Crystal notation - HM symbol and Schoenflies notation

Crystal notation - HM symbol and Schoenflies notation

1. Crystallography

Module 2

2. •Each of the 32 crystal classes has its own characteristic set of symmetry elements.
These elements are depicted by International symbols formulated by C.Hermann and
Ch.Mauguin. These symbols contain the minimum number of symmetry elements
necessary to define a class.
•Each Hermann Mauguin symbol is made of three or less parts.
•Each part refers to definite axis within the crystal. The parts are always arranged in a
definite order.
•The first part of the symbol refers to a prinicipal axis ,rotation or inversion e.g. 3 or 3
•The second part refers to another symmetry which is at some angle to the principal
axis.
•Mirror planes are designated by m .If the mirror plane is perpendicular to the axis, it is
added to that axis with a stroke /. For eg 3/m indicates an axis of three fold symmetry
with a mirror plane perpendicular to it.
HERMANN MAUGUIN SYMBOLS

3. •A mirror plane parallel or through the axis will be written without
stroke, for eg. 2m .
•Using the general symbol X to denote principal axis of any degree ,we
may have the following combinations.
•X - Rotation axis alone
•X - Inversion axis alone
•X/m - Rotation axis normal to a mirror plane.
•Xm(m) - Rotation axis with vertical mirror plane
•X2(2) - Rotation with 2-fold axis normal to it.
•Xm - Inversion axis with a vertical mirror plane
•X2 - Inversion axis with a 2-fold axis normal to it
•X/m2/m2/m - Rotation axis with both kinds of MIRROR PLANES.

5. SCHOENFLIES NOTATIONS
1.First he considered the classes with single axis of symmetry
(monogonal axis) C
1
.eg Triclinic system
2.2 fold axis - C
2

3.3 fold axis – C
3

4.4 fold axis – C
4

5.6 Fold axis – C
6


These C1 ,C2,C3,C4,C6 are known as cyclic classes

6. •To the cyclic classes ,he added diagonal and generated 4 new crystal
classes.
•C1 + diagonal =C2 ( C2 -Already derived)

6.C2 + diagonal = D2
7.C3 + diagonal = D3
8.C4 + diagonal = D4
9.C6 + diagonal = D6
These classes D2,D3,D4,D6 are called dihedral classes.

7. •He then added mirror planes to the cyclic classes to obtain more
crystal classes. By adding a horizontal plane (HP) ,he obtained 5 new
classes.
10.C1 + HP = CS
11.C2+ HP = C2h
12.C3 + HP= C3h
13.C4 + HP = C4h
14.C6 + HP= C6h

8. •He next added vertical planes (VP) to the cyclic classes ,produced 4
new classes.
•C1 + VP =CS
15. C2 + VP = C2v
16. C3 + VP = C3v
17. C4 + VP = C4v
18. C6 + VP = C6v

9. •By adding only centre of symmetry to C2 AND C3 classes,he obtained 2
more new classes.
19.C1+ Centre of Symmetry = Ci
20.C3 + Centre of Symmetry = C3i

He added horizontal plane to the 4 dihedral elements D2,D3,D4,D6.
21.D2 + HP = D2h
22.D3 + HP= D3h
23.D4 + HP = D4h
24.D6 + HP = D6h

10. •He derived crystal classes of cubic system by combining the four 3
fold axis with other axis of symmetry and found that two or more
classes were possible with the following combinations.
25.3-2 fold axes and 4-3 fold axes .ie 3 diagonals and 4 triagonals = T
26.3-4 fold,4-3 fold and 6-2 fold = O

Classes C1,C2,C3,C4,C6 ,D2,D3,D4,D6,T and O are called Holoaxial
classes

11. 29.He next added horizontal plane to T and O,he obtained another new
class Td
•Since D2h &D3h do not have diagonal planes,2 more classes
generated by addig only diagonal planes to D2 & D3.
30.D2 + Diagonal plane = D2d
31.D3 + Diagonal plane = D3d
32.Finally by a combination of rotation and inversion ,He derived S4
class where there is rotation by 90 ˚ and Inversion 4 times.