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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
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