Simplify tr.op. expression. Scattered field in periodic system.
Former-commit-id: 0b93f3bdf78a3a3d6a9a2cf628012f80790daebd
This commit is contained in:
Marek Nečada
parent
16f12c041b
commit
b80e7607f8
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@ -395,6 +395,11 @@ status open
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\tropcoeff}{C}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\truncated}[2]{\left[#1\right]_{l\le#2}}
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\end_inset
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@ -1919,10 +1919,27 @@ outside.
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\begin_layout Standard
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In our convention, the regular translation operator elements can be expressed
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explicitly as
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\begin_inset Note Note
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status collapsed
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\begin_layout Plain Layout
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\begin_inset Formula
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\begin{align}
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\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 \\
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\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}
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\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}
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\end{align}
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\end_inset
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\end_layout
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\end_inset
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\begin_inset Formula
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\begin{align}
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\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}
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\end{align}
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\end_inset
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@ -1936,10 +1953,27 @@ and analogously the elements of the singular operator
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\end_inset
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) in the radial part instead of the regular bessel functions,
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\begin_inset Note Note
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status collapsed
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\begin_layout Plain Layout
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\begin_inset Formula
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\begin{align}
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\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 \\
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\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}
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\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}
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\end{align}
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\end_inset
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\end_layout
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\end_inset
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\begin_inset Formula
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\begin{align}
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\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}
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\end{align}
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\end_inset
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@ -2135,15 +2169,26 @@ m & -m' & m'-m
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\end_inset
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\begin_inset Formula
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\[
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\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}=\begin{cases}
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A_{lm;l'm'}^{\lambda} & \tau=\tau',\\
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B_{lm;l'm'}^{\lambda} & \tau\ne\tau',
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\end{cases}
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\]
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\end_inset
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\begin_inset Formula
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\begin{multline}
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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\\
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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\\
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\times\begin{pmatrix}l & l' & \lambda\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right),\\
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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\\
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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\\
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\times\begin{pmatrix}l & l' & \lambda-1\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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@ -2222,16 +2267,6 @@ todo different notation for the complex conjugation without transposition???
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, which has to be taken into consideration when evaluating quantities such
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as absorption or scattering cross sections.
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Similarly, the full regular operators can be composed
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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better wording
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\end_layout
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\end_inset
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\begin_inset Formula
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\begin{equation}
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\troprp ac=\troprp ab\troprp bc\label{eq:regular translation composition}
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@ -140,6 +140,13 @@ Topology anoyne?
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\begin_layout Subsection
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Formulation of the problem
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\begin_inset CommandInset label
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LatexCommand label
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name "subsec:Quasiperiodic scattering problem"
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\end_inset
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\end_layout
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\begin_layout Standard
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@ -1061,6 +1068,10 @@ W_{\alpha\beta}(\vect k)\equiv\sum_{\vect m\in\ints^{d}}\left(1-\delta_{\alpha\b
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\end_inset
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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\begin_inset Formula
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\begin{align*}
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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),\\
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@ -1070,6 +1081,19 @@ W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k) & =\sum_{\lambda=\left|l-l'\right|+1
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\end_inset
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\end_layout
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\end_inset
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\begin_inset Formula
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\[
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W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k)=\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r_{\beta}-\vect r_{\alpha}\right),\quad\tau'\ne\tau,
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\]
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\end_inset
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\begin_inset Note Note
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status open
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@ -1913,7 +1937,8 @@ Whatabout different geometries?
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\end_inset
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However, at greater wavelengths
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However, at larger wavelengths, TODO BLA BLA BLA which is detrimental for
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accuracy in floating point arithmetics.
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\end_layout
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\begin_layout Standard
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@ -2015,14 +2040,14 @@ noprefix "false"
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Fortunately, these can be obtained easily from the expressions for the
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translation operator:
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\begin_inset Formula
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\begin{align*}
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\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),\\
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\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),
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\end{align*}
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\begin{align}
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\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 \\
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\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}
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\end{align}
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\end_inset
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where we used eqs.
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which follows from eqs.
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\begin_inset CommandInset ref
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LatexCommand eqref
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@ -2048,6 +2073,102 @@ noprefix "false"
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\end_inset
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vanish at origin.
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For the quasiperiodic scattering problem formulated in section
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "subsec:Quasiperiodic scattering problem"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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, the total electric field scattered from all the particles at point
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\begin_inset Formula $\vect r$
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\end_inset
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located outside all the particles' circumscribing sphere reads, using eqs.
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:translation operator singular"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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,
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:sigma lattice sums"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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,
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:scalar spherical wavefunctions"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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,
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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\begin_inset Formula
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\begin{align}
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\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}
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\end{align}
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\end_inset
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\begin_inset Formula
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\begin{equation}
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\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}
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\end{equation}
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\end_inset
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\begin_inset Formula
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\begin{align*}
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\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)\\
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& =\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)\\
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& =\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}\\
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& =\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)\\
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& =\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)
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\end{align*}
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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TODO fix signs and exponential phase factors
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\end_layout
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\end_inset
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\begin_inset Formula
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\begin{align*}
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\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)\\
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& =\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).
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\end{align*}
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\end_inset
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\end_layout
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\end_body
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