Earth Materials. John O'Brien

the crystal structure of the mineral. That is, they depend on the basic motif and the symmetry operations that produce the three‐dimensional crystal lattice. The nature of the crystal forms and cleavage surfaces depends on the crystal system and crystal class in which the mineral crystallized.

      4.5.1 Crystallographic axes

To identify, describe and distinguish between different planes in minerals, including cleavage planes, crystal faces, and X‐ray diffraction planes, a comprehensive terminology has been developed that relates each set of planes to the three crystallographic axes (Figure 4.13). For all but the rhombohedral (trigonal) division or system, the three crystallographic axes, designated a, b, and c, are chosen to correspond to the three unit cell translation vectors (t_a, t_b, and t_c). With the exception noted, the three crystallographic axes have lengths and angular relationships that correspond to those of the three sets of unit cell edges (Table 4.4). The rules for the three crystallographic axes are specific to each system; some systems have multiple sets of rules labeling. For details, see Klein and Hurlbut (1985).

When referencing crystallographic planes to the crystallographic axes, a standard set of orientation rules is used (Table 4.5). To indicate their similarity, crystallographic axes with the same length are labeled a₁, a₂ and/or a₃ instead of a, b and/or c. In the isometric, tetragonal, and orthorhombic systems (Figure 4.14), the b‐axis (or a₂‐axis) is oriented from left to right, with the right end designated as the positive end of the axis (b or a₂) and the left end designated as the negative end of the axis (b&c.ovline; or ). The a‐axis (or a₁‐axis) is oriented so that it trends from back to front toward the observer. The end of the a‐axis toward the observer is designated as the positive end of the axis (a) and the end of the a‐axis away from the observer is designated as the negative end (ā). The c‐axis is oriented vertically with the top end designated as the positive end of the axis (c) and the bottom end designated as the negative end (c&c.macr;). There are small exceptions to these rules in the hexagonal, monoclinic, and triclinic systems that result from the fact that unit cell edges are not perpendicular to one another so that not all crystallographic axes intersect at right angles. Even in the hexagonal and monoclinic systems, the b‐axis is oriented horizontally with the positive end to the right, and in all systems the c‐axis is vertical with the positive end toward the top and the negative end toward the bottom. The orientations, lengths, and intersection angles between crystallographic axes in each of the major crystal systems are illustrated in Figure 4.14. The characteristics of the crystallographic axes in each system and the standard rules for orienting them are summarized in Table 4.5.

Figure 4.13 Conventional labeling of crystallographic axes, illustrating the positive and negative ends of the three crystallographic axes and the angles between the axes for crystals in the orthorhombic system.

Table 4.5 The relationships of crystallographic axes and the rules for orienting crystals in each of the crystal systems. Note that the trigonal system (division) is listed independently of the hexagonal system (division) in this table.

Crystal system	Verbal description	Symbolic description
Isometric (cubic)	Three mutually perpendicular axes (a1, a2, a3) of equal length that intersect at right angles	(a1 = a2 = a3) (α = β = γ = 90°)
Tetragonal	Three mutually perpendicular axes; axes (a1, a2) are of equal length; the c axis may be longer or shorter	(a1 = a2 ≠ c) (α = β = γ = 90°)
Orthorhombic	Three mutually perpendicular axes of different lengths (a, b, c); two axial length ratios have been used to identify the axes: c > b > a (older) or b > a > c (newer)	(a ≠ b ≠ c) (α = β = γ = 90°)
Monoclinic	Three unequal axes lengths (a, b, c) only two of which are perpendicular. The angle (β) between a and c is not 90°. The a‐axis is inclined toward the observer. The b‐axis is horizontal and the c‐axis is vertical	(a ≠ b ≠ c) (α = γ = 90^o; β ≠ 90°)
Triclinic	Three unequal axes, none of which are generally perpendicular. The c axis is vertical and parallel to the prominent zone of crystal faces	(a ≠ b ≠ c) (α ≠ β ≠ γ ≠ 90°)
Hexagonal	Four crystallographic axes; three equal horizontal axes (a1, a2, a3) intersecting at 120°. One longer or shorter axis (c) perpendicular to the other three. a1 oriented to front left of observer; a2 to right; a3 to back left; c vertical. Sixfold axis of rotation or rotoinversion	(a1 = a2 = a3 ≠ c) (α = β = 90°; γ = 120°)
Trigonal (or rhombohedral)	Axes and angles are similar to the hexagonal system; crystal symmetry is different with the c‐axis a threefold axis of rotation or rotoinversion	(a1 = a2 = a3 ≠ c) (α = 120°; β = γ = 90°)

Figure 4.14 Crystallographic axes (positive ends labeled) and intersection angles for the major crystal systems: isometric, tetragonal, orthorhombic, monoclinic, triclinic, and hexagonal systems.

      4.5.2 Crystal forms

      Each of the crystal systems has an associated set of common (and rarer) crystal forms. Crystal forms consist of a three‐dimensional set of one or more crystal faces that possess similar spatial relationships to the crystallographic axes. Some natural crystals possess only one crystal form; others possess multiple or combined crystal forms. Crystal forms can be subdivided into two major groups: closed forms and open forms.