (here the purple one) pairing up
with one nearest neighbor, while its former partner

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


16

Fundamental Properties of Defects in Metals

(the blue atom) occupies the available lattice site.
Computer simulations of this migration process have
shown25 that 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 selfinterstitials in bcc metals has revealed that a rotation
may or may not accompany the migration, and these
two diffusion mechanisms are depicted in Figure 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 temperatures that the Frenkel pairs are retained. Their concentration 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 are
trapped at impurities. Table 8 lists the Stage I

temperature,7 TIm , for pure metals as well as two
alloys that represent ferritic and austenitic steels.
For a few cases, an associated activation energy EIm
is known, and in even fewer cases, a preexponential
factor, D0I , has been estimated.

1.01.5 Interaction of Point Defects
with Other Strain Fields
1.01.5.1

The Misfit or Size Interaction

Many different sources of strain fields may exist in
real solids, and they can be superimposed linearly
if they satisfy linear elasticity theory. If this is the
case, we need to consider here only the interaction
between one particular defect located at rd and an
extraneous displacement field u0(r) that originates
from some other source than the defect itself. In
particular, it may be the field associated with external
forces 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 the
defect under consideration is modeled by applying a
set of Kanzaki26 forces f(a)(R(a)), a ¼ 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.

Fundamental Properties of Defects in Metals

Table 8
Annealing temperatures for Stage I and migration properties estimated from them for self-interstitials
Metal

Stage I Tm
I (K)

Cr
Fe
K
Li
Mo
Na
Nb
Ta
V
W
CrxFe1Àx
Ag
Al
Au
Cu
Ir
Ni
Pb
Pd
Pt

Rh
Th
Cr Fe Ni
Be
Cd
Co
Mg
Re
Sc
Ti
Zn
Zr

36
23–144
<6
<6
33À39
<6
5.5–6.0
4–6
3.8
11–38
100
15
37
<0.3
38
50
56

4
35
22
32
10
100–200
30–50
<3.6
45–50
13
90
105
120–130
13
102

Em
I (eV)

DI0
(10À6 m2 sÀ1)

0.25–0.32

0.054, 0.085

5.0

0.117

0.5–1.0

Pij ¼ OCijkl ekl

0.06–0.07

W ¼ ÀOCijkl ekl e0ij ¼ ÀO ekl s0kl

ekl ¼

0.5–0.92

0.015
0.26, 0.30

and obtain

¼1

0
À ui;j
ðrd Þ

z
X
¼1

1 DV
dkl
3 O

and

0.1–0.15
0.029
0.16

If the displacement field varies slowly from one
atom position to the next, we may employ the Taylor
expansion for each displacement component,


ðÞ
0
ðrd ÞRj þ Á Á Á ½40Š
ui0 rd þ RðÞ % ui0 ðrd Þ þ ui;j

ðÞ

½44Š

Finally, for the simplest model of a point defect as a
misfitting spherical inclusion, as treated in Appendix A,

0.13

¼1

fi

½43Š

and the interaction energy can be written as
0.14, 0.15
0.01

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
z


X
f ðÞ Á u0 rd þ RðÞ
W ¼À
½39Š

z
X

½42Š

where e0ij ðrd Þ is the extraneous strain field at the
location of the defect.
Point defects can also be modeled as inclusions,
as we have seen, and they can be characterized by a
transformation strain tensor, ekl. In Appendix B it is
shown, with eqn [B25], that the dipole tensor is then
given by

0.083

Source: Ehrhart, P.; Schultz, H. In Landolt-Bo¨rnstein; SpringerVerlag: Berlin, 1991; Vol. III/25.

W % Àui0 ðrd Þ

The first term vanishes because the sum of Kanzaki
forces is zero, as pointed out in eqn [B4], and
the second term can be expressed in terms of the
dipole tensor defined in [B6]. Furthermore, since
the dipole tensor is symmetric, we obtain finally for
the interaction energy
W1 % ÀPij e0ij ðrd Þ

0.085, 0.088
0.112, 0.115

17

fi

ðÞ ðÞ
Rj À Á Á Á

½41Š

1
½45Š
W ¼ À V rel s0kk

3
In this last form, we used the notation for the relaxation
volume for the misfit or transformation volume, that is,
DV ¼ V rel, to emphasize the fact that the interaction
energy of vacancies and self-interstitials does not
depend on their formation volumes, but on their relaxation volumes. This dependence on the misfit volume
or defect size gave this energy the name of misfit or size
interaction.

1.01.5.2 The Diaelastic or Modulus
Interaction
The definition of the dipole tensor and of multipole
tensors with Kanzaki forces assumes that they are
applied to atoms in a perfect crystal and selected
such that they produce a strain field that is identical
to the actual strain field in a crystal with the defect
present. In particular, the dipole tensor reproduces
the long-range part of this real strain field, and it can
be determined from the Huang scattering measurements of crystals containing the particular defects.
The actual specification of the exact Kanzaki forces
is therefore not necessary. However, if the crystal is


18

Fundamental Properties of Defects in Metals

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 aijkl e0kl

½46Š

The tensor aijkl has been named diaelastic polarizability by Kro¨ner27 based on the analogy with diamagnetic materials.
When the change of the dipole tensor is included
in the derivation of the interaction energy performed in the previous section, an additional contribution arises, namely
1
W2 ¼ À aijkl e0ij e0kl
2

½47Š

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 fraction n of well-separated defects and is subject to
external deformation will have an enthalpy of


1
n
n f
E À Pij e0ij ½48Š
H ðcÞ ¼ Cijkl À aijkl e0ij e0kl þ
2
O
O
per unit volume.
It follows from this formula that the presence of
defects changes the effective elastic constants of the
sample by
n
½49Š
DCijkl ¼ À aijkl
O
Such changes have been measured in single crystal
specimens of only a few metals that were irradiated at
cryogenic temperatures by thermal neutrons or electrons. Significant reductions of the shear moduli C44
and C0 ¼ (C11–C12)/2 are observed from which the
corresponding diaelastic shear polarizabilities listed

in Table 9 are derived.25,28 These values are per
Frenkel pair, and hence each one is the sum of the

shear polarizabilities of a self-interstitial and a
vacancy. By annealing these samples and observing
the recovery of the elastic constants to their original
values, one can conclude that the shear polarizabilities of vacancies are small and that the overwhelming
contribution to the values listed in Table 9 comes
from isolated, single self-interstitials.
The softening of the elastic region around the selfinterstitial to shear deformation is not intuitively
obvious. However, the theoretical investigations by
Dederichs and associates29 on the vibrational properties of point defects have provided a rather convincing series of results, both analytical and by computer
simulations. According to these results, the selfinterstitial dumbbell axis is highly compressed, up
to 0.6 of the normal interatomic distance between
neighboring atoms. Therefore, the dumbbell axis
can be easily tilted by shear of the surrounding lattice
and thereby release some of this axial compression.
The weak restoring forces associated with this tilt
introduce low-frequency vibrational modes that are
also responsible for the low migration energy of selfinterstitials in pure metals.
Computer simulations carried out by Dederichs
et al.30 with a Morse potential for Cu gave the results
presented in Table 10.
While the shear polarizabilities compare favorably with the experimental results for Cu listed in
Table 9, the bulk polarizability in the last column of
Table 10 is too large, and most likely of the wrong
sign for the self-interstitial. The experimental results
for Cu indicate that the bulk polarizability for the
Frenkel pair is close to zero. Atomistic simulations

Table 9

Diaelastic shear polarizabilities per Frenkel pair

Metal

Al

Cu

Mo

a44 (eV)
(a11Àa12)/2 (eV)

269 Æ 20
125 Æ 16

377 Æ 44
111 Æ 14

162 Æ 60
298 Æ 43

Table 10
Diaelastic polarizabilities from computer
simulations of Cu
Diaelastic
polarizability

a44 (eV)

(a11Àa12)/2

(eV)

(a11þ2a12)/3
(eV)

Frenkel pair
Self-interstitial
Vacancy

481.6
443.9
37.7

109.5
77.7
31.8

117.7
90.3
27.4


Fundamental Properties of Defects in Metals

have also been reported by Ackland31 using an effective 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 particular 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 pioneered by Eshelby21 may be instructive to explain
the diaelastic polarizabilities of vacancies and selfinterstitials. A defect is viewed as a region with elastic
constants different from the surrounding elastic continuum. We suppose that this region occupies a
spherical volume of NO, has isotropic elastic constants 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 inhomogeneity, then an interaction is induced upon forming it
that is given by


1
W2 ¼ À N O K A e0ii e0jj þ 2GB e~ij0 e~ij0
½50Š
2
Here,
1
e~ij0 ¼ e0ij À dij e0kk
3
is the deviatoric strain tensor, and
A¼

3ðK Ã À K Þð1 À nÞ
ðK À K Ã Þð1 þ nÞ À 3K ð1 À nÞ

15ðG Ã À GÞð1 À nÞ
B¼
2ðG À G Ã Þð4 À 5nÞ À 15Gð1 À nÞ

½51Š

½52Š
½53Š

The Poisson’s ratio n in the above equations is that of
the matrix.
For an isotropic crystal, eqn [47] assumes the same
form as eqn [50], and the diaelastic polarization tensor has then only two components. These can now be
identified with the two coefficients in eqn [50] to give
the bulk polarizability
1
aK ¼ N OK A ¼ ða11 þ 2a12 Þ
3
and the shear polarizability

½54Š

19

3
1
½55Š
aG ¼ N OGB % a44 þ ða11 À a12 Þ
5

5
The approximation in the last equation is based on
Voigt’s averaging of the shear moduli of cubic materials to obtain an isotropic value.
Let us first apply the formulae [52] to [55] to a
vacancy. It seems plausible to select N ¼ 1 and assume
that K * ¼ G * ¼ 0. Then
AV ¼

3ð1 À nÞ
2ð1 À 2nÞ

½56Š

15ð1 À nÞ
½57Š
7 À 5n
With these expressions, it is easy to compute the bulk
and shear polarizabilities for vacancies, and their
values are listed in the second and third columns of
Table 11.
Next, we consider the bulk polarizability of selfinterstitials. The two atoms that form the dumbbell
are under compression, and the local bulk modulus
that controls their separation distance may be estimated as follows:
BV ¼

KÃ % K þ

dK dp 0
dK Du0
Du ¼ K À K

dp dV
dp O

½58Š

Here, Du0 is the volume compression of the dumbbell
exerted on it by the surrounding material, and as
shown in Section 1.01.4, it is given by
Du0 ¼ Du À DV ¼ ð1=gE À 1ÞDV

½59Š

where DV is the relaxation volume of the selfinterstitial as evaluated for the linear elastic medium.
As we have seen in Section 1.01.4, DV ¼ 1.10164O
for fcc and DV ¼ 0.6418O for bcc crystals. With the
relations [58] and [59] we obtain
DV
ðgE À 1ÞdK
dp O

AI ¼ À
DV
ð1 À 1=gE ÞdK
dp O þ gE

½60Š

and with it the bulk polarizability of self-interstitials as
aKI ¼ 2OKAI

½61Š

Numerical values for it are listed in the third
column of Table 11. We note that these values
are negative, meaning that self-interstitials increase
the effective bulk modulus of irradiated samples, and
this is in contrast to the results from atomistic simulations by Dederichs et al.29 obtained with a Morse
potential for Cu.
To rationalize the shear polarizability of selfinterstitials, we recall that the crystal lattice that


20

Fundamental Properties of Defects in Metals

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

aKV

aG
V

aKI

aG
I

aG
FP (from experiment)

dVI/V

Al
Cu
Ni
Cr
Fe
Mo

38.12
31.19
31.46
24.71
30.02
65.20

6.02
6.86
12.00
17.45
12.43
23.41

À15.93
À18.09
À27.47
À16.80

À15.06
À27.41

264.8
301.8
527.4
174.5
124.3
234.1

211 Æ 19
271 Æ 32

0.93
0.77
0.88
0.57
0.79
0.62

216 Æ 53

The last column is the nonlinear elastic contribution to the relaxation volume of self-interstitials.

surrounds the dumbbell becomes significantly dilated
due to nonlinear elastic effects. This additional dilatation, 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 adjacent 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 ¼

15ð1 À nÞ
7 À 5n

and
aG
I ¼ NG OGBI

½62Š

If the dilated region extends out to the first, second, or third nearest neighbors, then NG ¼ 14, 20, or
44, respectively, for fcc crystals, and NG ¼ 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 NG ¼ 44. However, the best match for bcc
Mo is obtained with NG ¼ 10, a region that only
includes the dumbbell and its first nearest neighbors.
These respective values for NG are also adopted for
the other fcc and bcc elements in Table 11, and the
shear polarizabilities so obtained are listed in the fifth
column.

To compare these estimates with experimental
results, the approximation given in eqn [55] is used
with the data in Table 9 for the shear polarizabilities
of Frenkel pairs. These isotropic averages are listed
in the sixth columnÀ of TableÁ 11, and they are to
G
be compared with aG
V þ aI . It is seen that the

inhomogeneity model is quite successful in explaining the experimental results, in spite of its simplicity
and lack of atomistic details.
1.01.5.3

The Image Interaction

This interaction arises not from the strain field of
other defects or from applied loads but is caused by
the changing strain field of the point defect itself as it
approaches an interface or a free surface of the finite
solid. We have shown in Section 1.01.4 that the strain
energy associated with a point defect is given by

2
2K mO V rel
2mð1 þ nÞ ðV rel Þ2
½63Š
¼
U0 ¼
3K þ 4m O
9ð1 À nÞ O

when the defect is in the center of a spherical body
with isotropic elastic properties or when the defect is
sufficiently far removed from the external surfaces of
a finite solid. This strain energy has been obtained by
integrating the strain energy density of the defect
over the entire volume of the solid, and since this
density diminishes as rÀ6, where r is the distance from
the 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 interaction
energy U im, and the actual strain energy of the defect
becomes
U ðhÞ ¼ U0 þ U im ðhÞ

½64Š

Here, 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 the
surface, the integration volume over regions of high
strain energy density diminishes, and second, the
strain field around the defect becomes nonspherical
and also smaller. The evaluation of both of these
contributions requires advanced techniques for solving elasticity problems.


Fundamental Properties of Defects in Metals

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 h0 is
obviously that for which U(h) ¼ 0, and it is given by
!
ð1 þ nÞ 1=3
r0
½66Š
h0 ¼
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
according to
"
1 þ nr0 3
US ðr Þ ¼ U0 1 À
R
4
%
1
X
ðn þ 1Þðn þ 1Þð2n þ 1Þð2n þ 3Þ r 2n
½67Š
n2 þ ð1 þ 2nÞn þ 1 þ n
R

n¼0
while the strain energy of a defect at a distance r from
the void center is given by
"
1 þ nr0 3
Uv ðr Þ ¼ U0 1 À
R
4
  %
1
X
nðn À 1Þð2n À 1Þð2n þ 1Þ R 2nþ2
½68Š
n2 þ ð1 À 2nÞn þ 1 À n r
n¼2
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(RÀh0) 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 r0 is 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

0.85

Closest distance/atomic radius

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


ð1 þ nÞ r03
½65Š
U ðhÞ ¼ U0 1 À
4 h3

21

Sphere
Halfspace
Void

0.8

0.75

0.7

0.65
Ni
Poisson's ratio = 0.287

0.6

0.55

1

10
100
Surface radius/atomic radius

1000

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

dislocations and interfaces in the material is the
most essential process that restores the material to
its almost normal state. The adjective of ‘almost normal’ is anything but a casual remark here, but it hints
at some subtle effects arising in connection with the
long-range diffusion that constitute the root cause for
the gradual changes that take place in crystalline
materials exposed to continuous irradiation at elevated temperatures. If these effects were absent, then
a steady state would be reached in the material subject to continuous irradiation at a constant rate and
temperature in which the rate of defect generation
would be balanced by their absorption at sinks, meaning the above-mentioned dislocations and interfaces.
As vacancies and self-interstitials are created as
Frenkel pairs in equal numbers, they would also be
absorbed in equal numbers at these sinks. At this
point, the microstructure of these sinks would also
be in a steady, unchanging state. While this steady

state would be different from the initial microstructure or the one reached at the same temperature but
in the absence of irradiation, it would correspond to
material properties that reached constant values.
The subtle effects alluded to in the above remarks
arise from the interactions of the point defects
with strain fields created both internally by the
sinks and externally by applied loads and pressures
on the materials that constitute the reactor components. The internal strain fields from sinks give rise to
long-range forces that render the diffusion migration
nonrandom, while the external strains induce anisotropic diffusion throughout the entire material. In the


22

Fundamental Properties of Defects in Metals

next section, we derive the diffusion equations for
cubic materials to clearly expose these two fundamental 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 E f(r), the formation
energy, but modified by the interactions with internal and external strain fields, which in general vary
with the defect location r. In order to move to the
adjacent energy minimum, the defect has to be
thermally activated over the saddle point that has
an energy

E S ðrÞ ¼ E f ðrÞ þ E m ðrÞ

½69Š

where E m ðrÞ is the migration energy. As the properties 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 in Figure 18. For a self-interstitial, we

Potential profile
Envelope for formation energies
Envelope for saddle points

must also consider the different orientations that it
may have in its stable configuration. Accordingly,
let Cm ðr; t Þ be the concentration of point defects at
the location r and at time t with an orientation m.
For instance, the point defect could be the selfinterstitial in an fcc crystal, in which case, there are
three possible orientations for the dumbbell axis
and m may assume the three values 1, 2, or 3 if the
axis is aligned in the x1, x2, or x3 direction, respectively. The elementary process of diffusion consists
now of a single jump to one adjacent site at r þ R,
where R is one of the possible jump vectors.
The rate of change with time of the concentration
Cm ðr; t Þ is now given by
@Cm X

¼
Cn ðr À R; t ÞLnm ðr À R j RÞ
@t
R;n
X
À
Cm ðr; t ÞLmn ðr j RÞ
½70Š
R;n

Here, the first term sums up all jumps from neighboring sites to site r thereby leading to an increase of
Cm ðr; t Þ, while the second term adds up all the jumps
(really the probabilities of jumps) out of the site r.
The frequency (or better the probability) of a particular jump from r to r þ R while changing the orientation from m to n is denoted by Lmn ðr j RÞ. The eqn
[70] applies now to each of the possible orientations,
and it appears that this leads to as many diffusion
equations as there are possible orientations, and these
equations may be coupled if the defect can change its
orientation between jumps.
To circumvent this complication, one considers an
ensemble of identical systems, all having identical
microstructures, and identical internal and external
stress fields. The ensemble average of the defect
concentration at each site, denoted simply as C(r,t)
without a subscript, is now assumed to be the thermodynamic average such that
expðÀbEmf Þ
Cm ðr; t Þ ¼ Cðr; t ÞP
expðÀbEnf Þ
n

¼ Cðr; t ÞexpðÀbEmf Þ=N ðrÞ

r

r + R/2

r+R

Figure 18 Schematic of the potential energy profile for a
migrating defect.

½71Š

where the normalization factor N only depends on
the location r as do the energies for the stable defect
configurations.
Substituting this into eqn [70] on both sides
constitutes another assumption. To see this, suppose
that the defect concentrations Cn ðr À R; t Þ on all
neighbor sites happen, at the particular instance t,
to be aligned in one direction. Since their new


Fundamental Properties of Defects in Metals

alignments after the jump to site r is correlated
with the jump vector R and the previous orientation, the added defect population does not possess
the equilibrium distribution of eqn [71]. However,
Kronmu¨ller et al.33 argue that after several subsequent jumps of defects from the neighbors to
this site r, the earlier deviation from the equilibrium

distribution will have died out. Thus, introducing
the thermodynamic averages on both sides of the
eqn [70] is a plausible approximation. To proceed
further requires a more specific form of the jump
probability. For a jump from the site r to r þ R, it is
assumed that
Lmn ðr j RÞ ¼ ðL0 =nÞexp

1
S
ÀbEmn
ðr þ RÞ þ bEmf ðrÞ
2

where L0 is a constant to be defined later and n is
the number of possible jump vectors. It is further
assumed that the saddle point is halfway between r
and r þ R, although different locations between
r and r þ R have no bearing on the final results.
The saddle point energy is then affected by the
strain fields at the location r þ R/2. For a reverse
jump of a defect initially at r þ R with orientation n
to the site r and with a new orientation m, the same
saddle point energy needs to be overcome. Hence,
S
S
 Enm
at the same location.
Emn
When eqns [71] and [72] are inserted in eqn [70],

and then the latter is summed over all orientations m,
one obtains
@C
¼ fCðr À R; t Þ À Cðr; t Þg
@t
with
L0 X
1
S
Cðr; t Þ ¼ Cðr; tÞ
exp ÀbEmn
ðr þ RÞ
N ðrÞn R;m;n
2

½73Š
!
½74Š

This latter function may be viewed as an analytic
function of r, since the saddle point energies vary
with strains that are obtained from continuum elasticity theory, and they are by definition analytic functions of r. Expanding the first term in eqn [73] into a
Taylor series up to second order, and then reverting
back to the real defect concentration C(r,t), one arrives
at the diffusion equation
@C
@
@
Dij ðrÞCðr; t Þ À
Fi ðrÞCðr; t Þ ½75Š

¼
@xi @xj
@t
@xi
2

with the diffusion tensor defined as
Dij ðrÞ ¼

and a drift force as
Fi ðrÞ ¼ À

!
L0 X
1
f
S
Xi exp ÀbEmn
ðr þ RÞ þ bE ðrÞ
n R;m;n
2

!
L0 X
1
f
S
Xi Xj exp ÀbEmn
ðr þ RÞ þ bE ðrÞ ½76Š
2n R;m;n

2

½77Š

The components of the jump vector R are denoted by
capital letters Xi, while the components of the location vector r are given by the lower case letters xi.
The normalization factor N(r) is replaced in the eqns
[76] and [77] with an exponential function of the
average defect formation energy according to
h
i
h
i
X
f
exp ÀbEmf ðrÞ ½78Š
exp ÀbE ðrÞ ¼ N ðrÞ ¼

!

½72Š

23

m

It is important to emphasize, as Dederichs and
Schro¨der34 first did, that the above Taylor expansion
does not remove the dependence of the saddle point
energy on the jump direction R. To what degree it

still depends on the jump direction is a function of
the crystal lattice and magnitudes of the elastic strains.
1.01.6.2 Stress-Induced Anisotropic
Diffusion in fcc Metals
To evaluate the diffusion tensor and the drift force,
we consider here the diffusion of self-interstitials and
vacancies in fcc crystals, and begin first with the ideal
case of a crystal free of any stresses other than those
produced by the migrating defect itself. The jump
vectors for both self-interstitials and vacancies coincide with the n ¼ 12 nearest neighbor locations of the
defect in its stable configuration. Table 12 lists the
components Xi of these jump vectors, and they are
divided into three groups according to the orientations of the dumbbell axes before and after the jump.
These are indicated in the first row of Table 12 by
the indices 1, 2, or 3 when the dumbbell axis is
aligned with the x1, x2, or x3 crystal coordinate direction, respectively.
In the absence of stress, all saddle point energies
f
are equal, say ES, and E S À E ¼ E m is just the defect
migration energy. It is then a simple matter to
Table
pffiffiffi 12
d0 = 2.

X1
X2
X3

Components of the jump vectors R in units of
1$2

2$3

3$1

1 1 À1 À1
1 À1 1 À1
0000

0000
1 1 À1 À1
1 À1 1 À1

1 1 À1 À1
0000
1 À1 1 À1

d0 is the nearest neighbor distance between atoms.


Fundamental Properties of Defects in Metals

perform the summations in the definitions of the
diffusion tensor and the drift force to show that
1
½79Š
Dij ¼ L0 d02 expðÀbE m Þdij ¼ D0 ðT Þdij
6
and Fi ¼ 0.
Next, let us consider the case of an applied spatially

uniform stress field. According to Section 1.01.5,
eqn [44], the interaction energy to linear order in the
applied stress will change the energy of the defect in its
stable configurations to
Emf ¼ E f þ Wmf ¼ E f À Oeijm s0ij

½80Š

and in its saddle point configurations to
S
S
Emn
¼ E S þ Wmn
¼ E S À Oeijmn s0ij

½81Š

Here, eijm is the transformation strain tensor of the defect
in its stable configuration with orientation m. We had
specified this tensor for self-interstitials in Section
1.01.4 for an orientation in the [001] direction, that is,
for m ¼ 3, although the nonlinear contributions were
not included. Below, it will be given with these contributions for self-interstitials in Cu and for the same
orientation. The corresponding transformation strain
tensors for the other two orientations, for m ¼ 1, 2, can
be obtained from eij3 by appropriate coordinate
rotations.
Similarly, eijmn is the transformation strain tensor of
the defect under consideration in its saddle point configuration while changing its orientation from m to n.
Specifying it for one particular jump is sufficient to

obtain the transformation strain tensors for all other
jumps by appropriate coordinate rotations. The inspection of the jump directions for a particular pair mn
reveals that there are equal but opposite jump directions
with the same saddle point interaction energy; the only
difference for such a pair of jump directions is that the
components of R and –R have equal and opposite signs.
As a result, the drift force F vanishes again. However, for
the diffusion tensor, the two opposite jump directions
make positive and equal contribution. Suppose, the
applied stress is uniaxial with the only nonvanishing
component s033 ¼ s. When the crystal is oriented such
that this uniaxial stress is perpendicular to a (001) plane,
then Chan et al.35 have shown that the diffusion within
the (001) plane and perpendicular to it are given by
ð001Þ

D11

ð001Þ

¼ D22

¼ D0

S sŠ þ exp½bOðe S þ e S Þs=2Š
3 exp½bOe33
11
22
f sŠ þ exp½bOe f sŠ
2

2 exp½bOe11
33

S þ e S Þs=2Š
3exp½bOðe11
ð001Þ
22
D33 ¼ D0
f sŠ þ exp½bOe f sŠ
2exp½bOe11
33

respectively.

½82Š

When the uniaxial stress is perpendicular to
a (111) crystal plane, then the diffusion tensor in
the reference frame of the stress tensor has the
components35
ð111Þ

D11

ð111Þ

¼ D22
¼ D0

ð111Þ

D33

¼ D0

S þ e S Þs=3Š þ exp½bOð2e S þ e S Þs=3Š
3exp½bOð2e22
33
11
33
f þ e f þ e f Þs=3Š
4exp½bOðe11
22
33
S þ e S Þs=3Š
exp½bOð2e11
33

½83Š

f þ e f þ e f Þs=3Š
exp½bOðe11
22
33

In order to obtain the transformation strains for the
saddle point configurations, Chan, Averback, and
Ashkenazy35 carried out molecular dynamics simulations of diffusion as a function of the applied stress.
Fitting their results to the above equations enabled
them to determine the tensors eijS and eijf . Their principal values are listed in Table 13 for self-interstitials

and vacancies in Cu.
The ratio of the diffusion coefficients in the plane
perpendicular to the uniaxial stress, namely D11 =D0,
is shown in Figures 19 and 20 as solid curves, while
Table 13
Principal transformation strain components
for self-interstitial atoms and vacancies in Cu

SIA, stable configuration
SIA, saddle point
Vacancy, stable configuration
Vacancy, saddle point

e11

e22

e33

0.66
0.73
À0.08
À0.64

0.66
0.32
À0.08
À0.46

0.48

0.75
À0.08
1.26

1.6
Stress in [001]
SIA in (100)
SIA out (100)
Vac in (100)
Vac out (100)

1.4

Diffusion ratio

24

1.2

1

0.8

0.6

–0.4

–0.2
0
0.2

Uniaxial stress (GPa)

0.4

Figure 19 Change of the diffusion coefficients within and
perpendicular to (001) crystal planes when a uniaxial
stress is applied.


Fundamental Properties of Defects in Metals

@spq L0 X
mn
Xi Xj Xk epq
@xk 4n R;m;n
h
i
f
S
exp ÀbEmn
ðrÞ þ bE ðrÞ

1.6

Dij ðrÞ þ bO

Stress in [111]
SIA in (111)
SIA out (111)
Vac in (111)

Vac out (111)

Diffusion ratio

1.4

1.2

1

0.8

0.6

–0.4

–0.2

0

25

0.2

0.4

Uniaxial stress (GPa)
Figure 20 Change of the diffusion coefficients within
and perpendicular to (111) crystal planes when a uniaxial
stress is applied.

diffusion parallel to the stress, D33 =D0 , is shown as
dashed curves. The enhancement of diffusion by
tensile (positive) stresses in the (001) planes is much
larger for vacancies than for self-interstitials. However,
when tensile stress is applied to (111) crystal planes,
diffusion in these planes is reduced for self-interstitials
but remains almost unaltered for vacancies.
1.01.6.3 Diffusion in Nonuniform
Stress Fields
Internal stress fields from dislocations and other
defects such as precipitates are spatially varying, that
is, they are functions of the location r of the migrating point defect. However, the stresses sij (r) may be
viewed as continuous functions except at material
interfaces such as grain boundaries and free surfaces,
and their differences between adjacent lattice sites
may be approximated by


1
1 @sij ðrÞ
½84Š
sij r þ R À sij ðrÞ % Xk
@xk
2
2
where a summation over the repeated index k is
implied. The saddle point energy as given in eqn
[81] is now replaced by



@spq
1
S
mn
mn 1
Emn
r þ R ¼ E S À Oepq
spq ðrÞ À Oepq
½85Š
Xk
2
2 @xk

When this is inserted into eqn [76], a new diffusion
tensor is given obtained that, to linear order in the
stress gradients, is given by

½86Š

Here, the first term is just the diffusion tensor obtained
for a uniform stress field, but now with the stress
replaced by the local stress field at r. The second
term is a correction linear in the stress gradient. However, this term vanishes for the following reasons. We
have seen that for fcc crystals there are pairs of opposite jump directions that have identical transformation
mn
, but they differ only in the signs of the
strain values epq
jump vector components Xk. Hence, these opposite
pairs of jump directions cancel each other’s contribution to the sum in eqn [86]. As a result, there is no

correction to the diffusion tensor that is linear in stress
gradients.
Spatially varying stress field, however, produce a
drift force. Inserting eqn [84] into the formula [77]
leads to


@spq L0 X
mn
Xi Xk epq
Fi ¼ ÀbO
@xk 2n R;m;n
h
i
f
S
½87Š
exp ÀbEmn
ðrÞ þ bE ðrÞ
The factor with the sum has a remarkable resemblance with the expression for the diffusion tensor.
Indeed, it is straightforward to show that
Fi ðrÞ ¼ À

@
f @
Dik ðrÞ À Dik ðrÞbOepq
spq ðrÞ ½88Š
@xk
@xk

Since the average energy of the point defect in its
stable configuration in the presence of a stress field is
given by
F

f
E ðrÞ ¼ E f À Oepq
spq ðrÞ

½89Š

we may also write eqn [88] as
h
i @ n h
i
o
f
f
Fi ðrÞ ¼ Àexp bE ðrÞ
exp ÀbE ðrÞ Dik ðrÞ ½90Š
@xk

In this last expression, the product of the two functions in the curly brackets depends now only on the
saddle point energies, and this shows that it is the
spatial dependence of the saddle point energy only
that gives rise to a drift force, while the variation of
the defect formation energy is not contributing to the
drift force.
The general formulae [88] or [90] reveal that stressinduced diffusion anisotropy affects the direction of the
drift force such that it is in general not collinear with

the stress-induced interaction force.