CHAPTER 3: SYMMETRY AND GROUPS, AND CRYSTAL …sarahlambart.com/teaching/mineralogy-03.pdf · CRYSTAL LATTICE IN TWO DIMENSIONS ! Unit cell: The unit cell of a mineral is the smallest

CHAPTER 3: SYMMETRY AND GROUPS, AND CRYSTAL STRUCTURES Sarah Lambart

� 2 different types of close packing:

� hcp: tetrahedral interstice (ABABA)

� ccp: octahedral interstice (ABCABC)

� Definitions: The coordination number or CN is the number of closest neighbors of opposite charge around an ion. It can range from 2 to 12 in ionic structures. These structures are called coordination polyhedron.

RECAP CHAP. 2

An ideal close-packing of sphere for a given CN, can only be achieved for a specific ratio of ionic radii between the anions and the cations.

RECAP CHAP. 2

Rx/Rz C.N. Type

1.0 12 Hexagonal or Cubic Closest Packing

1.0 - 0.732 8 Cubic 0.732 - 0.414 6 Octahedral

0.414 - 0.225 4 Tetrahedral (ex.: SiO4

4-)

0.225 - 0.155 3 Triangular <0.155 2 Linear

� Pauling’s rule:

� #1: the coodination polyhedron is defined by the ratio Rcation/Ranion

� #2: The Electrostatic Valency (e.v.) Principle: ev = Z/CN

� #3: Shared edges and faces of coordination polyhedra decreases the stability of the crystal.

� #4: In crystal with different cations, those of high valency and small CN tend not to share polyhedral elements

� #5: The principle of parsimony: The number of different sites in a crystal tends to be small.

RECAP CHAP. 2

� Definitions: unit cell and lattice

� 7 Crystal systems

� 14 Bravais lattices

� Element of symmetry

CONTENT CHAP. 3 (2-3 LECTURES)

� A crystal consists of atoms, molecules, or ions in a pattern that repeats in three dimensions.

CRYSTAL LATTICE IN TWO DIMENSIONS

� The geometry of the repeating pattern of a crystal can be described in terms of a crystal lattice, constructed by connecting equivalent points throughout the crystal.

� Step1: 2D crystal lattice

� Lattice point: A crystal lattice is constructed by connecting adjacent equivalent points (lattice points) throughout the crystal.


�  The environment about any lattice point is identical to the environment about any other lattice point.

� The choice of reference lattice point is arbitrary.

One choice of a reference point

� Lattice point: A crystal lattice is constructed by connecting adjacent equivalent points (lattice points) throughout the crystal.


� The basic parallelogram (parallelepiped in three dimensions) constructed by connecting lattice points defines a unit cell.

A lattice constructed from the chosen point.

� Lattice Points and Unit Cell Because the choice of reference lattice point is arbitrary, the location of the lattice relative to the contents of the unit cell is variable.


� Regardless of the reference point chosen, the unit cell contains the same number of atoms with the same geometrical arrangement.

The same lattice and unit cell defined from a different reference point.


The same lattice and unit cell defined from a different reference point.

� Unit Cell - The unit cell is the basic repeat unit from which the entire crystal can be built. - A primitive unit cell contains only one lattice point.

CRYSTAL LATTICE IN TWO DIMENSIONS � Unit cell: The unit cell of a mineral is

the smallest divisible unit of a mineral that possesses the symmetry and properties of the mineral.

The unit cell is defined by three axes or cell edges, termed a, b, and c and three inter-axial angles alpha, beta, and gamma, such that alpha is the angle between b and c, beta between a and c, and gamma between a and b.

CRYSTAL LATTICE IN TWO DIMENSIONS � Alternate lattice and choice of the unit cell

CRYSTAL LATTICE IN TWO DIMENSIONS � Rules to choose a unit cell

� Smallest repeat unit

� Highest possible symmetry (with the most 90° angles)




CRYSTAL LATTICE IN TWO DIMENSIONS � Unit cell: The unit cell of a mineral is

the smallest divisible unit of a mineral that possesses the symmetry and properties of the mineral.

The unit cell is defined by three axes or cell edges, termed a, b, and c and three inter-axial angles alpha, beta, and gamma, such that alpha is the angle between b and c, beta between a and c, and gamma between a and b.

CRYSTAL LATTICE

� Unit cell in 3 D: 4 type of unit cells:

� P: primitive

� I: Body-centered

� F: Face-centered

� C: Side-centered

CRYSTAL LATTICE


� P: primitive




CRYSTAL LATTICE


� P: primitive




CRYSTAL LATTICE


� P: primitive




CRYSTAL LATTICE


� P: primitive




CRYSTAL LATTICE


� P: primitive




� Each corner is shared between 8 cells

CRYSTAL LATTICE


� P: primitive





� P: primitive




CRYSTAL LATTICE


� P: primitive




CRYSTAL LATTICE


� P: primitive




CRYSTAL LATTICE

� Each face is shared between 2 cells

7 CRYSTAL SYSTEMS

a

b

c

Monoclinic α = γ = 90 ο ≠ β

a ≠ b ≠ c

a b

Triclinic α ≠ β ≠ γ a ≠ b ≠ c

c

c

a Orthorhombic α = β = γ = 90 ο a ≠ b ≠ c

b a 1

c

Tetragonal α = β = γ = 90 ο a 1 = a 2 ≠ c

a 2

a1

c

P or C

a2

RHexagonal Rhombohedralα = β = 90ο γ = 120ο

a1 = a2

≠ cα = β = γ ≠ 90ο

a1 = a2 = a3

a 1

a 3

Isometric (or cubic) α = β = γ = 90 ο a 1 = a 2 = a 3

a 2

7 CRYSTAL SYSTEMS

a

b

c

Monoclinic α = γ = 90 ο ≠ β

a ≠ b ≠ c

a b

Triclinic α ≠ β ≠ γ a ≠ b ≠ c

c

c

a Orthorhombic α = β = γ = 90 ο a ≠ b ≠ c

b a 1

c

Tetragonal α = β = γ = 90 ο a 1 = a 2 ≠ c

a 2

a1

c

P or C

a2

RHexagonal Rhombohedralα = β = 90ο γ = 120ο

a1 = a2

≠ cα = β = γ ≠ 90ο

a1 = a2 = a3

a 1

a 3

Isometric (or cubic) α = β = γ = 90 ο a 1 = a 2 = a 3

a 2

14 BRAVAIS LATTICES

P

F I

Isometric or cubic α = β = γ = 90 ο a 1 = a 2 = a 3

a 1

a 3

a 2

CN = 6 CN = 8 CN = 12

14 BRAVAIS LATTICES

P

F I


a 1

a 3

a 2

CN = 6 CN = 8 CN = 12

14 BRAVAIS LATTICES

P

F I


a 1

a 3

a 2

CN = 6 CN = 8 CN = 12

14 BRAVAIS LATTICES

P

F I


a 1

a 3

a 2

CN = 6 CN = 8 CN = 12

14 BRAVAIS LATTICES

14 BRAVAIS LATTICES

� Example: Sphalerite (ZnS)

14 BRAVAIS LATTICES

� Example: Sphalerite (ZnS)

� Basis: S2- + Zn2+

14 BRAVAIS LATTICES

� Example: Halite (NaCl)

14 BRAVAIS LATTICES

� Example: Halite (NaCl)

SYMMETRIES “… is the hardest thing for student to understand, appreciate or visualize.”

SYMMETRIES

4-fold rotational symmetry.

SYMMETRY OPERATIONS

� A Symmetry operation is an operation on an object that results in no change in the appearance of the object.

� There are 3 types of symmetry operations: rotation, reflection, and inversion.

ROTATIONAL SYMMETRY

� 1 fold rotation axis = no rotational symmetry

� 2 fold rotation axis: identical after a rotation of 180° (360/180 = 2) symbol: filled oval or A2

1

1

2

2

2

ROTATIONAL SYMMETRY

� 3 fold rotation axis = identical after a rotation of 120° (360/120 = 3) symbol: filled triangle or A3

� 4 fold rotation axis: identical after a rotation of 90° (360/90 = 4) symbol: filled square or A4

3

3

4

4

ROTATIONAL SYMMETRY

� 6 fold rotation axis = identical after a rotation of 60° (360/60 = 6) symbol: filled hexagon or A6

6

6

IMPROPER ROTATIONAL SYMMETRY � 5 fold, 7 fold, 8 fold or higher: does not exist in crystals because

cannot fill the space

MIRROR SYMMETRY � A mirror plan is something that gives you the reflection that

exactly reflects the other side: same distance, same component. The plane of the mirror is an element of symmetry referred to as a mirror plane, and is symbolized with the letter m.

MIRROR SYMMETRY

MIRROR SYMMETRY

MIRROR SYMMETRY

CENTER OF SYMMETRY � A center of symmetry is an inversion through a point,

symbolized with the letter "i".

ROTOINVERSIONS Combinations of rotation with a center of symmetry. � 1 fold rotoinversion axis =

center of symmetry symbol: A1

� 2 fold rotoinversion axis = 1) 180° rotation, 2) center of symmetry = mirror perpendicular to the axis symbol: A2

__ A1

m

ROTOINVERSIONS

1) Rotation 360° 2) Inversion

ROTOINVERSIONS Combinations of rotation with a center of symmetry. � 1 fold rotoinversion axis =

center of symmetry symbol: A1

� 2 fold rotoinversion axis = 1) 180° rotation, 2) center of symmetry = mirror perpendicular to the axis symbol: A2

__ A1

m

ROTOINVERSIONS

1) Rotation 180° 2) Inversion

ROTOINVERSIONS

� 3 fold rotoinversion axis = 1) 120° rotation, 2) center of symmetry symbol: A3


ROTOINVERSIONS



1

1

ROTOINVERSIONS



2

2

ROTOINVERSIONS

� 6 fold rotoinversion axis = 1) 60° rotation, 2) center of symmetry = 3 fold rotation axis + 1 perpendicular mirror plan symbol: A6




COMBINATIONS OF SYMMETRICAL OPERATIONS In crystals there are 32 possible combinations of symmetry elements: the 32 Crystal Classes.

a1 a2

c

a = b ≠ c

α = β = γ = 90°

COMBINATIONS OF SYMMETRICAL OPERATIONS In crystals there are 32 possible combinations of symmetry elements: the 32 Crystal Classes.

a1 a2

c

a = b ≠ c

α = β = γ = 90°

tetragonal

COMBINATIONS OF SYMMETRICAL OPERATIONS

� Square-shaped top

� 4 fold rotation axis




� A 2-fold axis that cuts diagonally through




� A 2-fold axis that cuts diagonally through

� Mirror plan through the diagonal


� Rectangular faces:

� 2-fold rotation axis perpendicular to the rectangular face.


� Square top + rectangular sides:

� mirror plan parallel to the 4-fold axis


� Square top + rectangular sides:

� mirror plan parallel to the 4-fold axis

� mirror plan perpendicular to the 4-fold axis

� One center of symmetry (not represented)


� 4-fold rotation axis: same face every 90°


� 4-fold rotation axis: same face every 90°

� 1 4-fold rotation axis

� 4 2 fold rotation axes

� 5 mirror plans

� 1 center of symmetry

A4, 4A2, 5m, i ⇔ 4/m2/m2/m

the ditetragonal dipyramidal class

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CHAPTER 3: SYMMETRY AND GROUPS, AND CRYSTAL …sarahlambart.com/teaching/mineralogy-03.pdf · CRYSTAL LATTICE IN TWO DIMENSIONS ! Unit cell: The unit cell of a mineral is the smallest