K vector in first brillouin zone

[Sophocles]

Hey guys,

I just realized that there is a gap somewhere in my understanding of K vectors and reciprocal space.
My question is how can we talk about K vectors "living" in the first Brillouin Zone, when these vectors cannot be expressed on the vector form of reciprocal space ( r*=ha*kb*+lc* , where h,k,l integers)? First Brillouin Zone as I understand it, extends until half the length of the first neighboring lattice point of the reciprocal space.

 

[DrDu]

If I interprete your question correctly, you seem to be confusing the reciprocal space and reciprocal lattice.
For example the electron density or wavefunction of some crystal is defined for any point in space and likewise is its Fourier transform defined for any k vector with real components in reciprocal space.
In direct space any location in the crystal can be described as a lattice vector plus some vector lying in the unit cell centered at the origin.
Likewise, any point in reciprocal space can be decomposed into a point inside the first brillouin zone and a reciprocal lattice vector.

 

[Sophocles]

Ok. So if I get it correctly, there are k vectors who belong to the reciprocal lattice, having periodicity the same as the real lattice (and the ones responsible for diffraction phenomena) and then there is K space for all the rest of the waves that can travel through the medium?

 

[DrDu]

Maybe it is helpful to have some example, where splitting a general wavevector into a vector k in the first BZ and a reciprocal lattice vector is useful:
http://en.wikipedia.org/wiki/Bloch_wave

 

[Sophocles]

Thank you very much DrDu. After all, I finally got it!