NEUTRON
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NEUTRON
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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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