6 4.6.6 Dislocations and stacking faults in

2 6 6 4.6.6 Dislocations and stacking faults in

The resultant super-dislocation is schematically

ceramics

At room temperature, the primary slip system in the Ni 3 Mn, Ni 3 Al, etc., the stacking fault ribbon is

shown in Figure 4.53b. In alloys such as Cu 3 Au,

fcc structure of magnesia, MgO, is f1 1 0gh1 1 0i. It too small to be observed experimentally but super-

is favoured because its Burgers vector is short and, dislocations have been observed. It is evident,

most importantly, because this vector is parallel to however, that the cross-slip of these super-dislocations

rows of ions of like electrostatic charge, permitting will be an extremely difficult process. This can lead to

the applied stress to shear the f1 1 0g planes past each

a high work-hardening rate in these alloys, as discussed other. Slip in the h1 0 0i directions is resisted at room in Chapter 7.

temperature because it involves forcing ions of like In an alloy possessing short-range order, slip will

charge into close proximity. If we consider the slip not occur by the motion of super-dislocations since

geometry of ionic crystals in terms of the Thompson there are no long-range faults to couple the dislocations

tetrahedron for cubic ionic crystals (Figure 4.54), six

Figure 4.54 Thompson tetrahedron for ionic crystals (cubic) .

116 Modern Physical Metallurgy and Materials Engineering primary f1 1 0g slip planes extend from the central

a h1 1 2 0 i-type direction, the movement and re- point O of the tetrahedron to its h1 1 0i edges. There is

registration of oxygen anions and aluminium cations no dissociation in the f1 1 1g faces and slip is only

must be in synchronism (‘synchro-shear’). Figure 4.55 possible along the h1 1 0i edges of the tetrahedron.

shows the Burgers vectors for slip in the [1 1 2 0] Thus, for each of the f1 1 0g planes, there is only one

direction in terms of the two modes of dissociation

h 1 1 0i direction available. This limiting ‘one-to-one’ proposed by M. Kronberg. These two routes are relation for a cubic ionic crystal contrasts with the

energetically favoured. The dissociation reaction ‘three-to-one’ relation of the f1 1 1gh1 1 0i slip system

for the oxygen anions is: 1 [1 1 2 0] ! 3 1 3 [1 0 1 0] C in cubic metallic crystals. As an alternative to the direct

3 [0 1 1 0]. The vectors for these two half-partials lie in ‘easy’ translation AC, we might postulate the route

close-packed directions and enclose a stacking fault. ! ! AB C BC at room temperature. This process involves

In the case of the smaller aluminium cations, further

dissociation of each of similar half-partials takes slip on plane OAB in the direction AB followed by slip

place (e.g. 1 [1 0 1 0] ! 1 [2 1 1 0] C 3 1 9 9 [1 1 2 0]). These on plane OCB in the direction BC. It is not favoured

quarter-partials enclose three regions in which stacking because, apart from being unfavoured in terms of

is faulted. Slip involves a synchronized movement of energy, it involves a 60 ° change in slip direction and

both types of planar fault (single and triple) across the the critical resolved shear stress for the second stage is

basal planes.

likely to be much greater than that needed to activate the first set of planes. The two-stage route is therefore

4.6.7 Defects in crystalline polymers

a difficult one. Finally, it will be noticed that the Crystalline regions in polymers are based upon long- central point lies at the junction of the h1 1 1i directions

chain molecules and are usually associated with at which, being in a cubic system, are perpendicular to

least some glassy (amorphous) regions. Although the four f1 1 1g faces. One can thus appreciate why

less intensively studied than defect structures in raising the temperature of an ionic crystal often allows

metals and ceramics, similar crystal defects, such the f1 1 0gh1 1 1i system to become active.

as vacancies, interstitials and dislocations, have been In a single crystal of alumina, which is

observed in polymers. Their association with linear rhombohedral-hexagonal in structure and highly

macromolecules, however, introduces certain special anisotropic, slip is confined to the basal planes.

features. For instance, the chain ends of molecules At temperatures above 900 °

C, the slip system is

can be regarded as point defects because they differ in

f 0 0 0 1gh1 1 2 0i. As seen from Figure 2.18, this chemical character from the chain proper. Vacancies, resultant slip direction is not one of close-packing.

usually associated with chain ends, and foreign atoms, If a unit translation of shear is to take place in

acting as interstitials, are also present. Edge and

Figure 4.55 Dissociation and synchronized shear in basal planes of alumina .

Defects in solids 117

screw dislocations have been detected. 1 Moir´e pattern

4.7 Stability of defects

techniques of electron microscopy are useful for revealing the presence of edge dislocations. Growth

4.7.1 Dislocation loops

spirals, centred on screw dislocations, have frequently During annealing, defects such as dislocation loops, been observed on surfaces of crystalline polymers,

stacking-fault tetrahedra and voids may exhibit with the Burgers vector, dislocation axis and chain

shrinkage in size. This may be strikingly demonstrated directions lying in parallel directions (e.g. polyethylene

by observing a heated specimen in the microscope. On crystals grown from a concentrated melt).

heating, the dislocation loops and voids act as vacancy Within crystalline regions, such as spherulites com-

sources and shrink, and hence the defects annihilate posed of folded chain molecules, discrepancies in fold-

themselves. This process occurs in the temperature ing may be regarded as defects. The nature of the

range where self-diffusion is rapid, and confirms that two-dimensional surface at the faces of spherulites

the removal of the residual resistivity associated with where chains emerge, fold and re-enter the crystalline

Stage II is due to the dispersal of the loops, voids, etc. region is of particular interest. Similarly, the surfaces

The driving force for the emission of vacancies from where spherulite edges impinge upon each other can

a vacancy defect arises in the case of (1) a prismatic

be regarded as planar defects, being analogous to grain loop from the line tension of the dislocation, (2) a boundary surfaces.

Frank loop from the force due to the stacking fault X-ray diffraction studies of line-broadening effects

on the dislocation line since in intermediate and high and transmission electron microscopy have been used

-metals this force far outweighs the line tension con- to elucidate crystal defects in polymers. In the latter

tribution, and (3) a void from the surface energy s . case, the high energy of an electron beam can damage

The annealing of Frank loops and voids in quenched the polymer crystals and introduce artefacts. It is

aluminium is shown in Figures 4.56 and 4.58, respec- recognized that the special structural features found

tively. In a thin metal foil the rate of annealing is generally controlled by the rate of diffusion of vacan-

in polymer crystals such as the comparative thinness cies away from the defect to any nearby sinks, usually of many crystals, chain-folding, the tendency of the

the foil surfaces, rather than the emission of vacan- molecules to resist bending of bonds and the great

cies at the defect itself. To derive the rate equation difference between primary intramolecular bonding

governing the annealing, the vacancy concentration and secondary intermolecular bonding, make them

at the surface of the defect is used as one bound- unique and very different to metallic and ceramic

ary condition of a diffusion-controlled problem and crystals.

the second boundary condition is obtained by assum- ing that the surfaces of a thin foil act as ideal sinks for vacancies. The rate then depends on the vacancy

concentration gradient developed between the defect, It is recognized that real glass structures are less homo-

4.6.8 Defects in glasses

where the vacancy concentration is given by geneous than the random network model might sug-

cDc 0 exp f⊲dF/dn⊳/kTg (4.14) gest. Adjacent glassy regions can differ abruptly in

composition, giving rise to ‘cords’, and it has been pro- with ⊲dF/dn⊳ the change of free energy of the defect posed that extremely small micro-crystalline regions

configuration per vacancy emitted at the temperature may exist within the glass matrix. Tinting of clear

T , and the foil surface where the concentration is the

equilibrium value c 0 .

glass is evidence for the presence of trace amounts of For a single, intrinsically-faulted circular dislocation impurity atoms (iron, chromium) dispersed throughout

loop of radius r the total energy of the defect F is given the structure. Modifying ions of sodium are relatively

by the sum of the line energy and the fault energy, i.e. loosely held in the interstices and have been known to

0 migrate through the structure and aggregate close to 2 free surfaces. On a coarser scale, it is possible for bub-

F'

In the case of a large loop ⊲r > 50 nm⊳ in a mate- bles (‘seeds’), rounded by surface tension, to persist

rial of intermediate or high stacking fault energy from melting/fining operations. Bubbles may contain

60 mJ/m 2 ⊳ the term involving the dislocation line gases, such as carbon dioxide, sulphur dioxide and sul-

energy is negligible compared with the stacking fault phur trioxide. Solid inclusions (‘stones’) of crystalline

energy term and thus, since ⊲dF/dn⊳ D ⊲dF/dr⊳ ð matter, such as silica, alumina and silicates, may be

⊲ dr/dn⊳, is given simply by B 2 , where B 2 is the present in the glass as a result of incomplete fusion,

cross-sectional area of a vacancy in the ⊲1 1 1⊳ plane. interaction with refractory furnace linings and localized

For large loops the diffusion geometry approximates crystallization (devitrification) during the final cooling.

to cylindrical diffusion 2 and a solution of the time- independent diffusion equation gives for the anneal- 1 Transmission electron microscopy of the organic

ing rate,

compound platinum phthalocyanine which has relatively large intermolecular spacing provided the first visual

2 For spherical diffusion geometry the pre-exponential evidence for the existence of edge dislocations.

constant is D/b.

118 Modern Physical Metallurgy and Materials Engineering

Figure 4.56 Climb of faulted loops in aluminium at 140 ° C. (a) t D 0 min, (b) t D 12 min, (c) t D 24 min, (d) t D 30 min (after Dobson, Goodhew and Smallman, 1967; courtesy of Taylor and Francis) .

(d) (e) Figure 4.58 Sequence of micrographs showing the

shrinkage of voids in quenched aluminium during isothermal annealing at 170 °

C. (a) t D 3 min, (b) t D 8 min, (c) t D 21 min, (d) t D 46 min, (e) t D 98 min. In all micrographs the scale corresponds to 0 .1 µ m (after Westmacott, Smallman and Dobson, 1968, 117; courtesy of the Institute of Metals) .

where the term containing the dislocation line energy can be approximated to ˛b/r. The annealing of Frank

Figure 4.57 Variation of loop radius with time of annealing loops obeys the linear relation given by equation (4.15) for Frank dislocations in Al showing the deviation from

at large r (Figure 4.57); at small r the curve devi- linearity at small r .

ates from linearity because the line tension term can no longer be neglected and also because the diffu-

sion geometry changes from cylindrical to spherical symmetry. The annealing of prismatic loops is much

slower, because only the line tension term is involved, and obeys an r 2 versus t relationship. where D D D 0 D /kT⊳ is the coefficient of self-

D const. [exp ⊲ B 2 1]

In principle, equation (4.15) affords a direct deter- diffusion and L is half the foil thickness. The annealing

mination of the stacking fault energy by substitution, rate of a prismatic dislocation loop can be similarly

but since U D is usually much bigger than B 2 this determined, in this case dF/dr is determined solely

method is unduly sensitive to small errors in U D . This by the line energy, and then

difficulty may be eliminated, however, by a compar- ative method in which the annealing rate of a faulted loop is compared to that of a prismatic one at the

D const. [˛b/r]

same temperature. The intrinsic stacking fault energy

Defects in solids 119

of aluminium has been shown to be 135 mJ/m 2 by this

D may be determined; Mg has been shown to have

D 125 mJ/m technique. 2 from such measurements. In addition to prismatic and single-faulted (Frank)

dislocation loops, double-faulted loops have also been

4.7.2 Voids

annealed in a number of quenched fcc metals. It is observed that on annealing, the intrinsic loop first

Voids will sinter on annealing at a temperature where shrinks until it meets the inner, extrinsically-faulted

self-diffusion is appreciable. The driving force for region, following which the two loops shrink together

sintering arises from the reduction in surface energy as one extrinsically-faulted loop. The rate of annealing

as the emission of vacancies takes place from the of this extrinsic fault may be derived in a way similar

void surface. In a thin metal foil the rate of annealing to equation (4.15) and is given by

is generally controlled by the rate of diffusion of vacancies away from the defect to any nearby sinks,

E B 2 1]

usually the foil surfaces. The rate then depends on the

vacancy concentration gradient developed between the

defect (where the vacancy concentration is given by from which the extrinsic stacking-fault energy may be

D const. fexp ⊲ E B 2

cDc 0 exp f⊲dF/dn⊳/kTg (4.21) determined. Generally E is about 10–30% higher in value than the intrinsic energy

with ⊲dF/dn⊳ the change in free energy of the defect Loop growth can occur when the direction of the

configuration per vacancy emitted at the temperature vacancy flux is towards the loop rather than away from

T ) and the foil surface where the concentration is the it, as in the case of loop shrinkage. This condition

equilibrium value c o .

can arise when the foil surface becomes a vacancy For a void in equilibrium with its surroundings source, as, for example, during the growth of a surface 2 , and since ⊲dF/dn⊳ D

s ⊳⊲/ oxide film. Loop growth is thus commonly found in 2 ⊳ where  is the Zn, Mg, Cd, although loop shrinkage is occasionally

atomic volume and n the number of vacancies in the observed, presumably due to the formation of local

void, equation (4.14), the concentration of vacancies cracks in the oxide film at which vacancies can be

in equilibrium with the void is annihilated. Figure 4.47 shows loops growing in Mg

c v Dc 0 exp ⊲2 s /rkT⊳

as a result of the vacancy supersaturation produced by oxidation. For the double loops, it is observed that a

Assuming spherical diffusion geometry, the diffu- stacking fault is created by vacancy absorption at the

sion equation may be solved to give the rate of shink- growing outer perimeter of the loop and is destroyed

age of a void as

at the growing inner perfect loop. The perfect regions expand faster than the outer stacking fault, since the

1g (4.22) addition of a vacancy to the inner loop decreases the

For large r⊲>50 nm⊳ the exponential term can be

approximated to the first two terms of the series vacancy to the outer loop increases the energy by the

energy of the defect by B 2 whereas the addition of a

expansion and equation (4.22) may then be integrated same amount. This effect is further enhanced as the

to give

two loops approach each other due to vacancy transfer from the outer to inner loops. Eventually the two loops

(4.23) coalesce to give a perfect prismatic loop of Burgers vector c D [0 0 0 1] which continues to grow under the

r 3 Dr 3 i

6D s /kT⊳t

where r i is the initial void radius at t D 0. By observing vacancy supersaturation. The outer loop growth rate is

the shrinkage of voids as a function of annealing time thus given by

at a given temperature (see Figure 4.58) it is possible to obtain either the diffusivity D or the surface energy

Pr 0 s /c 0 exp ⊲ B 2 /kT⊳ ]

s . From such observations, s for aluminium is shown

to be 1.14 J/m 2 in the temperature range 150–200 ° C, when the vacancy supersaturation term ⊲c s /c o ⊳ is larger

than the elastic force term tending to shrink the loop. to determine s for Al by zero creep measurements The inner loop growth rate is

because of the oxide. This method of obtaining s

2 Pr has been applied to other metals and is particularly i s /c 0 /kT⊳ ] useful since it gives a value of s in the self-diffusion

temperature range rather than near the melting point.

2 /kT⊳ −

1, and the resultant prismatic

loop growth rate is

4.7.3 Nuclear irradiation effects

Pr p

s /c 0 [⊲˛b/r⊳ C 1]g

4.7.3.1 Behaviour of point defects and

dislocation loops

where ⊲˛b/r⊳ < 1 and can be neglected. By measuring Electron microscopy of irradiated metals shows that these three growth rates, values for , ⊲c s /c o ⊳ and

large numbers of small point defect clusters are formed

120 Modern Physical Metallurgy and Materials Engineering loops, which eventually appear as dislocation tangles.

Neutron bombardment produces similar effects to ˛- particle bombardment, but unless the dose is greater

than 10 21 neutrons/m 2 the loops are difficult to resolve. In copper irradiated at pile temperature the density of loops increases with dose and can be as high as

10 14 m 2 in heavily bombarded metals. The micrographs from irradiated metals reveal, in addition to the dislocation loops, numerous small cen- tres of strain in the form of black dots somewhat less than 5 nm diameter, which are difficult to resolve (see Figure 4.59a). Because the two kinds of clusters differ in size and distribution, and also in their behaviour on annealing, it is reasonable to attribute the presence of one type of defect, i.e. the large loops, to the aggrega- tion of interstitials and the other, i.e. the small dots, to the aggregation of vacancies. This general conclusion has been confirmed by detailed contrast analysis of the

defects.

(a) The addition of an extra ⊲1 1 1⊳ plane in a crys- tal with fcc structure (see Figure 4.60) introduces two faults in the stacking sequence and not one, as is the case when a plane of atoms is removed. In conse- quence, to eliminate the fault it is necessary for two partial dislocations to slip across the loop, one above the layer and one below, according to a reaction of the form

a a a a [1 1 1] C [1 1 2] C [1 2 1] ! [0 1 1]

3 6 6 2 The resultant dislocation loop formed is identical

to the prismatic loop produced by a vacancy cluster but has a Burgers vector of opposite sign. The size of the loops formed from interstitials increases with the irradiation dose and temperature, which suggests that small interstitial clusters initially form and sub- sequently grow by a diffusion process. In contrast, the vacancy clusters are much more numerous, and although their size increases slightly with dose, their

0.3 µ number is approximately proportional to the dose and equal to the number of primary collisions which occur.

(b) This observation supports the suggestion that vacancy Figure 4.59 A thin film of copper after bombardment with

clusters are formed by the redistribution of vacancies 1 .4 ð 10 21 ˛ -particles m 2 . (a) Dislocation loops (¾40 nm

created in the cascade.

dia) and small centres of strain (¾4 nm dia); (b) after a Changing the type of irradiation from electron, to 2-hour anneal at 350 ° C showing large prismatic loops

light charged particles such as protons, to heavy ions (after Barnes and Mazey, 1960) .

such as self-ions, to neutrons, results in a progres- sive increase in the mean recoil energy. This results in an increasingly non-uniform point defect generation due to the production of displacement cascades by pri-

on a finer scale than in quenched metals, because mary knock-ons. During the creation of cascades, the of the high supersaturation and low diffusion dis-

interstitials are transported outwards (see Figure 4.7),

tance. Bombardment of copper foils with 1.4 ð 10 21

most probably by focused collision sequences, i.e.

along a close-packed row of atoms by a sequence of location loops as shown in Figure 4.59a; a denuded

38 MeV ˛-particles m 2 produces about 10 21 m 3

dis-

replacement collisions, to displace the last atom in this region 0.8 µ m wide can also be seen at the grain

same crystallographic direction, leaving a vacancy-rich boundary. These loops, about 40 nm diameter, indi-

region at the centre of the cascade which can collapse

to form vacancy loops. As the irradiation tempera- defects have precipitated in this form. Heavier doses

cate that an atomic concentration of ³1.5 ð 10 4 point

ture increases, vacancies can also aggregate to form of ˛-particle bombardment produce larger diameter

voids.

Defects in solids 121 Interstitial defects in the form of loops are

commonly observed in all metals. In fcc metals Frank loops containing extrinsic faults occur in Cu, Ag, Au, Ni, Al and austenitic steels. Clustering of interstitials on two neighbouring ⊲1 1 1⊳ planes to produce an intrinsically faulted defect may also occur, as shown in Figure 4.60. In bcc metals they are predominantly

B B perfect a/2h1 1 1i.

The damage produced in cph metals by electron

A irradiation is very complex and for Zn and Cd ⊲c/a > 1.633⊳ several types of dislocation loops, interstitial in nature, nucleate and grow; thus c/2 loops, i.e. with b D [c/2], c-loops, ⊲c/2 C p⊳ loops, i.e. with

bD 1 6 h 2 0 2 3i, [c/2] C [c/2] loops and hc/2 C pi C

2 pi loops are all formed; in the very early stages of irradiation most of the loops consist of [c/2] dislocations, but as they grow a second loop of b D [c/2] forms in the centre, resulting in the formation of a [c/2] C [c/2] loop. The hc/2 C pi C hc/2 pi loops form either from the nucleation of a hc/2 C pi loop inside a hc/2 pi loop or when a [c/2] C [c/2] loop

h c/

(a) shears. At low dose rates and low temperatures many of the loops facet along h1 1 2 0i directions.

Normal Extrinsic Intrinsic

In magnesium with c/a almost ideal the nature of

f.c.c fault fault

the loops is very sensitive to impurities, and interstitial

A loops with either b D

1 1 2 0i on non-basal planes

C C B C or basal loops with b D ⊲c/2 C p⊳ have been observed

A C A B A in samples with different purity. Double loops with C A

A C b D ⊲c/ C 2 C p⊳ C ⊲c/2 p⊳ also form but no c/2-loops

B C B C B have been observed.

A electrons or neutrons both vacancy and interstitial In Zr and Ti ⊲c/a < 1.633⊳ irradiated with either (b)

loops form on non-basal planes with b D 1 h 1 1 2 0i.

Figure 4.60 (a) Single (A) and double (B) dislocation loops Loops with a c-component, namely b D 3 h 1 1 2 3i on

in proton-irradiated copper (ð43 000). (b) Structure of a

f 1 0 1 0g planes and b D c/2 on basal planes have also double-dislocation loop (after Mazey and Barnes, 1968;

been observed; voids also form in the temperature courtesy of Taylor and Francis) .

range 0.3–0.6T m . The fact that vacancy loops are formed on electron irradiation indicates that cascades are not essential for the formation of vacancy loops.

Frank sessile dislocation loops, double-faulted Several factors can give rise to the increased stability loops, tetrahedra and voids have all been observed

of vacancy loops in these metals. One factor is the pos- in irradiated metals, but usually under different

sibility of stresses arising from oxidation or anisotropic irradiation conditions. Results from Cu, Ag and Au

thermal expansion, i.e. interstitial loops are favoured show that cascades collapse to form Frank loops, some

perpendicular to a tensile axis and vacancy loops par- of which dissociate towards stacking fault tetrahedra.

allel. A second possibility is impurities segregating to The fraction of cascades collapsing to form visible

dislocations and reducing the interstitial bias. loops, defined as the defect yield, is high, ³0.5 in

4.7.3.2 Radiation growth and swelling the fraction of vacancies taking part in the collapse

Cu to 1.0 in Au irradiated with self-ions. Moreover,

In non-cubic materials, partitioning of the loops on process, expressed as the cascade efficiency, is also

to specific habit planes can lead to an anisotropic high (³0.3 to 0.5). Vacancy loops have been observed

dimensional change, known as irradiation growth. The on irradiation at R.T. in some bcc metals (e.g. Mo,

aggregation of vacancies into a disc-shaped cavity Nb, W, ˛-Fe). Generally, the loops are perfect with

which collapses to form a dislocation loop will give

b D a/ 2h1 1 1i although they are thought to nucleate rise to a contraction of the material in the direction of as a/2h1 1 0i faulted loops on f1 1 0g but unfault at an

the Burgers vector. Conversely, the precipitation of a early stage because of the high stacking-fault energy.

plane of interstitials will result in the growth of the Vacancy loops have also been observed in some cph

material. Such behaviour could account for the growth metals (e.g. Zr and Ti).

which takes place in ˛-uranium single crystals during

122 Modern Physical Metallurgy and Materials Engineering neutron irradiation, since electron micrographs from

thin films of irradiated uranium show the presence of clusters of point defects.

The energy of a fission fragment is extremely high ⊲³ 200 MeV⊳ so that a high concentration of both vacancies and interstitials might be expected. A dose

of 10 24 nm 2 at room temperature causes uranium to grow about 30% in the [0 1 0] direction and contract in the [1 0 0] direction. However, a similar dose at the temperature of liquid nitrogen produces ten times this growth, which suggests the preservation of about

10 4 interstitials in clusters for each fission event that occurs. Growth also occurs in textured polycrystalline ˛ -uranium and to avoid the problem a random texture has to be produced during fabrication. Similar effects can be produced in graphite.

During irradiation vacancies may aggregate to form voids and the interstitials form dislocation loops. The voids can grow by acquiring vacancies which can be provided by the climb of the dislocation loops. How-

Figure 4.61 Plots of void swelling versus irradiation temperature for 1050 °

ever, because these loops are formed from interstitial C solution-treated Type 316 irradiated with 1 MeV electrons and 46 MeV Ni 6C atoms they grow, not shrink, during the climb process to a dose of 40 dpa and eventually become a tangled dislocation network. (after Nelson and Hudson, p. 19) .

Interstitial point defects have two properties important in both interstitial loop and void growth.

The formation of voids leads to the phenomenon First, the elastic size interaction (see Chapter 7) causes

of void swelling and is of practical importance in dislocations to attract interstitials more strongly than

the dimensional stability of reactor core components. vacancies and secondly, the formation energy of an

The curves of Figure 4.61 show the variation in total

f is greater than that of a vacancy E f void volume as a function of temperature for solution- so that the dominant process at elevated temperatures

interstitial E I v

treated Type 316 stainless steel; the upper cut-off is vacancy emission. The importance of these factors

arises when the thermal vacancy emission from the to loop stability is shown by the spherical diffusion-

voids exceeds the net flow into them. Comparing the controlled rate equation

ion- and electron-irradiated curves shows that increas- dr

1 ing the recoil energy moves the lower threshold to

D b D v c v Z i D i c dt i higher temperatures and is considered to arise from the removal of vacancies by the formation of vacancy

⊲F C ⊳b 2 D loops in cascades; cascades are not created by electron c v el 0 exp ⊲ 4.24⊳ kT

irradiation.

Voids are formed in an intermediate temperature For the growth of voids during irradiation the spherical

range ³0.3 to 0.6T m , above that for long-range single diffusion equation

vacancy migration and below that for thermal vacancy 

emission from voids. To create the excess vacancy dr

1/2 ]D v c v [1 C ⊲Z i 1/2 ]D i c i 

concentration it is also necessary to build up a critical

D 1/2

[⊲2 s /r⊳ P ]

dt r 

]D v c 0 exp

dislocation density from loop growth to bias the

interstitial flow. The sink strength of the dislocations, has been developed, where c i and c v are the interstitial

kT

i.e. the effectiveness of annihilating point defects, is and vacancy concentrations, respectively, D

given by K v 2 and D i i DZ i for interstitials and K v 2 DZ for their diffusivities,

s the surface energy and Z i is a vacancies where ⊲Z i Z v ⊳ is the dislocation bias for bias term defining the preferred attraction of the loops

for interstitials. voids form they also act as sinks, and are considered At low temperatures, voids undergo bias-driven

neutral to vacancies and interstitials, so that K 2 i D growth in the presence of biased sinks, i.e. dislocation 2 K v D v C v , where r v and C v are the void radius

and concentration, respectively. tures when the thermal emission of vacancies becomes

The rate theory of void swelling takes all these fac- important, whether voids grow or shrink depends on

tors into account and (1) for moderate dislocation den- the sign of [⊲2 s /r⊳ P ]. During neutron irradiation

sities as the dislocation structure is evolving, swelling when gas is being created continuously the gas pres-

is predicted to increase linearly with irradiation dose, sure P > 2 s /r and a flux of gas atoms can arrive at the voids causing gas-driven growth.

increase as (dose) 3/2 , and (3) when the void density is

Defects in solids 123 very high, i.e. the sink strength of the voids is greater

likely to be less effective than solute atoms since pin-

ning will only occur at intervals along the dislocation the rate of swelling should again decrease. Results

than the sink strength of the dislocations ⊲K 2 × K v 2 d ⊳ ,

line. Precipitates in the matrix which are coherent in from electron irradiation of stainless steel show that the

nature can also aid swelling resistance by acting as swelling rate is linear with dose up to 40 dpa (displace-

regions of enhanced vacancy–interstitial recombina- ment per atom) and there is no tendency to a (dose) 3/2

0 precipitates in Al –Cu law, which is consistent with dislocation structure con-

alloys have confirmed that as these precipitates lose tinuing to evolve over the dose and temperature range

coherency during irradiation, the swelling resistance examined.

decreases.

In the fuel element itself, fission gas swelling can occur since uranium produces one atom of gas (Kr

4.7.3.3 Radiation-induced segregation, diffusion and Ze) for every five U atoms destroyed. This leads

and precipitation

Radiation-induced segregation is the segregation under only 0.3% of the U atoms.

to ³2 m 3 of gas (stp) per m 3 of U after a ‘burnup’ of

irradiation of different chemical species in an alloy In practice, it is necessary to keep the swelling

towards or away from defect sinks (free surfaces, small and also to prevent nucleation at grain bound-

grain boundaries, dislocations, etc.). The segregation aries when embrittlement can result. In general,

is caused by the coupling of the different types of variables which can affect void swelling include alloy-

atom with the defect fluxes towards the sinks. There ing elements together with specific impurities, and

are four different possible mechanisms, which fall into microstructural features such as precipitates, grain size

two pairs, one pair connected with size effects and the and dislocation density. In ferritic steels, the intersti-

other with the Kirkendall effect. 1 With size effects, tial solutes carbon and nitrogen are particularly effec-

the point defects drag the solute atoms to the sinks tive in (1) trapping the radiation-induced vacancies and

because the size of the solute atoms differs from the thereby enhancing recombination with interstitials, and

other types of atom present (solvent atoms). Thus inter- (2) interacting strongly with dislocations and therefore

stitials drag small solute atoms to sinks and vacan- reducing the dislocation bias for preferential annihila-

cies drag large solute atoms to sinks. With Kirkendall tion of interstitials, and also inhibiting the climb rate

effects, the faster diffusing species move in the oppo- of dislocations. Substitutional alloying elements with

site direction to the vacancy current, but in the same

a positive misfit such as Cr, V and Mn with an affinity direction as the interstitial current. The former case is for C or N can interact with dislocations in combi-

usually called the ‘inverse Kirkendall effect’, although nation with interstitials and are considered to have a

it is still the Kirkendall effect, but solute atoms rather greater influence than C and N alone.

than the vacancies are of interest. The most impor- These mechanisms can operate in fcc alloys with

tant of these mechanisms, which are summarized in specific solute atoms trapping vacancies and also elas-

Figure 4.62, appear to be (1) the interstitial size effect tically interacting with dislocations. Indeed the inhibi-

mechanism–the dragging of small solute atoms to tion of climb has been advanced to explain the low

sinks by interstitials–and (2) the vacancy Kirkendall swelling of Nimonic PE16 nickel-based alloys. In this case precipitates were considered to restrict dislocation

1 The Kirkendall effect is discussed in Chapter 6, climb. Such a mechanism of dislocation pinning is

Section 6.4.2.

Figure 4.62 Schematic representation of radiation-induced segregation produced by interstitial and vacancy flow to defect sinks .

124 Modern Physical Metallurgy and Materials Engineering

Figure 4.63 Variation in the degree of long-range order S for initially (a) ordered and (b) disordered Cu 3 Au for various irradiation temperatures as a function of irradiation time. Accelerating voltage 600 kV (after Hameed, Loretto and Smallman, 1982; by courtesy of Taylor and Francis) .

effect –the migration away from sinks of fast-diffusing the eqilibrium concentration. Irradiation introduces atoms.

point defects and their effect on the behaviour of Radiation-induced segregation is technologically

ordered alloys depends on two competitive processes, important in fast breeder reactors, where the high

i.e. radiation-induced ordering and radiation-induced radiation levels and high temperatures cause large

disordering, which can occur simultaneously. The effects. Thus, for example, in Type 316 stainless steels,

interstitials do not contribute significantly to ordering at temperatures in the range 350–650 °

but the radiation-induced vacancies give rise to on the position in the reactor) silicon and nickel

C (depending

ordering by migrating through the crystal. Disordering segregate strongly to sinks. The small silicon atoms

is assumed to take place athermally by displacements. are dragged there by interstitials and the slow diffusing

Figure 4.63 shows the influence of electron irradiation nickel stays there in increasing concentration as the

time and temperature on (a) initially ordered and other elements diffuse away by the vacancy inverse

(b) initially disordered Cu 3 Au. The final state of the Kirkendall effect. Such diffusion (1) denudes the

alloy at any irradiation temperature is independent of matrix of void-inhibiting silicon and (2) can cause

the initial condition. At 323 K, Cu 3 Au is fully ordered precipitation of brittle phases at grain boundaries, etc.

on irradiation, whether it is initially ordered or not, Diffusion rates may be raised by several orders of

but at low temperatures it becomes largely disordered magnitude because of the increased concentration of

because of the inability of the vacancies to migrate point defects under irradiation. Thus phases expected

0.1 eV⊳ can from phase diagrams may appear at temperatures

and develop order; the interstitials ⊲E l m ³

migrate at low temperatures. where kinetics are far too slow under normal

circumstances. Many precipitates of this type have been seen in stainless steels which have been in

Further reading

reactors. Two totally new phases have definitely been

Hirth, J. P. and Lothe, J. (1984). Theory of Dislocations. and Pd 8 V) and others appear likely (e.g. Cu–Ni

produced and identified in alloy systems (e.g. Pd 8 W

McGraw-Hill, New York.

miscibility gap). Hume-Rothery, W., Smallman, R. E. and Haworth, C. W. This effect relates to the appearance of precipitates (1969). Structure of Metals and Alloys, Monograph No. 1.

Institute of Metals.

after irradiation and possibly arises from the two Kelly, A. and Groves, G. W. (1970). Crystallography and effects described above, i.e. segregation or enhanced