The Five Distinct 2D 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.

? ? ? ?
2
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, ?.

? ? ? ?
a ? b ? ?
a
b
?
3
The Five Distinct 2-D Lattices

• primitive rectangular lattice. define ? 90o,
  but a and b not related.

a
?
b
4
The Five Distinct 2-D Lattices

• primitive rectangular lattice. define ? 90o,
  but a and b not related.
• square lattice. define ? 90o, and a b.

a
?
b
a
?
b
5
The Five Distinct 2-D Lattices
a

• Next, require a b, but let ? 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.

?
b
6
The Five Distinct 2-D Lattices
a

• Next, require a b, but let ? 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 ? 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.

?
b
a
?
b
7
The Five Distinct 2-D Lattices

• hexagonal lattice. another special case of d),
  when ? 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.

a
?
b
8
The Five Distinct 2-D Lattices

• hexagonal lattice. another special case of d),
  when ? 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
?
b
9
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

? ? ?
2p 4
a
note if 90o, cos 0 and lama
2p n
2p n
2p n
a cos
2p n
2p 4
la
B C
? ?
? ? ? (?)x ? ?
a
2p n
2p 2
2p n
a
A B C D
ma
10
Limitations on Lattice Symmetry in Crystals
2p 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.

11
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.

12
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.

13
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.

14
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.

15
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.

16
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.

17
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.

18
Seventeen 2-D Space Symmetries

• p1. reflections pass through all 6-fold axes.

19
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.

20
Which Space Group?
21
Which Space Group?
pg
22
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
23
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
24
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