Scattering on crystals beyond Ewald and Bragg

So I whenever we consider the diffraction and scatting using crystals, powders, basically anything crystalline in the lab using X-rays we get a bunch of theories and models. The most simple one would be Bragg's law. And although Bragg's law tells you something about scattering it still does not contain information like intensities. I tend to tell my students in the lab that they can imagine the different miller planes and in order to get a relation between the incoming wave, the scattering angles and the detector position with each lattice plane, the angles have to be changed and therefore each angle represents a lattice plane. That helps them to understand the origin of the various reflections in their powder pattern and also shows why it's not a spectrum.

But this is only a scalar equation and in order to describe it a bit better we should introduce a wave-vector and describe the scattering using vectors. This leads to the relationship: $$\vec{G} = \vec{k} - \vec{k_s}$$ Where $$\vec{G}$$ is perpendicular to {hkl} and the two other vectors denote the incoming and scattered wave-vector.

A graphical representation is the Ewald-construction. Where we turn the crystal around the origin of the real-space and get a rotation of the reciprocal lattice around the reciprocal origin until one reciprocal lattice point intersects the Ewald-sphere.

What I was looking for then was how this can be presented within the regular drawing you get for Bragg's law. And I found an interesting representation in Robert Dinnerbier's new book Rietveld Refinement

Unfortunately there is no real explanation to the picture. But usually you would think, using Bragg's law that we are scattering on the lattics points that cross the black bars at the bottom. That is how you usually draw it. But he highlighted the (130) reflection and if you take the blue lattice point as origin of the reciprocal lattice you will have to draw it 'backwards', so a scattering vector that goes into the center of the Ewald sphere and then to an intersection of a reciprocal lattice point (red) with the sphere. At least I guess so. That would be how I would explain the picture but I might be wrong here.

Now this seems like a good represenation to me, we get the dynamic or kinematic effect of turning the crystal into different positions with the stattic picture of Bragg's law.

But then the authors writes 'although this isn't a good picture of what is happening physically [...]'. This left me wondering, what does he mean with that? What would be a better representation here? I was reading a bit into Brillouin zones and there seem to be relations to scattering as well. Then of course there is the physical point of Fourier-transformations. Perhaps this is what he means with 'physically'. But I was heading more for a geometrical perspective. It's been a while since I visited my last lecture on scattering so I won't bother with the why or how here but my question is more on the relationships.

So leading to my question. Does the author here indicate that perhaps those graphical represenations give a wrong impression here? Because it should basically be that per angle we get a different Miller index in a relationship where the scattered beam directly hits the detector.

• I would guess that he is saying that the graphical tool doesn't help you understand why diffraction occurs physically, but only serves as a useful way to remember when it occurs. – KF Gauss Apr 3 at 14:50