I. Lecture 1: Minerals and Symmetry Operations Definition of a mineral A mineral is a naturally occurring homogeneous solid usually formed by inorganic processes. It has a definite chemical composition and a highly ordered atomic arrangement. In detail— 1. Naturally occurring: Minerals are not man-made, but are produced by natural processes. A diamond found in the earth, for example, is a mineral, but a diamond synthesized in a lab is not. 2. Homogeneous: A mineral cannot be physically separated into simpler chemical compounds. 3. Solid: Liquids and gases are not minerals. Water, for example, is not a mineral, but ice is. 4. Formed by inorganic processes: Petroleum and coal are not minerals. 5. Definite (but generally not fixed) composition: Minerals can be expressed by chemical formulas, such as SiO2 for quartz. However, some minerals like feldspars have variable compositions. The composition of plagioclase, for example, ranges from NaAlSi3O8 to CaAl2Si2O8. 6. Ordered atomic arrangement: Minerals have long-range 3D atom order. Solids like glass that lack long-range atomic order are called amorphous and are not minerals. Symmetry The long-range 3D order of minerals may be defined in terms of molecule-scale building blocks that are arranged in a regular manner. The way these building blocks are arranged is described by symmetry. Symmetry not only describes a mineral’s atomic order, but also controls physicals properties such as cleavage and habit (Neumann’s Principle). 2D Symmetry Operations Symmetry operations consist of an element and an operator. Operation Translation Rotation Reflection Element vector axis mirror Operator distance angle distance A. Translation Translations of molecule-scale building blocks lead to long-range atomic order. This symmetry operation will be explored in more detail in Lecture 2; this lecture focuses on symmetry operations that don’t require translations. B. Reflection Mirror planes are designated by m. C. Rotation The angle of rotation α about an axis must be the quotient of 360° and an integer n. α = 360°/n When n = 1, an object is completely rotated about the rotation axis and exhibits 1-fold symmetry. When n = 2, an object is rotated α = 180° around the rotation axis and exhibits 2-fold symmetry. D. Combined Operations n α 3 120 3-fold 4 90 4-fold 5 *will be addressed later 6 60 Symmetry 6-fold 1 m 2 2mm 3 3m 4 4mm 6 6mm (Adapted from Klein and Hurlbut, Jr. Manual of Mineralogy. New York: John Wiley and Sons, Inc., 1993. 117) D. Inversion The inversion operation inverts an object through an inversion center i. The inverted object is formed by drawing imaginary lines from the original object through the inversion center and out an equal distance on the other side. In two dimensions, the operation is equivalent to a 2-fold rotation. Symmetry about a point in three dimensions will be covered in a later lecture.