Why do beams interact in scattering?

[aaaa202]

Our condition for scattering is based on the idea that the amplitude of the outcoming wave is maximal, when all atoms contribute to a constructive interference. By using the attached drawing a simple relationship between the phase change of the two beams are derived. But my, maybe stupid, question is: Why do these beams interact? After all they travel through quite different points in space?
So I thought well, actually to draw a beam is misleading, because what happens in x-ray diffraction is that plane waves hit the crystal(I would think so at least). But still based on this, the wavefront is spread out in space, so why is it that everywhere on the wavefront, where constructive interference occurs, should somehow contribute to a total amplification of the wave?
I am probably misunderstanding some things here. Maybe the relative size of the wave compared to the crystal has something to say...

 

[Claude Bile]

I think you misunderstand the diagram. It describes one plane wave scattering into another plane wave - of course, plane waves extend throughout all space and cannot be properly described as beams.

Here is what the diagram is depicting;

- Take a single volume element dV.
- Define the plane wave spectrum A(k) incident upon dV.
- The scattered plane wave spectrum is A'(k').
- The link between A and A' is provided by the scattering matrix (typically denoted S).
- The total scattered field is done by summing over all dV.

This problem is usually quite intractable, and so certain approximations (such as the Born approximation) are made to simplify the form of A(k).

Claude.

 

[aaaa202]

I see I misinterpreted the diagram now. But I still need some help. On the modified drawing below I have drawn some dots to indicate different points on the wave front. If we consider the red one, this point is not retarded by a phase of 2∏k, so does this point on the wave front not interfere destructively with the green points, which are in phase?
That seems weird because what is then the resulting plane wave? Maybe I am to think that there are only atoms spaced lrl apart, so that nothing happens on the wave front in between O and the point indicated by r. Is that the right understanding?

 

[Claude Bile]

There are some gaps in your understanding (Scattering problems are not easy, so I can sympathise).

The key point is that points do not interfere, plane wave components interfere.

Thus it is unhelpful to describe the relative phase of different points in real space, because the relative phase depends on the incident and exitant plane waves - of which there are many!

I hope this is of some help.

Claude.