Crystal Chemistry :

Chemical composition of minerals is of fundamental importance..as many properties depend on it

Properties also depend on :Geometric arrangement of constituent atoms or ionsNature of electric forces that bind them

Therefore for the understanding of minerals : - chemistry, bonding and structure

Mineralogical information available from :1. Earths Crust2. Extraterrestrial samples meteorites and lunar materials

Eight elements make up approx. 99 % of the earths crust..On recalculation of wt% atomic percent and finally in terms of volume percent

Oxygen constitute > 90% of the total volume of the crustOn atomic scale earths crust essentially of a packing of oxygen anions with interstitial metal ions.

Therefore oxygen containing minerals such as- silicates, oxides and carbonates abundant minerals

Understanding crystal chemistry :

Relating chemistry of minerals to their structure, crystallography and physical properties.Before this -----we need to know basics of elementary aspects/concepts of atoms, ions and their bondingforces in crystalline materials

Composition of the Earths Crust

Weight % Atom % Ionic Radius Volume %O 46.60 62.55 1.40 93.8Si 27.72 21.22 0.42 0.9Al 8.13 6.47 0.51 0.5Fe 5.00 1.92 0.74 0.4Ca 3.63 1.94 0.99 1.0Na 2.83 2.64 0.97 1.3K 2.59 1.42 1.33 1.8Mg 2.09 1.84 0.66 0.3

Total 98.59 100.00 100.00Most common silicates are from theseMost common silicates are from theseO alone = 94 vol. % of crustO alone = 94 vol. % of crust

ATOM & COMPOUND

ATOM: Smallest subdivision of matter retaining the characteristics of the elementssmall, massive nucleus, protons neutrons surrounded by electrons

Size are expressed as atomic radii in ngstron () units

* ATOMS OF AN ELEMENT ARE ESSENTIALLY IDENTICAL* ATOMS OF ONE ELEMENT DIFFER FROM ATOMS OF EVERY Of ELEMENT

COMPOUNDS: ONE OR MORE ELEMENTS COMBINE IN SPECIFIC PROPORTIONS TO FORM CHEMICAL COMPOUNDS

e.g : QUARTZ SiO2 (one Si ATOM and 2 Oxygen atoms )

Danish Physicist Niels Bohr accepted picture of planetary atom electrons circling the nucleus inorbits or energy levels at distances from nucleus depending on the energies of the electrons

Niels Bohrs model of an atom

Electrons and nucleus are both smallmovement of electrons give rise to large effective diameter.Though Bohr theory gained wider acceptanceexperiments based on adsorption and emission of energyof atoms revealed e-s cannot exist in one orbit around the nucleusbut are confined to specific orbitalswith discrete energy gaps between them.

Observation - Quantum properties of energy Theory of Quantum MechanicsIt does not allow e- movement in well defined orbits, but portrays motion of e- only in terms of finding aCertain e- within a small volume unit

In quantum theory each electrons in an atoms is assigned four principal quantum numbers.These are essentially to identify and describe electron orbitals :

1. The Principal Quantum number (n) reflects the effective volume of an electron or orbitalpositive integer value : one to infinity 1,2,3

2. Subsidary Quantum number or Orbital shape quantum number (I) determines the general shape ofthe region in which an electron moves . Values of 0,1,2n-1 (for any value of n)

3. Magnetic Quantum number (m) which refers to the orientation of the electrons orbital space.values either positive or negative, but not exceeding I such as 0, 1, 2,.I

4. Spin Quantum number (s) , which is assumed to define the direction of spin of the electron in spaceValues of .

Different values of n describes various energy levels of the electrons. The energy levels were designatedas K, L, M, N, O. K energy level n = 1; L is the energy with n = 2 and so on.

I are designated by the letters s, p, d, f respy (sharp, principal, diffuse and fundamental)s orbitals (I = 0) spherical probability of finding e- is same for all directionsfor I > 0; have strong directionalities outward from the nucleus.

I = 1 ; the wave function is dumbbell shapedI = 2 ; it is generally 4-fold symmetry

n = 1, l = 0 is a 1s orbital and n = 3, I = 2 indicates 3d orbitals

Quantized energy levels

R

e

l

a

t

i

v

e

E

n

e

r

g

y

R

e

l

a

t

i

v

e

E

n

e

r

g

y

n = 1 K 2 L 3 M 4 N 5 O 6 P 7 Qn = 1 K 2 L 3 M 4 N 5 O 6 P 7 Qss

ssss

ss

ssss

ss

pppp

pp

pppp

pp

dd

dddd

dddd

ffff

ff

Note that the energy does not Note that the energy does not necessarily increase K necessarily increase K L L M M N etc.N etc.4s < 3d4s < 3d

Shells and Subshells

Innermost - K (n = 1) 2e- s

L (n = 2) 8e- s, p

M (n = 3) 18e- s, p, d

outer - N (n = 4) 32e- s, p, d, f

(generally higher E)

higher levels not filled

IONS :

Atoms may give up or borrow electrons to become ions.Results in becoming Cations which have a net positive charge or Anions net negative chargeBecause there are more electrons than protons.

The ionic charge also called Valence is the number of protons less the number of electronsIndicated by a superscript number after the element symbol (Fe2+or Fe3+ indicates iron with two more protons than electrons)

Call ions with charge of +1 or -1 monovalent

Charge with 2 as divalent or trivalent = 3 or teravalent = 4Elements such as Helium in which electrons completely fill the s-orbitals and p-orbitals are calledNoble or inert elements as they are extremely stable and unreactive (Ne, Af, Kr, Xe and Ra).

The process of losing an electron is called Oxidation and gaining an electron reduction.Oxidation leaves metals with a positive chargecombine with oxygen anions (O2-) to form oxides.

MODERN PERIODIC TABLE

Bonding forces in Crystals :

Forces that bind together the atoms (ions or ionic group) are electrical in nature. Type and intensity are responsible for their physical and chemical properities _ Hardness, cleavageelectrical and thermal conductivity and coefficient of thermal expansion..

Stronger the bond the harder is the mineral..higher is the MPGreater hardness of diamond is due to very strong electrical forces linking the constituentC atoms

Periclase MgO and NaCl are similar yet Periclase melts = 2800oC and NaCl at 801oCThe greater amount heat energy required indicates it has a stronger electrical bond than halite.

Chemical Bonds and can be described belonging to one of the five principal bond types :1. Ionic2. Covalent3. Metallic4. Vander waals5. Hydrogen bonds

Transition may exists between all types of bonds.

It is the electrical interaction of the ions or atoms constituting the structural units determine the propertiesof the resulting crystal.

SiO2 display equal amount of characteristics of ionic and covalent bond Pbs mineral exhibits characteristics related to metallic bond electrical conductivity and ionic bond

features of excellent cleavage and brittleness

Similar kind bond types are called Homodesmic where as variable is called Hetrodesmic

Ionic Bonds : Exchange of electrons from metal to cationNacl : Na metal achieves ionic state (Na+) by loss of an e- and Cl- has become ionised through the gainof an e-formation of ionic bond between Na+ and Cl- as a result of exchange of e- between metal tothe anion. The attraction between unlike electrostatic charges holds ions together in a crystal. Type of bonding is called ionic or electrostatic

Physically ionic bonded crystals are generally of moderate hardness and specific gravity

Have fairly high melting point and poor conductors of electricity and heat because the electrostaticcharge constituting the ionic bond is evenly spread over the ions a cation will tend to get surroundedItself with many anions.

Ionic bonds are non-directional.Strength of ionic bonds depends on _ two factors1. Centre to centre spacing between the ions and 2. their total charge.

Effect of increased atomic distance on the strength of ionic bond is readily seen in the halides of Na

Strength of the bonds as measured by the melting temperature is inversely proportional to thelength of the bond

Covalent Bonds : Sharing of Electrons between metal and anion

A single valence Cl- atom with an incomplete valence orbital is highly reactive and there for combines withalmost anything in its neighborhood generally it is another Cl- and two unite in a way that e- shares.

Electron sharing bonds is the strongest of the chemical bonds. Minerals bonded covalently have :1. Insolubility2. Great stability3. High MP4. Do yield ions in the dissolved state 5. Non- conductor both solid and liquid as electrical forces constituting the bond are sharply localized

Element of middle periodic table (C, Si, Al and S) have two three or four vacancies in their outerorbitals and since they have low CN usually have enough e-s in the valence orbitals to form threeor four covalent bonds between each atom and its closest neighbour.thus forms large aggregates

C is the best example (four valence e- in each carbon are sufficient to fill the bonding orbitals by e-sharing with four other carbon atoms..hardest material

Covalent Atomic Radii:

In covalent bonded structures the interatomic distances in crystal is equal to the arithmetic mean of the Interatomic distances. Thus diamond the C-C is 1.54 and silicon the Si-Si = 2.34

SiC = 1.94 arithmetic mean of element spacing

Mixed Covalent and Van der Waals bonds

3.35

1.54

Character of Bonding Mechanism :

It is generally recognized that there is some electron sharing in most ionic bonded crystalsand in the same way atoms in covalent bonded substances often display electrostatic charges.

This assessment of character is based on the polarising power and polarizibility of the ions involved.Compound of highly polarising cation with easily polarised anion AgI show strong covalent character

Si O bond is 50% ionic; Al O is 63%

Linus Pauling gernalised this concept and rendered it quantitative by assessing to each element a numerical value of electronegativityThis arbitrary scale represents the power of atom to attract electrons.

Cs = 0.7 and F =4.0

Compounds made up of elements with very different values of electronegativity are more ionic than compounds made of elements close to each other

Metallic Bonds :

The attractive force between the nuclei( with filled electrons orbitals, but lacking valenceelectrons) ad the cloud of negative charges hold the structures together and the bond is called as Metallic bonds.

- High plasitcity, tenacity, ductitlity and conductivity , generally low hardness and MP and BP

- Electrons are free to move through out the structure

- Photoelectric effects :It's been determined experimentally that when light shines on a metal surface, the surface emits electrons. For example, you can start a current in a circuit just by shining a light on a metal plate. Why do you think this happens ??

Metallic bonding is the bonding within metals.

It involves the delocalized sharing of free electrons among a lattice of metal atoms. Metal atoms typically contain a high number of electrons in their valence shell compared to their period or energy level. These become delocalized and form a Sea of Electrons surrounding a giant lattice of positive ions.Metals seem to have higher boiling and melting points which might suggest stronger bonds between the atoms.

Metallic bonding is non-polar, in that there is no (for pure elemental metals) or very little (for alloys) electronegativitydifference among the atoms participating in the bonding interaction, and the electrons involved in that interaction are delocalized across the crystalline structure of the metal.

The metallic bond accounts for many physical characteristics of metals, such as strength, malleability, ductility, conduction of heat and electricity, and luster.Metallic bonding is the electrostatic attraction between the metal atoms or ions and the delocalised electrons. This is why atoms or layers are allowed to slide past each other, resulting in the characteristic properties of malleability and ductility.

Metal atoms have at least one valence electron, they do not share these electrons with neighboring atoms, nor do they lose electrons to form ions. Instead the outer energy levels of the metal atoms overlap. They are like covalent bonds

Metals are also known as being good conductors of heat, or thermal conductors. Heat is kinetic energy. In order for a substance to conduct heat, it must be able to transmit kinetic energy.

If heat is applied to one side of a piece of metal, then the kernels will start to vibrate. Because they are so loosely held into the crystal structure, they will be able to vibrate freely. With the increase in the amount of their vibration, they will run into the kernels located next to them. That will start more kernels to vibrate.

In this way, the process continues until all of the kernels in the system are vibrating. Any material that has highly rigid structures because of strong, rigid bonding will not have the freedom of motion that is needed for the transmission of the kinetic energy.

van der Waals Bond : weak dipole attraction

Molecules such as N2, O2, F2 and Cl2 form molecular soilds because all the valence orbitalsare occupied by nonbonding electrons or form coherent molecules

Take away energy from Cl2 gas by cooling ---molecules would collapse into a chaotic liquid stateIf further the energy is reducedamplitude of vibrations is still reduced then only minute strayelectrical field exists and serve to lock the sluggish moving molecules into a orderly structure of solidState. Solidification of chlorine takes place at -102oC

Electrons in the orbit may synchronies their motions in such a way that an instantaneous and weakDiople attraction is produced between two atoms. This weal dipole can induce similar effect inneighboring atoms which will cause the whole molecular structure to be bonded together by weakDipole effect.

This weak bond with ties molecules and essentially uncharged structure units into a cohesive structure by virture of small residual charges on surfaces is called van der Waals or residual bond

Hydrogen Bonds :

Polar molecules can form crystalline structures by the attraction between the oppositely charged endsof molecules. The hydrogen bond is an electrostatic bond between a positively charged hydrogenIon and negatively charged ion such as O-2 and N-3

Ice is the perfect example of an intermolecular structure

Atomic and Ionic Radii :The radius of an atom is defined by the radius of the maximum radial charge density of the charge of the outermost shells of the atomthe effective radius of and atom or ion is also dependent upon the type and number of neighboring atoms or ions.

Atomic Radii : The radius of an atom can only be found by measuring the distance between the nuclei of two touching atoms, and then halving that distance.

As you can see from the diagrams, the same atom could be found to have a different radius dependingon what was around it.The atoms are pulled closely together and so the measured radius is less than if they are just touching. Bonded.Metallic radii is half the distance between two the ceters of two adjoining metal atoms.

The right hand diagram shows what happens if the atoms are just touching. The attractive forces are much less, and the atoms are essentially "unsquashed". This measure of atomic radius is called the van der Waals radius after the weak attractions present in this situation.If crystals with two appositively charged ions are bonded together, the distance between (+) and (-) ions is the sum of the two different radii.

IONIC RADIUSIons aren't the same size as the atoms they come from. Compare the sizes of sodium and chloride ions with the sizes of sodium and chlorine atoms.

Positive ionsPositive ions are smaller than the atoms they come from. Sodium is 2,8,1 e-s; Na+ is 2,8. lost a whole layer of electrons, and the remaining 10 electrons are being pulled in by the full force of 11 protons.

Negative ionsNegative ions are bigger than the atoms they come from. Chlorine is 2,8,7 e-s; and Cl- ion is 2,8,8. Although the electrons are still all in the 3-level, the extra repulsion produced by the incoming electron causes the atom to expand. There are still only 17 protons, but they are now having to hold 18 electrons

0.97 1.85 1.07 1.81

Any pair of oppositely charged ions there exists an attractive electrostatic force which is directlyproportional to the product of the charges and inversely proportional to the square of the distance between their centers- Coulombs law

Ions approach each other under the influence of electrostatic forces are setup due to oppositely chargedelectron cloud with positively charged nucleus there decreasing the internuclear distance. The distanceat which these repulsive forces balance the attractive forces is the characteristic interanionic spacing or bond length

If one of the ionic radii in an ionic bond length is known from prior experimental measurement, the radius of the other ion can be obtained

e.g. radius of 6-coordinated O2- ions to be 1.40 , then the radii of many cations that are boned to Oxygen can be obtained by subtracting the value of 1.40 from the measured bond length betweencation-oxygen pairs

The atomic radius of an atom and ionic radius of an ion may vary from one crystal str to other due to changes in bond type and coordination number -- the atomic and ionic radii in tables as average values

Shanont and Prewit (1969) evaluated variation in ionic size as a function of coordination numbere.g. :1. Ionic size of O2- : 1.35 in 2-fold coordination; 1.42 8-fold.2. Stishovite : Si 6-fold coordination with oxygen, measured bond length Si-O bond : 1.76 to 1.81

Changes in ionic size : a) CN; b) bond type and c. Polarisation(change in the shape of some atoms and ions dilation and deformation)

Generally large monovalent ions that lack a noble gas configuration are most easily polarised covalent

Addition or loss of electrons causes a change in size from that of atom. Cations tend to be smaller andanions larger than their respective atoms

In same group (Periodic table) ionic radii increase as the atomic number increasesContradiction is seen in REE. The trivalent ions of these elements decrease in radius with increasingatomic number from La3+ (Z = 57) with radius of 1.14 to Lu3+ (Z = 71) 0.85 lanthanide Contractiondue to building up inner electron orbitals before adding to a new outer orbitalresults increasing nuclearcharge and increased attraction is exerted to outer electron decrease in ionic radii.

As shown in Figure, the ionic radii decrease smoothly across the series. This decrease in size is the famous 'lanthanide contraction'

The lanthanide contraction is caused by the increase in effective nuclear charge across the series due to the poor shielding ability of 4f electrons.

This is seen in every period as a shell is filled. It is particularly important for the lanthanides, however, because of the:

1) length of an f-series. (There are 2 electrons in a s-block, 6 in a p-block, 10 in a d-block but 14 in an f-block

2) directional characteristics of f-orbitals - see FigureThe f-orbitals are 'angularly diffuse'; the electrons are able to occupy different volumes of space (different lobes) and so avoid each other

Seven f - orbitals

Coordination Principle :

In an ionic structure each cation tends to surround itself with anion.

The number that can be grouped around it will depend upon the relative sizes of the cations and anions.The coordinated ions always cluster about the central coordinated ions in such a way that their centres lie at the apices of a polyhedron stable form of arrangement

The number of anions that can fit around each other is known as the Coordination number of the cation

NaCl Na+ has six closest Cl- neighbours and is said to be in 6 coordination with Cl- (C.N.6)

Fluorite (CaF2) each calcium ions is at the centre of a coordination polyhedron consisting of 8 F- (C.N.8)

Ca has 8 F- neighbours and F- has 4 Ca neighboursthere are twice of F-ions as Ca ions in the structureIn accordance with the formula CaF2

In a hypothetical 2-dimensional structure we can illustrate the notion of the coordination polyhedra using various regular polygons.

In naturally occurring 3-D silicate structures the main coordination polyhedra are tetrahedral or 4-fold(with four bounding oxygens), octahedral or 6-fold (with six bounding oxygens), cubic or 8-fold(with eight bounding oxygens) and 12-fold. The shape of the polyhedra formed by a particular cation is defined the relative size (as reflected in their radii) of the metal cations and the bounding anions (2-D example the hexagon clearly allows a bigger relative central "atom" than the triangle).

Radius Ratio :ion in a crystal affects every other ion to some extent and the strongest forces exist between

Ions that are nearest neighbours.. Said to constitute first coordination shellThe geometry and of arrangement of this shell and hence the coordination number are dependent onthe relative sizes of the coordinated ions.

Relative sizes of the ions is expressed as radius ratio Rx : Rz ; Rx = radius of the cation and Ry= radius of the anion in units

e.g. NaCl RNa+ = 0.97 and RCl- = 1.81 RNa+ : RCl- = 0.97/1.81 = 0.54

Similarly CaF2 the radius ration : 0.99/1.33 = 0.74

When two or more cations are present in a structure, coordinated with the same anion, separatedradius ratio needs to calculated.e.g. MgAl2O4 - both Mg and Al are coordinated to oxygen anions and hence

RMg2+ : RO2- = 0.66/1.40 = 0.47RAl3+ : RO2- = 0.51/1.40 = 0.36

When coordinating and coordinated ions are the same size, the radius ratio is 1

Radius ratio and predicted coordination number with respect to oxygen for the commoner cations together withthe coordination actually observed in minerals. Close correlation between obs and prediction confirms theassumption that ions do infact as spheres of definite radius[

Assuming that ions acts as arigid spheres of fixed radii, the stable arrangements of cationsand anions for particular radiusratio can be calculated frompurely geometrical considerations

See previous table

Many cations occur exclusively in a particular coordination.e.g. Aluminum whose radius ratio lies near the theoretical boundary between two types of coordinationTherefore can occur in both

Such cases the coordination is to some extent is controlled by the temperature and pressure at whichCrystallisation took place.

High temperatures and low pressures favor low coordination and low temperatures and high pressuresfavor high coordination

Al : in a typical high temperature minerals tends to assume four fold coordination and substitute for Si,whereas in minerals formed at lower temperatures it occurs more often in six-fold coordination

These principles are the basic for the crystal chemistry of minerals. They express the conditions forLow potential energy of the atoms and hence high stability.

Only very stable compounds can occur as minerals; less to tend to either do not form in nature or soon decompose..

Coordination number, C.N. depends on the relative size of the ions. If all of the atoms in a crystal are the same size, then there are two ways to pack the atoms to form a crystal structure. In this case, the maximum number of atoms that be coordinated around any individual is 12. We call this 12-fold coordination. There are two ways that atoms can be packed in 12-fold coordination.

First, examine a single layer of atoms of equal size. Note that there are two kinds of voids between the atoms, those that have a sort of triangular shape with the triangles pointing up we'll call B voids, and those with the triangles pointing down we'll call C voids.

If we add the next layer of atoms so that they occupy the space above the B voids, and then add the next layer above the A atoms, this will result in a stacking sequence that runs AB ABAB..etc.

This type of closest packing is referred to as hexagonal closest packing. It results in a hexagonal lattice with the c-axis oriented perpendicular to the AB AB layers.

If after adding the layer of B atoms we place the next layer so that the atoms occupy positions over the C voids in the A layer, and continue the process upward, we get a stacking sequence that runs ABC ABC ABC.... etc. This type of packing is referred to as cubic closest packing. It results in a cubic or isometric lattice with the axis perpendicular to the layers.

Hexagonal lattice that underliesHCP arrangement and Cubic lattice that underlies arrangement.

Space group HCP : P63/m2/m2/m

Space group for CCP : Fm3m

For Rx/Rz < 0.414 the structure goes into 4-fold coordination.Planes through the centers of the larger atoms in this case will form a tetrahedron, so 4-fold coordination is also called tetrahedral coordination

The calculation to determine the "no rattle" limit for tetrahedral coordination is complex.The result shows that the limit is reached when Rx/Rz = 0.225.As the radius ratio becomes smaller than this, triangular coordination becomes the stable configuration.

For triangular coordination, the coordination number is three, that is 3 anions surround the smaller cation.The "no rattle" limit is reached for triangular coordination when Rx/Rz becomes less than 0.155.

At values of Rx/Rz < 0.155 the only way the smaller ion can be coordinated by the larger ions is to have 2 of the larger ions on either side. This 2-fold coordination is termed linear coordination.

TetrahedraSi4+

Pythagorean Theorem

Theorem - claims that the sum of (the areas of) two small squares equals (the area of) the large one.

In algebraic terms, a2 + b2 = c2

where c is the hypotenuse while a and b are the sides of the triangle.

The theorem is of fundamental importance in the Euclidean Geometry where it serves as a basis for the definition of distance between two points.

Six-fold coordination is also called octahedral coordination because the shape defined by drawing planes through the center of the larger ions is an octahedron. Octahedral coordination is stable when Rx/Rz , 0.732, but decreasing the radius of cation, Rx, will eventually reach a limit where again the smaller ion will rattle in its site.

The no rattle limit can be determined by looking at thehorizontal plane running through the ions labeled C and D.

In this case we can write:

(2Rz + 2Rx)2 = (2Rz)2 + (2Rz)2= 2(4Rz2)

2Rz +2Rx = 22RzRz + Rx = 2RzRx = (2 - 1)RzRx/Rz = 0.414

To see what happens when one of the involved ions or atoms becomes smaller, we need to examine the relative sizes of the atoms. The relative sizes are indicated by the radius ratio of the coordinating atoms or ions. In crystal structures we usually look at cations surrounded by anions, so the radius ratio is defined as Rx/Rz, where Rx is the radius of the cation, and Rz is the radius of the surrounding anions.

Since the anions are usually the larger ions, this results in decreasing values of Rx/Rz as the size of the cation decreases.

If we decrease the size of the cation in such an arrangement, still allowing for the surrounding anions to touch each other and touch the cation, with decreasing size of the cation, the coordination will first result in 8 anions surrounding the cation.

This is called 8-fold coordination or cubic coordination because the shape of the object constructed by drawing lines through the centers of the larger ions is a cube.

If the size of the coordinated cation becomes smaller, it will become too small to touch the surrounding anions. Thus, there is limiting radius ratio that will occur when the Rx/Rzbecomes too small. To see what this limit is, we must look at the vertical plane running through the centers anions labeled A and B

In this construction we can determine the radius ratio for the limiting condition, often called the "no rattle limit because if the radius ratio becomes smaller than this the cation will "rattle" in its site.

Using the Pythagorean theorem we can write:

(2Rz +2Rx)2 = (2Rz)2 + (22Rz)2

2Rz +2Rx = (4Rz2 + 8Rz2)2Rz +2Rx = (12Rz2)2Rz +2Rx = 3.464Rz

2Rx = 1.464Rz

Giving Rx/Rz = 0.732

Thus, for Rx/Rz < 0.732 the cation will be too small or will rattle in its site and the structure will have to change to 6-fold coordination.

Pauling's RulesLinus Pauling studied crystal structures and the types of bonding and coordination that occurs

within them. His studies found that crystal structures obey the following rules, now known as Pauling's Rules.

Rule 1Around every cation, a coordination polyhedron of anions forms, in which the cation-

anion distance is determined by the radius sums and the coordination number is determined by the radius ratio.

This rule simply sets out what we have discussed above, stating that the different types of coordination polyhedra are determined by the radius ratio, Rx/Rz, of the cation to the anion.

Rule 2, The Electrostatic Valency PrincipleAn ionic structure will be stable to the extent that the sum of the strengths of the

electrostatic bonds that reach an ion equal the charge on that ion.In order to understand this rule we must first define electrostatic valency, e.v. e.v = Charge on the ion/C.N.

For example, in NaCl each Na+ is surrounded by 6 Cl- ions. The Na is thus in 6 fold coordination and C.N. = 6. Thus e.v. = 1/6. So 1/6 of a negative charge reaches the Na ion from each Cl. So the +1 charge on the Na ion is balanced by 6*1/6 =1 negative charge from the 6 Cl ions.

Similarly, in the CaF2 structure, each Ca+2 ion is surroundedby 8 F- ions in cubic or 8-fold coordination.The e.v. reaching the Ca ion from each of the F ions is thus 1/4.Since there are 8 F ions, the total charge reaching the Ca ion is 8*1/4 or 2. So, again the charge is balanced.

Notice that in NaCl, each Cl ion is also surrounded by 6 Na ions in octahedral coordination.So, again, the 1/6 of a positive charge from each Na reaches the Cl ion and thus the Cl ion sees 6*1/6 = 1 positive charge, which exactly balances the -1 charge on the Cl.In the case of NaCl the charge is exactly balanced on both the cations and anions. In such a case, we say that the bonds are of equal strength from all directions.When this occurs the bonds are said to be isodesmic.

This is not the case for C+4 ion in triangular coordination with O-2. Here, e.v. = 4/3 (C has a charge of +4 and is coordinated by 3 oxygens). Thus, the 3 Oxygens each contribute 4/3 charge to the Carbon ion, and the charge on the carbon is balanced. But, each Oxygen still has 2/3 of a charge that it has not used. Thus, a carbonate structural group is formed - CO3-2.In cases like this, where the electrostatic valency is greater than 1/2 the charge on the anion (4/3 > 1/2*2), the anion will be more strongly bound to the central coordinating cation than it can be bonded to other structural groups.When this occurs the bonding is said to be anisodesmic

A third case arises when the e.v. reaching the cation is exactly 1/2 the charge on the anion. This is the case for Si+4 in tetrahedral coordination with O-2. Here, the e.v. reaching the Si is 4/4 =1. This leaves each Oxygen with a -1 charge that it has not shared. Since this -1 is exactly 1/2 the original charge on O-2, the Oxygens in the SiO4-4 group can be just as tightly bound to ions outside the group as to the centrally coordinated Si.

In this case the bonding is said to be mesodesmic.The SiO4-4 group is the basic building block of the most common minerals in the Earth's crust, the silicates.

Rule 3Shared edges, and particularly faces of two anion polyhedra in a crystal structure decreases

its stability.

The reason for this is that sharing of only corners of polyhedra places the positively charged cations at the greatest distance from each other. In the example shown here, for tetrahedral coordination, if the distance between the cations in the polyhedrons that share corners is taken as 1, then sharing edges reduces the distance to 0.58, and sharing of faces reduces the distance to 0.38.

Rule 4

In a crystal structure containing several cations, those of high valency and small coordination number tend not to share polyhedral elements.

Sharing of polyhedral elements for cations of high charge will place cations close enough together that they may repel one another. Thus, if they do not share polyhedral elements they can be better shielded from the effects of other positive charges in the crystal structure.Rules 1 through 5 maximize the cation - anion attractions and minimize the anion-anion and cation-cation repulsions.

Rule 5, The Principle of Parsimony

The number of different kinds of constituents in a crystal tends to be small.

This means that there are only a few different types of cation and anion sites in a crystal. Even though a crystal may have tetrahedral sites, octahedral sites, and cubic sites, most crystals will be limited to this small number of sites, although different elements may occupy similar sites.

Composition of the Earths Crust