In a crystalline material, a displacement cascade can be visualized as a series of elastic collisions that is initiated when a given atom is struck by a high-energy neutron (or incident ion in the case of ion irradiation). The initial atom, which is called the primary knock-on atom (PKA), will recoil with a given amount of kinetic energy that it dissipates in a sequence of collisions with other atoms. The first of these are termed sec­ondary knock-on atoms and they will in turn lose energy to a third and subsequently higher ordered knock-ons until all of the energy initially imparted to the PKA has been dissipated. Although the physics is slightly different, a similar event has been observed on billiard tables for many years.

Perhaps the most important difference between billiards and atomic displacement cascades is that an atom in a crystalline solid experiences the binding forces that arise from the presence ofthe other atoms. This binding leads to the formation of the crystalline lattice and the requirement that a certain minimum kinetic energy must be transferred to an atom before it can be displaced from its lattice site. This minimum energy is called the displacement threshold energy (Ed) and is typically 20 to 40 eV for most metals and alloys used in structural applications.16

If an atom receives kinetic energy in excess of Ed, it can be transported from its original lattice site and come to rest within the interstices of the lattice. Such an atom constitutes a point defect in the lattice and is called an interstitial or interstitial atom. In the case of an alloy, the interstitial atom may be referred to as a self-interstitial atom (SIA) if the atom is the primary alloy component (e. g., iron in steel) to distinguish it from impurity or solute interstitials. The SIA nomen­clature is also used for pure metals, although it is somewhat redundant in that case. The complemen­tary point defect is formed if the original lattice site remains vacant; such a site is called a vacancy (see Chapter 1.01, Fundamental Properties of Defects in Metals for a discussion of these defects and their properties). Vacancies and interstitials are created in equal numbers by this process and the name Frenkel pair is used to describe a single, stable interstitial and its related vacancy. Small clusters of both point defect types can also be formed within a displace­ment cascade.

The kinematics of the displacement cascade can be described as follows, where for simplicity we consider the case of nonrelativistic particle energies with one particle initially in motion with kinetic energy E0 and the other at rest. In an elastic collision between two such particles, the maximum energy transfer (Em) from particle (1) to particle (2) is given by

Em = 4EoA1A2/ (A1 + A2)2 I1]

where Ax and A2 are the atomic masses of the two particles. Two limiting cases are of interest. If particle 1 is a neutron and particle 2 is a relatively heavy element such as iron, Em ~ 4E0/A. Alternately, if A1 = A2, any energy up to E0 can be transferred. The former case corresponds to the initial collision between a neutron and the PKA, while the latter corresponds to the collisions between lattice atoms ofthe same mass.

Beginning with the work of Brinkman mentioned above, various models were proposed to compute the total number of atoms displaced by a given PKA as a function of energy. The most widely cited model was that of Kinchin and Pease.17 Their model assumed that between a specified threshold energy and an upper energy cut-off, there was a linear rela­tionship between the number of Frenkel pair pro­duced and the PKA energy. Below the threshold, no new displacements would be produced. Above the high-energy cut-off, it was assumed that the addi­tional energy was dissipated in electronic excitation and ionization. Later, Lindhard and coworkers devel­oped a detailed theory for energy partitioning that could be used to compute the fraction of the PKA energy that was dissipated in the nuclear system in elastic collisions and in electronic losses.18 This work was used by Norgett, Robinson, and Torrens (NRT) to develop a secondary displacement model that is still used as a standard in the nuclear industry and elsewhere to compute atomic displacement rates.19

The NRT model gives the total number of dis­placed atoms produced by a PKA with kinetic energy epka as

Vnrt = 0.8Td(EPKA)/2Ed [2]

where Ed is an average displacement threshold energy.16 The determination of an appropriate average displacement threshold energy is somewhat ambiguous because the displacement threshold is strongly depen­dent on crystallographic direction, and details of the threshold surface vary from one potential to another. An example of the angular dependence is shown in Figure 1,20 for MD simulations in iron obtained using the Finnis-Sinclair potential.21 Moreover, it is not obvious how to obtain a unique definition for the angular average. Nordlund and coworkers22 provide a comparison of threshold behavior obtained with 11 different iron potentials and discusses several different possible definitions of the displacement threshold energy. The factor Td in eqn [2] is called the damage energy and is a function of EPKA. The damage energy is the amount of the initial PKA energy available to cause atomic displacements, with the fraction of the PKA’s initial kinetic energy lost to electronic excitation being responsible for the difference between EPKA and Td. The ratio of Td to EPKA for iron is shown in Figure 2 as a function of PKA energy, where the analytical fit to Lindhard’s theory described by Norgett and coworkers19 has been used to obtain Td.

Note that a significant fraction of the PKA energy is dissipated in electronic processes even for energies

as low as a few kiloelectronvolts. The factor of 0.8 in eqn [2] accounts for the effects of realistic (i. e., other than hard sphere) atomic scattering; the value was obtained from an extensive cascade study using the binary collision approximation (BCA).23,2

The number of stable displacements (Frenkel pair) predicted by both the original Kinchin-Pease model and the NRT model is shown in Figure 3 as a function of the PKA energy. The third curve in the figure will be discussed below in Section 1.11.3. The MD results presented in Section 1.11.4.2 indi­cate that vNRT overestimates the total number of

Frenkel pair that remain after the excess kinetic energy in a displacement cascade has been dis­sipated at about 10 ps. Many more defects than this are formed during the collisional phase of the cascade; however, most of these disappear as vacan­cies and interstitials annihilate one another in spon­taneous recombination reactions.

One valuable aspect of the NRT model is that it enabled the use of atomic displacements per atom (dpa) as an exposure parameter, which provides a common basis of comparison for data obtained in different types of irradiation sources, for example, different neutron energy spectra, ion irradiation, or electron irradiation. The neutron energy spec­trum can vary significantly from one reactor to another depending on the reactor coolant and/or moderator (water, heavy water, sodium, graphite), which leads to differences in the PKA energy spec­trum as will be discussed below. This can confound attempts to correlate irradiation effects data on the basis of parameters such as total neutron fluence or the fluence above some threshold energy, commonly 0.1 or 1.0 MeV. More importantly, it is impossible to correlate any given neutron fluence with a charged particle fluence. However, in any of these cases, the PKA energy spectrum and corresponding damage energies can be calculated and the total number of displacements obtained using eqn [2] in an integral calculation. Thus, dpa provides an environment — independent radiation exposure parameter that in

many cases can be successfully used as a radiation damage correlation parameter.25 As discussed below, aspects of primary damage production other than simply the total number of displacements must be considered in some cases.