17-plane groups When the three symmetry elements, mirrors, rotation axis and glide planes are shown...

17-plane groups

When the three symmetry elements, mirrors, rotation axis and glide planes are shown on the five nets, 17-plane groups are derived.

17-plane groups

17-plane groups

t1

17-plane groups

t2

t1

17-plane groups

The 14 Bravavis Lattices.

• There are 14 ways to combine to stack the 5 nets in 3D to give us 14 unique ways to translate a point in 3 dimensions.

The 14 Bravavis Lattices.

Stacking of the five nets (plane lattices) in various ways leads to the 14-possible lattices. These 14 lattices types are known as 14-Bravais lattices.

The orthogonallity gained by the use of these (14-Bravais lattices) is of considerable aid in visualizing and describing space lattices.

Multiple cells have lattice points on their: faces, interiors (body), and their corners; Cells with interior points are called—Body centered lattice—

(I) or (R) space lattice• Cells with Face centered lattice are called A-B-C-or F-centered space lattice.

The 14 Bravais Lattices

• When we add a third translation t3 to the 17 plane groups, we only get 14 space lattice that are know as 14 Bravais Lattices.

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The 14 Bravais Lattices• Monoclinic and Triclinic



The 14 Bravais Lattices• Tetragonal



The 14 Bravais Lattices• Orthorhombic



The 14 Bravais Lattices• Cubic



The 14 Bravais Lattices

• The 14 Bravais Lattices can also be grouped into 6(7) groups that are known as Crystal systems.QuickTime™ and a

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The 6 (7) Crystal Systems

• The seven crystal systems will be very important in our discussion of optical properties (and other physical properties) of crystals and in our discussion of phase transitions. You must know this table very well.

Relation between, Crystal System, Space and Plane Lattices.

Space lattices Plane Lattices Crystal SystemCube Square- Square- Square Isometric

Square prism Square-rectangle-rectangle Tetragonal

Hexagonal prism 120o rhombus-rectangle-rectangle

Hexagonal

Brick shaped Cell Rectangle-Rectangle-Rectangle Orthorhombic

Brick shaped Celldeformed to make 1 face aparallelogram

Parallelogram-rectangle-rectangle

Monoclinic

A cell whose faces are allgeneral parallelogram

Parallelogram-Parallelogram-Parallelogram

Triclinic

The Six Crystal SystemsName Unit Cell Dimentions AnglesIsometric(cubic) a=b=c α=β=γ=90ο

Tetragonal = ≠a b c α=β=γ=90ο

Orthorhombic ≠ ≠a b c α=β=γ=90ο

Monoclinic ≠ ≠a b c α=γ=90ο↑β

Ticlinic ≠ ≠a b c α, β,γ↑90ο

Hexagonal a1=a2=a3≠c ai^aj=120 , o a 1 ^ =90c o

The Six Crystal SystemsIsometric(cubic) a=b=

ca=b=g=90o

Tetragonal a=b≠c a=b=g=90o

Orthorhomb-ic

a≠b≠c

a=b=g=90o

Hexagonal a1=a2=a3≠c ai aj=120o, a1 c=90o

Monoclinic a≠b≠c

a=g=90o≠b

Ticlinic a≠b≠c

a, b,g≠90o

The 32 Crystallographic Point Groups

• There are 32 ways to combine the symmetry elements, 1, -1, m, 2, 3, 4, and 6 that are internally consistent. Each combination is called a point group.

• The 32 point groups fall into the seven different crystal systems. These you must know.

Significance of the Unit Cell Point Groups

• The External symmetry of the crystal will have the same symmetry as the Unit Cell. (Note: The form of shape of the crystal may be different from that of the unit cell, however).

Summary

• The 14 Bravais Lattices

• The 6 (7) Crystal Systems

• The 32 Point Groups (“Crystal Classes”)

Summary

• The 32 point groups are combinations of inversion, rotation and mirrors

• If in addition, we allow glide and screw we come up with 230 space groups and form the basis of mineralogical description.

Summary

• Space Groups, therefore reflect the point group and lattice type. Space group notation includes reference to both.

Summary

• P432 primitive, 4-fold, 3-fold, 2-fold, isometric• In as much as X-ray work is needed to determine

space groups we will not dwell on it here.

Relation of the crystal lattice to the crystal

• A) The 32 crystal classes (point groups) correspond to 32 unique combinations of symmetry elements (n, m, i).

• B) From observations of natural crystals find that only 32 possible combinations of symmetry elements are needed to describe their morphology. (Calcite overhead)

Relation of the crystal lattice to the crystal

• Infer:– A) That crystal morphology is an expression of

the point group (crystal class).– B) The plane surfaces that bound natural

crystals develop parallel to certain sets of net planes in the crystal lattice of a specific mineral.

Summary

• 1. morphology tells us of point group (crystal class).

• 2. crystal class tells us of crystal system (iso, ortho, mono, etc.)

• 3. Crystal system specifies certain possible lattices (P, I, F, R etc.)

• 4. X-rays needed to identify lattice type.

Summary

• But even without x-rays we have learned a lot! We can break down miriads of crystals into 32 crystal classes, and these into 6 (7) crystal systems.

17-plane groups When the three symmetry elements, mirrors, rotation axis and glide planes are shown...

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