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[ewald] pokračování 

Former-commit-id: c9a2cc5c707dbb90521c93bfa3e894ca956cb5ca
This commit is contained in:
Marek Nečada 2017-08-09 19:53:20 +03:00
parent 46138df4fe
commit 83fed81e24
1 changed files with 177 additions and 13 deletions
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182
notes/ewald.lyx
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@ -172,6 +172,11 @@
\end_inset
\begin_inset FormulaMacro
\newcommand{\nats}{\mathbb{N}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\reals}{\mathbb{R}}
\end_inset
@ -643,7 +648,7 @@ The translation operator
for compact scatterers in 3d can be expressed as
\begin_inset Formula
\[
S_{l',m',t'\leftarrow l,m,t}\left(\vect r\leftarrow\vect 0\right)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect r},\phi_{\vect r}\right)z_{p}^{(J)}\left(\left|\vect r\right|\right)
S_{l',m',t'\leftarrow l,m,t}\left(\vect r\leftarrow\vect 0\right)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect r},\phi_{\vect r}\right)z_{p}^{(J)}\left(k_{0}\left|\vect r\right|\right)
\]
\end_inset
@ -673,14 +678,14 @@ where
The spherical Hankel functions can be expressed analytically as (REF DLMF
10.49.6, 10.49.1)
\begin_inset Formula
\[
h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\right)!}{2^{k}k!\left(n-k\right)!},
\]
\begin{equation}
h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\right)!}{2^{k}k!\left(n-k\right)!},\label{eq:spherical Hankel function series}
\end{equation}
\end_inset
so if we find a way to deal with the radial functions
\begin_inset Formula $s_{q}(r)=e^{ir}r^{-q}$
\begin_inset Formula $s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
\end_inset
,
@ -699,18 +704,176 @@ h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\righ
\end_layout
\begin_layout Standard
Assume that all scatterers are placed in the plane
\begin_inset Formula $\vect z=0$
\end_inset
, so that the 2d Fourier transform of the long-range part of the translation
operator in terms of Hankel transforms, according to
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Fourier v. Hankel tf 2d"
\end_inset
, reads
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{multline*}
\uaft{S_{l',m',t'\leftarrow l,m,t}^{\textup{L}}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\\
\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\frac{\pi}{2},0\right)e^{i(m'-m)\phi}i^{m'-m}\pht{m'-m}{h_{p}^{(1)\textup{L}}\left(k_{0}\vect{\bullet}\right)}\left(\left|\vect k\right|\right)
\end{multline*}
\end_inset
Here
\begin_inset Formula $h_{p}^{(1)\textup{L}}=h_{p}^{(1)}-h_{p}^{(1)\textup{S}}$
\end_inset
is a long range part of a given spherical Hankel function which has to
be found and which contains all the terms with far-field (
\begin_inset Formula $r\to\infty$
\end_inset
) asymptotics proportional to
\begin_inset Formula $\sim e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
\end_inset
,
\begin_inset Formula $q\le Q$
\end_inset
where
\begin_inset Formula $Q$
\end_inset
is at least two in order to achieve absolute convergence of the direct-space
sum, but might be higher in order to speed the convergence up.
\end_layout
\begin_layout Standard
Obviously, all the terms
\begin_inset Formula $\propto s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
\end_inset
,
\begin_inset Formula $q>Q$
\end_inset
of the spherical Hankel function
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:spherical Hankel function series"
\end_inset
can be kept untouched as part of
\begin_inset Formula $h_{p}^{(1)\textup{S}}$
\end_inset
, as they decay fast enough.
\end_layout
\begin_layout Standard
The remaining task is therefore to find a suitable decomposition of
\begin_inset Formula $s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
\end_inset
,
\begin_inset Formula $q\le Q$
\end_inset
into short-range and long-range parts,
\begin_inset Formula $s_{q}(r)=s_{q}^{\textup{S}}(r)+s_{q}^{\textup{L}}(r)$
\end_inset
, such that
\begin_inset Formula $s_{q}^{\textup{L}}(r)$
\end_inset
contains all the slowly decaying asymptotics and its Hankel transforms
decay desirably fast as well,
\begin_inset Formula $\pht n{s_{q}^{\textup{L}}}\left(k\right)=o(z^{-Q})$
\end_inset
,
\begin_inset Formula $z\to\infty$
\end_inset
.
The latter requirement calls for suitable regularisation functions—
\begin_inset Formula $s_{q}^{\textup{L}}$
\end_inset
must be sufficiently smooth in the origin, so that
\begin_inset Formula
\begin{equation}
\pht n{s_{q}^{\textup{L}}}\left(k\right)=\int_{0}^{\infty}s_{q}^{\textup{L}}\left(r\right)rJ_{n}\left(kr\right)\ud r=\int_{0}^{\infty}s_{q}\left(r\right)\rho\left(r\right)rJ_{n}\left(kr\right)\ud r\label{eq:2d long range regularisation problem statement}
\end{equation}
\end_inset
exists and decays fast enough.
\begin_inset Formula $J_{\nu}(r)\sim\left(r/2\right)^{\nu}/\Gamma\left(\nu+1\right)$
\end_inset
(REF DLMF 10.7.3) near the origin, so the regularisation function should
be
\begin_inset Formula $\rho(r)=o(r^{q-n-1})$
\end_inset
only to make
\begin_inset Formula $\pht n{s_{q}^{\textup{L}}}$
\end_inset
converge.
The additional decay speed requirement calls for at least
\begin_inset Formula $\rho(r)=o(r^{q-n+Q-1})$
\end_inset
, I guess.
At the same time,
\begin_inset Formula $\rho(r)$
\end_inset
must converge fast enough to one for
\begin_inset Formula $r\to\infty$
\end_inset
.
\end_layout
\begin_layout Standard
The electrostatic Ewald summation uses regularisation with
\begin_inset Formula $1-e^{-cr^{2}}$
\end_inset
.
However, such choice does not seem to lead to an analytical solution for
the current problem
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:2d long range regularisation problem statement"
\end_inset
.
But it turns out that the family of functions
\begin_inset Formula
\[
\uaft{S_{l',m',t'\leftarrow l,m,t}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\frac{\pi}{2},0\right)e^{i(m'-m)\phi}i^{m'-m}\pht{m'-m}{z_{p}^{(J)}}\left(\left|\vect k\right|\right)
\rho_{\kappa,c}(r)\equiv\left(1-e^{-cr}\right)^{\text{\kappa}},\quad c>0,\kappa\in\nats
\]
\end_inset
leads to satisfactory results, as will be shown below.
\end_layout
\begin_layout Subsubsection
Hankel transforms of the long-range parts
\end_layout
\begin_layout Subsection
@ -719,9 +882,10 @@ h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\righ
\begin_layout Standard
\begin_inset Formula
\[
\uaft{S_{l',m',t'\leftarrow l,m,t}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect k},\phi_{\vect k}\right)\left(-i\right)^{p}\usht p{z_{p}^{(J)}}\left(\left|\vect k\right|\right)
\]
\begin{multline*}
\uaft{S_{l',m',t'\leftarrow l,m,t}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\\
\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect k},\phi_{\vect k}\right)\left(-i\right)^{p}\usht p{z_{p}^{(J)}}\left(\left|\vect k\right|\right)
\end{multline*}
\end_inset