Translational Symmetry:
Must specify Distances and Directions
Replace the motif being repeated by a point and you have a lattice.
"An imaginary pattern of points (or nodes) in which every point (node) has an environment that is identical to every other point (node) in the pattern. A lattice has no specific origin..." p. 110
Specify direction and repeat distance (Draw examples)
Can describe patterns by identifying the translation vectors (which have direction and magnitude). (See figures from text.)
Many choices of translations, some are more logical or conventional.
Smallest repeatable unit is the unit cell.
Only 5 possible and distinct plane lattices (or nets). (Fig 3.5, O.H. 1)
Oblique net where a not= b and gamma not= 90°
Rectangular net where a not= b and gamma = 90°
Centered rectangular net where a not= b and cos gamma =a/2b
(or diamond net) with a' = b' and gamma not= 90, 60, 120°
Hexagonal net with a = b and gamma = 60°
Square net with a = b and gamma = 90°
Again, remember that multiple choices of the unit cell are possible.
Consider ownership of the various lattice points.
Rotation Angle Restrictions:
See pgs. 115-116 for the proof that 1,2,3,4, and 6-fold axes are the only possible ones consistent with 2-D periodic repeats that fill space.
Symmetry Content of Planar Motifs: (Fig 3.10, O.H. 2)
10 possible combinations: 1, 2, 3, 4, 6, m, 2mm, 3m, 4mm, 6mm (Fig 3.10) (Remember the relationships of the symbology to the mirrors and the axes.)
Symmetry Content of Plane Lattices: (Fig 3.11, O.H. 2)
Note the symmetry which is explicit and implicit in each of the 5 plane lattices.
EX. p4 or p2
Glide planes appear as combinations of translation and reflection.
2-D Plane Groups: (Fig 3.13, 3.14, O.H. 3, 4)
These are the 17 combinations of the 5 plane lattices and the 10 translation free point groups.