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# Group Theory PHY 745

WFU

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This 9 page Class Notes was uploaded by Miss Lucienne Hamill on Wednesday October 28, 2015. The Class Notes belongs to PHY 745 at Wake Forest University taught by Eric Carlson in Fall. Since its upload, it has received 9 views. For similar materials see /class/230731/phy-745-wake-forest-university in Physics 2 at Wake Forest University.

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Date Created: 10/28/15

Group Theory I Basic mathematical theory With point group and space group examples N A W Holzvvarth II Lie groups and their application to particle physics Eric Carlson Lecture 1 0 De nition of a group 0 Example of the group of a triangle 0 What can group theory do for you Character Table example Quantum mechanical selection rules Relationships between crystal and atomic molecular structures Properties of a Group A group is a collection of elements A B C and a multiplication process The abstract multiplication pairs two group elements and associates the result with a third element For example A B The elements and the multiplication process must have the following properties 1 The collection of elements is closed under multiplication That is if elements A and B are in the group and A B C element C must be in the group 2 One of the members of the group is a unit element That is for any element A of the group A E E A A 3 For each element A of the group there is another element A 1 which is its inverse That is A A 1 A 1 A E 4 The multiplication process is associative That is for sequential mulplication of group elements A B and C A B C A B C 2 Group multiplication table Group of order 6 HEABCDF EABCDF WUQUJDgtE1 WUQUJCD WQUWH QgtWHU CDUUHUW HWCDQUJ UHUJCDQ Example of group theory applied to space groups Ref L P Bouckaert R Smoluchowski and E Wigner Phys Rev 50 58 1936 7 Theory of Brillouin zones and symmetry properties of wave functions in crystals I I 41 l A 5 i Tc t R I a Im quot x quotquotx g lt Z M 639 13102 Brillouin zone of simple cubic lattice Brillouin zone of face centered cubic lattice Example of group theory applied to space groups continued Ref G A Burdick7 Phys Rev 129 138 1963 7 Energy band structure of copper 05 u A1 T RYDBERGS Example of group theory applied to space groups continued Ref BSW Some appropriate character tables TABLE 1 Characters of small representations of 1 R H I39HR39 E 3C12 6C4 6C2 863 J 3JC42 6JC4 6JC2 8JC3 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 12 2 2 0 0 1 2 2 0 0 1 1 15 3 1 1 1 0 3 l 1 1 0 25 3 1 1 1 0 3 1 l 1 0 1 1 1 1 1 1 1 1 1 1 1 1 239 1 1 1 1 1 1 l 1 1 1 1239 2 2 0 0 1 2 2 0 0 1 1 15 3 1 1 1 0 3 1 1 1 0 F25 3 1 1 1 0 3 1 1 1 0 TABLE 11 Characters for the small representations of A T A T E C I 2C4 ZJCA2 7JC A1 1 39 1 1 l 1 Ag 1 1 1 1 1 A 1 1 1 1 1 AJ 1 1 1 1 5 2 2 0 0 0 TABLE V Characters of small representations of M X M I 2C12 2191 ZL39AJ 2L3 J ZJCF JCaU ZJCLL 39J 2 X E 2C12J WI 2C1 2639 J 2JC 3939J JCMI 2JC1II ZJCZ M 1 1 1 1 1 1 1 1 1 1 M2 1 1 1 e1 1 1 1 1 1 1 M 1 1 1 1 1 1 1 1 1 1 M1 1 1 1 1 1 1 1 1 1 1 M 1 1 1 1 1 1 1 1 1 1 My 1 1 1 1 1 71 v1 1 1 1 My 1 1 1 1 1 1 1 1 1 1 1111 1 1 1 1 1 1 1 1 1 1 ME 2 o 2 o 0 2 0 2 0 0 My 2 0 2 0 o 2 0 2 0 0 Example of group theory applied to space groups continued Ref BSW Some appropriate compatability tables TABLE VII Compatibility relations between P and A A 2 1 1 1 2 P12 I 1539 1 25 A A2 A1132 A1735 A2 A5 A A2 A3 A2A3 A1A3 21 x 274 P3124 222324 2122233 1 I g 1 12 F15 13925 A A2 Al A2 A1135 A2A5 A2 A1 A3 AIAS A2A3 272 233 2223 2312324 23122234 TABLE IX Compatibility relations between X and A Z S X1 X2 X3 X4 X139 X239 X3 X439 X5 Xs A1 A2 A239 All Al A2 A2 A1 A5 A5 n Example of group theory applied to space groups continued Analysis of transitions between quantum mechanical states 2 Transition probability oc lMP E d3r 1132r I r M QC ZNCXfCXoCX C 0 Some examples 0 Optical transitions absorption emission polarization effects 0 Analysis of phonon modes Infrared transitions Raman transitions Example of group theory applied to point groups Analysis of crystal eld effects on atomic states Spherical Cubic Symmetry Symmetry

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