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Lyx document structure Former-commit-id: 92b5d1d22d1447209145f6b062fcef74b1884721
2019-06-30 21:30:54 +03:00
#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\pdf_author "Marek Nečada"
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\begin_body
\begin_layout Section
Infinite periodic systems
\end_layout
Paper start single-particle part Former-commit-id: 287e4e9848528b84df91395522e34236f4d47535
2019-07-01 14:50:46 +03:00
\begin_layout Subsection
\lang english
Formulation of the problem
\end_layout
\begin_layout Standard
\lang english
Assume a system of compact EM scatterers in otherwise homogeneous and isotropic
medium, and assume that the system, i.e.
both the medium and the scatterers, have linear response.
A scattering problem in such system can be written as
\begin_inset Formula
\[
A_{α}=T_{α}P_{α}=T_{α}(\sum_{β}S_{α\leftarrowβ}A_{β}+P_{0α})
\]
\end_inset
where
\begin_inset Formula $T_{α}$
\end_inset
is the
\begin_inset Formula $T$
\end_inset
-matrix for scatterer α,
\begin_inset Formula $A_{α}$
\end_inset
is its vector of the scattered wave expansion coefficient (the multipole
indices are not explicitely indicated here) and
\begin_inset Formula $P_{α}$
\end_inset
is the local expansion of the incoming sources.
\begin_inset Formula $S_{α\leftarrowβ}$
\end_inset
is ...
and ...
is ...
\end_layout
\begin_layout Standard
\lang english
...
\end_layout
\begin_layout Standard
\lang english
\begin_inset Formula
\[
\sum_{β}(\delta_{αβ}-T_{α}S_{α\leftarrowβ})A_{β}=T_{α}P_{0α}.
\]
\end_inset
\end_layout
\begin_layout Standard
\lang english
Now suppose that the scatterers constitute an infinite lattice
\end_layout
\begin_layout Standard
\lang english
\begin_inset Formula
\[
\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{\vect aα}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=T_{\vect aα}P_{0\vect aα}.
\]
\end_inset
Due to the periodicity, we can write
\begin_inset Formula $S_{\vect aα\leftarrow\vect bβ}=S_{α\leftarrowβ}(\vect b-\vect a)$
\end_inset
and
\begin_inset Formula $T_{\vect aα}=T_{\alpha}$
\end_inset
.
In order to find lattice modes, we search for solutions with zero RHS
\begin_inset Formula
\[
\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=0
\]
\end_inset
and we assume periodic solution
\begin_inset Formula $A_{\vect b\beta}(\vect k)=A_{\vect a\beta}e^{i\vect k\cdot\vect r_{\vect b-\vect a}}$
\end_inset
, yielding
\begin_inset Formula
\begin{eqnarray*}
\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect a\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b-\vect a}} & = & 0,\\
\sum_{\vect bβ}(\delta_{\vect{0b}}\delta_{αβ}-T_{α}S_{\vect 0α\leftarrow\vect bβ})A_{\vect 0\beta}\left(\vect k\right)e^{i\vect k\cdot\vect r_{\vect b}} & = & 0,\\
\sum_{β}(\delta_{αβ}-T_{α}\underbrace{\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}}_{W_{\alpha\beta}(\vect k)})A_{\vect 0\beta}\left(\vect k\right) & = & 0,\\
A_{\vect 0\alpha}\left(\vect k\right)-T_{α}\sum_{\beta}W_{\alpha\beta}\left(\vect k\right)A_{\vect 0\beta}\left(\vect k\right) & = & 0.
\end{eqnarray*}
\end_inset
Therefore, in order to solve the modes, we need to compute the
\begin_inset Quotes eld
\end_inset
lattice Fourier transform
\begin_inset Quotes erd
\end_inset
of the translation operator,
\begin_inset Formula
\begin{equation}
W_{\alpha\beta}(\vect k)\equiv\sum_{\vect b}S_{\vect 0α\leftarrow\vect bβ}e^{i\vect k\cdot\vect r_{\vect b}}.\label{eq:W definition}
\end{equation}
\end_inset
\end_layout
\begin_layout Subsection
\lang english
Computing the Fourier sum of the translation operator
\end_layout
\begin_layout Standard
\lang english
The problem evaluating
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W definition"
\end_inset
is the asymptotic behaviour of the translation operator,
\begin_inset Formula $S_{\vect 0α\leftarrow\vect bβ}\sim\left|\vect r_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect r_{\vect b}\right|}$
\end_inset
that makes the convergence of the sum quite problematic for any
\begin_inset Formula $d>1$
\end_inset
-dimensional lattice.
\begin_inset Foot
status open
\begin_layout Plain Layout
\lang english
Note that
\begin_inset Formula $d$
\end_inset
here is dimensionality of the lattice, not the space it lies in, which
I for certain reasons assume to be three.
(TODO few notes on integration and reciprocal lattices in some appendix)
\end_layout
\end_inset
In electrostatics, one can solve this problem with Ewald summation.
Its basic idea is that if what asymptoticaly decays poorly in the direct
space, will perhaps decay fast in the Fourier space.
I use the same idea here, but everything will be somehow harder than in
electrostatics.
\end_layout
\begin_layout Standard
\lang english
Let us re-express the sum in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W definition"
\end_inset
in terms of integral with a delta comb
\end_layout
\begin_layout Standard
\lang english
\begin_inset Formula
\begin{equation}
W_{\alpha\beta}(\vect k)=\int\ud^{d}\vect r\dc{\basis u}(\vect r)S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})e^{i\vect k\cdot\vect r}.\label{eq:W integral}
\end{equation}
\end_inset
The translation operator
\begin_inset Formula $S$
\end_inset
is now a function defined in the whole 3d space;
\begin_inset Formula $\vect r_{\alpha},\vect r_{\beta}$
\end_inset
are the displacements of scatterers
\begin_inset Formula $\alpha$
\end_inset
and
\begin_inset Formula $\beta$
\end_inset
in a unit cell.
The arrow notation
\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})$
\end_inset
means
\begin_inset Quotes eld
\end_inset
translation operator for spherical waves originating in
\begin_inset Formula $\vect r+\vect r_{\beta}$
\end_inset
evaluated in
\begin_inset Formula $\vect r_{\alpha}$
\end_inset
\begin_inset Quotes erd
\end_inset
and obviously
\begin_inset Formula $S$
\end_inset
is in fact a function of a single 3d argument,
\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect 0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
\end_inset
.
Expression
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W integral"
\end_inset
can be rewritten as
\begin_inset Formula
\[
W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0))\left(\vect k\right)}
\]
\end_inset
where changed the sign of
\begin_inset Formula $\vect r/\vect{\bullet}$
\end_inset
has been swapped under integration, utilising evenness of
\begin_inset Formula $\dc{\basis u}$
\end_inset
.
Fourier transform of product is convolution of Fourier transforms, so (using
formula
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Dirac comb uaFt"
\end_inset
for the Fourier transform of Dirac comb)
\begin_inset Formula
\begin{eqnarray}
W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)(\vect k)\nonumber \\
& = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\recb{\basis u}}^{(d)}\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)\left(\vect k\right)\nonumber \\
& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W sum in reciprocal space}\\
& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}e^{i\left(\vect k-\vect K\right)\cdot\left(-\vect r_{\beta}+\vect r_{\alpha}\right)}\left(\uaft{S(\vect{\bullet}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\nonumber
\end{eqnarray}
\end_inset
\begin_inset Note Note
status open
\begin_layout Plain Layout
\lang english
Factor
\begin_inset Formula $\left(2\pi\right)^{\frac{d}{2}}$
\end_inset
cancels out with the
\begin_inset Formula $\left(2\pi\right)^{-\frac{d}{2}}$
\end_inset
factor appearing in the convolution/product formula in the unitary angular
momentum convention.
\end_layout
\end_inset
As such, this is not extremely helpful because the the
\emph on
whole
\emph default
translation operator
\begin_inset Formula $S$
\end_inset
has singularities in origin, hence its Fourier transform
\begin_inset Formula $\uaft S$
\end_inset
will decay poorly.
\end_layout
\begin_layout Standard
\lang english
However, Fourier transform is linear, so we can in principle separate
\begin_inset Formula $S$
\end_inset
in two parts,
\begin_inset Formula $S=S^{\textup{L}}+S^{\textup{S}}$
\end_inset
.
\begin_inset Formula $S^{\textup{S}}$
\end_inset
is a short-range part that decays sufficiently fast with distance so that
its direct-space lattice sum converges well;
\begin_inset Formula $S^{\textup{S}}$
\end_inset
must as well contain all the singularities of
\begin_inset Formula $S$
\end_inset
in the origin.
The other part,
\begin_inset Formula $S^{\textup{L}}$
\end_inset
, will retain all the slowly decaying terms of
\begin_inset Formula $S$
\end_inset
but it also has to be smooth enough in the origin, so that its Fourier
transform
\begin_inset Formula $\uaft{S^{\textup{L}}}$
\end_inset
decays fast enough.
(The same idea lies behind the Ewald summation in electrostatics.) Using
the linearity of Fourier transform and formulae
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W definition"
\end_inset
and legendre
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:W sum in reciprocal space"
\end_inset
, the operator
\begin_inset Formula $W_{\alpha\beta}$
\end_inset
can then be re-expressed as
\begin_inset Formula
\begin{eqnarray}
W_{\alpha\beta}\left(\vect k\right) & = & W_{\alpha\beta}^{\textup{S}}\left(\vect k\right)+W_{\alpha\beta}^{\textup{L}}\left(\vect k\right)\nonumber \\
W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect 0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\
W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S^{\textup{L}}(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition}
\end{eqnarray}
\end_inset
where both sums should converge nicely.
\end_layout
Lyx document structure Former-commit-id: 92b5d1d22d1447209145f6b062fcef74b1884721
2019-06-30 21:30:54 +03:00
\end_body
\end_document