All real crystals contain some dislocations except in the case of tiny, carefully pre­pared whiskers. Nevertheless, dislocations are not produced like intrinsic point defects such as vacancies or self-interstitials. The net free energy change due to the presence of dislocations is positive. Dislocations in freshly grown crystals are formed for various reasons: (a) Dislocations that may appear as preexisting in the seed crystal to grow the crystal. (b) “Accidental nucleation” — (i) heterogeneous nucleation of dislocations due to internal stresses generated by impurity, thermal contraction, and so on, (ii) impingement of different parts of the growing interface, and (iii) formation and subsequent movement of dislocation loops formed by the collapse of vacancy platelets. When a dislocation is created in a region of the crystal that is free from any defects, the nucleation is called “homogeneous.” This occurs only under extreme conditions and requires rupturing of atomic bonds, which need very high stresses. Nucleation of dislocations at stress concentrators is important — generation of prismatic dislocations at precipitates/inclusions (Figure 4.22a), misfit dislocations during coherency loss, dislocation generation at other surface irregularities and cracks. As a matter of fact, grain boundary irregu­larities such as grain boundary ledges/steps (Figure 4.22b) are considered to be important sources of dislocations, especially in the early stages of deformation.

Regenerative multiplication of dislocations is key to sustaining large plastic strains. Frank-Read (F-R)-type sources and multiple cross-glide can participate as a multiplication mechanism of dislocations. It requires a preexisting dislocation (DD0) pinned down at two ends (dislocation intersections or nodes, composite jogs, precipitates, etc.) with distance between pinning points being L, as illustrated in Figure 4.23. An applied resolved shear stress (t) makes the dislocation bow out and the radius of curvature R depends on the stress according to Eq. (4.17). When R

reaches a minimum value of L/2 and taking a = 0.5, the stress required to produce the following configuration is given by

Gb

t = = rFR — (4-19)

At this point in semicircular configuration, the segment becomes unstable and further increase in the dislocation loop does not require any more additional stress. As the loop expands, the dislocation at places such as m and n in Figure 4.23a(iv) attracts and annihilates leaving the loop expanding further while the original dislo­cation segment L is recovered. Under the applied stress (tFR), the dislocation seg­ment expands again resulting in another loop, while the first loop keeps on expanding due to the applied stress. The process repeats itself until the lead disloca­tion loop gets stuck at some obstacle such as a grain boundary or another similar dislocation loop on a parallel glide plane. Since the dislocation loops generated are all alike, the second loop will be repelled by the first one, while the third loop will experience repulsive force arising from both the first and second loops, and so on, and finally the force due to all the dislocation loops in this “pileup” results in the original dislocation segment unable to produce any further dislocation loops until the lead dislocation climbs out of the glide plane. This is the Frank-Read source that is responsible for dislocation multiplication, and the stress (tFR) required for F-R source to operate is given by Eq. (4.19).

Total dislocation length can also be increased by climb: (a) the expansion of pris­matic loops and (b) spiraling of a dislocation with a predominantly screw character. A regenerative multiplication known as the “Bardeen-Herring” source (Figure 4.24) can occur by climb in a way similar to the Frank-Read mechanism.

At temperatures above 0.4-0.5Tm (Tm is the melting point of the material), plastic deformation occurs as a function of time at constant load or stress. The phenome­non is known as creep “defined as time dependent plastic strain at constant temper­ature and stress.” Note that lead (Pb) may creep at room temperature, but iron (Fe) does not because for lead room temperature represents a higher homologous tem­perature (i. e., T/Tm = ~0.5) than that of iron (~0.16). Because creep occurs as a function of temperature, it is a thermally activated process. Creep must be taken into account when a load-bearing structure is exposed to elevated temperatures for a long duration of time.

A creep test is generally done under uniaxial tensile stress. There are different variants of creep tests, such as compressive creep, double shear creep, impression creep, and indentation creep. However, here we will deal with tensile creep only. A creep curve is basically plotted as a function of creep strain as a function of time at a constant load (or constant stress depending on the availability of such instrument set up to keep stress constant during creep deformation) and temperature. Figure 5.26a shows a creep equipment with a furnace and strain measuring device, a linear variable differential transformer (LVDT). A typical creep curve as shown in Figure 5.26b has three stages: (i) primary stage (transient creep) — work hardening during plastic deformation is more than recovery (softening) exhibiting decreasing strain rate with time; (ii) secondary or steady-state stage (minimum creep rate) — the rate of work hardening and softening balance each other, and (iii) tertiary stage — characterized by an accelerating creep rate where softening mechanisms predomi­nate. The third stage of tertiary creep is often considered as fracture stage rather

(b) (c)

than deformation. A real example of creep rate versus time plot for a Grade 91 steel is shown in Figure 5.26c where the steady-state creep rate occurs as a minimum rate that is typical for creep under constant load. Increase in temperature and/or stress tends to enhance creep strains and rates, as illustrated in Figure 5.27 where we note that increased creep rates are accompanied by decreased time to rupture (tr).

Chadwick discovered neutron in 1932. Generally, neutrons are generated during radioactive chain reactions in a power reactor. Neutron is subatomic particle pres­ent in almost all nuclides (except normal hydrogen isotope or protium) with a mass of 1.67 x 10-27 kg and has no electrical charge.

Neutrons are classified based on their kinetic energies. Although there is no clear boundary between the categories, the following limits can be used as a useful guideline:

Cold neutrons (<0.003 eV), slow (thermal) neutrons (0.003-0.4 eV), slow (epithermal) neutrons (0.4-100 eV), intermediate neutrons (100eV-200keV), fast neutrons (200keV-10 MeV), high-energy (relativistic) neutrons (>10 MeV). Note: 1 eV = 1.6 x 10~19J.

Generally, thermal neutrons are associated with a kinetic energy of 0.025 eV that translates into a neutron speed of 2200 m s-1!

1.4

The method devised by the British mineralogist, William H. Miller, in 1839 is still used to denote crystallographic planes and directions. That is why the method named after him uses the Miller indices labeling technique. Besides closed-packed planes, there could be a number of crystallographic planes of interest. Their orien­tation, arrangement, and atom density could be different. It may be cumbersome to name a crystallographic plane as “cube face plane,” “octahedral plane,” and so on. The same may be applicable to the issue of defining crystallographic directions. One would soon run out of names just to refer to the directions as “cube edge,” “face diagonal,” “body diagonal,” and so on. This type of nomenclature also lacks adequate specificity to be seriously considered. That is why a more convenient and systematic technique, such as Miller indices, is used to denote crystallographic planes and directions. As a matter of fact, each point on a crystal can be reached by the translation vector composed of the sum of the multiples of the crystal lattice vectors.

For labeling a plane in a crystal with Miller indices, the following general proce­dure needs to be followed:

1) Select an origin at a lattice point that is not on the crystallographic plane to be indexed.

2) Fix the three orthogonal axes (a, b, and c or x, y, and z) from the selected origin.

3) Find the intercepts (in multiples of the unit lattice vector) that the plane makes on the three coordinate axes.

4) Take reciprocals of these multiples.

5) Convert the fractions (if any) to a set of integers and reduce the integers by dividing by a common integer factor. However, care should be exercised so that the atom configuration of the original plane remains the same.

6) Enclose the final numbers in parentheses such as (hkl), which is the Miller index of the particular plane. A family of equivalent planes is given by numbers enclosed in curly brackets {hkl}. For example, the faces of a cube are given by {100}, whereas an individual plane is denoted by (100), (010), and so on. A nega­tive intercept is denoted by a “bar” on the index such as (100).

Figure 2.10a-d shows four planes in a cubic unit cell. In Figure 2.10a, the hatched plane makes an intercept on the x-axis for a one lattice vector (take it as a unity 1 in place of lattice constant a), however it does not intersect the y — and z-axes making the intercept of 1. Therefore, the reciprocals of the intercepts become 1/1, 1/i, and 1/i. Hence, the Miller index of the plane is (100). Similarly, for the plane shown in Figure 2.10b, the intercepts are 1, 1, and 1 on the x-, y-, and z-axes respectively, and therefore, the reciprocals to the intercepts become 1, 1, and 0. Thus, the Miller index of the plane becomes (110). In Figure 2.10c, the plane with hatch marks portends positive intercepts of 1 each on all the three axes. When the reciprocals of the intercepts are taken, they remain the same, and thus the Miller index of the plane becomes (111). In Figure 2.10d, the plane denoted by the hatch

marks cuts the x-, y-, and z-axes creating positive intercepts of 1, 1, and 1/2, respec­tively. We take the reciprocals of these intercepts to obtain 1, 1, and 2. Thus, the Miller index of the plane would be (112). Even though we have not given a direct example where one needs to reduce the Miller index to lower integers, the issue merits some discussion. This is related to the step 5 in the above-mentioned proce­dure. The operation needs to be carried out depending on the specific case. For example, (220) and (110) are equivalent planes in an FCC crystal, so be treated so. On the other hand, even though (200) and (100) planes in FCC and BCC crystals are equivalent, that is not the case in a simple cubic crystal. Hence, in these types of cases, individual attention needs to be paid to the atom configurations of the planes in question to ascertain whether further reduction is going to be permitted or not.

Another important issue for denoting crystallographic planes is the situation where negative indices become essential. This can be easily illustrated referring to Figure 2.10a. Let us say one wishes to index the leftmost cube face in Figure 2.10a. As the plane passes through the previous origin, it needs to be shifted to some other position, say shifted to the right side by one lattice spacing. In this case, the intercepts the plane is making to x-, y-, and z-axes can be interpreted as 1, —1, and 1, respectively. So by taking reciprocals, we get 0, —1, and 0. This means the Miller

index of the plane can be represented as (010) where 1 is written in place of —1, and called “bar 1.”

So we understand pretty much how to label a plane using Miller indices. We now turn our attention to labeling a crystallographic direction with a Miller index. Let us label the “direction A” in the cubic unit cell on the right-hand side in Figure 2.11. For denoting directions, the direction must pass through the origin (O). In a situa­tion, where the original direction does not go through the origin, one needs to a draw a parallel line that passes through the origin. This parallel line will have the same Miller index as the original one. For direction A, the components along the x-, y-, and z-axes need to be resolved. In this case, direction A can be resolved half the lattice constant along X-axis, a unit lattice vector distance along Y-axis, and no com­ponent (i. e., 0) along the Z-axis. Now these components need to be converted to the smallest whole integers and then put into the square brackets to obtain the Miller index of the direction. Hence, the Miller index of direction A would be [1/2 1 0], that is, [120]. General notation used for a crystallographic direction is [uvw]. The Miller index of “direction B” on the left-hand side unit cell can be found out in the same way. Note that the component of “direction B” can be resolved into a positive unit distance along the X-axis (i. e., 1), a lattice vector in the negative Y-axis (i. e., —1), and no component (i. e., 0) along the Z-axis. Thus, the Miller index of “direction B” would be [110]. Note that the face diagonal of the base face of the cube on the right — hand side unit cell is parallel to “direction B” and will have the same Miller index [110]. A class of equivalent directions is called a “family of directions” and denoted by {uvw). In the case of face diagonal of the cubic unit cell, all face diagonals belong to a single family of directions, {110), meaning directions [110], [101], [011], [110], and so on.

An alternative procedure is quite commonly used where we first determine the coordinates of two points on the direction; the points are specified with respect to a defined set of orthogonal coordinate system (x, y z or a, b, c), as shown in Figure 2.12a. Next, subtract the coordinates of the “tail” from the “head” and clear the fractions and/or reduce to lowest integers. Enclose the numbers in brack­ets [ ] with the negative sign by “bar” above the number. Three examples are

illustrated in Figure 2.12b. The Miller indices of the directions A, B, and C in Figure 2.12b are thus [100], [111], and [122], respectively.

We note that the definitions of the Miller indices for planes and directions in cubic structures are such that the direction with Miller index [hkl] is normal to the plane (hkl).

“Nothing in Life is to be Feared. It is Only to be Understood.”

—Marie Curie

Interactions of high-energy radiation such as a-, b-, and, y-rays as well as subatomic particles such as electrons, protons, and neutrons with crystal lattices give rise to defects/imperfections such as vacancies, self-interstitials, ionization, electron exci­tation, and so on. Fission fragments and neutrons cause the bulk of the radiation damage. Other types of radiation either do not have enough energy or are not pro­duced in sufficient number density to cause any major radiation damage. In a nuclear reactor scenario, the microscopic defects produced in materials due to irradiation are referred to as radiation damage. These defects result in changes in physical, mechanical, and chemical properties, and these macroscopic material property changes in aggregate are referred to as radiation effects. Before discussing the effects of radiation on various properties, we need to know how to describe the radiation damage in a quantitative fashion. The timescales in which the damage and effects take place are quite different. While radiation damage events take place within a short time period of around 10~n s or less, the radiation effects occur in a relatively large timescale ranging from milliseconds to months. Radiation effects range from the migration of defects to sinks that takes place in milliseconds to changes in physical dimensions due to swelling and so on with much longer dura­tion. Quantitative characterization of radiation damage is covered in this chapter, while radiation effects are discussed in Chapter 6 after descriptions of various prop­erties of materials. We mainly consider neutron irradiation here and the reader is referred to other monographs for damage calculations when charged particles (such as heavy ions and protons) and photon (such as y-ray) irradiations are involved.

The binding energy of lattice atoms is very small (~10-60 eV) compared to the energy of the impinging particles so that a scattering event between them results in the lattice atom getting knocked off from its position. The atom will generally have such a high energy that it can interact with another lattice atom that will also get knocked off from the lattice position. The atom that was knocked off by the incoming high-energy particle is known as “primary knock-on atom” or PKA, which in turn knocks off a large number of atoms before it comes to rest in an interstitial position, thereby creating a Frenkel pair in which case the atom is

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

K. Linga Murty and Indrajit Charit.

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

considered to have been “displaced.” However, if the atom is in proximity to a vacancy, it would occupy the vacant lattice position in which case it becomes a “replacement collision.” Thus, in general, a PKA can lead to a large number of higher order knock-on atoms (also known as recoil atoms and/or secondary knock — on atoms) resulting in many vacant lattice sites and this conglomeration of point defects is known as “displacement cascade.” If during these collision processes many nuclei go into higher energy states at their lattice position, thermal spike is created. Before the particle-lattice atom interactions, the incoming particle may interact with electrons leading to ionization. As will be seen later, there is an elec­tron energy cutoff above which no additional atomic displacements take place until the particle energy becomes lower than this cutoff value, as envisioned in the Kin­chin-Pease (K-P) model.

Brinkman [1] first came up with the displacement spike model, as shown in Figure 3.1a. In this model, PKA motion creates a core consisting of several vacan­cies surrounded by a periphery rich in interstitials. Later, Seeger [2] further refined the concept and showed that vacant lattice sites in proximity lead to zones devoid of atoms commonly referred to as depleted zones (Figure 3.1b). However, if these regions are large enough, they form voids that lead to decreased density or volume increase known as “swelling.” In cases where elements such as B, Ni, and Fe are present, (n, a)[1]* reactions will lead to the production of He that will stabilize these voids, in which case they are referred to as “cavities”. These cavities once formed are stable and cannot be removed by thermal annealing.

The following are the radiation defects induced by intense nuclear radiation, in particular high-energy {E > 0.1 MeV} neutrons:

• Vacancies.

• Interstitials.

• Impurity atoms — produced by transmutation.

• Thermal spikes — regions with atoms in high-energy states.

• Displacement spikes — regions with displaced atoms, vacancies, self-interstitials (Frenkel pairs) produced by primary and secondary knock-on atoms.

• Depleted zones — regions with vacancy clusters (depleted of atoms).

• Voids — large regions devoid of atoms.

• Bubbles — voids stabilized by filled gases such as He produced from (n, a) reactions with B, Ni, Fe, and so on.

• Replacement collisions — scattered (self) interstitial atoms falling into vacant sites after collisions between moving interstitial and stationary atoms and dissi­pating their energies through lattice vibrations.

A flux of neutrons then results in a large number of PKAs, which in turn produce higher order knock-on atoms, as illustrated in Figure 3.1. Our goal is to first calculate the number of PKAs produced due to a flux of neutrons with a range of energies comprising the neutron spectrum. Next step is to find the number of knocked-on atoms due to these PKAs with varied energies. Integration through the

whole neutron spectrum then yields the number of atoms displaced. The displace­ments per atom or dpa will give us a measure of quantitative radiation damage that can be later related to changes in the macroscopic properties of materials due to the given neutron spectrum. Earlier, the total fluence or dose (flux x time) in units of n cm-2 was commonly used, but this does not take into account the different spectral variations, and so dpa is a far better unit. It is commonly observed that the properties of irradiated materials depend on the specific neutron spectrum to which they are exposed and thus will be different for the same total neutron

dose. Figure 3.2 clearly demonstrates this behavior. The example here shows the correlation for mechanical properties, while other such relations can be found for other properties such as physical, thermal, electrical, and so forth, which are gener­ally referred to as radiation effects. It is important to note that many atomic dis­placements occur due to neutron-lattice interactions, but only a small fraction (~1%) survive since most of the defects anneal out in situ during irradiation mainly due to the proximity of the defects to the appropriate sinks and mutual recombination of vacancies and interstitials.

3.1

Electrical interaction arises from the fact that solute atoms always have some charge on it due to the dissimilar valences and the charge remains localized around the solute atoms. The solute atoms can then interact with dislocations with electri­cal dipoles. This interaction contributes negligibly compared to the elastic and modulus interaction effects. It becomes significant only when there is large differ­ence in valence between the solute and matrix and the elastic misfit is small.

There were different theories proposed and debated over several decades. Some have found that the strength increment varies as the square root of the solute atom fraction (c1/2) or c1/3, or even c. One of the examples of solid solution strengthening is the martensite strengthening. Martensite phase in steels has a body-centered tetragonal (BCT) lattice structure and is considered as a supersaturated solid solu­tion in the host lattice of iron. Carbon atoms present in the lattice strain the crystal creating nonspherical (tetragonal) distortion, which affects the dislocation move­ment severely. This is one of the reasons why the martensite phases are so hard and brittle. Figure 4.34 shows the variation in the yield strength of iron as a func­tion of different solute concentrations. Note that the solid solution strengthening imparted by the interstitial elements like carbon and nitrogen is much higher than that imparted by substitutional solutes.

4.4.4

Corrosion is a form of surface degradation of metallic materials via electrochemical means. Corrosion properties are not always an intrinsic property of the material since it is influenced by the chemical environment in which the material or the materials system exists. Corrosion is regarded as a life-limiting issue for nuclear reactor components that remain in contact with some kind of fluid for a long dura­tion of time. For example, in LWRs, the core materials remain submerged under the coolant (pressurized water in PWRs or water plus steam environment in BWRs). In modern reactors, coolant chemistry is carefully controlled in order to cause minimum disruption coming from corrosion. However, corrosion is a natu­ral process in a harsh environment such as a nuclear reactor. The cost of corrosion runs into several billions of dollars in the United States in the form of both direct and indirect costs. Note that although until 1960s corrosion was exclusively consid­ered for surface degradation of metallic materials, the corrosion is now generically applied to describe environmental attacks on a wide variety of materials, including ceramics, polymers, composites, and semiconductors.

5.3.1

As the name implies, LWRs use light water as the coolant and the moderator, and in many cases as the reflector material. These are typical thermal reactors as they utilize thermalized neutrons to cause nuclear fission reaction of the U235 atoms. The thermal efficiency of these reactors hover around 30%. Two main types of LWR are PWR and BWR. These two types are created mainly because of the differ­ence in approaches of the steam generating process (good quality steam should not contain more than 0.2% of condensed water). LWRs have routinely been designed with 1000 MWe capacity.

Pressurized Water Reactor PWRs were designed and implemented commercially much sooner than the BWRs due to the earlier notion that the pressurized liquid water would somehow be much safer to handle than the steam in the reactor core and would add to the stability of the core during the operation. That is why the first commercial reactor in Shippingport was a PWR. PWRs are designed and installed by companies such as Westinghouse and Areva.

A schematic design of a typical PWR plant is shown in Figure 1.9a. A PWR plant consists of two separate light water (coolant) loops, primary and secondary. The PWR core is located inside a reactor pressure vessel (RPV) made of a low-alloy fer­ritic steel (SA533 Gr. B) shell (typical dimensions: outside diameter ~5 m, height ~12 m, and wall thickness 30 cm), which is internally lined by a reactor cladding of 308-type stainless steel or Inconel 617 to provide adequate corrosion resistance against coolant in contact with the RPV. The PWR primary loop works at an average pressure of 15-16 MPa with the help of a set of pressurizers so that the water does not boil even at temperatures of 320-350 °C. The PWR core contains an array of fuel elements with stacks of a slightly enriched (2.5-4%) UO2 fuel pellets clad in Zircaloy-4 alloy (new alloy is Zirlo or M5). Individual cladding tubes are generally about ~10mm in outer diameter and ~0.7 mm in thickness. The fuel cladding tube stacked with fuel pellets inside and sealed from outside is called a fuel rod or fuel pin. About 200 of such fuel rods are bundled together to form a fuel element. Then, about 180 of such fuel elements are grouped together to form an array to create the reactor core of various shapes — square, cylindrical, hexagonal, and so on (Figure 1.9b). The reactor core is mounted on a core-support structure inside the RPV. Depending on the specific design, the above-mentioned dimensions of the various reactors may vary.

The control rod used is typically an Ag-In-Cd alloy or a B4C compound, which is used for rapid control (start-up or shutdown). Boric acid is also added to the primary loop water to control both the water chemistry acting as “poi­son” and the long-term reactivity changes. This primary loop water is trans­ported to the steam generator where the heat is transferred to the secondary loop system forming steam. The steam generator is basically a heat exchanger containing thousands of tubes made from a nickel-bearing alloy (Incoloy 800) or nickel-based superalloy (e. g., Inconel 600) supported by carbon steel plates (SA515 Gr.60).

Boiling Water Reactor BWR design embodies a direct cycle system of cooling, that is, only one water loop, and hence no steam generator (Figure 1.10a). Early boiling water experiments (BORAX I, II, III, etc.) and development of experi­mental boiling water reactor (EBWR) at the Argonne National Laboratory were the basis of the future commercial BWR power plants. Dresden Power Station (200 MWe), located at the south of Chicago, IL, was a BWR power plant that started operating in 1960. It is of note that this was a Generation-I BWR reactor. However, most BWRs operating today are of Generation-II type and most significant features are discussed below. The reactors (Fukushima Daii — chi) that underwent core melting following the unprecedented earthquake and tsunami in Japan during 2011 were all of BWR type.

The BWR reactor core is located near the bottom end of the reactor pressure ves­sel. Details of various components in a typical BWR are shown in Figure 1.10b. The BWR RPV is more or less similar to the PWR one. The BWR core is made of fuel assembly consisting of slightly enriched UO2 fuels clad with recrystallized Zircaloy-

BWR/6

REACTOR ASSEMBLY

1. VENT AND HEAD SPRAY

2. STEAM DRYER LIFTING LUG

3. STEAM DRYER ASSEMBLY 4 STEAM OUTLET

5. CORE SPRAY INLET

6. STEAM SEPARATOR ASSEMBLY

7. FEEDWATER INLET

8. FEEDWATER SPARGER

9 LOW PRESSURE COOLANT INJECTION INLET

10. CORE SPRAY LINE

11. CORE SPRAY SPARGER

12. TOP GUIDE

13. JET PUMP ASSEMBLY

14. CORE SHROUD

15. FUEL ASSEMBLIES

16. CONTROL BLADE

17. CORE PLATE

18. JET PUMP/RECIRCULATION WATER INLET

19. RECIRCULATION WATER OUTLET

20. VESSEL SUPPORT SKIRT

21. SHIELD WALL

22. CONTROL ROD DRIVES

23. CONTROL ROD DRIVE HYDRAULIC LINES

24. IN CORE FLUX MONITOR GENERAL ELECTRIC

2 cladding tubes (about 12.5 mm in outer diameter). For a BWR core of 8 x 8 type, each fuel assembly contains about 62 fuel rods and 2 water rods, which are sealed in a Zircaloy-2 channel box. The control material is in the shape of blades arranged through the fuel assembly in the form of a cruciform and is generally made of B4C dispersed in 304-type stainless steel matrix or hafnium, or a combination of both. Water is passed through the reactor core producing high-quality steam and dried at the top of the reactor vessel. The BWR operates at a pressure of about 7 MPa and the normal steam temperature is 290-330 ° C.

Pressurized Heavy Water Reactor The PHWR reactors were mainly developed by a partnership between the Atomic Energy of Canada Limited (AECL) and Hydro­Electric Power Commission of Ontario in 1960s. The reactors were of Generation — II type. Notably, these reactors are also called CANDU reactors (Figure 1.11a). They are so named because they use heavy water (deuterium oxide) as the moderator and natural uranium as the fuel. These reactors are located mainly in Canada, India, China, and few other countries. The CANDU reactor design does not require a reactor pressure vessel as in LWRs, and hence not a single CANDU reactor oper­ates in the United States since the nuclear safety regulations of the US Nuclear Regulatory Commission specifically call for an RPV in a compliant reactor design.

Unlike LWRs, the natural uranium (0.7% U235) oxide fuel clad in zirconium alloy tubes (known as pressure tubes, made of Zr-2.5Nb) is used in this reactor. These hundreds of pressure tubes are kept inside a calandria shell made of an austenitic stainless steel and reinforced by outer stiffening rings. The shell also keeps channels for the pressurized coolant (hot heavy water or light water) and moderator (heavy water). If light water is used to moderate the neutrons, it would adversely affect the neutron economy due to the absorp­tion of neutrons. That is why cold heavy water is used as the moderator. The pressure tubes along with moderator and cooling tubes are arranged in a hori­zontal fashion (not vertical as in LWRs) and the fuels can be replaced and reloaded without shutting down the whole reactor. Note the fuel and associated structural configuration in Figure 1.11b. The pressurized coolant stays typically at about 290 °C. This reactor system requires a steam generator to produce steam as does a conventional PWR. The control rods are dropped vertically into the fuel zones in case total shutdown or controlling of reactivity is neces­sary. Gadolinium nitrate is added in the moderator system that acts as a secondary shutdown system.

Liquid Metal Fast Breeder Reactor Liquid metal (generally sodium) is used in liquid metal fast breeder reactors (LMFBRs) to transport the heat generated in the core. These reactors are called “breeder” because more new fuels are pro­duced than consumed during its operation. The reactor can convert fertile material (containing U238 and Th232) into fissile materials (Pu239 and U233), respectively. The concept of this reactor type is very practical from the fuel utilization viewpoint. The natural uranium contains only about 0.7% of fissile U235. The majority of the natural uranium contains U238 isotope. These reactors are characterized by high power density (i. e., power per unit volume) due to the lack of a moderator (i. e., much more improved neutron economy). The reactor cores are typically small because of the high power density requirements. The temperature attained in this type of reactors is higher and thus leads to higher efficiency of electric power generation (~42% in LMFBRs versus ~30% in LWRs). The use of Th232 in LMFBRs is particularly advanta­geous for countries like India that do not have a large deposit of uranium, but has plenty of thorium. It should be noted that sodium used in this type of reactor transfers heat to the steam generators. The system containing sodium should be leak-proof since sodium reacts with oxygen and water vapor very fast. Furthermore, it becomes radioactive as the coolant passes through the reactor core. The whole primary coolant system should be put in the shielded protection to keep the operating personnel safe.

The first prototype LMFBR reactor named EBR-I (Experimental Breeder Reactor) was built at the present-day site of the Idaho National Laboratory near Idaho Falls, ID. This was the first reactor to demonstrate that the electricity can be generated using the nuclear energy. Also, it was the first “fast breeder” reactor. It used sodium-potassium (NaK) as coolant. It started its operation in 1951, and was decommissioned by 1964. By this definition, it was a Generation-I fast reactor

design. Following the decommissioning of EBR-I, another fast breeder reactor (EBR-II) was installed and started operation in 1963. EBR-II (Figure 1.12) was oper­ated very successfully before it was shut down in 1994. The fuel used was a mixture of 48-51% of U235, 45-48% U238, and the rest a mixture of fissium metals (Mo, Zr, Ru, etc.) and plutonium.

2.2.1.1 Point Defects in Metals/Alloys

Point (or zero-dimensional) defects are associated with imperfections involving an atom or only a few atoms in a localized region. They are often described as “zero­dimensional” defects. There are many types of point defects that one should be aware of. They are schematically shown in Figure 2.25:

Vacancies

Vacancy is simply a vacant lattice site. Vacancies are present in all crystalline solids, pure or impure, under almost all conditions, predictable by the laws of thermo­dynamics. Vacancies were first imagined to explain the diffusion phenomenon in solids (see Section 2.3). Here we develop a derivation for calculating the equili­brium vacancy concentration. A basic knowledge of thermodynamics is required to understand this approach. We assume that the process of vacancy formation must obey dG = 0 for maintaining a thermodynamic equilibrium condition. One can clearly note here that due to the existence of vacancies, the enthalpy and the entropy of the crystal would be greater than it would have been without the vacancies. Thus, we can write from the definition of the Gibbs free energy,

DGv = DHv — TASv, (2.2)

where DGv is the Gibbs free energy change, DHv is the enthalpy increase, and DSv is the entropy increase due to the presence of vacancies in the crystal.

Therefore, if G(o) is the Gibbs free energy of a perfect lattice and G(n) is the Gibbs free energy of the lattice with n number of vacancies, we can simply write

G(n) — G(o) = AGv. (2.3)

Now we definitely need some discussion on the entropy term. The entropy term is broadly categorized in two ways: vibrational entropy and configurational entropy. The vibrational entropy term is readily intuitional in that the atoms present in the neighborhood of a vacancy are less restrained than the atoms in the perfect portion of the crystal. Thus, each vacancy can provide a very small contribution to the total entropy of the crystal due to the more irregular or ran­dom vibration. Although a detailed theoretical treatment of vacancies would require consideration of the small vibrational entropy contribution, it is gener­ally not considered in this type of derivation as it is of secondary importance. Instead, we should take into account the effect of configurational entropy that arises due to the probabilistic nature of the vacancy creation process. If N is the total number of lattice atoms, there are W different ways of arranging the N atoms and n vacancies on (N + n) lattice sites. Hence, W can be given by the following expression following the combinatorial rules:

where N! = N(N — 1)(N — 2) … 3-2-1, and so forth. On the other hand, the config­urational entropy term is given by Sconf = kln W, where k is the Boltzmann’s con­stant. This constant appears in the science equations quite a bit and has the value on the order of the thermal energy per atom.

Thus, Eq. (2.2) becomes

AGv = ngv — kT ln W’

where gv is the Gibbs free energy associated with forming a vacancy. For thermal equilibrium, the following relation should hold:

@(AGV) @(ln W)

dn gv dn

For large numbers of N and n (in reality their values are in millions), Stirling’s approximation ln(N!) = Nln(N) — N can be used to show that @(ln W)/dn — ln ((N + n)/n).

Thus, Eq. (2.6) becomes

Since N ^ n, we can also write

(2.7)

where Cv is defined as the equilibrium vacancy concentration.

We know from the definition of Gibbs free energy,

gv — Hv — TSv.

So, Eq. (2.7) becomes

Cv — exp (— І)65*© • (2’8)

The above equation presents the compromise between the enthalpy (i. e., energy, Ev, in this case as PV term is negligible, that is, Hv « Ey) and entropy. The vacancy formation energy can also be defined as the energy needed to remove one atom from the lattice and place it on the crystal surface. However, it does not give us any indication of how much time it would take to accomplish it. That is why the ther­mal vacancy formation is guided by the thermodynamic principles, not kinetic ones. It is clear from this that a large vacancy concentration is favored with a decrease in the vacancy formation energy, whereas a large entropy of vacancy for­mation tends to increase the vacancy concentration. Furthermore, it is not only the vacancy formation energy alone that is solely important, but the ratio of Ev to ther­mal energy is also important. At higher temperatures, the thermal energy is high causing a significant probability of strong thermal fluctuations leading to the for­mation of vacancies. Conversely, at low temperatures, the probability of large ther­mal fluctuations is so low that less number of vacancies is created. So, the vacancy concentration strongly depends on the temperature and would increase exponen­tially. In other words, the term exp(—Ev/kT) represents a probability term giving

the chances that a crystal with thermal energy (kT) has to create thermal fluctua­tions sufficient enough to provide the energy needed to produce vacancies. It has been found that the contribution of exp(Sv/k) is very small at all temperatures com­pared to the exp(—Ev/kT) contribution, and hence will not be considered further. Hence, Eq. (2.7) can also be written as

Cv = exp (- І) • (2’9)

The results of experimental and theoretical studies have shown that the vacancy formation energies are typically on the order of 1 eV. There are several methods of measuring vacancy concentration — one being the electrical resistiv­ity measurement. The electrical resistivity of a metal generally increases because of the presence of vacancies, and the change in resistivity is proportional to the vacancy concentration. The vacancy formation energy is generally obtained from the slope of the semilog plot of Cv versus 1/T. Vacancies play a major role in the diffusion processes and thus affect various phase transformation, deformation, and physical processes. Sometimes, the nonequilibrium concentration of vacan­cies can be sustained at room temperature by heating a metal followed by quenching (i. e., fast cooling). Quenching ensures that the high concentration of vacancies characteristic of higher temperature is retained at room temperature without being depleted (as the migration of the vacancies become slower at lower temperatures).

The type of vacancies that we discussed previously should be called “monova­cancy” since only one (i. e., mono) lattice atom is missing (Figure 2.26a). At very high temperatures, many vacant lattice sites find the neighboring sites vacant, too.

If two vacancies come side by side, a “divacancy” is formed (Figure 2.26b). The divacancy formation energy (E®) can be expressed as (2Ev — B), where B is the binding energy of a divacancy (this energy is basically the energy required to sepa­rate a divacancy into two isolated monovacancies). It is generally hard to measure or calculate the binding energy. One estimate for copper has shown values of the

binding energy of a divacancy in the range of 0.3-0.4 eV. The equilibrium divacancy concentration (C®) is given by the following equation:

(С21) — b exp ; (2.10)

where b is the coordination number. Equation (2.10) can also be expressed in the following form:

Self-Interstitial Atom (SIA)

A self-interstitial atom is a type of point defect where a lattice atom occupies an interstitial site instead of its regular position (recall the interstitial space as dis­cussed in Section 2.1). For example, if a copper atom is in one of the interstitial positions, a self-interstitial type of point defect would be created. It can also be thought of as removing one atom from the crystal surface and moving it to an inter­stitial site. Any interstitial site is smaller than its own atom size, and the presence of such an interstitial atom could badly strain the lattice surrounding it. As a result, the formation energy of an SIA (£;) in copper even under equilibrium conditions is quite high (~4eV) compared to the similar quantity for a monovacancy (~1 eV). The equilibrium concentration of SIA (C;) is given by

(2.12)

where ni is the number of interstitial atoms and N; is the number of interstitial sites. Thus, thermal energy (kT) is not sufficient to create self-interstitials. The self­interstitials can be generated more readily under energetic particle radiation, such as fast neutron irradiation. Furthermore, the actual configuration of SIAs could be much different from the simple model that we just discussed (see Chapter 3).

■ Example 2.4

Calculate the concentrations of thermal monovacancies, divacancies, and self-interstitials in copper (FCC crystal structure, coordination number 12) at 20 °C, 500 °C, and 1073 °C. Comment on the results. Assume Ev — 20 kcal mol-1, B — 7 kcal mol-1, and £; — 90 kcal mol-1.

Solution

Since the activation energies are given as per mole, it is convenient to use the gas constant R (—1.987 cal mol-1 K-1) rather than Boltzmann’s constant. We use the above formulations to evaluate the concentration of monovacancies (C), divacancies (C2)), and self-interstitials (C;). It will be easier to use a standard spreadsheet software in which you can write in the equations, and then calculate for all three temperatures. The results are summarized in the table below.

Note that the melting temperature of copper is 1083 °C. So, if the temper­ature rises to that level or above, the concentration of point defects will lose their physical meaning as the liquid state of copper would not contain any crystal point defects.

4.2.1

Types of Dislocation Loops

We have noted that pure edge and pure screw dislocations are rarely observed. Most dislocation lines are of mixed type. Many of them stay in the form of disloca­tion loops. Let us discuss two types of dislocation loops here.