pauling’s rules - mitsnebulos.mit.edu/egoeke/iowalectures/chp4-handouts.pdfcrystal structure...
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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
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Exercise• Plot the following points on a ternary diagram
– ABC– A3B2
– A2BC– AB2C3
– A2B2C5
– B2C– AB2C2
– A4B8C10
– A0.5BC3
– A4C