PHY 745 Group Theory
11-11:50 AM MWF Olin 102
Plan for Lecture 16:
Introduction to space groups
Reading: Chapter 9 in DDJ
1. Definition and scope of space groups
2. Terminology
3. Examples
This lecture contains some materials from an electronic
version of the DDJ text.
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Comment on character analysis of iCn for angular momentum
basis functions
Ylm (rˆ )  (1)l Ylm (rˆ )
1
sin
l

 


2 
l
  (Cn ) 

sin( / 2) 
  2 / n
1

sin
l






l
2
  l (iCn )   1

sin( / 2) 
  2 / n
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Space groups
NaCl
t
R τ
Point operation
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t
Translation
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Space group operation on position vector r
R τ r  r'  R r  τ
Multiplication of space group elements
R τ 'R τ r  R  R r   R τ  τ '  R R








R τ  τ ' r
Special elements
Identity:
 0
Pure point group operation:
 τ
Pure translation:
Inverse:
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R τ
R 0
1

 R
1
1

 R τ
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More details
τ  τ  T
Lattice translation
Non primitive
translation
If, for a suitable choice of the origin in the lattice, all
of the group tranlation vectors are lattice translations T,
the group is called "symmorphic", otherwise the group
is called "non-symmorphic".
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Example of symmorphic lattice Fe (body
centered cubic lattice)
t=T
Primitive translations
Conventional cell
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Example of symmorphic lattice Cu (face
centered cubic lattice)
t=T
Conventional cell
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Primitive translations
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Non-symmorphic space group -- diamond
τ
Primitive translations
( 14 , 14 , 14 ) center of symmetry
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Some non-symmorphic operations and their descriptions
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Some examples in 2 dimensions
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Multiplicity
Wyckoff letter
Site symmetry
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Centered cell
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Example with higher symmetry
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Square lattice
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Hexagonal
b
-a
a
-b
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