This textbook was first conceived from our observation that there had been no suit­able textbook that caters towards the needs of undergraduate students in nuclear engineering for learning about nuclear reactor materials from materials science perspective. We have come across some books which are heavy on the ‘nitty gritty’ details that assume significant prior knowledge of basics of materials science, whereas others miss to highlight important materials science principles for stu­dents’ perspective to grow into the field of nuclear materials. Moreover, most of them are out-of-print. Thus, the book has primarily been written for undergraduate students; however it can also be used by beginning graduate students and profes­sionals who are interested in learning about nuclear materials from a rudimentary level but have very little background in materials science. We also hope that materi­als science and engineering (MSE) students will take delight into learning the topics if they wish to enter the nuclear materials field even though they may be quite conversed with the contents of some of the chapters involving materials sci­ence fundamentals.

Instructors who teach nuclear materials always have a difficult task of teaching the fundamentals of nuclear materials. Origin of the contents of the book has roots in the course on Nuclear Materials taught during more than three decades by the senior author (KLM) at NC State and recent teaching by the co-author (IC) at the University of Idaho. Copies taken of various chapters from different books have been used as course notes with various references that the students found not very convenient especially since only a few copies are available in the library. In addition, they were found to be disjointed especially in the varied nomenclatures used by different authors.

It has been our experience that many of the nuclear engineering students find that it is required to repeat basic materials science aspects including crystal struc­tures especially because hcp and fluorite structures are not well covered in the first course on Elements of Materials Science. Along with all basic phenomena of mate­rials science, the book deals with dislocation theory in more detail because of the significance of dislocations in the understanding of radiation damage and radiation effects. During the class teaching, we emphasize from the very beginning the sig­nificance of these fundamental materials science principles in our ability to be able

to predict/estimate effects of radiation using detailed quantitative microstructural features.

Following an introductory chapter on Nuclear Reactor Systems and Fundamen­tals, Chapter 2 starts with Crystal Structure followed by Crystal Imperfections (2.2) and Diffusion in Solids (2.3). Radiation Damage fundamentals are covered in Chapter 3 with the major aim of dpa (displacements per atom) calculation. Disloca­tion Theory comprises the Chapter 4 while Mechanical Properties, Fracture, Fatigue, Creep; some fundamentals of thermophysical properties; and Corrosion and SCC are all included in Chapter 5. Chapter 6 covers both Radiation Effects and Reactor Materials while the book is concluded with the Chapter 7 on Reactor Fuels. For first year graduate course, these sections were followed up with a term-long project on topics related to materials issues in reactor systems with in-class presen­tations by the students. Generally, we did not restrict on the students’ selection of topics; a list given in the semester-beginning as well as the students’ choice but on materials issues related to reactor applications.

We are not saying that this book is going to be a ‘panacea’ for curing all the prob­lems faced by these instructors. Also, we would like to accept that many issues of importance to nuclear field remained untouched because we did not wish the book to be so voluminous making it difficult to cover the subject matter in a single semester. Moreover, many of the aspects such as radiation creep, embrittlement of pressure vessel steels, etc., were dealt in a rather simplistic fashion albeit they have extensive pedagogic advantages. Our experience indicated that even the contents of chapters included here are quite large for a one-semester course and often the last chapters on reactor materials and fuels are covered rather briefly. For making the course manageable for one semester, we could not include/cover phase diagrams and phase transformations in this course and the text.

We would like to gratefully acknowledge several past and present colleagues and students whose work has been incorporated in this book in one way or the other. Special thanks go to Mr. Brian Marple for giving inputs on the Gen-III+ reactors for Chapter 1. We would also like to acknowledge the support of Drs. Louis Mansur, Donald Olander, and Sheikh T Mahmood for supporting our book proposal at the early stage. The authors are indebted to many authors and publishers who gave consent to the reproduction of appropriate figures etc. We would like to also acknowledge incredible efforts by Anja Tschortner of Wiley-VCH for driving us to complete the book and we commend her sustained efforts in this regard. Acknow­ledgements are due to our current students and colleagues who went through the proofs for corrections and many comments. Finally, the book would not have been written without the much needed emotional support of our families, in particular to our spouses, Ratnaveni Murty and Mohar B. Charit.

K. Linga Murty Indrajit Charit May 5, 2012

One-third of the hexagonal unit cell shown in Figure 2.2 can be considered as the primitive hexagonal unit cell since all crystal structures with a hexagonal symmetry can be created out of it, such that it leads to the creation of a HCP unit cell as well as the graphite crystal structure. A HCP unit cell (Figure 2.7) is quite different from the cubic ones that we discussed until now. Nonetheless, an ideal HCP crystal structure is as close-packed as the FCC one (packing factor of 0.74). This crystal structure is created by alternate layers of atoms on top of each other (see Section 2.1.3). This crystal structure has two characteristic lengths, a, in the base of the unit cell and the height of the unit cell along the c-axis (Figure 2.7). In fact, there are three separate coplanar axes with lattice vectors of a1, a2, and a3 at 120° apart even though they are all equal in length. The upper, middle, and lower

horizontal planes are called the basal planes. Planes (six vertical faces in a HCP unit cell) perpendicular to the basal planes are known as the prismatic planes. Planes other than the basal and prismatic planes are known as the pyramidal planes. This crystal structure has a coordination number of 12 (note the same as for FCC).

With a close examination of the HCP unit cell, the effective number of atoms can be easily calculated. Two atoms present on the upper and lower bases are shared by 2 HCP unit cells, whereas the corner atoms (12 of them) are shared by 6 unit cells, and 3 atoms reside entirely inside the unit cell. So, the count becomes (2 x 1/2) + (12 x 1/6) + (1 x 3) = 6.

The relation between the lattice constant a and the atom radius (r) is simple, a = 2r. However, in order to get a relation between the other lattice constant c and r, one needs to know little further. The HCP unit cell is inherently anisotropic. This implies that certain fundamental characteristics along the a — and c-axes are different. One measure of this anisotropy is given by the c/a ratio. An ideal HCP unit cell has a c/a ratio of 1.633 (can be shown from the geometry) based on a hard sphere model. But in reality, HCP metals are hardly ideal in dose packing, and thus the packing factors also change from metals to metals although slightly. Table 2.2 lists a number of HCP metals with different c/a ratios. The different c/a ratios are thought to be a result of the specific electronic structures of the atoms. Earlier it was thought that the c/a ratios influence the primary deformation mode, and that is why the corresponding dominant deforma­tion modes on the right-hand side column of Table 2.2 are also included. However, later this conjecture has been proved otherwise noting the glaring exception of Be (low c/a ratio, yet the primary deformation mode is basal). This issue will be further dis­cussed in Chapter 4 when we learn more about the slip deformation mode.

■ Example 2.2

Beryllium (Be) is used in LWRs as reflectors. Be has a HCP crystal structure at temperatures <1250 °C. Be has the lattice constants — a = 0.22858 nm and c = 0.35 842 nm (i. e., c/a ratio of 1.568) during normal conditions. The atomic radius of a Be atom is 0.1143 nm. What is the atomic packing effi­ciency of the unit cell? Discuss the significance of the computed result. Solution

Volume of atoms in the unit cell
Volume of the unit cell

Beryllium has a HCP crystal structure. The volume of a hexagonal prism (unit cell) is given by V HCP = (3/3a2/2)c, where a is the length of the prism base edge and c is the height of the hexagonal prism.

As a = 0.22858 nm and c = 0.35842 nm, VHCP = 0.04 865 nm3 (this is the input value in the denominator of the above expression for the PE).

Now we need to find out how much of the volume of the hexagonal prism is occupied by the atoms. The effective number of atoms in a HCP unit cell is 6. The atomic radius of each Be atom is 0.1143 nm. Therefore, the total volume of the atoms in the unit cell is 6 x (4/3)4(0.1143 nm)f = 0.03751 nm3.

Therefore, PE = 0 03751 nm = 0.77. Hence, the atoms must be spheroidal 0.04865 nm3

rather than spherical if they are assumed to be in contact.

The packing efficiency of a HCP unit cell is 0.74 only when the ideal c/a ratio is maintained. However, in reality, it is not and the packing efficiency of the HCP metals will always be little off from the ideal value. In the case of Be, the packing efficiency is slightly higher than the ideal.

2.1.3

Solute diffusion in a dilute alloy can be treated with a simple assumption that the environment the solute sees while diffusing almost entirely consists of host lattice atoms. The same is not true for a diffusion couple, say metal A and metal B brought together and held at elevated temperatures for longer time. Diffusion across the interface (A/B) will take place, and diffusion parameters such as the jump frequency and vacancy concentration will depend on the position and time. For explaining such a case, Darken defined a diffusivity term, chemical interdiffu­sion coefficient (D), to describe the diffusion that takes place in the diffusion cou­ple. It is given by the following relation:

D — xaDb + xb Da, (2.49)

where xa and xb are the atom fractions of A and B, respectively, at the point the interdiffusion coefficient is measured, and the intrinsic diffusion coefficients of A and B are DA and DB at the same point, and are not necessarily constant. More refinement of this model has been done by incorporating activity coefficients, known as the Darken-Manning relation. Readers are referred to Refs [3, 7, 9-11] for more information.

Intuitively, it is clear now that the diffusion rate of A into B is in general different from the diffusion rate of B into A. Kirkendall has conducted a famous experiment to elucidate the operation of vacancy diffusion in metals. A number of experimental and theoretical research studies have since then followed and expanded the under­standing of diffusion in a significant way. In this experiment, molybdenum wires were wound around an alpha-brass (70Cu-30Zn, wt%) block and then plated with a copper coating of appreciable thickness (Figure 2.50). The molybdenum wires act

as inert marker to locate the original interface. When the sample is kept in a fur­nace and sufficient diffusion is allowed, Kirkendall noticed that the wire markers present on the opposite sides ofbrass moved toward each another. This observation implied that more material has moved away from brass to copper than what entered from copper to brass. Now let us think about the atomic picture of the situ­ation. Direct interchange mechanism and ring mechanism of diffusion require that the net number of atoms crossing the interface is zero. But this is clearly not the case. The vacancy mechanism of diffusion is the only plausible explanation for this behavior. When zinc diffuses by a vacancy mechanism, there is a net flux of zinc atoms going in the opposite direction. That is, an equal number of vacancies are entering the brass block. However, this enhanced concentration of vacancies is thermodynamically unstable. There are many vacancy sinks (such as grain bounda­ries and dislocations) in the material, so the vacancy concentration does not go above the equilibrium vacancy concentration. Thus, it means that zinc leaves brass and the excess vacancies in brass get annihilated at the preexisting sinks in brass. So the natural result of the event is that the volume ofbrass decreases and the wire markers move closer together. It is true that similar event is also happening in cop­per plating, but because the diffusion rate of zinc is higher than that of copper, the net effect of diffusion of the latter does not show up.

2.3.6

We know about solid solutions (substitutional and interstitial types) and the requirements for forming one or the other. Now let us get into little more details about solid solution strengthening. This mode of strengthening is present in both single and polycrystalline materials. Here, dislocations encounter barriers on their path from the solutes present in the host lattice.

Solute atoms can interact with dislocations through the following mechanisms based upon various situations: (a) elastic interaction, (b) modulus interaction,

(c) long-range order interaction, (d) stacking fault interaction, (e) electrical interac­tion, and (f) short-range order interaction. The first three interactions are long — range barriers, and they are relatively insensitive to temperature and continue to act upto 0.6Tm. However, the last three are short-range barriers that contribute strongly to the flow stress only at low temperatures.

The materials characteristics that are not governed directly by mechanical forces and chemical environment are known as physical properties. There are many phys­ical properties, such as electrical conductivity, magnetic susceptibility, specific heat, thermal conductivity, thermal expansion coefficient, and so forth. These properties are usually structure-insensitive and intrinsic properties of materials. There are, however, a number of exceptions. Thermophysical properties (i. e., physical propert­ies affected by heat) are of great interest for nuclear reactors. We focus here exclu­sively on the thermophysical properties of solid materials, even though these properties pertaining to liquids and gases that may be used as coolants and moder­ators in nuclear reactors are also important. Here we attempt to understand the basic thermophysical properties along with some specific examples.

5.2.1

A good point to start the discussion on various examples of nuclear reactors is to understand the evolution of nuclear power over the past six to seven

decades. As shown in Figure 1.7, there are different generations of reactors. Thus, a chronological approach has been taken to start an overview of various nuclear power reactors. However, it is, by no means, exhaustive account. The first nuclear reactor known as Chicago Pile 1 (CP-1) was built during the hey­days of the World War II (criticality was achieved on December 2, 1942) at the University of Chicago, Chicago, IL. It was designed and built by a team led by Enrico Fermi. This reactor was a thermal reactor with graphite moderators and natural uranium dioxide fuel. No coolant or shielding was used. The reactor could produce only ~200W of heat. However, the primary aim of the reactor was to demonstrate the occurrence of the fission chain reaction. It was dis­mantled in February 1943, and CP-2 reactor was installed at the Argonne National Laboratory based on the experience gained with CP-1.

Oxides and sulfides (such as ZnO, ZnS, and BeO) containing smaller cations tend to have tetrahedral coordination (i. e., the coordination number of both the cation and the anion is 4). The same structure is also shown by the covalent compounds, such as SiC, BN, and GaAs. This structure class is named after the mineral zinc — blende in which the anions (S2~) form the FCC sublattice, and only half of the available tetrahedral sites are filled with the cations (Zn2+), as depicted in Figure 2.23. Note the similarity between the zincblende and diamond structures (refer back to Figure 2.19).

We have noted in the preceding chapters that the dislocation movement under applied stress creates plastic strain (permanent deformation) in crystals. This is dif­ferent from the elastic deformation under applied stress where the atomic bonds are stretched without the help of a dislocation. It is not simple to relate plastic strain to applied stress as it depends on temperature, strain rate, and micro­structural factors. However, Orowan developed a simple expression, known as Oro — wan’s equation, relating the macroscopic strain rate to the microscopic parameters such as dislocation density, velocity, and Burgers vector. This derivation is based on the recognition of the fact that when a single dislocation moves, it creates a dis­placement of the magnitude equal to the Burgers vector (b).

Consider a crystal of volume HLD containing a number of straight edge disloca­tions (for the sake of simplicity, dislocation type is assumed to be edge), as shown in Figure 4.5. Now let us consider an applied shear stress causing a dislocation move by di. We know that if the dislocation moves the distance D along the Burgers vector, it produces b displacement. Therefore, it can be said that the contribution of the dislocation moving by di distance to the displacement is (b/D) x di. The dis­placement generated by a single dislocation is pretty small. To produce plastic strain of engineering significance, there are a large number of dislocations which

when move produce the total displacement A given by

b N

A dj.

D j=i

where N is the total number of dislocations that have moved in the crystal volume. The macroscopic plastic shear strain (y) is given by

(4.4)

Equation (4.4) can be further simplified by defining the average distance x trav­eled by a dislocation as

x 1 N

X = N -1 d>’

Hence, by replacing Eq. (4.5) into Eq. (4.4), we obtain

The expression NL/HLD gives the total dislocation line length (NL) per unit vol­ume (HLD), that is, mobile dislocation density. Let us denote the mobile dislocation density by gm. Note that the term gm is not the total dislocation density as the immobile dislocation does not contribute to the plastic strain. Therefore, Eq. (4.6) can be rewritten as

C = Qmbx.

Equation (4.7) can be further expressed in terms of shear strain rate by taking differential of both sides of the equation with respect to time t.

. dc, dx

C = d = rmb d = PmbVd’

where Vd is the average velocity of dislocations. The equation is universal in nature and also applicable for climb of edge dislocations. Furthermore, it can be universally applied to screw dislocations and mixed dislocations. Equation (4.8) is a nice exam­ple of relating macroscopic behavior of a material described by a set of microscopic parameters. The Burgers vector (b) is determined by crystal structure of the material. The other two quantities gm and Vd are the parameters that depend on several other factors such as stress, temperature, prior processing history, and so on.

4.1.4

Tension (or tensile) test is a popular method of studying the short-term mechanical behavior (strength and ductility) of a material under quasi-static uniaxial tensile state of stress. This test provides important design data for load-bearing components. Tension test involves stretching of an appropriately designed tensile specimen under monotonic uniaxial tensile loading condition. Generally, a constant displace­ment rate is used during the test. The load and sample elongation are measured simultaneously using load cells and strain gage/LVDT (linear variable differential transformer), respectively. Figure 5.2 shows a schematic of a basic tensile tester.

The test generally uses a standard specimen size shown in Figure 5.3. But for irradiated materials (considering the induced radioactivity among other factors), the use of the subsize tensile specimens is quite common. One important thing to observe is that the transition region between the gauge length and the shoulders is smooth so that stress concentration effect does not set in creating flaws. It is gener­ally recognized that in order to compare elongation measurements with a good approximation in different sized specimens, the specimens must be geometrically similar. In this regard, Barba’s law (1880), which states that І0Д/A0 (L0 is the gauge length and A0 is the cross-sectional area) needs to be a constant, is useful. In the United States, І0Д/A0 is taken as ~4.5 for round specimens. The relevant ASTM standard for tension testing is E8 (standard test method for tension testing of metallic materials).

Following the primary damage induced during a few picoseconds, the irradiated material goes through several stages of evolution, as described in Chapter 3, over a

An Introduction to Nuclear Materials: Fundamentals and Applicatioins, First Edition.

K. Linga Murty and Indrajit Charit.

© 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.

Figure 6.1 A flowchart showing various causes of radiation damage and their consequent effects. Courtesy: US Department ofEnergy.

long period of time. We need to now discuss how the higher order defects are formed from the primary damage defects in the irradiated materials. For this, let us revisit the specific characteristics of two primary damage defects: vacancies and self-interstitial atoms summarized in Table 6.1.

volume assume lattice as an elastic contin­uum. The relaxation volume for SIA is consid­ered positive (because ofthe volume increase), while the relaxation volume for vacancies is considered negative for monovacancies (because of the volume shrinkage).

The high relaxation volume due to SIAs causes large lattice distortions, which create a strong interaction with other SIAs and other lattice defects (dislocations, impurity atoms, etc.). As we noted, the classical picture of self-interstitial like the interstitial impurity atoms is untenable energetically. Thus, the single SIAs become stable only in a dumbbell or split interstitial configuration around a single lattice site, as shown in Figure 6.2. The dumbbell axis is generally found to be along (100) in FCC metals, (110) direction in BCC metals, and (0001) in HCP metals. Multiple interstitials (interstitial clusters) are created by the aggregation of mobile SIAs at higher temperatures. Multiple interstitials have a high binding energy (~1 eV). Figure 6.3(a) and (b) show two di-interstitial configurations in FCC and BCC lat­tices, respectively.

Impurity atoms can act as effective traps for SIAs. Stable interstitial-impurity complexes having undersized impurity atoms (with respect to the host lattice atom) do not dissociate thermally under a certain temperature range where vacan­cies become mobile. Binding energy of the interstitial-impurity complexes vary typically between 0.5 and 1.0 eV. However, weaker trapping is generally observed with oversized impurity atoms. Figure 6.4 shows an interstitial-undersized atom complex. It has got a mixed dumbbell configuration that is stable. This configura­tion however can reorient itself through jumping of the undersize impurity atom across the vertices of the octahedron (forming a cage-like structure) shown in Figure 6.4. The activation energy associated with this type of motion, known as cage motion, is quite small, on the order of 0.01 eV.

Smaller binding energies (~0.1eV) are associated with multiple vacancies com­pared to multiple interstitials, often observed in irradiated metals. Various configu­rations of vacancies are shown in Figure 6.5. The migration energy of divacancies is

less than that for two monovacancies (0.9 versus 1.32 eV for Ni). We have com­mented on the energetics of divacancy formation in Section 2.2. Tetravacancies can only migrate by dissociation. Nonetheless, it can act as the first stable nucleus for further clustering.

Vacancies can bind with oversize solute/impurity atoms in order to lower the overall free energy of the solid. Estimates of the binding energy of a vacancy to an oversize solute in an FCC lattice range from ~0.2-1.0eV. These solutes can act as efficient traps for vacancies in the lattice.

6.1.1