Explain crystallinity, relate crystal structure to cleavage, hardness and density, and categorize the seven crystal systems

Learning goal: Explain what it means for a mineral to be crystalline; explain why different minerals have different crystal structures; use atomic or crystal models to illustrate the structures of common minerals; relate a structure to a quantifiable property such as cleavage, hardness or density; and categorize the seven crystal systems, naming minerals found in each.

What crystalline means. A crystalline solid is one whose atoms or ions are arranged in a pattern that repeats in all three directions, over and over, throughout the solid. The smallest block of that pattern, which when stacked reproduces the whole crystal, is the unit cell. In halite the unit cell is a cube about 0.56 nanometers on a side; a grain of table salt one millimeter across contains nearly two million unit cells along each edge. A solid whose atoms are jumbled with no long-range repeat is amorphous: window glass, obsidian and opal are amorphous, and that is why opal and obsidian are called mineraloids rather than minerals.

Crystallinity is not the same as having pretty crystal faces. Faces form only when a crystal grows into open space, in a cavity or from a solution, and they are the outward expression of the internal pattern: the angles between equivalent faces of a given mineral are always the same, which was noticed for quartz in 1669, long before anyone knew why. Most mineral grains in rocks grew against their neighbors and have no faces at all, yet X-ray diffraction, first used on crystals in 1912, shows that their atoms are as perfectly ordered as a museum specimen. Internal order is the criterion; external shape is evidence of it when it is present.

Why structures differ. Four factors decide the arrangement. First, the sizes of the ions: each positive ion surrounds itself with as many negative ions as can touch it. A small silicon ion fits inside four oxygens, giving the tetrahedron; a larger sodium ion fits inside six chloride ions, giving the halite cube; a large potassium ion needs eight or more neighbors. Second, the charges must balance in every unit cell. Third, the bond type: non-directional ionic and metallic bonds let ions pack as tightly as their sizes allow, while directional covalent bonds fix the angles, so diamond's carbons sit at 109.5° from each other even though a closer packing would be possible. Fourth, the temperature and pressure of formation. The same composition can adopt different structures under different conditions; these are polymorphs. Carbon is diamond at the very high pressures of the deep mantle and graphite at the surface. Calcium carbonate is calcite at ordinary pressure and aragonite at higher pressure. Al₂SiO₅ is andalusite, kyanite or sillimanite depending on temperature and pressure, which makes it a useful thermometer in metamorphic rocks. A polymorph found in a metamorphic rock, where minerals grew under the pressure and temperature of burial, is therefore a record of those conditions.

The silicate structures. Because most minerals are silicates, how tetrahedra link is the most important structural idea in the course. Tetrahedra can be isolated, sharing no oxygens, each held to the next only by metal ions: olivine and garnet, silicon-to-oxygen ratio 1 : 4. They can share two corners to form single chains: the pyroxenes, ratio 1 : 3. Two chains can join into double chains: the amphiboles, ratio 4 : 11. Chains can join sideways into sheets, each tetrahedron sharing three corners: the micas and clay minerals, ratio 2 : 5. Every corner can be shared to make a three-dimensional framework: quartz, ratio 1 : 2, and the feldspars, in which aluminum replaces some silicon and potassium, sodium or calcium ions fill the spaces to balance the charge. Each step of linking uses more of the strong silicon-oxygen bonds and fewer of the weaker metal-oxygen bonds, and that directly predicts the properties.

Structure to property: cleavage. Cleavage is the tendency to break along flat planes, and the planes are always the planes of weakest bonding. Mica has one perfect cleavage because its silicate sheets are strong but are held to each other only by potassium ions; it peels into flexible transparent sheets. Pyroxene has two cleavages at about 90° and amphibole two at about 60° and 120°, because the breaks run between the chains, and the cross-section of a single chain differs from that of a double chain. Quartz has no cleavage, because every direction is crossed by the same strong bonds, so it fractures along curved surfaces. Halite cleaves into cubes along the planes on which the bonds are fewest per unit area, and galena, with the same structure, does the same. Calcite cleaves into rhombohedra, three directions none at 90°. Diamond, despite its hardness, has a perfect octahedral cleavage, because fewer bonds cross those planes than any other. The number of cleavage directions and the angles between them are a measurable signature of structure.

Structure to property: hardness and density. Hardness rises with bond strength and with how tightly the atoms are packed. Diamond (10) and graphite (1 to 2) show that structure can outweigh composition. Among silicates, quartz (7), with all its bonds strong, is harder than mica (2 to 3), where the test scratches between sheets. Density rises with the mass of the atoms and with the closeness of packing. Diamond, at about 3.5 g/cm³, is denser than graphite, about 2.2 g/cm³, because its atoms are packed more closely; aragonite (about 2.95) is denser than calcite (about 2.71) for the same reason, which is why the high-pressure polymorph is the denser one. Lead makes galena dense; tight packing makes diamond dense. Both effects are visible in a measured specific gravity.

Here is a model worked through: counting the halite unit cell. Draw a cube. Put a chloride ion at each of the 8 corners and at the center of each of the 6 faces; put a sodium ion at the midpoint of each of the 12 edges and one at the body center. Now count how much of each ion actually belongs to this cell, since corners are shared by 8 cells, edges by 4 and faces by 2. Chloride: 8 × ⅛ + 6 × ½ = 1 + 3 = 4. Sodium: 12 × ¼ + 1 = 3 + 1 = 4. The cell holds four sodium and four chloride ions, ratio 1 : 1, exactly the formula NaCl, and every ion has six nearest neighbors of the opposite charge. The model also predicts the cubic cleavage, since planes parallel to the cube faces pass between equal numbers of positive and negative ions and cut the fewest bonds. What the model cannot show: the actual size (it is drawn a hundred million times too large), the electron sharing that makes the bonds, the constant thermal vibration of the ions, and the defects, missing or misplaced ions, that every real crystal contains.

The seven crystal systems. Crystal systems are defined by symmetry, the rotations and mirror planes that leave the atomic pattern unchanged; there are seven kinds of that symmetry, and each is conventionally described by the lengths of three unit-cell axes (a, b, c) and the angles between them. The axis descriptions below are what each system requires at least; a lower-symmetry crystal can happen to have nearly equal axes or a right angle without joining a higher system. Cubic (isometric): three equal axes, all at 90°; halite, galena, pyrite, fluorite, garnet, magnetite, diamond, gold. Tetragonal: two equal axes and one different, all at 90°; zircon, rutile, chalcopyrite. Orthorhombic: three unequal axes, all at 90°; olivine, topaz, sulfur, barite, aragonite. Hexagonal: two equal axes at 120° and a third at 90° to them; beryl, apatite, graphite. Trigonal (rhombohedral): the same axes as hexagonal but with a three-fold rather than six-fold symmetry; quartz, calcite, dolomite, corundum, hematite, tourmaline. Monoclinic: three unequal axes with one angle not 90°; gypsum, orthoclase, the micas, augite, hornblende. Triclinic: three unequal axes, with no angle required to be 90°; plagioclase, microcline, kyanite, turquoise. Some textbooks merge trigonal into hexagonal and count six systems; Ohio's course lists seven, and the seven-system scheme is the one used here. The system shows in the shapes crystals take when they can grow freely: cubes and octahedra from the cubic system, six-sided prisms from the hexagonal and trigonal systems, tilted prisms from the monoclinic system.

Growing crystals to see order appear. A safe investigation: dissolve potassium alum (sold in the spice aisle as a food-grade pickling additive; not "pickling salt", which is sodium chloride) in hot water until no more dissolves, with an adult handling the hot water; pour the clear solution into a clean jar, cover loosely with paper and leave it undisturbed for several days. Alum belongs to the cubic system and grows as octahedra, eight triangular faces. Run table salt beside it, which grows cubes, and, for contrast, table sugar, which is not a mineral but crystallizes in the monoclinic system as tilted prisms. Record daily with a ruler in the photograph. Measure the angle between adjacent faces on several alum crystals with a protractor: the same angle every time is the evidence that the external shape is controlled by an internal repeating pattern. What the investigation cannot show: the atoms themselves, the speed of natural crystal growth (which is usually far slower), or growth from a melt rather than a solution. Goggles are worn, nothing is tasted, and hands are washed.

Worked example, predicting cleavage from structure: A specimen is a sheet silicate. Prediction: one perfect cleavage, thin flexible flakes, low hardness across the sheets. Test: the sample splits into transparent leaves with a fingernail. Result matches; the specimen is consistent with a mica. A second specimen is a framework silicate with aluminum substitution and potassium in the framework spaces. Prediction: two good cleavages at roughly 90°, since the framework has planes of somewhat weaker bonds where the large potassium ions sit, and hardness about 6. Test: two flat faces meet at a near right angle, a steel nail (about 5.5) does not scratch it, and it scratches glass only with difficulty. Consistent with orthoclase feldspar. The structure predicted the numbers before the tests were run.

Worked example, assigning a crystal to a system: A crystal has six faces, all rhombus-shaped, with every edge the same length and angles of about 75° and 105° between adjacent faces (each rhombus face has corner angles of about 78° and 102°). Its symmetry has one three-fold axis (through two opposite corners) but not the four three-fold axes of cubic symmetry, so it is not cubic even though all its edges are equal; three equal axes with equal non-right angles define a rhombohedron, which belongs to the trigonal system. (Cubic minerals do not have to grow as cubes: octahedra and dodecahedra are cubic too.) Calcite grows and cleaves this way. A second crystal is a prism with a six-sided cross-section ending in a pyramid, with all six prism faces equal: hexagonal or trigonal system; if it scratches glass and has no cleavage, quartz fits. A third crystal has all faces square and all edges equal: cubic system; a metallic luster, brass color and greenish-black streak would make it pyrite.

Worked example, density from structure: Diamond and graphite are both carbon, so any density difference must come from packing. Diamond, about 3.5 g/cm³, is about 1.6 times as dense as graphite, about 2.2 g/cm³, because the wide gaps between graphite's sheets are absent in the three-dimensional network. Apply the same reasoning to the calcium carbonate polymorphs: aragonite (about 2.95 g/cm³) is denser than calcite (about 2.71 g/cm³), so aragonite is the form favored by high pressure, which squeezes atoms closer together. A rule follows: among polymorphs, higher pressure favors the denser structure, other conditions being equal. It does not work backwards on its own: aragonite also forms at the surface in many shells and warm shallow seas for chemical and biological reasons, so finding aragonite does not by itself prove high pressure.

Worked example, using the unit-cell count: A model of a hypothetical cubic mineral has a metal ion at each corner and a non-metal ion at the body center. Count: corners 8 × ⅛ = 1 metal; center 1 non-metal. Ratio 1 : 1. If the metal ion is +2, the non-metal must be −2 for the cell to be neutral. The formula is MX, with a structure in which each ion has eight nearest neighbors rather than halite's six, which is what happens when the positive ion is large enough to touch eight negative ions at once.

Representation task: Using marshmallows or clay balls of two sizes and toothpicks, build (1) an isolated tetrahedron, (2) a single chain of at least four tetrahedra sharing corners, and (3) a sheet fragment of at least six tetrahedra. For each, count silicon and oxygen atoms and write the ratio; then explain why your finite chain and sheet have a few extra oxygens at their ends and edges compared with the ratio of an endless chain (1 : 3) or sheet (2 : 5), and count one repeating unit in the middle to find that ratio, then state the number of cleavage directions you predict and the mineral family it represents. Photograph the models beside a card that names the silicate structure type (isolated, chain, sheet or framework) and the ratio.

Crystal-growth record: Run the alum, salt and sugar growth investigation. Keep a dated table of crystal count, largest crystal size in millimeters and face shape. On the final day measure the angle between adjacent faces on at least three alum crystals and report the mean and the range. State one thing the result supports (constant face angles show internal order) and one thing the investigation cannot show.

Independent performance: Prepare a one-page classification card for a mineral of your choice from the list in this lesson. Success criteria: the card gives the formula and the compositional group; names the crystal system with its axis and angle description; draws or describes the unit cell or the silicate linkage; predicts cleavage (number of directions and approximate angles), hardness range and approximate specific gravity from the structure; names one polymorph or structural relative if one exists and the condition that favors it; and ends with one property test whose failure would show the classification was wrong.