Vanadium alloys potentially have low-induced acti­vation characteristics, high-temperature strength, and high thermal stress factors. For the optimization of the composition, both major alloying elements and minor impurities need to be controlled. For main­taining the low activation properties, use of Nb and Mo, which used to be the candidate alloying elements for application to LMFBR, need to be avoided.

Cr was known to increase the strength of vanadium at high temperature and Ti was known to enhance ductility of vanadium by absorbing interstitial impu­rities, mostly oxygen. However, excess Cr or Ti can

lead to loss of ductility. Hence, optimization of Cr and Ti levels for V—xCr-yTi has been investigated. It was known that with x + y > 10%, the alloys became brittle6 as shown in Figure 2. With systematic efforts, V-4Cr-4Ti has been regarded as the leading candi­date. For low activation purposes, the level of Nb, Mo, Ag, and Al needs to be strictly controlled.

Large and medium heats of V-4Cr-4Ti have been made in the United States, Japan, and Russia.

Figure 2 DBTT as a function of Cr + Ti (wt%) of V-Cr-Ti alloy for various annealing temperatures. Reproduced from Zinkle, S. J.; Matsui, H.; Smith, D. L.; Rowcliffe, A. L.; van Osch, E.; Abe, K.; Kazakov, V. A. J. Nucl. Mater. 1998, 258-263, 205-214, with permission from Elsevier.

An especially high-purity V-4Cr-4Ti ingot pro­duced by the National Institute for Fusion Science (NIFS) in collaboration with Japanese Universities (NIFS-HEAT-1 and 2) showed superior properties in manufacturing due to their reduced level of oxygen

4

impurities.

Figure 3 compares the contact dose rate after use in the first wall of a fusion commercial reactor for four reference alloys. The full-remote and full-hands-on recycle limits are shown to indicate the guideline for recycling and reuse.1 SS316LN-IG (the reference ITER structural material) will not reach the remote­recycling limit after cooling and hence the recycling is not feasible. F82H (reference RAFM steel) and NIFS-HEAT-2 behave similarly, but NIFS-HEAT-2 shows significantly lower dose rate before the 100- year cooling. The dose rate of F82H and NIFS- HEAT-2 reached a level almost two orders lower than the remote-recycle limit by cooling for 100 and 50 years, respectively. The dose rate of SiC/SiC com­posites (assumed to be free from impurities because of lack of reference composition) is much lower at <1year cooling, but slightly higher at >100 year cooling relative to F82H and NIFS-HEAT-2.

Rudy Konings is currently head of the Materials Research Unit in the Institute for Transuranium Elements (ITU) of the Joint Research Centre of the European Commission. His research interests are nuclear reactor fuels and actinide materials, with particular emphasis on high temperature chemistry and thermodynamics. Before joining ITU, he worked on nuclear fuel-related issues at ECN (the Energy Research Centre of the Netherlands) and NRG (Nuclear Research and Consultancy Group) in the Netherlands. Rudy is editor of Journal of Nuclear Materials and is professor at the Delft University of Technology (Netherlands), where he holds the chair of ‘Chemistry of the nuclear fuel cycle.’

Roger Stoller is currently a Distinguished Research Staff Member in the Materials Science and Technology Division of the Oak Ridge National Laboratory and serves as the ORNL Program Manager for Fusion Reactor Materials for ORNL. He joined ORNL in 1984 and is actively involved in research on the effects of radiation on structural materials and fuels for nuclear energy systems. His primary expertise is in the area of computa­tional modeling and simulation. He has authored or coauthored more than 100 publications and reports on the effects of radiation on materials, as well as edited the proceedings of several international conferences.

Todd Allen is an Associate Professor in the Department of Engineering Physics at the University of Wisconsin — Madison since 2003. Todd’s research expertise is in the area of materials-related issues in nuclear reactors, specifi­cally radiation damage and corrosion. He is also the Scientific Director for the Advanced Test Reactor National Scientific User Facility as well as the Director for the Center for Material Science of Nuclear Fuel at the Idaho National Laboratory, positions he holds in conjunction with his faculty position at the University of Wisconsin.

vi Editors Biographies

There are essentially three primary energy sources for the billions of people living on the earth’s surface: the sun, radioactivity, and gravitation. The sun, an enormous nuclear fusion reactor, has transmitted energy to the earth for billions of years, sustaining photosynthesis, which in turn produces wood and other combustible resources (biomass), and the fossil fuels like coal, oil, and natural gas. The sun also provides the energy that steers the climate, the atmospheric circulations, and thus ‘fuelling’ wind mills, and it is at the origin of photovoltaic processes used to produce electricity. Radioactive decay of primarily uranium and thorium heats the earth underneath us and is the origin of geothermal energy. Hot springs have been used as a source of energy from the early days of humanity, although it took until the twentieth century for the potential of radioactivity by fission to be discovered. Gravitation, a non-nuclear source, has been long used to generate energy, primarily in hydropower and tidal power applications.

Although nuclear processes are thus omnipresent, nuclear technology is relatively young. But from the moment scientists unraveled the secrets of the atom and its nucleus during the twentieth century, aided by developments in quantum mechanics, and obtained a fundamental understanding of nuclear fission and fusion, humanity has considered these nuclear processes as sources of almost unlimited (peaceful) energy. The first fission reactor was designed and constructed by Enrico Fermi in 1942 in Chicago, the CP1, based on the fission of uranium by neutron capture. After World War II, a rapid exploration of fission technology took place in the United States and the Union of Soviet Socialist Republics, and after the Atoms for Peace speech by Eisenhower at the United Nations Congress in 1954, also in Europe andJapan. Avariety of nuclear fission reactors were explored for electricity generation and with them the fuel cycle. Moreover, the possibility of controlled fusion reactions has gained interest as a technology for producing energy from one of the most abundant elements on earth, hydrogen.

The environment to which materials in nuclear reactors are exposed is one of extremes with respect to temperature and radiation. Fuel pins for nuclear reactors operate at temperatures above 1000 °C in the center of the pellets, in fast reactor oxide fuels even above 2000 °C, whereas the effects of the radiation (neutrons, alpha particles, recoil atoms, fission fragments) continuously damage the material. The cladding of the fuel and the structural and functional materials in the fission reactor core also operate in a strong radiation field, often in a dynamic corrosive environment of the coolant at elevated temperatures. Materials in fusion reactors are exposed to the fusion plasma and the highly energetic particles escaping from it. Furthermore, in this technology, the reactor core structures operate at high temperatures. Materials science for nuclear systems has, therefore, been strongly focussed on the development of radiation tolerant materials that can operate in a wide range of temperatures and in different chemical environments such as aqueous solutions, liquid metals, molten salts, or gases.

The lifetime of the plant components is critical in many respects and thus strongly affects the safety as well as the economics of the technologies. With the need for efficiency and competitiveness in modern society, there is a strong incentive to improve reactor components or to deploy advanced materials that are continuously developed for improved performance. There are many examples of excellent achievements in this respect. For example, with the increase of the burnup of the fuel for fission reactors, motivated by improved economics and a more efficient use of resources, the Zircaloy cladding (a Zr-Sn alloy) of the fuel pins showed increased susceptibility to coolant corrosion, but within a relatively short period, a different zirconium-based alloy was developed, tested, qualified, and employed, which allowed reliable operation in the high burnup range.

Nuclear technologies also produce waste. It is the moral obligation of the generations consuming the energy to implement an acceptable waste treatment and disposal strategy. The inherent complication of radioactivity, the decay that can span hundreds of thousands of years, amplifies the importance of extreme time periods in the issue of corrosion and radiation stability. The search for storage concepts that can guarantee the safe storage and isolation of radioactive waste is, therefore, another challenging task for materials science, requiring a close examination of natural (geological) materials and processes.

The more than 50 years of research and development of fission and fusion reactors have undoubtedly demonstrated that the statement ‘technologies are enabled by materials’ is particularly true for nuclear technology. Although the nuclear field is typically known for its incremental progress, the challenges posed by the next generation of fission reactors (Generation IV) as well as the demonstration of fusion reactors will need breakthroughs to achieve their ambitious goals. This is being accompanied by an important change in materials science, with a shift of discovery through experiments to discovery through simulation. The progress in numerical simulation of the material evolution on a scientific and engineering scale is growing rapidly. Simulation techniques at the atomistic or meso scale (e. g., electronic structure calculations, molecular dynam­ics, kinetic Monte Carlo) are increasingly helping to unravel the complex processes occurring in materials under extreme conditions and to provide an insight into the causes and thus helping to design remedies.

In this context, Comprehensive Nuclear Materials aims to provide fundamental information on the vast variety of materials employed in the broad field of nuclear technology. But to do justice to the comprehensiveness of the work, fundamental issues are also addressed in detail, as well as the basics of the emerging numerical simulation techniques.

R. J.M. Konings European Commission, Joint Research Centre, Institute for Transuranium Elements, Karlsruhe, Germany

T. R. Allen

Department ofEngineering Physics, Wisconsin University, Madison, WI, USA

R. Stoller

Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA

S. Yamanaka

Division of Sustainable Energy and Environmental Engineering, Graduate School of Engineering, Osaka University, Osaka, Japan

‘Nuclear materials’ denotes a field of great breadth and depth, whose topics address applications and facilities that depend upon nuclear reactions. The major topics within the field are devoted to the materials science and engineering surrounding fission and fusion reactions in energy conversion reactors. Most of the rest of the field is formed of the closely related materials science needed for the effects of energetic particles on the targets and other radiation areas of charged particle accelerators and plasma devices. A more complete but also more cumbersome descriptor thus would be ‘the science and engineering of materials for fission reactors, fusion reactors, and closely related topics.’ In these areas, the very existence of such technologies turns upon our capabilities to understand the physical behavior of materials. Performance of facilities and components to the demanding limits required is dictated by the capabilities of materials to withstand unique and aggressive environments. The unifying concept that runs through all aspects is the effect of radiation on materials. In this way, the main feature is somewhat analogous to the unifying concept of elevated temperature in that part of materials science and engineering termed ‘high-temperature materials.’

Nuclear materials came into existence in the 1950s and began to grow as an internationally recognized field of endeavor late in that decade. The beginning in this field has been attributed to presentations and discussions that occurred at the First and Second International Conferences on the Peaceful Uses of Atomic Energy, held in Geneva in 1955 and 1958. Journal of Nuclear Materials, which is the home journal for this area of materials science, was founded in 1959. The development of nuclear materials science and engineering took place in the same rapid growth time period as the parent field of materials science and engineering. And similarly to the parent field, nuclear materials draws together the formerly separate disciplines of metallurgy, solid-state physics, ceramics, and materials chemistry that were early devoted to nuclear applications. The small priest­hood of first researchers in half a dozen countries has now grown to a cohort of thousands, whose home institutions are anchored in more than 40 nations.

The prodigious work, ‘Comprehensive Nuclear Materials, captures the essence and the extensive scope of the field. It provides authoritative chapters that review the full range of endeavor. In the present day of glance and click ‘reading’ of short snippets from the internet, this is an old-fashioned book in the best sense of the word, which will be available in both electronic and printed form. All of the main segments of the field are covered, as well as most of the specialized areas and subtopics. With well over 100 chapters, the reader finds thorough coverage on topics ranging from fundamentals of atom movements after displacement by energetic particles to testing and engineering analysis methods of large components. All the materials classes that have main application in nuclear technologies are visited, and the most important of them are covered in exhaustive fashion. Authors of the chapters are practitioners who are at the highest level of achievement and knowledge in their respective areas. Many of these authors not only have lived through a substantial part of the history sketched above, but they themselves are the architects. Without those represented here in the author list, the field would certainly be a weaker reflection of itself. It is no small feat that so many of my distinguished colleagues could have been persuaded to join this collective endeavor and to make the real sacrifices entailed in such time-consuming work. I congratulate the Editor, Rudy Konings, and the Associate Editors, Roger Stoller, Todd Allen, and Shinsuke Yamanaka. This book will be an important asset to young researchers entering the field as well as a valuable resource to workers engaged in the enterprise at present.

Dr. Louis K. Mansur Oak Ridge, Tennessee, USA

F. Onimus and J. L. Bechade

Commissariat a I’Energie Atomique, Gif-sur-Yvette, France © 2012 Elsevier Ltd. All rights reserved.

4.01.1 Irradiation Damage in Zirconium Alloys 2

4.01.1.1 Damage Creation: Short-Term Evolution 2

4.01.1.1.1 Neutron-zirconium interaction 2

4.01.1.1. 2 Displacement energy in zirconium 2

4.01.1.1.3 Displacement cascade in zirconium 2

4.01.1.2 Evolution of Point Defects in Zirconium: Long-Term Evolution 4

4.01.1.2.1 Vacancy formation and migration energies 4

4.01.1.2.2 SIA formation and migration energies 4

4.01.1.2.3 Evolution of point defects: Impact of the anisotropic diffusion of SIAs 6

4.01.1.3 Point-Defect Clusters in Zirconium Alloys 7

4.01.1.3.1 (a) Dislocation loops 7

4.01.1.3.2 (a) Loop formation: Mechanisms 8

4.01.1.3.3 (c) Component dislocation loops 9

4.01.1.3.4 (c) Loop formation: Mechanisms 9

4.01.1.3.5 Void formation 10

4.01.1.4 Secondary-Phase Evolution Under Irradiation 10

4.01.1.4.1 Crystalline to amorphous transformation of Zr-(Fe, Cr, Ni) intermetallic precipitates 10

4.01.1.4.2 Irradiation effects in Zr-Nb alloys: Enhanced precipitation 13

4.01.2 Postirradiation Mechanical Behavior 14

4.01.2.1 Mechanical Behavior During Tensile Testing 14

4.01.2.1.1 Irradiation hardening: Macroscopic behavior 14

4.01.2.1.2 Irradiation hardening: Mechanisms 14

4.01.2.1.3 Post-yield deformation: Macroscopic behavior 16

4.01.2.1.4 Post-yield deformation: Mechanisms 16

4.01.2.2 Effect of Postirradiation Heat Treatment 17

4.01.2.3 Postirradiation Creep 18

4.01.3 Deformation Under Irradiation 19

4.01.3.1 Irradiation Growth 19

4.01.3.1.1 Irradiation growth: Macroscopic behavior 19

4.01.3.1.2 Irradiation growth: Mechanisms 21

4.01.3.2 Irradiation Creep 24

4.01.3.2.1 Irradiation creep: Macroscopic behavior 24

4.01.3.2.2 Irradiation creep: Mechanisms 25

4.01.3.3 Outlook 26

References 27

An important contribution to the damage in nuclear fuels derives from fission fragments. There are two groups of fission products: one group with atomic number near 42 (Mo) and energy «100 MeV and the other with atomic number near 56 (Ba) and energy «70 MeV. The maximum electronic stopping powers of these energetic particles, «18keVnm~ for the heavier and 22 keV nm-1 for the lighter, are far greater than their respective nuclear stopping powers. Similar to ion irradiation studies described above, where the primary recoil spectrum can be systematically varied, the masses and energies of ions can be varied to examine effects of electronic stopping power. An example is shown in Figure 18 where the electronic stopping power is plotted as a function of energy (per nucleon) for different ion irradiations of UO2. The two boxes in the figure indicate stopping powers associated with the fission fragments and the heavy particle recoils of a emit­ters. One of the questions addressed by such studies

40 cN D) C Ф
20 15 10 5 0
20
10
20
10
0
10 20 HiapP^bP3
0
(b)
(a)

Figure 17 Cavity volume fraction (a) and cavity density (b) in pure vanadium irradiated with 12 MeV Ni3+ ions to 30dpa at 873 K with and without simultaneous irradiation of He and H. Reproduced from Sekimura, N.; Iwai, T.; Arai, Y.; etal.

J. Nucl. Mater. 2000, 283-287, 224-228.

has been the formation of fission fragment tracks. Tracks have not yet been observed in the bulk of UO2 due to fission; however, by using ion irradiation, the stopping powers could be increased. The dashed line at 29keVnm-1 in Figure 18 represents the threshold stopping power for track formation.37 This value is «30% greater than the maximum for fission fragments, thus helping to explain why fission fragment tracks are not seen in the bulk. Such tracks are observed, however, close to the surface. They are explained by fission products passing near or parallel to the surface and creating shock waves which inter­act with the surface.38 These studies have also been useful in gaining important data for understanding fission gas evolution in nuclear fuels. For example, 72MeV iodine ions (see Figure 18), approximate very closely the stopping power of fission fragments. Such studies have shown that 72 MeV I irradiations cause Kr atoms preimplanted into UO2 to nucleate into bubbles, and preformed bubbles to undergo res­olution. A radiation-enhanced diffusion coefficient for the Kr was estimated from these studies to be D « 1.2 x 10~30cm5 x F, where F is the fission rate per cubic centimeter, and found independent of tem­perature below «500 °C (see Matzke etal.37 for details). The importance of such studies as these is that the basic processes in complex nuclear fuels can be elucidated by studies that carefully control singly the irradiation conditions and materials parameters in the fuel, such as fission gas concentration, damage, etc.

Before the aforementioned work by Bockstedte and coworkers106 almost no work was devoted to migra­tion properties of point defects in SiC. We should, however, cite previous preliminary works by the same group,111,112 a work on the mechanisms of formation of antisite pairs,113 and a work on vacancy migration published in 2003.114 The comprehensive study of migration barriers in Bockstedte et a/.106 showed, first of all, that vacancies have much higher migration energies than those of interstitials: higher than 3 eV for the former in the neutral state, around 1 eV for the latter (0.5 for IC, 1.4 for ISi). Another

remarkable finding is the strong variation of the migration energy with the charge state; indeed, the migration energy for the carbon vacancy is raised by almost 2 eV going from the neutral to the 2+ charged state, whereas the silicon vacancy finds its migration barrier reduced by 1 eV when its charge goes from neutral to 2—. Interstitials are reported to have their lowest migration barriers in the neutral state, except for the ISTiC configuration, which is expected to have an almost zero energy barrier of migration in the 2+ and 3+ charge states. Such large changes in the migration energies of defects with their charge should induce tremendous variations in their kinetic behavior under different charge states.

The energy barriers of recombinations of close interstitial vacancy pairs have also been tack — led.115-117 It appears that the energetic landscape for the recombination of Frenkel pairs is extremely complex. One should distinguish the regular recom­bination of an homo interstitial-vacancy pairs from those of hetero interstitial-vacancy pairs, which leads to the formation of an antisite. Recent works tend to suggest that the latter may, in certain con­ditions, have a lower energy than the recombination of a regular Frenkel pair. A kinetic bias for the formation of antisites, preliminary to decomposition, may thus be active in SiC under irradiation.118

Calculations of threshold displacement energies from first-principles molecular dynamics29 have also been reported. Their results show that this quantity is strongly anisotropic, and they found average values (38 eV for Si and 19 eV for C) that are in agreement with currently accepted values (coming from experi­mental evaluations that are, however, largely dis­persed). These calculations prove that available CPU power is now large enough to calculate TDE from ab initio molecular dynamics. This is good news as empirical potentials are basically not reliable in the prediction of TDE.

1.08.5.1.1 Defect complexes

Several defect complexes have been studied by first-principles calculations in silicon carbide. The identification of EPR signals, deep level transient spectroscopy (DLTS), or photoluminescence (PL) experiments based on calculated properties have been attempted for some of them. Crucial to these identifications is the reliability of the predictions of charge transition levels (for the position of DLTS peaks) and of annealing temperatures, through more or less complicated mechanisms.

One of the first, and simplest, defect complex iden­tified through comparison of theory and experiment was the VC-CSi coming from the annealing of silicon vacancies in 6H-SiC, as previously mentioned. More complex antisite defects or antisite complexes119,120 as well as divacancy complexes121-123 were called upon for the attribution of PL or EPR peaks.

Various kinds of carbon clusters were studied in detail theoretically.124-127 The cited works deal with the stability, electrical properties, and local vibrational modes (LVM) of several structures. It was shown that the aggregation of carbon interstitials with carbon antisites can lead to various bound configurations. In particular, two, three, or even four carbon atoms can substitute one silicon atom forming very stable structures. The binding energy of these structures is high: from 3.9 to 5 eV, according to the charge state, for the (C2)Si, and further energy is gained when adding further carbon atoms. Silicon clusters did not raise as much interest as carbon ones; however, a recent work107 deals with the stability and dynamics of such silicon clusters (see Figure 12).

The modified embedded atom method (MEAM) is an empirical extension of EAM by Baskes, which

includes angular forces. As in the EAM, there are pairwise repulsions and an embedding function. In the EAM, the pi is interpreted as a linear supposition of species-dependent spherically averaged atomic elec­tron densities (here designated by f (r)); in MEAM pj is augmented by angular terms. The spherically sym­metric partial electron density p(0) is the same as the electron density in the EAM:

r(0)= X fm(ri)

j

where the sum is over all atoms j, not including the atom at the specific site of interest i. The angular contributions to the density are similar to spherical harmonics: they are given by similar formulas weighted by the x, y, and z components of the distances between atoms (labeled by a, b, g):

2

(r(1))2=E

a

(r(2))2 —

a, b

(r(3))2=E

a. p.g

The f(l) are so-called ‘atomic electron densities,’ which decrease with distance from the site of interest, and the a, b, and g summations are each over the three coordinate directions (x, y z). The functional forms for the partial electron densities were chosen to be trans­lationally and rotationally invariant and are equal to zero for crystals with inversion symmetry about all atomic sites. Although the terms are related to powers of the cosine of the angle between groups of three atoms, there is no explicit evaluation of angles, and all the information required to evaluate the MEAM is available in standard MD codes. Typically, atomic electron densities are assumed to decrease exponen­tially, that is, f(l) (R) — exp [—b(l) (R/re — 1)] where the decay lengths (re and b(l)) are constants.

While there is no derivation of the MEAM from electronic structure, it also introduced the physically reasonable idea ofmany-body screening, which is miss­ing in pair-functional forms such as EAM. Thus, f(Rj) is reduced by a screening factor determined by the other atoms k forming three-body triplets with i and j: primar­ily those lying between j and j. This eliminates the need for an explicit cut-off in the ranges of V (r) and f(l) (r).

For close-packed materials, the improvement of MEAM over standard EAM is marginal; the angular terms come out to be small. For sp-bonded materials, a large three-body term can stabilize tetrahedrally coor­dinated structures, but since the physics arises from preferred 109° angles rather than preferred fourfold coordination, it suffers problems similar to Stillinger- Weber type potentials (discussed below). Very high angular components enable one to fit the complex phases of lanthanides and actinides. It is tempting to attribute this to the correct capture of the f-electron physics, although the additional functional freedom may play a role in enabling fits to low symmetry structures.

Atomic-scale methods and particularly MD can pro­vide a wide range of valuable information on the processes simulated. The most important are

1. Information on the physical state of the system.

This includes temperature and stress and their distribution; displacement of atoms and their transport; interaction energy and therefore force between defects; and evolution of internal, elastic, and free energies. Extraction of this information is well understood and procedures can be found in Chapter 1.04, Effect of Radiation on Strength and Ductility of Metals and Alloys;

Chapter 2.13, Properties and Characteristics of ZrC; Chapter 5.01, Corrosion and Compatibility and Chapter 1.09, Molecular Dynamics.

2. Detail of atomic mechanisms. This includes anal­ysis of the position and environment of individual atoms based on calculation of their energy, site stress, or local atomic configuration. Atoms can then be identified with particular features such as constituents of defect clusters, stacking faults, dis­location cores, and so on. Having this information at particular times provides unique knowledge of
defect structure, motion, interactions, and transformation.

The information summarized in 1 and 2 can be used to determine how the mechanisms involved depend on parameters such as obstacle type and size and dislocation type, material temperature, and applied stress or strain.

By omitting the recombination term, eqns [10] —[12] for mobile defects can be rewritten as

= Gv + Gvc — DvCv(i2 + Zdpd + ZVclk2cl + zvclk2cl)

^ = g — Dici(k2 + zvclpd + Ziclki2d + zvclk2cl) dcg fpr 2

= Gi — 2DgCg 2d + prcNc + svcNvcl + °icNid [[15] [16]]

It has been shown that, under conditions in which swelling is observed, the vacancy and SIA clusters produced by cascades reach steady-state size distri­butions at relatively small doses.22 This is because vacancy clusters have far lower thermal stability than voids. The growth of sessile SIA clusters is restricted on account of the high vacancy supersaturation, which builds up due to rapid 1D diffusion of mobile SIA clusters to sinks. Consequently, at relatively low doses, the SDF of the sessile SIA clusters achieves steady state. After reaching steady state, both types of sessile clusters start to serve as recombination centers for PDs and glissile SIA clusters. Analytical expres­sions for the steady-state SDFs of vacancy and SIA clusters can be found (see eqns [23] and [24] in Singh et at22) and the corresponding sink strengths of the clusters at the steady state are given by (eqn [25] in the same reference) 4G
produced are reduced in size due to the high vacancy supersaturation. Fluctuations in the defect arrival to the clusters produce a tail in the SDF extending beyond the maximum size formed in cascades. The tail is characterized by very small concentrations and cannot describe the observed nucleation and growth of SIA clusters and the consequent formation of the dislocation network (see, e. g., Garner[17] and Garner et a/.[18]). The most probable reason for this failure is that the cluster-cluster interaction leading to their coalescence is neglected in the current theoretical framework.

Sessile interstitial clusters are produced in cas­cades at rates comparable to those of PDs. The evo­lution of concentrations of mobile species (PDs and glissile clusters) in this case may be described by nonstationary equations because of the very fast evo­lution of the sessile cluster population. High vacancy supersaturation will drive the evolution of the sessile SIA clusters toward quasisaturation state, beyond which the steady-state equations for the mobile spe­cies become valid. Similar steady state for vacancy clusters will be achieved because of the thermal instability of the clusters.22

Gv = DC (k2 + ZvVd) + DvCvZvlk2d

+ Di Ci z;cl4l + Dg CgLxgffidNid [133]

Gi = DiCi(k2 + Zfpd) + D^Zf^

+ ACiZ;dk2d — 2Dg CgLxg ffidNd [[19]]

Gg = 2DgCg + nr2cNc + svcNvd + ^Nd [135]

In the framework of PBM, the balance equations for PDs depend on the concentration of glissile clusters and, thus, are very different from those in the FP3DM.

The vacancy supersaturation is obtained from the difference between DvCv and DiCi as given by eqns

[133] and [134]

DVCV — DiCi

B Zvrd D C = її d DvCv

k2 + ZVd

eg gnrt (1 — er)

k2 + ZJPd

where Lg = k|/2 = Prdpd/2 + Prc2Nc + ffvdNvd+

SidNid. The first and the second terms on the right-hand side of eqn [136] correspond to the actions