Symmetry

Motif: the fundamental part of a symmetric design that, when repeated, creates the whole pattern

Operation: some act that reproduces the motif to create the pattern

Element: an operation located at a particular point in space

• External Shape of Crystals reflects Internal Structure

• External Shape is best described by Symmetry

Symmetry

• Repetitive arrangement of features (faces, corners and edges) of a crystal around imaginary lines, points or planes

• Reflects internal ordering of atoms in the mineral structure

CRYSTALLOGRAPHY FOR MINERALOGIST

RotationTwo-fold rotation

= 360o/2 rotation to reproduce a motif in a symmetrical pattern

= the symbol for a two-fold rotation

OperationOperation

Motif

Element6

6

Symmetry Elements

6

6

first operation step

second operationstep

Reflection (m)

Reflection across a “mirror plane”reproduces a motif

m = symbol for a mirror plane

m

Rotational Symmetry

• Rotation of x degrees with respect to a line called a rotation axis leaves the image or shape unchanged

• If an object looks the same after rotation of 360o/n, that object is said to have n-fold rotational symmetry or an n-fold axis

• Called n-fold, because it takes n rotations to return to its original position

• Only certain angles (“-folds”) of rotational symmetry are possible in minerals

Rotational Symmetry in Minerals

• Name Short-hand Angle Symbol

5-fold, 7-fold and other symmetries are not possible because one cannot fill space with 5 - sided OR 7- sided

inversion through a center to reproduce a motif in a symmetrical pattern

i = symbol for an inversion center

66

Inversion centre

For every point or face on one side of the center of symmetry, there is similar point or face at an equal distance on the opposite side of the center

Inversion (i)

The Cube and Octahedron are simple, common Isometric Forms

We now have 6 unique 2-D symmetry operations:

1 2 3 4 6 m

Rotations are congruent operations reproductions are identical

Inversion and reflection are enantiomorphic operationsreproductions are “opposite-handed”

2-D Symmetry

Combinations of symmetry elements are also possible

To create a complete analysis of symmetry about a point in space, we must try all possible combinations of these symmetry elements

In the interest of clarity and ease of illustration, See more 2-D examples

2-D Symmetry

2-fold rotation Step -1: reflect

Step -3: Reflect & rotate

Second mrequired

Step -4: Mirror plane

The result is Point Group 2mm

4-fold rotation with m 6- fold rotation with m

The result is Point Group 4mm The result is Point Group 6mm

3-fold rotation axis with a mirror creates point group 3m &6–fold rotation axis with mirror creates point group 6mm

3-D Symmetry

Rotoinversion

a) 1-fold rotoinversion ( 1 )

Step 1: rotate 360o/1

Step 2: invert

This is the same as i, so not a new operation

2-fold rotoinversion ( 2 )

The result:

Step 2: invertStep 1: 2-fold 360/2

This is the same as m, so not a new operation

m

c. 3-fold rotoinversion ( 3 ) = 3fold + i

1

Step 1: rotate 360o/3

Step 2: invert through center Completion of the sequence

1

2

4-fold rotoinversion ( 4 )

Rotate 360/4

Invert

Rotoinversion

4-fold operation ( 4 )

4-fold rotoinversion ( 4 )This is also a unique operation

4-fold

4

Note: 2 faces (or pairs of faces) on top and 2 faces on the bottom off-set by 90o

Rotoinversion ( 6 )

This is the same as a 3-fold rotation axis perpendicular to a mirror plane

Rotate 360o/6 = 60o

Types of symmetry possible in Minerals

• 1, 2, 3, 4, 6 : “proper rotations”

• m : mirror planes

• 1 or i : center of symmetryor inversion

•3 : bar 3 rotoinversion

• 4 : bar 4 rotoinversion

• 6 : bar 6 rotoinversion

PLUS OTHER COMBINATION OF ROTATION, MIRROR

These can be combined in 32 ways tomake crystal shapes

The 32 3-D Point GroupsEvery 3-D pattern must conform to one of them.

This includes every crystal, and every point within a crystal

Rotation axis only 1 2 3 4 6

Rotoinversion axis only 1 (= i ) 2 (= m) 3 4 6 (= 3/m)

Combination of rotation axes 222 32 422 622

One rotation axis ⊥ mirror 2/m 3/m (= 6) 4/m 6/m

One rotation axis || mirror 2mm 3m 4mm 6mm

Rotoinversion with rotation and mirror 3 2/m 4 2/m 6 2/m

Three rotation axes and ⊥ mirrors 2/m 2/m 2/m 4/m 2/m 2/m 6/m 2/m 2/mAdditional Isometric patterns 23 432 4/m 3 2/m

2/m 3 43m

Increasing Rotational Symmetry

32 3-D Point Groups

Regrouped by Crystal System (H & M symbol)Crystal System No Center Center

Triclinic 1 1

Monoclinic 2, 2 (= m) 2/m

Orthorhombic 222, 2mm 2/m 2/m 2/m

Tetragonal 4, 4, 422, 4mm, 42m 4/m, 4/m 2/m 2/m

Hexagonal 3, 32, 3m 3, 3 2/m

6, 6, 622, 6mm, 62m 6/m, 6/m 2/m 2/m

Isometric 23, 432, 43m 2/m 3, 4/m 3 2/m

3-D SymmetryThe 32 3-D Point Groups

After Bloss, Crystallography and Crystal Chemistry. © MSA

+c

+a

+b

γγ

ββ

αα

Axial convention:“right-hand rule”

Crystal Axes

FORMSThe term form is used to indicate general outward appearance

In Crystallography external shape is denoted by the word habit, where as form is used in a special and restricted sense. Thus forms consists of group of crystal faces, all of which have the same relation to element of symmetry and display the same chemical and physical properties

The number of faces that belongs to a form is determined by the symmetry of the crystal class.

Miller Indices enclosed in parenthesis as (hkl) or (010) indicate crystal faceMiller Indices enclosed in braces as {hkl} or {010} indicate form symbols

In each crystal there is form, faces of which intersect all the crystallographic axes at different lengths – general form {hkl}.All other forms are called as special forms

The concept of a general form can also be related to the symmetry elements of a specific crystal class. An (hkl) face will not be parallel or perpendicular to a single crystal symmetry elements regardless of the crystal class, where as special form consists of faces that are parallel or perpendicular anySymmetry elements in the crystal class.

Nomenclature of crystallographic forms was initially proposed by Groth, 1895 and later modified by A. F. Rogers in 1935 – It recognizes 48 forms of which 32 are the general forms of 32 crystal classes. 10 are special, closed forms of isometric system and 6 special open forms (prisms- hex and tetragonal)

Different scheme : Fedrov Institute of Leningrad, 1925 : 47 instead of 48 forms (Dihedrons)

NON – ISOMETRIC FORMS

ISOMETRIC FORMS

MALFORMATION OF CRYSTAL FACES

NON - ISOMETRIC FORMS

ISOMETRIC FORMS

Pedion (Monohedron) : A single face comprising form

Pinacoid (Parallehedron) : An open form made up of two parallel faces

Dome (Dihedron) : Two non parallel faces symmetric w.r.t a mirror plane (m).

Sphenoid (Dihedron) : Two non parallel faces symmetric w.r.t a 2-fold rotation axis

Prism : An open form composed of 3,4,6,8 or 12 faces, all of which are parallel to the same axis

Pyramid : An open form composed of 3,4,6,8 or 12 non parallel faces that meet at a point

Dipyramid : A closed form having 6,8,12,16 or 24 faces.

Trapezohedron : A closed form that has 6,8 or 12 faces in all, with 3,4 or 6 upper faces offset with 3,4 or 6 faces lower faces. These faces are the result of 3,4 or 6 fold axiscombined with perpendicular 2-fold axes.

There is Isometric trapezohedron (tetragon-trisoctahedron) – 24 face form

• Scalenohedron : A closed form with 8 or 12 faces grouped in symmetrical pairs. In the tetragonalscalenohedron (rhombic scalenohedron) pairs of upper faces are related by anaxis of 4-fold rotoinverstion to pairs of lower faces. The 12 faces of hexagonalscalenohedron display three pairs of upper faces and three pairs of lower faces in alternating positions. The pairs are related by the centre of symmetry, coexist with a 3-fold axis of rotoinverstion.

• Rhombohedron : A closed form composed of six faces of which three faces at the top alternateswith three faces at the bottom, the two sets of faces being offset by 60o.

Only seen in point groups• Disphenoid : A closed form consisting of two upper faces that alternate with two lower faces, offset

by 90o .