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Inelastic Neutron Scattering Selection Rules for group 62 (P n m a)



Space group  G62 , number 62
Lattice type : oP

Number of generators : 4

        1                  2                  3                  4          
  1  0  0     0     -1  0  0   1/2     -1  0  0     0     -1  0  0     0    
  0  1  0     0      0 -1  0     0      0  1  0   1/2      0 -1  0     0    
  0  0  1     0      0  0  1   1/2      0  0 -1     0      0  0 -1     0    



Number of elements : 8

        1                  2                  3                  4          
  1  0  0     0     -1  0  0   1/2     -1  0  0     0      1  0  0  -1/2    
  0  1  0     0      0 -1  0     0      0  1  0   1/2      0 -1  0   1/2    
  0  0  1     0      0  0  1   1/2      0  0 -1     0      0  0 -1  -1/2    

        5                  6                  7                  8          
 -1  0  0     0      1  0  0  -1/2      1  0  0     0     -1  0  0   1/2    
  0 -1  0     0      0  1  0     0      0 -1  0  -1/2      0  1  0  -1/2    
  0  0 -1     0      0  0 -1  -1/2      0  0  1     0      0  0  1   1/2    



K-vector GM :
  in primitive basis :   0.000  0.000  0.000
  in standart dual basis      :   0.000  0.000  0.000

The little group of the k-vector has the following 8
elements as translation coset representatives :

        1                  2                  3                  4          
  1  0  0     0     -1  0  0   1/2     -1  0  0     0      1  0  0  -1/2    
  0  1  0     0      0 -1  0     0      0  1  0   1/2      0 -1  0   1/2    
  0  0  1     0      0  0  1   1/2      0  0 -1     0      0  0 -1  -1/2    

        5                  6                  7                  8          
 -1  0  0     0      1  0  0  -1/2      1  0  0     0     -1  0  0   1/2    
  0 -1  0     0      0  1  0     0      0 -1  0  -1/2      0  1  0  -1/2    
  0  0 -1     0      0  0 -1  -1/2      0  0  1     0      0  0  1   1/2    



The little group of the k-vector has  8  allowed irreps.
The matrices, corresponding to all of the little group elements are :

 Irrep (GM)(1) ,  dimension 1
      1               2               3               4         
(1.000,  0.0)   (1.000,  0.0)   (1.000,  0.0)   (1.000,  0.0)   

      5               6               7               8         
(1.000,  0.0)   (1.000,  0.0)   (1.000,  0.0)   (1.000,  0.0)   


 Irrep (GM)(2) ,  dimension 1
      1               2               3               4         
(1.000,  0.0)   (1.000,  0.0)   (1.000,  0.0)   (1.000,  0.0)   

      5               6               7               8         
(1.000,180.0)   (1.000,180.0)   (1.000,180.0)   (1.000,180.0)   


 Irrep (GM)(3) ,  dimension 1
      1               2               3               4         
(1.000,  0.0)   (1.000,  0.0)   (1.000,180.0)   (1.000,180.0)   

      5               6               7               8         
(1.000,  0.0)   (1.000,  0.0)   (1.000,180.0)   (1.000,180.0)   


 Irrep (GM)(4) ,  dimension 1
      1               2               3               4         
(1.000,  0.0)   (1.000,  0.0)   (1.000,180.0)   (1.000,180.0)   

      5               6               7               8         
(1.000,180.0)   (1.000,180.0)   (1.000,  0.0)   (1.000,  0.0)   


 Irrep (GM)(5) ,  dimension 1
      1               2               3               4         
(1.000,  0.0)   (1.000,180.0)   (1.000,  0.0)   (1.000,180.0)   

      5               6               7               8         
(1.000,  0.0)   (1.000,180.0)   (1.000,  0.0)   (1.000,180.0)   


 Irrep (GM)(6) ,  dimension 1
      1               2               3               4         
(1.000,  0.0)   (1.000,180.0)   (1.000,  0.0)   (1.000,180.0)   

      5               6               7               8         
(1.000,180.0)   (1.000,  0.0)   (1.000,180.0)   (1.000,  0.0)   


 Irrep (GM)(7) ,  dimension 1
      1               2               3               4         
(1.000,  0.0)   (1.000,180.0)   (1.000,180.0)   (1.000,  0.0)   

      5               6               7               8         
(1.000,  0.0)   (1.000,180.0)   (1.000,180.0)   (1.000,  0.0)   


 Irrep (GM)(8) ,  dimension 1
      1               2               3               4         
(1.000,  0.0)   (1.000,180.0)   (1.000,180.0)   (1.000,  0.0)   

      5               6               7               8         
(1.000,180.0)   (1.000,  0.0)   (1.000,  0.0)   (1.000,180.0)   



The Q-vector in general is   Q = ( h, k, l )



There are 7 nontrivial allowed types of Q-vectors.


--------------------------------------------------------------------------------

H[1] = ( 0, 0, l )  ,   Q[1] = ( 0, 0, l )
where parameters are : l=any

The elements of the little group, which leaves Q invariant (GQ group) are :
1 2 7 8 

The sum over all GQ elements is :
X_j(1)  +  X_j(2).exp[-i.2.Pi.(0.50*l)]  +  X_j(7)  +  X_j(8).exp[-i.2.Pi.(0.50*l)]
Where j indexes the representations

Condition  l=any

For  l = 1

Rep.             Sum           Allowed
GM_1         (0.000,  0.0)      NO
GM_2         (0.000,  0.0)      NO
GM_3         (0.000,  0.0)      NO
GM_4         (0.000,  0.0)      NO
GM_5         (4.000,  0.0)      YES
GM_6         (0.000,  0.0)      NO
GM_7         (0.000,  0.0)      NO
GM_8         (4.000,  0.0)      YES

For  l = 2

Rep.             Sum           Allowed
GM_1         (4.000,  0.0)      YES
GM_2         (0.000,  0.0)      NO
GM_3         (0.000,  0.0)      NO
GM_4         (4.000,  0.0)      YES
GM_5         (0.000,  0.0)      NO
GM_6         (0.000,  0.0)      NO
GM_7         (0.000,  0.0)      NO
GM_8         (0.000,  0.0)      NO

--------------------------------------------------------------------------------

H[2] = ( 0, k, 0 )  ,   Q[2] = ( 0, k, 0 )
where parameters are : k=any

The elements of the little group, which leaves Q invariant (GQ group) are :
1 3 6 8 

The sum over all GQ elements is :
X_j(1)  +  X_j(3).exp[-i.2.Pi.(0.50*k)]  +  X_j(6)  +  X_j(8).exp[-i.2.Pi.(- 0.50*k)]
Where j indexes the representations

Condition  k=any

For  k = 1

Rep.             Sum           Allowed
GM_1         (0.000,  0.0)      NO
GM_2         (0.000,  0.0)      NO
GM_3         (4.000,  0.0)      YES
GM_4         (0.000,  0.0)      NO
GM_5         (0.000,  0.0)      NO
GM_6         (0.000,  0.0)      NO
GM_7         (0.000,  0.0)      NO
GM_8         (4.000,  0.0)      YES

For  k = 2

Rep.             Sum           Allowed
GM_1         (4.000,  0.0)      YES
GM_2         (0.000,  0.0)      NO
GM_3         (0.000,  0.0)      NO
GM_4         (0.000,  0.0)      NO
GM_5         (0.000,  0.0)      NO
GM_6         (4.000,  0.0)      YES
GM_7         (0.000,  0.0)      NO
GM_8         (0.000,  0.0)      NO

--------------------------------------------------------------------------------

H[3] = ( h, 0, 0 )  ,   Q[3] = ( h, 0, 0 )
where parameters are : h=any

The elements of the little group, which leaves Q invariant (GQ group) are :
1 4 6 7 

The sum over all GQ elements is :
X_j(1)  +  X_j(4).exp[-i.2.Pi.(- 0.50*h)]  +  X_j(6).exp[-i.2.Pi.(- 0.50*h)]  +  X_j(7)
Where j indexes the representations

Condition  h=any

For  h = 1

Rep.             Sum           Allowed
GM_1         (0.000,  0.0)      NO
GM_2         (0.000,  0.0)      NO
GM_3         (0.000,  0.0)      NO
GM_4         (4.000,  0.0)      YES
GM_5         (4.000,  0.0)      YES
GM_6         (0.000,  0.0)      NO
GM_7         (0.000,  0.0)      NO
GM_8         (0.000,  0.0)      NO

For  h = 2

Rep.             Sum           Allowed
GM_1         (4.000,  0.0)      YES
GM_2         (0.000,  0.0)      NO
GM_3         (0.000,  0.0)      NO
GM_4         (0.000,  0.0)      NO
GM_5         (0.000,  0.0)      NO
GM_6         (0.000,  0.0)      NO
GM_7         (0.000,  0.0)      NO
GM_8         (4.000,  0.0)      YES

--------------------------------------------------------------------------------

H[4] = ( 0, 0, 0 )  ,   Q[4] = ( 0, 0, 0 )

The elements of the little group, which leaves Q invariant (GQ group) are :
1 2 3 4 5 6 7 8 

The sum over all GQ elements is :
X_j(1)  +  X_j(2)  +  X_j(3)  +  X_j(4)  +  X_j(5)  +  X_j(6)  +  X_j(7)  +  X_j(8)
Where j indexes the representations

Rep.             Sum           Allowed
GM_1         (8.000,  0.0)      YES
GM_2         (0.000,  0.0)      NO
GM_3         (0.000,  0.0)      NO
GM_4         (0.000,  0.0)      NO
GM_5         (0.000,  0.0)      NO
GM_6         (0.000,  0.0)      NO
GM_7         (0.000,  0.0)      NO
GM_8         (0.000,  0.0)      NO

--------------------------------------------------------------------------------

H[5] = ( h, k, 0 )  ,   Q[5] = ( h, k, 0 )
where parameters are : h=any, k=any

The elements of the little group, which leaves Q invariant (GQ group) are :
1 6 

The sum over all GQ elements is :
X_j(1)  +  X_j(6).exp[-i.2.Pi.(- 0.50*h)]
Where j indexes the representations

Condition  h=any, k=any

For  h = 1 ,  k = 1

Rep.             Sum           Allowed
GM_1         (0.000,  0.0)      NO
GM_2         (2.000,  0.0)      YES
GM_3         (0.000,  0.0)      NO
GM_4         (2.000,  0.0)      YES
GM_5         (2.000,  0.0)      YES
GM_6         (0.000,  0.0)      NO
GM_7         (2.000,  0.0)      YES
GM_8         (0.000,  0.0)      NO

For  h = 2 ,  k = 1

Rep.             Sum           Allowed
GM_1         (2.000,  0.0)      YES
GM_2         (0.000,  0.0)      NO
GM_3         (2.000,  0.0)      YES
GM_4         (0.000,  0.0)      NO
GM_5         (0.000,  0.0)      NO
GM_6         (2.000,  0.0)      YES
GM_7         (0.000,  0.0)      NO
GM_8         (2.000,  0.0)      YES

--------------------------------------------------------------------------------

H[6] = ( h, 0, l )  ,   Q[6] = ( h, 0, l )
where parameters are : h=any, l=any

The elements of the little group, which leaves Q invariant (GQ group) are :
1 7 

The sum over all GQ elements is :
X_j(1)  +  X_j(7)
Where j indexes the representations

Rep.             Sum           Allowed
GM_1         (2.000,  0.0)      YES
GM_2         (0.000,  0.0)      NO
GM_3         (0.000,  0.0)      NO
GM_4         (2.000,  0.0)      YES
GM_5         (2.000,  0.0)      YES
GM_6         (0.000,  0.0)      NO
GM_7         (0.000,  0.0)      NO
GM_8         (2.000,  0.0)      YES

--------------------------------------------------------------------------------

H[7] = ( 0, k, l )  ,   Q[7] = ( 0, k, l )
where parameters are : k=any, l=any

The elements of the little group, which leaves Q invariant (GQ group) are :
1 8 

The sum over all GQ elements is :
X_j(1)  +  X_j(8).exp[-i.2.Pi.(- 0.50*k + 0.50*l)]
Where j indexes the representations

Condition  k=any, l=any

For  k = 1 ,  l = 1

Rep.             Sum           Allowed
GM_1         (2.000,  0.0)      YES
GM_2         (0.000,  0.0)      NO
GM_3         (0.000,  0.0)      NO
GM_4         (2.000,  0.0)      YES
GM_5         (0.000,  0.0)      NO
GM_6         (2.000,  0.0)      YES
GM_7         (2.000,  0.0)      YES
GM_8         (0.000,  0.0)      NO

For  k = 1 ,  l = 2

Rep.             Sum           Allowed
GM_1         (0.000,  0.0)      NO
GM_2         (2.000,  0.0)      YES
GM_3         (2.000,  0.0)      YES
GM_4         (0.000,  0.0)      NO
GM_5         (2.000,  0.0)      YES
GM_6         (0.000,  0.0)      NO
GM_7         (0.000,  0.0)      NO
GM_8         (2.000,  0.0)      YES

For  k = 2 ,  l = 1

Rep.             Sum           Allowed
GM_1         (0.000,  0.0)      NO
GM_2         (2.000,  0.0)      YES
GM_3         (2.000,  0.0)      YES
GM_4         (0.000,  0.0)      NO
GM_5         (2.000,  0.0)      YES
GM_6         (0.000,  0.0)      NO
GM_7         (0.000,  0.0)      NO
GM_8         (2.000,  0.0)      YES

For  k = 2 ,  l = 2

Rep.             Sum           Allowed
GM_1         (2.000,  0.0)      YES
GM_2         (0.000,  0.0)      NO
GM_3         (0.000,  0.0)      NO
GM_4         (2.000,  0.0)      YES
GM_5         (0.000,  0.0)      NO
GM_6         (2.000,  0.0)      YES
GM_7         (2.000,  0.0)      YES
GM_8         (0.000,  0.0)      NO


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