# Diffraction condition and the Fourier transform

1. Homework Statement
My book uses the following equation to derive the diffraction condition for electromagnetic waves scattering in a crystal lattice:

$$F= \int dV n(\mathbf{r}) \exp \left[i\Delta\mathbf{k}\cdot \mathbf{r} \right]$$

F is the scattering amplitude and n is the electron density. I just don't understand where that comes from. How does the Fourier transform relate to the diffraction? I have studied Fourier expansions in calculus and I understand how (virtually) every periodic function can be represented as a Fourier series.
2. Homework Equations

3. The Attempt at a Solution

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olgranpappy
Homework Helper
so, for example, does the following equation for elastic scattering of a wave off a collection of atoms located at R_1, R_2, R_3, etc, make more sense to you?
... the scattering amplitude is roughly
$$F\sim \sum_{i} e^{i(\vec k_0 - \vec k_f)\cdot\vec R_i}$$

can you derive the above equation? can you see how it relates to the equation you have written down?

so, for example, does the following equation for elastic scattering of a wave off a collection of atoms located at R_1, R_2, R_3, etc, make more sense to you?
... the scattering amplitude is roughly
$$F\sim \sum_{i} e^{i(\vec k_0 - \vec k_f)\cdot\vec R_i}$$

can you derive the above equation? can you see how it relates to the equation you have written down?
Well, I see how my equation is just the integral version of your equation, but no your equation does not make sense to me

olgranpappy
Homework Helper
okay. so. let's see... so, if you have an incoming plane wave $$e^{i\vec k_0\cdot\vec r}$$and it scatters off some point scatterer at a position $$R_1$$, the amplitude of the wave at the position of
the scatterer, but before it scatters is
$$e^{i\vec k_0 \cdot \vec R_1}$$
the wave scatters off the point scatterer and is thus now given by the original amplitude times a spherical wave of the form $$f e^{ik|\vec r-\vec R_1|}/|\vec r - \vec R_1|$$, where for simplicity f is just some number indep of scattering angle (cf. something like Jackson Third Ed. Eq. 10.2, im writing the same thing for a scalar theory with a much simpler scattering f, and also I'm keeping around the initial phase factor for latter, cf. also Jackson's section on scattering by a collection of scatterers, like Eq. 10.19).

So, the amplitude is
$$e^{i\vec k_0\cdot \vec R_1} \frac{fe^{ik|\vec r - \vec R_1|}}{|\vec r-\vec R_1|}$$

So, that's for a scatter at R_1

Next, add the amplitude from another scatterer at R_2, and another at R_3, etc.

Next take the limit where r >> R for all the R's and you can use
$$|\vec r-\vec R| \approx r - \vec R\cdot\frac{\vec r}{r}$$
('member this from scattering theory, maybe in quantum... you know what to do)

and get my previous result up to the factors I left off and compensated for with the $$\sim$$ symbol which is certainly the lazy man's good friend.