Formalism I - Resonant Magnetic Xray Scattering Intensity

The resonant magnetic scattering intensity is calculated as outlined for example in [29] 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

			$\displaystyle f_{nE1}^{RXMS}=$	(23)
			$\displaystyle =\left ( \begin{array}{cc} \sigma \rightarrow \sigma & \pi \right... ...ft ( \begin{array}{cc} 1 & 0 \\ 0 & \cos2\Theta \end{array} \right ) -iF^{(1)}$
			$\displaystyle \times \left ( \begin{array}{cc} 0 & z_1 \cos \Theta + z_3 \sin \... ...sin \Theta - z_1 \cos \Theta & -z_2 \sin 2\Theta \end{array} \right ) + F^{(2)}$
			$\displaystyle \times \left ( \begin{array}{cc} z_2^2 & -z_2(z_1 \sin \Theta - z... ...cos \Theta) & -\cos^2 \Theta (z_1^2 \tan^2 \Theta + z_3^2) \end{array} \right )$

Here, dropping the site index $n$

, $\Theta$ is the Bragg scattering angle and $z_1, z_2$

and

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 $\mu_x,\mu_y$ and $\mu_z$ of the magnetic moment along the orthogonal euclidean coordinate system defined by y||b, z||(a x b) and x perpendicular to y and z.

$\displaystyle z_1$	$\textstyle =$	$\displaystyle \mu_x \sin \alpha_1 \cos(\Psi+\delta_1)+\mu_y \sin \alpha_2 \cos(\Psi +\delta_2)$
		$\displaystyle +\mu_z \sin \alpha_3 \cos (\Psi +\delta_3),$
$\displaystyle z_2$	$\textstyle =$	$\displaystyle \mu_x \sin \alpha_1 \sin (\Psi +\delta_1)+\mu_y \sin \alpha_2 \sin (\Psi+\delta_2)$
		$\displaystyle +\mu_z \sin \alpha_3 \sin (\Psi +\delta_3)$
$\displaystyle z_3$	$\textstyle =$	$\displaystyle \mu_x \cos \alpha_1 + \mu_y \cos \alpha_2 + \mu_z \cos \alpha_3$	(24)

with angles $\alpha_i=\angle( \hat \mathbf x_i \cdot \mathbf u_3)_{\Psi=0}$ and $\delta_i=\angle(\mathbf x_i^{\perp}\cdot \mathbf u_1)_{\Psi=0}$ , where $\hat \mathbf x_{1,2,3}=\hat \mathbf x,\hat \mathbf y,\hat \mathbf z$ and $\mathbf x_i^{\perp}$ is the projection of $\hat \mathbf x_i$ onto the plane perpendicular to $\mathbf Q$ . In the chosen experimental geometry $\Psi=0$ when $\mathbf x$ points to the x-ray source (and for the special case of $\mathbf Q\vert\vert \mathbf x$ when $\mathbf y$ points to the x-ray source).