Mineralogy & Crystallography

2.1 Crystal Systems and Bravais Lattices

A crystal is a solid with atoms arranged in a periodic three-dimensional pattern. The mathematical framework for describing this periodicity begins with the concept of a lattice — an infinite array of points generated by three linearly independent translation vectors.

The Translation Lattice

A lattice point $\mathbf{R}$ is defined by:

where $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3$ are the primitive lattice vectors and $n_1, n_2, n_3$ are integers. The unit cell is the parallelepiped defined by these vectors, with volume:

The 7 Crystal Systems

The constraints on unit cell parameters $(a, b, c, \alpha, \beta, \gamma)$ define seven crystal systems, organized by decreasing symmetry:

The 14 Bravais Lattices

Auguste Bravais (1848) proved that there are exactly 14 distinct lattice types in three dimensions. These arise from the 7 crystal systems combined with centering operations: P (primitive), I (body-centered), F (face-centered), and C (base-centered). Not all centering types produce distinct lattices in every system. The cubic system has P, I, and F; monoclinic has P and C; triclinic has only P.

Point Groups and Space Groups

Point group symmetry operations leave at least one point fixed. The allowed operations are rotations ($1, 2, 3, 4, 6$-fold), mirrors ($m$), inversions ($\bar{1}$), and rotoinversions ($\bar{3}, \bar{4}, \bar{6}$). The crystallographic restriction theorem limits rotational symmetry to $n = 1, 2, 3, 4, 6$ because only these can tile space periodically:

This gives $n = 1, 2, 3, 4, 6$ only. The number 5-fold symmetry (as in quasicrystals) is forbidden for periodic lattices. Combining these operations yields exactly 32 crystallographic point groups.

Adding translational symmetry elements (screw axes and glide planes) to the 32 point groups and 14 Bravais lattices yields the 230 space groups, first enumerated independently by Fedorov, Schoenflies, and Barlow in 1891. Every crystalline mineral belongs to exactly one space group.

2.2 X-ray Diffraction

Bragg's Law

When X-rays of wavelength $\lambda$ impinge on crystal planes separated by distance $d$, constructive interference occurs when the path length difference equals an integer number of wavelengths. Consider two parallel planes: the extra path length for the ray reflecting from the second plane is $2d\sin\theta$, giving Bragg's law:

where $\theta$ is the angle of incidence (Bragg angle) and $n$ is the order of diffraction. For cubic crystals, the interplanar spacing for Miller indices $(hkl)$ is:

For the general case (orthorhombic):

Derivation: Structure Factor

The intensity of a diffracted beam depends not just on the geometry (Bragg condition) but on the arrangement of atoms within the unit cell. The structure factor is:

where $f_j$ is the atomic scattering factor of atom $j$ and $(x_j, y_j, z_j)$ are its fractional coordinates. The diffracted intensity is $I \propto |F(hkl)|^2$.

Example: BCC structure with atoms at $(0,0,0)$ and $(\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$:

This is a systematic absence (extinction rule). For BCC, reflections with$h + k + l$ odd are forbidden. Similarly, for FCC, reflections require $h, k, l$ all even or all odd.

FCC structure factor (atoms at $(0,0,0)$, $(\frac{1}{2},\frac{1}{2},0)$, $(\frac{1}{2},0,\frac{1}{2})$, $(0,\frac{1}{2},\frac{1}{2})$):

For NaCl (rock salt), with Na$^+$ at FCC positions and Cl$^-$ offset by $(\frac{1}{2},0,0)$:

2.3 Silicate Mineral Classification

Silicate minerals constitute over 90% of Earth's crust. Their classification is based on the degree of polymerization of $[\text{SiO}_4]^{4-}$ tetrahedra — the fundamental building block. Each tetrahedron has a central Si$^{4+}$ cation bonded to four O$^{2-}$ anions.

Polymerization and Si:O Ratio

When tetrahedra share oxygen atoms (bridging oxygens, BO), the Si:O ratio changes systematically. Define the number of bridging oxygens per tetrahedron as $n_{\text{BO}}$:

The charge balance requirement means that increasing polymerization reduces the negative charge per formula unit, requiring fewer charge-balancing cations. In tectosilicates, all oxygens are bridging and the framework is electrically neutral unless Al$^{3+}$ substitutes for Si$^{4+}$ (as in feldspars: $\text{NaAlSi}_3\text{O}_8$).

Olivine Solid Solution

Olivine is a key nesosilicate with complete solid solution between forsterite ($\text{Mg}_2\text{SiO}_4$, Fo) and fayalite ($\text{Fe}_2\text{SiO}_4$, Fa):

The composition is reported as Fo content: Fo$_{90}$ means $x = 0.10$ (90% Mg). Mantle olivine is typically Fo$_{88-92}$. The Mg-Fe substitution is possible because the ionic radii are similar ($r_{\text{Mg}^{2+}} = 0.72$ A, $r_{\text{Fe}^{2+}} = 0.78$ A).

2.4 Phase Diagrams and Mineral Stability

The Clausius-Clapeyron Equation

The slope of a phase boundary in P-T space is governed by the Clausius-Clapeyron equation. At equilibrium between two phases, the Gibbs free energies are equal:

Along the phase boundary, $dG_1 = dG_2$. Since $dG = VdP - SdT$:

Since $\Delta S = \Delta H / T$ at equilibrium, this can also be written:

Application: Al$_2$SiO$_5$ polymorphs. The aluminosilicate system has three polymorphs — andalusite, kyanite, and sillimanite — related by phase boundaries whose slopes are determined by $\Delta S / \Delta V$. Kyanite (triclinic, dense) is stable at high P; sillimanite (orthorhombic) at high T; andalusite at low P, low T. The triple point is near 500°C, 4 kbar.

Gibbs Phase Rule

The number of degrees of freedom $F$ in a system is constrained by:

where $C$ is the number of independent chemical components and $P$ is the number of phases. The $+2$ accounts for temperature and pressure as intensive variables.

Gibbs Free Energy and Stability

The thermodynamic criterion for equilibrium at constant P and T is minimization of the Gibbs free energy:

At a given P and T, the stable assemblage is the one with the lowest total Gibbs energy. For a reaction $A \rightarrow B$, the reaction proceeds if:

2.5 Optical Mineralogy and Identification

Optical mineralogy uses the interaction of polarized light with thin sections (30 $\mu$m thick) to identify minerals. The key optical properties arise from the crystal structure.

Birefringence and the Indicatrix

In anisotropic crystals, light travels at different speeds depending on its polarization direction. The refractive index $n$ is related to the dielectric constant by:

Birefringence is the difference between the maximum and minimum refractive indices:

The optical indicatrix is a 3D ellipsoid whose semi-axes are the principal refractive indices. Isotropic minerals (cubic) have a sphere; uniaxial minerals (tetragonal, hexagonal) have an ellipsoid of revolution; biaxial minerals (orthorhombic, monoclinic, triclinic) have a triaxial ellipsoid.

Michel-Levy Interference Color Chart

When light passes through a birefringent mineral in a thin section between crossed polars, the retardation $\Gamma$ produces interference colors:

where $d = 30$ $\mu$m is the standard thin section thickness. Common birefringences for rock-forming minerals:

Extinction Angle

When a crystal's vibration directions align with the polarizer and analyzer, the mineral goes dark (extinction). The angle between extinction and a crystallographic reference direction (cleavage, twin boundary) is diagnostic:

For plagioclase, the extinction angle on albite twins in sections perpendicular to$(010)$ varies systematically with An content, from ~20° (albite) to ~37° (anorthite), enabling composition determination by the Michel-Levy method.

2.6 Crystal Chemistry and Bonding

Pauling's Rules

Linus Pauling (1929) formulated five rules governing the structure of ionic crystals:

Ionic Substitution

Ions can substitute for one another in crystal structures if they have similar size and charge. Goldschmidt's rules state that substitution is favored when:

Common substitutions in silicates include: Mg$^{2+}$ ↔ Fe$^{2+}$ (olivine, pyroxene), Al$^{3+}$ ↔ Si$^{4+}$ (with charge balance by Na$^+$ ↔ Ca$^{2+}$ in feldspar), and OH$^-$ ↔ F$^-$ (in micas and amphiboles).

The Reciprocal Lattice

The reciprocal lattice is essential for understanding diffraction. Given real-space lattice vectors $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3$, the reciprocal lattice vectors are:

and cyclically for $\mathbf{b}_2, \mathbf{b}_3$. A reciprocal lattice vector is:

The Laue condition for diffraction is $\mathbf{k'} - \mathbf{k} = \mathbf{G}_{hkl}$, where $\mathbf{k}$ and $\mathbf{k'}$ are the incident and diffracted wavevectors. This is equivalent to Bragg's law, since $|\mathbf{G}_{hkl}| = 2\pi/d_{hkl}$. The Ewald sphere construction provides a geometric interpretation: diffraction occurs when a reciprocal lattice point lies on the sphere of radius $|\mathbf{k}| = 2\pi/\lambda$.

Mineral Hardness and Bonding

Mohs hardness scale reflects bond strength and crystal structure. The relationship between Mohs hardness and absolute hardness (Vickers, in kg/mm$^2$) is approximately exponential:

Computational Simulations