whereas in future fusion reactors employing deute­rium (D) and tritium (T) as fuel, the neutrons are the result of DT fusion. Spallation neutron sources, which are used for a variety of material research purposes, generate neutrons as a result of spallation reactions between a high-energy proton beam and a heavy metal target. Neutron exposure can lead to substantial changes in the microstructure of the materials, which are ultimately manifested as observ­able changes in component dimensions and changes in the material’s physical and mechanical properties as well. For example, radiation-induced void swelling can lead to density changes greater than 50% in some grades of austenitic stainless steels1 and changes in the ductile-to-brittle transition temperature greater than 200 °C have been observed in the low-alloy steels used in the fabrication of reactor pressure vessels.2,3 These phenomena, along with irradiation creep and radiation-induced solute segregation are discussed extensively in the literature4 and in more detail elsewhere in this comprehensive volume (e. g., see Chapter 1.03, Radiation-Induced Effects on Microstructure; Chapter 1.04, Effect of Radiation

on Strength and Ductility of Metals and Alloys; and Chapter 1.05, Radiation-Induced Effects on Mate­rial Properties of Ceramics (Mechanical and Dimensional)). The objective of this chapter is to describe the process of primary damage production that gives rise to macroscopic changes. This primary radiation damage event, which is referred to as an atomic displacement cascade, was first proposed by Brinkman in 1954.5,6 Many aspects of the cascade dam­age production discussed below were anticipated in Brinkman’s conceptual description.

In contrast to the time scale required for radiation — induced mechanical property changes, which is in the range of hours to years, the primary damage event that initiates these changes lasts only about 10~ns. Similarly, the size scale of displacement cascades, each one being on the order of a few cubic nan­ometers, is many orders of magnitude smaller than the large structural components that they affect. Although interest in displacement cascades was initi­ally limited to the nuclear industry, cascade damage production has become important in the solid state processing practices of the electronics industry also.7 The cascades of interest to the electronics industry arise from the use of ion beams to fabricate, modify, or analyze materials for electronic devices. Another related application is the modification of surface layers by ion beam implantation to improve wear or corrosion resistance of materials.8 The energy and mass of the particle that initiates the cascade provide the principal differences between the nuclear and ion beam applications. Neutrons from nuclear fission and DT fusion have energies up to about 20 MeV and 14.1 MeV, respectively, while the peak neutron energy in spallation neutron sources reaches as high as the energy of the incident proton beam, ~1 GeV in modern sources.9 The neutron mass of one atomic mass unit (1 a. m.u.^1.66 X 10-27 kg) is much less than that of the mid-atomic weight metals that comprise most structural alloys. In contrast, many ion beam applications involve relatively low-energy ions, a few tens of kiloelectronvolts, and the mass of both the incident particle and the target is typically a few tens of atomic mass unit. The use of somewhat higher energy ion beams as a tool for investigating neutron irradiation effects is discussed in Chapter 1.07, Radiation Damage Using Ion Beams.

This chapter will focus on the cascade energies of relevance to nuclear energy systems and on iron, which is the primary component in most of the alloys employed in these systems. However, the description of the basic physical mechanisms of displacement cascade formation and evolution given below is gen­erally valid for any crystalline metal and for all of the applications mentioned above. Although additional physical processes may come into play to alter the final defect state in ionic or covalent materials due to atomic charge states,10 the ballistic processes observed in metals due to displacement cascades are quite similar in these materials. This has been demon­strated in molecular dynamics (MD) simulations in a range of ceramic materials.11-15 Finally, synergistic effects due to nuclear transmutation reactions will not be addressed; the most notable of these, helium production by (n, a) reactions, is the topic of Chapter

1.6, The Effects of Helium in Irradiated Struc­tural Alloys.

The RDT is frequently but inappropriately called ‘the rate theory.’ This is due to the misunderstanding of the role of the transition state theory (TST) or (chemical reaction) RT (see Laidler and King38 and Hanggi et at39 for reviews) in the RDT. The TST is a seminal scientific contribution of the twentieth cen­tury. It provides recipes for calculating reaction rates between individual species of the types which are ubiquitous in chemistry and physics. It made major contributions to the fields of chemical kinetics, diffu­sion in solids, homogeneous nucleation, and electri­cal transport, to name a few. TST provides a simple way of formulating reaction rates and gives a unique insight into how processes occur. It has survived considerable criticisms and after almost 75 years has not been replaced by any general treatment compa­rable in simplicity and accuracy. The RDT uses TST as a tool for describing reactions involving radiation — produced defects, but cannot be reduced to it. This is true for both the mean-field models discussed here, and the kinetic Monte Carlo (kMC) models that are also used to simulate radiation effects (see Chapter 1.14, Kinetic Monte Carlo Simulations of Irradia­tion Effects).

The use of the name RT also created an incorrect identification of the RDT with the models that emerged in the very beginning, which assumed the production of only FPs and 3D migrating PDs to be the only mobile species, that is, FP3DM. It failed to appreciate the importance of numerous contradicting experimental data and, hence, to produce significant contribution to the understanding of neutron irra­diation phenomena (see Barashev and Golubov35 and Section 1.13.6). A common perception that the RDT in general is identical to the FP3DM has devel­oped over the years. So, the powerful method was rejected because of the name of the futile model. This caused serious damage to the development of RDT during the last 15 years or so. Many research proposals that included it as an essential part, were rejected, while simulations, for example, by the kMC etc. were aimed at substituting the RDT. The simula­tions can, of course, be useful in obtaining information on processes on relatively small time and length scales but cannot replace the RDT in the large — scale predictions. The RDT and any of its future developments will necessarily use TST.

An important approximation used in the theory is the MFA. The idea is to replace all interactions in a

Interaction of energetic particles with a solid target is a complex process. A detailed description is beyond the scope of the present paper (Robinson41). However, the primary damage produced in collision events is the main input to the RDT and is briefly introduced here. Energetic particles create primary knock-on (or recoil) atoms (PKAs) by scattering either incident radiation (electrons, neutrons, protons) or accelerated ions. Part of the kinetic energy, EPKa, transmitted to the PKA is lost to the electron excitation. The remaining energy, called the damage energy, Td, is dissipated in elastic collisions between atoms. If the Td exceeds a threshold displacement energy, Ed, for the target material, vacancy-interstitial (or Frenkel) pairs are produced. The total number of displaced atoms is proportional to the damage energy in a model proposed by Norgett et at42 and known as the NRT standard

Epka(£)
2 Ed

The Monte Carlo method was originally developed by von Neumann, Ulam, and Metropolis to study the diffusion of neutrons in fissionable material on the Manhattan Project9,10 and was first applied to simulate radiation damage of metals more than 40 years ago by Besco,11 Doran,12 and later Heinisch and coworkers.13,14

Monte Carlo utilizes random numbers to select from probability distributions and generate atomic configurations in a stochastic process,15 rather than the deterministic manner of MD simulations. While different Monte Carlo applications are used in com­putational materials science, we shall focus our atten­tion on KMC simulation as applied to the study of radiation damage.

The KMC methods used in radiation damage studies represent a subset of Monte Carlo (MC) methods that can be classified as rejection-free, in contrast with the more classical MC methods based on the Metropolis algorithm.9,10 They provide a solu­tion to the Master Equation which describes a physi­cal system whose evolution is governed by a known set of transition rates between possible states.16 The solution proceeds by choosing randomly among the various possible transitions and accepting them on the basis of probabilities determined from the corresponding transition rates. These probabilities are calculated for physical transition mechanisms as Boltzmann factor frequencies, and the events take place according to their probabilities leading to an evolution of the microstructure. The main ingredients of such models are thus a set of objects (which can resolve to the atomic scale as atoms or point defects) and a set of reactions or (rules) that describe the manner in which these objects undergo diffusion, emission, and reaction, and their rates of occurrence.

Many of the KMC techniques are based on the residence time algorithm (RTA) derived 50 years ago by Young and Elcock17 to model vacancy diffusion in ordered alloys. Its basic recipe involves the following: for a system in a given state, instead of making a number of unsuccessful attempts to perform a transi­tion to reach another state, as in the case of the Metropolis algorithm,9, 0 the average time during which the system remains in its state is calculated. A transition to a different state is then performed on the basis ofthe relative weights determined among all possible transitions, which also determine the time increment associated with the selected transition. According to standard transition state theory (see for instance Eyring1 ) the frequency Гx of a ther­mally activated event x, such as a vacancy jump in an alloy or the jump of a void can be expressed as:

Гх =exp(-йг) [11

where nx is the attempt frequency, kB is Boltzmann’s constant, Г is the absolute temperature, and Ea is the activation energy of the jump.

During the course of a KMC simulation, the probabilities of all possible transitions are calculated and one event is chosen at each time-step by extract­ing a random number and comparing it to the relative probability. The associated time-step length dt and average time-step length At are given by:

= and At = 1 [2]

Гх Г x И

n n

where r is a random number between 0 and 1. The RTA is also known as the BKL (Bortz, Kalos, Lebowitz) algorithm.19 Other techniques are possible, as described by Chatterjee and Vlachos.20 The basic steps in a KMC simulation can be summarized thus:

1. Calculate the probability (rate) for a given event to occur.

2. Sum the probabilities of all events to obtain a cumulative distribution function.

3. Generate a random number to select an event from all possible events.

4. Increase the simulation time on the basis of the inverse sum of the rates of all possible events

At — , where — is a random deviate that

N, R,

assures a Poisson distribution in time-steps and N and R are the number and rate of each event i.

5. Perform the selected event and all spontaneous events as a result of the event performed.

6. Repeat Steps 1-4 until the desired simulation condition is reached.

A common formalism for excess energy expressions is the Redlich-Kister-Muggianu relation, which for a binary system can be written as

Gex = X, XjX Lk, ij(X, — Xj)k [11]

k

where L is the coefficient of the expansion in k, which can also have a temperature dependence typically of the form a + bT. Thus, a regular solu­tion is defined as k equals zero leaving a single energetic term. This approach is related to the Bragg-Williams description, with random mixing of constituents yet with enthalpic energetic terms such that

Gex = XaXbEbw [12]

Here, XAXB represents a random mixture of A and B components and is thus the probability that A-B is a nearest-neighbor pair, and EBW is the Bragg — Williams model energetic parameter.

In a relevant example, Kaye eta/.16 have generated a solution model for the five-metal white phase noted above and more extensively discussed in Chapter 2.20, Fission Product Chemistry in Oxide Fuels. A binary constituent of the model is the fcc-structure Pd-Rh system, which at elevated temperatures forms a single solid solution across the entire compositional range. The phase diagram of Figure 2 also shows a low — temperature miscibility gap, that is, two coexisting identically structured phases rich in either end member. The excess Gibbs free energy expression for

the fcc phase was determined from an optimization using tabulated thermochemical information together with the phase equilibria and yielded

Gex = XPdXRh [21247 + 2199XRh

-(2.74 — 0.56XRh)T] [13]

The knowledge of the phenomenological coefficients L, including their dependence on the chemical compo­sition, allows the prediction of RIS phenomena. Unfor­tunately, in practice, it is very difficult to get such information from experimental measurements, espe­cially for concentrated and multicomponent alloys, and for the diffusion by interstitials. As we have seen, it is also quite difficult to establish the exact relation­ship between the phenomenological coefficients and the atomic jump frequencies because of the compli­cated way in which they depend on the local atomic configurations and because correlation effects are very difficult to be fully taken into account in diffusion theories. An alternative approach to analytical diffusion equations, then, is to integrate point defect jump mechanisms, with a realistic description ofthe frequen­cies in the complex energetic landscape of the alloy, in atomistic-scale simulations such as mean-field equa­tions, or Monte Carlo simulations (molecular dynam­ics methods are much too slow — by several orders of magnitude — for microstructure evolution governed by thermally activated migration of point defects).

Atomic-scale methods are appropriate techniques to simulate nanoscale phenomena like RIS. They are all based on an atomic jump frequency model. From this point of view, the difficulties are the same as for the modeling of other diffusive phase transformations (such as precipitation or ordering during thermal aging), complicated by the point defect formation and annihilation mechanisms and by the self-interstitial jump mechanisms, which are usually more complex than the vacancy ones.76

1.07.3.1 Defect Production

The parameter commonly used to correlate the dam­age produced by different irradiation environments is the total number of displacements per atom (dpa). Kinchin and Pease7 were the first to attempt to deter­mine the number of displacements occurring during irradiation and a modified version of their model known as the Norgett-Robinson-Torrens (NRT) model8 is generally accepted as the international standard for quantifying the number of atomic dis­placements in irradiated materials.9 According to the NRT model, the number of Frenkel pairs (FPs), nNRT(T), generated by a primary knock-on atom (PKA) of energy T is given by

kEd(T )
2 Ed

where ED(T) is the damage energy (energy of the PKA less the energy lost to electron excitation), Ed is the displacement energy, that is, the energy needed to displace the struck atom from its lattice position, and к is a factor less than 1 (usually taken as 0.8). Integration of the NRT damage function over recoil spectrum and time gives the atom concentration of displacements known as the NRT displacements per atom (dpa): f(E)vNRT(T)a(E, T)dTdE where f(E) is the neutron flux and s(E, T) is the proba­bility that a particle of energy E will impart a recoil energy Tto a struck atom. The displacement damage is accepted as a measure of the amount of change to the solid due to irradiation and is a much better measure of an irradiation effect than is the particle fluence. As shown in Figure 1, seemingly different effects of

Figure 1 Comparison of yield stress change in 316 stainless steel irradiated in three facilities with very different neutron energy flux spectra. While there is little correlation in terms of neutron fluence, the yield stress changes correlate well against displacements per atom (dpa). Reprinted, with permission, from ASTM, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428.

irradiation on low temperature yield strength for the same fluence level (Figure 1 (a)) and disappear when dpa is used as the measure of damage (Figure 1(b)).

A fundamental difference between ion and neu­tron irradiation effects is the particle energy spectrum that arises because of the difference in the way the particles are produced. Ions are produced in accel­erators and emerge in monoenergetic beams with very narrow energy widths. However, the neutron energy spectrum in a reactor extends over several orders of magnitude in energy, thus presenting a much more complicated source term for radiation damage. Figure 2 shows the considerable difference in neutron and ion energy spectra and also between neutron spectra in different reactors and at different locations within the reactor vessel.

Distance into solid (m)

Another major difference in the characteristics of ions and neutrons is their depth of penetration. As shown in Figure 3, ions lose energy quickly because of high electronic energy loss, giving rise to a spa­tially nonuniform energy deposition profile caused

by the varying importance of electronic and nuclear energy loss during the slowing down process. Their penetration distances range between 0.1 and 100 pm for ion energies that can practically be achieved by laboratory-scale accelerators or implanters. By virtue of their electrical neutrality, neutrons can penetrate very large distances and produce spatially flat dam­age profiles over many millimeters of material. Further, the cross-section for ion-atom reaction is much greater than for neutron-nuclear reaction giving rise to a higher damage rate per unit ofparticle fluence. The damage rate in dpa per unit of fluence is propor­tional to the integral of the energy transfer cross-section and the number of displacements per PKA, nNRT(T):	recoil atom to other target atoms. The fraction of recoils between the displacement energy Ed, and T is
T (eV) Figure 4 Integral primary recoil spectra for 1 MeV particles in copper. Curves plotted are the integral fractions of primary recoils between the threshold energy and recoil energy, Tfrom eqn [6]. Reproduced from Averback, R. S. J. Nucl. Mater. 1994, 216, 49.
a(E, r>NRT(T)dr
A N f
[3]
Ed
where Rd is the number if displacements per unit vol­ume per unit time, N is the atom number density, and f is the particle flux (neutron or ion). In the case of neutron-nuclear interaction described by the hard — sphere model, eqn [3] becomes
_Rd_ N f
[4]
where g = 4mM/(m + M)2, M is the target atom mass, m is the neutron mass, E is the neutron energy, and ss is the elastic scattering cross-section. For the case of ion — atom interaction described by Rutherford scattering, eqn [3] becomes Rd_ TCZ^e4 /Mj gE NI = 4EEd M2 Ej ’ []
E 106 105
1 MeV electrons T =60 eV e = 50-100%
where e is the unit charge, M1 is the mass of the ion, and M2 is the mass of the target atom. As shown in Figure 3, for comparable energies, 1.3 MeV protons cause over 100 times more damage per unit of fluence at the sample surface than 1 MeV neutrons, and the factor for 20 MeV C ions is over 1000. Of course, the damage depth is orders of magnitude smaller than that for neutron irradiation.
1 MeV protons T=200eV e = 25%
103
102’Tp Te -101
1 MeV heavy ions T =5keV e = 4%
E
E
Read More 22.08.2015   Comprehensive nuclear materials Ab Initio and Empirical Potentials Ab initio calculations are often compared to and sometimes confused with empirical potential calcu­lations. We will now try to clarify the differences between these two approaches and highlight their point of contacts. The main difference is of course that ab initio calculations deal with atomic and elec­tronic degrees of freedom. Empirical potentials depend only on the relative positions of the consid­ered atoms and ions. They do not explicitly consider electrons. Thus, roughly speaking, ab initio calcula­tions deal with electronic structure and give access to good energetics, whereas empirical potentials are not concerned with electrons and give approximate energetics but allow much larger scale calculations (in space and time). Going into some details, we have shown that ab initio gives access to very diverse phenomena. Some can be modeled with empirical potentials, at least partly; others are completely outside the scope of such potentials. In the latter category, one will find the phenomena that are really related to the electronic structure itself. For instance, the calculations of electronic excitations (e. g., optical or X-ray spectra) are concep­tually impossible with empirical potentials. In the same way, for insulating materials, the calculation of the relative stability of various charge states of a given defect is impossible with empirical potentials. Other phenomena that are intrinsically electronic in nature can be very crudely accounted for in empir­ical potentials. The electronic stopping power of an accelerated particle is an example. As indicated above, it can be calculated ab initio. Conversely, from the empirical potential perspective one can add an ad hoc slowing term to the dynamics of fast mov­ing particles in solids whose intensity has to be estab­lished by fitting experimental (or ab initio) data. In a related way, some forms of empirical potentials rely on electronic information; for instance, the Finnis-Sinclair36 or Rosato et a/.37 forms. In the same spirit, a recent empirical potential has been designed to reproduce the local ferromagnetic order of iron.38 However, this potential assumes a tendency for ferromagnetic order, while ab initio calculation can (in principle) predict what the magnetic order will be. Therefore, ab initio is very often used as a way to get accurate energies for a given atomic arrangement. This is the case for the formation and migration energies of defects, the vibration spectra, and so on. These phenomena are conceptually within reach of empirical potentials (except the ones that reincorpo­rate electronic degrees of freedom such as charged defects). Ab initio is then just a way to get proper and quantitative energetics. Their results are often used as reference for fitting empirical potentials. However, the fit of a correct empirical remains a tremendous task especially with the complex forms of potentials nowadays and when one wants to correctly predict subtle, out of equilibrium, properties. Finally, one should always keep in mind that cohesion in solids is quantum in nature, so classical interatomic potentials dealing only with atoms or ions can never fully reproduce all the aspects of bonding in a material. Read More 22.08.2015   Comprehensive nuclear materials Necessary Results with Pair Potentials Apart from the specific difficulties with Morse and LJ potentials, there are other general difficulties that are common to all pair potentials, which make them unsuitable for radiation damage studies. Expanding the energy as a sum of pairwise inter­actions introduces some constraints on what data can be fitted, even in principle. It is important to distin­guish this problem from a situation where a particular parameterization does not reproduce a feature of a material. There are many features of real materials that cannot be reproduced by pair potential whatever the functional form or parameterization used. 1.10.4.2.1 Outward surface relaxation For a single-minimum pair potential, the nearest neigh­bors repel one another, while longer ranged neighbors attract. When a surface is formed, more long-range bonds are cut than short-range bonds, so there is an overall additional repulsion. Hence, the surface layer is pushed outward. But in almost all metals, the surface atoms relax toward the bulk, because the bonds at the surface are strengthened. Similarly, pair potentials Figure 2 Pressure versus temperature phase diagram for the crystalline region of the Lennard-Jones system in reduced units where p is pressure and p is density. The equilibrium density is at ps3 = 1.0915. Filled squares are the harmonic free energy integrated to the thermodynamic limit from Salsburg, Z. W.; Huckaby, D. A. J. Comput Phys. 1971, 7, 489-502. All other points are from lattice-switch Monte Carlo simulations with N atoms, lines showing the phase boundary deduced from the Clausius-Clapeyron equation, from Jackson, A. N. Ph. D. Thesis, University of Edinburgh, 2001; Jackson, A. N.; Bruce, A. D.; Ackland, G. J. Phys. Rev. E 2002, 65, 036710. give too large a ratio of surface to cohesive energy, again consistent with the failure to describe the strengthening of the surface bonds. 1.10.4.2.2 Melting points With LJ, the relation between cohesive energy and melting is Ec/kBTm « 13, other pair potentials being similar. Real metals are relatively easier to melt, with values around 30. One can fit the numerical value of the e parameter to the melting point, and accept the discrepancy as a poor description of the free atom. Read More 22.08.2015   Comprehensive nuclear materials Influence of grain boundaries Depending on the complexity of the microstructure, internal interfaces such as grain boundaries, twins, Figure 26 Time dependence of displaced atoms in 10 keV cascades, three typical cascades initiated near the center of the cell are compared with a cascade initiated by an atom on a free surface and one initiated by an atom 10a0 below the free surface.   and lath and packet boundaries (in ferritic/martensitic steels) can provide a significant sink in the material for point defects. As such, they may play a significant role in radiation-induced microstructural evolution. For example, the effect of grain size on austenitic stainless steels was observed as early as 1972.112-114 The swelling effect was more closely associated with damage accumulation than damage production, but current understanding of the role of mobile inter­stitial clusters has provided a link to damage produc­tion as well (Singh and coworkers115 and Chapter Vacancy cluster size (4-NN) To date, there have been a limited number of stud­ies carried out to investigate whether and how primary damage formation would be altered in nanograined metals,116-121 and quite strong effects have been observed.116 The work from Stoller and coworkers122 will be used here to illustrate the phenomenon because the results of that study can be directly compared with the existing single crystal database that has been dis­cussed above. A sufficient number of simulations were carried out at cascade energies of 10 and 20keV and temperatures of 100 and 600 K to obtain a statistically significant comparison. The results demonstrate that the creation of primary radiation damage can be sub­stantially different in nanograined material due to the influence of nearby grain boundaries. To create the nanocrystalline structure, grain nucleation sites were chosen, and the grains were filled using a Voronoi technique.123 A 2 x 2 x 2 lattice parameter face-centered cubic (fcc) unit cell system was used to obtain the grain nucleation sites, resulting in 32 grains in the final sample. Each Voronoi poly­hedron was filled with atoms placed on a regular bcc iron crystalline lattice, with the lattice orientation randomly selected. Grain boundaries occur natu­rally when the atomic plains in adjacent polyhedra impinge on one another, and overlapping atoms at the grain boundaries were removed. The final system was periodic and had an average grain size of 10 nm, system box length of 28.3 nm, and contained roughly 1.87 million atoms. More details on the procedure can Figure 29 MD simulation cell, 32 ~10nm grains. Shaded red circle and green ellipse indicate approximate size of 5 and 10keV cascades, respectively. be found in Stoller and coworkers.122 The system was equilibrated for over 200 ps including a heat treat­ment up to 600 K. Figure 29 illustrates a typical grain structure with each grain shown in a different color. The approximate sizes of 5 and 20 keV cascades have been projected on to the face of the simulation cell. MD cascade simulations were carried out in the same manner discussed above, although the analysis was somewhat more difficult due to the need to differentiate cascade-produced defects from the defect structure associated with the grain boundaries. It was common for the cascade volume to cross from one grain to another.1 2 The number of vacancies and interstitials surviving in the nanograin simulations is compared to the single crystal results in Figure 30. A wider range of cascade energies is included in Figure 30(a) to show the trend in the single crystal data, while Figure 30(b) highlights the differences at the temperature and energy of the nanograin simulations. Mean values are indicated by the sym­bols in Figure 30(a) and the height of the bars in Figure 30(b), and the error bars indicate the standard error in both cases. Similar to the case for surface — influenced cascades, the number of surviving vacan­cies and interstitials is not the same for cascades in nanograined material. The number of vacancies sur­viving in the nanograined material is similar to the single crystal data for 10 keV cascades, but higher at 20 keV. Much lower interstitial survival is observed in nanograined material under all conditions. Consistent with the overall reduction in interstitial survival shown in Figure 30(a), the number of inter­stiials in clusters is dramatically reduced in nano­grained material for all the conditions examined. As the number of surviving point defects, particularly interstitials, is so strongly reduced in the nanograin material, it is helpful to compare the fraction of defects in clusters in addition to the absolute number. Such a comparison is shown in Figure 31 where the fractions of surviving interstitials and vacancies contained in clusters in both nanograined and single crystal iron are compared for all the conditions simulated. The relative change in the clustering fraction is somewhat less than the change in the total number of defects in clusters, but are still substantial for interstitial defects. Notably, the temperature dependence of clustering between 100 and 600 K observed in the single crystal 20 keV cascades is reversed in nanograined material. Between 100 and 600 K, the fraction of interstitials in clusters increases for single crystal iron but decreases for nanograined iron. Conversely, the vacancy cluster fraction decreases for single crystal iron and increases for nanograined iron. Although the range of this study was limited in temperature and cascade energy, the results have demonstrated a strong influence of microstructural length scale (grain size) on primary radiation damage production in iron. Both the effects and the mechan­isms appear to be consistent with previous work in nickel,116,120 in which very efficient transport of interstitial defects to the grain boundaries was observed. In both iron and nickel, this leads to an asymmetry in point defect survival. Many more vacancies than interstitials survive at the end of the cascade event in nanograined material while equal numbers of these two types of point defects survive in single grain material. Similar to single crystal iron,59,64 few of the vacancies have collapsed into compact clusters on the MD timescale. The vacancy clusters in both single and nanograined iron tend to be loose 3D aggregates of vacancies bound at the first and second NN distances as shown above in Figure 20. The size distribution of such vacancy clusters was not significantly different between the single and nanograin material. In contrast, the interstitial cluster size distribution was altered in the nanograined iron, with the number of large clus­ters substantially reduced. There appears to be both a reduction in the number of large interstitial clusters formed directly in the cascade and less coalescence of small mobile interstitial clusters as the latter are being transported to the grain boundaries. The changes in defect survival observed in these simulations are qualitatively consistent with the lim­ited available experimental observations.117-119 For example, Rose and coworkers117 carried out room — temperature ion irradiation experiments of Pd and ZrO2 with grain sizes in the range of 10-300 nm, and observed a systematic reduction in the number of visible defects produced. Chimi and coworkers118 measured the resistivity of ion irradiated gold speci­mens following ion irradiation and found that resis­tivity changes were lower in nanograined material after room-temperature irradiation. However, they observed an increased change in nanograined material following irradiation at 15 K. The low-temperature results could be related to the accumulation of excess vacancy defects as they would be immobile at 15 K. Read More 22.08.2015   Comprehensive nuclear materials Sink strengths of other defects The sink strengths of other defects can be obtained in a similar way. For dislocation loops of a toroidal shape97
kL(v, i) = 2pRlZL4
[62]
where Rl and r^e are the loop radius and the effec­tive core radii for absorption of vacancies and SIAs, respectively. Similar to dislocations, the capture effi­ciency for SIAs is larger than that of vacancies, ZL > ZL, for loops. For a spherical GB of radius Rg (see, e. g., Singh et a/.98)
m is the shear modulus, n the Poisson ratio and DO the dilatation volume of the PD under consideration. The solution of eqn [35] in this case was obtained by Ham95 but is not reproduced here because of its complexity. It has been shown that a reasonably accurate approximation is obtained by treating the dislocation as an absorbing cylinder with radius R, = Aeg/4kBT, where g = 0.5772 is Euler’s con — stant.95 The solution is then given by
K (kr) Ko(kRd)
G Dk2
C (r)	[57]
where K0(x) is the modified Bessel function of zero order. Using eqns [47] and [57], one obtains the total flux of PDs to a dislocation and the dislocation sink strength as I = -2nRdPdDJ (Rd) = k1iD(C1 — Ceq) [58]
k2 kL/2coth(kL/2) — 1
k2 kfoil
[65]
In the limiting case of kL ^ 1, that is, when the foil surfaces are the main sinks,
k2 = ra Z Zd 2p ln(l/kRa)
12 L2
[59]
kf2oil
[66]