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Final ansatz
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Hi everyone,
So, I've been trying to derive a general form for momentum Fourier transform of the polarizability [itex]\chi(\mathbf{r}t,\mathbf{r}'t')[/itex] for a crystalline material, i.e. a material with a Bravais lattice. Since the material is not translationally invariant, it won't be enough with just one momentum Fourier transform, and in general I should transform in both [itex]\mathbf{r}\rightarrow\mathbf{q}[/itex] and [itex]\mathbf{r}'\rightarrow\mathbf{q}'[/itex]. Due to crystallinity, however, I should be able to end up with a polarizability tensor [itex]\chi_{\mathbf{G}\mathbf{G}'}(\mathbf{q})[/itex] (where [itex]\mathbf{G}[/itex] and [itex]\mathbf{G}'[/itex] are reciprocal lattice vectors). I want to find the form of this tensor as an average of a commutator, i.e. I want to find something like (ignoring time-dependences):
[tex] \chi_{\mathbf{G}\mathbf{G}'}(\mathbf{q}) \sim \big\langle\big[ \rho(\ldots_{\mathbf{q}\mathbf{G}\mathbf{G}'}), \rho (\ldots_{\mathbf{q}\mathbf{G}\mathbf{G}'})\big]\big\rangle ,[/tex] Where I've written '[itex]\ldots_{\mathbf{q}\mathbf{G}\mathbf{G}'}[/itex]', when I'm unsure of what the facts should be (although I suspect something like [itex]\mathbf{q}+\mathbf{G}[/itex] for the first element and [itex]-\mathbf{q}-\mathbf{G}'[/itex] for the second).
This kind a derivation has been done in the momentum and frequency domain before (Refs. [1] and [2], see below), specifically in 1962 and 1963 by Adler and Wiser seperately, but the derivation starts from the self-consistent field method - of which I'm not a big fan. I'd much rather be able to do the derivation starting from Kubo's formula. I've tried, but I seem not to be able to finish the derivation. This is where I hope that someone on this board (if, amazingly, someone should have red this far in spite of everything!) might be able to help me!
I've thought about a way of presenting the problem shortly, but I'm afraid I've failed. Consequently, I've appended a 2 page document of my current derivation (sadly, I cannot append it to this forum, since I don't have 10 posts yet - I'm somewhat afraid that it also won't be OK to link, but ...): www.dropbox.com/s/57903d3q7rm0vrj/LocaFieldEffects_Polarizability_PF.pdf.
It's very understandable if no one should decide to read it - but I'll cross my fingers and hope someone will :). If you are in a hurry, you can perhaps jump immediately to Eq. (9) in that document, and see if you have any brilliant ideas for me.
In any case, whether you have any ideas or not, thanks for reading this far :)!
[1] http://[/U][U]link.aps.org/doi/10.1103/PhysRev.129.62[/U]
[2] http://[/U][U]link.aps.org/doi/10.1103/PhysRev.126.413[/U]
So, I've been trying to derive a general form for momentum Fourier transform of the polarizability [itex]\chi(\mathbf{r}t,\mathbf{r}'t')[/itex] for a crystalline material, i.e. a material with a Bravais lattice. Since the material is not translationally invariant, it won't be enough with just one momentum Fourier transform, and in general I should transform in both [itex]\mathbf{r}\rightarrow\mathbf{q}[/itex] and [itex]\mathbf{r}'\rightarrow\mathbf{q}'[/itex]. Due to crystallinity, however, I should be able to end up with a polarizability tensor [itex]\chi_{\mathbf{G}\mathbf{G}'}(\mathbf{q})[/itex] (where [itex]\mathbf{G}[/itex] and [itex]\mathbf{G}'[/itex] are reciprocal lattice vectors). I want to find the form of this tensor as an average of a commutator, i.e. I want to find something like (ignoring time-dependences):
[tex] \chi_{\mathbf{G}\mathbf{G}'}(\mathbf{q}) \sim \big\langle\big[ \rho(\ldots_{\mathbf{q}\mathbf{G}\mathbf{G}'}), \rho (\ldots_{\mathbf{q}\mathbf{G}\mathbf{G}'})\big]\big\rangle ,[/tex] Where I've written '[itex]\ldots_{\mathbf{q}\mathbf{G}\mathbf{G}'}[/itex]', when I'm unsure of what the facts should be (although I suspect something like [itex]\mathbf{q}+\mathbf{G}[/itex] for the first element and [itex]-\mathbf{q}-\mathbf{G}'[/itex] for the second).
This kind a derivation has been done in the momentum and frequency domain before (Refs. [1] and [2], see below), specifically in 1962 and 1963 by Adler and Wiser seperately, but the derivation starts from the self-consistent field method - of which I'm not a big fan. I'd much rather be able to do the derivation starting from Kubo's formula. I've tried, but I seem not to be able to finish the derivation. This is where I hope that someone on this board (if, amazingly, someone should have red this far in spite of everything!) might be able to help me!
I've thought about a way of presenting the problem shortly, but I'm afraid I've failed. Consequently, I've appended a 2 page document of my current derivation (sadly, I cannot append it to this forum, since I don't have 10 posts yet - I'm somewhat afraid that it also won't be OK to link, but ...): www.dropbox.com/s/57903d3q7rm0vrj/LocaFieldEffects_Polarizability_PF.pdf.
It's very understandable if no one should decide to read it - but I'll cross my fingers and hope someone will :). If you are in a hurry, you can perhaps jump immediately to Eq. (9) in that document, and see if you have any brilliant ideas for me.
In any case, whether you have any ideas or not, thanks for reading this far :)!
[1] http://[/U][U]link.aps.org/doi/10.1103/PhysRev.129.62[/U]
[2] http://[/U][U]link.aps.org/doi/10.1103/PhysRev.126.413[/U]
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