Equatorial-Transparency aberration in Powder XRD data collected in quasi-Bragg-Brentano geometry

\( \mbox{The instrumental function for the goniometer radius } R, \, \)
\( \mbox{divergence slit angle } \Phi_{\rm DS}, \)
\( \mbox{effective width of 1D X-ray detector along the equatorial direction }2W_{\rm 1D} \),
\( \mbox{specimen width along the equatorial direction } W \),
\( \mbox{specimen thickness }t \),
\( \mbox{penetration depth of specimen } \mu^{-1} \),
\( \mbox{penetration depth of sample holder } \mu’^{-1} \),
\( \mbox{beam stopper angle } 2\Theta_{\rm BS} \)\( \mbox{, is given by} \)

\[ \omega^{\rm (E)}(\Delta 2\Theta) = \frac{1}{\Phi_{\rm DS} \Psi } \int\limits_{-\frac{\Psi}{2}}^{\frac{\Psi}{2}} \int\limits_{-t}^{0} \int\limits_{\phi_{\rm L} }^{\phi_{\rm U} } \delta\left( \Delta 2\Theta – f(2\Theta,\phi,z,\psi) \right) g(2\Theta,\phi,z,\psi) \,{\rm d}\phi\,{\rm d}z\,{\rm d}\psi \]

\[ \Psi = \arctan \frac{W_{\rm 1D}}{2R} \]

\[ \phi_{\rm L} = \max\left\{ -\frac{\Phi_{\rm DS}}{2},\Phi_{\rm UB} \right\} \]

\[ \phi_{\rm U} = \min\left\{ \frac{\Phi_{\rm DS}}{2}, \Phi_{\rm DF} \right\} \]

\[ \Phi_{\rm UB} = \Theta_{\rm G} – \arctan\frac{\sin\Theta_{\rm G} + t/R }{ \cos\Theta_{\rm G} – W/2R } \]

\[ \Phi_{\rm DF} = \Theta_{\rm G} – \arctan\frac{\sin\Theta_{\rm G}}{\cos\Theta_{\rm G} – W / 2R} \]

\[ f(2\Theta,\phi,z,\psi) = 2\Theta – 2\theta \]

\[ g(2\Theta,\phi,z,\psi) = \left\{ \begin{matrix} \displaystyle\frac{2\mu \,{\rm e}^{-\mu l – \mu’l’} }{\sin\Theta} & \displaystyle\left[ |x^{\rm (r)}| \le \frac{W}{2} \mbox{ and } 2\Theta_{\rm BS} < 2\Theta_{\rm G} \right] \\ 0 & \left[ \rm otherwise \right]\end{matrix} \right. \]

\[ 2\theta = \Theta^{\rm (i)} + \Theta^{\rm (e)} \]

\[ 2\Theta_{\rm G} = 2\Theta – 2\psi \]

\[ \Theta^{\rm (i)} = \Theta_{\rm G} – \phi \]

\[ \Theta^{\rm (e)} = \arctan \frac{ z^{\rm (d)} – z }{ x^{\rm (d)} – x^{\rm (r)}} \]

\[ x^{\rm (i)} = R\left( \frac{ \sin\Theta_{\rm G} }{ \tan \Theta^{\rm (i)} } – \cos\Theta_{\rm G} \right) \]

\[ x^{\rm (r)} = x^{\rm (i)} -\frac{ z }{ \tan \Theta^{\rm (i)} } \]

\[ x^{\rm (d)} = R (\cos \Theta_{\rm G} – \sin \Theta_{\rm G} \tan 2\psi ) \]

\[ z^{\rm (d)} = R (\sin \Theta_{\rm G} + \cos \Theta_{\rm G} \tan 2\psi ) \]

\[ l = l^{\rm (i)} + l^{\rm (e)} \]

\[ l’ = l’^{\rm (i)} + l’^{\rm (e) }\]

\[ l’^{\rm (i)} = \left\{ \begin{matrix} 0 & \displaystyle \left[ -\frac{W}{2} \le x^{\rm (i)}\right] \\ \displaystyle \frac{-W/2-x^{\rm (i)}}{\cos\Theta^{\rm (i)}} & \left[ \rm otherwise \right] \end{matrix} \right. \]

\[ l^{\rm (i)} = \left\{ \begin{matrix} \displaystyle -\frac{z}{\sin\Theta^{\rm (i)}} & \displaystyle \left[ -\frac{W}{2} \le x^{\rm (i)} \right] \\ \displaystyle \frac{x^{\rm (r)} + W/2}{\cos\Theta^{\rm (i)}} & \left[ \rm otherwise \right] \end{matrix} \right. \]

\[ l^{\rm (e)} = \left\{ \begin{matrix} \displaystyle -\frac{z}{\sin\Theta^{\rm (e)}} & \displaystyle \left[ \frac{W}{2} \le x^{\rm (e)} \right] \\ \displaystyle \frac{W/2 – x^{\rm (r)} }{\cos\Theta^{\rm (e)}} & \left[ \rm otherwise \right] \end{matrix} \right. \]

\[ l’^{\rm (e)} = \left\{ \begin{matrix} 0 & \displaystyle \left[ x^{\rm (e)} \le W/2 \right] \\ \displaystyle \frac{ x^{\rm (e)} – W/2}{\cos\Theta^{\rm (e)}} & \left[ \rm otherwise \right] \end{matrix} \right. \]

\( \mbox{where } 2\Psi \mbox{ is the view angle of the detector from the center of the specimen,} \)
\( \phi_{\rm U} \mbox{ and } \phi_{\rm L} \mbox{ are the upper and lower limits of the equatorial deviation angle, } \)
\( 2\Theta \mbox{ the apparent diffraction angle, an instrument reports to users, } \)
\( 2\theta \mbox{ the true diffraction angle defined by physical and mathematical theories,} \)
\( \Theta^{\rm (i)} \mbox{ incident glancing angle for a reflection point, } \)
\( \Theta^{\rm (e)} \mbox{ emission glancing angle for a reflection point, } \)
\( \Theta_{\rm G} \mbox{ mechanical goniometer angle, if the mechanism is well adjusted, } \)
\( x^{\rm (i)} \mbox{ x-coordinate of the incident point on the specimen surface} \)
\( x^{\rm (r)} \mbox{ x-coordinate of the reflection point, } \)
\( x^{\rm (d)} \mbox{ x-coordinate of the detection point, } \)
\( z^{\rm (d)} \mbox{ z-coordinate of the detection point, } \)
\( l \mbox{ total path length in sample,} \)
\( l’ \mbox{ total path length in sample holder,} \)
\( l’^{\rm (i)} \mbox{ path length of incident beam in sample holder} \)
\( l^{\rm (i)} \mbox{ path length of incident beam in sample} \)
\( l^{\rm (e)} \mbox{ path length of emitting beam in sample} \)
\( l’^{\rm (e)} \mbox{ path length of emitting beam in sample holder} \)