Part 3 - The 14 Bravais Lattices - uni-saarland.de · Part 3 - The 14 Bravais Lattices. Functional...

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3.1 The plane lattice

3.2 The primitive space lattice (P-lattices)

3.3 The symmetry of the primitive lattices

3.4 The centered lattices

3.5 14 Bravais lattices

3.6 The unit cell of Bravais lattices

Part 3 - The 14 Bravais Lattices

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Crystal strukture

=

Basis

+

Lattice

AtomABC

Die Kristallstruktur ist durch die Raumkoordinaten der atomaren Bausteinebestimmt. Die Kenntnis der Symmetrie vereinfacht die Beschreibung.

ab

Lattice-konstant:

Lattice structure - basic conception

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⇓⇓⇓⇓Die Translationssymmetrie schränkt

die Zahl denkbarerSymmetrieelemente drastisch ein.

Symmetry basis

Allen Gittern gemeinsam ist dieTranslationssymmetrie.

(Einwirkung von 3 nicht komplanarenGitter-Translationen auf einen Punkt

⇒⇒⇒⇒ Raumgitter)

Andere Symmetrieeigenschaftentreten nicht notwendigerweise in

jedem Gitter auf.

⇓⇓⇓⇓

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General plane lattice

General lattice has no point symmetry elements except inversion centers.

.1

.2

2-fold

..

1

2

a

.3

Lattice translation a

Generation of the general plane lattice with an oblique unit mesh:parallelogram: a0 ≠ b0 and γ ≠ 90°:

2-fold

..

1

2.a

3

.bγ

4

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Special plane lattices - 1

c) a0=b0, γ = 90°. square unit

..

1

2.a3

.b90°

4 ....

d) a0=b0, γ = 120°. hexagonal

..

1

2.a3

.bγ

4

...

....

.

a) γ = 90°.

..

1

2.a3

.b90°

4 ....

mirror

mirror

b) a0=b0, γ ≠ 60°, 90°, 120°. rhombic unit

..

1

2.a3

.bγ

4

.

..

.

..

Centered rectanglular mech.

..

.

.

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Symmetry elements of specific lattice planes

• • ••

• • ••

• • ••

• • ••

Primitive rectangular

• • ••

• • ••

• • ••

• • ••

• ••

• ••

• ••

Centered rectangular

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

square

• • • •

• • • •

• • • •

• • • •

hexagonal

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Special plane lattices - summary

Shape of unit mesh

Lattice parameters

Characterizatic symmetry elements

General planes lattices

Parallelogram a0≠b0 γ≠90°

2

Special plane lattice

Rectangle (primitive)

a0≠b0 γ=90°

M

Rectangle (centered)

a0≠b0 γ=90°

M

Square a0=b0 γ=90°

4

120° Rhombus a0=b0 γ=120°

6(3)

Plane lattices:

a

b

c

d

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Primitive space lattices (P-Lattices)

For general space lattice, special space lattices may be derived, where congruentlattice planes are stacked above one another.

If the symmetry of the lattice planes is not changed, the five space lattices withprimitive unit cells (P-lattices) are produced.

Shape of unit mesh in stacked layers

Interplanar spacing

Characterizatic symmetry elements

Parallelogram (a0≠c0)

b0

Monoclinic P

Rectangle (a0≠b0)

c0

Orthorhombic P

Square (a0=b0)

c0≠(a0=b0)

Tetragonal P

Square (a0=b0)

c0=(a0=b0)

Cubic P

120° Rhombus (a0=b0)

c0

Hexagonal P

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Primitive space lattices - Triclinic crystal system

Triclinic P-lattice: a0 ≠ b0 ≠ c0α ≠ β ≠ γ

Stacking:no coincide with 2-fold axes

Triclinic axial system

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Primitive space lattices - Monoclinic

Monoclinic P-lattice: a0 ≠ b0 ≠ c0α = γ =90° β>90°

Stacking:directly one another

90°

90°

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Primitive space lattices - Orthorhombic

Orthorhombic P-lattice: a0 ≠ b0 ≠ c0α = β = γ =90°

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Primitive space lattices - Tetragonal

Tetragonal P-lattice: a0 = b0 ≠ c0α = β = γ =90°

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Primitive space lattices - Trigonal

Two kinds of unit cell:

1) Trigonal R-lattice: a0 ≠≠≠≠ b0 ≠≠≠≠ c0 αααα = ββββ =90° γγγγ = 120°

2) Rhombohedral P-lattice:a’0 ≠≠≠≠ b’0 ≠≠≠≠ c’0αααα’ = γγγγ’ = ββββ’

Stack on center of three

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Primitive space lattices - Hexagonal

Hexagonal P-lattice (120° rhombus unit cell): a0 = b0 ≠ c0α = β = 90°, γ =120°

Direct stack

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Primitive space lattices - Cubic

Cubic P-lattice: a0 = b0 = c0α = β = γ =90°

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14 Bravais lattice

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Symmetry rule

2⊥ m→1 1 on m→2 1 on 2→m

m’⊥ m”→2 2 on m”→m’⊥ m” 2 on m’→m’⊥ m”

The presence of any two of the given symmetry elements implies the presenceof the third:

Rule 1: A rotation axis of evenorder (Xe=2, 4 or 6), a mirrorplane normal to Xe, and aninversion center at the point ofintersection of Xe and m.

Rule 2: The mutuallyperpendicular mirror planesand a 2-fold axis along theirline of intersection.

3

1

1

23

1

2 3

1

2 3

2

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Space group - Symmetry of Triclinic P-lattice

Space group:The complete set of symmetry operations in a lattice or a crystal structure,or a group of symmetry operations including lattice translations.

The space group of a primitive lattice which has only 1 is called P1and the conditions for its unit cell parameters: a0≠b0≠c0; α≠β≠γP: space group

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Symmetry of Monoclinic P-Lattice

Space group of highestsymmetry: P2/m: 2 normal to m

b

b-axis is symmetry direction

Projection x,y,0Projection x,0,z

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Symmetry - Orthorhombic P-Lattice 1

Space group of highest symmetry: P2/m 2/m 2/m (P4/mmm)

a b c

1. Symmetry elements are in order of axes: a, b, c2. Each axis has a 2-fold rotation axis parallel to it and mirror planes normal to it.

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Symmetry - Orthorhombic P-Lattice 2

Mirror plane parallel tothe plane of the page atheights of 0 and 1/2

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Symmetry - Tetragonal P-Lattice 1

<uvw> denotes the lattice direction [uvw] and all equivalent directionsIn group symbol, the symmetry elements are given in the order:

c, <a>, diagonal of the <a>-axis = <110>they are called symmetry directions

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Symmetry - Tetragonal P-Lattice 2

P4/m 2/m 2/m

c <a> <110>

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Symmetry - Hexagonal P-Lattice-1

Projection on (001) ⊥ c

The symmetry directions for tetragonal lattice inthe order of: c, <a>, diagonals of the <a> axes = <210>

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Symmetry - Hexagonal P-Lattice-2

P6/m 2/m 2/m

c <a> <210>

a

c

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Symmetry - Cubic P-Lattice 1

P4/m 3 2/m

<a> <111> <110>

Symmetry directions:<a>, <111>, <110>

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Centered Lattices - Monoclinic 1Is it possible to import into P-lattices one or more lattice planes withoutdestroying the symmetry?

For monoclinic P-lattice, P2/m,Each point of the lattice has 2/m symmetry, which implies the presence

of an inversion center in the point.Insertion of new lattice planes parallel to (010) into the lattice is only

possible if the lattice point fall on a position which also has symmetry 2/m, i.e. on1/2,0,0; 0,1/2,0; 0,0,1/2; 1/2,1/2,0; 1/2,0,1/2; 0,1/2,1/2; or 1/2,1/2,1/2.

project onto x,0,z

90°

90°>90°

x

z

y

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Centered Lattices - Monoclinic 2C-lattice or C-face centered lattice:

a,b-face 1/2,1/2,0

A-lattice or A-face centered lattice: b,c-face 0,1/2,1/2

B-lattice or face centered lattice: a,c-face 1/2,0,1/2

Monoclinic withsmaller unit cell

90°>90°

x

z

y

x

z

y x

z

y

A-lattice converted into C-lattice

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Centered Lattices - Monoclinic 3

F-lattice or all-face centered lattice: 1/2,1/2,0 and 0,1/2,1/2

Reduce F-lattice into C-lattice

C-lattice with halfthe volume

=

I-lattice or body centered lattice: 1/2,1/2,1/2

90°>90°

x

z

y

I-lattice convert to C-lattice

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Centered Lattices

Monoclinic lattice:A, I, F →C; B →P

Orthorhombic lattice:A, B →C

Tetragonal lattice:A, B, C, F →I

Cubic lattice:A, B, C →I;

Space group symbols for the 14 Bravais lattice

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Crystal systemAll crystals, all crystal structures and all crystal morphologies which can bedefined by the same axial system belong to the same crystal system.

14 Bravais-lattice, 7 primitiveCrystal structure = Lattice + Basis

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Principle to choise the unit cell:

Bravais lattice - Bravais-Regeln

• Maximal Symmetry

• Smallest Volume

• Orthogonality

• Shortest Basis vector

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Axis system:

Unit cell:

a = b = cα = β = γ = 90°

Würfel

Crystal system - Cubic

ab

c

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Axis system:

Unit cell:

a = b ≠ cα = β = γ = 90°

Tetragonal Prisma

Crystal system - Tetragonal

ab

c

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Beispiel: Topas

Crystal system - Orthorhombisch

Axis system:

Unit cell:

a ≠ b ≠ cα = β = γ = 90°

Quader

ab

c

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Axis system:

Unit cell:

a = b ≠ cα = β = 90°, γ = 120°

oder a1 = a2 = a3 ≠ c

1/3 hexagonal Prisma

a1

a2a3

Crystal system - Hexagonal

a b

c

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Crystal system - Trigonal / Rhomboedrisch

Axis system:

Unit cell:

Rhomboedrisch: a = b = c α = β = γ ≠ 90°oder a = b ≠ c α = β = 90°, γ = 120°

Rhomboder

a bc

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Axis system:

Unit cell:

a ≠ b ≠ c α = γ = 90°, β > 90°

oder α = β = 90°, γ > 90°

Parallelpiped

Crystal system - Monoklin

a b

c

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Axis system:

Unit cell:

a ≠ b ≠ c α ≠ β ≠ γ

Parallelpiped

Crystal system - Triklin

a b

c

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There are 14 possible Bravais-lattice, whose 3-dimensionalperiodic structure could be constructured from one point.

These translation lattice can be primitiveor

centered.

There are 7 primitive und 7 centred Bravais-lattice.

Bravais Lattice - Summary