Lecture 4: Directions and Planes in Space

Point Groups -> Crystal Systems

Coordinate Axes: For Unit Cells; For Crystals

[1] 32 combinations of symmetry elements around a point
	rotation, mirror, inversion, rotoinversion
	see chart or table from book

[2] Remember the symbology of the stereographic projections: vertical,
     inclined, horizontal axes, mirrors

[3] These 32 crystal classes can be grouped into crystal systems based on
     common symmetry characteristics.

	Isometric (Cubic): three 4-fold or bar 4 axes  (or four 3-fold or bar 3 axes)
	Hexagonal: one 6-fold axis
	Trigonal: one 3-fold or bar 3 axis
	Tetragonal: one 4-fold or bar 4 axis
	Orthorhombic: three orthogonal 2-fold axes or mirrors
	Monoclinic: one 2-fold axis or mirror
	Triclinic: one center or 1-fold axis

Refer the external morphology or internal symmetry to 3 or 4 reference axes.

• a, b, c; positive and negative ends
• alpha, beta, gamma ; angles between axes
• See Fig. 2.27 for summary of relations between angles and lengths
• Revisit the symmetry of the crystal systems

Axial Ratios

• Measure of lengths of unit cell, ratioed to one another.
• The `b' axis is commonly set equal to `1'
• Thus a:b:c may be expressed as 0.813:1:1.903 for sulfur.

• Crystal faces are defined by indicating their intercepts on the crystallographic axes.
  • Ex: 1a: 1b: infinite c or infinite a: 2b: 1c
• IMPORTANT: these measures are RELATIVE and not absolute

Miller Indices

"A series of whole numbers that have been derived from the intercepts by their inversions and, if necessary, clearing fractions."

• Note difference between intercept of zero and infinity: definition of parallel
• Reported in parentheses: (100)
• Unknown distances reported using (hkl)
• Commas only if needed to deal with double digit numbers (1,12,3)
• Hexagonal system: needs 4 numbers: (10í0)
•   however the sum of the first 3 digits (hki) is always zero

Form

Outward appearance of a crystal is its `habit'; the word "form" has a special and restricted use.

Form: group of crystal faces, all of which have the same relation to the elements of symmetry.

Ex: difference between (111) in the triclinic and cubic systems

The number of faces that belongs to a form is controlled by the symmetry.

The group of faces that comprise a form are designated by braces: {111}

General Form: depends on class

• Triclinic, monoclinic and orthorhombic: {111}
• Tetragonal, hexagonal: {121}
• Isometric: {123}

Open vs closed forms.

Names and descriptions of forms: see figures

Zones

A group of faces whose lines of intersection are parallel. [100]

Direct vector sum of directions, no inversion as was done with Miller Indices.