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: $(q_{x},q_{y},q_{z})$. 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 elastic 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 $(q_{x},q_{y},q_{z})$. 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 $\theta _{i}$ with respect to the beam reciprocal-space coordinates. We denote the sample's reciprocal coordinate system by uppercase, $(Q_{x},Q_{y},Q_{z})$, and note that the equation of the Ewald sphere becomes (the center of the sphere is at $(0,k\cos \theta _{i},k\sin \theta _{i})$):

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