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.

Mathematics

In TSAXS of an isotropic sample, we only probe the magnitude (not direction) of the momentum transfer:

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 = \frac{4 \pi}{\lambda} \sin(\theta) }

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 2\theta} is the full scattering angle. In GISAXS, we must take into account the vector 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 \begin{alignat}{2} q_x & = \frac{2 \pi}{\lambda} \cos(\alpha_f)\sin(2\theta_f) \\ q_y & = \frac{2 \pi}{\lambda} \left[ \cos(\alpha_f)\cos(2\theta_f) - \cos(\alpha_f) \right ]\\ q_x & = \frac{2 \pi}{\lambda} \left[ \sin(\alpha_f) + \sin(\alpha_f) \right ] \end{alignat} }

Derivation

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 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{k_i} = \langle 0,-k,0 \rangle} , 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 k = \frac{2 \pi}{\lambda}}

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 \lambda} is, of course, the wavelength of the incident beam. An elastic scattering event has an outgoing momentum (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{k_f}} ) of the same magnitude as the incident radiation (i.e. 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{k_i}| = |\mathbf{k_f}| = k} ). Consider a momentum vector, and resultant momentum transfer, 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{q}} , of:

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{k_f} & = \begin{bmatrix} 0 \\ -k \cos(2 \theta_s) \\ +k \sin(2 \theta_s) \rangle \\ \end{bmatrix} \\ \mathbf{q} & = \mathbf{k_f} - \mathbf{k_i} \\ & = \begin{bmatrix} 0 \\ -k \cos(2 \theta_s) \\ +k \sin(2 \theta_s) \\ \end{bmatrix} - \begin{bmatrix} 0 \\ -k \\ 0 \\ \end{bmatrix} \\ & = \begin{bmatrix} 0 \\ k( 1 -\cos(2 \theta_s) ) \\ +k \sin(2 \theta_s) \rangle \\ \end{bmatrix} \\ & = \begin{bmatrix} 0 \\ 2 k \sin^2(\theta_s) \\ 2 k \sin(\theta_s) \cos(\theta_s) \rangle \\ \end{bmatrix} \\ \end{alignat} }

The magnitude of the momentum transfer is thus:

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 & = | \mathbf{q} | \\ & = \sqrt{ [2 k \sin^2(\theta_s))]^2 + [ 2 k \sin(\theta_s)\cos(\theta_s)]^2 } \\ & = \sqrt{ 4k^2 [ \sin^4(\theta_s) + \sin^2(\theta_s)\cos^2(\theta_s)] } \\ & = 2 k \sqrt{ \sin^4(\theta_s) + \sin^2(\theta_s)\cos^2(\theta_s) } \\ & = 2 k \sqrt{ \sin^2(\theta_s)} \\ & = 2 k \sin(\theta_s) \\ & = \frac{4 \pi}{\lambda} \sin(\theta_s) \\ \end{alignat} }

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 2 \theta_s} is the full scattering angle. The Ewald sphere is centered about the point 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,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 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{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 we use the form of q derived above:

$q={\frac {4\pi }{\lambda }}\sin(\theta _{s})$

In the more general case of probing an anisotropic material (e.g. CD-SAXS), one must take into account the full q-vector, and in particular the relative orientation of the incident beam and the sample: i.e. the relative orientation of the Ewald sphere and the reciprocal-space.

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 convert to the sample's reciprocal coordinate space; still denoted 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 (Q_x,Q_y,Q_z)} . 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 \langle 0,k \cos\theta_i, k \sin\theta_i \rangle} ):

${\begin{alignedat}{2}&q_{x}^{2}+\left(q_{y}-k\cos \theta _{i}\right)^{2}+\left(q_{z}-k\sin \theta _{i}\right)^{2}-k^{2}=0\\&q_{y}=+k\cos \theta _{i}+{\sqrt {k^{2}-q_{x}^{2}-\left(q_{z}-k\sin \theta _{i}\right)^{2}}}\end{alignedat}}$

Reflectivity

Consider for a moment that the q-vector is confined to 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 (q_y,q_z)} 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 \begin{alignat}{2} \mathbf{q} = \begin{bmatrix} 0 \\ - q \sin(\alpha_f - \alpha_i) \\ + q \cos(\alpha_f - \alpha_i) \\ \end{bmatrix} \end{alignat} }

Obviously the specular condition is when 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 \alpha_f=\alpha_i} , in which case one obtains:

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{q} = \begin{bmatrix} 0 \\ 0 \\ + q \\ \end{bmatrix} \end{alignat} }

That is, reflectivity is inherently only probing the out-of-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 q_z} ) component:

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_z = \frac{4 \pi}{\lambda} \sin \alpha_i }

Out-of-plane scattering only

Simplifying the above expression yields:

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{q} = \begin{bmatrix} 0 \\ - q \sin(\alpha_f - \alpha_i) \\ + q \cos(\alpha_f - \alpha_i) \\ \end{bmatrix} \end{alignat} }

Literature

Conceptual Understanding of Ewald sphere

• Simulating X-ray diffraction of textured films D. W. Breiby, O. Bunk, J. W. Andreasen, H. T. Lemke and M. M. Nielsen J. Appl. Cryst. 2008, 41, 262-271. doi: 10.1107/S0021889808001064
• Quantification of Thin Film Crystallographic Orientation Using X-ray Diffraction with an Area Detector Jessy L. Baker, Leslie H. Jimison, Stefan Mannsfeld, Steven Volkman, Shong Yin, Vivek Subramanian, Alberto Salleo, A. Paul Alivisatos and Michael F. Toney Langmuir 2010, 26 (11), 9146-9151. doi: 10.1021/la904840q

Equations of GISAXS Geometry

• gisaxs.de: Brief introduction to the theory of GISAXS.
• IsGISAXS Manual: Provides a complete description (including equations) of the GISAXS theory required to model data.
• Probing surface and interface morphology with Grazing Incidence Small Angle X-Ray Scattering Gilles Renaud, Rémi Lazzari, Frédéric Leroy, Surface Science Reports 2009, 64 (8), 255-380. doi: 10.1016/j.surfrep.2009.07.002
• A Basic Introduction to Grazing Incidence Small-Angle X-Ray Scattering P. Müller-Buschbaum DOI 10.1007/978-3-540-95968-7 (Chapter 3 in Applications of Synchrotron Light to Scattering and Diffraction in Materials and Life Sciences)