Formalism I - Resonant Magnetic Xray Scattering Intensity

The resonant magnetic scattering intensity is calculated as outlined for example in [28] for the magnetic electric dipole scattering according to the squared absolute value of the scattering amplitude given for the ion $n$ in the lattice as

$$f_{nE1}^{RXMS}=$$ (22)

$$=\left ( \begin{array}{cc} \sigma \rightarrow \sigma & \pi \right... ...ft ( \begin{array}{cc} 1 & 0 \\ 0 & \cos2\Theta \end{array} \right ) -iF^{(1)}$$

$$\times \left ( \begin{array}{cc} 0 & z_1 \cos \Theta + z_3 \sin \... ...sin \Theta - z_1 \cos \Theta & -z_2 \sin 2\Theta \end{array} \right ) + F^{(2)}$$

$$\times \left ( \begin{array}{cc} z_2^2 & -z_2(z_1 \sin \Theta - z... ...cos \Theta) & -\cos^2 \Theta (z_1^2 \tan^2 \Theta + z_3^2) \end{array} \right )$$

Here, dropping the site index $n$, $\Theta$ is the Bragg scattering angle and $z_1, z_2$ and $z_3$ aree components of the magnetization vector with respect to the coordinate system $[\mathbf u_1,\mathbf u_2,\mathbf u_3]$. The scattering plane, defined by the direction of the incident and final wave vectors $\mathbf k$ and $\mathbf k'$, contains $\mathbf u_1$ lying perpendicular and in the sense of $\mathbf k$ and $\mathbf u_3$ parallel to the scattering vector $\mathbf Q=\mathbf k-\mathbf k'$. The scattering channels for $F^{(1)}$ and $F^(2)$ terms are considered separately and no interference is considered.

To determine the azimuthal dependence the $z_i$ are expanded in terms of the components $\mu_a,\mu_b$ and $\mu_c$ of the magnetic moment along the orthogonal crystallographic unit cell vectors $\mathbf a$,$\mathbf b$ and $\mathbf c$ (for non orthogonal crystal lattices currently magnetic xray scattering intensities cannot be calculated):

$$z_1 = \mu_a \sin \alpha_1 \cos(\Psi+\delta_1)+\mu_b \sin \alpha_2 \cos(\Psi +\delta_2)$$

$$+\mu_c \sin \alpha_3 \cos (\Psi +\delta_3),$$

$$z_2 = \mu_a \sin \alpha_1 \sin (\Psi +\delta_1)+\mu_b \sin \alpha_2 \sin (\Psi+\delta_2)$$

$$+\mu_c \sin \alpha_3 \sin (\Psi +\delta_3)$$

$$z_3 = \mu_a \cos \alpha_1 + \mu_b \cos \alpha_2 + \mu_c \cos \alpha_3$$ (23)

with angles $\alpha_i=\angle(\mathbf a_i \cdot \mathbf u_3)_{\Psi=0}$ and $\delta_i=\angle(\mathbf a_i^{\perp}\cdot % \mathbf u_1)_{\Psi=0}$, where $\mathbf a_{1,2,3}=\mathbf a,\mathbf b,\mathbf c$ and $\mathbf a_i^{\perp}$ is the projection of $\mathbf a_i$ onto the plane perpendicular to $\mathbf Q$. In the chosen experimental geometry $\Psi=0$ when $\mathbf a$ points to the x-ray source.