GS 122 – Elementary Mineralogy

Why should we be interested?

● Important physical properties depend on crystal structure:

– Conductivity

– Magnetic properties

– Stiffness

– Strength

● These properties also often depend on crystal orientation

Some Important Terms for Crystals

– Crystal Structure – Bravais lattices – Symmetry operations

– Miller Indices – Stereographic – Standard Projection

● Planes Projection

● Directions – Principal Metal Structures

BCC, FCC, HCP

– Ionic Crystals – Isomorphism – Polymorphism

Atomic arrangement

SOLID

Crystalline – periodic arrangement of atoms: definite repetitive pattern

Non-crystalline or Amorphous – random arrangement of atoms

The periodicity of atoms in crystalline solids can be described by a network

of points in space called lattice. A space lattice can be defined as a three

dimensional array of points, each of which has identical surroundings.

Fourteen Bravais Lattices

Symmetry

The seven symmetry operators are:

1)Translation

2) Rotation

3) Reflection

4) Inversion (center of symmetry)

5) Roto-inversion (inversion axis)

6) Glide (translation + reflection)

7) Screw (rotation + translation)

– Spatial transformations or the spatial relationships between objects in a pattern

Symmetry Elements

– Primary function is to specify the reference point about which an action occurs

The first five symmetry elements that we consider are:

1) Translation vectors

2) Rotation axes

3) Mirror planes

4) Centers of symmetry (inversion points)

5) Inversion axes

Symmetry operations:

1. Translation

- Replication of an object at a new spatial coordinate

- Shift in a specified direction by a specified length

- Used to build a crystal structure by replicating an object (the basis) at each of

the Bravais lattice points

- Based on a, b and c vectors of unit cell, a translation vector t can be expressed

as t = ua + vb + wc, where u, v and w are positive or negative integers.

2. Rotation

- Motion through an angle about an axis

- Symmetry element is an N-fold rotation axis

- The multiplicity N is an integer.

- After having performed the rotation N times the object returns to its

original position

Symmetry operations:

2. Rotation

Symmetry operations:

Figure. The five rotation operators that are consistent with translational symmetry.

- The large circles are lines of construction to guide the eye.

- The solid object in the center shows the position of the rotation axis and the small

circle is the object which is repeated to form the pattern.

The 2 axis is referred to as a diad, the 3 axis as a triad, the 4 axis as a tetrad,

and the 6 axis as a hexad.

1-Fold Rotation Axis - An object that

requires rotation of a full 360° to repeat

itself has no rotational symmetry

2-fold Rotation Axis - If an object

appears identical after a rotation of 180°,

that is twice in a 360° rotation, then it is

said to have a 2-fold (2 /180) rotation

symmetry

Question: Is it possible to have 5, 7 or 8-fold rotation symmetry?

2. Rotation

Symmetry operations:

Objects with 5, 7 and 8 or higher

order symmetry exist in nature.

However, these are not possible in crystallography

as they cannot fill the space completely.

3. Reflection

- Describes the operation of a mirror

- Symmetry element is a reflection plane.

- Hermann– Mauguin symbol: m.

Figures with the axes of symmetry drawn

in. The figure with no axes is asymmetric.

Symmetry operations:

3. Reflection

Symmetry operations:

- In 2D there is a line of symmetry, in 3D a plane of symmetry.

- An object or figure which is indistinguishable from its transformed

image is called mirror symmetric.

Reflection converts a right-handed object into a left-handed

or enantiomorphous (in opposite shape) replica.

Symmetry operations:

4. Inversion

- “Reflection” through a point.

- This point is the symmetry element and is called inversion center or

center of symmetry.

5. Rotoinversion

-The symmetry element is a rotoinversion axis or, for short, an inversion axis

- This refers to a coupled symmetry operation which involves two motions: take a rotation

through an angle of 360/N degrees immediately followed by an inversion at a point located

on the axis

Symmetry operations:

6. Screw rotation

- The symmetry element is a screw axis.

- It can only occur if there is translational symmetry in the direction of the axis.

- The screw rotation results when a rotation of 360/N degrees is coupled with a

displacement parallel to the axis.

The Hermann–Mauguin symbol is NM („N subM‟); N expresses the

rotational component and the fraction M/ N is the displacement

component as a fraction of the translation vector.

Symmetry operations:

6. Screw rotation

Symmetry operations: Screw axes and their graphical symbols. The axes 31, 41,

61, and 62 are right-handed; 32, 43, 65, and 64 are the

corresponding left-handed screw axes.

The operation of a 42 screw axis

parallel to the z direction;

(a) Atom A at z=0 is repeated at

z=T and then rotated counter

clockwise by 90°;

(b) The atom is translated parallel

to z by a distance of t =2T/4, i.e.

T/2 to create atom B;

(c) Atom B is rotated counter

clockwise by 90° and translated

parallel to z by a distance of t =

2T/4, i.e. T/2, to give atom C;

Construction of 42 screw operation:

6. Screw rotation

Symmetry operations:

(d) atom C is at z=T, the lattice

repeat, and so is repeated at z

= 0;

(e) repeat of the symmetry

operation produces atom D at z

= T/2;

(f) standard crystallographic

depiction of a 42 screw axis

viewed along the axis

Construction of 42 screw operation:

6. Screw rotation

Symmetry operations: In this figure, the motif is represented by

a circle, the + means that the motif is

situated above the plane of the paper

and ½+ indicates the position of a motif

generated by screw operation.

7. Glide reflection

- The symmetry element is a glide plane. It can only occur if translational symmetry

is present parallel to the plane. At the plane, reflections are performed, but every

reflection is coupled with an immediate displacement parallel to the plane.

Hermann–Mauguin symbol is a, b, c, n, d or e, the letter designating the

direction of the glide referred to the unit cell. a, b and c refer to

displacements parallel to the basis vectors a, b and c, the

displacements amounting to 1/2a, 1/2b and 1/2c, respectively.

- The glide planes n and d involve displacements in a diagonal direction by

amounts of ½ and ¼ of the translation vector in this direction, respectively. The

letter e designates two glide planes in one another with two mutually

perpendicular glide directions.

Symmetry operations:

7. Glide reflection

Symmetry operations:

Top left:

Perspective illustration of a

glide plane.

Other images:

Printed and graphical

symbols for glide planes

perpendicular to a and c

with different glide

directions.

z = height of the point in

the unit cell