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 ofsinks.

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,
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.

4. The regions occupied by the atomic defects within the crystal lattice possess a distorted, if not totally different, arrangement of atoms. As a result, these regions are endowed with different elastic 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.

5. Both the dipole tensors and elastic polarizabilities determine the strengths of interactions with both internal and external stress fields as well as their mutual interactions.

6. When the stress fields vary, the gradients of the interactions impose drift forces on the diffusion migration of the atomic defects that influences their reaction rates with each other and with the sinks.

7. At these sinks, vacancies can also be generated by thermal fluctuations and be released via diffusion to the crystal lattice. Each sink therefore possesses a vacancy chemical potential, and this potential determines both the nucleation of vacancy defect clusters and their subsequent growth to become another defect sink and part of the changing microstructure of extended defects.

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

further elaborated in other chapters.

where mc2 is the rest energy of an electron and Л — 4 mM/(m + M) . 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

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

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

volume O, and melt temperature of elemental metals approximately satisfy the rule

BO « 100kBTm

discovered by Leibfried3 and shown in Figure 3.