Surface spherical harmonics and intensity and strain pole figures of cubic textured materials

The equations for diffraction strain pole figures measured on textured cubic materials exhibit an hkl dependence. This is expressed by an hkl-permutation-invariant 'surface' spherical harmonic. Four types of new harmonics are defined. These harmonics differ in essence from those for the hkl-dependent expression obtained for diffraction intensity pole figures. In the latter case associated Legendre polynomials arise whereas in the former (Jacobi type) generalizations of these polynomials occur. Equations exhibiting diffraction intensity and diffraction strain expressions are given for all cubic point groups. Structure factors arise in the expressions. The treatment is given for both anomalous and normal scattering modes. Surface spherical harmonics do not satisfy Laplace's equation. This only occurs upon conversion into 'solid spherical harmonics'. Then the harmonics associated with intensity pole figures satisfy Laplace's equation. The 'diffraction strain harmonics' do not, however. Orthonormalization is also different from the case of conventional hkl-permutation-invariant surface spherical harmonics. Stereographic projections are given for a few examples of harmonics.