1. Point defects
and diffusion

2. Point defects in ionic solids
Frenkel defect:
anion vacancy-
interstitial cation
pair.
Schottky defect:
anion-cation-
vacancy pair.
Anti-Schottky
defect: anion-
cation-vacancy
plus interstial pair.
Isovalent
substitute
atom e- e- e-
F-center: anion
vacancy with
excess electron
in alkalihalides.
M-center: two
anion vacancies
with one excess
electron each.

3. Thermodynamics of point defects
hf
T=const.
- Free energy of formation for n Schottky
defect pairs in NaCl: G G
G0: free energy of the perfect crystal
Dh: enthalpy of formation of a defect Ds : entropy change
n: number of defect pairs G0
Gmin
-TDs
Consequence of the thermodynamic analyses:
Above 0K, point defects will always be present in a crystal that is in thermodynamic equilibrium. The number of point defects will increase with increasing temperature.
neq n

4. Point defects and diffusion I
Exchange mechanism
Ring rotation mechanicsm
Vacancy mechanism
Interstitial mechanism
The jump of an atom from one site to an adjacent empty side needs
the "cooperation" of the neighboring atoms, e.g. the green atoms have to "open" a gap (through termal vibrations) exactly when the red atom tries to jump.
that the jumping atom has enough energy to do the movement (activation energy EA )
The number of atoms which have enough energy and the number of ideal geometries increase with increasing temperature.

5. Point defects and diffusion II
d2C
dx2
< 0 x C
> 0 xi dC dx C x
The diffusion rate J in direction x is proportional to a property of the system diffusant/diffusor ( diffusion constant D) and the concentration gradient along x. For a constant gradient the flux (e.g. migration of particles accross a unit area in unit time) is given by Fick's first law as:
In most cases, however, the concentration gradient is not constant, but changes with time. Fick's second law describes diffusion with non steady-state concentration gradients:
Qualitatively, the equations indicates, that in regions where the concentration gradient is convex, the flux (and the concentration) will decrease with time, for concave gradients it will increase.
The order of magnitude of the diffusion distance
can be calculated by the following thumb rule:

6. Point defects and diffusion III
In most cases the diffusion of one species above is not independent of the diffusion of a second species. For example the speed with which green atoms above can move into the red phase depend on the speed with which red atoms move to the left = interdiffusion. The arrangement above, often used to measure interdiffusion, is called a diffusion couple
Fo100
Fo82 c*
1 mm
T = 1200 °C, 254h
DEMP:9 x 10-13
DTEM:6 x 10-13
norm. conc.
EMP
TEM p1
TEM p2
distance mm
T = 980 °C, 336h
TEM
EMP
norm. conc.
Determination of a diffusion profile in a olivine diffusion couple (Meissner et al., 1998)
distance mm

7. Point defects and diffusion IV
ln D
1/T
The diffusion constant increases
with temperature.
D diffusion constant
D0 preexponential factor
EA activation energy
K Bolzmann constant
The dependency is shown in a lnD vs. 1/T (= inverse temperature) plot called Arrhenius diagram.
Units of D: m2s-1
Self- and interdiffusion coefficient in garnets (Schwandt et al., 1996)

8. Solid solutions

9. Solid solutions
Elements present in a pure phase with an ideal structure can be replaced by other elements. Such a replacement is called a substitution. A solid solution is a single phase which exists over a range in chemical compositions. Pure phases are calle endmembers, while partially substituted phases are intermediate members.
Endmember
100% red atoms
0% green atoms
Solid solution
89% red atoms
11% green atoms
Endmember
0% red atoms
100% green atoms
Almost all minerals are able to tolerate variations in their chemistry (some more than others). Chemical variation greatly affects the stability and the behaviour of the mineral. Therefore it is crucial to understand: - the factors controlling the extent of solid solution tolerated by a mineral - the variation in enthalpy and entropy as a function of chemical composition - different types of phase transition that can occur in solid solutions

10. Olivine solid solution
Following rules apply for substitutions: - conservation of charge neutrality
- similar size of substituting ion
- preference for the same coordination of the substituting ion
- similar electronic configuration
Olivin solid solution: Forsterite Fayallite
Mg2SiO VI Mg2+ => VI Fe2+ Fe2SiO4
Magnesium in octahedral
coordination can be replaced
by Fe and Mn
Cation radii: Mg2+ : 0.72Å vs. Fe2+ : 0.78Å
Vector notation for the substitution: Mg-1 Fe
All intermediate compositions between pure Mg-olivine (forsterite) and pure Fe-olivine (fayallite)
are possible => complete solid solution

11. Structural sites and substitutions
Cation (anion) positions which are related by symmetry are called sites. Positions with the same polyhedral environment may not represent the same site. In the olivine structure f.ex. there are two octahedral cation positions, which are not related by symmetry, and represent,therefore, two different sites called M1 and M2. The two sites are not regular octahedra but are slightly distorted. The distorsion is not equal for both sites.
M1 - site
M2- site
T-site
av. M-O bond lengths in M1: 2.101
av. M-O bond lengths in M2: 2.135
Polyhedral representation of a Complete unit cell
fragment of the olivine structure of the olivine structure
viewed down the a-axis.
In forsterite all octahedra are filled by magnesium, but can be replaced by other cations like Fe, Mn, Ca.Whereas Fe has a weak preference for M1, Mn and Ca enter preferentially the M2 site. The latter two are larger than Mg.

12. Limited solid solution
Calcite-dolomite solid solution: Calcite Dolomite
Ca2(CO3) VI Ca2+ => VI Mg2+ CaMg(CO3)2
Cation radii: Ca2+ : 1.0 Å vs. Mg2+ : 0.72Å !! large difference !!
Vector notation for the substitution: Ca-1 Mg
Large difference in size => only limited substitution possible => limited solid solution
Composition range which is not possible: solution gap, miscibility gap
The size of the miscibility gap is temperature dependent:
200
400
600
800
1000
1200
1400
T ( C)
[Mg]
XMg : molar fraction =
[Mg] + [Ca]
The higher the temperature, the smaller the solutiongap. The diagram on the right is valid if the carbonates are heated in a pure CO2 atmosphere.
Miscibility gap
0.0
0.1
0.2
0.3
0.4
0.5
Calcite
Dolomite
XMg

13. Exsolution
The system NaCl-KCl has, due to the difference in cation radii (1.0 vs 1.4Å) a miscibility gap. Above 410°C a crystal with a composition Na0.75K0.25Cl will be homogeneous, below this temperature, e.g. when cooling the crystal below the solvus, potassium rich exsolution lamellae will form, which grow and become richer in K with further cooling, whereas the matrix will become Na richer.
Microstructures
T=590°C
composition of
crystal: M0
T=410°C
composition of
matrix (99%) : M1
exsolution lamellae (1%): Ex1 M0
T=350°C
composition of
matrix (86%) : M2
Exsol. lamellae (14%): Ex2 M1
99%
Ex1
100% M2
Ex2
14%
86%
Ex3
T=200°C
composition of
matrix (76%) : M3
Exsol. lamellae (24%): Ex3 M3
24%
76 %
Halite
Sylvite

14. Exsolution in orthoamphiboles
Between amphibole endmembers there are often miscibility gaps, f.ex. between cummingtonite and glaucophane. Cooling of a glaucophane (endmember: Na2Mg3Al2Si8O22(OH)2) with excess Mg and Ca will lead to exsolution of cummingtonite lamellae. The exsolution lamellae are very fine and bearly visible in the optical microscope
Cummingtonite lamellae formed within a glaucophane host seen in the TEM. The lamellae form in two symetrically related orientations.

15. Coupled substitutions
Albite-Anorthite solid solution
Albite
NaAlSi3O8
Anorthite
CaAlSi3O8
IX Na+ => IXCa2+
Cation radii: Ca2+ : 1.18 Å vs. Na+ : Å ok! but violation of charge neutrality!!
conservation of charge neutrality => second exchange
Albite Anothite
NaAlSi3O8 IX Na+ => IX Ca2+ CaAl(AlSi2)O8
IV Si4+ => IV Al3+
Two or more substitution at once: coupled substitutions
Vector notation for the substitution: Na-1Si-1 Ca AlIV
charge balance: Na1+ => Ca2+ + Si4+ => Al3+ : =2+3 ok!!
Vectors are often named after the most prominent occurence. The above vector is called plagioclase vector.
Tschermak’s exchange:
Diopside “Cats”-module
CaMgSi2O6 VI Mg2+ => VI Al3+ CaAl(AlSi)O
IV Si4+ => IV Al3+
AlIVAlVI Mg-1Si-1 Tschermak’s exchange

16. Omission solid solutions
Substitutions where charge balance is maintained by substituting only a part of the ions taken out is called omission substitution. Such a substitution will leave some sites empty, which are filled in the endmember.
Perspective views of the kalifeldspar structure down the c-axis.
potassium atoms
potassium vacancy
silicium tetrahedra
lead cation Pb2+
aluminium tetrahedra
K1+ =>
K1+ => Pb2+ Vector notation: K-21+ Pb2+
1 + 1 = 2+0
Lead containing kalifeldspar has a green-blue color and is called amazonite.

17. Substoichiometry Omission substitution through oxidation
Iron layer in pyrrhotite Fe1-xS
Iron layer in troilite FeS
metallic Fe
Ferric iron (Fe2+ )
Ferrous iron (Fe3+ )
Oxidation reaction:
3Fe2+ => 2Fe Fe0 (metallic) ( = empty site, vacancy)
Vector notation: Fe2+-3 Fe2+2
The stoichiometry of pyrrhotite can hardly be expressed by integral numbers, such phases are said to be substoichiometric
The introduction of trivalent iron occurs often already during the growth of the crystal. If the oxidation occurs later, the reduced iron has to be expelled from the structure.

18. Interstitial solid solution
Substituting ions can also go into interstitial sites (interstitial substitution), not occupied in the pure phase.
Example: In the ring silicate beryl (Be3Al2Si6O18) (Fig. 2.27) alcali cations can lodge into the channels formed by silicate rings, which are empty in the pure phase. Charge balance is reestablished by replacing one silicon by aluminum:
Exchange vector: Si Al Na
Beryl structure seen
along the open channels
= c-axis
Instead of alkali cations, neutral noble gas atoms or water molecules can occupy the channels, which require no charge balancing.
Silica tetrahedra
tetrahedral aluminum
Beryllium tetrahedra
alkali cation (e.g. Na+)
Exchange vector: (H2O)
octahedral aluminum
water molecule

19. Order-disorder I
The order in a structure with two cations sharing the same site can be described by an order parameter Q.
Ordered substitution
Disordered substitution
site: potential position for an atom
Site 
Atom A
Site 
Atom B
Four possibilities:
atom A on site  atom B on site 
atom A on site  atom B on site 
Probability of A on :
Probability of B on  1 - p

20. Order-disorder II Complete order : p = 1 Complete disorder: p = 0.5
Normalizing: long range order Q = 2p – 1
Another way to define Q is: where the terms Aa is the fraction of atoms A on site a.
For a phase with stoichiometry AmBn , Q is given as: and
Disordering occurs not suddenly, but over a certain temperature range below the critical temperature. In the disordered structure it is impossible to predict what atom (green or red) occupies a certain site. Results from bulk property measurements of a disordered crystals are the same as the results from a hypothetical crystal with each site filled with half a green and half a red atom. Disordering is thus accompanied by a change in symmetry. For the example shown on the left, the symmetry would change from face-centered cubic to primitiv cubic.

21. Order-disorder III Long - vs. short range order
bad long, bad short r. o.
good long, good short r. o.
bad long, good short r. o.
boundary between perfectly ordered areas = anti-phase boundaries. The domains are related by a translation that is a fraction of the lattice translation of the ordered structure.

22. Ordering vs. exsolution
Cation ordering

23. Basic elements of the pyroxene structure
The pyroxene structure is characterized by single tetrahedral chains (right). Two of the them sandwich an octahedral chain (top)

24. Ortho- vs. clinopyroxenes
C2/c
e.g. diopside CaMgSi2O6
Pbca
e.g. enstatite Mg2Si2O6

25. Octahedral sites in pyroxenes

26. Pyroxene quadrilateral

27. Ordering in OPX I
I-beam representation of the orthopyroxene structure. view is parallel to the silicon tetrahedra.
Octahedral strip of the OPX structure.

28. M1 M2 Ordering in OPX II Low temperature Mg Fe2+
Fe2+ is slightly larger than Mg and prefers to sit on the larger M2 site (i.e. the crystal has a lower enthalpy when Fe2+ is sitting on M2).
Example: For a composition (Mg0.5Fe2+0.5)SiO3 M1 M2
Low temperature Mg
Fe2+
Intermediate temperature
“Infinite” temperature
Mg1-xFe2+x
MgxFe2+1-x
Mg0.5Fe2+0.5
We can measure “x” experimentally and use it to determine what the cooling rate and effective equilibration rate of the mineral was (geospeedometry).

29. Ordering in OPX III