GEOL 333 #6 - Crystal Structure (Radius Ratios, Polymorphism, Symmetry, Crystal Systems)

Crystal structure of minerals (and size of ions) is determined mainly using X-ray diffraction. Some minerals (e.g., halite) consist of cations and anions held together with ionic bonds. Like charges repel, so anions stay away from other anions and cations stay away from other cations. This repulsion results for halite in crystal structure with regularly alternating Na+ (purple balls in figure) and Cl- (green balls in figure) ions (ball and stick model of halite structure). Each Na+ is surrounded by 6 Cl- ions (i.e., 6-fold or octahedral coordination).

Ionic Radii - Ionic size (radius) is critical as to whether ionic substitutions (solid solutions) occur.

Controls on Size of Ionic Radii (link #2) (1) What kinds of cations are shown in Table 13.1 (Perkins, 2011)? (Hint: Consider their position in periodic table.)

What is trend in ionic radius for these cations (bonded to oxygen)?


(2) What kinds of ions are shown in Table 13.2 (Perkins, 2011)? (Hint: Consider their position in periodic table.)

What is trend in ionic radius for these ions (averaged over variety of minerals)?


(3) Which is larger S6+ or S2- (Table 13.2)?


(4) What kinds of ions (and neutral atom) are shown in Table 13.3 (Perkins, 2011)?

What is trend in radius for these ions (and neutral atom)?



What are primary controls on ionic size?

Coordination Number (CN) - Most minerals (except pure metals) contain anions (O2- is most common, also S2- and CO32-). Because of size differences between anions and cations, most minerals consist of large anions with small cations in spaces between anions. Number of anions bonded to specific cation = coordination number (CN) of cation. Different CN's give rise to different kinds of geometric shapes (polyhedra), shown in Fig. 13.3. Cation is in middle surrounded by various numbers of anions.

Ratio of size of cation/anion (radius ratio = Rc/Ra in Table 13.5) controls which CN occurs. In general, small cations fit into small holes associated with low CN and larger cations fit into larger holes associated with higher CN. Radius ratio of Silicon/Oxygen = 0.3. What is CN for Si in O?

Table 13.6 (Perkins, 2002) gives ionic radii for common cations in Earth's crust and expected CN for each cation assuming oxygen is only anion (true for silicates and oxides/hydroxides). Al can occur in either 4-fold or 6-fold coordination, transition metals (e.g., Fe and Mn) are commonly in 6-fold coordination, large alkali's and alkaline earths (Na, K, and Ca) are in 8-fold or 12-fold coordination. Periodic table at beginning of Perkins (2002) textbook lists ionic radii and coordination numbers for many more cations and anions.

Polymorphism - definition and geologic examples




Crystal Symmetry - Crystal = mineral with smooth flat surfaces (crystal faces, see beautiful photos of crystals of quartz, feldspar, calcite, halite, mica) that can develop if mineral grows without obstructions.

Particular set of crystal faces for mineral depends on its crystal structure (internal arrangement of atoms). Many minerals occur with crystal shapes that are rather unique to that mineral.

Crystallography = study of internal arrangement of atoms in crystals and how this is reflected by external shape (morphology). To characterize different crystals, we must recognize symmetry using following parameters: mirror plane - imaginary plane that passes through object such that images on opposite sides of plane are same (mirror images of each other); rotation axis - imaginary line through object in which motif (appearance of object) can be repeated during full 360° rotation - image can be repeated 2 times (2-fold rotation axis, A2), 3 times (3-fold rotation axis, A3), 4 times (4-fold rotation axis, A4), and 6 times (6-fold rotation axis, A6); center of symmetry - exists if face, edge or point on crystal can be projected through center of crystal and appear on other side in reversed position. Consider symmetry of human being:


Consider symmetry of cube (Fig. 9.8, Perkins, 2011):


Unit cell (link #2) = smallest unit of crystal that still retains all physical, chemical and crystallographic properties of mineral (collection of several to many atoms that represent basic building blocks of crystals). Fig. 10.31 shows crystal structure of calcite with 3 different unit cells given (2 pushed-over cubes = rhombohedra and elongated 6-sided figure).

Need to describe size and shape of unit cell, which is box (regular cube or irregularly shaped) that has dimensions a, b, and c (lengths of each unit cell edge) and Greek letters alpha (angle between b and c), beta (angle between a and c), and gamma (angle between a and b), which represent angles between two different axes. Values for a, b, and c (unit cell dimensions) typically range from 2 ­ - 20 Å (Å = 10-10 m).

Minerals are divided in 6 (or 7) different crystal systems (link #2) based on shared symmetry.

Crystal Systems

Isometric (cubic) symmetry (highest symmetry), a = b = c; alpha= beta = gamma = 90°
Tetragonal symmetry, a = b ≠ c; alpha= beta = gamma = 90°
Orthorhombic symmetry, a ≠ b ≠ c; alpha= beta = gamma = 90°
Hexagonal (+/- Trigonal/Rhombohedral) symmetry a = b ≠ c; alpha= beta = 90°, gamma = 120°
Monoclinic symmetry, a ≠ b ≠ c; alpha= gamma = 90°, beta ≠ 90°
Triclinic symmetry (lowest symmetry), a ≠ b ≠ c; alpha ≠ beta ≠ gamma ≠ 90°

Different minerals have different levels of symmetry and unique set of unit cell parameters (values for a, b, c, alpha, beta, and gamma). V = unit cell volume (a x b x c) and Z = # of structural formula units in unit cell (Z = 4 for fluorite, Ca4F8). K-feldspar has monoclinic symmetry and albite (plagioclase feldspar) has triclinic symmetry. Calcite has hexagonal symmetry.