20082008년년 22학기학기 결정학결정학 특강특강Prof. KiProf. Ki--Bum KimBum Kim

Derivation of 14 Bravais latticesDerivation of 14 Bravais lattices

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The symmetrical spaceThe symmetrical space--lattice typeslattice types

NANO FABRICATION LABORATORY SEOUL NATIONAL UNIVERSITYThe distribution of rotation axes and mirrors in the five plane lattice types

The symmetrical spaceThe symmetrical space--lattice typeslattice types

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Principles of DerivationPrinciples of Derivation

33--dimensional latticedimensional lattice

Periodic repetition of a 2-D lattice by a third translation

Plane-lattice type and t3 have been selected

General procedureGeneral procedure

→ space lattice is determined

General procedureGeneral procedure

1st level t

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

2t

3t

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

1t

normal to the plane of the plane lattice

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Derivation of Derivation of Derivation of Derivation of

the spacethe space--lattice lattice

typestypes

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Derivation of the generalspaceDerivation of the generalspace--lattice typeslattice types

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

per cellP primitive 1P 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 spaceDerivation of the space--lattice typeslattice types

Symmetry Symmetry 11y yy y

1-fold axis :

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

Derivation of space lattice type 1P2t3t

Derivation of space-lattice type 1P1t

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Derivation of the spaceDerivation of the space--lattice typeslattice types

Symmetry Symmetry 22y yy y

112t

)00(3 zt (a) )21

21( (b) 3 zt

1t

t1t

2t

t 2P 2I

3t3t ′

2t

1t 3t

t 1t

2P

NANO FABRICATION LABORATORY SEOUL NATIONAL UNIVERSITY2t 1t

Derivation of the spaceDerivation of the space--lattice typeslattice types

)01((c) zt )10( zand / or)02

( (c) 3 zt )2

0( zand / or

2t

1t3t ′

3t1t

′2t

→ 2A

1

321 ,, ttt ′

3211 ,, tttt ′+ → 2I

)210( z 2B, 2IBy the same taken

∴ Possible types : 2P and 2I

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Derivation of the spaceDerivation of the space--lattice typeslattice types

Symmetry Symmetry 222 (2mm)222 (2mm)y yy y ( )( )Two choices : rectangular, diamond

• 2mm symmetry occurs at

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

• Displacement vector t3

:)00(at (a) 3 zt

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

)(( ) 3

2t

3t

1t2t

1t primitive 222P

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Derivation of the spaceDerivation of the space--lattice typeslattice types

:)21

21(at (b) 3 zt

22

2t1t 222I

3t

2t 1t3t ′

2t

:)210(at (c) 3 zt

2t 1

3t′A face centered3

2t 1t3t ′

:)01(at(d) zt Obtain B face Centered → same as 222A

222A

Why not I222?

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:)02

(at (d) 3 zt Obtain B face Centered → same as 222A

Derivation of the spaceDerivation of the space--lattice typeslattice types

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)

:)00(at (a) 3 zt

t

3t

2t ′

2t1t

1t ′

C t d ll 222C 222A

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C centered cell 222C → same as 222A

Derivation of the spaceDerivation of the space--lattice typeslattice types

:)11(at (b) 3 zt )22

(( ) 3

t 3t

2t1t

1t ′2t ′

222FThe 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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, , ,

Derivation of the spaceDerivation of the space--lattice typeslattice types

Symmetry Symmetry 444 f ld t ti i

y yy y• 4-fold rotation axis :

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

• Displacement vector t3Displacement vector t3

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

:)00(at (a) 3 zt

3t

2t

1t4P : primitive orthorhombic

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Derivation of the spaceDerivation of the space--lattice typeslattice types

:)11(at(b) 3 zt :)22

(at (b) 3 zt

3t ′

2t1t

3t

4I : body-centered orthorhombic

4I can be 4F, also

not used

Th 4P 4I

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There are 4P, 4I

Derivation of the spaceDerivation of the space--lattice typeslattice types

Symmetry Symmetry 33 Three locations for 3 fold a isy yy y • Three locations for 3-fold axis :

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

• Displacement vector t3p 3

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

:)00(at (a) 3 zt

3t

t

NANO FABRICATION LABORATORY SEOUL NATIONAL UNIVERSITY2t

1t

Derivation of the spaceDerivation of the space--lattice typeslattice types

)31

32(,)3

13

2(at (b) 3 zzt same thing)33(,)33(( ) 3

prismRhombohedral

3R p

There are 3P, 3R

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,

Derivation of the spaceDerivation of the space--lattice typeslattice types

Symmetry Symmetry 66y yy y• 6-fold location : (0 0)

• Displacement vector t3 (0 0 z)

same lattice as in 3P

Symmetry Symmetry 2323y yy y

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Derivation of the spaceDerivation of the space--lattice typeslattice types

Cubic SymmetryCubic Symmetryy yy yCubic symmetry is not associated with 4-fold symmetry

Central symmetry of cubic system ( four 3-fold axis )

Special type of rhombohedral

cubicprimitive90When(a) =α cubicprimitive ,90 When (a) =α

90

==

β

cba

90=== γβα

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Derivation of the spaceDerivation of the space--lattice typeslattice types

(FCC) cubic centered-face ,60 When (b) =α

6060 60===

==

γβα

cba

6060 60=== γβα

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Derivation of the spaceDerivation of the space--lattice typeslattice types

(BCC) cubic centered-body ,091 When (c) =α

)100(

)21

21

21( −− )2

12

12

1( −−)010( −

)010()21

21

21(

)010(

)000( )010()222(

)111(− )21

21

21(

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Derivation of Derivation of Derivation of Derivation of

the spacethe space--lattice lattice

typestypes

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