chapter 4 point defects in solids - penn state engineering ... defects aug09.pdf · chapter 4 point...

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Chapter 4 Point Defects in Solids 4.1. Classification of Point Defects.............................................................................. 2 4.2. Equilibrium Concentrations of Point Defects in Elemental Crystals ................. 3 4.3 Point Defects in Ionic Crystals.............................................................................. 8 4.4 Point Defects in Stoichiometric UO2.................................................................... 10 4.5. Effects of Cation Impurities, Multiple cation Valence States, and Nonstoichiometry........................................................................................................ 13

4.5.1. Doping of an MX-type Crystal .......................................................................... 14 4.5.2 Nonstoichiometry in UO2 ............................................................................... 15

Problems ...................................................................................................................... 18 References ................................................................................................................... 21

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4.1. Classification of Point Defects

Contrary to the perfect lattices discussed in the preceding chapter, all real crystals contain defects differentiated according to their dimension. The one-dimensional defect is called a point defect, implying that it involves only one atom surrounded by an otherwise perfect lattice. However, the presence of a point defect may affect the properties of its nearest neighbors, and by elastic interactions, a sizable spherical region of the lattice around the defect.

Two point defects are intrinsic to the material, meaning that they form

spontaneously in the lattice without any external intervention. These two are the vacancy and the self-interstitial, shown schematically in a 2-D representation in the upper panels of Fig. 4.1. The vacancy is simply an atom missing from a lattice site, which would be occupied in a perfect lattice. The self-interstitial is an atom lodged in a position between normal lattice atoms; that is in an interstice. The qualification “self” indicates that the interstitial atom is the same type as the normal lattice atoms.

vacancy self interstitial

substitutionalimpurityatom(e.g.,Sn in Zr)

interstitial impurityatom (small, H, or C)

Fig. 4.1 Point Defects in an Elemental Crystal The lower two panels in Fig. 4.1 show the two basic mechanisms by which a

foreign or impurity atom exists in the crystal lattice of a host element. Large impurity atoms, usually of the same category as the host atoms (e.g., both metals, as nickel in iron), replace the host atoms on regular lattice positions. These are called substitutional impurities. The structure of the lattice is not disturbed, only the identity of the atoms occupying the lattice sites are different. Small atoms that are also chemically dissimilar from the host atoms occupy interstitial positions and do not appreciably distort the surrounding host crystal. They are termed interstitial impurities. Typical examples are carbon in iron and hydrogen in zirconium.


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Aside from their identities relative to the host atoms, the self-interstitial and the interstitial impurity differ in the way that they reside in the lattice. As shown in Fig. 4.2

Fig. 4.2 Interstitials in the bcc lattice. Left: self interstitials; Right: interstitial impurities

using the bcc lattice as an example, the self interstitial, because of its size, displaces a host atom off of its normal lattice position, creating a dumbbell-shaped pair. This configuration is also called a split interstitial. The orientation of the dumbbell and the distance between the two atoms are determined by the condition that potential energy of the lattice be a minimum.

The small interstitial impurity atoms, on the other hand, occupy definite sites without significant distortion of the host lattice. These sites are named after the shape of the polyhedron formed by joining the host atoms surrounding the interstitial. The examples shown in Fig. 4.2 for the bcc lattice are octahedral and tetrahedral sites. These two interstices offer the most space for the impurity atom to reside in, but which site is occupied is a sensitive function of the interaction energy between the impurity atom and the host atom.

4.2. Equilibrium Concentrations of Point Defects in Elemental Crystals


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In Section 2.3 of Chap. 2, it was shown that the criterion of chemical equilibrium is the minimization of the Gibbs free energy of the system at constant temperature and pressure. The system in this case consists of atoms on regular lattice sites and the intrinsic point defects randomly distributed on some lattice sites. Even though there is only one element involved, thermodynamically the system consists of two components, the regular atoms and the empty sites in case of a vacancy or atoms on irregular sites in case of the self interstitial. In thermodynamic analysis, the two types of point defects are treated as independent entities.

The reason that point defects spontaneously form lies in the components of the

Gibbs free energy, which, according to Eq. (2.1) of Chap. 2, contains enthalpy and entropy contributions, G= H – TS. Each of the two properties on the right also contain two terms. The enthalpy change accompanying creation of the point defects is written as H = EPD + pV, where EPD, the dominant contribution, is the energy required to create the point defect from the perfect lattice. It is always positive. The pV term represents the work involved in the change in system volume as the atom is moved between the interior site and the surface. Because of EPD, the enthalpy component acts to make G more positive than that of the perfect lattice, and thus tends to suppress the formation of the point defect.

The two parts of the entropy involved in formation of point defects are expressed

by the equation S = SPD + Smix. The minor component SPD results from the change in the vibrational motion of the atoms around the point defect from that of atoms in the perfect lattice. This term is positive for vacancies and negative for interstitials. The major component of the entropy change accompanying point defect formation is Smix, the entropy of mixing. This term, given by Eq (2.27) of Chap. 2, is a positive quantity that contributes negatively to the system free energy, and thus promotes the formation of the point defects. The reason for this behavior is that Smix is a measure of randomness, and the introduction of point defects into a perfect crystal reduces the system’s state of order. The entropy of mixing is responsible for the spontaneous existence of point defects at equilibrium; the magnitude of the positive energy of point defect formation, EPD, governs the concentration of these species at thermal equilibrium.

Quantitative application of thermodynamics to point defect formation is illustrated

using the vacancy. The system consists of NV vacancies and N atoms. The vacancy, even though it is literally nothing, is a bona fide component in a thermodynamic sense. Together, these two components occupy NS = N + NV lattice sites.

The process to which the thermodynamic properties apply is the creation of a

vacancy by moving an interior atom to the surface of the solid, as shown in Fig. 4.4. The process can be described the equilibrium reaction:

Null = V + Asurf (4.1)

Where “null” denotes the perfect crystal, V is a single vacancy, and Asurf is an atom on the surface of the solid. The equilibrium is maintained because of the equality of the


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formation and removal rates depicted in Fig. 4.4. The components of the free energy equation, G = H – TS, are considered as property differences between the right and left hand sides of Eq (4.2). The enthalpy to create NV vacancies is HV = NVhV, where hV is the

vacancy

formation removal

atom on surface

Fig. 4.3 Vacancy formation

enthalpy of formation of a single vacancy. The creation of a vacancy from a region of perfect lattice involves breaking bonds between the atom to be moved and its neighbors and recouping about half of these in its surface position. The common process of vaporization entails breaking of bonds of surface atoms to form free gas atoms. Based on this simplistic picture, it is understandable that the energy of vacancy formation is found to be roughly equal to the heat of vaporization (per atom).

Similarly, the minor pV and SV terms can be written as pΩNV and sVNV, respectively, where Ω is the volume per atom and sV is the change in entropy associated with the formation of a single vacancy.

The principal contributor to the entropy change is that due to randomly mixing NV

vacancies with N atoms. For this situation, Eq (2.27) of Chap. 2 takes the form:

ln ln Vmix B V

V V

NNS k N NN N N N

⎡ ⎛ ⎞ ⎛= − +⎢ ⎜ ⎟ ⎜+ +⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

⎤⎞⎥⎟ (4.2)

Where k is Boltzmann’s constant. The Gibbs free energy of the system becomes:

G (T) = G(0) + NV(hV + pΩ -ΔTsV) - TΔSmix (4.3) where G(0) is the free energy of a reference perfect lattice with N atoms and no vacancies. Even though the term in parentheses is always positive, the ΔTSmix term provides a negative component to G, effectively assuring that the minimum value of G will be one that has some vacancies present. The equilibrium vacancy concentration is Light Water Reactor Materials, Draft 2006 © Donald Olander and Arthur Motta 8/31/2009 5


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obtained from Eq (4.3) using the general criterion for equilibrium in a system at constant temperature and pressure, namely, that G be a minimum:

0 ln VV V B

V VN

NdG h p Ts k TdN N N

⎛ ⎞= = + Ω− + ⎜ +⎝ ⎠

⎟

The last term on the right in this equation is the derivative of Eq (4.2) with respect to NV, holding N constant. The sum of N and NV is equal to NS, so the parenthetical term is the site fraction of vacancies, denoted by xV. This concentration unit is the solid-state analog of the mole fraction unit that appears in the equilibrium equations in gases and liquids. The above equation is rewritten as:

/ -h / - Ω // V B V B Bs k k T p k TV V Sx N N e e e= = (4.4)

sV is usually approximated as zero, principally because it is unknown for most

elements, and is small for those elements for which it has been measured. The term involving the pressure can be estimated using the following values: p = 100 MPa, Ω = 4x10-29 m3/atom, kB = 1.38x10-23 J/K-atom (8.62x10-5 eV/K) and T = 1200 K. This combination yields pΩ/kT = 0.25, so that this factor in Eq (4.4) is ~ 0.8. For most applications, this factor is sufficiently close to unity to be ignored. There are, however, phenomena in which this term in Eq (4.4) is essential to a correct description of the process. Barring such situations, a common approximation to Eq (4.4) is:

-h /V Bk TVx e= (4.5)

Example: The formation energy of vacancies in copper is 100 kJ/mole (about 1 eV/atom). At 1300 K (which is just below the melting point) what fraction of the lattice sites are empty? Using these values in Eq (4.5), the site fraction of vacancies is xV = 10-4. This value is too small to influence properties such as the density, but even smaller values of xV are critical in determining the transport property of self-diffusion (see following chapter). The process analogous to that shown in Fig. 4.3 for forming self-interstitials in elemental crystals is shown in Fig. 4.4

split interstitial

formation

atom on surface

removal

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Fig. 4.4 Formation of self interstitials

The formation of self-interstitials is totally independent of vacancy creation. However, the thermodynamic analysis is formally identical to that described above for vacancies. Aside from changes in the sign of the entropy si compared to sV and of the pΩ term, the equilibrium concentration (site fraction) of self interstitials is given in approximate form by the following:

-h /i Bk Tix e= (4.6)

For copper, the interstitial formation enthalpy is hi ~ 300 kJ./mole (~3eV/atom), which is about three times larger than the energy required to create a vacancy in this metal. As a result, the equilibrium concentration of interstitials is very much smaller than that of vacancies (by 8 orders of magnitude at 1300 K). This is true of all elements. Thermally generated interstitials can usually be neglected in most applications. However, in the presence of high-energy radiation, the two types of point defects are created at equal rates, and interstitials cannot be ignored. An approximate but extremely simple method for treating point defect equilibria is afforded by regarding the process as a pseudo chemical reaction and directly utilizing the well-known theory of chemical reaction equilibria. The “reaction” that produces vacancies is given by Eq (4.1). The law of mass action expresses the equilibrium constant for this reaction by:

latticeperfectinatomsofactivityatomssurfaceofactivityvacanciesofactivityK V

×=

Because atoms in the undisturbed lattice and on the surface are at concentrations much greater than that of the point defect, they are undisturbed by the formation of the latter. Consequently, their activities can be taken to be unity. The activity of the vacancies, however, is equal to its site fraction xV. This choice is based on the results of the previous exact analysis that produced Eq (4.4). The other feature of chemical reaction theory that is adapted for the vacancy formation process is the relation between the equilibrium constant and the free energy change of the reaction (see Eq (1.58 of Chap. 1). The Gibbs free energy change for forming a single vacancy, gV, is equal to the sum of the three terms in parentheses in Eq (4.3). Applying these adaptations of chemical reaction equilibrium theory to the process of vacancy formation yields:

/ / h / Ω /V V B V B Bg kT s k k T p k TV VK x e e e e− − −= = = (4.4a)

which is identical to Eq (4.4) obtained by the exact method. A similar application of chemical equilibrium theory applies to self-interstitials as well.


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4.3 Point Defects in Ionic Crystals Self interstitials and vacancies occur naturally in ionic crystals as well as in elemental solids. However, because the cations and anions carry electrical charges, vacancy and interstitial formation are not independent processes. To create a vacancy on the anion sublattice by moving the anion to the surface, for example, would leave the surface negatively charged and the interior around the vacancy with a net positive charge. This violation of local electrical neutrality precludes such a process. Similar arguments apply to cation vacancies or self interstitials of either ionic type. Point defects that preserve local electrical neutrality are shown in Fig. 4.5 in a

Fig. 4.5 Point defects in an MX-type ionic crystal

simplified two-dimensional representation. For simplicity, an MX type ionic solid is depicted. The Schottky defect involves simultaneous movement of a cation and an


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anion to the surface. In an MX2 type crystal, two anion vacancies would have to be created for each cation vacancy. The other important point defect in ionic crystals is called a Frenkel defect. It can occur either on the cation sublattice or on the anion sublattice. As shown in Fig. 4.5, an ion is moved from a regular lattice site to an nearby interstitial site, thereby maintaining local electrical neutrality* Schottky and Frenkel defects are created independently. In any particular crystal, one will dominate, while the other will either be absent or a minor contributor. In UO2, for example, the dominant defect is anion Frenkel, but Schottky defecting occurs to a lesser extent. The chemical equilibrium treatment of the thermodynamics of defect formation in ionic crystals is a straightforward extension of the method applied to elemental solids. The major advantage of this simplified approach is the avoidance of the complication of calculating the entropy of mixing for a two-component system with its associated point defects. The reaction producing a Schottky pair is:

Null = VM + VX + ions on surface (4.7)

Where null denotes the perfect crystal and VM and VX are vacancies on the cation and anion lattice sites, respectively. Since the activities of the ions in the perfect crystal and on the surface are both unity, the law of mass action for Eq (4.7) is:

kT/k/sVMVXS eexxK S Sε−== (4.8)

KS is the Schottky equilibrium constant expressed in terms of the entropy sS and energy εS of formation of a Schottky pair. The pressure-volume term that appears in Eq (4.4) for elemental crystals has been neglected for simplicity in Eq (4.8). The vacancy site fractions are defined in terms of the numbers of defects and numbers of regular sites: NVX = number of vacant anion sites NSX = number of anion sites NVM = number of vacant cation sites NSM = number of cation sites So that:

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* The interstitial ion that charge-compensates a nearby vacant site need not have originated from that site. All that is required is a distribution of vacancies and interstitials of the same type so as to maintain local electrical neutrality.

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xVX = NVX/NSX and xVM = NVM/NSM (4.9)

For an MX-type crystal, NSX = NSM, so that charge neutrality is expressed by:

NVX = NVM or xVX = xVM (4.10)

Combining Eqs (4.8) and (4.10) yields the individual vacancy fractions:

SVMVX Kxx == (4.11) Anion Frenkel defects are produced by the reaction:

Null = VX + IX (4.12)

where IX is an interstitial anion. The law of mass action for this reaction is:

kT/k/sIXVXFX

FXFX eexxK ε−== (4.13)

where the subscript FX denotes Frenkel defects on the anion sublattice. The condition of charge neutrality is:

NVX = NIX (4.14)

This condition must first be converted to site fractions before combining with Eq (4.13). The site fraction of vacancies, xVX, is the same as in Eq (4.9). The site fraction of interstitials, however, depends on the number of anion interstitial sites:

xIX = NIX/NSIX (4.15) where NIX = number of anion interstitials

NSIX = number of anion interstitial sites

When counted on the same basis (say per unit volume, or per unit cell), the numbers of anion interstitial sites need not be equal to the number of regular anion sites. However, assuming for simplicity that NSIX = NSX, Eq (4.14) becomes:

xVX = xIX (4.16)

Substituting Eq (4.16) into Eq (4.13) gives the final result:

FXIXVX Kxx == (4.17) An entirely analogous derivation applies to cation Frenkel defects. 4.4 Point Defects in Stoichiometric UO2


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Uranium dioxide is a complicated material insofar as point defect formation is

concerned. First, the uranium cation can assume oxidation states of 3+, 4+, 5+ and 6+. The mixture of oxidation states depends upon the prevailing oxygen pressure. Thus, the nature and concentrations of the point defect in UO2 depends on a gas-solid equilibrium as well as on the point defect equilibria treated in the previous two sections. In addition, UO2 as a reactor fuel is never a pure material; as a result of the fission process, sites on the uranium sublattice contain fission products of a multiplicity of valences, ranging from Ba2+, La3+, Zr4+ to Nb5+.

In this section, we analyze point defect production in uranium dioxide with three

important restrictions: first, all uranium atoms are in their IV oxidation state; second, no impurities are present on the cation (or anion) sublattice; third, the oxide is exactly stoichiometric, meaning that the O/U atom ratio is 2.

Extensive research has demonstrated that the principal defect in UO2 is the Frenkel

anion defect of the type shown generically in Fig. 4.5. Understanding of the defect type goes further than simply this identification. The unit cell of the perfect UO2 lattice is shown in Fig. 2.10. While there is no ambiguity concerning the anion vacancy (it is simply a missing oxygen ion), the nature of the oxygen interstitial ions is more complicated. Neutron diffraction studies have shown that oxygen interstitial ions occupy the unit cell in pairs, as shown in Fig. 4.6.

Fig. 4.6 Location of anion (oxygen) interstitials in UO2

The two oxygen interstitial ions added to the unit cell are labeled “1” in the diagram. They are disposed along a [110] direction. To reduce the repulsive Coulomb interaction between the interstitial pair and the nearby corner O2- on normal lattice sites, the latter Light Water Reactor Materials, Draft 2006 © Donald Olander and Arthur Motta 8/31/2009 11

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(labeled “2” in the diagram) are pushed off in [111] directions. The separation distances between the “1” interstitials and the “2” original anions from their normal lattice sites are determined by minimizing the energy of the configuration.

Even though UO2 is an MX2-type crystal, the anion Frenkel defect formation reaction is still given by Eq (4.12) with the equilibrium constant given by Eq (4.13). For the restrictions placed on this process (pure UO2 with an O/U ratio of exactly 2), the electrical neutrality condition is given by Eq (4.14). Where the UO2 analysis differs from the simple anion Frenkel analysis of the preceding section is in the relation between the numbers of point defects and sites to the site fractions that are required in Eq (4.13). Replacing the anion identification X by that for oxygen, O, the mass action law for the anion Frenkel equilibrium can be written as:

SO

VO

SIO

IOVOIOFO N

NNN

xxK •=•=

where the defect and site numbers on the right have been defined in the text above Eq (4.9) and above Eq (4.15). The numbers in the denominators of the above equation are to be related to the number of cation (uranium ion) sites, NSU. From the stoichiometry of UO2, NSO = 2 NSU. Examination of Fig. 4.6 shows that the oxygen interstitial pairs can be located on any of the 12 edges of the small cube. Since there are 4 cations in the unit cell, NSIO = 3 NSU. Using these ratios to convert the denominators of the above equation to NSU and taking account of the electrical neutrality condition of Eq (4.14) yields the following result:

FO2SU

VO

SU

IO K6UOofmole

pairsdefectofmolessitecationpairsdefect

NN

NN

==== (4.18)

Example: The properties of the anion Frenkel defect in UO2 are: sFO = 63 J/mole-K and εFO = 297 kJ/mole. What is the concentration of anion Frenkel pairs at 2000K? KFO is computed from Eq (4.13) using the gas constant R = 8.314 J/mole-K instead of Boltzmann’s constant. The result is KFO = 4.4x10-5. From Eq (4.18), the concentration of defect pairs is:

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2104.1104.36

UOofmolepairsdefectofmoles −− ×=××=

As will be seen in the next chapter, the anion Frenkel defects greatly influence the diffusivity of oxygen ions in UO2. The secondary defecting process in UO2 is the Schottky process. This defect produces vacancies on the cation sublattice, and is responsible for the diffusion coefficient of uranium ions in UO2.

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Another property affected by the creation of point defects in ionic crystals is the heat capacity. At relatively low temperatures (but not approaching absolute zero), the heat capacity is nearly temperature-independent with a value of 3R per gram atom (R is the gas constant). Per mole of UO2, this classical value of the heat capacity is (CV)lattice =

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9R. This contribution to the heat capacity arises from the vibrations of the atoms in the lattice. The creation of point defects by the anion Frenkel process provides an additional component to the heat capacity, which can be expressed by:

CV = (CV)lattice + (CV)defects (4.19)

The extra energy in the solid provided by the presence of the anion Frenkel defects is the concentration given by Eq (4.18) times the energy to produce a mole of defects, which is εFO (expressed in molar units). The excess energy due to the point defects is:

eex = εFO FOK6

The excess heat capacity arising from the point defects is the derivative of eex with respect to temperature, or, using Eq (4.13) for KFO with k replaced by R:

( ) RT2/R2/s2

FOexdefectV

FOFO eeRT

R26

dTde

C ε−⎟⎠

⎞⎜⎝

⎛ ε== (4.20)

Example: What is the contribution of anion Frenkel defects to the heat capacity of UO2 at 2000 K? Using the values of sFO and εFO given in the previous example, the above equation gives: (CV)defect =2.2R. This is nearly 25% of the classical value of 9R, and is an important contribution to CV. 4.5. Effects of Cation Impurities, Multiple cation Valence States, and Nonstoichiometry

Thermodynamic computation of point defect concentrations becomes more complicated than the examples treated in the previous two sections if one or more of the following factors arise: • Substitutional impurity cations with a valence different from the host cation are

introduced into the cation sublattice. Examples are: i) doping of NaCl with small amounts of CaCl2. Ca2+ replaces Na+ on the cation sublattice. ii) Introduction of trivalent rare earth cations into the uranium sublattice as a result of uranium fission

• The host cation has several accessible valence states. Sodium in ionic solids always forms Na+, but uranium in the oxide form can exist in several oxidation states, principally U3+, U4+, and U5+.

• The ionic compound can deviate from exact stoichiometry. For instance, instead of stoichiometric MX, the solid can form MX1±y, where y < 1. Such deviations from exact stoichiometry require multiple cation valence states.

• Two types of defect processes occur simultaneously (see Problem 4.2).


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4.5.1. Doping of an MX-type Crystal

We treat the first of these complications without considering the last three. The case considered is an MX-type crystal where both M and X are univalent. A divalent dopant D2+ replaces M+ on a fraction xD of the cation lattice sites. The effect of doping is superimposed in the normal defecting process, which is assumed to be of the Schottky type (Eq (4.7)). Because of the presence of the dopant, the electrical neutrality condition is no longer given by Eq (4.10). Instead, we must first proceed through several formal steps relating the numbers of species in the solid. The first is the lattice structure condition, which relates the numbers of cation and anion sites. For the MX-type crystal, this condition is:

NSM = NSX (4.21)

Next are the site-filling conditions, which account for all species present on

normal lattice sites: For the cation sites: NSM = NM + ND + NVM (4.22)

For the anion sites: NSX = NX + NVX (4.23)

Where NM = number of host cations on regular cation sites NX = number of host anions on regular anion sites The condition of electrical neutrality takes the form:

2ND + NM = NX (4.24)

NM and NX are first eliminated from Eq (4.24) using Eqs (4.22) and (4.23) and the result converted to site fractions by division by Eq (4.21) to yield:

xVX = xVM - xD (4.25) This equation is combined with the mass action law for Schottky defects, Eq (4.8),

and solved for the cation vacancy fraction:

⎟⎠⎞

⎜⎝⎛ ++= S

2DDVM K4xx

21x (4.26)

The anion vacancy site fraction follows from Eq (4.8): xVX = KS/xVM (4.27)


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Equations (4.26) and (4.27) are graphed in Fig. 4.7. As the dopant concentration approaches zero, the vacancy fractions approach the pure-material or intrinsic limit given by Eq (4.11). At the opposite, or extrinsic, limit, the Schottky defect equilibrium is negligible and xVM = xD. This is simply the charge balance when no anion vacancies are present.

Fig 4.7 Vacancy site fractions for Schottky defecting in a doped crystal

4.5.2 Nonstoichiometry in UO2 When the atomic ratio of anions to cations is a noninteger value, the compound is said to be nonstoichiometric. In uranium oxide, UO2+y is termed hyperstoichiometric while UO2-y is hypostoichiometric. In either case, the stoichiometry deviation parameter y is generally << 1. The value of this parameter is dependent on the temperature and ambient oxygen pressure, and is determined by the thermochemistry of the oxide of uranium (see Chap. ). The degree of nonstoichiometry affects the concentrations of point defects and consequently the macroscopic properties dependent on them. The method of calculating the concentrations of anion Frenkel defects in hyperstoichiometric urania is outlined below. The equilibrium constant KFO and the nonstoichiometry parameter y are assumed to be specified. There are two structural relationships: Regular lattice sites: NSO = 2NSU (4.28)

Anion interstitial sites: NSIO = 3NSU (4.29)

Site filling of the regular lattice sites: Cation: NSU = N4U + N5U (4.30)

Anion: NSO = NO + NVO (4.31)


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Where N4U and N5U are the numbers of uranium ions in the 4+ and 5+ valence states, respectively. The excess oxygen in the lattice implied by the nonstoichiometry parameter y requires partial oxidation of the original tetravalent uranium ions. Electrical neutrality is expressed by:

4N4U + 5N5U = 2NO + 2 NIO (4.32) Note that both regular anions and interstitial anions must be included in the charge balance. The specified oxygen-to-uranium ratio (O/U) is related to the numbers of ions by:

SU

IOO

U5U4

IOON

NNNNNNy2

UO +

=++

=+= (4.33)

The anion Frenkel mass action law is:

SO

VO

SIO

IOFO N

NNNK = (4.34)

In Eq (4.33), eliminate NO using Eq (4.31) wherein NSO is expressed in terms of NSU by Eq (4.28). This yields:

SU

VOION

NNy −= (4.33a)

Let f = fraction of uranium ions in the 5+ valence state. Then N4U = (1-f)NSU and N5U = fNSU. Using this in Eq (4.32) along with the method used above for expressing NO reduces the electrical neutrality condition to:

f = 2y (4.32a)

Finally, eliminating NVO from the equilibrium of Eq (4.34) using Eq (4.33a) and expressing the regular and anion interstitial sites in terms of the uranium sites via Eqs (4.28) and (4.29) yields a quadratic equation that can be solved for the anion interstitial concentration:

2K24yy

)y(gNN FO

2

SU

IO ++== (4.35)

This result can be used to determine the site fractions of anion interstitials and anion vacancies:

)y(g31

NNxSIO

IOIO == (4.36a)


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[ y)y(g21

NNx

SO

VOVO −== ] (4.36b)

The oxygen interstitial and vacancy concentrations are tabulated in Table 4.1 for a temperature of 2000 K at which KFO = 4.4x10-5.

y xIO x 103 xVO x 103 0 4.8 7.1 0.01 6.7 5.1 0.05 18.0 1.9

Table 4.1 Variation of Frenkel defect concentrations in uranium dioxide with stoichiometry deviation at 2000 K

The values in the first row are equivalent to the defect concentration in exactly

stoichiometric UO2 calculated in Sect. 4.4. At y = 0.01, the intrinsic defects produced by the anion Frenkel equilibrium dominate the point defect population, but the excess oxygen in the lattice exerts a noticeable effect. At y = 0.05, the relatively high concentration of excess oxygen causes a significant imbalance in the concentrations of the two types of point defects; the defects are basically extrinsic and the thermally-generated contribution is small


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Problems 4.1. A substance with the alumina-type(M2X3) crystal structure forms Schottky defects with an equilibrium constant Ks. This material is doped with a cation impurity having a valence of +2. The fraction of the cation sites containing the impurity ion is xDM. Derive the equation for the equilibrium fraction of vacancies on the anion sublattice in the doped crystal. Solve this equation for ss = 4R, εx = 4 eV, T = 1800 K, and xDM = 10-2. Compare this result to the anion vacancy fraction in the undoped crystal and physically explain the difference. 4.2 (a) Anion Frenkel and Schottky defects exist in comparable amounts in an M2X type crystal. The anion X is divalent and the cation M is univalent. The defect equilibrium constants are KFX and KS, respectively. The number of anion interstitial sites is equal to the number of regular anion sites. What fraction of the anion and cation lattice sites are vacant?

(b) Repeat part (a) for an MX type crystal. 4.3 An M2X type Crystal forms Schottky defects with equilibrium constant KS. The crystal is doped with a cation site fraction xDM of an impurity of the DX type. The common anion has a valence of -2. Derive the equation giving the cation vacancy fraction xVM. 4.4 The energy stored in graphite as displacements(i.e., vacancy-interstitial pairs, or Frenkel pairs) produced by low-temperature irradiation can be released if the temperature is raised beyond a critical point. The phenomenon is called “Wigner release” after the physicist who first predicted its existence. The phenomenon was responsible for overheating of an early British reactor at Windscale, UK. The magnitude of the stored energy, Q J/g, for irradiation of a graphite specimen to a particular fast-neutron fluence is measured in the following experiment. Ten g of irradiated graphite are placed in a furnace and held at 200oC and the temperature is monitored as a function of time. the upper curve on the graph shows the temperature history. The sample is then cooled to room temperature and the anneal repeated. The lower curve shows the time-temperature behavior of the second anneal. There is no stored energy to be released in the second anneal, so the lower curve represents simple heat transfer between the sample and the furnace. The rate of heat addition is found to be represented by the function 0.44(200-T) J/min, where T is in oC. The temperature at which the two curves diverge is 128oC and the maximum temperature achieved in the first anneal is 270oC. The area between the two curves is 4200 oC-min. The heat capacity of graphite is 1.26 J/g-oC.


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(a) Determine the value of the stored energy per gram of graphite. Hint: the rate of release of stored energy is written as a temperature-dependent function q(T), which can be used in developing equations from which Q can be determined. (b) If the stored energy is due to recombination of radiation-produced Frenkel pairs, what was the atomic fraction of these pairs prior to annealing. the formation energies of vacancies and interstitials in graphite are 500 and 1340 kJ/moles of defect, respectively.

The figure above is the time vs. temperature plot of graphite placed in a furnace held at 200oC. Upper curve; sample previously irradiated at 55oC. Bottom curve: same sample after annealing. (after G.J.Dienes and G.H. Vineyard, Radiation Effects in Solids, p.100, Wiley-Interscience, Inc., New York, 1957) 4.5 In addition to single vacancies, a small concentration of divacancies (denoted by V2) is present naturally in elemental solids. The equilibrium concentration of V2 is maintained by formation from single vacancies (V) and thermal decomposition of the divacancies. The divacancy has a negative formation energy with respect to the single vacancies of which it is composed (i.e., εV2 < 0). The entropy of formation of the divacancy from single vacancies is sV2.


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Using the “chemical reaction” equilibrium approach (notes, pp. 5 and 6): (a) What is the “chemical reaction” that describes formation of the divacancies

from single vacancies? (b) Write the law of mass action that relates the equilibrium constant KV2 for the

reaction of part (a) to the site-fraction concentrations xV of single vacancies and xV2 of divacancies.

(c) Express the equilibrium constant KV2 in terms of εV2 and sV2. (d) Is sV2 positive or negative? Give reasons (e) Does xV2 increase or decrease as the temperature is increased? Show why

with appropriate equations. 4.6 An ionic crystal of the MX2 type forms Schottky defects, for which the energy of formation is 200 kJ/mole. By how much should the temperature be reduced from 2000K in order to reduce the concentration of anion vacancies by a factor of 10? 4.7 The vacancy formation enthalpy of a metal is 1.5 eV. The material is under a pressure of 850 MPa, and the lattice parameter of this simple cubic metal is 0.286 nm. Make a plot of the equilibrium vacancy concentration versus the inverse of absolute temperature (K-1) (a) for the case where no pressure is applied and (b) for the case where the pressure above is applied. How much does the vacancy concentration change at 1500 K when pressure is applied?


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References 1. R. Szwarc, J. Phys. Chem. Solids, 30 (1969) 705



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chapter 4 point defects in solids - penn state engineering ... defects aug09.pdf · chapter 4 point...

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