# 没有幻灯片标题 - sess.pku.edu.cn

: Next section: crystal symmetry 1 ( )

2 Symmetry 3 ( ) 4

5 (symmetry operation) ( ) ( )

some acts that reproduce the motif to create the pattern Motif: the fundamental part of a symmetric design that, when repeated, creates the whole pattern 6 (symmetry element) (center of symmetry) (symmetry plane) (symmetry axis)

(rotoinversion axis) (rotoreflection axis) 3-1 7-1 7

360 L1 1 180 L2 2 120 L3 3 90 L4 4

60 L6 6 C 1 1 Li C P m L2I

120 90 60 L3I L4i L6i 3 4 6 L3+C L3+P 8

(n) n : n = 360/ a + 2a cos = ma cos = (m-1)/2 1 m = 3, 2, 1, 0, -1 = 0, 60, 90, 120, 180 n = 1, 6, 4, 3, 2 9 (x, y, z) X a11 x a12 y a13 z Y a 21 x a 22 y a 23 z Z a x a y a z 31

32 33 (X, Y, Z) or X x Y y Z z a11 a 21

a 31 a12 a 22 a32 a13 a 23 a33 10 (Ln) (two-fold rotation) = 360o/2 rotation to reproduce a motif in a

symmetrical pattern A Symmetrical Pattern 6 6 11 (Ln) Operation (two-fold rotation) = 360o/2 rotation to reproduce a motif in a

symmetrical pattern = the symbol for a twofold rotation A Symmetrical Pattern 6 Motif Element 6 12 (Ln) (two-fold rotation)

= 360o/2 rotation to reproduce a motif in a symmetrical pattern = the symbol for a twofold rotation A Symmetrical Pattern 6 Motif Element 6

13 (Ln) (two-fold rotation) A Symmetrical Pattern cos sin 0 sin cos 0 0

0 1 6 6 14 (Ln) (two-fold rotation)

15 (Ln) (two-fold rotation) 16 (Ln) (two-fold rotation) 17

(Ln) (two-fold rotation) 18 (Ln) (two-fold rotation) 19 (Ln) (two-fold rotation)

20 (Ln) (two-fold rotation) 1st 180o rotation makes it coincident 2nd 180o brings the object back to its original position 21 (Ln) (three-fold rotation)

6 6 = 360o/3 rotation to reproduce a motif in a symmetrical pattern A Symmetrical Pattern 6 22 (Ln)

A Symmetrical Pattern (three-fold rotation) 6 6 step 3 6 = 360o/3 rotation to reproduce a motif in a symmetrical pattern step 1

step 2 23 (Ln) 6 6 6 6 6 6 6 2-fold 3-fold

cos sin 0 4-fold sin cos 0 0 0 1 6 6 6

1-fold 6 6 6 6 6 6 ( 5-fold > 6-fold ) 6-fold 24

(m) (mirror) Reflection across a mirror plane reproduces a motif = symbol for a mirror m 25 (m) (mirror) ( m x y ) x

x y y z z m 1 0 0 0 1 0 0 0 1

m x z ? m y z ? m ? 26 (C, 1) (x, y, z) (-x, -y, -z) 1 0 0 0 1 0 0 0 1

27 (C, 1) (x, y, z) (-x, -y, -z) 28 (C) (x, y, z) (-x, -y, -z) Step 1: rotate 360o/1

(identity)? 29 (C) (x, y, z) (-x, -y, -z) Step 1: rotate 360o/1 Step 2: invert 30 (Lin)

= sin cos 0 Li1 = C Li2 = P Li3 = L3 +C cos sin 0 0

0 1 Li4 Li6 = L3 +P 31 (Lin) Li4 32 (Lin)

Li4 33 (Lin) Li4 Step 1: Rotate 360/4 34 (Lin) Li4 Step 1: Rotate 360/4 Step 2: Invert

35 (Lin) Li4 Step 1: Rotate 360/4 Step 2: Invert 36 (Lin) Li4 Step 1: Rotate 360/4 Step 2: Invert

Step 3: Rotate 360/4 37 (Lin) Li4 Step 1: Rotate 360/4 Step 2: Invert Step 3: Rotate 360/4 Step 4: Invert 38 (Lin)

Li4 Step 1: Rotate 360/4 Step 2: Invert Step 3: Rotate 360/4 Step 4: Invert 39 (Lin) Li4 Step 1: Rotate 360/4 Step 2: Invert Step 3: Rotate 360/4 Step 4: Invert Step 5: Rotate 360/4 40

(Lin) Li4 Step 1: Rotate 360/4 Step 2: Invert Step 3: Rotate 360/4 Step 4: Invert Step 5: Rotate 360/4 Step 6: Invert 41 (Lin) Li4 4-fold rotoinversion (4)

42 (Lin) Li3 3-fold rotoinversion (3) 5 4 3 1 6 2

43 (Lin) Li6 6-fold rotoinversion (6) Top View 44 (Lin) 45

(Lsn) ? L1i = L2s = C L2i = L1s = P L4i = L4s L3i = L6s = L3 + C L6i = L3s = L3 + P 46 (L2)+ (P)

47 (L2)+ (P) Step 1: reflect 48 (L2)+ (P) Step 1: reflect Step 2: rotate 49

(L2)+ (P) Step 1: reflect Step 2: rotate Is that all?? No! A second mirror is required ! So, L2 + P = L2 2P (2-D) 50 (L4)+ (P) 51

(L4)+ (P) Step 1: reflect 52 (L4)+ (P) Step 1: reflect Step 2: rotate 1 53

(L4)+ (P) Step 1: reflect Step 2: rotate 1 Step 3: rotate 2 54 (L4)+ (P) Step 1: reflect Step 2: rotate 1 Step 3: rotate 2 Step 4: rotate 3 55

(L4)+ (P) Any other elements? Yes, two more mirrors So, L4 + P = L4 4P (2-D) 56 (L3)+ (P) L3 + P = L3 3P (2-D) 57

(L6)+ (P) L6 + P = L6 6P (2-D) 58 Ln P(||) Ln n P Ln L2() Ln nL2 Ln P() = Ln C Ln P C (n = ) Lni P(||) = Lni L2() Ln i nL2 nP (n = ) Lni P(||) = Lni L2() Ln i n/2L2 n/2P (n = ) 59

60 ( ) (point group)? 10 unique 3-D symmetry elements: 1 2 3 4 6 i m 3 4 6 And 22 possible combinations of these elements Totally, 32 point groups 61 ( ) Increasing Rotational Symmetry Rotation axis only

1 2 3 4 6 1 (= i ) 2 (= m) 3 4 6 (= 3/m) Combination of rotation axes

One rotation axis mirror 222 32 422 622 2/m 3/m (= 6) 4/m 6/m One rotation axis || mirror 2mm

3m 4mm 6mm 3 2/m 4 2/m 6 2/m 2/m 2/m 2/m 4/m 2/m 2/m 6/m 2/m 2/m 23 432

4/m 3 2/m 2/m 3 43m Rotoinversion axis only Rotoinversion with rotation and mirror Three rotation axes and mirrors Additional Isometric patterns 62 ( ) ? ? 32 (

.gif ) ? ? (H-M) (Schoenflies) : P33 3 3 63 P35 36 64

, mmm, 432, 4/m (!)32 65 1 2

3 4/mmm P91, 7 5 ( ) 2 1 3 1 2 3

c a+b+c a+b [001] [111] [110] c a a+b

[001] [100] [110] a b c [100] [010] [001]

b [010] c [100] [210] 66 a 2a+b

[001] (crystal category) (higher category) (intermediate category) (lower category) : ? ? 67 (crystal system)

, (isometric system), (cubic system) (hexagonal system) (tetragonal system) (trigonal system) (orthorhombic system), (monclinic system) (triclinic system) : P34 3 4 68 : : P34 3 4 --- the below and next page Crystal System No Center Center

1 1 Monoclinic 2, 2 (= m) 2/m Orthorhombic 222, 2mm 2/m 2/m 2/m Tetragonal 4, 4, 422, 4mm, 42m 4/m, 4/m 2/m 2/m

Hexagonal 3, 32, 3m 3, 3 2/m 6, 6, 622, 6mm, 62m 6/m, 6/m 2/m 2/m 23, 432, 43m 2/m 3, 4/m 3 2/m Triclinic Isometric 69

L2 P L2 P 2 L P

L3 4L3 L3

L1 **C L2 P **L2PC 3L2 L22P **3L23PC 1 1 2 m 2/m 222 mm2 mmm

L3 *L3C *L33L2 L33P **L33L2 3PC L4 L4 i *L4PC L44L2 L44P L4 i2L2 2P **L44L2 5PC L6 +L6I *L6PC L66L2 L66P L6 i3L2 3P **L66L2 7PC 3L2 4L3 *3L2 4L3 3PC 3L4 3L36L2

*3L4 4L3 6P **3L44L3 6L2 9PC 3 3 32 3m 3m 4 4 4/m 422 4mm 42m 4/mmm 6 6 6/m 622 6mm 6m2 6/mmm

23 m3 432 43m m3m -

- () 70 32 Wulff See java applet

71

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