Derivation of 14 Bravais lattices

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2008

2008년년 22학기학기 결정학결정학 특강특강 Prof. Ki

Prof. Ki--Bum KimBum Kim

Derivation of 14 Bravais lattices

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The symmetrical space

The symmetrical space--lattice typeslattice types

The distribution of rotation axes and mirrors in the five plane lattice types

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The symmetrical space

The symmetrical space--lattice typeslattice types

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Principles of Derivation Principles of Derivation

3

3--dimensional latticedimensional lattice

Periodic repetition of a 2-D lattice by a third translation Plane-lattice type and t₃ have been selected

General procedure General procedure

→ space lattice is determined

General procedure General procedure

1^st level

tG

Relative placement of the zero and first levels can be described by giving the components of

tG2

t3

z can be described by giving the components of

t₃ in terms of fractions, x and y, of the vectors t₁ and t₂ respectively, plus a distance parameter z normal to the plane of the plane lattice

Zero level

x y

tG1

normal to the plane of the plane lattice

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Derivation of Derivation of Derivation of Derivation of the space

the space--lattice lattice types

types

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Derivation of the generalspace

Derivation of the generalspace--lattice typeslattice types

Symbol Name Locations of additional points Total number of Symbol Name Locations of additional points Total number of

lattice points per cell

P primitive 1

P primitive - 1

I body-centered center of cell 2

A A-centered center of A, or (100) face 2

B B-centered center of B, or (010) face 2

C C-centered center of C, or (001) face 2

F face-centered centers of A, B, and C faces, , 4 R rhombohedral at 2/3 1/3 1/3 and 1/3 2/3 2/3, i.e. two points

along the long body diagonal of cell

3

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Derivation of the space

Derivation of the space--lattice typeslattice types

Symmetry Symmetry 1yy yy 1

1-fold axis :

No restriction on the perpendicular plane: general parallel-pipe No restriction on the perpendicular plane: general parallel pipe

Derivation of space lattice type 1P tG2

tG3

Derivation of space-lattice type 1P tG1

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1

2 1

tG

) 0 0

3 ( z

tG

(a) )

2 1 2 (1

(b) tG₃ z

tG1

tG tG¹

t2

G

tG

2P 2I

tG3

tG3′

t2

tG1

tG3

tG

tG1

2P

t2 t¹

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Derivation of the space--lattice typeslattice types )

1 0 (

(c) tG z

1 ) 0

( z

and / or

) 2 0

(

(c) t₃ z )

0 2

( z

and / or

tG2

tG1

tG3′

tG3

tG1

G′ G G

tG2 → 2A

1

3 2 1, t , t tG G G′

3 2

1

1, t t , t tG G + G G′

→ 2I

2 ) 0 1

( z 2B, 2I

By the same taken

∴ Possible types : 2P and 2I

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Symmetry

Symmetry 222 (2mm)yy yy 222 (2mm)(( ))

Two choices : rectangular, diamond

• 2mm symmetry occurs at (0 0) , (1/2 1/2) , (1/2 0) , (0 1/2)

• Displacement vector t₃

: ) 0 0 ( at

(a) tG₃ z

(0 0 z) , (1/2 1/2 z) , (1/2 0 z) , (0 1/2 z)

) (

( ) ₃

tG2

tG3

tG1

tG2

tG1

primitive 222P

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: 2 ) 1 2 (1 at

(b) tG₃ z 2 2

G

tG2 tG¹

222I

tG3

tG2

tG1

tG3′

t2

: 2 ) 0 1 ( at

(c) tG₃ z

t2 ¹

tG3

G′ ₃ A face centered

tG2

tG1

tG3′

: ) 1 0

( at

(d) tG z

222A

9 Why not I222?

: ) 0 ( at

(d) t z

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Diamond Plane Lattice

• The locations of symmetry 2mm (0 0) , (1/2 1/2)

• Displacement vector t₃ (0 0 z) , (1/2 1/2 z)

: ) 0 0 ( at

(a) tG₃ z

tG

tG3

tG2′ tG2

t1

tG1′

C t d ll 222C 222A

C centered cell 222C → same as 222A

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Derivation of the space--lattice typeslattice types :

1 ) (1 at

(b) tG₃ z

2 ) (2 ( ) ₃

tG tG₃ tG2

t1

tG1′ tG₂′

222F

The primitive cell is very difficult to deal with,

and it is customary to choose a new cell with orthogonal edges → 222F

There are 222P, 222I, 222F, 222C, , ,

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Symmetry Symmetry 44

4 f ld t ti i

y y

• 4-fold rotation axis : (0 0) , (1/2 1/2)

• Displacement vector tDisplacement vector t₃₃ (0 0 z) , (1/2 1/2 z)

: ) 0 0 ( at

(a) tG₃ z

tG3

tG2

tG1 4P

: primitive orthorhombic

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: 1 )

(1 at

(b) tG₃ z

: 2 )

(2 at

(b) t₃ z

tG3′

G

tG2

tG1 tG³

4I : body-centered orthorhombic

4I can be 4F, also not used

4P 4I 4P, 4I

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Symmetry Symmetry 33

Three locations for 3 fold a is

y y

• Three locations for 3-fold axis : (0 0) , (2/3 1/3) , (1/3 2/3)

• Displacement vector tp ₃₃

(0 0 z) , (2/3 1/3 z) , (1/3 2/3 z)

: ) 0 0 ( at

(a) t^G₃ z

tG3

tG t2

tG1

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Derivation of the space--lattice typeslattice types 3 )

3 1 (2 , 3 )

3 1 (2 at

(b) tG₃ z z

same thing

3 ) ( 3

, 3 )

( 3 ( ) ₃

prism Rhombohedral

3R p

There are 3P, 3R,

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• 6-fold location : (0 0)

• Displacement vector t₃ (0 0 z)

same lattice as in 3P

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Cubic Symmetry Cubic Symmetryyy yy

Cubic symmetry is not associated with 4-fold symmetry Central symmetry of cubic system ( four 3-fold axis )

Special type of rhombohedral

cubic primitive

90 When

(a) When α = 90^D ,primitivecubic

(a) α =

90D

=

= β

c b a

90D

=

= β γ α

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(FCC) cubic

centered -

face , 60 When

(b) α = ^D

60D

60D = = = 60^D

=

γ β α

c b a 60D

60D α = β = γ = 60

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(BCC) cubic

centered -

body ,

09 1 When

(c) α = ^D

) 1 0 0 ( 2) 2 1 2 1

(−1 − 12)

12 12

(− −

) 0 1 0 ( −

) 0 1 0

) (

12 12 12 (

) 0 1 0 (

) 0 0 0

( ⁽ ² ² ²⁾ (010)

1 ) 1

(−1 12) 12

12 (

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Derivation of Derivation of Derivation of Derivation of the space

the space--lattice lattice types

types