GE 321: Crystal Chemistry

*Nonuniform Bond strength: Anisodesmic and mesodesmic*

*Anisodesmic* because they contain bonds of differing strengths. When the strengths of bonds between the central cation and its coordinating anions equal exactly half the charge of the anion the term is *Mesodesmic*

*Substitutional*

*Complete binary solid solution series*, meaning that substitution of one ion by another occurs over the total possible compositional range as defined by two end member compositions.

*Twin Classification* •Contact Twins

*Contact twins* have a regular composition surface separating the two individuals. The twin is defined by a twin plane (A).

*Deformation Twinning*

*Deformation twinning* is another type of secondary twinning and results when a crystal is deformed by the application of a mechanical stress. If the stress produces atomic slippage on a small scale, gliding or deformation twins result.

*Growth Twins*

*Growth Twins* are the result of an emplacement of atoms, or ions on the outside of a growing crystal in such a way that the regular arrangement of the original crystal structure (and, therefore, its lattice) is interrupted.

*Isostructuralism*

*Isostructuralism* This forms isostructural groups of minerals related to each other by analogous structures, generally having a common anion but different cations. Crystals in which the centers of the constituent atoms occupy geometrically similar positions, regardless of the size of the atoms or the absolute dimensions of the structure, are said to belong to the same structure type. •for example, all isometric crystals in which there are equal numbers of cations and anions in a 6-fold coordination belong to halite structure type(KCL, NACL, AGCL, MGO, PBS, MNS)

*Omission Solid Solution*

*Omission solid solution* occurs when a more highly charged cation replaces two or more lower-charged cations. To maintain charge balance, another site (or sites) is left unfilled, or vacant.

*Twin Classification* •Penetration Twins

*Penetration twins* are made up of interpenetrating individuals having an irregular composition surface, and the twin law is usually defined by a twin axis direction (D to F).

*Nonuniform Bond strength: Polymerize*

*Polymerize to form chains, sheets or network by sharing oxygen.

*Polymorphs*

*Polymorphs*When two minerals have the same chemical formula but different structures. When radius ratios are near limiting values, the cation may occur in structures in one of two different coordination polyhedra. At such limiting values, the cation has a close fit for either coordination polyhedron.

*Twin Classification* •Polysynthetic Twins

*Polysynthetic Twin* (A, B, C). When a large numbers of individuals in a polysynthetic twin are closely spaced, crystal faces or cleavages crossing the composition planes show striations because of the reversed positions of adjacent individuals.

K&D Chapter 14 *Qualitative vs. quantitative*

A *qualitative* analysis involves the detection and identification of all the chemical constituents of a compound (what is present). A *quantitative* analysis involves the determination of the weight percentages (or parts per million composition) of all elements in a compound (how much of each is present).

*Reconstructive Polymorphism*

A *reconstructive* polymorphic reaction involves the breaking of atomic bonds and a reassembly of the structural units into different arrangements (Fig). This type of transformation requires a large amount of energy, is not readily reversed and proceeds sluggishly. Therefore, one polymorph may exist metastably in the stability field of another. The persistence of metastable minerals testifies to the fact that high energy is required to activate a reconstructive polymorphic transformation. •Such as Andalusite, sillimanite, and kyanite •Tridymite or cristobolite to low quartz (high temperature) coesite and stishovite are high temperature.

*Mineral Identification process*

A diffractometer tracing for low quartz is given in figure. The 2θ positions of the diffraction peaks in such a tracing can be read off directly or they can be tabulated as 2θ positions by an online computer. The interplanar spacings giving rise to them are calculated using the equation nλ=2dsinθ. Once a diffractometer tracing has been obtained and the various diffraction peaks have been tabulated in a sequence of decreasing interplanar spacing (d), together with their relative intensities (I, with the strongest peak represented by 100 and all other peaks scaled with respect to 100), the investigator can begin the mineral identification process.

*This diffraction process is the basis for single crystal and powder X-ray techniques*

A monochromatic X-ray beam is parallel to a cleavage plate of halite, and the plate is supported in such a way that it can be rotated about an axis at right angles to the X-ray beam. As the halite is slowly rotated, there is no "reflection" until the incident beam makes an angle θ that satisfies the Bragg equation, with n=1. on continued rotation, there are additional reflections only when the equation is satisfied at certain θ angles with n=2, 3,.... these are known as first-, second-, and third-order reflections. These reflections are the diffraction effects that occur when the three diffraction cones about three noncoplanar rows of atoms intersect in a common direction.

*Calculation of Mineral Formulae From Metal Percentages*

A quantitative analysis is reported in weight percentages (wt. %) of metals, or oxides, and is a listing of which elements are present in what concentrations. Should add up to 1% of 100%.

*Polytypism*

A special kind of polymorphism, known as *polytypism*, occurs when two polymorphs differ only in the stacking of identical, two-dimensional sheets or layers. As a consequence, the unit cell dimensions parallel to the sheets will be identical in the two polytypes. However, the atomic spacing between the sheets (or layers) will be related to each other as multiples or submultiples.

*Order-Disorder Polymorphism*

A third type of polymorphism is referred to as an *order-disorder transformation*. Recall that perfect order in a mineral structure occurs only at absolute zero. An increase in temperature disturbs a perfectly ordered structure, until at some high temperature, a totally disordered (random) state may occur. As such, there is no definite transition point between perfect order and complete disorder; a continuum of structural states. Slow cooling of such a high temperature mineral allows the randomized ions (at high temperatures) to select specific sites in the structure and become more ordered as temperature decreases. Therefore, a mineral may exist in various states of disorder, of which a totally disordered and perfectly ordered state are two extreme conditions.

*Exsolution and diffusion*

An increase in temperature will increase mobility of an atom and its chances of breaking away from its neighbors and moving to a new position. This means diffusion rate is strongly dependent on upon temperature. As temperature is lowered, as in the unmixing process (Fig), the mobility of the atoms may become so limited that the original homogeneous mineral cannot split into the pure end-member compositions. The phase/assemblage with the lowest free energy will be the most stable.

*Interstitial Solid Solution*

Between atoms, ions, or ionic groups of a crystal structure, interstices may exist that normally are empty. Occasionally, ions or atoms occupy these structured sites resulting in what is known as *interstitial substitution*

*Bragg's Law*

Bragg showed that a "reflection" takes place from a family of parallel planes only under certain conditions. These conditions must satisfy the equation nλ=2dsinθ, where n is an integer (1,2,3,....n), λ is the wavelength, d is the distance between successive planes, and θ is the angle of incidence and "reflection" of the X-ray beam fro the given atomic plane.

*Mineral Formulae for Hydrous Silicates* Complex hydrous silicates, such as amphiboles, are recalculated by the same sequence of steps as olivine and pyroxenes, but the H₂O content is evaluated as (OH) groups in the amphibole structure.

Column 1 lists weight percentages for H₂O(+) and H₂O(-). The H₂O(+) is considered part of the amphibole structure, but the H₂O(-) is not, and is neglected. Column 2 lists the molecular proportions, column 3 the cation proportions, and column 4 gives total contribution of (O,OH) for each of the "molecules" in column 2. An amphibole has 24 (O+OH) and the sum in column 4 (2.8301) divided into 24 gives the ratio by which the entire analysis must be multiplied by the factor of 8.4803, giving the results in column 5 on the basis of 24(O,OH)

*Variability of Mineral Compositions* •Solid Solution

Compositional variations is known as *solid solution* and occur when minerals as a result of chemical substitution in the crystal structure. One ion or ionic group can exchange or substitute with another ion or ionic group occupying a specific structure site in the mineral.

*Common Ions: C⁴⁺*

Coordiation Number with oxygen: 3 •triangular •Ionic Radius in A: -.08(3)

*Common Ions: Al³⁺*

Coordiation Number with oxygen: 4, •Tetrahedral •Ionic Radius in A: .39(4)

*Common Ions: S⁶⁺*

Coordiation Number with oxygen:4, •Tetrahedral •Ionic Radius in A: .12(4)

*Common Ions: P⁵⁺*

Coordiation Number with oxygen:4, •Tetrahedral •Ionic Radius in A: .17(4)

*Common Ions: Si⁴⁺*

Coordiation Number with oxygen:4, •Tetrahedral •Ionic Radius in A: .26(4)

*Common Ions: Al³⁺*

Coordiation Number with oxygen:6, •Octahedral •Ionic Radius in A: .54(6)

*Common Ions: Ca²⁺*

Coordination Number with oxygen: 8-6,Cubic to octahedral •Ionic Radius in A: 1.12(8)-1.0(6)

*Common Ions: Fe²⁺*

Coordination Number with oxygen: 6, Octahedral •Ionic Radius in A: .78(6)

*Common Ions: Mn₂⁺*

Coordination Number with oxygen: 6, Octahedral •Ionic Radius in A: .83(6)

*Common Ions: K⁺*

Coordination Number with oxygen: 8-12 •ionic Radius in A: 1.51(8)-1.64(12)

*Common Ions: Na⁺*

Coordination Number with oxygen: 8-6, Cubic to octahedral •Ionic Radius in A: 1.18(8)-1.02(6)

*Common Ions: Fe³⁺*

Coordination Number with oxygen: 6, •Octahedral •Ionic Radius in A: .65(6)

*Common Ions: O²⁻*

Coordination Number with oxygen: N/A •ionic Radius in A: 1.36(3)

*Common Ions: Ti⁴⁺*

Coordination Number with oxygen:6, •Octahedral •Ionic Radius in A: .61(6)

*Uniform Bond Strength*

Crystals in which all bonds are equal strength are called *Isodesmic*. This the case for ionically bonded crystals that have a single cation and anion.

*Epitaxis*

During growth of a mineral, offsets to the atomic arrangement of the structure may occur that there are non random. This results in the development of relatively common intergrowth patterns of well-formed crystals. These intergrowths may be of minerals with different compositions, or they may be minerals of the same composition. A type of non-random, crystallographically oriented growth of one crystalline substance on another of different composition is known as *epitaxis*.

*Radioactivity and Metamictization*

During mineral formation, radioactive elements may be incorporated into the crystal structure. The nuclei of these elements are unstable and decay spontaneously to different kinds of nuclei, with the release of radioactive energy in the process. Examples of geologically important unstable nuclei are ⁴⁰K, ⁸⁷Rb, ²³²Th, ²³⁸U. In the decay process such unstable nuclei, nuclear particles (alpha and beta particles) are emitted as gamma rays.

*Relation between energy and wavelength*

E=h•f=h•c/λ Where E is energy, v is frequency, c is velocity of light, λ is wavelength, h is planks constant. This equation shows that the shorter the wavelength the greater its energy, thus, the greater its powers of penetration.

*Coupled substitution*

Electrical neutrality is maintained by substitution of more than one ion. Such as Fe²⁺ and Ti⁴⁺ substituting for 2 Al³⁺ in corundum to form sapphire.

*Triangular Diagrams of more than three components* Triangular diagrams limit representation to only three components, which may be simple elements, compound oxides, or more complex components as expressed by mineral formulae. To represent more than three components on one triangle, some components are commonly combined, and some may not be considered in the graphical representation.

For example, to represent carbonate compositions in the system CaO-MgO-FeO-MnO-CO₂, the possible number of compositional variables must be reduced. Because all carbonates contain CO₂, no additional information is grained in using the CO₂, components to show the small CO₂ variations that exist. CO₂, therefore, is ignored in the graphical representation. Because Fe²⁺ and Mn²⁺ substitute easily for each other in the carbonate structure, and because Mn²⁺ typically is far less abundant in most environments than Fe²⁺, FeO and MnO are combined. This leaves three components CaO, MgO, and (FeO + MnO).

*Triangular Diagrams* Mineral analysis can be very complex and typically suggest substitution of several elements in the same atomic site of the structure. To portray the variation of three components, a triangular diagram is used. Triangular diagrams are basically three linear diagrams, which share components, combined.

For plotting in weight percent, the corners of the triangle are marked as 100 wt.% SiO₂, 100 wt.% MgO, and 100 wt.% FeO (B). The four minerals can be plotted because they contain only these three oxides. The MGO scale extends 0% to 100% on the left side of the triangle, and it is along this side that the MgO values are directly plotted. Both minerals contain no FeO, so that they lie on the edge directly between SiO₂ and MgO. The right side of the triangle extends from 100% to SiO₂ to 100% FeO; with FeO (0% to 100%) increasing along the right side from top to bottom. Along this side lie the compositions of the Fe-end members (A) on the basis of their FeO weight percentage values. The end members of both series are joined by a dashed line to portray the locations where intermediate compositions plot. The final plot, in weight percent, is easy enough to achieve if weight percentage values are available.

*Twin Classification* •Cyclic Twins

If successive composition planes are not parallel a *cyclic twin* results (D and E).

*X-ray Powder Diffraction and Mineral Identification*

If the rays forming these cones are permitted to fall on a flat photographic plate at right angles to the incident beam, a series of concentric circles will result. However, only reflections with small values of the angle 2θ can be recorded in this manner. In order to record all possible diffraction cones that may occur a film method is used in which the film is wrapped around the inside of a cylindrical camera.

*Characteristic Radiation*

If the voltage across the X-ray tube is increased to a critical level, which is dependent on the element of the target, a line spectrum of *characteristic radiation*specific to the target material becomes superimposed on the continuous spectrum. The characteristic X-ray spectrum is produced when the bombarding electrons have sufficient energy to dislodge inner shell electrons. The electrons transition from outer to inner shells, are accompanied by the emission of X-radiation with specific energy/wavelengths producing Kβ and Kα peaks (B). •L to K produce Kα •M to K produce Kβ

*Cubic Closest Packing*

In *cubic closest packing* every fourth layer repeats. Here, the third layer is stacked on top of the second layer of the AB sequence in the dimples that directly overlie the C voids in the first layer. This third layer, C, is not equivalent in position to layer A or B so that a three-layer sequence ABC is created as in (C). When this stacking sequence repeats infinitely, it results in an ABCABCABC sequence. •Max spheres=12 •Min Spheres= 6

*Displacive Polymorphism*

In a *displacive* polymorphic transformation, only a slight displacement of atoms or ions and readjustment of bond angles (kinking) between ions is needed. The structure is generally left completely intact and no bonds between ions are broken so that only a small amount of energy is required. This type of transformation occurs instantaneously and is easily reversible, that is, one polymorph can easily transform to other and vice versa.

*Rule 4: Sharing of polyhedral elements ii*

In a crystal containing different cations, those of high valence and small coordination number tend not to share polyhedral elements with each other. This maintains maximum distance between between cations.Small, highly charged cations form low-coordination number anionic groups where more than half of the charge is neutralized within the group.

*In a row of regularly spaced atoms that is bombarded by X-rays*

In a row of regularly spaced atoms that is bombarded by X-rays, every atom can be considered the center of radiating, spherical wave shells (Fig). When the scattered waves interfere constructively they produce wave front that are in phase, and diffraction will occur.

*Paulings Rules*

In both closest packing arrangements, interstices exist. These empty spaces can accommodate smaller spheres. It follows then, that most minerals are composed of numerous elements, with various ionic or atomic sizes. The bonding in many of these minerals, such as those between oxygen and other common elements (Al, Na, K, Ca, Fe, Mg) is primarily ionic. Therefore, to a first approximation mineral structures may considered to consist of different size spheres packed tightly together. In general, it is noted that anions form closest packed array, and the smaller cations fill the voids in this closest-packed array,

*Commontwin laws* •Isometric

In the 4/m3⁻2/m class of the isometric system, the twin axis, with a few rare exceptions, is a three-fold symmetry axis, and the twin plane is parallel to a face of the octahedron. This type of twin is common in spinels and is called spinel twin. IN the class 2/m3⁻, two pyritohedrons may form penetration twins with a 90° rotation about the twin axis [001]. This twin is known as the iron cross. The morphology expressed by twinned crystals can be highly diagnostic in the identification of a mineral species.

*Commontwin laws* •Hexagonal

In the hexagonal system, the carbonates, especially calcite, serve as excellent examples of three twin laws. Several examples are shown in (Fig). The twin plane may be {0001}, with c, the twin axis (A), or it may be the positive rhombohedron [101⁻1]. However, twinning on the negative rhombohedron [011⁻2] is most common (B) and may yield contact twins or polysynthetic twins as the result of pressure. The ease of twinning according to this law can be demonstrated by the artificial twinning of a cleavage fragment of transparent calcite by the pressure a knife blade (B). Quartz shows several types of twinning. (C) illustrates the brazil law with the twin plane parallel to [112⁻0]. Here, right- and left-handed individuals have formed a penetration twin. (D) shows a Dauphine twin, which is penetration twin with c, the twin axis. Such twins are composed either of two right-handed or two left-handed individuals. (Fig) also illustrates the Japan Law with the twi plane [112⁻]. The reentrant angles usually found on twinned crystals are not present in either Brazil or Dauphine Twins.

*Commontwin laws* •Monoclinic

In the monoclinic system, twinning on {100} and {001} is most common. (Fig) illustrates gypsum with {100} as the twin plane producing what is known as a shallow-tail twin. This same figure also shows three twin laws that occur in the mineral orthoclase. Two of these are contact twins: Manebach twin with {001} as the twin plane, and a Baveno twin with {021} as the twin plane. The most common twin in orthoclase is the Carlsbad twin, an interpenetration twin in which the c axis, [001], is the twin element. IN this case, the two individuals are united along an irregular surface roughly parallel to (010).

*Commontwin laws* •Orthorhombic

In the orthorhombic system, the twin plane is most commonly parallel to a prism face. The contact twin of aragonite and the cyclic twins of aragonite and cerusite are all twinned on {110} (A and B). The pseudo-hexagonal appearance of the cyclically twinned aragonite results from the fact that (110) ^ (11⁻0) is nearly 60°. The twinning of such a crystal is recognized by the presence of reentrant angles that occur between individual crystals (A and B). The mineral staurolite, which is monoclinic with a β angle close to the 90°, is pseudo-orthorhombic and morphologically appears orthorhombic. It is commonly found displaying two types of penetrations twins. In one, with {031} as the twin plane, a 90° cross results; in the other, with twin plane {231}, a 60° cross is formed (C).

*Driving force of exsolution*

Is the minimization of the free energy(gibbs) in the crystal. Free energy varies as a function of composition at constant T. •Gibbs=E+PV-TS E=internal energy PV=Pressure volume TS= temperature-entropy term

*Diffraction Effects and the Bragg Equation*

Minerals consist of an ordered three-dimensional structure with characteristic periodicities along the crystallographic axes. When an X-ray beam strikes such a three-dimensional orderly arrangement, it causes electrons in its path to vibrate with a frequency of the incident X-radiation. These vibrating electrons absorb some of the X-ray energy and, acting as a source for new wave fronts, emit (scatter) this energy as X-radiation of the same frequency and wavelength. In general, the scattered waves interfere destructively, but in some specific directions, they reinforce one another (interfere constructively) to produce a cooperative scattering known as *diffraction*

*Non-random intergrowths of the same composition and substance*

Non-random intergrowths of the same composition and substance are *parallel growths* and *twinned crystals*. An aggregate of similar crystals with their crystallographic axes and faces parallel to each other is called *parallel growth*.

*Other Coordination Geometries*

Other coordination numbers are known, such as 5-, 7-, 9-, 10-fold coordination. An example of 5-fold coordination is found in the mineral andalusite. Such coordination numbers are possible only in complex structures in which the anions are not closely packed.

*4-fold coordination*

Radius of .225 to .414, four anions touch each other and the central cation. This results in a changeover from triangular coordination to 4-fold coordination. The centers of the anions are at the corners of a tetrahedron, thus, 4-fold coordination is termed *tetrahedron coordination*.

*6-fold coordination*

Radius ratio of .414 to .732, 6-fold coordination becomes stable. The centers of the coordinated ions lie at the corners of an octahedron, thus, it is termed *octahedral coordination*. Remember, an octahedron has eight faces but only six apices, hence *C.N.* is 6

*8-fold coordination*

Radius ratio of .732 to 1.0 results in eight anions touching each other as well as the central cation and 8-fold coordination results. The coordinating ions lie at the eight corners of a cube, thus, 8-fold coordination is termed *cubic coordination*.

*12-fold coordination*

Radius ratio of 1, when cations and anions have similar sizes, 12-fold coordination is stable. Each cation can now be in contact with 12 anions. *C.N.*=12

*Miscibility Gap*

Represents a region of immiscibility in temperature-composition space. For a specific composition (X1), in the region above the miscibility gap at T₂, the structure of the silicate is in a high-energy state that allows accommodation of A and B (although radius size differs >25%) in the same atomic sites

*Solid Solution* •Availability

The *availability* of the ion(s). For solid solution to occur, substituting ions must be readily available.

*Solid Solution* •Charges

The *charges* of the ions involved in the substitution. Mineral structure must maintain electrical neutrality. If the charges of the substituting ions are the same, as in Mg²⁺ and Fe²⁺, the ionic replacement remains electrically neutral and is, therefore, more likely. If the charges are not the same, as in the case of Al³⁺ substituting for Si⁴⁺, an additional ionic substitution in another structural site must take place in order to maintain overall electrostatic neutrality. When two or more ions substitute in different structure sites to maintain charge balance, it is called *coupled substitution*.

*Solid Solution* •Relative Size

The *relative sizes* of the ions, atoms, or ionic groups that are substituting of reach other. Generally, a wide range of substitutions is possible if the size difference between the ions replacing one another is >~15%, If the radii of the two elements differ by 15 to 30%, substitution is limited or rare, and if the radii differ by more than 30%, substitution is poor to nonexistent.

*Polymorphism*

The ability of a specific chemical substance to occur with more than one type of structure (as a function of temperature and pressure, or both) is known as *polymorphism*. The various structures of such a chemical element or compound are known as *polymorphs*.

*Mineral Formulae from oxide weight percentages* The majority of minerals, such as silicates, oxides, carbonates etc. are compounds containing large quantities of oxygen. By convention, the analysis of these minerals are reported as percentages of oxides, rather than as percentages of elements. •Gypsum

The analytically determine oxide components in column 1 are divided by the molecular weights of the corresponding oxides (column 2) to arrive at molecular proportions (column 3). The molecular ratios in column 4 we see that CaO: SO₃:H₂O=1:1:2. This composition can be written as CaO•SO₃•2H₂O, or as CaSO₄•2H₂O.

*2-fold coordination*

The cation only maintains contact with two anions. Here the radius <.155 •*Linear Coordination*

*Radius Ratio*

The coordination number, and the resulting geometric arrangement is a function of the relative size of the cation and anion. The relative size of ions is generally expressed as a *radius ratio (R.R.)=cation/anion* The radius ratio dictates how many anions can fit snuggly around the cation.

*X-ray Powder Diffraction and Mineral Identification*

The diffraction maxima from a given set of planes form cones with the incident beam as axis and the internal angle 4θ. Any set of atomic planes yields a series of nested cones corresponding to "reflections" of the first, second, third, and high orders (N=1, 2, 3..). Different families of planes with different interplanar spacings will satisfy the Bragg Law at appropriate values of θ for different sets of nested cones of reflected rays

*Rule 3: Sharing of Polyhedral elements i*

The existence of edges, and particularly of faces, common to two anions polyhedra in a coordinated structure, decreases the stability of ionic structures. This, in effect, is a reminder that cations like to be as far apart as possible because of cation repulsion. Recall that every ion in a crystal structure has some effect on every other ion-it is attracted if the charges are opposite, repulsed if the charges are the same. Yet consistent with the coordination of the anions that result in electrical neutrality.when coordination polyhedra share corners, the cations are farthest apart, followed by sharing an edge *large cations with have valency and small CN, ESPESCIALLY WITH LARGE RADIUS RATIOS DO NOT WANT TO SHARE FACES*

*Commontwin laws* •Triclinic System

The feldspar best illustrate twinning in the triclinic system. They are almost universally twinned according to the albite law, along the {010} twin plane, as shown in (A and B). Another important type of twinning in triclinic feldspar is according to the pericline law, along [010], the twin axis. When albite and pericline twins are commonly interwoven, as frequently occurs in microcline, a typical cross-hatched/tartan pattern can be seen. •Triclinic feldspars also twin according to the same laws as monoclinic feldspars.

*Calculation of Mineral Formulae From Metal Percentages*

The formula of chalcopyrite is CuFeS₂; thus, the gram-formula weight is 183.52 (column 2). Dividing the weight of each element by the total weight and converting these values to percentages, yields the calculated Cu, Fe, and S weight percentage values, very similar to the measured percentages reported in column 1.

*Commontwin laws* •Tetragonal

The most common type of twin in the tetragonal system has {011} as the twin plane. Crystals of cassiterite and rutile, twinned according to this law, are seen in (Fig).

*Rule 5: The principle of parsimony*

The number of essentially different kinds of constituents in crystal tend to be small. There are only a few types of contrasting cation and anion sites. Thus, in structures with complex chemical composition, a number of different ions may occupy the same structural position. These ions are then considered as single "constituent". This phenomenon leads to what is termed solid solution

The twin operation is known as a *Twin Law*, which states whether there is a center, an axis, or a plane of twinning.

The operation that relate a crystal to its twinned counterpart are symmetry operations: •(1) reflection by mirror plane, *twin plane* •(2) rotation about a crystal direction common to both, *twin axis*, with the angular rotation normally 180° •(3) inversion about a point, *twin center*. The surface on which two individuals are is known as the *composition surface* or *composition plane*.

*Unmixing*

The original homogenous mineral segregates into two chemically different minerals. Region of unmixing represented by the miscibility gap continually widens as the temperature is lowered from T4-T1, structure becomes less tolerant of ionic size differences.

*X-ray Powder Diffraction and Mineral Identification*

The original specimen is prepared by grinding it to a fine powder, which is bonded wiht an amorphous material into a small spindle, or surface of a slide, or a special rectangular sample holder.

*Coordination of Ions*

The packing of atoms compromising metals is first discussed. In any given metal consisting of a single element, all atoms are assumed to be the same size and all are spherical in shape based on electron density maps.These metal atoms pack together in an ordered arrangement that minimized void space. There are two primary packing arrangements, and the structure are termed collectively, *closest packing*. These structures are based on *hexagonal closest packing* and *cubic closest packing*

*Why does constant chemical composition have different structural arrangements?

The reason why a constant chemical composition may have different structural arrangements is the tendency of a crystal structure to minimize its internal energy. A higher internal energy, as a function of increasing temperature, is caused by higher frequencies of atomic vibrations. Pressure also can be a major driving force in polymorphic transformations. Increasing pressure favors the development of structural arrangements that result in an increase in the density of atomic packing.

*Mineral Formulae from oxide weight percentages* The majority of minerals, such as silicates, oxides, carbonates etc. are compounds containing large quantities of oxygen. By convention, the analysis of these minerals are reported as percentages of oxides, rather than as percentages of elements. •Olivine

The steps in going from columns 1 to 3 are the same as in the gypsum analysis. Column 4 lists the values for the atomic proportions of the various atoms, based on the molecular proportions determined in column 3. To arrive at the numbers in column 4 and 5, the molecular proportion of each element is multiplied by the number of cations and anions in the oxide. For example, one molecule of SiO₂ contributes one Si (1 X column 3= atomic proportion of cation in column 4). SiO₂ contributes two oxygens (2 X column 3 = atomic proportions of column 5. The total number of oxygens, contributed by the atomic proportions of each oxide in column 5, is 2.3535. At this point, a mineral formula of Mg.₆₇₀₈Mn.₀₀₇₈Fe.₅₁₁₈O₂.₃₅₃₅ has been calculated. Olivine, with the general formula (Mg,Fe)₂SiO₄ has four oxygens (per formula unit). To arrive at the cation proportions in terms of four oxygens instead of 2.3535, one must multiply each cation number in column 4 with 1.699 (ratio of 4/2.3535), referred to as the oxygen factor. This leads to the numbers in columns 6 and 7. If the oxygens are multiplied by this amount arrive at simple whole numbers, then the cation proportions must be multiplied by the same amount to maintain the correct ratios. The final chemical formula for olivine is Mg₁.₁₄Fe.₈₇Mn.₀₁SiO₄

*Hexagonal Closest Packing*

The third layer is positioned with the spheres resting resting in the dimples of the second layer, directly over the first layer (A). The spheres alternate between only two positions (A and B) and this sequence of stacking can be represented by the combination of AB (B). This extends upward by another layer of spheres on top of the B voids, giving rise to an infinite stacking sequence, ABABAB.

*Solid Solution* •Temperature and Pressure

There is, in general, a greater tolerance toward ionic substitution at higher temperatures when thermal vibration are greater resulting in expanded structure. At elevated temperatures, the size of available atomic sites are larger and more tolerant of size differences. Therefore, in a given structure, one expects a larger variability in mineral composition at higher temperatures than that at lower temperature. The converse occurs with increasing pressure. As pressure increases, crystal structures compress and are less tolerant to size discrepancies. Temperature is typically the overriding factor.

*Rule 1: The coordination Principle*

This principle states that the relative sizes of the cation and anion determine how they pack together, or coordinate. A *coordination polyhedron* of anions is formed about each cation, the cation-anion distance is determined by the radius sum, and the *coordination number* of the cation is by the radius ratio. When oppositely charged ions unite to form a crystal structure with dominantly ionic bonding, each ion tends to gather to its self as many ions of opposite sign as size permits. The maximum number of coordinating ions is limited by the requirement that the cation must maintain contact with all of the surrounding anions. in accordance with this rule, the cation will bond with as many anions as possible size permitting. This keeps bond strength equal

*Calculation of Mineral Formulae From Metal Percentages*

To arrive at the relative proportions of the elements, the weight percentage in each element is divided by the atomic weight of the element. This gives the atomic proportions (Column 3) from which the atomic ratios can be quickly derived (column 4). In the analysis of chalcopyrite, these ratios are Cu:FE:S=1:1:2; resulting in CuFeS₂ as the chemical formula.

*Transformation Twinning*

Twinning also can occur in minerals in the solid state after growth of the crystals is complete. *Transformation twinning* results when a crystal that formed at high temperature is cooled and, subsequently, rearranges its structure to a symmetry different from that of the high temperature form.

*Twinning*

Under certain conditions of growth, two or more crystals may form a rational, symmetrical intergrowth. Such a crystallographically controlled intergrowth is called a twin. The lattice directions of one crystal in a twin bear a definite crystallographic relation to the lattice directions of the other crystals. Twinning can be considered a type of planar feature.

*X-ray Powder Diffraction and Mineral Identification*

When a beam of monochromatic X-rays strikes the mount, all possible diffractions take place simultaneously. If the orientation of the crystalline particles in the mount is truly random, for each family of atomic planes with its characteristic interplanar spacing (d), there are many particles whose orientation is such that they make the proper angle to with the incidence beam to satisfy Bragg Law

*Exsolution*

When an originally homogeneous high-temperature mineral containing ions of considerably different sizes cools, thermal vibrations decrease and the original structure becomes unstable. This results in a structural reorganization in which exsolution occurs. The process whereby an initially homogeneous solid solution separates into two or more distinct crystalline minerals with out the addition or removal of material to or from the system. This means that there is no change in bulk composition.

*Nonuniform Bond strength: Radicals*

When small, highly charged cations coordinate larger and less highly charged anions, compact, firmly bonded groups result. If the strength of the bonds within such groups is calculated, the numerical value of the electrostatic valency is always greater than 1/2 of the total charge on the anion. This means that in such groups, the anions are more strongly bonded to the central coordinating cation than any other ion. If a radical exists, the mineral cleaves through the weaker bonds leaving the radical intact.

*Monolayer

When spheres are arranged as compactly as possible, each sphere touches six other spheres in a closepacked layer, called a *monolayer* (A). Two types of voids are created based on the orientation of their triangular shapes; voids with point up (B), and voids that point down (C). These voids create two equivalent positions, differing only in orientation, in which a second layer can snuggly fit into the first (lower A) layer. When a second layer is added, nestled within the B voids, it is termed the B layer. •The difference between hexagonal closest packing lies in the position taken by a third layer with respect to that of the first two layers.

*3-fold coordination*

With an increase in the relative size of the central ion, three anions can fit around the central ion and *triangular coordination* becomes stable *C.N.*=3 Radius between .155 to .225

*continuous spectrum*

X-rays are generated when the source electrons impact the target (anode). The wavelength of the X-rays depends on the metal of the target and the applied voltage. No X-rays are produced until the voltage reaches a minimum value dependent on the target material. At that voltage, a *continuous spectrum* is generated. With increasing accelerating potential, the intensity of all wavelengths increases and the minimum wavelength of the continuous spectrum decreases (A). The continuous spectrum is caused by stepwise loss of energy of bombarding electrons in a series of encounters with atoms of the target material.

*X-rays occupy only a small portion of the electromagnetic spectrum*

X-rays occupy only a small portion of the electromagnetic spectrum, with wavelengths ranging between slightly more than 100 angstoms and 0.02 angstroms (Fig). X-rays used in investigation of crystals have wave-lengths on the order of 1 angstrom, similar in mangitude to the size of a unit cell.

*Common ions: Mg²⁺*

coordination Number with oxygen: 6,Octahedral •Ionic Radius in A: .72(6)

*The faces that are most likely to appear on crystals are those parallel*

←The lines p, p1, and p2 represent the traces of a family of atomic planes with spacing d. X-rays striking the outer plane pp would be reflected at incident angle θ. To reinforce one another in order to give a reflection that can be recorded, all reflected rays must be in phase. The path of the waves along DEF reflected at E is longer than the path of waves along ABC reflected at B. If the two sets of waves are to be in phase, the path difference of ABC and DEF must be a whole number of wavelngths (nλ) The faces that are most likely to appear on crystals are those parallel to atomic planes with the greatest density of lattice nodes. Parallel to each face is a family of equally spaced identical planes. When an X-ray beam strikes a crystal, it penetrates it, and the resulting diffraction effect is not from a single plane but from an almost infinite number of parallel planes, each contributing a small bit to the total diffraction effect (reflection) to be of sufficient intensity to be recorded, the individual reflections must be in phase with one another.

*Rule 2: The electrostatic valency principle*

←in the example above the 1/6 ratio is the bond strength to the cation. It is simply the valence charge/C.N. All bond strengths should be equal In a stable crystal structure, the total strength of the valency bonds that reach an anion from all the neighboring cations is equal to the charge of the anion.This rule is a statement of bond strength. The strength of an *electrostatic valency* (E.V) may be defined as an ions valence charge (Z). •Bond Strength(E.V)=Z/C.N