Diffraction condition and the Fourier transform

[ehrenfest]

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.

Homework Equations

The Attempt at a Solution

 

[olgranpappy]

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?

 

[ehrenfest]

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]

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, I am 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.