I read from book that:
the macroscopic dielectric function only depends on the position difference, the integral relation between E and ε in the real space can be transfered to momentum space simply as
E(q,ω) = ε−1mac(q,ω)Eext(q,ω).
Can any one give me some clear explanation why the properties like dielectric function depends on many points in real space can be presented by only few high-symmetric points in the momentum space?
And why the bigger the supercell i use, the fewer points needed in the reciprocal space?
Thanks.
Hi,
the electrons "live" in the reciprocal space. This means that if we want to plot their wavefunction in real space - this is difficult. If we do it in reciprocal space - this is simply a grid. Mind that the reciprocal lattice has the units of inverse angstroms which is the units of wavevector which describes the electron. In a crystal reciprocal lattice also has periodicity and symmetry. So, if we describe only a Brillouin zone - we describe all the points in the space. The relation b=2pi/a between reciprocal and direct lattice vectors gives that bigger the real cell - smaller the corresponding reciprocal cell, so less points are needed to sample it with the same accuracy.
Can any one give me some clear explanation why the properties like dielectric function depends on many points in real space can be presented by only few high-symmetric points in the momentum space?
You switch from the real to the reciprocal space by Fourier transform (and the reverse by inverse fourier transform) so the space where you describe the "behavior" doesn't really matter i would say (one will correct me if I'm wrong) as you can get the behavior in the other space simply by normal or reverse Fourier transform.
About your specific question, it is related to the so called invariance by symmetry.
For example, considering a simple school case : the diffusion of a specie A in a simple cubic crystal. the diffusion within the [-100], [010], [0-10], [001] and , [00-1] directions will be exactly the same than the diffusion within the [100] so you will only study the diffusion within [100] and say that in the other direction it's the same : due to the invariance by rotation.
Now, if we consider a case closer to our preoccupations, when we perform a simulation we need to provide a grid of points (generally in the reciprocal space) to calculate or integer the quantities we want because it is computationally not possible to integer on the entire space (infinity of points !!). And it's appeared sometimes that the systems we studying have some symmetries, so it is not necessary to calculate the quantities over certain points of the grid because they have the same value and behavior than another points. Then the grid are simplified by symmetry, which lead to less points so less calculations to perform (which decrease the time needed to do the simulation). We rebuild the behavior of the quantity by applying the symmetry operations which we used to simply the grid.
to resume :
If P1 is the symmetric of P2, We just calculate F(P1) and we know that F(P2)= F(P1).
For the question : Why the bigger the supercell i use, the fewer points needed in the reciprocal space ?
The idea is more or less expose by Aram,reciprocal space is a Fourier transform of the real space and it appears that to a distance d in a real space correspond 2pi/d.
Then bigger the real space, smaller the reciprocal one is (sounds like Yoda)
BUT the grid does not give you the behavior at each point of the space (reciprocal or real) actually you can't know the behavior at each point of the space, but you assume that local behavior around a point of the grid is the same : your space is divide into region (number of part = number of grid points of course) each centered on a point of the grid. The behavior in each region is considered to be the same than the behavior calculated at the point of the region. (I think, I don't know if it's that or if you do an interpolation actually, ones we correct me for that).
best wishes,
Matth'
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