20 November 2016

The Ewald sphere

The Ewald sphere is a widely used concept, but one that is quite difficult to grasp in the beginning (at least it was for me, as well as for some of my colleagues.) It can be seen as a way of converting vectors between the "real" space, in which the experiment is performed, and "reciprocal" space.


In the Figure below, the incident beam is given by \(\mathbf{k}_i\) (making an angle \(\alpha_i\) with the horizontal), and the elastically scattered signal is described by \(\mathbf{k}_f\), which is equal in length to \(\mathbf{k}_i\) but can assume all orientations, parameterized by the in-plane angle \(\alpha_f\) and an out-of-plane angle \(\beta\) (not represented.)

Figure: reciprocal space diagram for a lamellar phase on a substrate. Note that the Ewald sphere (shown in red) crosses the z axis in two points: the origin (position of the primary beam) and the specular position. From an older paper of mine (the preprint is also available).
The locus described by the extremity of \(\mathbf{k}_f\) as its orientation varies is a sphere, that we can materialize in real space by a two dimensional detector, as shown by the image on the right in the Figure (contrary to the scattering diagram, the image is seen from the point of view of the incoming beam.) A pixel placed at angles \((\alpha_f,\beta)\) records the scattering with a \(\mathbf{k}_f\) described by the same angles.

The most important quantity in scattering experiments is the scattering vector \(\mathbf{q} = \mathbf{k}_f - \mathbf{k}_i\). As \(\mathbf{k}_f\) describes a sphere, so does \(\mathbf{q}\). There are however two important differences:
  • \(\mathbf{q}\) lives in "reciprocal space", where it probes the reciprocal lattice of the sample (if it is crystalline).
  • The modulus of \(\mathbf{q}\) is no longer constant: when \(\mathbf{k}_f\) aproaches \(\mathbf{k}_i\) (close to the primary beam) \(|\mathbf{q} |\) vanishes.
I suspect the Ewald sphere is hard to grasp intuitively for three reasons. I have already mentioned two of them above: the ambiguity of the construction between real and reciprocal space (which is in fact its most useful feature) and the fact that, although \(\mathbf{q}\) describes a sphere, its modulus is not constant, as the sphere is not centered at the origin of reciprocal space.

The third reason is that, while both the sphere and the reciprocal lattice of the crystal reside in the same "reciprocal space", they are attached to the scattering setup and to the sample, respectively. Thus, they do not behave in the same way under rotation.

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