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physics_chemistry:point_groups:d3d:orientation_zx_a [2018/09/06 13:06]
Maurits W. Haverkort
physics_chemistry:point_groups:d3d:orientation_zx_a [2018/09/06 13:28] (current)
Maurits W. Haverkort
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 ====== Orientation Zx_A ====== ====== Orientation Zx_A ======
  
-###                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         ​The point group D3d is a subgroup of Oh. Many materials of relevance have near cubic symmetry with a small D3d distortion. It thus makes sense to label the states in D3d symmetry according to the states they branch from. For d orbitals the eg orbitals in Oh symmetry branch to orbitals that belong to the eg irreducible representation in D3d symmetry. The t2g orbitals in Oh symmetry branch to an orbital that belongs to the a1g irreducible representation and two that belong to the eg irreducible representation. We label the eg orbitals that descend from the eg irreducible representation in Oh symmetry eg$\sigma$ and the eg orbitals that descend from the t2g irreducible representation eg$\pi$ orbitals. (The mixing is given by the parameter Meg.)+### 
 +The point group D3d is a subgroup of Oh. Many materials of relevance have near cubic symmetry with a small D3d distortion. It thus makes sense to label the states in D3d symmetry according to the states they branch from. For d orbitals the eg orbitals in Oh symmetry branch to orbitals that belong to the eg irreducible representation in D3d symmetry. The t2g orbitals in Oh symmetry branch to an orbital that belongs to the a1g irreducible representation and two that belong to the eg irreducible representation. We label the eg orbitals that descend from the eg irreducible representation in Oh symmetry eg$\sigma$ and the eg orbitals that descend from the t2g irreducible representation eg$\pi$ orbitals. (The mixing is given by the parameter Meg.)
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-The parameterisation A of the orientation Zx is related to the orientation ​Sqrt[2]1z of the Oh pointgroup.+The parameterisation A of the orientation Zx is related to the orientation ​0Sqrt[2]1z of the Oh pointgroup.
 ### ###
- 
  
 ===== Symmetry Operations ===== ===== Symmetry Operations =====
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  ​$$A_{k,​m} = \begin{cases}  ​$$A_{k,​m} = \begin{cases}
- ​\frac{1}{5} (\text{Ea1g}+2 (\text{Eeg$\pi $}+\text{Eeg$\sigma ​$})) & k=0\land m=0 \\ + ​\frac{1}{5} (\text{Ea1g}+2 (\text{Eeg}\pi +\text{Eeg}\sigma )) & k=0\land m=0 \\ 
- ​\text{Ea1g}-\text{Eeg$\pi $}-2 \sqrt{2} \text{Meg} & k=2\land m=0 \\ + ​\text{Ea1g}-\text{Eeg}\pi -2 \sqrt{2} \text{Meg} & k=2\land m=0 \\ 
- -i \sqrt{\frac{7}{10}} \left(-2 \text{Eeg$\pi $}+2 \text{Eeg$\sigma ​$}+\sqrt{2} \text{Meg}\right) & k=4\land (m=-3\lor m=3) \\ + -i \sqrt{\frac{7}{10}} \left(-2 \text{Eeg}\pi +2 \text{Eeg}\sigma +\sqrt{2} \text{Meg}\right) & k=4\land (m=-3\lor m=3) \\ 
- ​\frac{1}{5} \left(9 \text{Ea1g}-2 \text{Eeg$\pi $}-7 \text{Eeg$\sigma ​$}+10 \sqrt{2} \text{Meg}\right) & k=4\land m=0+ ​\frac{1}{5} \left(9 \text{Ea1g}-2 \text{Eeg}\pi -7 \text{Eeg}\sigma +10 \sqrt{2} \text{Meg}\right) & k=4\land m=0
 \end{cases}$$ \end{cases}$$
  
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 <code Quanty Akm_D3d_Zx_A.Quanty>​ <code Quanty Akm_D3d_Zx_A.Quanty>​
  
-Akm = {{0, 0, (1/5)*(Ea1g + (2)*(Eeg\[Pi] ​Eeg\[Sigma]))} ,  +Akm = {{0, 0, (1/5)*(Ea1g + (2)*(EegPi EegSigma))} ,  
-       {2, 0, Ea1g + (-1)*(Eeg\[Pi]) + (-2)*((sqrt(2))*(Meg))} ,  +       {2, 0, Ea1g + (-1)*(EegPi) + (-2)*((sqrt(2))*(Meg))} ,  
-       {4, 0, (1/​5)*((9)*(Ea1g) + (-2)*(Eeg\[Pi]) + (-7)*(Eeg\[Sigma]) + (10)*((sqrt(2))*(Meg)))} ,  +       {4, 0, (1/​5)*((9)*(Ea1g) + (-2)*(EegPi) + (-7)*(EegSigma) + (10)*((sqrt(2))*(Meg)))} ,  
-       ​{4,​-3,​ (-I)*((sqrt(7/​10))*((-2)*(Eeg\[Pi]) + (2)*(Eeg\[Sigma]) + (sqrt(2))*(Meg)))} ,  +       ​{4,​-3,​ (-I)*((sqrt(7/​10))*((-2)*(EegPi) + (2)*(EegSigma) + (sqrt(2))*(Meg)))} ,  
-       {4, 3, (-I)*((sqrt(7/​10))*((-2)*(Eeg\[Pi]) + (2)*(Eeg\[Sigma]) + (sqrt(2))*(Meg)))} }+       {4, 3, (-I)*((sqrt(7/​10))*((-2)*(EegPi) + (2)*(EegSigma) + (sqrt(2))*(Meg)))} }
  
 </​code>​ </​code>​
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