Extra:Intersecting planes

A common problem in scattering is to consider the intersection of various planes (representing the Ewald sphere, reciprocal space, etc.).

Angle between two planes

The general case for the angle between two planes is well known. Consider a particular case where we want to know how the angle between two planes depends on the direction/orientation of a third plane/vector that intersects the first two. I.e. what is the minimal angle between two planes along a 'certain direction' (what is the angle between two vectors that both lie on the third plane, and which lie on planes 1 and 2, respectively).

One of the planes represents reciprocal-space scattering (e.g. mostly localized to a plane); the other represents the detector. We are interested in the angle between them so that we can calculate the distance between them, so that we can compute 'how much' scattering is seen on the detector. To make this concrete, plane 1 lies in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle xy} plane, and thus has normal vector:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{n}_1 = \begin{bmatrix} 0 & 0 & 1 \end{bmatrix} }

The first plane intersects the origin. The second plane also intersects the origin, but is tilted about the $\scriptstyle x$-axis by $\scriptstyle \alpha$, such that its normal is:

$\mathbf {n} _{2}={\begin{bmatrix}0&-\sin \alpha \ &\cos \alpha \end{bmatrix}}$

We are interested in quantities that are a particular distance ($\scriptstyle q$) from the origin. Imagine a vector of length $\scriptstyle q$ lying in plane 1, rotated about the $\scriptstyle z$ axis by $\scriptstyle \phi$ (i.e. the angular distance from the $\scriptstyle y$-axis is $\scriptstyle \phi$):

$\mathbf {v} _{1}={\begin{bmatrix}q\sin \phi &q\cos \phi \ &0\end{bmatrix}}$

The second vector ($\scriptstyle \mathbf {v} _{2}$) is lying in plane 2. We call $\scriptstyle \alpha _{r}$ the angle between $\scriptstyle \mathbf {v} _{1}$ and $\scriptstyle \mathbf {v} _{2}$. The specified geometry uniquely defines $\scriptstyle \alpha _{r}$ in terms of the angle between the planes ($\scriptstyle \alpha$) and the amount of rotation of the vectors ($\scriptstyle \phi$) within their respective planes. In particular, $\scriptstyle \mathbf {v} _{2}$ can be thought of as $\scriptstyle \mathbf {v} _{2b}$ rotated about $\scriptstyle \mathbf {n} _{2}$ by $\scriptstyle \phi$, where $\scriptstyle \mathbf {v} _{2}b$ is the vector in plane 2 without any $\scriptstyle \phi$ rotation (i.e. lying in the $\scriptstyle yz$ plane):

$\mathbf {v} _{2b}={\begin{bmatrix}0&q\cos \alpha \ &q\sin \alpha \end{bmatrix}}$

In general, rotation of a vector $\scriptstyle \mathbf {v} _{\mathrm {start} }={\begin{bmatrix}x&y&z\end{bmatrix}}$ about an arbitrary unit-vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{n} = \begin{bmatrix} u & v & w \end{bmatrix}} gives (1, 2):

$\mathbf {v} _{\mathrm {end} }={\begin{bmatrix}u(ux+vy+wz)(1-\cos \theta )+x\cos \theta +(-wy+vz)\sin \theta \\v(ux+vy+wz)(1-\cos \theta )+y\cos \theta +(wx-uz)\sin \theta \\w(ux+vy+wz)(1-\cos \theta )+z\cos \theta +(-vx+uy)\sin \theta \end{bmatrix}}$

In this particular case, we thus expect:

${\begin{alignedat}{2}\mathbf {v} _{2}&={\begin{bmatrix}(-wy+vz)\sin \phi \\v(vy+wz)(1-\cos \phi )+y\cos \phi \\w(vy+wz)(1-\cos \phi )+z\cos \phi \end{bmatrix}}\\&={\begin{bmatrix}(-\cos \alpha q\cos \alpha +-\sin \alpha q\sin \alpha )\sin \phi \\-\sin \alpha (-\sin \alpha q\cos \alpha +\cos \alpha q\sin \alpha )(1-\cos \phi )+q\cos \alpha \cos \phi \\\cos \alpha (-\sin \alpha q\cos \alpha +\cos \alpha q\sin \alpha )(1-\cos \phi )+q\sin \alpha \cos \phi \end{bmatrix}}\\&=q{\begin{bmatrix}-(\cos ^{2}\alpha +\sin ^{2}\alpha )\sin \phi \\\sin ^{2}\alpha (\cos \alpha -\cos \alpha )(1-\cos \phi )+\cos \alpha \cos \phi \\\cos ^{2}\alpha (-\sin \alpha +\sin \alpha )(1-\cos \phi )+\sin \alpha \cos \phi \end{bmatrix}}\\&=q{\begin{bmatrix}-\sin \phi \\\cos \alpha \cos \phi \\\sin \alpha \cos \phi \end{bmatrix}}\end{alignedat}}$

Note that we replace $\scriptstyle \phi$ by $\scriptstyle -\phi$ to force the same orientation convention in the definition of rotating Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{v}_1} and $\scriptstyle \mathbf {v} _{2}$:

${\begin{alignedat}{2}\mathbf {v} _{2}&=q{\begin{bmatrix}\sin \phi \\\cos \alpha \cos \phi \\\sin \alpha \cos \phi \end{bmatrix}}\end{alignedat}}$

The angle between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{v}_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{v}_2} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \alpha_r} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \cos \alpha_r & = \frac{\mathbf{v}_1 \cdot \mathbf{v}_2}{|\mathbf{v}_1| |\mathbf{v}_2|} \\ & = \frac{(q^2 \sin \phi \sin \phi)+(q^2\cos \phi \cos \alpha \cos\phi)+(0)}{(q) (q)} \\ & = \sin^2 \phi + \cos^2 \phi \cos \alpha \\ \alpha_r & = \cos^{-1}\left[ \cos^2 \phi \cos \alpha + \sin^2 \phi \right ] \end{alignat} }

Distance between two planes

The distance between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{v}_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{v}_2} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle d} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \sin \left( \frac{\alpha_r}{2} \right ) & = \frac{d/2}{q} \\ d & = 2 q \sin \left( \frac{\alpha_r}{2} \right ) \\ & = 2 q \sin \left( \frac{1}{2} \cos^{-1} \left[ \cos^2 \phi \cos \alpha + \sin^2 \phi \right ] \right ) \end{alignat} }

Alternatively:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} d^2 & = 2 q^2 - 2 q^2 \cos \alpha_r \\ d & = q \sqrt{ 2 \left ( 1 - \cos \alpha_r \right ) } \\ d & = q \sqrt{ 2 \left ( 1 - \cos \left( \cos^{-1} \left[ \cos^2 \phi \cos \alpha + \sin^2 \phi \right ] \right ) \right ) } \\ d & = q \sqrt{ 2 \left ( 1 - \cos^2 \phi \cos \alpha + \sin^2 \phi \right ) } \end{alignat} }

If the two vectors do not have equal length:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} d^2 & = q_1^2 + q_2^2 - 2 q_1q_2 \cos \alpha_r \\ d & = \sqrt{ q_1^2 + q_2^2 - 2 q_1q_2 \left( \cos^2 \phi \cos \alpha + \sin^2 \phi \right) }\\ \end{alignat} }

Generalized distance between two vectors

Imagine reciprocal-space scattering that is a ring; more specifically a pseudo-toroid with Gaussian-like decay. The intensity overall is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I = \exp \left [ -(q_{rx}^2 + q_{ry}^2)/(2 \sigma_q^2) \right ] \exp \left [ -q_{rz}^2/(2 \sigma_q^2) \right ] }

Where we use the subscript r to denote the reciprocal-space coordinate system. The plane of the detector (i.e. the Ewald plane) is denoted by d:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}_{d} = \begin{bmatrix} q_{dx} & q_{dy} & 0 \end{bmatrix} }

We set the symmetry axis in realspace (detector coordinate system) to be the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle x} -axis. The reciprocal-space is tilted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \chi_0} (about the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle y} -axis), before the 'powder' rotation about the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle x} -axis (where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \alpha} goes from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle -\pi} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle +\pi} ). Consider an initial vector:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}_{d0} = \begin{bmatrix} q \sin \phi & q \cos \phi & 0 \end{bmatrix} }

The 1st rotation (about Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle y} -axis by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \chi_0} ) involves:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathbf{v}_{d1} & = R_y(\chi_0) \mathbf{v}_{d0} \\ & = \begin{bmatrix} \cos \chi_0 & 0 & \sin \chi_0 \\ 0 & 1 & 0 \\ -\sin \chi_0 & 0 & \cos \chi_0 \\ \end{bmatrix} \begin{bmatrix} q \sin \phi \\ q \cos \phi \\ 0 \\ \end{bmatrix} \\ & = \begin{bmatrix} q \sin \phi \cos \chi_0 \\ q \cos \phi \\ - q \sin \phi \sin \chi_0 \\ \end{bmatrix} \end{alignat} }

The 2nd rotation (about the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle q_{dx}} -axis by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \alpha} ) occurs with respect to the vector:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathbf{n}_{d1} & = R_y(\chi_0) \mathbf{n}_{d0} \\ & = \begin{bmatrix} \cos \chi_0 & 0 & \sin \chi_0 \\ 0 & 1 & 0 \\ -\sin \chi_0 & 0 & \cos \chi_0 \\ \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 1 \\ \end{bmatrix} \\ & = \begin{bmatrix} \sin \chi_0 \\ 0 \\ \cos \chi_0 \\ \end{bmatrix} \end{alignat} }

So the second rotation yields:

${\begin{alignedat}{2}\mathbf {v} _{d2}&={\begin{bmatrix}u(ux+wz)(1-\cos \alpha )+x\cos \alpha +-wy\sin \alpha \\y\cos \alpha +(wx-uz)\sin \alpha \\w(ux+wz)(1-\cos \alpha )+z\cos \alpha +uy\sin \alpha \end{bmatrix}}\\&={\begin{bmatrix}\sin \chi _{0}(\sin \chi _{0}x+\cos \chi _{0}z)(1-\cos \alpha )+x\cos \alpha +-\cos \chi _{0}q\cos \phi \sin \alpha \\q\cos \phi \cos \alpha +(\cos \chi _{0}x-\sin \chi _{0}z)\sin \alpha \\\cos \chi _{0}(\sin \chi _{0}x+\cos \chi _{0}z)(1-\cos \alpha )+z\cos \alpha +\sin \chi _{0}q\cos \phi \sin \alpha \end{bmatrix}}\\\end{alignedat}}$