There are two types of point defects, Frenkel and Schottky defects that can be found

in ionic crystals.

a) Frenkel defect or disorder forms when an ion leaves its original place in the lattice creating a vacancy, and becomes an interstitial by moving into a nearby intersti­tial space not usually occupied by another ion. This type of defect was first dis­covered by the Soviet physicist Yakov Frenkel (1926). This defect is actually a

pair of vacancy and interstitial (Figure 2.28). Only a negligible volume expansion occurs when the Frenkel defects are formed. Frenkel defects can form on both the cation sublattice and the anion sublattice. Anion Frenkel defects are also known as anti-Frenkel. It is to be noted that a vacancy-interstitial pair (known as Frenkel pairs) can be formed in metals under irradiation damage. For an MX-type of ionic crystal, the concentrations of these defects are given by the following:

CvM = CiM = exp (-£F/2fcT) (2.13)

and

CvX = CiX = exp (-Ef/21cT), (2.14)

where CvM and CiM are the concentrations of the cation Frenkel, CvX and CiM are the anti-Frenkel concentrations, EF is the formation energy for a Frenkel defect, and k, T have the usual meaning. In a given ionic crystal, either Frenkel or anti — Frenkel defects are created but never both kinds.

b) Schottky defect or disorder was named after Walter H. Schottky. A Schottky defect is composed of differently charged pairs of vacancies, that is, missing Na+ and CT ions in the NaCl crystal (Figure 2.29). However, in TiO2, the Schottky defect consists of one titanium ion vacancy and two oxygen ion vacancies in order to maintain the electrical neutrality of the crystal. Unlike Frenkel defect, Schottky defect is unique to ionic compounds only. Since the number of ions has to stay constant, no matter how many Schottky defects are present, the surplus of the ions must be thought of as sitting on the crystal surface. That is why the crystal

Figure 2.29 A schematic of a Schottky defect in an MX-type ionic lattice. After Olander

expands measurably when Schottky defects are formed. The concentration of Schottky defect pair is expressed as the following:

CvX = CvM = exp (-Es =2kT), (2.15)

where ES is the Schottky defect formation energy.

2.2.2

Assume an elastic distortion in a hollow cylinder, as shown in Figure 4.13a, creating a straight screw dislocation. This type of construction is known as Vol — terra dislocation, named after the Italian mathematician who first considered such distortions even though the concept of dislocations was not introduced. In the figure, MN appears as a screw dislocation due to the way radial slit (LMNO) is cut in the hollow cylinder and displaced by a distance of b, which is the magnitude of the Burgers vector of the screw dislocation in the z-direction. We can confirm it as a screw dislocation since the Burgers vector and the dislo­cation line are parallel to each other. Without getting into the details of the derivation, only the stress components of the stress field that are nonzero are found to be shear components. There is no dilatational (i. e., tensile or compres­sive) stress component associated with the screw dislocation. The stress field around a screw dislocation is given by

Gb

The strain field is given by the following equation: b

^ = 2РГ • (4:10b)

The above equations imply that the stress and strain fields consist of pure shear components. The stress field exhibits complete radial symmetry. As one can see, the stress varies inversely with the radius of the cylinder. The stress then becomes infinite when r tends to zero (i. e., approaching the center of the cylinder). The cyl­inder cannot physically sustain infinite stress. That is why a hollow cylinder (with a radius of r0) assumption is justified. Also, the region in the hollow region does not follow the linear elastic theories. This region is akin to the dislocation core. In order to know the stress field in this region, one needs to employ nonlinear atomistic theories leading to complex derivations. In most cases, the dislocation core dimen­sion remains approximately at <1 nm.

This test uses a square-based pyramid indenter (as shown in Figure 5.11) whose angle between the opposite faces is 136°. The Vickers Hardness Number (VHN) is

Figure 5.11 (a) Vickers indenter and (b) the corresponding impression created by the Vickers

indenter.

determined by the load divided by the surface area of the indentation. The area is calculated from the average length of the diagonals (d1 and d2) of the impression (Figure 5.11). The following relationship is generally used:

where P is the applied load in kgf, d is the average length (i. e., (d1 + d2 )/2) of the diagonals in mm, and 0 is the angle between the opposite faces of the indenter (136°). So, VHN is in the unit of kgfmm~2. Vickers test is extremely useful as it can range from a scale of 5-1500. This can be used as a microhardness technique if the loads used are small (as discussed later). The disadvantages associated with Vickers hardness technique are the requirement of good surface preparation and errors in the determination of the diagonal length.

Rockwell Hardness Test

In 1908, Professor Ludwig of Vienna, Austria, first described a method by which a hardness of a material can be measured by a differential depth measurement tech­nique. This mode of hardness test technique consisted of measuring the increment of depth of a cone-shaped diamond indenter (known as Brale) or spherical steel indenters of various diameters forced into the material by a minor load and a major load. The test first uses a minor load of 10 kg to set the indenter onto the material surface, and then a major load (as determined by the particular scale chosen) is applied and the penetration depth is shown instantly on the scale with 100 divisions (usually each division corresponds to 0.00008 in.). Although the required surface preparation is minimum, depending on the hardness of the material, there are sev­eral Rockwell scales that represent a particular combination of the indenters and major loads used; some of them are included in Table 5.1. Note that there is a super­ficial Rockwell test mode that uses lower load compared to that used in the standard Rockwell tests and is used for thinner material or probing surface-hardened mate­rials. The martensite formed in eutectoid (0.77 wt% C) plain carbon steel can be very high (Rockwell hardness of up to 65 Rc).

5.1.2.1 Microhardness Testing

Microhardness technique is generally employed for measuring indentation hard­ness of very small objects, thin sheet materials, surface hardened materials, electro­plated materials, structural phases in multiphase alloys, and so forth. There are two

standard indenters with which microhardness tests are done. One is Vickers indenter, the same one used in macro-Vickers test (square-based diamond pyramid with 136° apex angle), already illustrated in Figure 5.11. Knoop indenter is also a pyramid-shaped indenter (an included transverse angle of 130° and a longitudinal angle of 172° 30′ such that it produces an indentation with a long to short diagonal ratio of ~7 : 1) (Figure 5.12). The Vickers indenter is pressed into the smooth, pol­ished surface of the specimen. The load is applied for a predetermined time and removed. After a specified load and time, precision objectives are used to view the indentation and using scales attached on the eyepiece, one can measure the size of the two diagonal lengths of the square-shaped impression. Then, this square length is used to measure VHN from Eq. (5.1). There are also standard charts available that relate diagonals to the VHNs. Generally, indentation should be made away from the edge of the sample as much as possible. The thickness of the specimen should be at least 1.5 times of the diagonal length. There should be adequate spac­ing between the indentations so that their deformation zones do not affect the deformation zone of other indentation. Generally, one takes several indents from a region and measure hardness and take the mean with measurement error.

In all indentation hardness tests, when the applied load is removed, there always occurs some elastic recovery. The amount of recovery and the distorted shape depend on the size and precise shape of the indenter. Due to the unique shape of the Knoop indenter, elastic recovery of the projected impression occurs in a trans­verse direction, that is, shorter diagonal length, rather than long diagonal. There­fore, the measured longer diagonal length will give a hardness value close to what is given by the uncovered impression.

The Knoop hardness number (KHN) is given by the load divided by the unrecovered projected area of the impression. It should be noted that the area referred to here is the projected area and not the surface area of the indentation as in the Vickers and Brinell hardness techniques. Hence, KHN is given by

P P

KHN = = 2, (5.27)

Ap Cl2 V ;

where P is the applied load in kgf, Ap is the unrecovered projected area of indenta­tion (mm2), l is the longer diagonal length (in mm), and C is the Knoop indenter constant that relates the longer diagonal length to the unrecovered projected area (generally 0.07028). Standard hardness charts are available where hardness values are provided against load and long diagonal values.

Note that Knoop hardness numbers are not independent of the load used. So, while reporting the hardness numbers, one should also report the load used (espe­cially, when load is less than 300 gf). Knoop hardness technique is much more sen­sitive to specimen surface preparation than the Vickers hardness in the low load range. Main advantages of Knoop hardness are its ability to measure near-surface hardness, hardness gradients, anisotropy in properties in thin sections, and so forth.

5.1.3

Let us be very clear that the origin of void swelling has been a topic of study for many years and still remains an active area of research interest. Fast neutrons tend to readily produce defects like vacancy-interstitial (Frenkel) pairs as a result of their interaction with the lattice atoms. Most of these defects recombine with each other or migrate to the sinks following their formation. The defects remain in a dynamic balance and maintain steady-state concentrations that are more than those of ther­mal equilibrium. Both vacancies and interstitials cluster together if the temperature

is high enough for diffusion, yet not so high that the defect supersaturation is main­tained by not allowing point defects to be annihilated by recombination or migration to sinks. A cluster of interstitials forms a dislocation loop, while vacancies can cluster in two different ways:

a) Vacancies can agglomerate into platelets that collapse into dislocation loops

b) Vacancies can form three-dimensional clusters known as voids.

The driving force for void nucleation is given by supersaturation of vacancies due to irradiation defined by

Sv = Cv/Cvo, (6.1)

where Cv and Cvo are the total vacancy concentration and thermal equilibrium con­centration of vacancies, respectively. If we work on the energetics of void formation, one could see that void is the stable form for small clusters of vacancies. However, as the number of vacancies in the cluster grows, the loops become energetically favorable configurations. But the collapse of void embryo into a vacancy loop is not favored by the presence of inert gas (such as He) in the void and that is why voids can survive and grow. Now the question remains why radiation-produced point defects form separate interstitial loops and voids. Since the vacancies and intersti­tials formed in equal numbers by fast neutron irradiation, it is expected that point defects of both types would diffuse to voids at equal rates and hence produce no net growth of the voids. Since voids represent accumulated excess vacancies, the inter­stitials must be preferentially absorbed elsewhere in the solid. The preferential interstitial sink is dislocation because dislocations interact more strongly with inter­stitials than vacancies because they distort the surrounding lattice more than vacan­cies. The preferred migration of interstitials to dislocations leaves the matrix metal depleted in interstitials relative to vacancies at somewhat greater rate than intersti­tials, and growth results.

The following are the conditions necessary for void swelling [11]:

a) Both vacancies and interstitials must be mobile in the solid.

b) At least one type of sink must differentiate between interstitials and vacancies, and should have more interactions with other types of defects.

c) The supersaturation of vacancies must be large enough to permit voids and dis­location loops to be nucleated and to grow.

d) Trace quantities of insoluble gases like He must be present to stabilize the embryo voids and prevent their collapse to vacancy loops.

A unit cell is the smallest building block of a crystal, which when repeated in trans­lation (i. e., with no rotation) in three-dimension can create a single crystal. There­fore, a single crystal or a “grain” in a polycrystalline material would contain many of these unit cells. A general unit cell can be created based on three lattice translation vectors (a, b, and c) on three orthogonal axes and interaxial angles (a, p, and y), which are also known as lattice parameters or lattice constants. Figure 2.3 illustrates the definitions of the lattice parameters and the angles. There are seven basic crystal systems. They are summarized in Table 2.1 with their lattice parameters. The least symmetric crystal structure is the triclinic and the most symmetric one is cubic.

There are a total of 14 Bravais lattices (space lattice) based on the atom positions in the unit cells of the basic crystal systems (Figure 2.4). The scheme consists of three cubic systems (simple, body-centered, and face-centered), four orthorhombic systems (simple, base-centered, body-centered, and face-centered), two tetragonal (simple and body-centered) and two monoclinic systems (simple and base-centered), and one each from triclinic, rhombohedral, and hexagonal systems. In each system, simple system is also known as primitive unit cells. A primitive unit cell contains 1 atom per unit cell. Note that during the discussion of crystal structure, we would not consider any crystal imperfection/defects, and we will introduce the topic in Chapter 2.3.)

2.1.2

Random walk diffusion refers to the situation where an atom can jump from one site to another neighboring site with equal probability. Interestingly, one can derive Fick’s first law from the theories of random walk diffusion. Note that the following derivation does not assume any particular micromechanism. Consider two adja­cent crystal planes (A and B) that are l distance apart, and there are nA number of diffusing species per unit area in plane A and nB in plane B with nA > nB in x-direc — tion only. Г is the number of atoms making jumps from one plane to the other neighboring plane per second (i. e., jump frequency). If we consider the fact that the atomic jump may be either along the forward direction (A to B) or along the backward direction (B to A), the following equations can be written:

jA! B ~ 2 nAC* (2.28)

Jb^a = 1 ПвГ. (2.29)

Then, the net flux crossing over from plane A to plane В is

Jnet = 2 (nA — nB)Г. (2.30)

Comparing Eq. (2.33) with Eq. (2.18), that is, Fick’s first law, we can write 1 2

D = 2 !2Г. (2.34)

Hence, a general form of diffusivity is obtained from this atomic theory, which could not be obtained from the continuum theory of diffusion. Diffusivity is the product of a geometric factor, square of diffusion jump distance, and the jump fre­quency. The geometrical factor for a simple cubic lattice would be 1/6 because of the atomic jump that can take place in three orthogonal directions and thus

1

D = — !2Г. (2.35)

6

If we wish to describe the geometric factor in a more general form, it can be described by x — 1/Nc, where Nc is the number of adjacent sites to which the atom may jump. So, l in general is the distance between such sites. We can take an example from UC that has a NaCl-type structure (see Section 2.1). In ionic crystals, cation can jump to equivalent cation sites and anion to equivalent anion sites, and these sites need to be neighboring sites. Now the coordination number for U cation and C anion in the UC crystal structure is 6 and the jump distance is the length associated with the vector, (1/2)(110). The distance between the two jump sites along (110) would be a0//2. Hence, the diffusivity term can be expressed by

D = ХІ2Г = Г, (2.36)

where a0 is the lattice constant.

Note that the geometric factor and the jump distance do not vary much; however, diffusivity value varies a lot among between different materials. This variability is attributed to the dependence of jump frequency on the energy of migration, tem­perature, and probability of adequate jump sites, which in turn depends on the defect concentration.

Einstein arrived at the similar expression by directly using random walk theory. He derived the following relations for mean square displacement (r2):

r2 = Гг!2. r2 — 6Dt,

where t is the time interval between jumps and other terms are as defined before. Comparing Eqs. (2.37) and (2.38), the relation D = (1/6)12Г is obtained (similar to the relation obtained in Eq. (2.34)). A more detailed account of this approach can be found in the books listed in Bibliography.

2.3.3

In BCC crystals, slip occurs in the direction of (111). The shortest lattice vector extending along the body diagonal is (a0/2)[111], which is the Burgers vector of a perfect dislocation in a BCC lattice. While the slip occurs normally in {110} planes, it is important to note that three {110}, three {112}, and six {123} can intersect along the same (111) direction. Hence, screw dislocations may move at random on these slip planes under the action of high resolved shear stresses. This is a reason for observing wavy slip lines in BCC crystals. Extended dislocation formation is not common in BCC metals as they are in FCC and HCP metals due to their relatively high stacking fault energy. Even though theoretical research indicates such forma­tion, it has not been substantiated well through experiments.

Cottrell (1958) proposed a dislocation reaction through which immobile dis­locations can be produced in BCC lattice. It is a mechanism through which a[001]-type dislocation networks are produced in the BCC lattice (experimen­tally observed). This type of dislocation is produced when dislocation 1 with Burgers vector (a0/2) [111] glides on (101) plane, while dislocation 2 with Bur­gers vector (a0/2)[111] glides on the slip plane (101). Figure 4.29a illustrates the dislocation reaction involved following the dislocation reaction given below:

0° [111]+0° [111]! *,[001]. (4.28)

The two dislocations react and form a pure edge dislocation with a Burgers vector ofa0[001], which resides on {100}-type planes that happen to be the cleavage plane, and thus do not take part in plastic deformation. It plays a role in the crack nuclea — tion and consequent brittle fracture.

4.3.3

Dislocation Reactions in HCP Lattices

Dislocation reactions in the HCP lattices are bit more complicated than those in the FCC and BCC ones. Some metals (such as magnesium and cobalt) with HCP lat­tices slip in their basal planes under normal deformation conditions. This is natu­ral to expect given that the close-packed plane is (0002). The slip direction is [1120], and the Burgers vector is given by (a0/3) [1120]. The magnitude of the Burgers vec­tor is a0. This perfect dislocation can dissociate into two Shockley partials (each has a Burgers vector magnitude of а0Д/3 as shown in Eq. (4.29). A stacking fault with an FCC crystal structure is produced in between the partials.

у [1120] !y [10T0]+a°[01І0]. (4.29)

Slip in some HCP lattices (alpha-Zr and alpha-Ti) with prismatic slip systems occurs along the close-packed direction (1120) and dislocations involved have Burgers vector of (а0/3)[1120]. Even though it is not very clear, the following types of disloca­tion reactions have been proposed that form stacking faults during prismatic slip:

a0 [1120] ! 18 [426 3] +18 [2463]. (4.30)

у [1120] ! I [1120] + ^ [1120]. (4.31)

Slip has also been observed with Burgers vector (a0/3) [1123] with a magnitude of (c2 + a2)1/2, but only under exceptional conditions. The glide planes are generally pyramidal planes {1011} and {1022}. Dislocation reactions are also possible, but they are outside the scope ofthis chapter.

4.3.4

Dislocation Reactions in Ionic Crystals

Dislocations in ionic crystals have unique electric charge effects that are not observed in the dislocations of simple metals or alloys. It stems from the very

[110]

nature of the ionic crystal structure. Table 4.2 summarizes the slip planes and directions in three ionic lattices. Figure 4.30 shows a pure edge dislocation with Burgers vector of (ao/2) [110] lying on (110) slip plane.

4.4

A steel specimen was subjected to two fatigue tests at two different stress ranges of 400 and 250 MPa and failure occurred after 2 x 104 and 1.2 x 106 cycles, respec­tively. Determine the fatigue life at 300 MPa stress range.

Solution

First find p and C in Basquin equation (Eq. (5.67)) and use them to find N for 300 MPa range to be 2.54 x 105 cycles.

Example Problem

A steel with an yield strength (Sy) of 30 000psi, tensile strength (Su) of 50 000psi, and true fracture strain of 0.3 is to be used under cyclic loading at a constant strain range. Evaluate the limit on the total cyclic strain range if the steel is to withstand

4.9 x 105 cycles; E = 30 x 106 psi.

Solution

Let us use the universal slopes equation: Ae = 3.5 —u Nf 012 + e°’6Nf 06.

E

In strain-controlled fatigue tests for life evaluation, it may be noted that the cyclic stress-strain curve leads to a hysterics loop, as depicted in Figure 5.47a where O-A-B is the initial loading curve and on unloading the yielding occurs at lower stress (point C compared to A), which is known as Bauschinger effect. The material may undergo cyclic hardening or softening and in rare cases it remains stable (Figure 5.47b) and this behavior depends on the initial metallurgical condition of the material. According to Figure 5.47b, cyclic hardening leads to decreasing peak strains, while the peak strains increase in the case of cyclic softening with increas­ing number of cycles. In general, the hysteresis loop stabilizes after about 100 cycles and the stress-strain curve obtained from cyclic loading will be different from that of monotonic loading (Figure 5.47c), but the stress-strain follows a power-law relationship similar to that in monotonic loading,

До = K'(As)n’, (5.72)

where the cyclic hardening coefficient n0 ranges from 0.1 to 0.2 for many metals and is given by the ratio of the parameters (b/c) (Eq. (5.70b)). In some cases, fatigue ratcheting occurs with a resulting increase in strain as a function of time when tested under a constant strain range that is often referred to as cyclic creep. In a stress-controlled test with nonzero mean stress, the shift in the hysteresis loop along the strain axis, as depicted in Figure 5.47d, is attributed to thermally activated

dislocation movement at stresses well below the yield stress and/or due to disloca­tion pileup resulting in stress enhancement. Fatigue ratcheting may also occur in the presence ofresidual stress and in cases where microstructural inhomogeneities exist such as in welded joints.

5.1.7.2 Miner’s Rule

In real situations, stresses change at random frequencies and, in general, the percent­age oflife consumed in one cyclic loading depends on the magnitude of stress in subsequent cycles. However, the linear cumulative damage rule, known as Miner’s rule, assumes that the total life of a component can be estimated by adding up the life fraction consumed by each of the loading cycles. If N is the number of cycles to fail­ure at ith cyclic loading and ni is the number of cycles experienced by the structure,

— = 1.

N;

Miner’s rule is too simplistic and fails to predict the life when notches are present. Furthermore, it fails to predict the life when mean stress and temperature are high enough or cyclic frequency is low where creep deformation dominates over fatigue loading. It turns out that many materials exhibit deviations from this linear addition depending on whether it is cyclically hardening or softening. In particular, the pre­dictions tend to be highly nonconservative for cyclically softening materials. How­ever, it is a very useful rule in fatigue life prediction and has pedagogic advantages.

5.1.7.3 Crack Growth

Crack growth tests are generally performed under stress control tests with r = 0 using CT specimens with surface notches, while fatigue life tests are performed

under r =—1 condition with smooth specimens. Crack propagation generally occurs in two stages (Figure 5.48):

Stage-1 Propagation: After initiation of a stable crack, the crack then propagates through the material slowly along crystallographic planes of high shear stresses. The fatigue fracture surface has a flat and featureless appearance during this stage. Stage-2 Propagation: Crack extension rate increases dramatically. At this point, there is also a change in propagation direction to one that is roughly perpendicu­lar to the applied tensile stress. Crack growth occurs through repeated plastic blunting and sharpening. After reaching a critical crack dimension that initiates the final failure step, catastrophic failure starts. The fracture surface may be char­acterized by two markings: beach marks/clamshell marks (macroscopic), as shown in Figure 5.49a schematically, and striations (microscopic), as illustrated by a SEM image in Figure 5.49b. The presence ofbeach marks and/or striations on a fracture surface confirms that the cause of failure is fatigue. However, the absence of either or both does not exclude fatigue as a cause of fatigue-type failure.

5-1-7-4 Paris Law

The kinetics of crack growth in the subcritical stage can be experimentally deter­mined. One of the commonly used empirical equations correlates the crack growth rate per cycle, da/dN, with the range of the stress intensity factor. The intermediate region of crack growth is governed by what is known as Paris law:

da = A(DK)m, (5.74)

where A and m (Paris exponent) depend on the particular material, as well as on environment, frequency, and the stress ratio, and DK is the range ofstress intensity factors [=До (pa)]. Figure 5.50 shows a log-log plot of da/dN versus ДК. For tension-tension loading (with minimum stress being more than zero), the range

of both K and maximum K are important. However, for tension-compression-type loading, only maximum K is relevant as no crack growth takes place under com­pressive part of the fatigue cycle. In stage-II known as Paris region, cracks propa­gate in stable manner exhibiting fatigue striations until a critical crack length (ac) is reached at the beginning of the stage-III during which unstable crack growth takes place with eventual fracture.

Paris law makes it possible to estimate the number of fatigue cycles (Nf) and fatigue life:

Thus, knowing Of along with the parameters in Paris law, one can calculate the number of cycles to failure. If the frequency (f) of cycling is known, the time to failure (tf) can be obtained:

t _Nf

tf — T.

The Paris exponent m has values from 2 to 4, and Eq. (5.67) will have a special case for m 2.

Stress intensity factor range, Да (log scale)

Figure 5.50 The crack growth rate during fatigue as a function of the stress intensity factor range shows a threshold stress intensity factor, a regime of subcritical crack growth, and a fast fracture regime.

Example Problem

A mild steel plate (Kic — 40MPam1=2) contains an initial through thickness edge crack of 0.5 mm and was subjected to uniaxial fatigue loading: Smax — 180 MPa and R — 0. If the material follows the Paris crack-growth relation, (da/dN) (m per cycle) — 5.185 x 10-13 (DK)2, with K in MPam1/2. Evaluate the number of cycles required to break the plate (assume Y — 1)?

Solution

da/dN — 5.185 x 10~13(DK)2, where DK — Yamsxs/Pa, da/dN = 5.185 x 10-13amaxPa.

Thus,

where a0 — 0.5 mm and af — KjC/na1msxL — px0802 — 15.72 mm, so Nf — 6.534 x 107 cycles.

Another important aspect of considering the crack growth versus DK is to exam­ine the effects of superimposed environment such as corrosion and radiation. The variation of da/dN with DK in these cases would be to shift the threshold stress intensity range to lower values and the critical crack length at fracture is now indi­cated by KISCC instead of KIC.

We have already discussed different ways in which a neutron can interact with nuclides (or specifically nuclei). This is indeed a probabilistic event that depends on the energy of the incident neutrons and the type of nuclei involved in the inter­action. Therefore, one can define this probability of interactions in terms of cross section that is a measure of the degree to which a particular material will interact with neutrons of a particular energy. But remember that the neutron cross section for a particular element has nothing to do with the actual physical size of the atoms. The range of neutrons (the distance traveled by the neutron before being stopped) is a function of the neutron energy (recall the classification based on neutron energy) as well as the capture cross section of the medium/material through which the neutrons traverse.

To understand it easily, one may consider a simple case shown in Figure 1.2 with a beam of neutrons impinging on a material of unit area (in cm2) and thickness x (in cm). Thus, the intensity of neutrons traveling beyond the material will be diminished depending on the number of nuclei per unit volume of the material (n0) and the “effective area of obstruction” (in cm2) presented by a single nucleus. This area of obstruction is generally called “microscopic cross section” (o) of the material. Like any other absorption equation, one can write

I — I0 exp (—n’ox), (1-5)

where the quantity n’o is called the macroscopic cross section or obstruction coefficient 22 (unit in cm-1). This represents the overall effect of nuclei (n0) in the neuron beam path and the power of the nuclei to take part in the interaction. Equation (1.5) is specifically for only one type of reaction when the absorber mate­rial contains only one type of pure nuclide. But actually the material could consist of several types of nuclei, and in that case, we should add all the neutron cross sections for all possible reactions to obtain the total neutron cross section.

Figure 1.2 Attenuation of an incident neutron beam of intensity I0 by an absorber material.

From Eq. (1.5), a half-value thickness (x1j2) for neutron beam attenuation can be derived using the following relation:

x1/2 = 0.693/(П ■ a). (1.6)

The values of neutron microscopic cross section (a) are typically between 10~22 and 10~26cm2, leading to the development of a convenient unit called barn (1 barn = 10~24cm2).

Note: Macroscopic neutron cross section Q^) can be calculated with the knowl­edge of П that can be calculated from the following relation:

П = (q/M) x (6.023 x 1023), (1.7)

where q is the density (g cm~3) and M is the atomic weight of the element.

Thus, ^ = (q/M) x (6.023 x 1023)a. (1.8)

In order of increasing cross section for absorption of thermal neutrons, various metals can be classified as follows (normalized to Be):

Be 1, Mg 7, Zr 20, Al 24, Nb 122, Mo 278, Fe 281, Cr 322, Cu 410, and Ni 512

We will see in later chapters that neutron capture cross section has a significant role to play in the selection of reactor materials coupled with other considerations. See Table 1.1 for some representative values of neutron capture cross sections for several important nuclides.

Most nuclides exhibit both the 1/v (v = velocity of neutron) dependence of neu­tron cross section and the resonance effects over the entire possible neutron energy spectrum. We should not forget that the neutron cross sections heavily depend on the type of reactions they take part in, such as alpha particle producing reaction (aa), fission reactions (af), neutron capture cross sections (ac), and so on. As dis­cussed above, the total cross section (at) is a linear summation of all neutronic reactions possible at the specific neutron energy level.

The inverse proportionality of total neutron cross section in elemental boron as a function of neutron energy is shown in Figure 1.3. However, this is not always the case. Many nuclides show abrupt increases in neutron cross section at certain narrow energy ranges due to resonance effects, which happens when the energy of the incident neutron corresponds to the quantum state of the excited compound nucleus. An example of such a situation for Mn-55 is shown in Figure 1.4.

10 I 1 Overview of Nuclear Reactor Systems and Fundamentals

1.7.1

Many of the nuclear fuels are in the form of ceramics. Thus, understanding the crystal structure of ceramics is of paramount importance. Ceramics gener­ally have the ionic bonding between their lattice entities. An ionic solid con­tains two or more ionic species (positive and negative). The crystal structures of an ionic solid can generally be described as comprising two or more inter­mingling simpler lattices (called sublattices). The stoichiometry of ionic solids is influenced by the fact that the nearest neighbors of a particular ion should be the ions with the opposite charge in order to maximize the Coulomb energy (attractive) associated with the crystal structure and minimize the repulsive energy that may destabilize the structure. This means a positive ion prefers to be surrounded by negative ions, and a negative ion needs to be surrounded by positive ions as their first nearest neighbors. The cations (+ive ions) also tend to keep the maximum separation between the other cations that are their second

nearest neighbors and vice versa. Usually, the larger of the ions would create an FCC or HCP ion array, and the corresponding interstitial sites are occupied in a regular manner by the opposite type of ions. However, any such arrangement must conform to the rule of local charge neutrality, which can be extended to the entire crystal and still maintains the stoichiometry (i. e., cation/anion ratio) of the crystal. Although some ceramic structures are very complex, we shall limit our discussion to a few common, simpler structures, some of which are of nuclear importance. The crystal structure of a ceramic is generally named after a common compound that shows that particular structure.