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14 bravais lattices and Screw axes

Point and Space Groups

• 7 crystal systems

• 14 Bravais lattices

• 230 non-Bravais lattices

• 32 point symmetries

– 2 triclinic

– 3 monoclinic

– 3 orthorhombic

– 7 tetragonal

– 5 cubic

– 5 trigonal

– 7 hexagonal

Space Groups

Combining translations and point groups

Plane Group Symmetry

p211

Plane Group Symmetry

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

17-plane groups

17-plane groups

17-plane groups

17-plane groups

17-plane groups

17-plane groups

17-plane groups

t

1

17-plane groups

t

2

t

1

17-plane groups

t

2

t

1

17-plane groups

t

2

t

1

17-plane groups

t

2

t

1

17-plane groups

t

2

t

1

17-plane groups

t

2

t

1

17-plane groups

t

2

t

1

17-plane groups

Symmetry 2 (00z)

Symmetry 2(.5 .5 z)

نقاط توپر الیه اول، نقاط تو خالی الیه دوم

تقارنهای صفحه ای و نقطه ای

Symmetry 2

Monoclinic

(00z) R (.5 .5 z) R

Symmetry 222

(.5 .5 z) R

222 symmetry in Diamond

222 symmetry in Diamond

Crystal Structure 37

ORTHORHOMBIC SYSTEM

Orthorhombic (Simple) a = ß = g = 90o

a b c

Orthorhombic (Base-centred)

a = ß = g = 90o

a b c

Orthorhombic (BC) a = ß = g = 90o

a b c

Orthorhombic (FC) a = ß = g = 90o

a b c

Symmetry 4

Symmetry 4

Crystal Structure 40

– TETRAGONAL SYSTEM

Tetragonal (P) a = ß = g = 90o

a = b c

Tetragonal (BC) a = ß = g = 90o

a = b c

Symmetry 3

Symmetry 3

Symmetry 6

Crystal Structure 51

1-CUBIC

Crystal Structure 52

a- Simple Cubic (SC)

Crystal Structure 54

Face Centered Cubic (FCC)

اتم در سلول واحدش وجود دارد 4•

•(Cu,Ni,Pb..etc) ساختارfcc. دارند

Crystal Structure 55

3 - Face Centered Cubıc

Atoms are all same.

Crystal Structure 56

2 - HEXAGONAL SYSTEM

دارد وجود واحدش سلول در اتم سه.

Crystal Structure 57

2 - HEXAGONAL SYSTEM

Atoms are all same.

Crystal Structure 58

a

b

c

P Monoclinic

a= g =90 o b a b c

a

b

c

I = C a

b

P Triclinic a b g a b c

c

c

a P

Orthorhombic a= b = g =90 o a b c

C F I b

3-D Symmetry

3-D Symmetry

3-D Symmetry

Contrasting glide and screw

screw axes

screw axes

monoclinic

Orthorhombic

http://img.chem.ucl.ac.uk/sgp/large/sgp.htm

89

Symbols for Symmetry Elements (2)

International Tables for Crystallography, Volume A, Table 1.4.5, p. 9 (2002)

90

Symbols for Symmetry Elements (3)

International Tables for Crystallography, Volume A, Table 1.4.2, p. 7 (2002)