1

1MATS 227 / MAE 251/CHEM 222Lecture notes 1

Atomic arrangement in solids

• Solids are classified by theway the atoms/ions arearranged

• In crystalline solids, there isa periodic, long range orderto the arrangement

• In amorphous solids (glass)there is no long range order

2MATS 227 / MAE 251/CHEM 222Lecture notes 1

2

3MATS 227 / MAE 251/CHEM 222Lecture notes 1

2-fold rotation leaves object unchanged

3-fold rotation leaves object unchanged

120˚ 120˚

4MATS 227 / MAE 251/CHEM 222Lecture notes 1

parallelogram

rectangle square

1 2

m,2mm

m,2mm

4,4mm

hexagonal

3,3m

6,6mm

centered rectangle

5 distinct 2D nets

3

5MATS 227 / MAE 251/CHEM 222Lecture notes 1

6MATS 227 / MAE 251/CHEM 222Lecture notes 1

Determination of 2D point groups

C6vC6

C4vC4

C3vC3

C2vC2

C1v = C1h = CsC1

4

7MATS 227 / MAE 251/CHEM 222Lecture notes 1

Determination of 2D point groups

6mm6

4mm4

3m3

2mm2

m1

8MATS 227 / MAE 251/CHEM 222Lecture notes 1

The 17 plane groups

p2mg, p4mg, p31m(6) Rotation, mirror and glide

p2gg(5) Rotation and glide

p2mm, c2mm, p4mm, p3m1,p6mm

(4) Rotation and mirror

pg(3) Glide

pm, cm(2) Mirror

p1, p2, p3, p4, p6(1) Rotation

Plane groupSet of symmetriesTranslation +

5

9MATS 227 / MAE 251/CHEM 222Lecture notes 1

p1

cm

pgpm

p2 p2mm

c2mm p2gg p2mg

10MATS 227 / MAE 251/CHEM 222Lecture notes 1

p4 p4mm p4gm

p3 p3m1 p31m

p6p6mm

6

11MATS 227 / MAE 251/CHEM 222Lecture notes 1

12MATS 227 / MAE 251/CHEM 222Lecture notes 1

p1 pm

p2gg

pg

cm

http://escher.epfl.ch/escher/

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13MATS 227 / MAE 251/CHEM 222Lecture notes 1

Determine the plane group

14MATS 227 / MAE 251/CHEM 222Lecture notes 1

Three dimension point symmetry operations

• We need to add someadditional operations– Inversion (i or )• x, y, z → -x, -y, -z

– Horizontal mirror plane

1

•+z-z

•

Ch or σh 6

Enantiomorphbelow the plane

8

15MATS 227 / MAE 251/CHEM 222Lecture notes 1

– Improper rotation (Sn = σhCn)• Rotation of 2π/n followed by reflection in a horizontal plane (plane ⊥

to rotation axis)– The only one we have to consider as a point group is S4

α

S4

S3 = C3h = 6_

S6 = 3 = C3i

_

S2 = i = 1_

16MATS 227 / MAE 251/CHEM 222Lecture notes 1

–Glide plane instead of glide axis•Glide is still translation + reflection

9

17MATS 227 / MAE 251/CHEM 222Lecture notes 1

– Rotoinversion (iCn or )• C2i = m C3i = S6 C4i = S4

4

n

3 6

18MATS 227 / MAE 251/CHEM 222Lecture notes 1

Adding another 2-fold axis

• We can add an additional 2-fold axis to some groupsforming the D groups

• If we take C4 and add a perpendicular 2-fold axis

Above the planeBelow the plane

C4 (4) D4 (422)

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19MATS 227 / MAE 251/CHEM 222Lecture notes 1

• Consider C6 and D6

2f axis

20MATS 227 / MAE 251/CHEM 222Lecture notes 1

11

21MATS 227 / MAE 251/CHEM 222Lecture notes 1

22MATS 227 / MAE 251/CHEM 222Lecture notes 1

= C3i

12

23MATS 227 / MAE 251/CHEM 222Lecture notes 1