1. Lecture Notes II Defect Chemistry
Ole Toft Sørensen
(Risoe National Laboratory)
Ceramic Materials Consultant

2. Electroceramics Electrical properties determined by defects
Knowledge of defect chemistry necessary to
understand Electroceramics!
Properties of electroceramics like diffusion and electrical properties like electronic and ionic conduction are highly dependent on the type of defects and their concentration. To understand electroceramics it is therefore important t have a prior knowledge of defect chemistry, that is the type of defects typically present and how these are designated, how the concentration of these depend on external conditions as for instance the oxygen pressure and how the electrical conduction depend on these concentrations. In this section we shall discuss these aspects in greater detail.

3. What is a defect?
Fundamental definition: Any deviation from the perfect crystal is a defect!
- Macroscopic defects (porosities, cracks)
Let us first consider whar a defect really is. As shown in the slide the fondamental definition is that any deviation from a perfect lattice is a defect. One type is the macroscopic defects like porosities and cracks, which have an overall negative influence of mechanical as well as electrical properties of a material. In the fabrication of electroceramics these macroscopic defects are diminished as much a possible i n the fabrication of high density and crack free materials.
The defects which directly influence the electrical properties and on which we shall focus here are of two types: the atomic defects and the electronic defects. In he following slides we consider these defects in greater detail.
Atomic defects
Electronic defects

4. Atomic (point) defects in Oxides!
Missing ions:
- oxygen ions,
oxygen vacancies
- cations,
cation vacancies
Important atomic defects or as they also are called, point defects, in oxides are shown in this slide. These comprises missing oxygen and cations, which give rise to vacancies, i.e. oxygen vacancies and cation vacances. The second class of point defects is the substituted cations and the effec of this substitution depend on the charge difference in charge and size between the host cation and the substituting cation.It is however seldom that the anoins are substituted in oxides. Finally there are the ions in nterstitial positions, which are empty in the pure lattice. Both anoins and cations can jump to interstitial positions.
Substituted ions
Interstitiel ions

5. Electronic defects in oxides
Atomic (point) defects – type, properties depend on position !
Electronic defects – type, properties depend on energy levels
available for the electrons
In Lecture Notes I – Electroceramics we have already introduced the electronic defects, the electrons and the positive holes.There we saw that electrons in the crystal are placed in energy levels, the so-called energy band model. The two bands determining the electrical pproperties are the valence band and the coduction band which are separeted with the so-calles forbidden band gap where the electrons are not allowed. We als saw that a positive hole is formed when an electron jump from the valence band across the band gap to the conduction band. A posive hole is thus a missing electron in the valence band wth a relative charge of +1.
Contrary to the point defects, where the type and properties depend on their position in the lattice, the electronic defects, i.e. electrons and positive holes, depend the energy levels available for the electrons which spread out over the whole crystal.
electrons
positive holes

6. Defect notations (Symbol for type)position
subscript
(Symbol for type)position
VO = vacancy on oxygen position
VM = vacancy on metal position, VFe
Oi = interstitial oxygen ion
In order to be able to express the formation of defects in the form of a chemical reaction we need have symbols to describe the defects. The notation used here is the so-called Kröger – Vink notation, introduced by the the two american scientists in 1956.
As shown in the slide a symbol decribes the type, V for vacancy for instance with a subscript added which gives the position in the lattice, it can be an oxygen position, a cation position, an interstitial position or the postion of a substituted ion sitting on the position of a host cation. Some examples are presented in the slide. Study these!
Mi = interstitial cation
YZr = Y-ion on Zr-ion position

7. Charges of defects
Relative charge Charge relative to the charge normally present in the position of the defect
Examples:
ZrZr – relative charge = zero,
but
YZr - relative charge = -1
FeO:
Fe2+ vacancy – rel. ch. = ?
Besides the symbol for the defect and its position it is also important to indicate the charge, or rather the relative charge of the defect. As indicated in the slide we talk always of relative charges, i.e. the charge relative to the charge normally present in the position where he defect is sitting.
Eaxamples are presented in the slide:
Zr4+ sitting on a Zr4+ site has of course a zero relative charge. But if Zr4+is substituted with Y3+ , which is expressed as YZr, then the relative charge becomes -1. Or if Y+3 is substituted with Zr4+ then the relative charge becomes +1.
Study also the example of an Fe2+ vacancy.
-2 of course!
ZrY ?
Rel. ch.+1

8. Charges of oxygen vacancies
Formation of oxygen vacancies:
Oxygen atoms are removed from the crystal
Oxygen ions – how many electron in outer orbital ? 8
Oxygen atoms – how many electrons in outer orbital ? 6
Oxygen vacancy – how many electrons left ? 2
Now that we have seen how to write the symbols and the relative charges for different types of defects, we shall in this slide study 1) how oxygen vacancies are formed, and 2) what relative charges xygen vacancies can have.
First the formation of oxygen vacancies: As explained on the slide, it is oxygen atoms which are removed from the crystal, for instance at low oxygen pressures (reduction). Well it is well known that the oxygen ions have 8 electrons in their outer orbital, valence electrons, which of course give an real charge of -2 (but a relative charge of zero as discussed previously). On the other hand an oxgen atom has only 6 electron in the outer orbital, and removing an oxygen atom will therefore leave two electrons in the oxygen vacancy. This oxygen vacancy will therefore have a relative charge of zero.
As also explained in the slide, the two electrons are only loosely bound and they can therefore relatively migrate to neighbouring ions, preferably cations. The relative charges of oxygen vacancies can therefore be zero, +1 or +2 dependin on whether 2,1 or zero electrons are presnent in thje vacancy.
Rel. Charge ?
Zero!
But these electrons can easily migrate to neighbouring ions forming vacancies with one or zero electrons present.
Thus VO with rel. charges of zero, +1 and +2 can be formed !

9. Relative charges of interstitial ions
O2- ions,Oi ? -2
Cl-1 – ions, Cli, rel. charge = -1
Na+1 – ions, Nai , ? +1
When O2-jumps to an intersitial position with th chage zero, an interstitial oxygen ion naturally will have a relative charge of -2.
Other examples are shown in the slide. Study these!
Zr4+ - ions, Zri, rel. charge = +4

10. Nomenclature: relative charges
Relative charges are indicated by a superscript:
neutral - x
positive charges – black dots
negative charges - apostrophes
Examples:
Neutral: VOx
Positive charges: VO•, VO••
But how do we now indicate the relative charges for the defects? Well this is done by adding a superscript to the symbol. As indicated x means netral (relative charge zero), black dots relative postive charges and apostrophes for relative negative charges.
Examples given on slide. Study these!
Negative charges: VFe″

11. Practise: Nomenclature
Here are some examples which you can use to check wether you have understood the nomcalture of defects correctly. Answers can be found on the notes for slide 12:
Write the symbol and the relative charges for the following defects:
1) Oxygen vacancies:
neutral
single positive charge
double positive charge
2) Metal vacancies in MO:
single negative charge
double negative charge
3) Interstitial oxygen ion
4) Interstitial Cl-1 ion
5) Interstitial Ca2+ion
6) Interstitial Y3+ion
7) Interstitial Zr4+ ion
8) Substituted cations
Zr4+ substituted with Y3+
9) Substituted cations
Y3+ substituted with Zr4+
10) Substituted cations
Mg2+ substituted with Ca2+
11) Substituted anions
Cl ion substituting an oxygen ion
12)Al –and oxygen vacancies in Al2O3
13) Ce- and oxygen vacancies in CeO2
14) Ce – and oxygen vacancies in Ce2O3
15) V –and oxygen vacancy in V2O5

12. Answers to practise Answers: VOx, VO•,VO••
2) VMx, VM′, VM″ 3) Oi″ 4) Cli′ 5) Cai•• 6) Yi••• 7) Zri•••• 8) YZr′ 9) ZrY• 10) CaMgx 11) ClO• 12) VAl′′′, VOx,VO•,VO•• 13) VCe′′′′, VOx,VO•,VO•• 14) VCe′′′′′, VOx,VO•,VO•• 15) VV′′′′′, VOx,VO•,VO••

13. Formation of defects Three typical areactions:
- ”high” temperatures, INTRINSIC DEFECTS
- reaction with surrounding atmosphere
- substitution
Let us now consider how defects are formed. As shown on this slide typical reactions are: 1)intrinsic reactions, i.e. reactions characteristic of the material itself without the influence of foreign substances, at high temperatures where the lattice becomes partially unstable due to thermal vibrations. These defects are also termed as ”thermal defects”; 2) reactions with the surrounding atmosphere, for instance reductions at low oxygen pressure or oxydations at high oxygen pressure, and 3) substitutions with aliovalent cations, i.e. Cations with a different valence than that of the host lattice.

14. Intrinsic defects Pair of defects: Frenkel defect:
cation vacancy and interstitial cation
Anti-Frenkel defect:
oxygen vacancy and interstitial oxygen ion
Schottky defect:
oxygen vacancy and cation vacancy
Let us first look at the intrinsic defects formed at high temperatures. As shown, these defects are generally formed in pairs, which means that the stoichiometric composition is always maintained.
Generally we have three types of defects traditionnaly named after the scientists which originally introduced these defects:
1)Frenkel defects, which are formed when a cation is jumping into an interstitial position leaving behind a cation vacancy. This can be expressed by the reaction in a MO oxide:
MMx = VM″ + Mi•• ;
2) Anti-Frenkel defects, which are formed when an oxygen ion is jumping into an interstitial position leaving beins an oxygen vacancy as expressed by the reaction (MO oxide)
OOx = VO•• + Oi″ ;
3) Schottkey defects, which are formed when both an oygen ion and a cation leaves the lattice, for imstance by evaporation from the surface of the crystal. In this case oxygen vacancy – cations vancancy pairs are formed. The reactions in respectively a MO an a M2O3 oxide are shown on the slide. Please note that stoichiometry is maintained in both reactions.
In all reactions it is also clear that electrical neutrality is stricly obeyed. It is of course not possble create an electrically charged material by these reactions.
Finally it may seen odd to use these designation for defect-pairs but this is nevertheless how they often are designated in the litterature.
MO: OOx + MMx = VO•• + VM″
M2O3: 3OOx + 2MMx = 3VO•• + 2VM″′
Stoichiomtry must be maintained !

15. Defects formed in an reaction with surronding atmosphere.
Reduction
MO2 = MO2-x+ x/2 O2
OOx + 2MMx = VO•• + 2MM′ + 1/2O2
In this slide we shall look at how oxygen vacancies are formed in a reaction (reduction) with the surrounding atmosphere at low oxygen pressure. First we can note that this reduction leads to the formation of a nonstoichiometric compound, an oxygen deficient compound as expressed in the overall reaction shown on top to the right.
As shown earlier (slide 8 ) it is the oxygen atoms which are removed from the lattice, not oxygen ions. This leaves a vanacy behind with two electrons, which creates a neutral oxygen vacancy VOx and which easily are ionized into a single charged vacancy with one electron, VO•, and an empty double charged vacancy, VO••. The electrons liberated from the vacancies are captured by the cations which therefore are reduced from MMx to MM′.
The reaction of formation of double charged oxygen vacancies and reduced cations, called compensating defects, are shown on the slide.
Finally it should be noted that this reaction only can take place in oxides where the cations are easily reduced. The dominating defect in these oxides are therefore oxygen vacancies.
Oxides with cations
easily reduced!

16. Defects formed in a reaction with a surrounding atmosphere - 2
Oxidation
MO + y/2O2 = M1-yO
Note – clusters !
But how will the reaction be in an oxidation in an oxide with easily oxidized cations, in this case a MO oxide.
Well, in this case oxygen atoms from the atmospere are first adsorbed to the surface of the crystal. For the formation of oxygen ions we need two extra electrons for each oxygen atom (see slide 8) which are provided from neighbouring cations which then are oxidized – in these cations a valence electron is removed forming a positive hole in the valence band for these cations. In such reactions it is however also necessary to maintain the ratio between the two types of sites, i.e. the oxygen ion and the cation sites. Empty cation sites, in fact cation vacancies, are therefore formed in the surface in positions adjacent to the adsorbed oxygen ions. By diffusion these surface-vacancies are however quickly filled up with cations from the interior, and the cation vacancies are movng inwards forming defect clusters of different types as well as free cation vacancies. The formation of VM″ is shown on the slide
As shown we can have the following types of defects:
VMx = [VM″ 2MM•]x, where one cation vacancy is attached to two oxidized cations;
VM′ = [VM″ MM•]′,where one cation vacancy is attached to one oidized cation;
VM″ which is an empty cation vacancy.
It is important to note the difference to oxygen vacancies, which can contain two, one or no electrons. In the case of cation vacancies it is nonsense to speak about vacancies containing one or two postive holes, as these always will de present in the valence band of the cations.
Finally it is interesting to note that the dominating defects in oxides which are easlily oxidized are cation vacancies. The general formula for such oxides is therefore M(1-y)O
Cations easily oxidized!

17. Formation of interstitial oxygen ions- Oi
High oxygen pressures !
1/2 O2 + 2MMx = Oi″ + 2MM•
Oxides where cations are easily oxidized – FeO
Oxygen ions can also be formed in interstitial positions at high oxygen pressure . The reaction is shown on this slide. In this case the electrons needed to form oxygen ions are provided from adjacent cations which becomes oxidized as for the formation of cation vacances discussed on the previous slide. He formation of interstitial oxygen ions thus preferentially takes place in osides which are easily oxidized.

18. Defects formed by substitution!
Substitution of cations !
Lower valency:
Higher valency:
ZrO2 doped with CaO:
Y2O3 doped with ZrO2:
CaO(ZrO2) = CaZr″ + VO•• + OOx
2ZrO2(Y2O3) =
2ZrY• + Oi″ + 3OO
Let us finally consider how defects can be formed by substitution of the cations. As shown on this slide the defects formed depend on the valency of the substituting cations relative to the valence of the host cations. If the valencies are different the substituting cations are called aliovalent cations.
If the valence is higher, for instance when ZrO2 is doped with CaO as shown to the left , Ca2+ will replace Zr4+ions forming the defect CaZr″. In order to maintain electrical neutrality it is necessary that a positively charged defect is formed, which in this case is an oxygen vacancy, VO••. Doping zirconia with an oxide with a cation with a lower valence is thus an efficient way to form good oxygen ion conductors as will be discussed later. Another dopant very much used in zirconia is yttria, Y2O3.
To the righ is shown what happens when the dopant has a higher valency, as for instance when yttria is doped with zirconia as discussed here. Well, the defect formed by the substitution will in this case be ZrY• and the compensating defect with an opporsite charge will in this case be interstitial oxygen ions, Oi″.
But what happens when the dopant has the same valency as that of the host lattice, for example in MgO doped with CaO?
No defects are of course formed, but it can have an effect if the two are ions have different sizes. This will create tensions in the lattice which can have an influence on for instance the mechanical properties of the material.
Oxygen vacancies formed to
maintain electrical neutrality !
Same valency?

19. Practice Formation of Defects
1) What is an intrinsic defect?
2) Mention four types of intrinsic defects? Explain how these are formed.
3) What types of defect are typically formed under reducing conditions in a MO-oxide? Write the reactions for their formation?
4) What types of defect are typically formed under oxidizing conditions in a M2O3 oxide? Write the reactions for their formation.
5) What is an interstitial ion? Write the reaction for its formation.
6) What type of defects are typically formed with doping of cations of a lower valency than that of the cations of the host lattice? Write the reactions for their formation.
7) What type of defects are typically formed with doping of cations with a higher valency than that of the cations of the host lattice. Write the reaction or their formation.

20. Answers Practice – Formation of Defects
An intrinsic defect is a defect formed by an intrinsic reactions, i.e. reactions characteristic of the material itself without the influence of foreign substances. Intrinsic defects are formed at high temperatures, and they are ften designated as ”thermal defects”.
The four types are: elctrons excited up in the conduction band forming electron holes in the valence band; Frenkel defects; Anti-Frenkel defects and Schottky defects. The three latter defects consists of defect pairs. Reactions of formation: a) electrons/holes – intrinsic ionization across the forbidden band gap; b) Frenkel defect – cation jumping into interstitial position leaving cation vacancy, MMx = VM″ + Mi•• c) Anti-Frenkel defect – oxygen ion jumping into interstitial position leaving oxygen ion vacancy, OOx = VO•• + Oi″ d) Schottky defect – oxygen vacancy – cation vacancy pairs formed when oxygen ions and cations are removed from the lattice: MO: OOx + MMx = VO•• + VM″ M2O3: 3OOx + 2MMx = 3VO•• + 2VM″′ N.B. Stoichiometry must be maintained.
3) Type of defects formed under reducing conditions – oxygen vancancies: VOx, VO• , VO•• and reduced cations ex MM′ (in MO). Reactions for their formation: a) OOx = VOx + ½ O2 b) OOx + MMx = VO• + MM′ c) OOx + 2MMx = VO•• + 2MM′
4) Type of defects formed under oxidicing conditions in M2O3– cation vacancies and oxidiced cations, VM″′, VM″ = [VM′″ MM•]″ , VM′ = [VM″′ 2MM•]′ and VMx = [VM″′ 3MM•]x Reactions: 3/2 O2 + 6 MMx = 2VM″′ + 6MM• + 3OOx > 2[VM″′ 3MM•]x + 3OOx > 2[VM″′ 2MM•]′ + 3OOx + 2MM• > 2[VM″′ MM•]″ + 3OOx + 4MM•
5) An interstitial ion is an ion sitting on an interstitial site. Both anions and carions can be present in these sites. As these sites are generally empty the real charges will be carried to the interstitial sites. An interstial oxygen ion thus have a relative charge of -2 and a cation a relative charge corresponding ro the charge of the cation sitting in the lattice positions, CaCaX thus becomes Cai••.
Reactions:
Formation of Oi″:
a) ½ O2 +2MMx = Oi″ + 2MM• (Oxidation)
b) OOx = VO•• + Oi″ (Anti-Frenkel)
Formation of Mi•• (MO)
MMx = VM″ + Mi•• (Frenkel)
6) Oxygen vacancies are formed when substitution is done with cations of a lower valence.
Reaction – ZrO2 doped with CaO: CaO(ZrO2) = CaZr″ + VO•• + OOx
7) Oxygen in interstitial positions is formed when substitution is done with cations of higher valences.
Reaction – Y2O3 doped with ZrO2: 2ZrO2(Y2O3) = 2ZrY• + Oi″ + 3OOx

21. Dependence on oxygen pressure1
Can defects in a solid be considered as ions in a solution?
In this section we shall use our knowledge about defect notations and defect charges first to express the formation of different types of defects, and then using the mass action law to find the dependece of defects concentrations on the oxygen pressure. One example is shown on this slide for the formation of oxygen vacancies and reduced cations in nonstoichiometric cerium oxide (Eq.1).
A fondamental question is of course, whether one can use the mass action law in a solid. Well if the three conditions indicated on the slide are fulfilled it is possible to use the mass action law, just as it is possible to use it on chemical reactions taking place in a liquid.
Yes if these conditions are fulfilled:
random distribution of defects
no interactions
high mobility
Law of mass action can be used.

22. Rules which must be obeyed
ratio between cation and anion positions – Constant!
the total number of positions can be changed,
but not the ratio!
In setting up reaction schemes for defect formation in a solid, the three rules given on the slide must however be obeyed. The two first rules naturally comes from the fact that the reactions are taking place in a solid with a fixed ratio of anion – and cation sites, which must be kept constant. If this rationis changed then another phase with another composition and structure is formed and this changes the conditions completely.
The last rule of maintaining neutrality is evident, it is not possible to create ”a charged material” with these processes.
neutrality must be maintained

23. Formation of oxygen vacancies
But let us now look at the formation of oxygen vacancies and reduced cations in CeO2-x.
First we have the reaction as explained previously. If we use the mass action law on this reaction, the K(T) expression is obtained – here K(T) means the equilibrium constant at constant temperature T, all the a(...) values the activities of the different species and p(O2) of course the partiel pressure of oxygen.
Now as indicated on the slide, the solid phase where the reaction is taking place can thermodynamically be consirered as a ”non-ideal solution”, which means that the activity of the defects is proportional to the concentration of the defect, c or [..], times he activity coefficient, γ (gamma), which can be considered constant within a relatively small compostion range.
Replacing the activity terms in the K(T) expression we obtain the K′(T) expression where all the γ values are included in the constant. As this reaction describes the formation of oxygen vancies, VO••, so the equilibrium constant is naturally designated as K(VO••).

24. [VO]  log pO2 1
In order to carry on we need to consider the neutrality condition. From the equation for the edefect formation (Eq.1) we can see, that eact time one oxygen vancancy is formed two reduced cations are formed to maintain the charge balance. As shown on the slide, the neutrality condition must be that the concentration of the reduced cations must be twice the concentration of the oxygen vacancies, as also shown in this simple neutrality condition.
Now replacing the concentration terms for the reduced cations with conccentration terms for the oxygen vacancies we get equation 2 from which it is easy to see that the concentration of oxygen vacancies is proportional to the oxygen pressure raised to the power of -1/6. This exponent is in fact characteristic for the formation of oxygen vancancies in any oxide. 2

25. Brouwer plots - VO
A good way to present the characteristic defect concentration – oxygen pressure relationship is to use a so-called Brouver plot named after an american scientist. In this plot the logarithme of defect concentration is plottet against the logarithme of the oxygen pressure which in the case ofoxygen vacancies will give a straight line with a slope depending on the type of oxygen vacancy. In the case of VO•• this slope will of course be -1/6. The characteristic slopes for neutral and single charged vacancies are also shown on this plot.
Next slide – practice!

26. Practise: Brouwer plots oxygen vacancies
1) Calculate the relationship between [VOx] and p(O2) and draw the Brouwer plot for this relationship.
2) Calulate the relationship between [VO•] and p(O2) and draw the Brouwer plot for this relationship.

27. Answers to practice: Brouwer plots oxygen vacancies
1) Formation of VOx:
Reaction: OOx = VOx + ½ O2
Equilibrium constant: K(VOx) = [VOx] p(O2)1/2
Thus [VOx] prop. to p(O2)-1/2
Brouwer plot see slide 25
2) Formation of VO•:
Reaction: OOx + MMx = VO• + MM′ + ½ O2
Neutrality condition: [MM′] = [VO•]
Equilibrium constant: K(VO•) = [VO•] p(O2)1/2 [MM′] = [VO•] p(O2)1/2
Thus [VO•] prop. to p(O2)-1/4
Brouwer plot see slide 25.

28. [VM]  log pO2 1 2
In this slide we shall look at the formation of VFe″ which as shown in slide 16 typically is formed by oxidation of an oxide which is easily oxidized as FeO.
As shown in equation 1 this double negatively charged iron vacancy iis formed together with two oxdized cations and an oxygen ion in an oxugen position in the lattice – as explained earlier, the oxygen atoms in the atmosphere need two electrons forthe formation of oxygen ions and these electron are delivered by adjacent Fe2+ ions, which become oxidized.
The resulting equilibrium constant K(Ve″) is shown in eq. 2, and the neutrality condition in eq. 3 – that the neutrality condition must be ike shown in eq. 3 is of course quite clear, as two oxidized cations are formed for each VFe″, and the concentration of oxized cations must therefore be twice that of VFe″.
Inserting this neutrality condition into eq.2 gives the expression for K(VFe″) (eq.4), which naturally leads to eq. 5 which shows that the concentration of VFe″ is proprtional to the oxygen pressure with an exponent of +1/6. The exponent for the formation of cation vacancies thus has the same value but with opposite sign as compared to the exponent for the formation of oxygen acancies.
Finally note that the concentration of the iron vancancies correspond to the deviation, y, from the stoichiometric composition. 3 4 5

29. Brouwer plots - VM
The Brouwer plots for the three types of iron vacancies will thus also be straight lines but with slopes of 1/6, 1/4 and ½. From this slide you should also note single charged and neutral vacancies in fact consists of clusters with the compositions shown.

30. Practice: Formation of cation vacancies
Find the oxygen pressure dependence for the concentration of neutral and single charged cation vacancies in FeO. Base the derivation on the cluster model for for these two defects.

31. Answers to practice: Formation of cation vacancies
The two types can in terms of the cluster model be written as (Slide 29):
(2FeFe•VFe″)x ~VFex and (FeFe•VFe″)′ ~VFe′
1) Formation of VFex:
½ O2 + 2FeFex = (2FeFe•VFe″)x + OOx
K(VFex) = [(2FeFe•VFe″)x] p(O2)-1/2
[(2FeFe•VFe″)x] prop.to p(O2)1/2
2) Formation of VFe′
1/2O2 +2 FeFex = (FeFe•VFe″)′ + OOx
K(VFe′) = [(FeFe•VFe″)′] [VFe•] p(O2)-1/2
Neutrality: [(FeFe•VFe″)′] = [VFe•]
[(FeFe•VFe″)′] prop. to p(O2)1/4

32. Brouwer plot for Oi 1 2
Let us finish by looking at the dependence of the concentration of interstitiel oxygen ions on the oxygen pressere.
The formation of interstitial oxygen ions by oxidation is shown in eq. 1, the neutrality condition in eq.2 and the dependence of the concentration of the interstitial oxygen ions in eq. 3.
Note that the oxygen pressure dependence is the same as that obtained for the concentration of VFe″.
Note also that the formation of the intrinsic anti-Frekel defects (slide 14) consisting of an oxygen vacancy – oxygen interstitial of course is independent of the oxygen pressure. 3

33. Brouwer plot: many defects
Construction:
- Log Conc. defect vs log(pO2) - 3 p(O2) regions;
one type of defect dominates in each region
- sharp transition between regions, approximation
Let us finally look at a Brouwer plot for a material in which many defect are possible and which is generally the case. This plot shown on the slide is clearly much more complicated than the simple Brouwer plots for a sigle defect.
As in the simple Brouwer plots the logaritme to the defect concentrations are plotted as a function of log (p(O2). Generally three oxygen pressure ranges are considered, a low pressure region, a medium pressure- and a high pressure region. For each region straight lines with characteristic slopes are obtained for each type of defect as in the simple plots.Note also that in this simple representaion the transition between the pressure regions for a specific defect is considered to be abrupt. This is of course an approximation, but it is rather close to reality. Finally it will also be observed that one type of defect is dominant in the three regions – here it is: low pressure – n, high pressure – p, medium pressure n and p.
This diagram therefore show the defects in an electronic conductor
For a detailed discussion how the relations ships are calculated for the different regions and how the diagram is constructed: Introduction to Defects in Nonstroichiometric Binary Oxides, O.Toft Sørensen should be consulted – can be obtained from the author

34. Calculation of defect concentrations
Deviation from the stoichiometric composition.
Fraction of defects
Number of defects per cm3
In dealing with defect it is of course important t be able to calculate the defect concentrations. In this slide the three most important mmeasures of defects concentrations are mentioned

35. Deviation from stoichiometric composition
First the deviation frm the soichiometric composition. In the case of a oxygen deficient oxide and in oxides this is indicated in the formula with an x, which can be negative (oxygen vacancies) or positive (oxygen in interstitial positions).
In the case of an oxide with cation vacancies, the presence of these are indicated with a y written for the cations in order to emphasize that there are vacant cation sites present.

36. Site Fractions MO2-x Fe1-yO MO2+x
The second method to indicate the defect concentration is also very simple – the site fraction. Examples of fractions of oxygen vacancies and cation vacancies are demonstrated on the slide. How the fractions depend on the deviation from stoichiometric compostion is also shown on the slide for MO2-x, MO2+x and M1-yO.
MO2+x

37. Number of defects per cm3
Finally the formulas for calculating the defect concentrations as number of defect per cm3 are shown in this slide for the thRee different types of oxides. These formulas are quite easy to derive and we shal not do this here. However the derivations are described in ”Introduction to defects in Nonstoichiometric Oxides by the author.

38. Practice Calculation of defect concentrations
Fe0.98O – density 5.70 g/cm3 , M of Fe is 55,85 g.
Type of defects?
Deviation from stoichiometric composition?
Calculate fraction of defects?
Calculate number of defects per cm3 ?

39. Answers Calculaion of defect concentrations
Type of defects: Fe-vancancies, VFe″
Dev. from stoichiometric comp. Y = 0,02
Fraction of defects = 0,02
Number of defects per cm3 = 9,7 • 1020 cm-3
Amazingly large number!

40. Content Lecture Notes II: Defect Chemistry
(Numbers indicate the slide numbers)
Electrocerramics – general -2
What is a defect – 3
Atomic (point) defect in Oxides – 4
Electronic defects in oxides – 5
Defect notations – 6
Charges of defects – 7
- Charges of oxygen vacancies -8
- Charges of interstitial ions – 9
- Nomenclature: relative charges – 10
Practise: Nomenclature – 11
Answers to practise: Nomenclature
Formation of defects – 13
- Intrinsic defects – 14
- Defects formed by reduction – 15
- Defects formed by oxidation – 16
- Interstitial oxygen ions – 17
- Defects formed by substitution – 18
Practice: Formation of defects -19
Answers to practice: Formation of defects – 20
Dependence of Oxygen pressure – general – 21/22
Oxygen vacancies – 23/24
- Brouwer plot – 26
- Practice – Brouwer plot Answer to practise – Brouwer plot – 27
Cation vacancies – 28
- Brouwer plots – 29
- Practice – cation vacancies – 30
- Answers to practice – cation vacancies – 31
Brouwer plot for interstitial oxygen – 32
Brouwer plot or may defects – 33
Calculation of defect concentrations - 34
- Deviaion from stoichiometric composition – 35
- Site fractions – 36
- Number of defects per cm3 – 37
- Practice: defect concentrations – 38
- Answer to practise: defect concentrations – 39
Content – 40.