Ewald sphere

The Ewald sphere is the surface, in reciprocal-space, that all experimentally-observed scattering arises from. (Strictly, only the elastic scattering comes from the Ewald sphere; inelastic scattering is so-called 'off-shell'.) A peak observed on the detector indicates that a reciprocal-space peak is intersecting with the Ewald sphere.

Geometry

Definitions

Consider reciprocal-space in the incident beam coordinate system: 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 (q_x,q_y,q_z)} . The incident beam is the vector $\mathbf {k_{i}} =\langle 0,-k,0\rangle$, where:

$k={\frac {2\pi }{\lambda }}$

where $\lambda$ is, of course, the wavelength of the incident beam. An inelastic scattering event has momentum vector, and resultant momentum transfer, $\mathbf {q}$, of:

${\begin{alignedat}{2}\mathbf {k_{f}} &=\langle 0,-k\cos(2\theta _{s}),+k\sin(2\theta _{s})\rangle \\\mathbf {q} &=\mathbf {k_{f}} -\mathbf {k_{i}} \\&=\langle 0,-k\cos(2\theta _{s}),+k\sin(2\theta _{s})\rangle -\langle 0,-k,0\rangle \\&=\langle 0,k(1-\cos(2\theta _{s})),+k\sin(2\theta _{s})\rangle \\&=\langle 0,2k\sin ^{2}(\theta _{s})),2k\sin(\theta _{s})\cos(\theta _{s})\rangle \\q&=|\mathbf {q} |\\&={\sqrt {[2k\sin ^{2}(\theta _{s}))]^{2}+[2k\sin(\theta _{s})\cos(\theta _{s})]^{2}}}\\&={\sqrt {4k^{2}[\sin ^{4}(\theta _{s})+\sin ^{2}(\theta _{s})\cos ^{2}(\theta _{s})]}}\\&=2k{\sqrt {\sin ^{4}(\theta _{s})+\sin ^{2}(\theta _{s})\cos ^{2}(\theta _{s})}}\\&=2k{\sqrt {\sin ^{2}(\theta _{s})}}\\&=2k\sin(\theta _{s})\\&={\frac {4\pi }{\lambda }}\sin(\theta _{s})\\\end{alignedat}}$

where $2\theta _{s}$ is the full scattering angle. The Ewald sphere is centered about the point $(0,-k,0)$ and thus has the equation:

$q_{x}^{2}+(q_{y}-k)^{2}+q_{z}^{2}-k^{2}=0$

TSAXS

In conventional SAXS, the signal of interest is isotropic: i.e. we only care about $|\mathbf {q} |=q$, and not the individual (directional) components 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 (q_x,q_y,q_z)} . In such a case:

GISAXS

Now let us assume that the incident beam strikes a thin film mounted to a substrate. The incident beam is in the grazing-incidence geometry (e.g. GISAXS, and we denote the angle between the incident beam and the film surface as $\theta _{i}$. The reciprocal-space of the sample is thus rotated 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 \theta_i} with respect to the beam reciprocal-space coordinates. We denote the sample's reciprocal coordinate system by uppercase, 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 (Q_x,Q_y,Q_z)} , and note that the equation of the Ewald sphere becomes (the center of the sphere is at 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 (0,k \cos\theta_i, k \sin\theta_i)} ):

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} & Q_x^2 + \left(Q_y-k \cos\theta_i \right)^2 + \left(Q_z-k \sin\theta_i\right)^2 - k^2 = 0 \\ & Q_y = +k\cos\theta_i + \sqrt{ k^2 - Q_x^2 - \left(Q_z-k\sin\theta_i\right)^2 } \end{alignat} }