From cae5cee97d3922a64a3742a9eb2ccc2248b4ea7f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Tue, 16 Jun 2020 00:35:03 +0300 Subject: [PATCH] =?UTF-8?q?Ewald=201D=20in=203D:=20jdu=20sp=C3=A1t?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Former-commit-id: 7ed762d676c8f7fb1c4988522d3fca51eb66cc9b --- notes/ewald_1D_in_3D.lyx | 81 ++++++++++++++++++++++++++++++++++++++-- 1 file changed, 77 insertions(+), 4 deletions(-) diff --git a/notes/ewald_1D_in_3D.lyx b/notes/ewald_1D_in_3D.lyx index 6dc67f8..b1bfad3 100644 --- a/notes/ewald_1D_in_3D.lyx +++ b/notes/ewald_1D_in_3D.lyx @@ -340,6 +340,49 @@ so \end_inset +\end_layout + +\begin_layout Standard +We also have +\begin_inset Formula +\begin{align*} +e^{-\left|\vect s_{\bot}+\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau} & =e^{-\left(\left|\vect s_{\bot}\right|^{2}+\left|\vect r_{\bot}\right|^{2}+2\vect r_{\bot}\cdot\vect s_{\bot}\right)\kappa^{2}\gamma_{\vect K}^{2}/4\tau}\\ + & =e^{-\left|\vect s_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau}\sum_{n=0}^{\infty}\frac{1}{n!}\left(-\frac{\left(\left|\vect r_{\bot}\right|^{2}+2\vect r_{\bot}\cdot\vect s_{\bot}\right)\kappa^{2}\gamma_{\vect K}^{2}}{4\tau}\right)^{n}, +\end{align*} + +\end_inset + +hence +\begin_inset Formula +\[ +\int\ud\Omega_{\vect r}\,G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)\ushD{l'}{m'}\left(\uvec r\right)=-\frac{1}{2\pi\mathcal{A}}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\frac{1}{2\pi\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect K\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect K\right|\left|\vect r\right|\right)\ushD lm\left(\uvec r\right)\ush lm\left(\uvec K\right)\sum_{n=0}^{\infty}\frac{1}{n!}\left(-\frac{\left(\left|\vect r_{\bot}\right|^{2}+2\vect r_{\bot}\cdot\vect s_{\bot}\right)\kappa^{2}\gamma_{\vect K}^{2}}{4}\right)^{n}\underbrace{\int_{\kappa^{2}\gamma_{\vect K}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect s_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau}\tau^{-1-n}\ud\tau}_{\Delta_{n+1/2}} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +BTW: +\begin_inset Formula +\begin{align*} +\left|\vect r_{\bot}\right|^{2} & =\left|\vect r\right|^{2}-\left|\vect r_{\parallel}\right|^{2}=\left|\vect r\right|^{2}-\left(\vect r\cdot\uvec K\right)^{2},\\ +\vect r_{\bot}\cdot\vect s_{\bot} & =\vect r\cdot\vect s_{\bot} +\end{align*} + +\end_inset + + +\end_layout + +\end_inset + + \end_layout \begin_layout Standard @@ -364,13 +407,13 @@ Now we set the conventions: let the lattice lie on the \end_inset , -\begin_inset Formula $\vect r_{\bot}=\uvec xr_{\bot}\cos\phi+\uvec yr_{\bot}\sin\phi$ +\begin_inset Formula $\vect r_{\bot}=\uvec xr_{\bot}\cos\phi+\uvec yr_{\bot}\sin\phi=\uvec xr\sin\theta\cos\phi+\uvec yr\sin\theta\sin\phi$ \end_inset , we have \begin_inset Formula \[ -\left|\vect s_{\bot}+\vect r_{\bot}\right|^{2}=s_{\bot}^{2}+r_{\bot}^{2}+2s_{\bot}r_{\bot}\cos\left(\phi-\Phi\right). +\left|\vect s_{\bot}+\vect r_{\bot}\right|^{2}=s_{\bot}^{2}+r^{2}\left(\sin\theta\right)^{2}+2s_{\bot}r\sin\theta\cos\left(\phi-\Phi\right). \] \end_inset @@ -411,7 +454,7 @@ The angular integral (assuming it can be separated from the rest like this) is \begin_inset Formula \[ -I_{l}^{m}\equiv\int\ud\Omega_{\vect r}\,\ushD lm\left(\uvec r\right)e^{-\left(r_{\bot}^{2}+2s_{\bot}r_{\bot}\cos\left(\phi-\Phi\right)\right)^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau} +I_{l'}^{m'}\equiv\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)e^{-\left(r_{\bot}^{2}+2s_{\bot}r_{\bot}\cos\left(\phi-\Phi\right)\right)^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau} \] \end_inset @@ -427,12 +470,37 @@ Let's further extract the azimuthal part \begin_inset Formula \[ -e^{-im\Phi}A_{l}^{m}\equiv\int_{0}^{2\pi}e^{-im\phi}e^{-w\cos\left(\phi-\Phi\right)}\ud\phi=e^{-im\Phi}\int_{0}^{2\pi}e^{-im\varphi}e^{-w\cos\varphi}\ud\varphi +e^{-im'\Phi}A_{l'}^{m'}\equiv\int_{0}^{2\pi}e^{-im'\phi}e^{-w\cos\left(\phi-\Phi\right)}\ud\phi=e^{-im'\Phi}\int_{0}^{2\pi}e^{-im'\varphi}e^{-w\cos\varphi}\ud\varphi \] \end_inset +Using [DLMF 10.9.2], +\begin_inset Formula $\int_{0}^{2\pi}e^{-im'\varphi}e^{-w\cos\varphi}\ud\varphi=\int_{0}^{2\pi}\cos\left(m'\varphi\right)e^{i(iw)\cos\varphi}=2\pi i^{m'}J_{m'}\left(iw\right)$ +\end_inset + we have +\begin_inset Formula +\[ +e^{-m'\Phi}A_{l'}^{m'}=2\pi i^{m'}J_{m'}\left(iw\right), +\] + +\end_inset + +assuming that +\begin_inset Formula $w$ +\end_inset + + is real (which does not necessarily have to be true!); numerical experiments + in Sage show that the result is valid also for complex +\begin_inset Formula $w$ +\end_inset + +. +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout \begin_inset Formula \begin{align*} A_{l}^{m} & =\int_{0}^{2\pi}e^{-im\varphi}\sum_{n=0}^{\infty}\frac{\left(-w\cos\varphi\right)^{n}}{n!}\ud\varphi\\ @@ -450,6 +518,11 @@ A_{l}^{m} & =\int_{0}^{2\pi}e^{-im\varphi}\sum_{n=0}^{\infty}\frac{\left(-w\cos\ \end_inset +\end_layout + +\end_inset + + \begin_inset Note Note status collapsed