Paulings Rules:

Lecture 5

Classification of Minerals; Crystal growth

Review from last time - Paulings Rules:

1. The Coordination (radius ratio) Principle – a coordination polyhedron of anions surrounds each cation. The cation-anion distance is determined by the sum of the cation and anion radii and the number of anions coordinating with the cation is determined by the relative size of the cation and anion.

2. Electrostatic Valency Principle – in a stable ionic structure, the total strength of the valency bonds that reach an anion from all neighboring cations is equal to the charge of the anion.

3. Sharing of Polyhedral Elements I – the existence of edges (and particularly faces) common to coordination polyhedra decreases the stability of ionic structures

4. Sharing of Polyhedral Elements II – in a crystal containing different cations, those with large valence and small coordination number tend not to share polyhedral elements with each other.

5. Principle of Parsimony – the number of essentially different kinds of constituents in a crystal tends to be small.

Chemical Variation in Minerals

Pauling’s rules can be used to explain not only crystal structure, but also allowed variations in mineral compositions.

Only a few minerals have a fixed composition – even quartz, which comes about as close as possible to being purely SiO2, has trace amounts of other elements, yielding varieties of quartz like

amethyst Fe³⁺

rose quartz Ti⁴⁺

smoky quartz structural defects

This raises an issue of terminology:

major elements are fundamental to the mineral, control its structure and gross physical properties

minor elements are present in small amounts (up to a few %), usually as substitutes for major elements

trace elements are present in extremely small amounts but are often responsible for mineral color.

We also need to introduce the idea of mineral formulas, which is how we describe mineral compositions. Conventions:

Cations are listed before anions and anion complexes

Subscripts outside of parentheses refer to everything inside

cations in same structural site are grouped together

elements separated by a comma inside parentheses are substitution pairs

Charges must balance

Largest cations are listed first (listed in order of decreasing coordination number)

Loosely bonded interstitial compounds are on the right; when water occupies an interstitial site it is often separated from the main formula by a dot.

EX: olivine (Mg, Fe)₂SiO₄

natrolite Na₂Al₂Si₃O₁₀.2H₂O

montmorillonite (Na, Ca)(Al,Mg)₂(Si₄O₁₀)(OH)₂.nH₂O

We can also write a single formula in different ways to convey different information. Let’s look at an oxide mineral that is a type of spinel.

Idealized formula Fe₂ZnO₄

The cations are Fe³⁺_{and Zn}²⁺_{, and they coordinate with O}^2-_{, which is the only anion.}

Based on knowledge of cation size, we can anticipate that Zn will be in 4-fold (tetrahedral) coordination and Fe in 6-fold (octahedral) coordination, consistent with the order in which they are written.

Structural formula ^VIFe₂^IVZnO₄

Often, a structural site may be interchangeably occupied by different cations as part of a solid solution series. In this case, the interchangeable cations are grouped within parentheses. The spinel formula shown above can be modified to show that the octahedral sites can hold either Fe³⁺ or Mn³⁺ ions and the tetrahedral sites can hold either Zn²⁺ or Fe²⁺ ions.

General formula (Fe, Mn)₂(Zn,Fe)O₄

Note that in this formula, the cations in parentheses are conventionally assumed to be listed in order of decreasing abundance… that is, Fe is more likely than Mn to occupy the octahedral site, while Zn is more likely than Fe to occupy the tetrahedral site.

Using certain analytical techniques, it is possible to determine the proportion or relative abundance of each type of cation occupying a substitution site in a given sample. This information yields the sample’s specific mineral formula, which could look something like this:

Specific formula (Fe_1.4 Mn_0.6)(Zn_0.8 Fe_0.2)O₄

In this example, Fe³⁺ ions proportionally across the structure occupy 1.4 of every two filled octahedral sites, while Mn³⁺ ions occupy the remaining 0.6 of every two filled octahedral sites. Similarly, Zn²⁺ ions proportionally occupy 0.8 of every filled tetrahedral site, while Fe²⁺ ions occupy 0.2 of every filled tetrahedral site.

HOMEWORK

Solid Solutions

The discussion above leads directly to a discussionof substitutions of one element for another within the stable mineral structure called isostructural substitutions. This process is known as solid solution, defined in a mineral structure as specific atomic sites that are occupied in variable proportions by two or more different chemical elements.

Three main factors determine whether or not solid solution is possible:

1. Comparative size of ions (atoms, molecules) that are substituting for one another This results directly from Pauling’s first rule of radius ratios, in that ions that substitute must be able to occupy the same interstitial site. Generally, for this to happen the radius ratios must be within 15%; substitution is unlikely when the radii differ by > 30%.

2. The valence state (charge) of the ions involved in the substitution. This stipulation relates to Pauling’s second rule, which involves electrical neutrality. If the substituting elements have the same charge (Fe²⁺ and Mg²⁺; Na⁺ and K⁺), then neutrality will be maintained. If the charges are different (Al³⁺ and Si⁴⁺; Na⁺ and Ca²⁺), then another ionic substitution must take place to maintain neutrality – this is called a coupled substitution, for example Ca²⁺Al³⁺for Na⁺Si⁴⁺

3. The temperature at which the substitution takes place. Substitution of ions of different size is favored by elevated temperatures, where the structure is expanded and there is greater tolerance for size variation.

Types of substitution

Simple cationic/anionic: Ions of similar size and charge substitute for each other. Examples:

K = Na	KCl – NaCl (sylvite - halite); KAlSi₃O₈-NaAlSi₃O₈ (orthoclase – albite)
Mg = Fe (= Mn)	Mg₂SiO₄ – Fe₂SiO₄ – Mn₂SiO₄ (forsterite – fayalite - tephroite; olivine) MgSiO₃ – FeSiO₃ (enstatite – ferrosilite; pyroxene)
Cl - Br	KCl - KBr
Fe = Zn	(Zn, Fe)S (sphalerite)

Depending on the relative sizes of the ions involved, the solid solution may be either partial (K = Na; ionic radii 1.46:1.08 in 6-fold coordination) or complete (Mg = Fe; ionic radii 0.77:0.80 in 6-fold coordination).

Coupled substitution: For electrical neutrality to be maintained, substitution of two elements requires an additional substitution. Examples:

Fe²⁺ + Ti⁴⁺ = 2Al³⁺	(Al, Ti)₂O₃ (corundum, var. sapphire)
Ca²⁺Al³⁺ = Na⁺Si⁴⁺	CaAl₂Si₂O₈-NaAlSi₃O₈ (plagioclase)
Mg²⁺ + 2Al³⁺= 2Fe²⁺ + Ti⁴⁺	(Mg, Fe)(Al, Ti)₂O₄ (spinel group)

Interstitial substitution: Between some ions or ionic groups there may exist structural voids. Particularly where these have the form of channels (as in beryl and some zeolites), they may be partially filled. Example:

BERYL Be₃Al₂Si₆O₁₈ may contain substantial amounts of Li, Na, K, Rb through coupled substitutions involving Si⁴⁺ and Al³⁺

Vacancy solid solution: remember that close packing of anions often creates more cation sites than can be filled. Partial filling of these sites forms another type of substitution. A common example is the mineral amphibole, which has the end member

TREMOLITE [] Ca₃Mg₅Si₈O₂₂(OH)₂

where [] represents a vacant site that may be filled using the coupled substitution

[] + Si⁴⁺ = Na⁺Al³⁺

Omission solid solution: this is the opposite of filling a vacancy, that is, creating one. An example is the substitution of the large Pb²⁺ cation for the equally large K⁺ cation as

K⁺ + K⁺ = Pb²⁺ + []

The result of these substitutions is a wide variety of mineral and mineral formulas!!!

Crystallization and polymorphs

So – how, and why, do crystals form?

Crystals typically form from a supersaturated solution, as we experimented with in lab. That solution may be an aqueous phase, a magma, or a gas. During metamorphism we also see examples of solid state crystallization (that is, one crystal growing from another solid).

We may create a supersaturated solution by changing the temperature, changing the pressure, or changing the composition (by either adding or subtracting components). We usually show mineral stability fields using phase diagrams, as shown below for the system SiO₂.

The figure shows graphically the ranges of pressure-temperature conditions under which each of the different polymorphs are stable. At low temperatures and pressures like those at the Earth's surface, low quartz is the thermodynamically stable polymorph, accounting for its great abundance. As P and/or T change however, the low quartz structure becomes unstable, and the SiO4 tetrahedra rearrange themselves to form new structures each of which has different symmetry:

Polymorph Symmetry

low quartz trigonal

high quartz hexagonal

tridymite orthorhombic

cristobalite tetragonal

coesite monoclinic

stishovite tetragonal

You will note from the figure that the reaction boundaries separating adjacent stability fields are either quite P-dependent (roughly parallel to the T-axis, e.g., coesite = stishovite), quite T-dependent (roughly parallel to the P-axis, e.g., high quartz = liquid), or dependent on both P and T. As a rule, if a reaction is very pressure dependent, this means that the polymorph on the high-P side of the boundary has a lower molar volume that that on the low-P side of the boundary. This is illustrated by the transition from coesite (density = 2.93 g/cc) to stishovite (density = 4.30 g/cc), which is accompanied by a nearly 50% increase in density or a nearly 50% decrease in molar volume.

The phase transitions between the various polymorphs are of two types with that between low quartz and high quartz being of a type referred to as displacive and all others being of a type known as reconstructive. The reconstructive transitions are most familiar and consist of actual breaking of bonds and formation of new bonds in a different configuration. Transitions of this type require considerable energy (enthalpy of reaction) to break the bonds and, as a consequence, these reactions are often sluggish leading to the common preservation of high-T or high-P forms at conditions way outside of their stability fields. By contrast, the displacive transitions do not require any breakage of bonds but instead, bonds and polyhedra rotate to new positions with different symmetry. Transitions of this type are extremely rapid and consequently, only the low-T, low-P forms are preserved at the Earth's surface for us to collect.