584 lines
16 KiB
Plaintext
584 lines
16 KiB
Plaintext
#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\pdf_author "Marek Nečada"
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\end_header
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\begin_body
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\begin_layout Section
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Infinite periodic systems
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\begin_inset FormulaMacro
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\newcommand{\dlv}{\vect a}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\rlv}{\vect b}
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\end_inset
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\end_layout
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\begin_layout Standard
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Although large finite systems are where MSTMM excels the most, there are
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several reasons that makes its extension to infinite lattices (where periodic
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boundary conditions might be applied) desirable as well.
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Other methods might be already fast enough, but MSTMM will be faster in
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most cases in which there is enough spacing between the neighboring particles.
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MSTMM works well with any space group symmetry the system might have (as
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opposed to, for example, FDTD with cubic mesh applied to a honeycomb lattice),
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which makes e.g.
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application of group theory in mode analysis quite easy.
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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Topology anoyne?
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\end_layout
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\end_inset
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And finally, having a method that handles well both infinite and large
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finite system gives a possibility to study finite-size effects in periodic
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scatterer arrays.
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\end_layout
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\begin_layout Subsection
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Notation
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\end_layout
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\begin_layout Standard
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TODO Fourier transforms, Delta comb, lattice bases etc.
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\end_layout
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\begin_layout Subsection
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Formulation of the problem
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\end_layout
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\begin_layout Standard
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Let us have a linear system of compact EM scatterers on a homogeneous background
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as in Section
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "subsec:Multiple-scattering"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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, but this time, the system shall be periodic: let there be a
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\begin_inset Formula $d$
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\end_inset
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-dimensional (
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\begin_inset Formula $d$
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\end_inset
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can be 1, 2 or 3) lattice embedded into the three-dimensional real space,
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with lattice vectors
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\begin_inset Formula $\left\{ \dlv_{i}\right\} _{i=1}^{d}$
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\end_inset
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, and let the lattice points be labeled with an
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\begin_inset Formula $d$
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\end_inset
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-dimensional integar multiindex
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\begin_inset Formula $\vect n\in\ints^{d}$
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\end_inset
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, so the lattice points have cartesian coordinates
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\begin_inset Formula $\vect R_{\vect n}=\sum_{i=1}^{d}n_{i}\vect a_{i}$
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\end_inset
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.
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There can be several scatterers per unit cell with indices
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\begin_inset Formula $\alpha$
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\end_inset
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from set
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\begin_inset Formula $\mathcal{P}_{1}$
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\end_inset
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and (relative) positions inside the unit cell
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\begin_inset Formula $\vect r_{\alpha}$
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\end_inset
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; any particle of the periodic system can thus be labeled by a multiindex
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from
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\begin_inset Formula $\mathcal{P}=\ints^{d}\times\mathcal{P}_{1}$
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\end_inset
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.
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The scatterers are located at positions
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\begin_inset Formula $\vect r_{\vect n,\alpha}=\vect R_{\vect n}+\vect r_{\alpha}$
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\end_inset
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and their
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\begin_inset Formula $T$
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\end_inset
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-matrices are periodic,
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\begin_inset Formula $T_{\vect n,\alpha}=T_{\alpha}$
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\end_inset
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.
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In such system, the multiple-scattering problem
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:Multiple-scattering problem"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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can be rewritten as
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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\outcoeffp{\vect n,\alpha}-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect n,\alpha\right)\right\} }\tropsp{\vect n,\alpha}{\vect m,\beta}\outcoeffp{\vect m,\beta}=T_{\alpha}\rcoeffincp{\vect n,\alpha}.\quad\left(\vect n,\alpha\right)\in\mathcal{P}\label{eq:Multiple-scattering problem periodic}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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Due to periodicity, we can also write
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\begin_inset Formula $\tropsp{\vect n,\alpha}{\vect m,\beta}=\tropsp{\alpha}{\beta}\left(\vect R_{\vect m}-\vect R_{\vect n}\right)=\tropsp{\alpha}{\beta}\left(\vect R_{\vect m-\vect n}\right)=\tropsp{\vect 0,\alpha}{\vect m-\vect n,\beta}$
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\end_inset
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.
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Assuming quasi-periodic right-hand side with quasi-momentum
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\begin_inset Formula $\vect k$
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\end_inset
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,
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\begin_inset Formula $\rcoeffincp{\vect n,\alpha}=\rcoeffincp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$
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\end_inset
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, the solutions of
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:Multiple-scattering problem periodic"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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will be also quasi-periodic according to Bloch theorem,
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\begin_inset Formula $\outcoeffp{\vect n,\alpha}=\outcoeffp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$
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\end_inset
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, and eq.
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:Multiple-scattering problem periodic"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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can be rewritten as follows
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\begin_inset Formula
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\begin{align*}
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\outcoeffp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect n,\alpha\right)\right\} }\tropsp{\vect n,\alpha}{\vect m,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}},\\
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\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect n,\alpha\right)\right\} }\tropsp{\vect 0,\alpha}{\vect m-\vect n,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m-\vect n}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),\\
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\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect 0,\alpha\right)\right\} }\tropsp{\vect 0,\alpha}{\vect m,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),\\
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\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\beta\in\mathcal{P}}W_{\alpha\beta}\left(\vect k\right)\outcoeffp{\vect 0,\beta}\left(\vect k\right) & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),
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\end{align*}
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\end_inset
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so we reduced the initial scattering problem to one involving only the field
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expansion coefficients from a single unit cell, but we need to compute
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the
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\begin_inset Quotes eld
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\end_inset
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lattice Fourier transform
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\begin_inset Quotes erd
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\end_inset
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of the translation operator,
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\begin_inset Formula
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\begin{equation}
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W_{\alpha\beta}(\vect k)\equiv\sum_{\vect m\in\ints^{d}}\left(1-\delta_{\alpha\beta}\right)\tropsp{\vect 0,\alpha}{\vect m,\beta}e^{i\vect k\cdot\vect R_{\vect m}}.\label{eq:W definition}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Subsection
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Computing the Fourier sum of the translation operator
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\end_layout
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\begin_layout Standard
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The problem evaluating
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:W definition"
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\end_inset
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is the asymptotic behaviour of the translation operator,
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\begin_inset Formula $\tropsp{\vect 0,\alpha}{\vect m,\beta}\sim\left|\vect R_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect R_{\vect b}\right|}$
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\end_inset
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that does not in the strict sense converge for any
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\begin_inset Formula $d>1$
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\end_inset
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-dimensional lattice.
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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\begin_inset Foot
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status open
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\begin_layout Plain Layout
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Note that
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\begin_inset Formula $d$
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\end_inset
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here is dimensionality of the lattice, not the space it lies in, which
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I for certain reasons assume to be three.
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(TODO few notes on integration and reciprocal lattices in some appendix)
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\end_layout
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\end_inset
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\end_layout
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\end_inset
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In electrostatics, this problem can be solved with Ewald summation [TODO
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REF].
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Its basic idea is that if what asymptoticaly decays poorly in the direct
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space, will perhaps decay fast in the Fourier space.
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We use the same idea here, but the technical details are more complicated
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than in electrostatics.
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\end_layout
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\begin_layout Standard
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Let us re-express the sum in
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:W definition"
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\end_inset
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in terms of integral with a delta comb
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\begin_inset FormulaMacro
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\renewcommand{\basis}[1]{\mathfrak{#1}}
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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W_{\alpha\beta}(\vect k)=\int\ud^{d}\vect r\dc{\basis u}(\vect r)S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})e^{i\vect k\cdot\vect r}.\label{eq:W integral}
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\end{equation}
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\end_inset
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The translation operator
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\begin_inset Formula $S$
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\end_inset
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is now a function defined in the whole 3d space;
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\begin_inset Formula $\vect r_{\alpha},\vect r_{\beta}$
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\end_inset
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are the displacements of scatterers
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\begin_inset Formula $\alpha$
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\end_inset
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and
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\begin_inset Formula $\beta$
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\end_inset
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in a unit cell.
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The arrow notation
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\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})$
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\end_inset
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means
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\begin_inset Quotes eld
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\end_inset
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translation operator for spherical waves originating in
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\begin_inset Formula $\vect r+\vect r_{\beta}$
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\end_inset
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evaluated in
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\begin_inset Formula $\vect r_{\alpha}$
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\end_inset
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\begin_inset Quotes erd
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\end_inset
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and obviously
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\begin_inset Formula $S$
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\end_inset
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is in fact a function of a single 3d argument,
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\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect 0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
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\end_inset
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.
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Expression
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:W integral"
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\end_inset
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can be rewritten as
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\begin_inset Formula
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\[
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W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0))\left(\vect k\right)}
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\]
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\end_inset
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where changed the sign of
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\begin_inset Formula $\vect r/\vect{\bullet}$
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\end_inset
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has been swapped under integration, utilising evenness of
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\begin_inset Formula $\dc{\basis u}$
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\end_inset
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.
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Fourier transform of product is convolution of Fourier transforms, so (using
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formula
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Dirac comb uaFt"
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\end_inset
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for the Fourier transform of Dirac comb)
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\begin_inset Formula
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\begin{eqnarray}
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W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)(\vect k)\nonumber \\
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& = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\recb{\basis u}}^{(d)}\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)\left(\vect k\right)\nonumber \\
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& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W sum in reciprocal space}\\
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& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}e^{i\left(\vect k-\vect K\right)\cdot\left(-\vect r_{\beta}+\vect r_{\alpha}\right)}\left(\uaft{S(\vect{\bullet}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\nonumber
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\end{eqnarray}
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\end_inset
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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Factor
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\begin_inset Formula $\left(2\pi\right)^{\frac{d}{2}}$
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\end_inset
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cancels out with the
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\begin_inset Formula $\left(2\pi\right)^{-\frac{d}{2}}$
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\end_inset
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factor appearing in the convolution/product formula in the unitary angular
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momentum convention.
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\end_layout
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\end_inset
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As such, this is not extremely helpful because the the
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\emph on
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whole
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\emph default
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translation operator
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\begin_inset Formula $S$
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\end_inset
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has singularities in origin, hence its Fourier transform
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\begin_inset Formula $\uaft S$
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\end_inset
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will decay poorly.
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\end_layout
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\begin_layout Standard
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However, Fourier transform is linear, so we can in principle separate
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\begin_inset Formula $S$
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\end_inset
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in two parts,
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\begin_inset Formula $S=S^{\textup{L}}+S^{\textup{S}}$
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\end_inset
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.
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\begin_inset Formula $S^{\textup{S}}$
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\end_inset
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is a short-range part that decays sufficiently fast with distance so that
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its direct-space lattice sum converges well;
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\begin_inset Formula $S^{\textup{S}}$
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\end_inset
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must as well contain all the singularities of
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\begin_inset Formula $S$
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\end_inset
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in the origin.
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The other part,
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\begin_inset Formula $S^{\textup{L}}$
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\end_inset
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, will retain all the slowly decaying terms of
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\begin_inset Formula $S$
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\end_inset
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but it also has to be smooth enough in the origin, so that its Fourier
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transform
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\begin_inset Formula $\uaft{S^{\textup{L}}}$
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\end_inset
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decays fast enough.
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(The same idea lies behind the Ewald summation in electrostatics.) Using
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the linearity of Fourier transform and formulae
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:W definition"
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\end_inset
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and
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:W sum in reciprocal space"
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\end_inset
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, the operator
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\begin_inset Formula $W_{\alpha\beta}$
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\end_inset
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can then be re-expressed as
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\begin_inset Formula
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\begin{eqnarray}
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W_{\alpha\beta}\left(\vect k\right) & = & W_{\alpha\beta}^{\textup{S}}\left(\vect k\right)+W_{\alpha\beta}^{\textup{L}}\left(\vect k\right)\nonumber \\
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W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect 0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\
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W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S^{\textup{L}}(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition}
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\end{eqnarray}
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\end_inset
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where both sums should converge nicely.
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\end_layout
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\end_body
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\end_document
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