From 76abecce48355d65082c8c6ab9960bb2c7361b8e Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Wed, 31 Jul 2019 11:43:24 +0300 Subject: [PATCH] Expressions for translation operators (to be checked) Former-commit-id: f19217f0f3f66a79d724840f633731f44bfd755f --- lepaper/arrayscat.lyx | 2 +- lepaper/finite.lyx | 57 +++++++++++++++++++++++++++++++------------ 2 files changed, 43 insertions(+), 16 deletions(-) diff --git a/lepaper/arrayscat.lyx b/lepaper/arrayscat.lyx index c318f56..71d17e5 100644 --- a/lepaper/arrayscat.lyx +++ b/lepaper/arrayscat.lyx @@ -713,7 +713,7 @@ Abstract. \end_layout \begin_layout Itemize -Translation operators: explicit expression, also in sph. +Translation operators: rewrite in sph. harm. convention independent form. \end_layout diff --git a/lepaper/finite.lyx b/lepaper/finite.lyx index 46c809f..4490db2 100644 --- a/lepaper/finite.lyx +++ b/lepaper/finite.lyx @@ -290,8 +290,8 @@ outgoing , respectively, defined as follows: \begin_inset Formula \begin{align*} -\vswfrtlm1lm\left(k\vect r\right) & =j_{l}\left(kr\right)\vsh1lm\left(\uvec r\right),\\ -\vswfrtlm2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krj_{l}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(kr\right)}{kr}\vsh3lm\left(\uvec r\right), +\vswfrtlm 1lm\left(k\vect r\right) & =j_{l}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\ +\vswfrtlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krj_{l}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right), \end{align*} \end_inset @@ -299,8 +299,8 @@ 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),\\ -\vswfouttlm2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(kr\right)}{kr}\vsh3lm\left(\uvec r\right),\\ +\vswfouttlm 1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\ +\vswfouttlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(kr\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(kr\right)}{kr}\vsh 3lm\left(\uvec r\right),\\ & \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l, \end{align*} @@ -326,9 +326,9 @@ 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,\\ -\vsh2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\\ -\vsh3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right). +\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,\\ +\vsh 2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\\ +\vsh 3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right). \end{align*} \end_inset @@ -452,7 +452,7 @@ noprefix "false" \end_inset inside a ball -\begin_inset Formula $\openball0{R^{>}}$ +\begin_inset Formula $\openball 0{R^{>}}$ \end_inset with radius @@ -470,7 +470,7 @@ noprefix "false" \end_inset to have a complete basis of the solutions in the volume -\begin_inset Formula $\openball0{R^{>}}\backslash B_{0}\left(R\right)$ +\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$ \end_inset . @@ -496,7 +496,7 @@ The single-particle scattering problem at frequency \end_inset and let the whole volume -\begin_inset Formula $\openball0{R^{>}}\backslash B_{0}\left(R\right)$ +\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$ \end_inset be filled with a homogeneous isotropic medium with wave number @@ -532,7 +532,7 @@ If there was no scatterer and \end_inset due to sources outside -\begin_inset Formula $\openball0R$ +\begin_inset Formula $\openball 0R$ \end_inset would remain. @@ -793,7 +793,7 @@ literal "true" . Let the field in -\begin_inset Formula $\openball0{R^{>}}\backslash B_{0}\left(R\right)$ +\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$ \end_inset have expansion as in @@ -812,7 +812,7 @@ noprefix "false" \end_inset to -\begin_inset Formula $\openball0{R^{>}}\backslash B_{0}\left(R\right)$ +\begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$ \end_inset via by electromagnetic radiation is @@ -1535,10 +1535,37 @@ outside. \begin_layout Standard In our convention, the regular translation operator can be expressed explicitly - as + as (TODO CHECK CAREFULLY FOR POSSIBLE +\begin_inset Formula $(-1)^{m'}$ +\end_inset + + AND SIMILAR FACTORS AND REWRITE IN TERMS OF SPHERICAL HARMONICS) +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Teďka jsou tam zkopírovány výrazy pro C a D z Kristenssona, chybějí fase +\end_layout + +\end_inset + + \begin_inset Formula \begin{multline} -\tropr_{\tau lm;\tau l'm'}\left(\vect d\right)=\dots.\label{eq:translation operator} +\tropr_{\tau lm;\tau l'm'}\left(\vect d\right)=\frac{\left(-1\right)^{m+m'}}{2}\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\ +\times\begin{pmatrix}l & l' & \lambda\\ +0 & 0 & 0 +\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ +m & -m' & m'-m +\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\ +\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\\ +\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right)=-i\frac{\left(-1\right)^{m+m'}}{2}\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\ +\times\begin{pmatrix}l & l' & \lambda-1\\ +0 & 0 & 0 +\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ +m & -m' & m'-m +\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\ +\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}}.\label{eq:translation operator} \end{multline} \end_inset