How do we know that the x-ray pattern is in the reciprocal space? I wonder if any one can tell that why do we consider the x-ray pattern (for example, a x-ray pattern on a film for a crystal) in the reciprocal space?
(I don't want any explanation about the Ewald bubble and so on... those are according the fact that we accept the data are in the reciprocal space...
but I ask for a step behind that and that is why should the data be in the reciprocal space?!)
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I mean, why do we say that the spots in the crystallographic films are in the reciprocal space?!

(Image: crystallographic picture of a sample)
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I mean, why dont we analyze the incident waves in terms of lambda (2pi/k) and so relate the points in xrd data to the direct lattice?!
 A: After x-rays hit a substance they will be scattered in all directions; if the material is a crystal then you will obtain a diffraction pattern where each point is created by the constructive interference of the scattered rays.
The connection between the diffraction pattern and the reciprocal space is readily found:
take a crystal and consider an atom located at the origin and another one at another lattice point $ \mathbf{R}_n $. Consider radiation with wave vector $ \mathbf{k}$ in the direction $ \mathbf{n} $ ($\mathbf{k} =2 \pi\mathbf{n}/ \lambda $) hits these points, the atoms will scatter the incident light in every direction. Let's consider only the light scattered elastically in a fixed direction, say along $ \mathbf{n'} $, with wave vector $\mathbf{k'} =2 \pi\mathbf{n'}/ \lambda $. The path difference of the rays scattered by these two different atoms will be: $ \mathbf{R}_n \cdot (\mathbf{n}-\mathbf{n'}) $. For having a point of the diffraction pattern the interference between the scattered rays should be a maximum; i.e. $$ \mathbf{R}_n \cdot (\mathbf{n}-\mathbf{n'}) = m \lambda  $$
with $ m = 1,2,... $ Multiplying by $2 \pi/ \lambda $ you get: 
$$ \mathbf{R}_n \cdot (\mathbf{k}-\mathbf{k'}) = 2 \pi m $$ and thus for having a spot on the diffraction pattern,  $ \mathbf{k}-\mathbf{k'} $ must be a vector of the reciprocal lattice; since for every vector $ \mathbf{g} $ in the reciprocal lattice we know that: 
$$ e^{i \mathbf{R}_n \cdot \mathbf{g}}=1 $$
This gives a basic relation between diffraction pattern and reciprocal space.
