Help understanding reciprocal space

phasor

Hello,

I am having a hard time understanding the concept of the reciprocal space. Here is my general understanding of it so far: the reciprocal space contains all of the points that light could be diffracted to from the real space. But I don't understand why this is the case from the mathematical definition of the reciprocal space. Why are all of the vectors in reciprocal space normal to planes in the real space? I guess I'm just not making the connection between the two spaces very well.

hokhani

Reciprocal space is the collections of those electron's wave vectors in crystals by which
electrons'wave functions have the periodicity of the crystal.
the electron's wave function is:
[itex]\psi[/itex](r)=exp(ik.r)
Now we want to have
[itex]\psi[/itex](r+R)=[itex]\psi[/itex](r)
that R is a lattice vector
So we should have exp(ik.R)=1
so all of k that satisfy this condition are members of reciprocal space and are showed by
K.
Do you need more explain?

marcusl

Reciprocal space is the spatial frequency dual of physical space. If you Fourier transform a waveform in time, you visualize it's spectrum in frequency space ω. If you FT a spatial arrangement of atoms, you visualize its 3D "spectrum" in spatial frequency space k. You are, in essence, finding the set of spatial frequency variations (in various directions) that characterize the atomic lattice. You can see why that is useful in understanding the directional properties of scattering.