Diffraction and $k$-space

Question

Regarding diffraction I am a little bit lost reading about reciprocal space and the space of $k$'s. As I understand it the Fourier relationship between a wavepacket $\Psi(\vec r,t)$ and the complex weighting factors of each constituent plane wave $A(\vec k)$ is given by: \begin{equation} \Psi(\vec r,t)=\frac{1}{\sqrt{2\pi}}\int ^{\infty}_{-\infty}A(\vec k)e^{i(\vec k\vec r-\omega t)}d\vec k \end{equation} demonstrating a sort of linear superposition of reflected plane waves from a diffraction grating (or crystal lattice).Further by Parseval's theorem the intensity of this reflected packet is given by: \begin{equation} \int^{\infty}_{-\infty}\big|\Psi(\vec r,t)\big|^2d\vec r=\int^{\infty}_{-\infty}\big|A(\vec k)\big|^2d\vec k\end{equation}

However I am not really sure how this relates to the other sort of understanding of $k$ space. That is to say the space that can give us meaningful information about crystal lattices and unit cells. Are they the same spaces?

Would this mean therefore that the intensity/position of the diffraction spots can be related to the structure of the solid's lattice. If so how can we understand distributions in terms of the Fourier relationship above?

I understand there have been several questions so far regarding the reciprocal k-space however so far I have not found anything that helps me particularly grasp this aspect of diffraction.

As you can see I am quiet confused in this matter and would greatly appreciate some help!

Answer 1

I think I understand it a little better now so I decided to post my thought and obviously won't accept my own answer.

My understanding is that A(k) is the space of wave vectors, however we know these only differ by direction not magnitude for the diffracted wavepacket (Rayleigh scattering). Therefore the function A(k) indicates the "spread" of the directionality of the wave vectors. So a narrow distribution of A(k) indicates a larger intensity $|\Psi(x)|^2$ and therefore a wider diffraction grating (or indeed lattice spacing).

The mathematical reasoning behind this is simple: $\Psi(x)$ and A(k) are Fourier conjugates.

The physical origin is where I am still struggling. My thoughts are that the wider slits scatter the waves less so more of the wavefront is unaffected by the lattice leading to an increase in the intensity $|\Psi(x)|^2. The reduction in scattering then means that more waves have the same direction for small perturbations around some central wave vector.

Is this the correct idea? Many thanks.