From a578b04a65bf03939e63b6604b8bd9e84a7f7100 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Thu, 7 Nov 2019 00:29:41 +0200 Subject: [PATCH] Fix sign in absorption cross section formula. Former-commit-id: 695731c1ab4934abf88c6603a696cf5855cd4582 --- lepaper/finite.lyx | 42 +++++++++++++++++++++--------------------- 1 file changed, 21 insertions(+), 21 deletions(-) diff --git a/lepaper/finite.lyx b/lepaper/finite.lyx index 15c97c2..f2e7433 100644 --- a/lepaper/finite.lyx +++ b/lepaper/finite.lyx @@ -135,7 +135,7 @@ Single-particle scattering In order to define the basic concepts, let us first consider the case of electromagnetic (EM) radiation scattered by a single particle. We assume that the scatterer lies inside a closed ball -\begin_inset Formula $\closedball{R^{<}}{\vect0}$ +\begin_inset Formula $\closedball{R^{<}}{\vect 0}$ \end_inset of radius @@ -144,16 +144,16 @@ In order to define the basic concepts, let us first consider the case of and center in the origin of the coordinate system (which can be chosen that way; the natural choice of -\begin_inset Formula $\closedball{R^{<}}{\vect0}$ +\begin_inset Formula $\closedball{R^{<}}{\vect 0}$ \end_inset is the circumscribed ball of the scatterer) and that there exists a larger open cocentric ball -\begin_inset Formula $\openball{R^{>}}{\vect0}$ +\begin_inset Formula $\openball{R^{>}}{\vect 0}$ \end_inset , such that the (non-empty) spherical shell -\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}=\openball{R^{>}}{\vect0}\setminus\closedball{R^{<}}{\vect0}$ +\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}=\openball{R^{>}}{\vect 0}\setminus\closedball{R^{<}}{\vect 0}$ \end_inset is filled with a homogeneous isotropic medium with relative electric permittivi @@ -173,7 +173,7 @@ ty \end_inset in -\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}$ +\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$ \end_inset must satisfy the homogeneous vector Helmholtz equation together with the @@ -278,8 +278,8 @@ outgoing , respectively, defined as follows: \begin_inset Formula \begin{align} -\vswfrtlm1lm\left(k\vect r\right) & =j_{l}\left(\kappa r\right)\vsh1lm\left(\uvec r\right),\nonumber \\ -\vswfrtlm2lm\left(k\vect r\right) & =\frac{1}{\kappa r}\frac{\ud\left(\kappa rj_{l}\left(\kappa r\right)\right)}{\ud\left(kr\right)}\vsh2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(\kappa r\right)}{kr}\vsh3lm\left(\uvec r\right),\label{eq:VSWF regular} +\vswfrtlm 1lm\left(k\vect r\right) & =j_{l}\left(\kappa r\right)\vsh 1lm\left(\uvec r\right),\nonumber \\ +\vswfrtlm 2lm\left(k\vect r\right) & =\frac{1}{\kappa r}\frac{\ud\left(\kappa rj_{l}\left(\kappa r\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(\kappa r\right)}{kr}\vsh 3lm\left(\uvec r\right),\label{eq:VSWF regular} \end{align} \end_inset @@ -287,8 +287,8 @@ outgoing \begin_inset Formula \begin{align} -\vswfouttlm1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh1lm\left(\uvec r\right),\nonumber \\ -\vswfouttlm2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(\kappa r\right)\right)}{\ud\left(kr\right)}\vsh2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(\kappa r\right)}{\kappa r}\vsh3lm\left(\uvec r\right),\label{eq:VSWF outgoing}\\ +\vswfouttlm 1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh 1lm\left(\uvec r\right),\nonumber \\ +\vswfouttlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(\kappa r\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(\kappa r\right)}{\kappa r}\vsh 3lm\left(\uvec r\right),\label{eq:VSWF outgoing}\\ & \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l,\nonumber \end{align} @@ -323,9 +323,9 @@ vector spherical harmonics \begin_inset Formula \begin{align} -\vsh1lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\times\left(\vect r\ush lm\left(\uvec r\right)\right)=\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\ush lm\left(\uvec r\right)\times\vect r,\nonumber \\ -\vsh2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\nonumber \\ -\vsh3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).\label{eq:vector spherical harmonics definition} +\vsh 1lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\times\left(\vect r\ush lm\left(\uvec r\right)\right)=\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\ush lm\left(\uvec r\right)\times\vect r,\nonumber \\ +\vsh 2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\nonumber \\ +\vsh 3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).\label{eq:vector spherical harmonics definition} \end{align} \end_inset @@ -473,7 +473,7 @@ noprefix "false" \end_inset inside a ball -\begin_inset Formula $\openball{R^{>}}{\vect0}$ +\begin_inset Formula $\openball{R^{>}}{\vect 0}$ \end_inset with radius @@ -483,7 +483,7 @@ noprefix "false" and center in the origin, were it filled with homogeneous isotropic medium; however, if the equation is not guaranteed to hold inside a smaller ball -\begin_inset Formula $\closedball{R^{<}}{\vect0}$ +\begin_inset Formula $\closedball{R^{<}}{\vect 0}$ \end_inset around the origin (typically due to presence of a scatterer), one has to @@ -492,7 +492,7 @@ noprefix "false" \end_inset to have a complete basis of the solutions in the volume -\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}=\openball{R^{>}}{\vect0}\setminus\closedball{R^{<}}{\vect0}$ +\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}=\openball{R^{>}}{\vect 0}\setminus\closedball{R^{<}}{\vect 0}$ \end_inset . @@ -514,11 +514,11 @@ The single-particle scattering problem at frequency \end_inset can be posed as follows: Let a scatterer be enclosed inside the ball -\begin_inset Formula $\closedball{R^{<}}{\vect0}$ +\begin_inset Formula $\closedball{R^{<}}{\vect 0}$ \end_inset and let the whole volume -\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}$ +\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$ \end_inset be filled with a homogeneous isotropic medium with wave number @@ -527,7 +527,7 @@ The single-particle scattering problem at frequency . Inside -\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect0}$ +\begin_inset Formula $\mezikuli{R^{<}}{R^{>}}{\vect 0}$ \end_inset , the electric field can be expanded as @@ -549,7 +549,7 @@ doplnit frekvence a polohy \end_inset If there were no scatterer and -\begin_inset Formula $\closedball{R^{<}}{\vect0}$ +\begin_inset Formula $\closedball{R^{<}}{\vect 0}$ \end_inset were filled with the same homogeneous medium, the part with the outgoing @@ -558,7 +558,7 @@ If there were no scatterer and \end_inset due to sources outside -\begin_inset Formula $\openball{R^{>}}{\vect0}$ +\begin_inset Formula $\openball{R^{>}}{\vect 0}$ \end_inset would remain. @@ -1114,7 +1114,7 @@ literal "true" \sigma_{\mathrm{ext}}\left(\uvec k\right) & = & -\frac{1}{\kappa^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)=-\frac{1}{2\kappa^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}+\Tp{}^{\dagger}\right)\rcoeffp{},\label{eq:extincion CS single}\\ \sigma_{\mathrm{scat}}\left(\uvec k\right) & = & \frac{1}{\kappa^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\left\Vert \outcoeffp{}\right\Vert ^{2}=\frac{1}{\kappa^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}\right)\rcoeffp{},\label{eq:scattering CS single}\\ \sigma_{\mathrm{abs}}\left(\uvec k\right) & = & \sigma_{\mathrm{ext}}\left(\uvec k\right)-\sigma_{\mathrm{scat}}\left(\uvec k\right)=-\frac{1}{\kappa^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\left(\Re\left(\rcoeffp{}^{\dagger}\outcoeffp{}\right)+\left\Vert \outcoeffp{}\right\Vert ^{2}\right)\nonumber \\ - & & =\frac{1}{\kappa^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}+\frac{\Tp{}^{\dagger}+\Tp{}}{2}\right)\rcoeffp{},\label{eq:absorption CS single} + & & =-\frac{1}{\kappa^{2}\left\Vert \vect E_{0}\right\Vert ^{2}}\rcoeffp{}^{\dagger}\left(\Tp{}^{\dagger}\Tp{}+\frac{\Tp{}^{\dagger}+\Tp{}}{2}\right)\rcoeffp{},\label{eq:absorption CS single} \end{eqnarray} \end_inset