TenXar
 Calculations with tensors and characters

M. Ephraim, T. Janssen, A. Janner and A. Thiers

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Subsections

Character tables

Invariant magnetic field

Point group m
Vector character 3 1
Determinant character 1 -1
Product(1,2) 3 -1
Decompose(3) 1 2

The multiplicity of the trivial representation in the pseudovector representation which is the product of the vector representation and the determinant representation is one. Therefore, there is one free parameter.

Metric tensor

Point group 3
Vector representation 3 0 0
Symmetrized square 6 0 0
Decompose 2 2 2
Physical representation of irrep 2 2 -1 -1

The metric tensor invariant under the three-dimensional group 3 has two free parameters. The six-dimensional space of symmetric rank-two tensors has a two-dimensional invariant subspace. The remaining four-dimensional space carries the irreducible representations 2 and 3 twice. This space is twice the physically irreducible representation 2+3.

Elastic tensor

Point group 4
Vector representation 3 1 -1 1
Symmetrized square 6 0 2 0
Symmetrized square of the former 21 1 5 1
Decompose 7 4 6 4

The elastic tensor is a rank-four tensor with intrinsic symmetry ((1 2)(3 4)). It can be obtained by taking twice the symmetrized square.

Vector product

Point group 4
Vector representation 3 1 -1 1
Antisymmetrized square 3 1 -1 1
Decompose 1 1 1

The vector product of two vectors corresponds to a rank-two tensor with intrinsic symmetry [1 2]. The number of free parameters if the symmetry group is 4 is equal to 1.

Magnetoelectric tensor

Point group mm2
Vector representation 3 -1 1 1
Determinant representation 1 1 -1 -1
Power $n=2$ of the vector representation 9 1 1 1
Product(former, determinant rep.) 9 1 -1 -1
Decompose 2 3 2 2

The multiplicity of the trivial representation in the decomposition being 2, the number of free parameters in a pseudotensor of rank two invariant under mm2 is 2.

Selection rules

Matrix elements $<k\vert A\vert m>$ of an operator $A$ transforming with an irreducible representation $\beta$ between states $\vert m>$ and $\vert k>$ transforming with irreducible representations $\gamma$ and $\alpha$, respectively, vanish if $\alpha$ is not a component in the decomposition of the tensor product of $\beta$ and $\gamma$. Take as example the group 432. The tensor products of the irreducible representations are given in the following table.

  1 2 3 4 5 $\gamma$
1 1 2 3 4 5  
2 2 1 3 5 4  
3 3 3 1+2+3 4+5 4+5  
4 4 5 4+5 1+3+4+5 2+3+4+5  
5 5 4 4+5 2+3+4+5 1+3+4+5  
$\beta$            

If $\alpha =3$ the selection rules for the matrix element $<k\vert A\vert m>$ are given in the following table.

  1 2 3 4 5 $\gamma$
1 0 0 * 0 0  
2 0 0 * 0 0  
3 * * * 0 0  
4 0 0 0 * *  
5 0 0 0 * *  
$\beta$            

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