Exercise: Calculation of the Magnetic Form Factor for the Spin Density of a 3d Shell in a Given Environment

Abstract

The calculation of a spin structure factor can be performed for any local symmetry using the general formulae (¹)

$$F\left( \overset{\to }{\mathop{H}}\, \right)=u\underset{K}{\mathop{\sum }}\,<{{j}_{K}}>\underset{Q}{\mathop{\sum }}\,C_{Q}^{K}Y_{Q}^{{{K}^{}}}(\theta ,\Phi )$$

$$C_{Q}^{K}={{i}^{K}}(2\ell +1){{\left[ 4\pi (2K+1) \right]}^{\frac{1}{2}}}\underset{m,{{m}^{?}}}{\mathop{\sum }}\,{{(-1)}^{m}}a_{m}^{}{{a}_{{{m}^{?}}}}\left( \overset{\ell }{\mathop{0}}\,\overset{K}{\mathop{0}}\,\overset{\ell }{\mathop{0}}\, \right)\left( \overset{\ell }{\mathop{-m}}\,\overset{K}{\mathop{Q}}\,\overset{\ell }{\mathop{{{m}^{?}}}}\, \right)$$

where $\left( \overset{{{j}_{1}}}{\mathop{{{m}_{1}}}}\,\overset{{{j}_{2}}}{\mathop{{{m}_{2}}}}\,\overset{{{j}_{3}}}{\mathop{{{m}_{3}}}}\, \right)$ are 3j coefficients, and $Y_{Q}^{K}(\theta ,\Phi )$ are spherical harmonics.