Point defects in ionic solids

Frenkel defect:anion vacancy-interstitial cationpair.

Schottky defect:anion-cation- vacancy pair.

Anti-Schottky defect: anion-cation-vacancyplus interstial pair.

F-center: anion vacancy with excess electronin alkalihalides.

e-

M-center: two anion vacancies with one excess electron each.

e-

e-

Isovalentsubstitute atom

- Free energy of formation for n Schottky defect pairs in NaCl:

G =G0 +n(Δh−TΔs)G0: free energy of the perfect crystal h: enthalpy of formation of a defect s : entropy change n: number of defect pairs

Thermodynamics of point defects

G

n

G0

hf

G

neq

-Ts

Gmin

T=const.

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.

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

1.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.

2.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.

dCdxC

xThe 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:

x≈ Dt

J =−DΔCΔx

The order of magnitude of the diffusion distancecan be calculated by the following thumb rule:

Point defects and diffusion II

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.

d2Cdx2 < 0

x

Cd2Cdx2 > 0

xi

€

∂C∂t= −D∂

2C∂x 2

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:

t

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

norm.

conc.

norm.

conc.

distance m

distance m

EMPTEM p1TEM p2

TEMEMP

Fo100

Fo82

c*

1 mDetermination of a diffusion profile in a olivine diffusion couple (Meissner et al., 1998)

T = 1200 °C, 254hDEMP:9 x 10-13

DTEM:6 x 10-13

T = 980 °C, 336h

D=D0expEAkT

⎛ ⎝ ⎜

⎞ ⎠ ⎟

ln D

1/TThe diffusion constant increases with temperature.

D diffusion constantD0 preexponential factorEA activation energyK Bolzmann constant

The dependency is shown in a lnD vs. 1/T (= inverse temperature) plot called Arrhenius diagram. Units of D: m2s-1

Point defects and diffusion IV

Self- and interdiffusion coefficient in garnets (Schwandt et al., 1996)

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 atoms0% green atoms

Solid solution89% red atoms11% green atoms

Endmember0% red atoms100% 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

Olivine solid solution

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

Magnesium in octahedralcoordination can be replacedby Fe and Mn

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 FayalliteMg2SiO4

VI Mg2+ => VI Fe2+ Fe2SiO4

M1 - siteM2- siteT-site

Structural sites and substitutionsCation (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.

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.

Polyhedral representation of a Complete unit cellfragment of the olivine structure of the olivine structureviewed down the a-axis.

av. M-O bond lengths in M1: 2.101av. M-O bond lengths in M2: 2.135

[Mg]XMg : molar fraction =

[Mg] + [Ca]

Calcite-dolomite solid solution: Calcite DolomiteCa2(CO3)2

VI Ca2+ => VI Mg2+ CaMg(CO3)2

Limited solid solution

DolomiteCalcite XMg

0.1 0.2 0.3 0.40.0 0.5

200400600800100012001400T ( C)

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.

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 solutionComposition range which is not possible: solution gap, miscibility gap

The size of the miscibility gap is temperature dependent:

Miscibility gap

Halite Sylvite

100%

14% 86%

24% 76 %

99% Ex1M1

M2

M3

Ex2Ex3

Microstructures

T=590°Ccomposition ofcrystal: M0

M0

T=410°Ccomposition ofmatrix (99%) : M1exsolution lamellae (1%): Ex1

T=350°Ccomposition ofmatrix (86%) : M2Exsol. lamellae (14%): Ex2

T=200°Ccomposition ofmatrix (76%) : M3Exsol. lamellae (24%): Ex3

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.

Exsolution in orthoamphiboles

Cummingtonite lamellae formed within a glaucophane host seen in the TEM. The lamellae form in two symetrically related orientations.

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

Albite-Anorthite solid solution

Coupled substitutions

AlbiteNaAlSi3O8

IX Na+ => IXCa2+

AnorthiteCaAlSi3O8

Cation radii: Ca2+ : 1.18 Å vs. Na+

: 1.24 Å ok! but violation of charge neutrality!!

conservation of charge neutrality => second exchange

Albite AnothiteNaAlSi3O8

IX Na+ => IX Ca2+ CaAl(AlSi2)O8

IV Si4+ => IV Al3+

charge balance: Na1+ => Ca2+ + Si4+ => Al3+ : 1 + 4 =2+3 ok!!

Two or more substitution at once: coupled substitutionsVector notation for the substitution: Na-1Si-1 Ca AlIV

Vectors are often named after the most prominent occurence. The above vector is called plagioclase vector.

Tschermak’s exchange:Diopside “Cats”-moduleCaMgSi2O6

VI Mg2+ => VI Al3+ CaAl(AlSi)O

IV Si4+

=> IV Al3+

AlIVAlVI Mg-1Si-1 Tschermak’s exchange

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.

silicium tetrahedra

Perspective views of the kalifeldspar structure down the c-axis.

lead cation Pb2+

K1+ => K1+ => Pb2+ Vector notation: K-2

1+ Pb2+

1 + 1 = 2+0Lead containing kalifeldspar has a green-blue color and is called amazonite.

potassium atoms

aluminium tetrahedra

potassium vacancy

Substoichiometry

Ferrous iron (Fe3+ )

Omission substitution through oxidation

Oxidation reaction:

3Fe2+ => 2Fe 3+ + + Fe0 (metallic) ( = empty site, vacancy)

Vector notation: Fe2+-3 Fe2+

2

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.

Ferric iron (Fe2+ )

Iron layer in pyrrhotite Fe1-xSIron layer in troilite FeS

metallic Fe

The stoichiometry of pyrrhotite can hardly be expressed by integral numbers, such phases are said to be substoichiometric

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:

Interstitial solid solution

tetrahedral aluminum

alkali cation (e.g. Na+)

Beryl structure seen along the open channels= c-axis

Silica tetrahedra

Beryllium tetrahedraoctahedral aluminum water molecule

Instead of alkali cations, neutral noble gas atoms or water molecules can occupy the channels, which require no charge balancing.

Exchange vector: Si-1 -1 Al Na

Exchange vector: -1 (H2O)

Substituting ions can also go into interstitial sites (interstitial substitution), not occupied in the pure phase.

Order-disorder I

The order in a structure with two cations sharing the same site can be described by an order parameter Q.

Disordered substitution

Atom ASite

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

p A=

on totl

Ordered substitution

site: potential position for an atom

Site Atom B

Complete order : p = 1Complete 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

For a phase with stoichiometry AmBn , Q is given as: and

Order-disorder II

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.

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.

Ordering vs. exsolution

Exsolution

Cation ordering

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)

C2/c

e.g. diopside CaMgSi2O6

Pbca

e.g. enstatite Mg2Si2O6

Ortho- vs. clinopyroxenes

Octahedral sites in pyroxenes

Pyroxene quadrilateral

Ordering in OPX I

I-beam representation of the orthopyroxene structure. view is parallel to the silicon tetrahedra.

Octahedral strip of the OPX structure.

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 M2Low temperature Mg Fe2+

Intermediate temperature

“Infinite” temperature

Mg1-xFe2+x MgxFe2+

1-x

Mg0.5Fe2+0.5 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).

Ordering in OPX II

Ordering in OPX III