When properties depend on orientation, the property is termed as anisotropic. Crystallographic anisotropy arises because of the crystallographic texture (preferred orientation of grains in a material). As we know, a single crystal is the most aniso­tropic. However, in polycrystalline materials, each grain tends to be oriented in dif­ferent ways, thus reducing the texture and thus anisotropy. However, polycrystalline materials can also become textured depending on the processing it undergoes. Yield strength may then be strongly dependent on crystalline texture. Polycrystalline materials with anisotropic crystal structure like HCP tend to have strong texture and thus mechanical anisotropy. One such example would be zirca — loy fuel cladding.

Dislocation loops resulting from vacancy and interstitial condensations are created from clusters of the respective defects, and either shrink or grow depending on the flux of defects reaching the embryo. Once they have reached a critical size, the loops become stable and grow until they unfault by interaction with other loops or with the network dislocations. Russell and Powell (1973) [5] have determined that interstitial loops nucleate much easier than vacancy loops because interstitial loop nucleation is less sensitive to vacancy involvement than is vacancy loop nucleation to interstitial involvement. The nucleation of loops is essentially a clustering pro­cess in which enough of one type of defect needs to cluster, in the presence of other types of defects, to result in a critical size embryo that will survive and grow.

The effects of irradiation temperature on the faulted Frank loop size and density in irradiated cubic silicon carbide (SiC) are shown in Figure 6.12(a) and (b), respec­tively. As the loops grew in size, their number density decreased. Figure 6.13 shows the effect of radiation dose (in dpa) on the faulted Frank loops in ion-irradiated cubic SiC at an irradiation temperature of 1400 °C.

Here, we summarize the criteria for materials selection for different nuclear components. Let us take the example of fuel cladding material for the LWR. As noted before, cladding materials are used to encapsulate the fuel and separate it from the coolant. The requirements for fuel cladding material are as follows: (a) low cross section for absorption of thermal neutrons, (b) higher melting point, adequate strength and ductility, (c) adequate thermal conductivity, (d) compati­bility with fuel, and (e) corrosion resistance to water. Following the first factor, we have discussed in Section 1.7 how different metals have different cross sec­tions for absorption of thermal neutrons. Although Be, Mg, and Al all have lower cross sections for absorption of thermal neutrons, other nonnuclear fac­tors become the impediment for their use in commercial power reactors. Even though Be has a high melting point (1278 °C), it is scarce, expensive, difficult to fabricate, and toxic. Mg has a low melting point (650 °C), is not strong at higher temperatures, and has poor resistance to hot water corrosion. Al has a low melt­ing point (660 °C) and poor high-temperature strength. Even though an Al — based alloy has been used as fuel cladding materials in reactors like ATR, and in the past a magnesium-based alloy was used in Magnox reactors, their use remains very limited. This leaves zirconium-based materials as the mainstay of fuel cladding materials for LWRs. Zirconium has various favorable features: (a) relatively abundant, (b) not prohibitively expensive, (c) good corrosion resist­ance, (d) reasonable high-temperature strength, and (e) good fabricability. Some of the properties could be further improved through appropriate alloying. More detailed discussion on the development of zirconium alloys is included in Appendix 1.A at the end of the chapter.

(Example:

The world’s first nuclear power plant was EBR-1.

It carried a coolant, an alloy of sodium (Na) and potassium (K), called Na—K (“nack”).

The following are the coolant characteristics:

• Stays liquid over a wide range of temperatures without boiling away.

• Transfers heat very efficiently taking heat away from the reactor core and keeping it cool.

• Allows neutrons from the reactor core to collide with U-238 in the breeding blan­ket and produce more fuels.)

1.9.3.5 Shielding Materials

1.10

If we consider random walk phenomenon as a general event in atomic diffusion, the net movement of atoms (or net motion of point defects) through a homoge­neous crystal will be zero. However, if one assumes that the impurity concentration is more in region A than in region B, a net movement of atoms occurs from region A to region B until the concentrations of impurities in both the regions become uniform. In a simple kind of picture, a net movement of atoms occurs along a concentration gradient. However, without knowing the details of atom movement, one can come up with the phenomenological diffusion theories as apparent in Fick’s laws for diffusion.

Dislocations generated from a Frank-Read source pile up at barriers such as grain boundaries, second phases, and sessile dislocations or a lead dislocation from

another such source on a parallel glide plane. The number of dislocations in the pileup would be directly proportional to the obstacle distance from the source (i. e., the pileup length Lpileup) and the applied stress. From the force balance between the dislocation-dislocation interactions and the external stress, one can show that

where n is the number of dislocations in the pileup with pileup length Lpileup under an applied shear stress tjy.

This leads to a high stress concentration ahead of the pileup that is usually relieved by plastic deformation. Just ahead of the pileup (i. e., r is small and в « 0 in Figure 4.25, which shows a dislocation pileup as a cross section of the loops com­ing out of the paper) the stress t is enhanced by the number of dislocations in the pileup:

At large r but smaller than L (such as at point P in Figure 4.25), there exists a stress concentration due to the pileup:

tat P&J pileu-rxy (true for q < 70°). (4.22)

One can show from the above Eq. (4.22) the relation between the yield stress and grain size (namely, the Hall-Petch relation; refer to, for example, Ref. [1]):

sy = so + -7=. (4.23)

4.3

Creep constitutive behavior is generally described using the minimum creep rate or in most cases the steady-state creep rate. Norton’s law is the equation that is used for describing the dependence of stress on strain rate at a given temperature:

e = A1an,

where Aj is a constant dependent on the test temperature and n is the stress expo­nent (basically it is the reciprocal of strain rate sensitivity m, as given in Eq. (5.23)). It is important not to confuse between strain hardening exponent and stress expo­nent, which have the same symbol, n. The increase of the steady-state creep rate with test temperature (Figure 5.27a) under a given stress follows an Arrhenius equation with a characteristic activation energy for creep (Qc):

e = A2e—Qc/RT (5.36)

where A2 is a constant dependent on the applied stress/load. For creep at high tem­peratures (T> 0.4Tm), the activation energy for creep (Q;) was shown to be equal to that for self-diffusion (Qd). Thus, the temperature and stress variations of creep rate can be combined into one equation:

e = A3e-Qc/RT an, (5.37)

where A3 is a material constant and R is the gas constant (1.987 cal mol-1 K-1). It has been shown that factors affecting self-diffusivity also influenced the creep rates similarly; for example, pressure dependence, effect of C on self-diffusion and creep of у-iron, influence of ferromagnetism and crystal structure (diffusion and creep of a-iron versus у-iron), influence of crystal structure on creep and diffusion in thal­lium, effect of composition (stoichiometry) on diffusion and creep, and so on. Thus, one can state that the steady-state creep rate is proportional to DL (lattice or self-diffusion) so that the creep rate equation becomes

/a n

e = A4Dan = A5D(jJ. (5.38)

Here, A4 and A5 are constants dependent on the material and microstructure such as the grain size. Thus, the steady-state creep rate results plotted as e/D versus a/E or a/G in double-logarithmic scale yield a straight line with a slope of n (stress exponent); this is known as Sherby plot and takes care of the temperature variation of the elastic modulus (E = E0 — E(T — T0)). At very high stresses (>10—3E), however, power-law breakdown is noted with creep rates increasing

more rapidly (Figure 5.28). This creep regime is due to dislocation glide-climb with climb of edge dislocations being the rate controlling process as exhibited by pure metals (as well as ceramics) and some alloys.

However, these relationships do not show the effect of other parameters in a use­ful way. In fact, the strain rate is a function of stress, temperature, and micro­structural factors. The Bird-Mukherjee-Dorn (BMD) equation is commonly used to express the constitutive behavior during creep deformation:

where A is a material constant, d is the grain size, and p is known as the grain size exponent. It should be noted that in the above Dorn equation, the strain rate, stress, and grain size are all normalized to dimensionless parameters. The values for A, n, and p along with appropriate diffusivity {D = D0exp(—Q/RT)} in Eq. (5.39) charac­terize the underlying creep mechanisms. The underlying creep mechanisms have been dealt with in detail in Section 5.1.6.4.

Various radiation types are produced in a nuclear environment. It could be alpha particles, beta particles, gamma rays, or neutrons. In this book, we are primarily concerned with the radiation damage and effects caused by neutrons. There could be several sources of neutrons, including alpha particle-induced fission, spontane­ous fission, neutron-induced fission, accelerator-based sources, spallation neutron source, photoneutron source, and nuclear fusion.

1.5

Interactions of Neutrons with Matter

Collision of neutrons with atom nuclei may lead to different scenarios — scattering of the neutrons and recoil of nuclei with conservation of momentum (elastic scat­tering) or loss of kinetic energy of the neutron resulting in gamma radiation (inelastic scattering). The capture of neutrons may result in the formation of new

nuclei (transmutation), or may lead to the fragmentation of the nucleus (fission) or

the emission of other nuclear particles from the nucleus. We shall discuss some of

the effects in more detail in Chapter 3.

a) Elastic Scattering

Elastic scattering refers to a neutron-nucleus event in which the kinetic energy and momentum are conserved.

b) Inelastic Scattering

This interaction refers to neutron-nuclide interaction event when the kinetic energy is not conserved, while momentum is conserved.

c) Transmutation

When a nuclide captures neutrons, one result could be the start of a sequence of events that could lead to the formation of new nuclide. The true examples of this type of reaction are (n, a), (n, p), (n, b+), (n, b), and (n, f). Reactions like (n, y) and (n, 2n) do not result in new elements, but only produce isotopes of the origi­nal nuclide.

d) Fission

Fission is a case of (n, f) reaction, a special case of transmutation reaction. Ura­nium is the most important nuclear fuel. The natural uranium contains about 0.7% U235, 99.3% U238, and a trace amount of U234. Here, we discuss the neu­tron-induced nuclear fission, which is perhaps the most significant nuclear reaction. When a slow (thermal) neutron gets absorbed by a U235 atom, it leads to the formation of an unstable radionuclide U236, which acts like an unstable oscillating droplet, immediately followed by the creation of two smaller atoms known as fission fragments (not necessarily of equal mass). About 2.5 neutrons on average are also released per fission reaction of U23 . An average energy of

193.5 MeV is liberated. A bulk of the energy (~160 MeV or ~83%) is carried out by the fission fragments, while the rest by the emitted neutrons, gamma rays, and eventual radioactive decay of fission products. Fission fragments rarely move more than 0.0127 mm from the fission point and most of the kinetic energy is transformed to heat in the process. As all of these newly formed parti­cles (mostly fission fragments) collide with the atoms in the surroundings, the kinetic energy is converted to heat. The fission reaction of U235 can occur in 30 different ways leading to the possibility of 60 different kinds of fission frag­ments. A generally accepted equation for a fission reaction is given below:

U235 + nj! Kr36 + Ba562 + 2n1 + Energy, (1.1)

which represents the fission of one U235 atom by a thermal neutron resulting into the fission products (Kr and Ba) with an average release of two neutrons and an average amount of energy (see above). It is clear from the atomic masses of the reactant and products, that a small amount of mass is converted into an equivalent energy following Einstein’s famous equation E = mc2.

U235 is the one and only naturally occurring radioisotope (fissile atom) in which fission can be induced by thermal neutrons. There are two other fissile atoms (Pu239 and U233) that are not naturally occurring. They are created during

the neutron absorption reactions of U238 and Th232, respectively. Each event con­sists of (n, c) reactions followed by beta decays. Examples are shown below:

The concept of the “breeder” reactors is based on the preceding nuclear reactions, and U238 and Th232 are known as “fertile” atoms. Heavy radioisotopes such as Th232, U238, and Np237 can also undergo neutron-induced fission, how­ever, only by fast neutrons with energy in excess of 1 MeV. That is why these radionuclides are sometimes referred to as “fissionable.”

1.5.1

For hexagonal crystals, it is convenient to use a reference system with four axes (a1, a2, a3, and c), as shown in Figure 2.13, to specify crystallographic planes and

directions. Three axes (a1t a2, and a3) are coplanar and lie on the base of the hexagonal prism of the unit cell with a 120° angle between them. The fourth axis (c) is perpendicular to the base. Thus, a four-digit notation (hkil), known as Miller-Bravais indices, can be used for denoting planes, and [uvtw] for direc­tions in a hexagonal crystal. The use of the Miller-Bravais indices enables denoting crystallographically equivalent planes by the same set of indices. How­ever, the basic procedure for the Miller-Bravais indices are the same as that of the Miller indices. Let us take a look at the plane shown in Figure 2.13a. The intercepts created by the four axes are 1, 1, —1, and 1. Thus, the Miller-Bravais indices of the plane is (1/1 1/i —1/1 1/1) or (1011). Similarly, Miller-Bravais indices of the other planes shown in Figure 2.13b-d can be derived. Note that in all the four cases, the condition h + k = — i is satisfied as a1, a2, and a3 axes are coplanar vectors.

For the Miller-Bravais indices of a direction in the hexagonal crystal, the basic procedure again remains the same. However, it is bit different. For example, the direction along or parallel to the axis a1 can be resolved into components along a2 and a3, each component being —1, as shown in Figure 2.14. The first index can be obtained from the relation u + v = — t or u = —(t + v) = —(—1 — 1) = 2. The direction does not have any component in the c direction. Thus, the Miller-Bravais index of the direction is [2110].

One can use the alternative method of using points as per the cubic, but in this case we first start from defining a1, a2, and c axes as points with the coordinates:

a1 — (100), a2 — (010), and c =(001).

As per the cubic system, find the direction as a subtraction of tail from the head and call it [h’k’l’]. Find the four-axes notation [hkil] by using the following relations and clearing of fractions and reducing to lowest integers:

h = 1 (2h’ — k’), k = 3(2k’- h’), i = —1(h’ + k’), ‘ =

Thus, the Miller index of а1, for example, is easily found as (100)-(000) so that h’ = 1, k = 0, and l’ = 0 or h = 2/3, k = —1/3, i = -1/3, and l = 0 or Miller index of aj is [2110]. This is a very useful procedure in deriving the Miller indices for direc­tions in HCP crystals.

2.1.6

Before starting discussions on various radiation damage models, let us first under­stand the concept of displacement threshold. While describing the primary damage events, it is essential to develop a clear understanding of the displacement energy or displacement threshold (denoted by Ed), which is defined as the minimum energy that must be transferred to a lattice atom in order for it to be dislodged from its lattice site. Generally, average displacement energy of 25 eV is used. How­ever, the fixed value of 25 eV is only an average of all the possible displacement energies calculated along different crystallographic directions in a given material. This value also agrees well with the experimentally measured displacement energy values. The specific values of displacement energy depend on the nature of the momentum transfer, trajectory of the knock-ons, crystallographic structure, and thermal energy of the atoms. It has been noted that higher melting point metals tend to have higher displacement energies (Figure 3.3). Even though all elements do not follow the trend as there are many other factors that influence displacement

o I— .—- ,—- .—- ,—- .—- ,—- .—-

750 1500 2250 3000 3750

Melting Temperature (K)

Figure 3.3 The variation of displacement energy as a function of melting temperature of metals.

energy, it is clear from Figure 3.3 that the refractory metals (such as Mo and W) tend to have much higher displacement energies compared to those of lower melt­ing point ones. It is most possibly related to the stronger binding energy present in higher melting point metals.

As regards to the lower limiting value of the displacement energy, it is close to the energy needed to produce Frenkel pairs (3-6 eV). If the energy transferred by the knock-on atom to the struck lattice atom is less than the displacement energy, the atom will not dislodge from its site. Rather it will vibrate around an equilibrium position, transfer the energy through the neighboring lattice atoms, and eventually dissipate as heat. In order to calculate the displacement energy, it is essential to know the description of the interatomic potential fields since this is the energy bar­rier that the struck atom needs to surmount to eject successfully from its regular lattice site. A simple example is shown below.

■ Example 3.1

Calculate the displacement energy along (110) direction (i. e., a face-centered position to an adjacent face-centered position) in an FCC metal unit cell (as shown in Figure 3.4a) in which the interatomic potential is given by a simple repulsion potential as described below:

V(r) = — U + У2 k(req — r)2 at r < req, (3.1a)

V(r) — 0 at r > req, (3.1b)

where U is the binding energy of atom (energy per atomic bond), k is a force constant indicative of the repulsive portion of the potential, req is the equili­brium atom separation, and r is the general separation distance. The force constant is k«0 and U values can be taken as 60 and 1 eV, respectively. Note that a0 is the lattice constant of the crystal.

Solution

The atom at position “M” after being dislodged from the site needs to follow the direction of the arrow (parallel to (110) direction) to the position “N.” During the trajectory motion of the atom M, it needs to go through a poten­tial field created by the four-atom barrier (1-1-1-1). The center of this four — atom barrier represents the saddle point (at O) along the trajectory of M atom with the maximum potential energy; the variation of potential energy as a function of position is shown in Figure 3.4b.

The energy of a single atom in the FCC crystal can then be given by Eeq = — 12 U, considering that an FCC atom is surrounded by 12 equidistant near­est neighbors (due to the coordination number of FCC crystal being 12). In turn, we can say that when M atom is at the equilibrium site, the energy of M atom is given by as shown above.

Now when the M atom moves to the center of four-atom barrier, it achieves the highest potential energy following the potential described in Eq. (3.1a). Therefore, the potential energy at the saddle point “O” is given by

E* — 4V(r)=4[- U + =, fe(req — r)2].

The displacement energy (Ed) is given by

Ed — E* — Eeq — 4[— U + =, k(req — r)2] — (-12 U)

— 8 U + 2k(r eq — r)2. (3.2)

Now we need to express req and r in terms of lattice constant (a0). This can be accomplished from the geometrical relations in the FCC unit cell, as shown in Figure 3.4a. In this case, req is the minimum distance from M to O. A simple geometric construction can show that the distance can be
determined by calculating half the hypotenuse of a triangle with other two sides being a/2 and a/2.

Hence, we get

and r (the impact parameter), the distance of each atom-1 of the four-atom barrier from the point O, is given by

r (i. e., O1) = 2 у/(Я0)2 + = y^. (3.4)

Now we take req and r relations from Eqs (3.3) and (3.4), respectively, and use them in Eq. (3.2). Thus, we obtain

Ed along (110}=8U + 2k(req — r)2 = 8U + 2k~ Jh^ = 8U + 2 x k

(0.213)2a2 = (8U) + (2 x 0.045)(k a2) = (8 x 1eV) + (0.09 x 60eV) = ~13.4eV.

As additional exercises, determine the displacement energy along (100} and (111} in FCC yourself using the same potential expression given in Eq. (3.1). It is very clear that accurate knowledge of interatomic potentials is quite important in the accuracy of the calculated displacement energy values.

Special Note: The use of displacement energy in ceramics is similar but bit complicated because of the presence of multiple atomic species (cations and anions). Table 3.1 summarizes some displacement energy values of cations and anions in some well-known ceramics.

In case of multicomponent ceramics, an effective displacement energy (Edff) is often used to calculate the extent of radiation damage. This is given by

where Si is the stoichiometric fraction and Ed is the displacement energy of the ith atomic species. Taking into account the nature of Coulombic interactions in

collision cascades, the scaling parameter is more appropriately expressed as (SiZi2)/Ai, where Z; and A; are the atomic number and atomic mass of the ith species.

3.2

Dispersed particles can impart significant obstacles to the dislocation motion, thus increasing the strength of the material. Particle hardening is a much stronger mechanism than the solid solution hardening. This type of strengthening could be effective even at higher temperatures based on the stability of the precipitates.

There are two types of fine particle strengthening mechanism: precipitation hard­ening and dispersion strengthening.

• For precipitation hardening (or age hardening) to happen, the second phase needs to be soluble at elevated temperatures and should have decreasing solubil­ity with decreasing temperatures. However, in dispersion hardening, the second phases have very little solubility in the matrix (even at higher temperatures).

• Generally, there is atomic matching (i. e., coherency) between the precipitates and the matrix. However, in dispersion hardening system, there is no coherency between the second-phase particles and the matrix.

• The number of precipitation hardening systems is limited, whereas it is theoreti­cally possible to create an almost infinite number of dispersion hardening systems.

As the particles are central to these kinds of strengthening mechanisms, the fol­lowing particle characteristics are important: (a) particle volume fraction,

(b) particle shape, (c) particle size, (d) nature of the particle-matrix interface, and (e) particle structure. Note that particle volume fraction can be related to the parti­cle size through the interparticle spacing (l):

4(1 — f)r

3f

where f is the volume fraction of spherical particles of radius r. There are other similar relations also, depending on the assumptions of derivation.