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Si $\Gamma $-phonon

Si possesses a face-centered cubic structure with two equivalent atoms per unit cell, at ($\pm$0.125, $\pm$0.125, $\pm$0.125). The site symmetry is -43m. For the $\Gamma $-phnon the two atoms are displaced in opposite direction along the (111) direction and cubic symmetry is lost. The output of SYMMETRY gives the following information:

Si         operation #  1     1                   
Si         operation # 13     m n -110            
Si         operation # 16     m n -101            
Si         operation # 17     m n 0-11            
Si         operation # 24     3 || 111            
Si         operation # 38     3 || 111            
  pointgroup is 3m (neg. iatnr!!)
     axis should be: 3 || z, m n y
lm: 0 0  1 0  2 0  3 0  3 3  4 0  4 3  5 0  5 3  6 0  6 3  6

Therefore the required local rotation matrix should rotate z into the (111) direction and thus the matrix in the ``struct`` file should be:

0.4082483 -.7071068 0.5773503 $\frac{\sqrt 6}{6}$ -$-\frac{\sqrt 2}{2}$ $\frac{\sqrt 3}{3}$
0.4082483 0.7071068 0.5773503 $\frac{\sqrt 6}{6}$ $\frac{\sqrt 2}{2}$ $\frac{\sqrt 3}{3}$
-.8164966 0.0000000 0.5773503 -2$-2\frac{\sqrt 6}{6}$ $\frac{\sqrt 2}{2}$ $\frac{\sqrt 3}{3}$




2000-04-11