Crystallography

1
Crystallography

• Crystal Forms

Orpiment (As2S3) Monoclinic
2
Crystal Classes

• Triclinic
• a?b?c
• a?ß??
• Monoclinic
• a?b?c
• a?90 ßlt90
• Orthorhombic
• a?b?c
• aß?90

• Hexagonal
• ab?c
• aß90 ?120
• Tetragonal
• ab?c
• aß?90
• Isometric
• abc
• aß?90

3
Subdivision of Classes

• (P) Primitive
• All nodes are located at corners
• (I) Body Centered
• Nodes are located at corners 1 in the center of
  the cell
• (C) Face Centered
• Nodes are located at corners in the middle of
  the face of 2 or more sides (Paired)
• (F) Face Centered
• Nodes are located at corners in the middle of
  all of the faces

4
Symmetry Operations

• Translation
• Movement along a vector
• Rotation
• Movement around a central point
• Reflection
• Mirror Image
• Inversion
• Identical features are located along the same
  line and equidistant from the center

Read handout for more information
5
Translation
Translations (Lattices) A property at the atomic
level, not of crystal shapes Symmetric
translations involve repeat distances The origin
is arbitrary 1-D translations a row
a
?
a is the repeat vector
6
Translation (Cont.)
Translations (Lattices) 2-D translations a net
Unit Cell the basic repeat unit that, by
translation only, generates the entire pattern
7
Rotation

• Rotation may be
• 1-fold
• 2-fold
• 3-fold
• 4-fold
• 6-fold

8
2-D Symmetry Rotation

• Symmetry Elements
• Two-fold rotation
• 360o/2 rotation
• to reproduce a motif in a symmetrical pattern
• the symbol for a two-fold rotation

Operation
A Symmetrical Pattern
6
Motif
Element
6
9
2-D Symmetry Rotation

• Symmetry Elements
• Two-fold rotation
• 360o/2 rotation
• to reproduce a motif in a symmetrical pattern
• the symbol for a two-fold rotation

• 2-fold Rotation Axis (A2)- If an object appears
  identical after a rotation of 180o, that is twice
  in a 360o rotation, then it is said to have a
  2-fold rotation axis (360/180 2).  Note that in
  these examples the axes we are referring to are
  imaginary lines that extend toward you
  perpendicular to the page or blackboard.  A
  filled oval shape represents the point where the
  2-fold rotation axis intersects the page.  

10
2-D Symmetry Rotation

• Symmetry Elements
• b. Three-fold rotation
• 360o/3 rotation
• to reproduce a motif in a symmetrical pattern

6
6
6
11
2-D Symmetry Rotation

• Symmetry Elements
• b. Three-fold rotation
• 360o/3 rotation
• to reproduce a motif in a symmetrical pattern

the symbol for a three-fold rotation
3-Fold Rotation Axis (A3) - Objects that repeat
themselves upon rotation of 120o are said to have
a 3-fold axis of rotational symmetry (360/120
3), and they will repeat 3 times in a 360o
rotation.  A filled triangle is used to symbolize
the location of 3-fold rotation axis.
12
2-D Symmetry Rotation

• 4-Fold Rotation Axis (A4) - If an object repeats
  itself after 90o of rotation, it will repeat 4
  times in a 360o rotation, as illustrated
  previously.  A filled square is used to symbolize
  the location of 4-fold axis of rotational
  symmetry.

13
2-D Symmetry Rotation

• 6-Fold Rotation Axis (A6) - If rotation of 60o
  about an axis causes the object to repeat itself,
  then it has 6-fold axis of rotational symmetry
  (360/606).  A  filled hexagon is used as the
  symbol for a 6-fold rotation axis. 

14
2-D Symmetry Rotation
A1 A2 A3 A4 A6
6
2-fold
3-fold
4-fold
1-fold
6-fold
?
Objects with symmetry
?
?
a
Z
identity
5-fold and gt 6-fold rotations will not work in
combination with translations in crystals (as we
shall see later). Thus we will exclude them
now.
15
2-D Symmetry Inversion

• Symmetry Elements
• Inversion (i)
• inversion through a center to reproduce a motif
  in a symmetrical pattern
• symbol for an inversion center
• inversion is identical to 2-fold rotation in 2-D,
  but is unique in 3-D (try it with your hands)

6
6
16
2-D Symmetry Reflection

• Symmetry Elements
• Reflection (m)
• Reflection across a mirror plane reproduces a
  motif
• symbol for a mirror
• plane

17
3-D Symmetry and Crystal Systems

• Combinations of Symmetry Operations
• Triclinic System One-fold rotation with or
  without inversion.
• Monoclinic System Two-fold rotation and/or a
  single mirror.
• Orthorhombic System Three two-fold rotation
  axes and/or three mirrors
• Tetragonal System Single four-fold rotation or
  rotoinversion axis.
• Hexagonal System Trigonal division One
  three-fold axis
• Hexagonal division One six-fold axis.
• Isometric System Four three-fold axes oriented
  so that they go diagonally through the center of
  the unit cell form corner to corner.

18
Hermann-Mauguin Symbols

• Hermann-Maugin Symbols are used as a shorthand
  notation in order to indicate the symmetry of a
  crystal
• Mirror symmetry is denoted by an m
• Rotation is denoted by 1,2,3,4,5,6 for 1-6 fold
  rotation
• RotoInversion (Rotation Inversion) is denoted
  by 1,m,3,4,6

19
Determining Hermann-Maugin

• Determine the Crystallographic axes for the
  mineral.
• Determine all of the symmetry elements for the
  mineral
• Determine which, if any, of the elements are
  perpendicular
• List the symmetry elements. Perpendicular
  elements should appear next to each other and
  separated by a /

20
H-M Notation Examples

• 4mm
• 4 fold rotation and 2 mirror planes
• 4/m
• 4 fold rotation axis - to a mirror plane
• 2/m2/m2/m
• 3 2-fold rotation axes all of which are - to a
  mirror plane