Simplify tr.op. expression. Scattered field in periodic system.

Former-commit-id: 0b93f3bdf78a3a3d6a9a2cf628012f80790daebd

lepaper/arrayscat.lyx

@@ -395,6 +395,11 @@ status open
 \end_inset
 
 
+\begin_inset FormulaMacro
+\newcommand{\tropcoeff}{C}
+\end_inset
+
+
 \begin_inset FormulaMacro
 \newcommand{\truncated}[2]{\left[#1\right]_{l\le#2}}
 \end_inset

lepaper/finite.lyx

@@ -1919,10 +1919,27 @@ outside.
 \begin_layout Standard
 In our convention, the regular translation operator elements can be expressed
  explicitly as 
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
 \begin_inset Formula 
 \begin{align}
 \tropr_{\tau lm;\tau l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|}^{l+l'}C_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)j_{\lambda}\left(d\right),\nonumber \\
-\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}D_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)j_{\lambda}\left(d\right),\quad\tau'\ne\tau,\label{eq:translation operator}
+\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}D_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)j_{\lambda}\left(d\right),\quad\tau'\ne\tau,\label{eq:translation operator-1}
+\end{align}
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula 
+\begin{align}
+\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)j_{\lambda}\left(d\right),\label{eq:translation operator}
 \end{align}
 
 \end_inset
@@ -1936,10 +1953,27 @@ and analogously the elements of the singular operator
 \end_inset
 
 ) in the radial part instead of the regular bessel functions,
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
 \begin_inset Formula 
 \begin{align}
 \trops_{\tau lm;\tau l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|}^{l+l'}C_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)h_{\lambda}^{(1)}\left(d\right),\nonumber \\
-\trops_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}D_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)h_{\lambda}^{(1)}\left(d\right),\quad\tau'\ne\tau,\label{eq:translation operator singular}
+\trops_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}D_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)h_{\lambda}^{(1)}\left(d\right),\quad\tau'\ne\tau,\label{eq:translation operator singular-1}
+\end{align}
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula 
+\begin{align}
+\trops_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)h_{\lambda}^{(1)}\left(d\right),\label{eq:translation operator singular}
 \end{align}
 
 \end_inset
@@ -2135,15 +2169,26 @@ m & -m' & m'-m
 \end_inset
 
 
+\begin_inset Formula 
+\[
+\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}=\begin{cases}
+A_{lm;l'm'}^{\lambda} & \tau=\tau',\\
+B_{lm;l'm'}^{\lambda} & \tau\ne\tau',
+\end{cases}
+\]
+
+\end_inset
+
+
 \begin_inset Formula 
 \begin{multline}
-C_{lm;l'm'}^{\lambda}=\left(-1\right)^{\frac{l'-l+\lambda}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\
+A_{lm;l'm'}^{\lambda}=\left(-1\right)^{\frac{l'-l+\lambda}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\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),\\
-D_{lm;l'm'}^{\lambda}=-i\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\
+B_{lm;l'm'}^{\lambda}=-i\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\
 \times\begin{pmatrix}l & l' & \lambda-1\\
 0 & 0 & 0
 \end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
@@ -2222,16 +2267,6 @@ todo different notation for the complex conjugation without transposition???
 , which has to be taken into consideration when evaluating quantities such
  as absorption or scattering cross sections.
  Similarly, the full regular operators can be composed
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-better wording
-\end_layout
-
-\end_inset
-
- 
 \begin_inset Formula 
 \begin{equation}
 \troprp ac=\troprp ab\troprp bc\label{eq:regular translation composition}

lepaper/infinite.lyx

@@ -140,6 +140,13 @@ Topology anoyne?
 
 \begin_layout Subsection
 Formulation of the problem
+\begin_inset CommandInset label
+LatexCommand label
+name "subsec:Quasiperiodic scattering problem"
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
@@ -1061,6 +1068,10 @@ W_{\alpha\beta}(\vect k)\equiv\sum_{\vect m\in\ints^{d}}\left(1-\delta_{\alpha\b
 \end_inset
 
  
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
 \begin_inset Formula 
 \begin{align*}
 W_{\alpha,\tau lm;\beta,\tau l'm'}(\vect k) & =\sum_{\lambda=\left|l-l'\right|}^{l+l'}C_{lm;l'm'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r_{\beta}-\vect r_{\alpha}\right),\\
@@ -1070,6 +1081,19 @@ W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k) & =\sum_{\lambda=\left|l-l'\right|+1
 \end_inset
 
 
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k)=\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r_{\beta}-\vect r_{\alpha}\right),\quad\tau'\ne\tau,
+\]
+
+\end_inset
+
+
 \begin_inset Note Note
 status open
 
@@ -1913,7 +1937,8 @@ Whatabout different geometries?
 
 \end_inset
 
- However, at greater wavelengths
+ However, at larger wavelengths, TODO BLA BLA BLA which is detrimental for
+ accuracy in floating point arithmetics.
 \end_layout
 
 \begin_layout Standard
@@ -2015,14 +2040,14 @@ noprefix "false"
  Fortunately, these can be obtained easily from the expressions for the
  translation operator: 
 \begin_inset Formula 
-\begin{align*}
+\begin{align}
-\vswfrtlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\tropr_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm 21{m'}\left(0\right),\\
+\vswfrtlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\tropr_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm 21{m'}\left(0\right),\nonumber \\
-\vswfouttlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm 21{m'}\left(0\right),
+\vswfouttlm{\tau}lm\left(\kappa\vect r\right) & =\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\vect r\right)\vswfrtlm 21{m'}\left(0\right),\label{eq:VSWFs expressed as translated dipole waves}
-\end{align*}
+\end{align}
 
 \end_inset
 
-where we used eqs.
+which follows from eqs.
  
 \begin_inset CommandInset ref
 LatexCommand eqref
@@ -2048,6 +2073,102 @@ noprefix "false"
 \end_inset
 
  vanish at origin.
+ For the quasiperiodic scattering problem formulated in section 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "subsec:Quasiperiodic scattering problem"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+, the total electric field scattered from all the particles at point 
+\begin_inset Formula $\vect r$
+\end_inset
+
+ located outside all the particles' circumscribing sphere reads, using eqs.
+ 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "eq:translation operator singular"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+, 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "eq:sigma lattice sums"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+, 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "eq:scalar spherical wavefunctions"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+,
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\begin{align}
+\trops_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\psi_{\lambda,m-m'}\left(\vect d\right),\label{eq:translation operator singular-1}
+\end{align}
+
+\end_inset
+
+
+\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(\kappa\left(\vect{R_{n}}-\vect s\right)\right),\label{eq:sigma lattice sums}
+\end{equation}
+
+\end_inset
+
+
+\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_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lme^{i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm 21{m'}\left(0\right)\\
+ & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lme^{-i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\trops_{\tau lm;21m'}\left(\kappa\left(\vect r+\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm 21{m'}\left(0\right),\text{FIXME signs}\\
+ & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lme^{-i\vect k\cdot\vect R_{\vect n}}\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\psi_{\lambda,m-m'}\left(\kappa\left(\vect r+\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\vswfrtlm 21{m'}\left(0\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*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+TODO fix signs and exponential phase factors
+\end_layout
+
+\end_inset
+
+
+\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*}
+
+\end_inset
+
+
 \end_layout
 
 \end_body