The Five Distinct 2-D Lattices

1. The Five Distinct 2-D Lattices ● ● ● ● • In 1-D symmetry there was only one type of lattice. • We will see that there are five types of lattice in a 2-D plane. • Not talking about symmetries of motifs, just lattice points. • These are the unique ways in which translation vectors can be related to one another. • oblique lattice. most general; no special relationship between a, b, and the angle, γ. a ≠ b γ = ? a b γ

2. The Five Distinct 2-D Lattices a • primitive rectangular lattice. define γ = 90o, but a and b not related. • square lattice. define γ = 90o, and a = b. γ b a γ b

3. The Five Distinct 2-D Lattices a • Next, require a = b, but let γ be arbitrary. • although this is a proper lattice, there is a different way to look at this and there are advantages to this different point of view. • centered rectangular lattice. here γ = 90o. • have the benefit of dealing with right angles. • no difference between the center lattice point and others; just our imposition of the 90o reference. γ b a γ b

4. The Five Distinct 2-D Lattices • hexagonal lattice. another special case of d), when γ = 60o or 120o. • differs from others because it has 6-fold rotational symmetry about each lattice point. • in addition to the primitive rhombic unit cell shown, a centered hexagonal cell can also be seen. • could also treat hexagonal lattice as though it were a centered rectangular lattice, however this would obscure, rather than illuminate, the highest symmetry of the lattice. a γ b

5. Limitations on Lattice Symmetry in Crystals • Although all point groups are permissable for isolated molecules, only certain symmetries possible for crystals. • Imagine a set of lattice points (with A, B, C & D labeled). An n-fold rotation axis through A generates B’; a similar axis though D generates C’. • D is at a distance ma from A (m = an integer). • B’ is at a distance la from C’ (l = an integer). • la = ma – 2acos • l = m – 2cos cos = (m – l)/2 ● ● ● a 2π 4 note: if 90o, cos = 0 and la=ma 2π n 2π n 2π n a cos 2π n 2π 4 la B’ C’ ● ● ● ● ● (●)x ● ● a 2π n 2π 2 2π n a A B C D ma

6. Limitations on Lattice Symmetry in Crystals 2π n • cos = (m – l)/2 Restrictions: cosine values must be between -1 and +1 m – l must be a whole number (because m & l are both integers) possible values of (m – l)/2 are 0, ±½, ±1 • So, although any symmetry is possible in a molecule, the point symmetry elements of a crystal are limited to 1-, 2-, 3-, 4-, & 6-fold rotations.

7. Seventeen 2-D Space Symmetries • p1. start with an array generated only by translations. Here the two unit translations are unequal at at a random angle (i.e. not 60o, 90o, or 120o). • the letter “p” indicatesthat the lattice is primitive (only 1 lattice point per unit cell); the “1” indicates that no rotation (other than C1) is present.

8. Seventeen 2-D Space Symmetries • p2-p6. several new space symmetries can be generated by adding rotational symmetry (only 2-, 3-, 4- & 6-fold possible) • in each case the combined effect of the explicitly introduced rotational axis and the translational operations generates further symmetry axes.

9. Seventeen 2-D Space Symmetries • We can create new 2-D space symmetries by introducing reflections; this can be done only for the rectangular, square, trigonal and hexagonal lattices. • pm. one set of mirror planes is introduced parallel to one of the translation direction (m = mirror). • pmm. two perpendicular sets of reflection lines introduced; C2 axis generated.

10. Seventeen 2-D Space Symmetries • We can do the same by introducing glide planes. • pg. one set of glide planes is introduced parallel to one of the translation direction in a rectangular lattice (g = glide plane). • pgg. two perpendicular sets of glide planes introduced; C2 axis generated.

11. Seventeen 2-D Space Symmetries • There are 3 combinations of m & g symmetry elements. • pmg. has mutually perpendicular m and g lines. • cm. has sets of parallel m & g lines; the combination yields a centered rectangualr lattice (c = centered). Note: pmg implies perpendicular planes. • cmm. has mutually perpendicular mirror planes and glide planes; C2 axes generated.

12. Seventeen 2-D Space Symmetries • The remaining five 2-D space symmetries are all obtained by adding reflections to the p3, p4 & p6 groups. • There are two ways to add reflections to p3. • p3m. reflections pass through all 3-fold axes. • p31m. reflections pass through alternate 3-fold axes.

13. Seventeen 2-D Space Symmetries • There are also two ways to add reflections to p4. • p4m. reflections pass through all 4-fold axes; there are also glide lines generated between reflection lines. • p4g. add reflection lines so that they pass through the 2-fold exes; generates glide lines which pass between the reflection lines.

14. Seventeen 2-D Space Symmetries • p1. reflections pass through all 6-fold axes.

15. Seventeen 2-D Space Symmetries • Diagram shows all seventeen 2-D Space Groups. • Can use these to determine space symmetry. • Can use these to generate 2-D patterns.

16. Which Space Group? pg

17. 2-D Space Symmetry Diagram • Diagram showing how an entire set of objects is generated from an initial motif (#1) at a general position (x,y) by the combined action of the various symmetry elements. • In this case, there are also C2 axes (not shown on diagram). pgg

18. 2-D Space Symmetry Diagram • Diagram showing how an entire set of objects is generated from an initial motif (#1) at a general position (x,y) by the combined action of the various symmetry elements. • In this case, there are also C2 axes (not shown on diagram). if we add a new motif here, where else would it also appear? pgg

19. 2-D Space Symmetry Diagram • Diagram showing how an entire set of objects is generated from an initial motif (#1) at a general position (x,y) by the combined action of the various symmetry elements. • In this case, there are also C2 axes (not shown on diagram). pgg