Mineralogy 12:041, Fall 2006, E. Goeke 1

E. Goeke, Fall 2006

Crystal Structure

Chapter 4

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Packing of Ions• If all of the ions are of the same size (e.g.

metallic bonding), then there are threetypes of packing possible:1. Hexagonal closest packing2. Cubic closet packing– Both use the same base layer– Differences are due to how you stack

the layers– Each ion is in contact with 12 other

ions3. Body centered cubic packing– Less dense packing– Each ion in contact with 8 ions

http://www.tulane.edu/~sanelson/eens211/paulingsrules.htm

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Body-centeredcubic packing

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Ionic Bonding Structures• Most minerals have anions &

cations packed together ofdifferent sizes

• For the majority of minerals,cations are the smaller ion andare surrounded by larger anions

• Radius ratio = RR = Rc / Ra =radius of the cation / radius ofthe anion

• Cations will attempt to have asmany anions around them aspossible, but it is limited by theneed of the anions to keep incontact with both the cation andthe other surrounding anions

Linear2< 0.155

Triangular30.225 -0.155

Tetrahedral40.414 -0.225

Octahedral60.732 -0.414

Cubic81.0 -0.732

Hexagonal orcubic closestpacking

121.0

TypeC.N.

Rc / Ra

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Hexagonal / cubicclosest

Cubic

Octahedral

Tetrahedral

Triangular

LinearE. Goeke, Fall 2006

Pauling’s Rules• Pauling’s Rules = 5 rules that outline the basic

assumptions for crystal structures formed through ionicbonding

1. Coordination Principle = around each cation, acoordination polyhedron of anions will form; the numberof anions is determined by the relative size of the cation &anion; the cation-anion distance = cation + anion radii

2. Electrostatic Valency Principle = for a stable ionicstructure, the total strength of the valency bonds thatconnect the anion to all the neighboring cations will beequal to the charge of the anion– evb = ion charge / CN– Three cases:

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i. Isodemic = bonds are equal strength in alldirections• Oxides, fluorides, chlorides all fall into

this categoryii. Anisodemic = the evb is > 0.5 the charge on

the anion• Anion is more strongly bonded to the

central cation than to other structuralgroups

• CO32- is a good example

iii. Mesodemic = the evb is exactly equal to 1/2the charge on the anion• The anion can be bound as tightly to ions

outside the group as to the centrallycoordinated cation

• SiO44- is the prime example

http://www.tulane.edu/~sanelson/eens211/paulingsrules.htm E. Goeke, Fall 2006

3. Sharing of Polyhedral Elements I = shared edges andparticularly faces of two anion polyhedra in a xtalstructure will decrease its stability– If anions only share one corner, the positively charged

cations are kept at the greatest distance from oneanother

– The closer together two cations are, the more they willrepel each other and make the structure more unstable

http://www.tulane.edu/~sanelson/eens211/paulingsrules.htm

E. Goeke, Fall 2006

4. Sharing of Polyhedral Elements II = when structures haveseveral different charged cations, the high-charged cationswill not be located adjacent to one another– Two high charged cations that share an anion will be

close enough together to repel one another5. Principle of Parsimony = nature will try to be as simple as

possible– The number of cation & anion sites in a given crystal

will be small– Several different cations/anions may occupy the same

kind of site

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Isostructural Minerals• Isostructural = isomorphism = isotypism = 2+ minerals

that have their cations & anions organized in the samemanner– e.g. halite & galena– Minerals may have very different physical and

chemical properties– Minerals will have the same symmetry, cleavage, and

crystal habits• Isostructural group = minerals that are isostructural and are

also chemically related by a common anion or anionicgroup– e.g. calcite, magnesite, siderite, rhodochrosite– Tend to have a large amount of ionic or atomic

substitution between the different phases

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Polymorphism• Polymorphism = same chemical formula, but different

structure– E.g. aluminosilicates: kyanite, andalusite, silliamnite;

SiO2: quartz, tridymite, coesite, stishovite– Polymorph = polymorphic form = different possible

structures– Different polymorphs will be stable at different

temperatures and pressures• High pressures = more dense structure• High temperatures = more vibration of atoms, so

larger structure– Polymorphic transformation = change that occurs

between two different polymorphs; 4 types:

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1. Reconstructive transformations = extensiverearrangement of the xtal structure isrequired– Normally involves a large change in

energy of the structure– May be a slow transformation, causing

the unstable polymorph to be presentfor a long time (metastable)

– e.g. carbon transformation fromdiamond to graphite

2. Displacive transformations = only smallchanges are needed to switch from onepolymorph to another– Normally no bonds are broken, the

angles between atoms simply change– No energy change is involved, so the

transformation is instantaneous &reversible

– e.g. α-quartz and β-quartzhttp://www.tulane.edu/~sanelson/eens211/twinning.htm

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3. Order-Disorder Transformations = gradual changeeither from a low-temperature ordered state to ahigh-temperature disordered state or vice versa– e.g. K-feldspars: microcline, orthoclase,

sanidine– If the temperature change is fast, a metastable

polymorph may continue to be present4. Polytypism = polymorphs only differ in the

stacking order of identical sheets– e.g. hexagonal vs. cubic closest packing– Important when we get to the sheet silicates

disordered

ordered

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Compositional Variation• Solid solution = ions substituting for each other in a xtal

structure; depends on:– Size of the ion & the crystallographic site involved– The overall charge must remain the same, so it is

easier for two ions of the same charge to swap,otherwise more than one ion will have to change tokeep the balance

– The temperature & pressure the substitution isoccurring at; e.g. at high T’s, garnet likes Mg betterthan Fe, but at higher P’s, Ca is preferred

• Three types of solid solution: substitutional, omission, andinterstitial

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Substitutional SolidSolution

• Divided into two types:1. Simple

– Ions of equal charge & ~equal sizesubstitute for one another

– End-member = compositionalextremes (e.g. Fe3Al2Si3O12 &Mg3Al2Si3O12)

– Complete = continuous =substitution occurs over the entirepossible range from one end-member to the other

– Incomplete = discontinuous =partial = solid solution occurs onlyover a limited range of the possiblecompositions http://tesla.jcu.edu.au/Schools/Earth/EA1001/Mineralogy/Minerals/Feldspars.html

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2. Coupled– Maintains charge balance by substituting 2+ ions--one with a

larger & one with a smaller charge than the ions they arereplacing

– Plagioclase is a good example of this:– Albite: NaAlSi3O8 - Anorthite: CaAl2Si2O8– Interstitial substitution is a type of coupled substitution

• Occurs in minerals that have large voids in theirstructures (e.g. beryl)

• Al3+ is often substituted for Si4+ to maintain chargebalance when adding an ion into the interstitial space

http://home.hetnet.nl/~turing/preparation_3dim_5.html

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Omission Solid Solution• Occurs when a higher charged ion replaces a lower

charged ion– Maintain charge balance by having the larger charged

ion fill the space originally held by two smaller chargedions

– The unfilled space will then become vacant or omitted– In blue microcline, one Pb2+ ion replaces two K+ ion

which leaves one space blank– A blank space is represented by: �

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Mineral Formulas• Few basic rules:

– Cations are grouped to the left and anions to the right(e.g. CaSiO3)

– Charges must balance (e.g. K+Al3+Si34+O8

2-)– Cations are grouped by structural groups (e.g.

(Ca,Mg,Fe,Mn)3VIII(Al,Cr,Fe)2

VI(Si,Ti)3IVO12)

– Cations are listed in order of decreasing CN (e.g.CaVIIIMgVISi2IVO6)

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Graphic Representation• Binary diagrams =

composition describedas somewhere betweentwo end-members;shown on a lineardiagram– Can plot either wt

% or molecular %• Ternary diagram =

composition can bedefined as a mix ofthree end-members

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http://www.eos.ubc.ca/courses/eosc221/ternary/ternary.html

E. Goeke, Fall 2006

Exercise• Plot the following points on a ternary diagram

– ABC– A3B2

– A2BC– AB2C3

– A2B2C5

– B2C– AB2C2

– A4B8C10

– A0.5BC3

– A4C