From 3ce1210d8aa89399377a228f3a1de33a717783f7 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Mon, 22 Jun 2020 15:25:55 +0300 Subject: [PATCH] Rework periodic part. Former-commit-id: 4b15d1e9342e7c6b049c5bd29fd98c9766bf38a4 --- lepaper/infinite.lyx | 313 +++++++++++++++++++++++++++++++------------ 1 file changed, 229 insertions(+), 84 deletions(-) diff --git a/lepaper/infinite.lyx b/lepaper/infinite.lyx index 5821ea1..0b54e0b 100644 --- a/lepaper/infinite.lyx +++ b/lepaper/infinite.lyx @@ -1063,11 +1063,28 @@ FP: Check signs. \begin_inset Formula \begin{equation} -\sigma_{l,m}\left(\vect k,\vect s\right)=\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},\vect s}\right)e^{i\vect{\vect k}\cdot\left(\vect R_{\vect n}-\vect s\right)}\sswfoutlm lm\left(\vect{R_{n}}-\vect s\right),\label{eq:sigma lattice sums} +\sigma_{l,m}\left(\vect k,\vect s\right)=\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},\vect s}\right)e^{i\vect{\vect k}\cdot\vect R_{\vect n}}\sswfoutlm lm\left(\kappa\left(\vect s+\vect{R_{n}}\right)\right),\label{eq:sigma lattice sums} \end{equation} \end_inset + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula +\begin{equation} +\sigma_{l,m}\left(\vect k,\vect s\right)=\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},\vect s}\right)e^{i\vect{\vect k}\cdot\left(\vect R_{\vect n}-\vect s\right)}\sswfoutlm lm\left(\vect{R_{n}}-\vect s\right),\label{eq:sigma lattice sums-1} +\end{equation} + +\end_inset + + +\end_layout + +\end_inset + we see from eqs. \begin_inset CommandInset ref @@ -1129,6 +1146,10 @@ W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k) & =\sum_{\lambda=\left|l-l'\right|+1 \end_inset +\begin_inset Note Note +status open + +\begin_layout Plain Layout \begin_inset Formula \[ W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k)=\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r_{\beta}-\vect r_{\alpha}\right),\quad\tau'\ne\tau, @@ -1137,6 +1158,19 @@ W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k)=\sum_{\lambda=\left|l-l'\right|+\lef \end_inset +\end_layout + +\end_inset + + +\begin_inset Formula +\[ +W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k)=\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\sigma_{\lambda,m-m'}\left(-\vect k,\vect r_{\alpha}-\vect r_{\beta}\right),\quad\tau'\ne\tau, +\] + +\end_inset + + \begin_inset Note Note status open @@ -1223,14 +1257,19 @@ FP: Check sign of s \begin_layout Standard \begin_inset Formula \begin{multline} -\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=-\frac{2^{l+1}i}{\kappa^{l+1}\sqrt{\pi}}\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},-\vect s}\right)\left|\vect{R_{n}+\vect s}\right|^{l}\ush lm\left(\vect{R_{n}+\vect s}\right)e^{i\vect k\cdot\left(\vect{R_{n}+\vect s}\right)}\\ -\times\int_{\eta}^{\infty}e^{-\left|\vect{R_{n}+\vect s}\right|^{2}\xi^{2}}e^{-\kappa^{2}/4\xi^{2}}\xi^{2l}\ud\xi\\ -+\delta_{\vect{R_{n}},-\vect s}\frac{\delta_{l0}\delta_{m0}}{\sqrt{4\pi}}\Gamma\left(-\frac{1}{2},-\frac{\kappa^{2}}{4\eta^{2}}\right)\ush lm\left(\vect{R_{n}+\vect s}\right).\label{eq:Ewald in 3D short-range part} +\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=-\frac{2^{l+1}i}{\kappa^{l+1}\sqrt{\pi}}\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},-\vect s}\right)\left|\vect s_{\vect n}\right|^{l}\ush lm\left(\uvec s_{\vect n}\right)e^{i\vect k\cdot\vect{R_{n}}}\\ +\times\int_{\eta}^{\infty}e^{-\left|\vect s_{\vect n}\right|^{2}\xi^{2}}e^{-\kappa^{2}/4\xi^{2}}\xi^{2l}\ud\xi\\ ++\delta_{\vect{R_{n}},-\vect s}\frac{\delta_{l0}\delta_{m0}}{\sqrt{4\pi}}\Gamma\left(-\frac{1}{2},-\frac{\kappa^{2}}{4\eta^{2}}\right)\ush lm\left(\uvec s_{\vect n}\right),\label{eq:Ewald in 3D short-range part} \end{multline} \end_inset -The formal +where we labeled +\begin_inset Formula $\vect s_{\vect n}\equiv\vect s+\vect R_{\vect n}$ +\end_inset + +. + The formal \begin_inset Formula $\left(1-\delta_{\vect{R_{n}},-\vect s}\right)$ \end_inset @@ -1355,7 +1394,19 @@ The explicit form of the long-range part of the lattice sum depends on the \end_inset . - + In the following, let us label +\begin_inset Formula $\vect k_{\vect K}\equiv\vect k+\vect K$ +\end_inset + +, where +\begin_inset Formula $\vect K$ +\end_inset + + is a point in the reciprocal lattice, and let +\begin_inset Formula $\mathcal{A}$ +\end_inset + + be the lattice unit cell volume (or area/length in the 2D/1D cases). \end_layout \begin_layout Paragraph @@ -1369,7 +1420,7 @@ Case \begin_layout Standard \begin_inset Formula \begin{equation} -\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)=\frac{4\pi i^{l+1}}{\kappa\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}\underbrace{e^{i\vect K\cdot\vect s}}_{\text{nemá tu být \ensuremath{\vect{k\cdot s}?}}}\frac{\left(\left|\vect k+\vect K\right|/\kappa\right)^{l}}{\kappa^{2}-\left|\vect k+\vect K\right|^{2}}e^{\left(\kappa^{2}-\left|\vect k+\vect K\right|^{2}\right)/4\eta^{2}}\ush lm\left(\vect k+\vect K\right)\label{eq:Ewald in 3D long-range part 3D} +\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)=\frac{4\pi i^{l+1}}{\kappa\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{-i\vect k_{\vect K}\cdot\vect s}\frac{\left(\left|\vect k_{\vect K}\right|/\kappa\right)^{l}}{\kappa^{2}-\left|\vect k_{\vect K}\right|^{2}}e^{\left(\kappa^{2}-\left|\vect k_{\vect K}\right|^{2}\right)/4\eta^{2}}\ush lm\left(\uvec k_{\vect K}\right)\label{eq:Ewald in 3D long-range part 3D} \end{equation} \end_inset @@ -1384,14 +1435,115 @@ regardless of chosen coordinate axes. \end_layout \begin_layout Paragraph -Case -\begin_inset Formula $d=2$ +Cases +\begin_inset Formula $d=1,2$ \end_inset \end_layout \begin_layout Standard +In the quasiperiodic cases, we decompose vectors into parallel and orthogonal + parts with respect to the linear subspace in which the Bravais lattice + lies (the reciprocal lattice lies in the same subspace), +\begin_inset Formula $\vect v=\vect v_{\perp}+\vect v_{\parallel}$ +\end_inset + +, and we label +\begin_inset Formula +\begin{equation} +\gamma_{\vect k_{\vect K}}\equiv\gamma_{\vect k_{\vect K}}\left(\kappa\right)\equiv\left(\left|\vect k_{\vect K}\right|^{2}-\kappa^{2}\right)^{\frac{1}{2}}/\kappa,\label{eq:lilgamma} +\end{equation} + +\end_inset + + +\begin_inset Formula +\begin{equation} +\Delta_{d;j}\left(x,z\right)\equiv\int_{x}^{\infty}t^{-\frac{d_{c}}{2}-n}\exp\left(-t+\frac{z^{2}}{4t}\right)\ud t,\label{eq:Delta_j} +\end{equation} + +\end_inset + +where +\begin_inset Formula $d_{c}=3-d$ +\end_inset + + is the complementary dimension of the lattice. + Then +\begin_inset Formula +\begin{multline} +\sigma_{l}^{m}\left(\vect k,\vect s\right)=\frac{-i}{2\pi^{d_{c}/2}\mathcal{A}\kappa}\frac{\left(2l+1\right)!!}{\kappa^{l}}\sum_{\vect K\in\Lambda^{*}}e^{-i\vect k_{\vect K}\cdot\vect s}\times\\ +\times\sum_{j=0}^{l}\frac{\left(-1\right)^{j}}{j!}\left(\frac{\kappa\gamma_{\vect k_{\vect K}}}{2}\right)^{2j}\Delta_{d;j}\left(\frac{\kappa^{2}\gamma_{\vect k_{\vect K}}^{2}}{4\eta^{2}},-i\kappa\gamma_{\vect k_{\vect K}}\left|\vect s_{\perp}\right|\right)\times\\ +\times\sum_{l'=\max\left(0,l-2j\right)}^{l-j}4\pi i^{l'}\left(2\left|\vect s_{\bot}\right|\right)^{2j-l+l'}\frac{\left|\vect k_{\vect K}\right|^{l'}}{\left(2l'+1\right)!!}\sum_{m'=-l'}^{l'}\ush{l'}{m'}\left(\uvec k_{\vect K}\right)\times\\ +\times\int\ud\Omega_{\vect r}\,\ush lm\left(\uvec r\right)\ushD{l'}{m'}\left(\uvec r\right)\left(\frac{\left|\vect r_{\perp}\right|}{\left|\vect r\right|}\right)^{l-l}\left(\frac{-\vect r_{\perp}\cdot\vect s_{\perp}}{\left|\vect r_{\perp}\right|\left|\vect s_{\perp}\right|}\right)^{2j-l+l'}.\label{eq:Ewald in 3D long-range part 1D 2D} +\end{multline} + +\end_inset + +The angular integral on the last line of +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:Ewald in 3D long-range part 1D 2D" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + gives a set of constant coefficients characteristic to a chosen convention + for spherical harmonics and coordinate axes; relatively simple closed-form + expressions are obtained for 2D periodicity if we choose the lattice to + lie in the +\begin_inset Formula $xy$ +\end_inset + + plane, so that both +\begin_inset Formula $\vect r_{\perp},\vect s_{\perp}$ +\end_inset + + are parallel to the +\begin_inset Formula $z$ +\end_inset + + axis, as done in +\begin_inset CommandInset citation +LatexCommand cite +key "kambe_theory_1968" +literal "false" + +\end_inset + +, see also Supplementary Material. + In the special case +\begin_inset Formula $\vect s_{\perp}=0$ +\end_inset + + the expressions can be considerably simplified as most of the terms vanish + and +\begin_inset Formula $\Delta_{d;j}\left(x,0\right)=\Gamma\left(1-d_{c}/2-j,x\right)$ +\end_inset + +, but the general case is needed for evaluating the fields in space (see + Section +\begin_inset CommandInset ref +LatexCommand ref +reference "subsec:Periodic scattering and fields" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +) or if there is an offset between two particles in a unitcell that is not + parallel to the lattice subspace. +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout Reasonable explicit forms assume that the lattice lies inside the \begin_inset Formula $xy$ \end_inset @@ -1462,12 +1614,17 @@ FP: check sign of j\le s\le\min\left(2j,l-\left|m\right|\right)\\ l-n+\left|m\right|\,\mathrm{even} } -}\frac{1}{\left(2j-s\right)!\left(s-j\right)!}\frac{\left(-\kappa s_{\perp}\right)^{2j-s}\left(\left|\vect k+\vect K\right|/\kappa\right)^{l-s}}{\left(\frac{1}{2}\left(l-m-s\right)\right)!\left(\frac{1}{2}\left(l+m-s\right)\right)!}\label{eq:Ewald in 3D long-range part 1D 2D} +}\frac{1}{\left(2j-s\right)!\left(s-j\right)!}\frac{\left(-\kappa s_{\perp}\right)^{2j-s}\left(\left|\vect k+\vect K\right|/\kappa\right)^{l-s}}{\left(\frac{1}{2}\left(l-m-s\right)\right)!\left(\frac{1}{2}\left(l+m-s\right)\right)!}\label{eq:Ewald in 3D long-range part 1D 2D-1} \end{multline} \end_inset +\end_layout + +\end_inset + + \begin_inset Note Note status open @@ -1486,23 +1643,12 @@ status open \end_inset -where -\begin_inset Formula -\begin{equation} -\gamma\left(z\right)=\left(z^{2}-1\right)^{\frac{1}{2}},\label{eq:lilgamma} -\end{equation} -\end_inset +\begin_inset Note Note +status open - -\begin_inset Formula -\begin{equation} -\Delta_{j}\left(x,z\right)=\int_{x}^{\infty}t^{\frac{-1}{2}-n}\exp\left(-t+\frac{z^{2}}{4t}\right)\ud t.\label{eq:Delta_j} -\end{equation} - -\end_inset - -If the normal component +\begin_layout Plain Layout +where If the normal component \begin_inset Formula $s_{\bot}$ \end_inset @@ -1538,12 +1684,20 @@ noprefix "false" \end_inset . - If +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +If \begin_inset Formula $s_{\bot}\ne0$ \end_inset , the integral -\begin_inset Formula $\Delta_{j}\left(x,z\right)$ +\begin_inset Formula $\Delta_{d;j}\left(x,0\right)$ \end_inset can be evaluated e.g. @@ -1555,7 +1709,7 @@ using the Taylor series \begin_inset Formula \[ -\Delta_{j}\left(x,z\right)=\sum_{k=0}^{\infty}\Gamma\left(\frac{1}{2}-j-k,x\right)\frac{\left(z/2\right)^{2k}}{k!} +\Delta_{d;j}\left(x,z\right)=\sum_{k=0}^{\infty}\Gamma\left(1-\frac{d_{c}}{2}-j-k,x\right)\frac{\left(z/2\right)^{2k}}{k!} \] \end_inset @@ -1586,27 +1740,19 @@ noprefix "false" \end_inset - by parts (note that the signs are wrong in -\begin_inset CommandInset citation -LatexCommand cite -key "kambe_theory_1968" -literal "false" - -\end_inset - -) + by parts (with signs corrected here): \begin_inset Formula \begin{equation} -\Delta_{j+1}\left(x,z\right)=\frac{4}{z^{2}}\left(\left(\frac{1}{2}-j\right)\Delta_{j}\left(x,z\right)-\Delta_{j-1}\left(x,z\right)+x^{\frac{1}{2}-j}e^{-x+\frac{z^{2}}{4x}}\right)\label{eq:Delta_j recurrent} +\Delta_{d;j+1}\left(x,z\right)=\frac{4}{z^{2}}\left(\left(\frac{1}{2}-j\right)\Delta_{d;j}\left(x,z\right)-\Delta_{d;j-1}\left(x,z\right)+x^{\frac{d_{c}}{2}-j}e^{-x+\frac{z^{2}}{4x}}\right)\label{eq:Delta_j recurrent} \end{equation} \end_inset -with the first two terms +with the first two terms for 2D periodicity \begin_inset Formula \begin{align*} -\Delta_{0}\left(x,z\right) & =\frac{\sqrt{\pi}}{2}e^{-x^{2}+\frac{z^{2}}{4x}}\left(w\left(-\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)+w\left(\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)\right),\\ -\Delta_{1}\left(x,z\right) & =i\frac{\sqrt{\pi}}{z}e^{-x^{2}+\frac{z^{2}}{4x}}\left(w\left(-\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)-w\left(\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)\right), +\Delta_{2;0}\left(x,z\right) & =\frac{\sqrt{\pi}}{2}e^{-x^{2}+\frac{z^{2}}{4x}}\left(w\left(-\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)+w\left(\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)\right),\\ +\Delta_{2;1}\left(x,z\right) & =i\frac{\sqrt{\pi}}{z}e^{-x^{2}+\frac{z^{2}}{4x}}\left(w\left(-\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)-w\left(\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)\right), \end{align*} \end_inset @@ -1717,9 +1863,8 @@ FP: I have some error estimates derived in my notes. \end_layout \begin_layout Standard -One pecularity of the two-dimensional case is the two-branchedness of the - function -\begin_inset Formula $\gamma\left(z\right)$ +One pecularity of the two-dimensional case is the two-branchedness of +\begin_inset Formula $\gamma_{\vect k_{\vect K}}\left(\kappa\right)$ \end_inset and the incomplete @@ -1735,7 +1880,7 @@ One pecularity of the two-dimensional case is the two-branchedness of the \end_inset the function -\begin_inset Formula $\gamma\left(z\right)$ +\begin_inset Formula $\gamma_{\vect k_{\vect K}}\left(\kappa\right)$ \end_inset appears with even powers, and @@ -1898,21 +2043,6 @@ Detailed physical interpretation of the remaining branch cuts is an open \end_inset - -\begin_inset Note Note -status open - -\begin_layout Plain Layout -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. -\end_layout - -\end_inset - \begin_inset Note Note status open @@ -1927,23 +2057,6 @@ status open \end_layout -\begin_layout Paragraph -Case -\begin_inset Formula $d=1$ -\end_inset - - -\end_layout - -\begin_layout Standard -For one-dimensional chains, the easiest choice is to align the lattice with - the -\begin_inset Formula $z$ -\end_inset - - axis. -\end_layout - \begin_layout Subsubsection Choice of Ewald parameter and high-frequency breakdown \end_layout @@ -2016,7 +2129,10 @@ Whatabout different geometries? However, in floating point arithmetics, the magnitude of the summands must be taken into account as well in order to maintain accuracy. - There is a particular problem with the +\end_layout + +\begin_layout Standard +There is a particular problem with the \begin_inset Quotes eld \end_inset @@ -2026,7 +2142,7 @@ central reciprocal lattice points in the long-range sums for which the real part of -\begin_inset Formula $\left|\vect k+\vect K\right|^{2}-\kappa^{2}$ +\begin_inset Formula $\left|\vect k_{\vect K}\right|^{2}-\kappa^{2}$ \end_inset is negative: the incomplete @@ -2077,7 +2193,7 @@ central \end_inset needs to be adjusted in a way that keeps the value of -\begin_inset Formula $\Gamma\left(a,\frac{\left|\vect k+\vect K\right|^{2}-\kappa^{2}}{4\eta^{2}}\right)$ +\begin_inset Formula $\Gamma\left(a,\left(\left|\vect k_{\vect K}\right|^{2}-\kappa^{2}\right)/4\eta^{2}\right)$ \end_inset within reasonable bounds. @@ -2197,6 +2313,13 @@ left out for the time being \begin_layout Subsection Scattering cross sections and field intensities in periodic system +\begin_inset CommandInset label +LatexCommand label +name "subsec:Periodic scattering and fields" + +\end_inset + + \end_layout \begin_layout Standard @@ -2391,14 +2514,36 @@ TODO fix signs and exponential phase factors \begin_inset Formula -\begin{align*} -\vect E_{\mathrm{scat}}\left(\vect r\right) & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect n,\alpha}{\tau}lm\vect u_{\tau lm}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\\ - & =\sum_{\alpha\in\mathcal{P}_{1}}e^{-i\vect k\cdot\left(\vect r-\vect r_{\alpha}\right)}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r-\vect r_{\alpha}\right). -\end{align*} +\begin{align} +\vect E_{\mathrm{scat}}\left(\vect r\right) & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect n,\alpha}{\tau}lm\vect u_{\tau lm}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\nonumber \\ + & =\sum_{\alpha\in\mathcal{P}_{1}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\vswfrtlm 21{m'}\left(0\right)\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(-\vect k,\vect r-\vect r_{\alpha}\right).\label{eq:Scattered fields in periodic systems} +\end{align} \end_inset +In the scattering problem, the total field intensity is obtained by adding + the incident field to +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:Scattered fields in periodic systems" +plural "false" +caps "false" +noprefix "false" +\end_inset + +; whereas in the lattice mode problem the total field is directly given + by +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:Scattered fields in periodic systems" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. \end_layout \end_body