InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 619-621
## Section 4.6.3.3.3.5. Relationships between structure factors at symmetry-related points of the Fourier image |

#### 4.6.3.3.3.5. Relationships between structure factors at symmetry-related points of the Fourier image

The weighted 3D reciprocal space exhibits the icosahedral point symmetry . It is invariant under the action of the scaling matrix : The scaling transformation leaves a primitive 6D reciprocal lattice invariant as can easily be seen from its application on the indices: The matrix leaves invariant, for any with all even or all odd, corresponding to a 6D face-centred hypercubic lattice. In a second case the sum is even, corresponding to a 6D body-centred hypercubic lattice. Block-diagonalization of the matrix *S* decomposes it into two irreducible representations. With we obtain the scaling properties in the two 3D subspaces: scaling by a factor τ in parallel space corresponds to a scaling by a factor in perpendicular space. For the intensities of the scaled reflections analogous relationships are valid, as discussed for decagonal phases (Figs. 4.6.3.36 and 4.6.3.37, Section 4.6.3.3.2.5).