There are many different kinds of high-energy radi- ation (e.g. neutrons, electrons, c~-particles, protons, deuterons, uranium fission fragments, y-rays, X-rays) and all of them are capable of producing some form of 'radiation damage' in the materials they irradiate.
While all are of importance to some aspects of the solid state, of particular interest is the behaviour of materials under irradiation in a nuclear reactor. This is because the neutrons produced in a reactor by a fis- sion reaction have extremely high energies of about 2 million electron volts (i.e. 2 MeV), and being elec- trically uncharged, and consequently unaffected by the electrical fields surrounding an atomic nucleus, can travel large distances through a structure. The resul- tant damage is therefore not localized, but is distributed throughout the solid in the form of 'damage spikes.'
The fast neutrons (they are given this name because 2 MeV corresponds to a velocity of 2 x 107 m s - l ) are slowed down, in order to produce further fission, (a) Zn 2" 0 2- Zn 2" 0 2- (b) Cu" Cu" Cu 2" Cu ~
...~.--~Zn 2.§ 2etectrons 0 2" 0 2" 0 2-
Zn z" 0 2- Zn 2. Cation ---~-ci Cu" Cu ~ Cu"
Cot ion vaconcy 0 2- 0 2- 0 2-
interstitial
" ~ , . 2 ~ 02" Zn2" 02" ~ u " Cu~ Cu2~ Cu*
Zn2~2 electrons "~2- 0 2- 0 2-
0 2- Zn 2" 0 z- Zn 2" ~ ' C u " E! Cu" Cu"
Figure 4.6 Schematic arrangement of ions in two typical oxides. (a) Zn> / O, with excess metal due to cation interstitials and (b) Cu <2 O, with excess non-metal due to cation vacancies.
88 Modern Physical Metallurgy and Materials Engineering by the moderator in the pile until they are in thermal equilibrium with their surroundings. The neutrons in a pile will, therefore, have a spectrum of energies which ranges from about 1/40 eV at room temperature (thermal neutrons) to 2 MeV (fast neutrons). However, when non-fissile material is placed in a reactor and irradiated most of the damage is caused by the fast neutrons colliding with the atomic nuclei of the material.
The nucleus of an atom has a small diameter (e.g.
10 -Z~ m), and consequently the largest area, or cross- section, which it presents to the neutron for collision is also small. The unit of cross-section is a barn, i.e.
10 -28 m 2 so that in a material with a cross-section of 1 barn, an average of 109 neutrons would have to pass through an atom (cross-sectional area 10 -19 m 2) for one to hit the nucleus. Conversely, the mean free path between collisions is about 109 atom spacings or about 0.3 m. If a metal such as copper (cross-section, 4 barns) were irradiated for 1 day (105 s) in a neutron flux of 1017 m -2 s -I the number of neutrons passing through unit area, i.e. the integrated flux, would be 1022 n m -2 and the chance of a given atom being hit (=integrated flux x cross-section) would be 4 • 10 -6, i.e. about 1 atom in 250000 would have its nucleus struck.
For most metals the collision between an atomic nucleus and a neutron (or other fast particle of mass m) is usually purely elastic, and the struck atom mass M will have equal probability of receiving any kinetic energy between zero and the maximum Emax = 4 E , M m / ( M + m) 2, where E, is the energy of the fast neutron. Thus, the most energetic neutrons can impart an energy of as much as 200000 eV, to a copper atom initially at rest. Such an atom, called a primary 'knock-on', will do much further damage on its sub- sequent passage through the structure often producing secondary and tertiary knock-on atoms, so that severe local damage results. The neutron, of course, also con- tinues its passage through the structure producing fur- ther primary displacements until the energy transferred in collisions is less than the energy Ed (~25 eV for copper) necessary to displace an atom from its lat- tice site.
The damage produced in irradiation consists largely of interstitials, i.e. atoms knocked into interstitial posi- tions in the lattice, and vacancies, i.e. the holes they leave behind. The damaged region, estimated to con- tain about 60000 atoms, is expected to be originally pear-shaped in form, having the vacancies at the cen- tre and the interstitials towards the outside. Such a displacement spike or cascade of displaced atoms is shown schematically in Figure 4.7. The number of vacancy-interstitial pairs produced by one primary knock-on is given by n "~ Emax/4Ed, and for copper is about 1000. Owing to the thermal motion of the atoms in the lattice, appreciable self-annealing of the damage will take place at all except the lowest tem- peratures, with most of the vacancies and interstitials
x
,~ Interstit~als
x ~ ~
Figure 4.7 Formation of vacancies and interstitials due to particle bombardment (after Cottrell, 1959; courtes~., of the hzstitute of Mechanical Engineers).
annihilating each other by recombination. However, it is expected that some of the interstitials will escape from the surface of the cascade leaving a correspond- ing number of vacancies in the centre. If this number is assumed to be 100, the local concentration will be 100/60 000 or -~2 x 10- 3
Another manifestation of radiation damage concerns the dispersal of the energy of the stopped atom into the vibrational energy of the lattice. The energy is deposited in a small region, and for a very short time the metal may be regarded as locally heated. To distinguish this damage from the 'displacement spike', where the energy is sufficient to displace atoms, this heat-affected zone has been called a 'thermal spike'. To raise the temperature by 1000~ requires about 3R x 4.2 kJ/mol or about 0.25 eV per atom. Consequently, a 25 eV thermal spike could heat about 100 atoms of copper to the melting point, which corresponds to a spherical region of radius about 0.75 nm. It is very doubtful if melting actually takes place, because the duration of the heat pulse is only about 10 - ~ to 10 -12 s. However, it is not clear to what extent the heat produced gives rise to an annealing of the primary damage, or causes additional quenching damage (e.g.
retention of high-temperature phases).
Slow neutrons give rise to transmutation products.
Of particular importance is the production of the noble gas elements, e.g. krypton and xenon produced by fis- sion in U and Pu. and helium in the light elements B, Li, Be and Mg. These transmuted atoms can cause severe radiation damage in two ways. First, the inert gas atoms are almost insoluble and hence in association with vacancies collect into gas bubbles which swell and crack the material. Second, these atoms are often created with very high energies (e.g. as a-particles or fission fragments) and act as primary sources of knock-on damage. The fission of uranium into two new elements is the extreme example when the fis- sion fragments are thrown apart with kinetic energy
~ 1 0 0 MeV. However, because the fragments carry a large charge their range is short and the damage restricted to the fissile material itself, or in materi- als which are in close proximity. Heavy ions can be
accelerated to kilovolt energies in accelerators to pro- duce heavy ion bombardment of materials being tested for reactor application. These moving particles have a short range and the damage is localized.
4 . 2 . 4 P o i n t d e f e c t c o n c e n t r a t i o n a n d a n n e a l i n g
Electrical resistivity p is one of the simplest and most sensitive properties to investigate the point defect concentration. Point defects are potent scatterers of electrons and the increase in resistivity following quenching (Ap) may be described by the equation
Ap = A exp [-Ef/kTQ] (4.4)
where A is a constant involving the entropy of formation, Ef is the formation energy of a vacancy and TQ the quenching temperature. Measuring the resistivity after quenching from different temperatures enables Ef to be estimated from a plot of Ap0 versus 1/TQ. The activation energy, Em, for the movement of vacancies can be obtained by measuring the rate of annealing of the vacancies at different annealing temperatures. The rate of annealing is inversely proportional to the time to reach a certain value of 'annealed-out' resistivity. Thus, 1/tl = A exp [ - E m / k T l ] and 1/t2 = exp [-Em/kT2] and by eliminating A we obtain In ( t 2 / t l ) - " E m [ ( 1 / T E ) - ( 1 / T l ) ] / k where Em is the only unknown in the expression. Values of Ef and Em for different materials are given in Table 4.1.
At elevated temperatures the very high equilibrium concentration of vacancies which exists in the structure gives rise to the possible formation of divacancy and even tri-vacancy complexes, depending on the value of the appropriate binding energy. For equilibrium between single and di-vacancies, the total vacancy concentration is given by
Cv -" Clv + 2C2v
and the di-vacancy concentration by C2v = Azclv 2 exp [B2/kT]
where A is a constant involving the entropy of forma- tion of di-vacancies, B2 the binding energy for vacancy pairs estimated to be in the range 0.1-0.3 eV and z a configurational factor. The migration of di-vacancies is Table 4.1 Values of vacancy formation (Ef) and migration (Em) energies for some metallic materials together with the self-diffusion energy (Eso)
Energy Cu Al Ni Mg Fe W NiAl
(eV)
Ef 1.0-1.1 0.76 1.4 0.9 2.13 3.3 1.05 Em 1.0-1.1 0.62 1.5 0.5 0.76 1.9 2.4 ED 2.0-2.2 1.38 2.9 1.4 2.89 5.2 3.45
Defects in solids 89 an easier process and the activation energy for migra- tion is somewhat lower than Em for single vacancies.
Excess point defects are removed from a mate- rial when the vacancies and/or interstitials migrate to regions of discontinuity in the structure (e.g. free sur- faces, grain boundaries or dislocations) and are annihi- lated. These sites are termed defect sinks. The average number of atomic jumps made before annihilation is given by
n = Azvt exp [-Em/kTa] (4.5)
where A is a constant (m l) involving the entropy of migration, z the coordination around a vacancy, v the Debye frequency ( ~ 1013/s), t the annealing time at the ageing temperature Ta and Em the migration energy of the defect. For a metal such as aluminium, quenched to give a high concentration of retained vacancies, the annealing process takes place in two stages, as shown in Figure 4.8; stage I near room temperature with an activation energy ~0.58 eV and n ~ 104, and stage II in the range 140-200~ with an activation energy of --~ 1.3 eV.
Assuming a random walk process, single vacancies would migrate an average distance ( ~ x atomic spac- ing b) ~ 3 0 nm. This distance is very much less than either the distance to the grain boundary or the spac- ing of the dislocations in the annealed metal. In this case, the very high supersaturation of vacancies pro- duces a chemical stress, somewhat analogous to an osmotic pressure, which is sufficiently large to create new dislocations in the structure which provide many new 'sinks' to reduce this stress rapidly.
The magnitude of this chemical stress may be estimated from the chemical potential, if we let dF represent the change of free energy when dn vacancies are added to the system. Then,
= kT In (c/co)
1.0
where c is the actual concentration andc0 the equilib- rium concentration of vacancies. This may be rewrit- ten as
1
oQ. 0.5
<1
~j
- 100 0 o ~oo 200
Temperature (~
dF/dn = Ef + k T In (n / N ) = - k T In co + kT In c
Figure 4.8 Variation of quenched-in resistivity with temperature of annealing for aluminium (after Panseri and Federighi, 1958, 1223).
90 Modern Physical Metallurgy and Materials Engineering d F / d V = Energy/volume = stress
-- (kT/b3)[ln (C/Co)] (4.6)
where dV is the volume associated with dn vacancies and b 3 is the volume of one vacancy. Inserting typical values, K T "-" 1/40 eV at room temperature, b = 0.25 nm, shows K T / b 3 ~ 150 MN/m 2. Thus, even a moderate 1% supersaturation of vacancies i.e. when ( c / c o ) = 1.01 and In (c/co)=O.O1, introduces a chemical stress oc equivalent to 1.5 MN/m 2.
The equilibrium concentration of vacancies at a temperature T2 will be given by c2 = exp [ - E f / k T 2 ] and at T1 by cl = exp [ - E f / k T I ] . Then, since
In (c2/cl) = ( E l / k ) Tl T2
the chemical stress produced by quenching a metal from a high temperature T2 to a low temperature T~
is
= ( k T / b 3 ) l n ( c ~ , / c . ) = (Ef/b 3) 1 - T-~2 ]
~rc
For aluminium, Er is about 0.7 e V so that quench- ing from 900 K to 300 K produces a chemical stress of about 3 GN/m 2. This stress is extremely high, sev- eral times the theoretical yield stress, and must be relieved in some way. Migration of vacancies to grain boundaries and dislocations will occur, of course, but it is not surprising that the point defects form addi- tional vacancy sinks by the spontaneous nucleation of dislocations and other stable lattice defects, such as voids and stacking fault tetrahedra (see Sections 4.5.3 and 4.6).
When the material contains both vacancies and interstitials the removal of the excess point defect concentration is more complex. Figure 4.9 shows the 'annealing' curve for irradiated copper. The resistivity decreases sharply around 20 K when the interstitials start to migrate, with an activation energy Em " 0 . 1 eV. In Stage I, therefore, most of the Frenkel (interstitial-vacancy) pairs anneal out. Stage II has been attributed to the release of interstitials from impurity traps as thermal energy supplies the necessary activation energy. Stage III is
1.0
l 0.8
0.6
~ O.4
0.2 - - -
0 I t
100 200 300 400 5OO 60O
Temperature (K)
Figure 4.9 Variation of resistivity with temperature produced by neutron irradiation for copper (after DieM).
around room temperature and is probably caused by the annihilation of free interstitials with individual vacancies not associated with a Frenkel pair, and also the migration of di-vacancies. Stage IV corresponds to the stage I annealing of quenched metals arising from vacancy migration and annihilation to form dislocation loops, voids and other defects. Stage V corresponds to the removal of this secondary defect population by self-diffusion.
4 . 3 L i n e d e f e c t s