Lecture II - Crystallography
Every Crystal possesses an internal 3-D order (unit cells) that are
repeated
The study of internal structure, external shape of crystals, and laws
governing their growth is called Crystallography
Crystal – A homogenous solid possessing long range 3-D internal
order
-
Euhedral – perfect planes (Eu = good, hedron = plane)
Subhedral – imperfect planes (Sub = less than)
Anhedral – no planes (An = without)
Microcrystalline – need microscope to see Crystals
Cryptocrystaline – need X-ray unit to show structure
Amorphous – No ordered internal atomic arrangement
Crystallization – minerals can crystallize from liquids, melts, vapors
and from other solids
e.g. NaCl (halite) precipitates from hypersaline lakes
- The ionic charge and radius of Na+ and Cl- ions dictates that
they are packed together in a Face Centered Cubic (FCC)
arrangement.
- This packing arrangement (FCC) dictates that Halite has a cubic
external morphology
- Haϋy (1743-1822) was the first to demonstrate that external
crystal form is an expression of internal atomic order
- “molécule” = unit cell – the smallest unit of a structure (or
pattern) that can be indefinitely repeated to generate the whole
structure.
Symmetry Elements
Include Rotational Axis, Mirror Planes and Centers of Symmetry
(1)
1
2
3
4
6
–
–
–
–
–
Rotational Symmetry – in crystals there are only 5 kinds of
rotational symmetry (around an axis).
Fold symmetry = no symmetry
Fold symmetry – 180°
Fold symmetry – 120°
Fold symmetry – 90°
Fold symmetry – 60°
No crystals have 5 – Fold symmetry. Starfish and Echinoderms
and some similar organisms have 5 – Fold symmetry
(2)
Planes of Symmetry = mirror (across a plane)
(symbol “m” used to indicate mirror operation)
(3)
Inversion through a Center or point (through a point)
(symbol “i” used to indicate inversion center operation)
(4)
Translational Symmetry
(symbol “t” is used)
- lattice – periodic arrangement of points in space (upon which
the unit cell is repeated)
*
There are only 32 ways to combine these symmetry
operations = 32 Point Groups or Crystal Classes
Rotational Operations – generate the same motif over and over
again. They cannot make a change in the hand (start with left
hand, always have left hand)
Mirror or Inversion Operations – produce a change in hand (start
with left – end up with right)
Inversion Axis (improper axes)
- combines rotation with inversion (aka roto-inversion)
(1, 2, 3, etc.)
_
_
1  i (1 is preferred)
_
2  m (m is  to the axes and also preferred)
_
33+i
_
4  unique
_
6  3m
There are 10 basic symmetry elements:
_
_
_ _
_
1, 2, 3, 4, 6, 1 (=i), 2 (=m), 3, 4, and 6 (3/m)
These symmetry elements can be represented by the symbols below
– from Klein Fig. 5.20 (p. 188)
Hermann-Maugin Symbols: Combination of numbers and letters to
designate symmetry of the 32 crystal classes (International symbols).
Examples: Symmetry of an ashtray
4-fold axis = (4)
mirror plane = (m)
diagonal mirror plane = (m)
Point Group Notation 4mm