diff --git a/lepaper/arrayscat.lyx b/lepaper/arrayscat.lyx index a96fa72..a7548ab 100644 --- a/lepaper/arrayscat.lyx +++ b/lepaper/arrayscat.lyx @@ -712,6 +712,12 @@ Consistent notation of balls. Abstract. \end_layout +\begin_layout Itemize +Translation operators: explicit expression, also in sph. + harm. + convention independent form. +\end_layout + \begin_layout Itemize Truncation notation. \end_layout diff --git a/lepaper/infinite.lyx b/lepaper/infinite.lyx index b84ea21..df709e3 100644 --- a/lepaper/infinite.lyx +++ b/lepaper/infinite.lyx @@ -136,7 +136,8 @@ Notation \end_layout \begin_layout Standard -TODO Fourier transforms, Delta comb, lattice bases etc. +TODO Fourier transforms, Delta comb, lattice bases, reciprocal lattices + etc. \end_layout \begin_layout Subsection @@ -277,12 +278,12 @@ noprefix "false" can be rewritten as follows \begin_inset Formula -\begin{align*} -\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}},\\ -\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),\\ -\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),\\ -\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), -\end{align*} +\begin{align} +\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}},\nonumber \\ +\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),\nonumber \\ +\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),\nonumber \\ +\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),\label{eq:Multiple-scattering problem unit cell} +\end{align} \end_inset @@ -299,16 +300,35 @@ lattice Fourier transform of the translation operator, \begin_inset Formula \begin{equation} -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} +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} \end{equation} \end_inset +evaluation of which is possible but quite non-trivial due to the infinite + lattice sum, so we explain it separately in Sect. + +\begin_inset CommandInset ref +LatexCommand ref +reference "subsec:W operator evaluation" +plural "false" +caps "false" +noprefix "false" +\end_inset + +. \end_layout \begin_layout Subsection Computing the Fourier sum of the translation operator +\begin_inset CommandInset label +LatexCommand label +name "subsec:W operator evaluation" + +\end_inset + + \end_layout \begin_layout Standard @@ -576,7 +596,178 @@ W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\bas \end_inset -where both sums should converge nicely. +where both sums expected to converge nicely. + We note that the elements of the translation operators +\begin_inset Formula $\tropr,\trops$ +\end_inset + + in +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:translation operator" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + can be rewritten as linear combinations of expressions +\begin_inset Formula $\ush{\nu}{\mu}\left(\uvec d\right)j_{n}\left(d\right),\ush{\nu}{\mu}\left(\uvec d\right)h_{n}^{(1)}\left(d\right)$ +\end_inset + + (TODO WRITE THEM EXPLICITLY IN THIS FORM), respectively, hence if we are + able evaluate the lattice sums sums +\begin_inset Note Note +status open + +\begin_layout Plain Layout +CHECK THE FOLLOWING EXPRESSION FOR CORRECT FUNCTION ARGUMENTS +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{equation} +\sigma_{\nu}^{\mu}\left(\vect k\right)=\sum_{\vect n\in\ints^{d}\backslash\left\{ \vect 0\right\} }e^{i\vect{\vect k}\cdot\vect R_{\vect n}}\ush{\nu}{\mu}\left(\uvec{R_{n}}\right)h_{n}^{(1)}\left(R_{n}\right),\label{eq:sigma lattice sums} +\end{equation} + +\end_inset + +then by linearity, we can get the +\begin_inset Formula $W_{\alpha\beta}\left(\vect k\right)$ +\end_inset + + operator as well. +\end_layout + +\begin_layout Standard +TODO ADD MOROZ AND OTHER REFS HERE. + +\begin_inset CommandInset citation +LatexCommand cite +key "linton_one-_2009" +literal "true" + +\end_inset + + offers an exponentially convergent Ewald-type summation method for +\begin_inset Formula $\sigma_{\nu}^{\mu}\left(\vect k\right)=\sigma_{\nu}^{\mu(\mathrm{S})}\left(\vect k\right)+\sigma_{\nu}^{\mu(\mathrm{L})}\left(\vect k\right)$ +\end_inset + +. + Here we rewrite them in a form independent on the convention used for spherical + harmonics (as long as they are complex +\begin_inset Note Note +status open + +\begin_layout Plain Layout +lepší formulace +\end_layout + +\end_inset + +). + The short-range part reads (UNIFY INDEX NOTATION) +\begin_inset Formula +\begin{multline} +\sigma_{n}^{m(\mathrm{S})}\left(\vect{\beta}\right)=-\frac{2^{n+1}i}{k^{n+1}\sqrt{\pi}}\sum_{\vect R\in\Lambda}^{'}\left|\vect R\right|^{n}e^{i\vect{\beta}\cdot\vect R}Y_{n}^{m}\left(\vect R\right)\int_{\eta}^{\infty}e^{-\left|\vect R\right|^{2}\xi^{2}}e^{-k/4\xi^{2}}\xi^{2n}\ud\xi\\ ++\frac{\delta_{n0}\delta_{m0}}{\sqrt{4\pi}}\Gamma\left(-\frac{1}{2},-\frac{k}{4\eta^{2}}\right)Y_{n}^{m},\label{eq:Ewald in 3D short-range part} +\end{multline} + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +NEPATŘÍ TAM NĚJAKÁ DELTA FUNKCE K PŮVODNÍMU +\begin_inset Formula $\sigma_{n}^{m(0)}$ +\end_inset + +? +\end_layout + +\end_inset + +and the long-range part (FIXME, this is the 2D version; include the 1D and + 3D lattice expressions as well) +\begin_inset Formula +\begin{multline} +\sigma_{n}^{m(\mathrm{L})}\left(\vect{\beta}\right)=-\frac{i^{n+1}}{k^{2}\mathscr{A}}\sqrt{\pi}2\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\times\\ +\times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}Y_{n}^{m}\left(\frac{\pi}{2},\phi_{\vect{\beta}_{pq}}\right)\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/k\right)^{n-2j}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(n-m\right)-j\right)!\left(\frac{1}{2}\left(n+m\right)-j\right)!}\left(\gamma_{pq}\right)^{2j-1}\label{eq:Ewald in 3D long-range part} +\end{multline} + +\end_inset + +where +\begin_inset Formula $\xi$ +\end_inset + + is TODO, +\begin_inset Formula $\beta_{pq}$ +\end_inset + + is TODO, +\begin_inset Formula $\Gamma_{j,pq}$ +\end_inset + + is TODO and +\begin_inset Formula $\eta$ +\end_inset + + is a real parameter that determines the pace of convergence of both parts. + The larger +\begin_inset Formula $\eta$ +\end_inset + + is, the faster +\begin_inset Formula $\sigma_{n}^{m(\mathrm{S})}\left(\vect{\beta}\right)$ +\end_inset + + converges but the slower +\begin_inset Formula $\sigma_{n}^{m(\mathrm{L})}\left(\vect{\beta}\right)$ +\end_inset + + converges. + Therefore (based on the lattice geometry) it has to be adjusted in a way + that a reasonable amount of terms needs to be evaluated numerically from + both +\begin_inset Formula $\sigma_{n}^{m(\mathrm{S})}\left(\vect{\beta}\right)$ +\end_inset + + and +\begin_inset Formula $\sigma_{n}^{m(\mathrm{L})}\left(\vect{\beta}\right)$ +\end_inset + + . + Generally, a good choice for +\begin_inset Formula $\eta$ +\end_inset + + is TODO; in order to achieve accuracy TODO, one has to evaluate the terms + on TODO lattice points. + (I HAVE SOME DERIVATIONS OF THE ESTIMATES IN MY NOTES; SHOULD I INCLUDE + THEM?) +\end_layout + +\begin_layout Standard +In practice, the integrals in +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:Ewald in 3D short-range part" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + can be easily evaluated by numerical quadrature and the incomplete +\begin_inset Formula $\Gamma$ +\end_inset + +-functions using the series TODO and TODO from DLMF. \end_layout \end_body