To span length and temporal scales, these methods can be linked into a multi-scale simulation. The objective of the lower length scale modeling is to provide constitutive properties to higher length scale, continuum level simulations, whereas these higher length scale simulations can provide boundary conditions to the lower length scale models, as well as input regarding the verification/validity of the predicted constitutive properties. Here some examples are provided of models and simulations of materials under irradiation. The emphasis is on the development of the model, the assumptions and the underlying physics that goes into model development.

1.1 Atomistic simulations of radiation damage cascades

Damage to materials caused by neutron irradiation is an inherently multiscale phenomenon. Macroscopic properties, such as plasticity, hardness, brittleness, and creep behavior, of structural reactor materials may change due to microstructural effects of radiation. Atomistic models can be a useful tool to generate data about the structure and development of defects, on length and time scales that experiments cannot probe. Data from simulations can be fed into larger scale models that predict the long term behavior of materials subject to irradiation.

The amount of energy that an incident particle can transfer to a lattice atom is a function of their masses and the angle at which the collision occurs. Energy can be lost through inelastic collisions, (n, 2n) or (n, y) reactions, and, most importantly, elastic collisions. Elastic collisions between neutrons and nuclei can be treated within the hard sphere model with the following equations:

T = Y Ei (1 — cos0)

where, T is the total energy transferred, Ei is the energy of the incident neutron, 0 is the angle of collision, and A is the mass of the lattice nucleus. With the assumption that scattering is isotropic in the center of mass system, the average energy transferred over all angles can be shown to be the average of the minimum and maximum possible transfer energies, i. e., .

For iron, the energy required to displace an atom is about 40 eV, depending on the direction from which it is struck. So, a neutron needs a minimum energy of about 581 eV to displace an iron atom. Neutrons produced from fission of uranium carry around 2 MeV of kinetic energy and so have potential to cause damage as they slow down. Additionally, deuterium — tritium fusion reactions produce neutrons with energy of 14.1 MeV.

The first attempt to create a model for defect production based on PKA energy comes from Kinchin and Pease (1955). In this model, above a certain threshold Ed, energy is lost only to electron excitation, while below it, energy is lost only in hard-sphere elastic scattering. Norgett, Robinson et al. (1975) proposed a revised model, taking into account a more realistic energy transfer cross section, based on binary collision model simulations, . Here, ND is the number of Frenkel pairs surviving relaxation and the damage energy ED is the amount of energy available for creating displacements through elastic collisions and is a function of T. Since some of the energy of the cascade is lost to electronic excitation, ED will be less than T; for the energy range considered in this paper ED can be estimated as equal to

T. This model is frequently used as the standard for estimating DP A, but many molecular dynamics simulations have shown that it tends to strongly overestimate the actual damage efficiency. Bacon, Gao et al. (2000)proposed an empirical relationship between ND and T, where A and n are weakly temperature dependent constants fit to particular materials, respectively equal to 5.57 and 0.83 for Fe at 100 K, and T is in keV.

The study of irradiation damage cascades has been a popular topic over the last fifteen or so years. A through literature review of the many different of damage cascade simulations, such as binary collision approximation and kinetic Monte Carlo, that have been performed in a variety of materials is beyond the scope of this paper and readers are referred to (Was 2007). The following brief review will concentrate solely on molecular dynamics simulations in a-Fe. A thorough review of results from many papers was written by Malerba (2006).

Malerba (2006) states that the first published MD study in alpha-iron was performed by Calder and co-workers (Calder and Bacon 1993; Calder and Bacon 1994). Eighty cascades with PKAs of up to 5 keV were analyzed for properties such as percent of defects surviving relaxation, channeling properties, temperature dependence, and clustering. The interatomic potential used was developed by (Finnis and Sinclair 1984) and stiffened by Calder and Bacon to treat small interatomic distance properly. This article established a large base of data for future papers to compare with.

Following this initial study, many papers came out which utilized both the modified FS potential mentioned above and competing multi-body potentials including those from Johnson and Oh (1989), Harrison, Voter et al. (1989), and Simonelli, Pasianot et al. (1994). These papers had three main motivations: to generate data from a new potential, to compare data between two or more potentials, or to compare damage in a-Fe with that in another material such as copper. The main difficulties in comparing results from different authors are defining what makes up a cluster of defects and non-reporting of exactly how cascades were generated.

Many authors contributed to generate databases; some papers of note are described here. Stoller, Odette et al. (1997), using the FS potential modified by Calder and Bacon, ran a number of cascades at energies up to 40 keV. They found evidence for vacancy clustering, a feature not seen in previous works. Bacon et al. (2000) performed a study comparing the cascade characteristics of bcc, hcp, and fcc metals. They found that there were no major differences in interstitial and vacancy production, so concluded that any differences observed experimentally must be due to evolution of the microstructure following the primary damage event. Caturla, Soneda et al. (2000) compared bcc Fe with fcc Cu, finding that clustering in Fe was at least an order of magnitude less than in Cu. Terentyev, Lagerstedt et al. (2006) produced a study looking solely at differences between four available potentials by applying the same defect counting criteria to each. They found that the stiffness of a potential, a somewhat arbitrary feature, was the most important factor in determining cascade properties.

In all primary damage cascade simulations, first, the incident radiation has an interaction with an atom in the crystal lattice, transferring enough energy to remove the atom from its site. This atom, the primary knock-on atom (PKA), goes on to interact with other atoms in the crystal, removing them from their sites and generating a displacement cascade in the thermal spike phase. Atoms that are removed from their perfect lattice sites and come to rest between other atoms are known as interstitials; the empty lattice sites they leave are called vacancies. At some time shortly after the PKA is created, some peak number of Frenkel pairs, Np, will exist in the crystal, where a Frenkel pair is defined as one vacancy plus one interstitial. After this point, the defects will begin to recombine as the energy is dissipated. After a few picoseconds, only a few defects, Nd will remain. This generally results in a core of vacancies surrounded by a shell of interstitials. A profile of the number of defects over time in a typical cascade can be seen in Figure 3.

The atomistic simulations provide information on the number of surviving defects after initial damage and are not able to simulate large time scale or length scales. They provide a good atomistic picture of the unit processes affecting the formation of defects and the evolution of the primary damage state. Experiments cannot yet access this small time and length scale of radiation damage processes, therefore, experimental corroboration is hard to find for such simulations. Results of atomistic simulations of radiation cascades can be used to develop higher length and time scale theories and simulation of radiation effects in materials. Parameters that can be passed to other simulations/theories include the number and spatial distribution of defects created at the conclusion of the radiation cascade phase.