Download - Watkins/Fronczek - Rotational Symmetry 1 Symmetry Rotational Symmetry and its Graphic Representation

Watkins/Fronczek - Rotational Symmetry 1

SymmetrySymmetryRotational Symmetry

and itsGraphic Representation


Rotational SymmetryRotational SymmetryA pattern is symmetricsymmetric if a

single motif is repeated in space.

A wheel is a repeating pattern of spokes; the motif is one spoke.

Each ccw rotation through angle is a symmetry symmetry operationoperation which produces similarity.

For n equally spaced spokes, = 360o/n

is the repetition angle or "throw" of the rotation axis.

4 is the "fold" of this rotation axis. It is also called the order of the axis.

90o

4-fold axis

24


The motif is the smallest part of a symmetric pattern, and can be any asymmetric chiralchiral* object.

Rotational SymmetryRotational Symmetry

To produce a rotationally symmetric pattern, place the same motif on each “spoke”.

This pattern is produced by a proper rotationproper rotation because it is a real rotation which produces similarity in the pattern.

*not superimposable on its mirror image, like a right hand.

The proper rotation operator is a geometric lineline.

1

2

3

0,4


One of the most important skills a student of crystallography must develop is the ability to discover the symmetry of a pattern. He or she must be able to


1. Locate the motif;2. name all symmetry operations which produce similarity

in the pattern;3. name the complete set of all such symmetry operations;4. represent the set of symmetry operations in both

diagramatic and mathematical terms.


Rotational SymmetryRotational SymmetryNormal crystals contain only five kinds of proper rotational symmetry:1. One foldOne fold, = 360o (IdentityIdentity)2. Two foldTwo fold, = 180o

3. Three foldThree fold, = 120o

4. Four foldFour fold, = 90o

5. Six foldSix fold, = 60o

The proper rotation axis is a line and is denoted by the symbol nn (Hermann-Maugin) or CCnn (Schoenflies). Thus, the five proper crystallographic rotation axes are called 11, 22, 33, 44, 66, or CC11, CC22, CC33, CC44, CC66.

Note: molecules can have proper rotation axes of any value up to


Rotational SymmetryRotational SymmetryThere is another, quite different way to produce a rotationally symmetric pattern:

put motifs of the opposite hand on every other spoke.

The imaginary operation required to do this is:

RotateRotate the motif through angle

InvertInvert the motif through a point on the rotational axis - this changes the chirality of the motif.

This “roto-inversion” is called an improper rotationimproper rotation

1 2


Rotational SymmetryRotational SymmetryNormal crystals contain only five kinds of improper rotational symmetry:1. One foldOne fold, = 360o

2. Two foldTwo fold, = 180o




The roto-inversion operator is a line and a point on the line, and is denoted by the symbol nn. Thus, the five improper rotation axes are called 11, 22, 33, 44, 66.


Rotational SymmetryRotational SymmetryThere is another, quite different way to produce a rotationally symmetric pattern:

put motifs of the opposite hand on every other spoke.

The imaginary operation could also be:

RotateRotate the motif through angle

ReflectReflect the motif through a plane perpendicular to the rotational axis - this changes the chirality of the motif.

This “roto-reflection” is equivalent to roto-inversion.







The roto-reflection operator is a line and a plane perpendicular to the line, and is denoted by the symbol nn. Thus, the five improper rotation axes are called 11, 22, 33, 44, 66.

~~ ~ ~ ~ ~







There is equivalence between n and n~

1 2 3 4 6

12 346~~ ~ ~ ~

1 = 2 is inversion or “center of symmetry” i (Sch)

~

2 = 1 is mirrorm (HM) or (Sch)

~


Rotational SymmetryRotational SymmetryOur patterns, which have 4 and 4 ( 4) symmetry looks like this:

Changing the motif would change how the pattern looks, but would not change the symmetry of the pattern.

We need a general representation of symmetric patterns which is independent of the motif.

_ ~


y

x

z

Rotational SymmetryRotational SymmetryThe stereographic projectionstereographic projection is a graphic representation of any kind of rotation about a fixed point (point group).

1. Start with a sphere and a 3-d coordinate system.

2. Place the rotation axis (proper or improper) of highest order along z.

3. The point of an improper rotation is at the center of the sphere.

4. Look down the rotation axis at the x-y (equatorial) plane.

n or n



1. Place the chiral motif on the surface of the sphere.

2. Project each motif onto the equatorial plane.

equatorial plane

x

y


Rotational SymmetryRotational Symmetryequatorial plane

x

yThe symbols are called equipointsequipoints because in a symmetry pattern, each one is equivalent to all the others by symmetry.


Rotational SymmetryRotational Symmetryequatorial plane

x

yIf all equipoints lie on one side of the equatorial plane, the pattern belongs to a 2-D planeplane groupgroup. Otherwise, the pattern belongs to a 3-D point grouppoint group.



1 1

2

11 or CC11 22 or CC22

One symmetry operatorTwo symmetry operations

one equipoint



1

2

1 1 or C Cii 22

12

= m m or C Css

inversion center horizontal mirror (h)

Two symopsTwo equipoints

Two symopsTwo equipoints



1

3

2

33 or CC33 3 3 or S S66

1

3

5

4

6

2

3 symops3 equipoints

6 symops6 equipoints



1 1

3

44 or CC44

2

3

4

4 4 or S S44

2

4



1

5

3

66 or CC66 6 6 or C C3h3h

6

4

2

14

52

36



• The 10 symmetry patterns 1, 2, 3, 4, 61, 2, 3, 4, 6 and 1, 2, 3, 4, 61, 2, 3, 4, 6 are called crystallographic point crystallographic point groupsgroups because these are patterns found in crystals.

• There are 22 other patterns (combinations of proper and improper rotations) also found in crystals, for a total of 32 crystal classes.

• The general nomenclature for these (and other) patterns is as follows:



• Cn – one n-fold proper rotation axis only(the primary axis)

– 1 2 3 4 6 (Hermann-Maugin)

– C1 C2 C3 C4 C6 (Schoenflies)

The primary axis is oriented along the z-direction of the stereographic projection.

Crystallographic Point GroupsCrystallographic Point GroupsCyclic Groups

Non-xtal objects can have n from 1 to ∞.

n



• Cnh – one n-fold proper rotation axis and one horizontal mirror.

– m 2/m 3/m 4/m 6/m

– C1h Cs C2h C3h C4h C6h

The proper rotation axis is along z; the improper rotation axis is also along z (the mirror is in the equatorial plane).


n 2



• Cnv – one n-fold proper rotation axis andn vertical mirrors.

– m mm2 m3 mm4 mm6

– C1v Cs C2v C3v C4v C6v

The primary axis is along z, the n mirror planes are perpendicular to the equatorial (x-y) plane (2 secondary axes).


n2



• Dn – one n-fold proper rotation axis andn secondary 2-fold axes (dihedral 2-folds).

– 2 222 23 224 226– D1 C2 D2 D3 D4 D6

The primary rotation axis is along z, with n 2-fold secondary axes in the equatorial plane perpendicular to the primary axis.

Crystallographic Point GroupsCrystallographic Point GroupsDihedral Groups

n2


• Dnh – one primary n-fold proper rotation axis, n dihedral (secondary) 2-folds, one horizontal mirror, and n vertical mirrors coincident with the secondary 2-folds.

Rotational SymmetryRotational SymmetryCrystallographic Point GroupsCrystallographic Point Groups

Dihedral Groups

mm2 2 2 2mmm 2 3

m2 2 4mmm

2 2 6mmm

– D1h C2v D2h D3h D4h D6h

n2

2

2



• Dnd – one primary n-fold proper rotation axis, n dihedral secondary 2-folds, n dihedral mirrors bisecting the secondary 2-folds

– 2/m 42m 62m

– D1d C2h D2d D3d (D4d D6d)

Crystallographic Point GroupsCrystallographic Point GroupsDihedral Groups

n2

2


• Sn – one n-fold improper (roto-reflection)

axis only.– m, 1, 3/m, 4, 3

– S1 = Cs, S2 = Ci, S3 = C3h, S4, S6


S Groups

n


Illustration of differencesbetween cyclic, dihedral, and S-type groups

(from Wikipedia)



The five cubic point groups are based on the two geometric solids which can be derived from a cube: the octahedronoctahedron and the tetrahedrontetrahedron.

Crystallographic Point GroupsCrystallographic Point GroupsCubic Groups



Cubic GroupsThe cube, octahedron and tetrahedron all have four secondary 3-fold axes inclined at 54.8o (half the tetrahedral angle) to the primary axis.



Add the four secondary 3-folds to the 2-fold dihedral groups:

Crystallographic Point GroupsCrystallographic Point GroupsTetrahedral Cubic Groups

D2 + 3 = T

D2h + 3 = Th

D2d + 3 = Td

23

m2 3

43m Full symmetry group of a regular tetrahedron



Add the four secondary 3-folds to the 4-fold dihedral groups:

Crystallographic Point GroupsCrystallographic Point GroupsOctahedral Cubic Groups

D4 + 3 = O

D4h + 3 = Oh

432

m4 3m

2

D4d + 3 = Od

Full symmetry group of a cube and a regular octahedron.



Crystals (and their point groups) are classified according to the order, n, of the primary (1o) and secondary (2o) axes, regardless of whether these axes are proper or improper.

There are seven broad classifications, called crystal systemscrystal systems.

Further classification is according to the specific point groups, called crystal classescrystal classes.



11oo

nn22oo

n’n’AngleAnglen-n’n-n’

CrystalCrystalSystems (7)Systems (7)

CrystalCrystalClasses (32)Classes (32)

1 1 - Triclinic C1, Ci

2 1 - Monoclinic C2, Cs, C2h

2 2 90 Orthorhombic D2, C2v, D2h

3 1 or 2 90 Trigonal C3, S6, D3, C3v, D3d

4 1 or 2 90 Tetragonal C4, S4, C4h, D4, C4v, D2d, D4h

6 1 or 2 90 Hexagonal C6, C3h, C6h, D6, C6v, D3h, D6h

2; 4 3 54.8 Cubic T, Th, Td ; O, Oh

Crystallographic Point GroupsCrystallographic Point Groups

Download - Watkins/Fronczek - Rotational Symmetry 1 Symmetry Rotational Symmetry and its Graphic Representation

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