1.10.6.1 Energy of a Part-Filled Band

An alternate starting point to defining potentials is tight-binding theory. As this already has localized orbitals, it gives a more intuitive path from quantum mechanics to potentials. Consider a band with a den­sity of electronic states (DOS) n(E) from which the cohesive energy becomes •Ef

(E — Eo)n(E) dE where E0 is the energy of the free atom, which to a first approximation lies at the center of the band.

For example, a rectangular d-band describing both spin states and containing N electrons, width W has n(E) = 10/ W, and Ef = W(N — 5)/10 + E0 whence (Figure 3),

This gives parabolic behavior for a range of energy — related properties across the transition metal group, such as melting point, bulk modulus, and Wigner — Seitz radius. For a single material, the cohesion is proportional to the bandwidth. Even for more com­plex band shapes, the width is the key factor in determining the energy.

The width of the band can be related to its second moment9 here: (E — Eo)2n(E) dE

E0+W/2

Eo—W/2

= 5 W2/6

To build a band in tight-binding theory, we set up a matrix of onsite and hopping integrals (Figure 4). For a simple s-band ignoring overlap, n(E)

Figure 4 Matrix of onsite and hopping integrals for a planar five-atom cluster — in tight binding this gives five eigenstates, each of which contributes one level to the ‘density of states’: five delta functions. In an infinite solid, the matrix and number of eigenstates become infinite, so the density of states becomes continuous. Of course, tricks then have to be employed to avoid diagonalizing the matrix directly.

S = (Ф,1 Vi |Ф,> h{rtJ ) = (Фі | Vi |фу>

The electron eigenenergies come from diagonalizing this matrix (there are, of course, cleverer ways to do this than brute force). Typically, we can use them to create a density of states, n(E), which can be used to determine cohesive energy (as above).

The width of this band depends on the off-diagonal terms (in the limit of h = 0, the band is a delta function). One can proceed by fitting S and h, or move to a further level of abstraction.

Interest in ferritic and ferritic-martensitic steels has stimulated the development of Fe-Cr potentials such as those discussed by Malerba and coworkers.129 These potentials have been applied to investigate the influence of Cr on displacement cascades13 , and on point defect diffusion.1 The MD cascade study by Malerba and coworkers130 involved cascade energies from 0.5 to 15keV at 300 K. In contrast to the Fe-C and Fe-Cu results discussed above, a slight increase in stable defect formation was observed in Fe-10%Cr relative to pure Fe. The asymptotic value of the defect survival ratio (relative to the NRT) at the highest energies was 0.28 for Fe and 0.31 for Fe-10%Cr. In a later study by the same

authors, which involved a larger number of simula­tions and energies up to 40 keV, they also concluded that the presence of 10%Cr did not lead to a change in the collisional phase of the cascade but rather reduced the amount of recombination during the cooling phase.1 Additional detailed studies per­

formed with more recent Fe-Cr potentials essentially confirmed the absence of any significant effect of Cr on primary damage in Fe-Cr alloys as compared to pure Fe.136-138

The lack of a Cr effect on the collisional or ballistic phase of the cascade may be expected because, like Cu, the mass of Cr is similar to Fe. The reduced recombination appears to be related to the formation of highly stable mixed Fe-Cr dumb­bell interstitials. About 60% of interstitial dumbbells contain a Cr atom, which is substantially higher than the overall Cr concentration of 10%. In spite of the strong mixed dumbbell formation, the fraction of point defects in clusters did not seem to be signifi­cantly different than in pure Fe. However, if the stability and mobility of the mixed dumbbells and clusters containing them proves to be appreciably different than pure iron dumbbells,1 2 there could be an influence on damage accumulation at longer times. Experimental results that are consistent with this hypothesis135 are mentioned in Terentyev and

coworkers.138

Equations [54] and [62] for the sink strengths of voids and dislocation loops for mobile PDs permit the calculation of rates P(x) and Q(x), which deter­mine the cluster evolution described by the ME (see Section 1.13.4.3.2). For example, the total rate of absorption of vacancies by voids is equal to k^DvCv (see eqns [10] and [45]). The same quantity is given

by Pc( x)fc(x). By equating these two rates one

x=2

obtains

 

Read More 22.08.2015   Comprehensive nuclear materials Irradiation-induced segregation on defect clusters In order to circumvent the problems raised in the previous paragraph and in Section 1.15.4.1 for the inclusion of defect clusters in a PFM, Badillo et al.146 have recently proposed a mixed approach that com­bines discrete and continuum treatments of the defect clusters, so that each cluster is treated as a separate entity. Point-defect cluster size is treated as a discrete quantity for cluster production, whereas the long-term fate of clusters is controlled by a continuum-based flux of free point defects. New field variables are thus introduced to describe the size of these clusters: Np Np Np Cp _ Nc, V; Cp _ Nc, A; Cp _ Nc, B c, V _ Nd ’ c, A _ Nd ’ c, B _ Nd where Nf v is the number of vacancies in the vacancy cluster in the cell p, and Nd is the number ofsubstitu- tional lattice sites per cell; N a and Np B are the numbers of A and B interstitial atoms, respectively, forming the interstitial cluster in the cell p. Each cell contains at most one cluster. The production of point defects by irradiation takes place at a rate dictated by the irradiation flux f in dpas-1 and by the simulated volume. In the case of irradiation conditions leading to the intracascade clustering of point defects, the total number of point defects created in a displacement cascade, the frac­tion of those defects that are clustered, and the size and spatial distribution of these clusters are used as input data. The production of Frenkel pairs is treated in the same way as defect clusters, except that the variables affected are the free vacancy and interstitial concentrations. This treatment of defect and defect cluster pro­duction makes it possible to compare irradiation con­ditions with identical total defect production rates, that is, identical dpas-1 values, but with varied frac­tions of intracascade defect clustering and varied spatial distribution of these clusters. Furthermore, it is also very well suited for system-specific modeling, since all the above information can be directly and accurately obtained from molecular dynamic simula­tions sampling the primary recoil spectrum.1 1 In particular, one can build a library of such displace­ment cascades, so that the PFM will inherit the stochastic character in space and time of the produc­tion of defect clusters by displacement cascades. The continuous flux of free point defects to the clusters results in the growth or shrinkage of these clusters, which translates into the continuous evolution of the cluster field variables CjP v, Cp a, and Cpp B. When any cluster field variable drops below 1 /Nd, this cluster is assumed to have dissolved, and the remaining one point defect is transferred to the corresponding free point-defect variable of that cell. For the sake ofsimplicity, defect clusters are treated as immobile, but the approach can be extended to include mobility, in particular for small interstitial clusters. Further details are available in Badillo eta/.146 The potential of the above approach is illustrated by considering a 2D A8B92 alloy with a zero heat of mixing, so that at equilibrium, it always forms a random solid solution. The production of interstitials is, however, biased so that only A interstitial atoms are created. This could, for instance, simulate an alloy where there is a rapid conversion ofB interstitial atoms into A interstitial atoms via an interstitialcy mecha­nism. The preferential transport of A interstitial atoms to defect clusters should lead to an enrichment of A species around defect clusters, since these clusters act as defect sinks. The effect of the primary recoil spectrum on this irradiation-induced segregation is studied by comparing two cases: the first one where a small fraction of cascades, 1/Ncas = 5 x 10~4, produces defect clusters, and the second one where that fraction is 100 times higher, 1/Ncas = 5 x 10~2. In both cases, however, the displacement rate per atom per second is the same, here 10~7 dpa s_1. Figure 14 shows instanta­neous concentration maps of the A solute atoms for 1/Ncas = 5 x 10~4. In this case, the PFM uses 64 x 64 cells, each containing 7 x 7 lattice sites. Segregation of A species is clearly observed at a few locations, typi­cally 2-5. This number is close to the average number of defect clusters. The sharp peaks with high levels of segregation correspond to segregation of existing defect clusters, either interstitial or vacancy ones. This is confirmed by visualizing the defect and defect cluster fields, see Figure 15. The broader segregation profiles in Figure 14 are the remnants of sharp segre­gation profiles after their corresponding clusters shrank A8B92 alloy with zero heat of mixing where all interstitials are created as A atoms. The two-dimensional model system contains 448 x 448 lattice sites, decomposed into 64 x 64 cells for defining the phase field variables, each cell containing 7 x 7 lattice sites. Irradiation displacement rate is 10~7 dpa s~1; the cascade frequency rate is1/Ncas = 5 x 10~4, and the irradiation dose is 6dpa. Reproduced from Badillo, A.; Bellon, P.; Averback, R. S. to be submitted. and disappeared. As a result, the nonequilibrium segre­gation that build up on those clusters is being washed out by vacancy diffusion, as expected for an A-B alloy system with zero heat of mixing. In the case where defect sinks have a finite lifetime, as in the present case, one should thus expect a dynamical formation and elimination of segregated regions. In the case of much higher defect cluster produc­tion rate, 1/Ncas = 5 x 10~2, a very different micro­structure is stabilized by irradiation, as illustrated in Figure 16. Now a high density of clusters is present, typically 40 interstitial clusters and 20 vacancy clus­ters, as seen in Figure 17(a) and 17(b), and the segregation measured on these clusters is reduced by about one order of magnitude compared to the previous case. These results are reminiscent of the experimental findings reported by Barbu and Ardell147 and Barbu and Martin,148 showing that, with irra­diation conditions producing displacement cascades (e. g., 500-keV Ni ion irradiation), the domain of irradiation-induced segregation and precipitation in undersaturated Ni-Si solid solutions is significantly reduced compared to the case where irradiation pro­duces only individual point defects (e. g., 1-MeV elec­tron irradiation). Read More 22.08.2015   Comprehensive nuclear materials Diffusion Equations: Nonequilibrium Thermodynamics In pure metals, the evolution of the average concen­trations of vacancies CV and self-interstitials Q are given by: ^ = K0 — RCiCv — X 4sDv(CV — CVq) s dd = K0 — RCi Cv — X 4 A Ci [1] s where K0 is the point defect production rate (in dpas-1) proportional to the radiation flux, R is the recombination rate, and DV and DI are the point defect diffusion coefficients. The third terms of the right hand side in eqn [1] correspond to point defect annihilation at sinks of type s. The ‘sink strengths’ kVs and k2s depend on the nature and the density of sinks and have been calculated for all common sinks, such as dislocations, cavities, free surfaces, grain bound­aries, etc.56,57 The evolution of point defect concen­trations depending on the radiation fluxes and sink microstructure can be modeled by numerical integra­tion of eqn [1], and steady-state solutions can be found analytically in simple cases.43 The evolution of concentration profiles of vacan­cies, interstitials, and chemical elements a in an alloy under irradiation are given by 0CV V =-div Jv + Ko — RCiCv ot -X kVsDv [Cv — C Vq] s = — div Ji + Ko — RCi Cv — 4 A Ci ot s ^ =-div Ja [2] The basic problem of RIS is the solution of these equations in the vicinity of point defect sinks, which requires the knowledge of how the fluxes Ja are related to the concentrations. Such macroscopic equations of atomic transport rely on the theory of TIP. In this chapter, we start with a general descrip­tion of the TIP applied to transport. Atomic fluxes are written in terms of the phenomenological coeffi­cients of diffusion (denoted hereafter by Lj or, sim­ply, L ) and the driving forces. The second part is devoted to the description of a few experimental procedures to estimate both the driving forces and the L-coefficients. In the last part, we present an atomic-scale method to calculate the fluxes from the knowledge of the atomic jump frequencies. 1.18.3.1 Atomic Fluxes and Driving Forces Within the TIP,58’59 a system is divided into grains, which are supposed to be small enough to be consid­ered as homogeneous and large enough to be in local equilibrium. The number of particles in a grain varies if there is a transfer of particles to other grains. The transfer of particles a between two grains is described by a flux Ja, and the temporal variation of the local a concentration is given by the continuity equation dCa — =-div Ja [3] The flux of species a between grains i and j is assumed to be a linear combination of the thermody­namic forces, Xp = (/p — mp)/kB T (i. e., of the gradi­ent of chemical potentials V/p) where /p is the chemical potential of species p on site i, T the tem­perature, and kB the Boltzmann constant. Variables Xp represent the deviation of the system from equi­librium, which tend to be decreased by the fluxes: Ja = ~^2 LapXp [4] p The equilibrium constants are the phenomenological coefficients, and the Onsager matrix (Lap) is sym­metric and positive. When diffusion is controlled by the vacancy mechanism, atomic fluxes are, by con­struction, related to the point defect flux: Jv = — X JV [5] a As gradients of chemical potential are indepen­dent, eqn [5] leads to some relations between the phenomenological coefficients and, if we choose to eliminate the LVp coefficients, we obtain an expres­sion for the atomic fluxes: Vacancy and interstitial contributions to the atomic fluxes are assumed to be additive: Ja = — E LVp (Xp — Xv) — X Lap (Xp + Xi) [8] pp While thermodynamic data is usually available for the determination of driving forces, it is very difficult to determine the whole set of the L-coefficients from diffusion data. In the first part of the chapter, we define the driving forces as a function of concentra­tion gradients. Then, we present the experimental diffusion coefficients in terms of the L-coefficients and thermodynamic driving forces. The last part of the section shows how to use first-principle calcula­tions for building atomic jump frequency models to calculate macroscopic fluxes of specific alloys. Read More 22.08.2015   Comprehensive nuclear materials Electron Irradiations The unique feature ofelectron irradiations in compari­son to ions and neutrons is that they create defects in very low-energy recoil events. As a consequence, nearly all FPs are produced in isolation. This has been of foremost importance in developing our understanding of radiation damage, as it made studies of defect crea­tion mechanisms as well as the fundamental properties of FPs possible. Recall that the properties of vacancies and vacancy clusters, for example, formation and migra­tion energies, stacking fault energies, etc., could be determined from quenching studies. It is not possible, however, to quench in interstitials in metals. Very little was therefore known about this intrinsic defect prior to about 1955 when irradiation experiments became widely employed. In this section, we highlight some of the key findings derived from these past studies. Read More 22.08.2015   Comprehensive nuclear materials Oxygen clusters Another point of interest beyond point defects is the clustering of oxygen interstitials. Indeed, oxygen interstitial clustering has been deduced from diffrac­tion experiments174 many years ago. However, a debate remains on the exact shape of such clusters. Two configurations are contemplated: the so-called Willis clusters174,175 or cubo-octahedral clusters that have been observed by neutron diffraction in U4O9176 and U3O7.177 These clusters are made of 12 oxygen and 8 uranium atoms and amount for 4 oxygen inter­stitials. An additional oxygen interstitial may reside in the center of the cluster, forming a so-called filled cube-octahedral cluster (with five interstitials). Recent calculations have proved that Willis clus­ters are in fact unstable and transform upon re­laxation into assemblies of three or four interstitials surrounding a central vacancy cluster (Figure 13).178 The three interstitial-1 vacancy cluster has been found independently by other authors173,180 who refer to it as split di-interstitials. These clusters prove in fact to have a formation energy higher than the cube-octahedral cluster (Figure 14), especially the filled one.178,179 Figure 14 Cubo-octahedral cluster of oxygen interstitials in UO2. Reprinted with permission from Geng, H. Y.; Chen, Y.; Kaneta, Y.; Kinoshita, M. Phys. Rev. B 2008, 77, 180101(R).180 Copyright (2008) by the American Physical Society. Read More 22.08.2015   Comprehensive nuclear materials Short-Range Interactions In radiation damage simulations, atoms can come much closer together than in any other application. Interatomic force models are often parameterized on data that ignore very short-ranged interactions, and the physics of core wavefunction overlap is seldom well described by extarpolation. In radiation damage, we are normally dealing with a case where two atoms come very close together, but the density of the material does not change. By anal­ogy with a free electron gas, we see that the energy cost of compressing the valence wavefunctions is important for high-pressure isotropic compression but absent in the collision scenario. In the first case, the ‘short range’ repulsion is a many-body effect, while in the second case, it is primarily pair­wise. Thus, we should not expect that fitting to the high-pressure equation of state will give a good representation of the forces in the initial stages of a cascade, or even for interstitial defects. Indeed, when there was no accurate measurement of interstitial formation energies, older potentials gave a huge range of values based on extrapolation of near-equilibrium data and uncertain partition of energy between pairwise and many-body terms. Accurate values for these forma­tion energies are now available from first principles calculations and are incorporated in the fitting; there­fore, a symptom of the problem is resolved. At very short range, the ionic repulsive interaction can be regarded as a screened-Coulombic interac­tion, and described by multiplying the Coulombic repulsion between nuclei with a screening function W(r/a): Z1 Z2e2 4ne0r where w(r) ! 1 when r! 0 and Z1 and Z2 are the nuclear charges, and a is the screening length. The most popular parameterization of w is the Biersack — Ziegler potential, which was constructed by fitting a universal screening function to repulsions calculated for many different atom pairs (Ziegler 1985). The Biersack-Ziegler potential has the form W(x) = 0.1818e~32x + 0.5099e~09423x + 0.2802e~0:4029x + 0.02817e~0:2016x where 0. 8854a0 a = zpTzp and x = r/a and a0 = 0.529 A is the Bohr radius. This potential must then be joined to the longer ranged fitted potential. There are many ways to do this, with no guiding physical principle except that the potential should be as smooth as possible. Typical implementations ensure that the potential and its first few derivatives are continuous. The short-range interactions arising from high pressure come mainly from isotropic compression, and should be fitted to the many-body part. To achieve this division in glue models, the function F(p) should become repulsive at large p, but f(r) should not become very large at small Rj. Read More 22.08.2015   Comprehensive nuclear materials Dislocation-Type Obstacles Extensive simulations of interactions between moving dislocations and dislocation-like obstacles such as DLs and SFTs has demonstrated that the reactions involved follow the general rules of dislocation — dislocation reaction, for example, Frank’s rule for Burgers vectors,1,2 even though the reacting segments are of the nanometer scale in length. Results of these interactions are in the range from no effect on both dislocation and obstacle to complete disappearance of the obstacle and significant modification of the dislo­cation. A detailed analysis of reactions was made for SFTs an fcc metal56 and later for SIA loops in Fe.57 In general, five types of reaction were identified, as sum­marized in Table 1. The outcomes in Table 1 were observed for different obstacles under different reac­tion conditions such as interaction geometry, strain rate, ambient temperature, and so on. We give some examples in the following section. 1.12.4.2.1 Stacking fault tetrahedra Reactions of type R1 have been observed for both screw and edge dislocations and all the defects with dislocation character. Interestingly, the strength effect of this reaction varies from minimum to maximum. For example, it is insignificant in the case of a 1/2(110) {111} edge dislocation interacting with an SFT58 and maximum for a screw dislocation inter­acting with a DL when the loop is fully absorbed into a helical turn on the dislocation.57 The mechanism for the way both the obstacle and dislocation remain unchanged is different for each case. An edge dislo­cation interacting with an SFT close to its tip creates a pair of ledges on its surface that are not stable and annihilate athermally.56,58 An example of this reaction is presented in Figure 10. If the dislocation slip plane is far enough from the SFT tip in the compressive region of the dislocation (for details of geometrical definitions see Bacon et a/.4), the ledges can be stabilized.56,58,59 This can be seen in Figure 11 (1). If the dislocation passes through the SFT several times in the same slip plane, it can detach the portion of the SFT above the slip plane, as shown in Figure 11 (2-4). Both the above mechan­isms are common for small SFTs, low T, fast disloca­tions, and the position of the SFT tip above the slip plane of an edge dislocation. If, however, the SFT tip is below the dislocation slip plane, and T is high enough and the dislocation speed low enough, reaction R3 can be activated. The stages of this reac­tion are presented in Figure 12. An example of effects of SFT orientation and temperature for the interaction of an edge dislocation with an SFT is presented in Figure 13. In this study, the dislocation slip plane intersected a 4.2 nm SFT through its geo­metrical center at the applied є = 5 x 106s-1 in a wide temperature range from 0 to 450 K.59 Reaction R1 was observed (see Figure 10) at all temperatures when the SFT was oriented with its tip up relative to the dislocation slip plane (orange triangles up in Figure 13) and at the two lowest temperatures when it was oriented in the opposite sense. At T = 300 K and orientation with tip down, a couple of ledges were formed on the SFT surface (see Figure 11 ). It may be noted that the R2 mechanism requires higher applied stress even though the tem­perature is increased. At higher T = 450 K, the inter­action mechanism is changed and the whole portion of the SFT above the slip plane is absorbed by the dislocation (Figure 12), that is, reaction R3 occurs, creating a pair of superjogs on the dislocation line. Some vacancies were also found to form to accom­modate the glissile configuration of the superjogs. This is discussed later in Section 1.12.4.3. More details on interactions between screw and edge dis­locations and SFT can be found elsewhere.60-63 In general, it can be concluded that the SFTs created under irradiation, that is, <4nm in size,6 are very stable objects and unlikely to be eliminated by a simple interaction with either edge or screw dislocations. Numerous attempts have been made to find a mechanism responsible for formation of clear, defect-free channels in irradiated fcc materials.64 One of the most used models considers absorption of an SFT by screw and mixed 60° dislocations.65 The absorption by conversion of an SFT into a heli­cal turn on a screw dislocation has been observed by in situ transmission electron microscope (TEM) deformation experiments,64,66-68 but only partial absorption has been found in MD modeling. As observed and discussed elsewhere,4,59 SFT separation into parts due to temporary absorption of part of an SFT as a helical turn and its expansion along a screw dislocation line can occur, but complete annihilation of an SFT has not been reproduced by atomic-scale modeling of bulk material. Possible reasons, including inability of MD to reproduce the whole set of exper­imental conditions such as stress state, scale, strain rate, and so on, were discussed in Matsukawa eta/.66-68 An alternative interpretation was suggested as a result of MD simulation of interaction between a screw dislocation and an SFT in a thin film,26 that is, the conditions realized experimentally for in situ TEM deformation. A thin film of fcc Cu was simulated and a 1/2(110) screw dislocation with an end on each surface was moved toward the SFT (size 12 and 18 nm) placed in the film center. A number of steps occurred that resulted in elimination of the vast portion of the SFT26: 1. The dislocation glided toward the SFT and par­tially absorbed it as a helical turn. 2. Edge segments of the turn glided toward the free surfaces and were annihilated there. 3. Glide of edge segments provided mass transport; in this particular case, transport of vacancies to the free surfaces. 4. The screw character of the dislocation was restored but in another slip plane. The dislocation then glided away leaving behind a small portion of the original SFT. Although this mechanism provides a mechanistic understanding of in situ TEM observations, it can operate only between surfaces or interfaces where the ends of the screw dislocation can cross-slip and, therefore, cannot be applied to bulk material and is unlikely to be responsible for the clear channel for­mation observed in bulk samples. We will return to this question later. 1.12.4.2.2 Dislocation loops DLs are common objects formed in metals under irra­diation. Depending on metal properties and irradiation conditions, loops ofdifferent types can be formed. They are mainly interstitial in nature although vacancy loops can appear in some metals under specific conditions. Here we consider only interstitial DLs. The shortest Burgers vector in bcc metals is bL = 1/2(111) and loops with this are the most common. In neutron and heavy-ion irradiated Fe, loops can have bL = (100),69 particularly at Tabove about 300 °C. Note that both have a perfect Burgers vector and are glissile. The most common loops in fcc metals have bL = 1/2(110) and are also perfect and glissile. In metals and alloys with a low stacking fault energy, Frank loops with bL = 1/3(111) can form. They are faulted and sessile. In the following section, we consider examples of reactions R1-R4 in fcc and bcc metals. Read More 22.08.2015   Comprehensive nuclear materials GB effects and void ordering As shown in the previous section, several striking observations of the damage accumulation observed in metals under cascade damage conditions can be rationalized in the framework of the PBM. This became possible because of the recognition of the importance of 1D diffusion of SIA clusters, which are continuously produced in cascades. The reaction kinetics in this case are a mixture of those for 1D and 3D migrating defects. Here, we emphasize that 1D transport is the origin of some phenomena, which are not observed in solids under FP irradiation. One such phenomenon is the enhanced swelling observed near GBs. It is well known that GBs may have significant effect on void swelling. For example, zones denuded of voids are commonly observed adja­cent to GBs in electron-, ion-, and neutron-irradiated materials.134-137 Experimental observations on the effect of grain size on void concentration and swelling in pure austenitic stainless steels irradiated with 1 MeV electrons were also reported.138,139 In these experiments both void concentration and swelling were found to decrease with decreasing grain size. Theoretical calculations are in good agreement with the grain-size dependence of void concentration and swelling measured experimentally in austenitic stain­less steel irradiated with 1 MeV electrons.139,140 However, there is a qualitative difference between grain-size dependences of void swelling for electron irradiation and that for higher recoil energies. In particular, in the latter case, in the region adjacent to the void-denuded zone, void swelling is found to be substantially enhanced.134,136,141-147 Furthermore, in neutron-irradiation experiments on high-purity aluminum, the swelling in the grain interior increases strongly with decreasing grain size.144 This is oppo­site to the observations under 1 MeV electron irradi — ation139 and to the predictions of a model based on the dislocation bias.1 0 An important feature of the enhanced swelling near GBs under cascade irradiation is its large length scale. The width of this enhanced-swelling zone is of the order of several micrometers, whereas the mean distance between voids is of the order of 100 nm. Thus, the length scale is more than an order of magnitude longer than the mean distance between voids. The MFP of 3D diffusing vacancies and single SIAs is given by L3D = = p2(ZdPd + 4nrcNc) —1/2 [140] and is of the order of the mean distance between defects. Hence, 3D diffusing defects cannot explain the length scale observed. In contrast, the MFP of 1D diffusing SIA clusters is given by L’D=yf = (pr2Pi+’"А’Г [141] and is of the order of several micrometers, hence, exactly as required for explanation of the GB effect (see Figure 6). A possible explanation for the obser­vations would be as follows. The SIA clusters pro­duced in the vicinity of a GB, in the region of the size ~ L1D, are absorbed by it, while 3D migrating vacancies give rise to swelling rates higher than that in the grain interior. The impact of the GB on the concentration of 1D diffusing SIA clusters can be understood by using local sink strength, that is, the sink strength that depends on the distance of a local area to the GB, /. It has been shown22 that the local sink strength in a grain of radius RClB is given by (/ (2rGB — 1 ))1/2 As can be seen from eqn [142], the sink strength has a minimum at the center of the grain, that is, at / = RgB, and increases to infinity near the GB, when 1! 0. The so-called grain-size effect, an increase of the swelling rate in the grain interior in grains of relatively small sizes (less than about 5 pm) with decreasing grain size, has the same origin as the GB effect discussed above. The swelling rate at the center of a grain may increase with decreasing grain size, when the grain size becomes comparable with the MFP of 1D diffusing SIA clusters and the zones of enhanced swelling of the opposite sides of GBs overlap. The swelling in the center of a grain as a function of grain size is presented in Figure 7 26 For comparison purposes, the values of the local void swelling (see Table 3 in Singh eta/.26) determined in the grain interiors by TEM are also shown. The PBM predicts a decrease of swelling with increasing grain size for grain radii bigger than 5 pm, which is in accordance with the experimental results. Note that the swelling values calculated by the FP3DM (broken curve in Figure 7) are magnified by a factor of 10. Another striking phenomenon observed in metals under cascade damage conditions is the formation of void lattices. It was first reported in 1971 by Evans148 in molybdenum under nitrogen ion irradi­ation, by Kulchinski et a/.149 in nickel under sele­nium ion bombardment, and by Wiffen150 in molybdenum, niobium, and tantalum under neutron irradiation. Since then it has been observed in bcc tungsten, fcc Al, hcp Mg, and some alloys.1 1-1 Jager and Trinkaus156 reviewed the characteristics of defect ordering and analyzed the theories pro­posed at that time, including those based on the elastic interaction between voids and phase instabil­ity theory. They concluded that in cubic metals, the void ordering is due to the 1D diffusion of SIA clusters along close-packed crystallographic direc­tions (first proposed by Foreman157). Two features of void ordering support this conclusion. First, the symmetry and crystallographic orientation of a void lattice are always the same as those of the host lattice. Second, the void lattices are formed under neutron and heavy-ion but not electron irradiation. This conclusion is also supported by theoretical analysis performed in Hahner and Frank15 and Barashev and Golubov.159 Read More Календарь Ноябрь 2024 Пн Вт Ср Чт Пт Сб Вс   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30   Рубрики 1 BIOFUELS A Look Back at the U. S. Department of Energy’s Aquatic Species Program: Biodiesel from Algae A. Worrall ABOUT THE AUTHOR ACCELERATOR DRIVEN SUBCRITICAL REACTORS ADVANCED BIOFUELS Advanced Biofuels and Bioproducts Advanced separation techniques for nuclear fuel reprocessing and radioactive waste treatment ADVANCES IN Advances in Biochemical Engineering/Biotechnology Advances in Biofuels Advances in Biorefineries Alcoholic Fuels Algae Energy All alternative energy Alternative transportation An Indispensable Truth An Introduction to Nuclear Materials AN INTRODUCTION. TO THE ENGINEERING. OF FAST NUCLEAR REACTORS ANNEX III. AHWR. Bhabha Atomic Research Centre, India ATOMIC ACCIDENTS BACKGROUND Bioenergy Bioenergy — Realizing the Potential Bioenergy from Wood BIOENERGY. RESEARCH:. ADVANCES AND. APPLICATIONS BIOETHANOL BIOFUEL'S ENGINEERING PROCESS TECHNOLOGY BIOFUELS BIOFUELS 1 Biofuels and Bioenergy Biofuels for Road Transport Biofuels from Agricultural Wastes and Byproducts BIOFUELS FROM ALGAE Biofuels Refining and Performance BIOGAS BIOGAS 1 BIOMASS — DETECTION, PRODUCTION AND USAGE Biomass and Biofuels from Microalgae Biomass Conversion Biomass Gasification and Pyrolysis BIOMASS NOW — CULTIVATION AND UTILIZATION BIOMASS NOW — SUSTAINABLE GROWTH AND USE Biomass Recalcitrance Biomasses identities Biotechnological Applications of Hemicellulosic Derived Sugars: State-of-the-Art Biotechnological Applications of Microalgae Biotechnology. for Fuels and Chemicals Cellulosic Energy Cropping Systems Comprehensive nuclear materials DESIGN FEATURES TO ACHIEVE. DEFENCE IN DEPTH IN SMALL AND. MEDIUM SIZED REACTORS Design of Reactor Containment Systems for Nuclear Power Plants ENERGY Energy Efficiency ENERGY SERVICES IN THE HINDU KUSH HIMALAYAN REGION Energy Storage Estimating Loss-of-Coolant Accident (LOCA) Frequencies Through the Elicitation Process EUROSUN 2008 EuroSun2004 EuroSun2008-10 EuroSun2008-11 EuroSun2008-13 EuroSun2008-2 EuroSun2008-3 EuroSun2008-4 EuroSun2008-5 EuroSun2008-7 EuroSun2008-9 EXAMPLES OF REACTIVITY-CONTROL SYSTEMS Exploitation des creurs REP Fast Breeder Reactor Programs: History and Status Fast Reactor Safety. (Nuclear science. and technology) Fertilization Frequency Response Testing. in Nuclear Reactors Fuels and Chemicals. from Biomass Gebaudetypologie Bayern Geothermal Energy GREEN BIORENEWABLE. BIOCOMPOSITES Handbook Nuclear Terms Handbook of biofuels production Handbook of Small Modular Nuclear Reactors Hydrogen — Fuel Cells Hydropower IAEA RADIATION TECHNOLOGY SERIES No Infrastructure and methodologies for the. justification of nuclear power programmes Innovative Biofibers from Renewable Resources Integral design concepts of advanced water cooled reactors Introduction to Nuclear Power Liquid Biofuels: Emergence, Development and LIQUID, GASEOUS AND SOLID BIOFUELS — CONVERSION TECHNIQUES Management von Biogas-Projekten Materials' ageing and degradation in. light water reactors Microbes and biochemistry of gas fermentation Modern Power Station Practice Natural circulation data and methods for advanced water cooled nuclear power plant designs Natural gas Neutron Scattering Applications and Techniques Nuclear and Radiochemistry Nuclear Back-end and Transmutation Technology for Waste Disposal NUCLEAR CHEMICAL ENGINEERING NUCLEAR ELECTRIC POWER Nuclear energy Nuclear Power and the Environment Nuclear power plant life management processes: Guidelines and practices for heavy water reactors NUCLEAR POWER PLANTS NUCLEAR POWER the PUBLIC Nuclear Reactor Design NUCLEAR REACTOR ENGINEERING Nuclear Reactors 1 NUCLEAR REACTORS 2 Others Particle Image Velocimetry (PIV) Passive Safety Systems and Natural Circulation in Water Cooled Nuclear Power Plants PHYSICS OF. HIGH-TEMPERATURE. REACTORS POWER Pretreatment Techniques for Biofuels and Biorefineries Principles of Fusion Energy PROCESS SYNTHESIS. FOR FUEL ETHANOL. PRODUCTION Production of Biofuels and Chemicals with Ionic Liquids Progress, Challenges, and Opportunities for. Converting U. S. and Russian Research Reactors Pumping Pyrolysis Radioactive waste management and contaminated site clean-up Recycling of Biomass Ashes Renewable Energy Products Rudiger Meiswinkel, Julian Meyer, Jurgen Schnell Second Generation Biofuels and Biomass Small modular reactors (SMRs) the case of Russia solar energy Solar Thermal and. Biomass Energy Sonar-Collecttors SonSolar Springer Series in Materials Science Study on Neutron Spectrum of Pulsed Neutron Reactor Sustainable Biotechnology Switchgrass Technologies for Converting Biomass to Useful Energy The Asian Biomass Handbook The Experimental Analyze Of The Solar Energy Collector The Future of Nuclear Power Thermal biomass. conversion and utilization —. Biomass information. system Thoria-based Nuclear Fuels Type of Membrane Contactor Suitable for CO2-O2 Exchanges in Culture Broth Vereinfachtes Verfahren WAS INTENTIONALLY LEFT BLANK Why We Need Nuclear Power wind power WORKSHOP ON NUCLEAR REACTION DATA AND. NUCLEAR REACTORS:. PHYSICS, DESIGN AND SAFETY Автономное энергообеспечение Альтерантивная энергия Альтернативные источники энергии и энергосбережение Без рубрики Биотопливо Ветровая энергия Ветрогенераторы, солнечные батареи и другие полезные конструк­ции Возобновляемые источники энергии ВОПРОСЫ ТЕОРИИ. И ИННОВАЦИОННЫХ РЕШЕНИЙ. ПРИ ИСПОЛЬЗОВАНИИ. ГЕЛИОЭНЕРГЕТИЧЕСКИХ СИСТЕМ ГЕЛИОЭНЕРГЕТИКА Другая энергия Зарядные станции Источники питания Источники энергии ИСУ Концентраторы солнечного излучения КРЕМНИЕВЫЕ СОЛНЕЧНЫЕ БАТАРЕИ Получение кремниевых пластин для солнечной энергетики Применение солнечной энергии Разное Солнечная электроэнергия Солнечная энергия Солнечная энергия — использование Солнечные батареи Солнечные коллекторы Солнечный ДОМ ТОНКОПЛЕНОЧНЫЕ. СОЛНЕЧНЫЕ ЭЛЕМЕНТЫ. НА ОСНОВЕ КРЕМНИЯ ФОТОЭЛЕКТРИЧЕСКОЕ. ПРЕОБРАЗОВАНИЕ. 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