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 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 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} , such that its normal 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 \mathbf{n}_2 = \begin{bmatrix} 0 & - \sin \alpha \ & \cos \alpha \end{bmatrix} }

We are interested in quantities that are a particular distance (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} ) from the origin. Imagine a vector of 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 \scriptstyle q} lying in plane 1, rotated 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 z} 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 \phi} (i.e. the angular distance from 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 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 \phi} ):

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}_1 = \begin{bmatrix} q \sin \phi & q \cos \phi \ & 0 \end{bmatrix} }

The second 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{v}_2} )is lying in plane 2. We call 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} 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} . The specified geometry uniquely defines 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} in terms of the angle between the planes (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} ) and the amount of rotation of the vectors (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 \phi} ) within their respective planes. In particular, 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} can be thought of as 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}_{2b}} rotated 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 \mathbf{n}_2} 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 \phi} , 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 \mathbf{v}_2b} is the vector in plane 2 without any 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 \phi} rotation (i.e. lying 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 yz} plane):

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

In general, rotation of a 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{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):

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}_{\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:

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}_{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{alignat} }

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} }