CH.3 Geoscience Basics: Understanding Minerals and Crystallography

Mineral Mysteries: A Study Guide

⚛ Atomic Substitution

Ionic Substitution: The replacement of one ion by another in a mineral's crystal structure. This happens when ions have similar sizes and charges, and are available during mineral formation. Think of it like LEGOs – similar-sized bricks can fit in the same spot.

Simple Substitution: Complete replacement of one ion by another in any proportion. Olivine ((Mg,Fe)₂SiO₄) is a great example: Magnesium (Mg²⁺) and iron (Fe²⁺) can substitute freely. This creates a solid solution series between the end members forsterite (Mg₂SiO₄) and fayalite (Fe₂SiO₄). End members are the pure forms of a mineral with only one type of ion in a specific site.

Coupled Substitution: Simultaneous substitution of ions with different charges in different sites to maintain electrical neutrality. Plagioclase feldspars are a classic example. Sodium (Na⁺) and calcium (Ca²⁺) substitute in one site, balanced by aluminum (Al³⁺) and silicon (Si⁴⁺) substitution in another. This keeps the overall charge balanced.

Limited Substitution: Partial replacement of one ion by another due to significant size or charge differences. The substitution between calcite (CaCO₃) and magnesite (MgCO₃) is limited because Ca²⁺ is much larger than Mg²⁺. This can lead to miscibility gaps, where certain compositions don't exist in nature.

Solid Solution Series: A continuous range of compositions between two end-member minerals due to complete substitution. The olivine series is a perfect example.

End Members: The pure forms of a mineral in a solid solution series. Forsterite and fayalite are the end members of the olivine solid solution series.

Miscibility Gaps: Ranges of compositions within a solid solution series that do not occur naturally due to limited substitution. These gaps represent compositions that are unstable.

📊 Phase Diagrams

Phase: A physically distinct and homogeneous part of a system. A mineral, a liquid, or a gas are all phases.

Phase Rule: A fundamental principle governing the number of phases that can coexist in equilibrium in a system. The rule is expressed as: P = C + 2 - F, where: - P = Number of phases - C = Number of components (chemical elements or compounds) - F = Degrees of freedom (variables like temperature and pressure that can change without altering the number of phases)

Liquidus: The temperature above which a substance is entirely liquid.

Solidus: The temperature below which a substance is entirely solid.

Eutectic: A point on a phase diagram where a liquid cools to form two solid phases simultaneously.

Peritectic: A point on a phase diagram where a solid phase reacts with a liquid to form a new solid phase.

Solvus: A boundary on a phase diagram separating regions of complete solid solution from regions of limited solid solution.

Lever Rule: A method used to determine the relative proportions of phases in a two-phase region of a phase diagram.

Continuous Reaction Series: A series of minerals that change composition continuously during crystallization or melting. Plagioclase feldspars are an example.

Discontinuous Reaction Series: A series of minerals that change abruptly during crystallization or melting. Bowen's reaction series is a classic example.

Silica Saturation: The relative abundance of silica (SiO₂) in a system, which influences mineral formation. Systems can be silica-undersaturated, silica-saturated, or silica-oversaturated.

⏳ Isotopes and Radiometric Dating

Isotopes: Atoms of the same element with different numbers of neutrons. This leads to different atomic masses. Some isotopes are stable, while others are radioactive.

Isotopic Ratios: The relative abundance of different isotopes of an element. These ratios are used in various geological applications. Examples include δ¹⁸O (oxygen isotopes) and δ¹³C (carbon isotopes).

Radioactive Decay: The spontaneous transformation of an unstable (radioactive) isotope into a more stable isotope. Three main types are: - Alpha (α) decay: Loss of two protons and two neutrons. - Beta (β) decay: Conversion of a neutron to a proton. - Electron capture: Addition of an electron to a proton, converting it to a neutron.

Half-life: The time it takes for half of the atoms in a radioactive sample to decay.

Decay Constant (λ): A measure of the rate of radioactive decay.

Age Calculation: The age of a rock can be calculated using the following equation: t = (1/λ)ln[(D+P)/P], where: - t = age - λ = decay constant - D = number of daughter atoms - P = number of parent atoms

Uranium-Lead Dating: A radiometric dating method using the decay of uranium isotopes (²³⁸U and ²³⁵U) to lead isotopes (²⁰⁶Pb and ²⁰⁷Pb).

Rubidium-Strontium Dating: A radiometric dating method using the decay of rubidium-87 (⁸⁷Rb) to strontium-87 (⁸⁷Sr).

Potassium-Argon Dating: A radiometric dating method using the decay of potassium-40 (⁴⁰K) to argon-40 (⁴⁰Ar).

Concordia Diagram: A graphical representation used in Uranium-Lead dating to determine the age of a sample.

Isochron: A line on a graph representing samples of the same age in Rubidium-Strontium dating.

💎 Crystallography

Crystallography: The study of the structure and symmetry of crystals.

Coordination Polyhedra: Geometric shapes formed by the arrangement of atoms or ions around a central atom or ion in a crystal structure.

Unit Cell: The smallest repeating unit of a crystal lattice. There are seven basic unit cell shapes in three dimensions.

Bravais Lattices: The 14 unique ways to arrange points in three-dimensional space while maintaining symmetry.

Crystal Systems: Seven categories of crystals based on their unit cell symmetry: cubic, tetragonal, orthorhombic, monoclinic, triclinic, hexagonal, and trigonal.

Miller Indices: A system of three numbers used to identify crystallographic planes.

Crystal Forms: The external shapes of crystals, determined by the arrangement of crystal faces. Examples include cubes, octahedra, prisms, pyramids, and rhombohedra.

Twinning: The intergrowth of two or more crystals of the same mineral with a symmetrical relationship.

💎 Crystalline Substances and Long-Range Order

Crystalline Substances:Definition: Materials where atoms or ions are arranged in a highly ordered, repeating three-dimensional pattern. This long-range order is what distinguishes them from amorphous substances (like glass) which lack this regular structure.

Importance of Long-Range Order: The repeating pattern dictates many physical properties of a crystal, including its shape, cleavage, hardness, and optical properties. Understanding this order is fundamental to understanding crystal behavior.

Advanced Concepts: The degree of long-range order can be affected by defects in the crystal lattice (discussed later). Also, the concept of "quasi-crystals" challenges the strict definition of long-range order, exhibiting aperiodic but ordered structures.

🔄 Symmetry Operations: Repeating Patterns

Symmetry Operations:Definition: A set of rules that describe how a motif (the basic repeating unit) can be repeated to generate the entire crystal structure. These operations create the long-range order.

Types of Symmetry Operations:

• Translation: Moving a motif a specific distance in a specific direction. This is the most fundamental operation.

  • Unit Translation Vector: The vector defining the length and direction of the translation.

• Rotation: Rotating a motif around an imaginary axis. The axis is defined by the rotational symmetry (n-fold rotation, where n is the number of times the motif is repeated in a 360° rotation).

• Reflection: Mirroring a motif across a plane (mirror plane). This creates mirror images.

• Inversion: Inverting a motif through a point (center of inversion). This creates an inverted mirror image.

Compound Symmetry Operations: These combine two or more simple operations:

• Glide Reflection: A combination of translation and reflection.

• Rotoinversion: A combination of rotation and inversion.

• Screw Rotation: A combination of rotation and translation along the axis of rotation.

Advanced Concepts: Space groups describe the complete symmetry of a crystal, including both translational and rotational symmetry elements. There are 230 possible space groups.

🗺 Two-Dimensional Motifs and Lattices

Motif:Definition: The smallest repeating unit of a pattern. In crystals, it's a cluster of atoms or coordination polyhedra.

Node:Definition: A point used to represent a motif, simplifying the visualization of lattice structures.

Plane Lattice (Mesh):Definition: A two-dimensional array of nodes or motifs where every node has an identical environment.

Types of Plane Lattices:

• Square: Unit translation vectors of equal length at 90° angles.

• Rectangular: Unit translation vectors of unequal length at 90° angles.

• Centered Rectangular: A rectangular lattice with an additional node at the center of the unit cell.

• Hexagonal: Unit translation vectors of equal length at 120° angles.

• Oblique: Unit translation vectors of unequal length at angles other than 90°, 60°, or 120°.

Plane Point Groups: These describe the rotational and reflection symmetry of two-dimensional patterns. There are ten plane point groups.

Advanced Concepts: The combination of plane point groups and plane lattices results in 17 plane lattice groups, representing all possible two-dimensional crystal symmetries.

🧊 Three-Dimensional Motifs and Lattices

Space Lattice:Definition: A three-dimensional array of nodes or motifs where every node has an identical environment. This is the three-dimensional equivalent of a plane lattice.

Unit Cell:Definition: The smallest repeating unit of a three-dimensional lattice. It defines the lattice parameters (a, b, c, α, β, γ).

Bravais Lattices:Definition: There are 14 unique three-dimensional Bravais lattices, representing all possible ways to arrange points in space with translational symmetry.

Crystal Systems: The 14 Bravais lattices are grouped into seven crystal systems based on the symmetry of their unit cells:

1. Cubic: a = b = c; α = β = γ = 90°

2. Tetragonal: a = b ≠ c; α = β = γ = 90°

3. Orthorhombic: a ≠ b ≠ c; α = β = γ = 90°

4. Monoclinic: a ≠ b ≠ c; α = γ = 90°, β ≠ 90°

5. Triclinic: a ≠ b ≠ c; α ≠ β ≠ γ ≠ 90°

6. Hexagonal: a = b ≠ c; α = β = 90°, γ = 120°

7. Trigonal (Rhombohedral): a = b = c; α = β = γ ≠ 90°

Space Point Groups: These describe the rotational and reflection symmetry of three-dimensional motifs. There are 32 space point groups, also known as crystal classes.

Advanced Concepts: The combination of Bravais lattices and space point groups defines the 230 space groups, representing all possible three-dimensional crystal symmetries.

📐 Indexing Planes in Crystals: Miller Indices

Miller Indices:Definition: A system of three integers (hkl) that uniquely identify a set of parallel planes in a crystal lattice. They are the reciprocals of the intercepts of the plane with the crystallographic axes, cleared of fractions.

Weiss Parameters:Definition: Represent the intercepts of a plane with the crystallographic axes. Miller indices are derived from Weiss parameters.

Calculating Miller Indices:

1. Determine the intercepts of the plane with the crystallographic axes (a, b, c).

2. Take the reciprocals of these intercepts.

3. Clear fractions by multiplying by the least common denominator.

4. The resulting integers are the Miller indices (hkl).

Form Indices: Curly brackets {hkl} denote a form, a set of crystal faces that are symmetrically equivalent.

Advanced Concepts: Miller-Bravais indices are used for hexagonal and trigonal systems, using four indices (hkil). More complex indexing systems exist for non-Bravais lattices.

Crystal Defects and Polymorphism

Crystal Defects:Definition: Imperfections in the regular arrangement of atoms in a crystal lattice. These can significantly affect the physical properties of the crystal.

Types of Crystal Defects:

• Point Defects: Zero-dimensional defects involving individual atoms (substitution, interstitial, omission).

• Line Defects (Dislocations): One-dimensional defects that disrupt the lattice along a line (edge, screw).

• Planar Defects: Two-dimensional defects involving planes within the crystal (grain boundaries, stacking faults, twin boundaries).

Polymorphism:Definition: The ability of a chemical compound to crystallize in more than one crystal structure. The different structures are called polymorphs.

Types of Polymorphic Transformations:

• Reconstructive Transformations: Involve breaking and reforming chemical bonds. These are slower and can lead to metastable polymorphs.

• Displacive Transformations: Involve only small shifts in atomic positions without bond breaking. These are faster and less likely to produce metastable polymorphs.

• Order-Disorder Transformations: Involve changes in the arrangement of atoms within the crystal structure, from ordered to disordered states.

Pseudomorphs:Definition: Minerals that have inherited the external form of a pre-existing mineral, but have a different chemical composition. They form through replacement, casting, or other processes.

Advanced Concepts: The study of crystal defects is crucial in materials science, influencing mechanical strength, electrical conductivity, and other properties. Polymorphism is important in understanding mineral stability and phase transitions in geological processes.

Facts to Memorize

1. Ionic substitution requires similar ion sizes and charges.

2. Simple substitution creates solid solution series between end members.

3. Coupled substitution maintains electrical neutrality.

4. Limited substitution leads to miscibility gaps.

5. The phase rule is P = C + 2 - F.

6. The liquidus is the temperature above which a substance is entirely liquid.

7. The solidus is the temperature below which a substance is entirely solid.

8. Eutectic points involve simultaneous crystallization of two solids.

9. Peritectic points involve a solid reacting with a liquid to form a new solid.

10. Isotopes have the same number of protons but different numbers of neutrons.

11. Radioactive decay transforms unstable isotopes into stable ones.

12. Half-life is the time for half of a radioactive isotope to decay.

13. Uranium-Lead dating uses the decay of uranium to lead.

14. Rubidium-Strontium dating uses the decay of rubidium to strontium.

15. Potassium-Argon dating uses the decay of potassium to argon.

16. Crystallography studies crystal structure and symmetry.

17. There are seven crystal systems and 14 Bravais lattices.

18. Miller indices describe crystallographic planes.

19. Twinning involves the symmetrical intergrowth of crystals.

20. The lever rule helps determine phase proportions in phase diagrams.

21. Crystalline substances exhibit long-range order in their atomic arrangement.

22. Symmetry operations (translation, rotation, reflection, inversion) generate crystal structures.

23. Compound symmetry operations combine simple operations (glide reflection, rotoinversion, screw rotation).

24. Plane lattices are two-dimensional arrays of motifs; five basic types exist (square, rectangular, centered rectangular, hexagonal, oblique).

25. Space lattices are three-dimensional arrays of motifs; 14 Bravais lattices represent all possible translational symmetries.

26. Seven crystal systems (cubic, tetragonal, orthorhombic, monoclinic, triclinic, hexagonal, trigonal) classify crystals based on unit cell symmetry.

27. Miller indices (hkl) represent the orientation of crystallographic planes.

28. Weiss parameters are the intercepts of a plane with crystallographic axes; Miller indices are their reciprocals.

29. Crystal defects (point, line, planar) disrupt the regular crystal structure.

30. Polymorphism is the existence of multiple crystal structures for the same chemical composition.

31. Reconstructive transformations break and reform bonds during polymorph transitions.

32. Displacive transformations shift atomic positions without bond breaking.

33. Order-disorder transformations change the arrangement of atoms within a crystal structure.

34. Pseudomorphs inherit the external form of a pre-existing mineral but have a different composition.

35. Hardness depends on bond strength and bond density.

36. Cleavage is the tendency of a mineral to break along planes of weakness.

37. Density depends on atomic mass and packing efficiency.

38. Specific gravity is the ratio of a mineral's density to the density of water.

39. Metallic luster is characteristic of opaque minerals that strongly reflect light.

40. Non-metallic luster is characteristic of translucent or transparent minerals.

Idiochromatic minerals have consistent color; allochromatic minerals have variable color due to impurities.