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.
Although the Ewald sphere may seem like an abstract concept, it is simply the geometric representation of the scattering equation (i.e. it is the surface that satisfies the scattering condition).
Contents
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_i) \right ]\\ q_z & = \frac{2 \pi}{\lambda} \left[ \sin(\alpha_f) + \sin(\alpha_i) \right ] \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 \alpha_i} is the incident angle (the grazing angle of the beam with respect to the substrate 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 \alpha_f} is the out-of-plane scattering angle (with respect to the substrate plane), 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 2\theta_f} is the full in-plane scattering angle.
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:
- 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^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:
- 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_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 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} . 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} ):
- 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} }
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
TBD
General form
TBD
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)
