"Rule of thumb" on cutoff

Former-commit-id: 79a98d83bd88baab80d454ec4147e4a0b739647c

lepaper/finite.lyx

@@ -746,9 +746,93 @@ A rule of thumb to choose the
 \begin_inset Formula $\delta$
 \end_inset
 
- can be obtained from the spherical Bessel function expansion around zero,
+ can be obtained from the spherical Bessel function expansion around zero
- TODO.
  
+\begin_inset CommandInset citation
+LatexCommand cite
+after "10.52.1"
+key "NIST:DLMF"
+literal "false"
+
+\end_inset
+
+ by requiring that 
+\begin_inset Formula $\delta\gtrsim\left(nR\right)^{L}/\left(2L+1\right)!!$
+\end_inset
+
+, where 
+\begin_inset Formula $R$
+\end_inset
+
+ is the scatterer radius and 
+\begin_inset Formula $n$
+\end_inset
+
+ its (maximum) refractive index.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\begin{align*}
+\left(2n+1\right)!! & =\frac{\left(2n+1\right)!}{2^{n}n!},\\
+\delta\gtrsim & \frac{R^{L}}{\left(2L+1\right)!!}=\frac{\left(2R\right)^{L}L!}{\left(2L+1\right)!}
+\end{align*}
+
+\end_inset
+
+Stirling 
+\begin_inset Formula $n!\approx\sqrt{2\pi n}\left(n/e\right)^{n}$
+\end_inset
+
+ so 
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula 
+\begin{align*}
+\delta & \gtrsim\left(2R\right)^{L}\frac{\sqrt{2\pi L}\left(\frac{L}{e}\right)^{L}}{\sqrt{2\pi\left(2L+1\right)}\left(\frac{2L+1}{e}\right)^{2L+1}}\\
+\delta & \gtrsim\left(2R\right)^{L}\frac{\sqrt{L}\left(\frac{L}{e}\right)^{L}}{\sqrt{2L+1}\left(\frac{2L+1}{e}\right)^{2L+1}}\\
+\log\delta & \gtrsim L\log2+L\log R+\frac{1}{2}\log L-\frac{1}{2}\log\left(2L+1\right)+L\log L-L\log e-\left(2L+1\right)\log\left(2L+1\right)+\left(2L+1\right)\log e\\
+\log\delta & \gtrsim L\log2+L\log R+\left(L+\frac{1}{2}\right)\log L-\left(2L+\frac{3}{2}\right)\log\left(2L+1\right)+\left(L+1\right)\\
+ & >L\log2+L\log R+\left(L+\frac{1}{2}\right)\log L-\left(2L+\frac{3}{2}\right)\log\left(2L\right)+\left(L+1\right)\\
+ & =L\log2+L\log R-\left(L+1\right)\log L-\left(2L+\frac{3}{2}\right)\log\left(2\right)+\left(L+1\right)\\
+ & =L\log2+L\log R-\left(L+1\right)\log L-\left(2L+\frac{3}{2}\right)\log\left(2\right)+\left(L+1\right)
+\end{align*}
+
+\end_inset
+
+too complicated, watabout
+\begin_inset Formula 
+\[
+\delta\gtrsim\left(2R\right)^{L}\frac{L^{L+1/2}e^{L}}{\left(2L\right)^{2L}}=\frac{R^{L}e^{L}}{2^{L}}L^{L+1/2}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\log\delta\gtrsim L\log\frac{ReL}{2}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\log\delta\gtrsim L\log\frac{ReL}{2}
+\]
+
+\end_inset
+
+yäk
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Subsubsection

lepaper/symmetries.tex

@@ -1,149 +0,0 @@
-\selectlanguage{finnish}%
-
-\section{Symmetries}\label{sec:Symmetries}
-
-If the system has nontrivial point group symmetries, group theory
-gives additional understanding of the system properties, and can be
-used to reduce the computational costs. 
-
-As an example, if our system has a $D_{2h}$ symmetry and our truncated
-$\left(I-T\trops\right)$ matrix has size $N\times N$, it can be
-block-diagonalized into eight blocks of size about $N/8\times N/8$,
-each of which can be LU-factorised separately (this is due to the
-fact that $D_{2h}$ has eight different one-dimensional irreducible
-representations). This can reduce both memory and time requirements
-to solve the scattering problem (\ref{eq:Multiple-scattering problem block form})
-by a factor of 64.
-
-In periodic systems (problems (\ref{eq:Multiple-scattering problem unit cell block form}),
-(\ref{eq:lattice mode equation})) due to small number of particles
-per unit cell, the costliest part is usually the evaluation of the
-lattice sums in the $W\left(\omega,\vect k\right)$ matrix, not the
-linear algebra. However, the lattice modes can be searched for in
-each irrep separately, and the irrep dimension gives a priori information
-about mode degeneracy.
-
-\subsection{Excitation coefficients under point group operations}
-
-In order to use the point group symmetries, we first need to know
-how they affect our basis functions, i.e. the VSWFs.
-
-Let $g$ be a member of orthogonal group $O(3)$, i.e. a 3D point
-rotation or reflection operation that transforms vectors in $\reals^{3}$
-with an orthogonal matrix $R_{g}$:
-\[
-\vect r\mapsto R_{g}\vect r.
-\]
-Spherical harmonics $\ush lm$, being a basis the $l$-dimensional
-representation of $O(3)$, transform as \cite[???]{dresselhaus_group_2008}
-\[
-\ush lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\ush l{m'}\left(\uvec r\right)
-\]
-where $D_{m,m'}^{l}\left(g\right)$ denotes the elements of the \emph{Wigner
-matrix} representing the operation $g$. By their definition, vector
-spherical harmonics $\vsh 2lm,\vsh 3lm$ transform in the same way,
-\begin{align*}
-\vsh 2lm\left(R_{g}\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 2l{m'}\left(\uvec r\right),\\
-\vsh 3lm\left(R_{g}\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 3l{m'}\left(\uvec r\right),
-\end{align*}
-but the remaining set $\vsh 1lm$ transforms differently due to their
-pseudovector nature stemming from the cross product in their definition:
-\[
-\vsh 3lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vsh 3l{m'}\left(\uvec r\right),
-\]
-where $\widetilde{D_{m,m'}^{l}}\left(g\right)=D_{m,m'}^{l}\left(g\right)$
-if $g$ is a proper rotation, but for spatial inversion operation
-$i:\vect r\mapsto-\vect r$ we have $\widetilde{D_{m,m'}^{l}}\left(i\right)=\left(-1\right)^{l+m}D_{m,m'}^{l}\left(i\right)$.
-The transformation behaviour of vector spherical harmonics directly
-propagates to the spherical vector waves, cf. (\ref{eq:VSWF regular}),
-(\ref{eq:VSWF outgoing}):
-\begin{align*}
-\vswfouttlm 1lm\left(R_{g}\vect r\right) & =\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vswfouttlm 1l{m'}\left(\vect r\right),\\
-\vswfouttlm 2lm\left(R_{g}\vect r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vswfouttlm 2l{m'}\left(\vect r\right),
-\end{align*}
-(and analogously for the regular waves $\vswfrtlm{\tau}lm$).  For
-convenience, we introduce the symbol $D_{m,m'}^{\tau l}$ that describes
-the transformation of both types (``magnetic'' and ``electric'')
-of waves at once:
-\[
-\vswfouttlm{\tau}lm\left(R_{g}\vect r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\vect r\right).
-\]
-Using these, we can express the VSWF expansion (\ref{eq:E field expansion})
-of the electric field around origin in a rotated/reflected system,
-\[
-\vect E\left(\omega,R_{g}\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\vect r\right)+\outcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\vect r\right)\right),
-\]
-which, together with the $T$-matrix definition, (\ref{eq:T-matrix definition})
-can be used to obtain a $T$-matrix of a rotated or mirror-reflected
-particle. Let $T$ be the $T$-matrix of an original particle; the
-$T$-matrix of a particle physically transformed by operation $g\in O(3)$
-is then 
-\begin{equation}
-T'_{\tau lm;\tau'l'm'}=\sum_{\mu=-l}^{l}\sum_{\mu'=-l'}^{l'}\left(D_{\mu,m}^{\tau l}\left(g\right)\right)^{*}T_{\tau l\mu;\tau'l'm'}D_{m',\mu'}^{\tau l}\left(g\right).\label{eq:T-matrix of a transformed particle}
-\end{equation}
-If the particle is symmetric (so that $g$ produces a particle indistinguishable
-from the original one), the $T$-matrix must remain invariant under
-the transformation (\ref{eq:T-matrix of a transformed particle}),
-$T'_{\tau lm;\tau'l'm'}=T{}_{\tau lm;\tau'l'm'}$. Explicit forms
-of these invariance properties for the most imporant point group symmetries
-can be found in \cite{schulz_point-group_1999}.
-
-If the field expansion is done around a point $\vect r_{p}$ different
-from the global origin, as in \ref{eq:E field expansion multiparticle},
-we have\foreignlanguage{english}{
-\begin{align}
-\vect E\left(\omega,R_{g}\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)+\outcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)\right).\label{eq:rotated E field expansion around outside origin}
-\end{align}
-}
-
-\begin{figure}
-\caption{Scatterer orbits under $D_{2}$ symmetry. Particles $A,B,C,D$ lie
-outside of origin or any mirror planes, and together constitute an
-orbit of the size equal to the order of the group, $\left|D_{2}\right|=4$.
-Particles $E,F$ lie on the $xz$ plane, hence the corresponding reflection
-maps each of them to itself, but the $yz$ reflection (or the $\pi$
-rotation around the $z$ axis) maps them to each other, forming a
-particle orbit of size 2. The particle $O$ in the very origin is
-always mapped to itself, constituting its own orbit.}\label{fig:D2-symmetric structure particle orbits}
-\end{figure}
-
-With these transformation properties in hand, we can proceed to the
-effects of point symmetries on the whole many-particle system. Let
-us have a many-particle system symmetric with respect to a point group
-$G$. A symmetry operation $g\in G$ determines a permutation of the
-particles: $p\mapsto\pi_{g}(p)$, $p\in\mathcal{P}$. For a given
-particle $p$, we will call the set of particles onto which any of
-the symmetries maps the particle $p$, i.e. the set $\left\{ \pi_{g}\left(p\right);g\in G\right\} $,
-as the \emph{orbit} of particle $p$. The whole set $\mathcal{P}$
-can therefore be divided into the different particle orbits; an example
-is in Fig. \ref{fig:D2-symmetric structure particle orbits}. The
-importance of the particle orbits stems from the following: in the
-multiple-scattering problem, outside of the scatterers  one has \foreignlanguage{english}{
-\begin{align}
-\vect E\left(\omega,R_{g}\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{\pi_{g}(p)}\right)\right)+\outcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)\right)\label{eq:rotated E field expansion around outside origin-1}\\
- & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)+\outcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)\right).
-\end{align}
-This means that the field expansion coefficients $\rcoeffp p,\outcoeffp p$
-transform as 
-\begin{align}
-\rcoeffptlm p{\tau}lm & \mapsto\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right),\nonumber \\
-\outcoeffptlm p{\tau}lm & \mapsto\outcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right).\label{eq:excitation coefficient under symmetry operation}
-\end{align}
-Obviously, the expansion coefficients belonging to particles in different
-orbits do not mix together. As before, we introduce a short-hand block-matrix
-notation for \ref{eq:excitation coefficient under symmetry operation}}
-
-\selectlanguage{english}%
-\begin{align}
-\rcoeff & \mapsto D\left(g\right)a,\nonumber \\
-\outcoeff & \mapsto D\left(g\right)\outcoeff.\label{eq:excitation coefficient under symmetry operation block form}
-\end{align}
-
-\selectlanguage{finnish}%
-
-\subsection{Irrep decomposition}
-
-\subsection{Periodic systems}
-
-\selectlanguage{english}%
-