Infinite lattices WIP
Former-commit-id: d7afe75b6a8dc2d4ad38b607446c7ab675391b0c
This commit is contained in:
Marek Nečada
parent
c51a567f6d
commit
a6e90b43ae
|
|
@ -149,11 +149,25 @@
|
|||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\dc}[1]{Ш_{#1}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\dc}[1]{|||_{#1}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\rec}[1]{#1^{-1}}
|
||||
\end_inset
|
||||
|
|
@ -379,6 +393,11 @@
|
|||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\antidelta}{\gamma}
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
|
@ -705,6 +724,10 @@ Example results.
|
|||
Concrete comparison with other methods.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Itemize
|
||||
Fix notation (mainly index) clashes in infinite lattices.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
\begin_inset CommandInset include
|
||||
LatexCommand include
|
||||
|
|
|
|||
|
|
@ -1149,7 +1149,7 @@ where
|
|||
\begin_inset Formula $I$
|
||||
\end_inset
|
||||
|
||||
is the identity matrix and
|
||||
is the identity matrix,
|
||||
\begin_inset Formula $T$
|
||||
\end_inset
|
||||
|
||||
|
|
@ -1157,7 +1157,16 @@ is a block-diagonal matrix containing all the individual
|
|||
\begin_inset Formula $T$
|
||||
\end_inset
|
||||
|
||||
-matrices.
|
||||
-matrices, and
|
||||
\begin_inset Formula $\trops$
|
||||
\end_inset
|
||||
|
||||
contains the individual
|
||||
\begin_inset Formula $\tropsp pq$
|
||||
\end_inset
|
||||
|
||||
matrices as the off-diagonal blocks, whereas the diagonal blocks are set
|
||||
to zeros.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
|
|
|||
|
|
@ -96,7 +96,7 @@
|
|||
\begin_layout Section
|
||||
Infinite periodic systems
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\dlv}{\vect b}
|
||||
\newcommand{\dlv}{\vect a}
|
||||
\end_inset
|
||||
|
||||
|
||||
|
|
@ -135,6 +135,10 @@ Topology anoyne?
|
|||
Notation
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
TODO Fourier transforms, Delta comb, lattice bases etc.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Subsection
|
||||
Formulation of the problem
|
||||
\end_layout
|
||||
|
|
@ -151,7 +155,7 @@ noprefix "false"
|
|||
|
||||
\end_inset
|
||||
|
||||
, but this time, system shall be periodic: let there be a
|
||||
, but this time, the system shall be periodic: let there be a
|
||||
\begin_inset Formula $d$
|
||||
\end_inset
|
||||
|
||||
|
|
@ -160,110 +164,131 @@ noprefix "false"
|
|||
\end_inset
|
||||
|
||||
can be 1, 2 or 3) lattice embedded into the three-dimensional real space,
|
||||
with lattice vectors.
|
||||
set of
|
||||
with lattice vectors
|
||||
\begin_inset Formula $\left\{ \dlv_{i}\right\} _{i=1}^{d}$
|
||||
\end_inset
|
||||
|
||||
, and let the lattice points be labeled with an
|
||||
\begin_inset Formula $d$
|
||||
\end_inset
|
||||
|
||||
(one to three) lattice vectorsAssume 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
|
||||
\[
|
||||
A_{α}=T_{α}P_{α}=T_{α}(\sum_{β}S_{α\leftarrowβ}A_{β}+P_{0α})
|
||||
\]
|
||||
|
||||
-dimensional integar multiindex
|
||||
\begin_inset Formula $\vect n\in\ints^{d}$
|
||||
\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
|
||||
...
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
\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
|
||||
Now suppose that the scatterers constitute an infinite lattice
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
\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 $T_{\vect aα}=T_{\alpha}$
|
||||
, so the lattice points have cartesian coordinates
|
||||
\begin_inset Formula $\vect R_{\vect n}=\sum_{i=1}^{d}n_{i}\vect a_{i}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
In order to find lattice modes, we search for solutions with zero RHS
|
||||
There can be several scatterers per unit cell with indices
|
||||
\begin_inset Formula $\alpha$
|
||||
\end_inset
|
||||
|
||||
from set
|
||||
\begin_inset Formula $\mathcal{P}_{1}$
|
||||
\end_inset
|
||||
|
||||
and (relative) positions inside the unit cell
|
||||
\begin_inset Formula $\vect r_{\alpha}$
|
||||
\end_inset
|
||||
|
||||
; any particle of the periodic system can thus be labeled by a multiindex
|
||||
from
|
||||
\begin_inset Formula $\mathcal{P}=\ints^{d}\times\mathcal{P}_{1}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
The scatterers are located at positions
|
||||
\begin_inset Formula $\vect r_{\vect n,\alpha}=\vect R_{\vect n}+\vect r_{\alpha}$
|
||||
\end_inset
|
||||
|
||||
and their
|
||||
\begin_inset Formula $T$
|
||||
\end_inset
|
||||
|
||||
-matrices are periodic,
|
||||
\begin_inset Formula $T_{\vect n,\alpha}=T_{\alpha}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
In such system, the multiple-scattering problem
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "eq:Multiple-scattering problem"
|
||||
plural "false"
|
||||
caps "false"
|
||||
noprefix "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
can be rewritten as
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\sum_{\vect bβ}(\delta_{\vect{ab}}\delta_{αβ}-T_{α}S_{\vect aα\leftarrow\vect bβ})A_{\vect bβ}=0
|
||||
\]
|
||||
\begin{equation}
|
||||
\outcoeffp{\vect n,\alpha}-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect n,\alpha\right)\right\} }\tropsp{\vect n,\alpha}{\vect m,\beta}\outcoeffp{\vect m,\beta}=T_{\alpha}\rcoeffincp{\vect n,\alpha}.\quad\left(\vect n,\alpha\right)\in\mathcal{P}\label{eq:Multiple-scattering problem periodic}
|
||||
\end{equation}
|
||||
|
||||
\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_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Due to periodicity, we can also write
|
||||
\begin_inset Formula $\tropsp{\vect n,\alpha}{\vect m,\beta}=\tropsp{\alpha}{\beta}\left(\vect R_{\vect m}-\vect R_{\vect n}\right)=\tropsp{\alpha}{\beta}\left(\vect R_{\vect m-\vect n}\right)=\tropsp{\vect 0,\alpha}{\vect m-\vect n,\beta}$
|
||||
\end_inset
|
||||
|
||||
, yielding
|
||||
.
|
||||
Assuming quasi-periodic right-hand side with quasi-momentum
|
||||
\begin_inset Formula $\vect k$
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Formula $\rcoeffincp{\vect n,\alpha}=\rcoeffincp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$
|
||||
\end_inset
|
||||
|
||||
, the solutions of
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "eq:Multiple-scattering problem periodic"
|
||||
plural "false"
|
||||
caps "false"
|
||||
noprefix "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
will be also quasi-periodic according to Bloch theorem,
|
||||
\begin_inset Formula $\outcoeffp{\vect n,\alpha}=\outcoeffp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}$
|
||||
\end_inset
|
||||
|
||||
, and eq.
|
||||
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "eq:Multiple-scattering problem periodic"
|
||||
plural "false"
|
||||
caps "false"
|
||||
noprefix "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
can be rewritten as follows
|
||||
\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*}
|
||||
\begin{align*}
|
||||
\outcoeffp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect n,\alpha\right)\right\} }\tropsp{\vect n,\alpha}{\vect m,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}},\\
|
||||
\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect n,\alpha\right)\right\} }\tropsp{\vect 0,\alpha}{\vect m-\vect n,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m-\vect n}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),\\
|
||||
\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect 0,\alpha\right)\right\} }\tropsp{\vect 0,\alpha}{\vect m,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),\\
|
||||
\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\beta\in\mathcal{P}}W_{\alpha\beta}\left(\vect k\right)\outcoeffp{\vect 0,\beta}\left(\vect k\right) & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),
|
||||
\end{align*}
|
||||
|
||||
\end_inset
|
||||
|
||||
Therefore, in order to solve the modes, we need to compute the
|
||||
so we reduced the initial scattering problem to one involving only the field
|
||||
expansion coefficients from a single unit cell, but we need to compute
|
||||
the
|
||||
\begin_inset Quotes eld
|
||||
\end_inset
|
||||
|
||||
|
|
@ -274,7 +299,7 @@ lattice Fourier transform
|
|||
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}
|
||||
W_{\alpha\beta}(\vect k)\equiv\sum_{\vect m\in\ints^{d}}\left(1-\delta_{\alpha\beta}\right)\tropsp{\vect 0,\alpha}{\vect m,\beta}e^{i\vect k\cdot\vect R_{\vect m}}.\label{eq:W definition}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
|
@ -295,14 +320,18 @@ reference "eq:W definition"
|
|||
\end_inset
|
||||
|
||||
is the asymptotic behaviour of the translation operator,
|
||||
\begin_inset Formula $S_{\vect 0α\leftarrow\vect bβ}\sim\left|\vect r_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect r_{\vect b}\right|}$
|
||||
\begin_inset Formula $\tropsp{\vect 0,\alpha}{\vect m,\beta}\sim\left|\vect R_{\vect b}\right|^{-1}e^{ik_{0}\left|\vect R_{\vect b}\right|}$
|
||||
\end_inset
|
||||
|
||||
that makes the convergence of the sum quite problematic for any
|
||||
that does not in the strict sense converge for any
|
||||
\begin_inset Formula $d>1$
|
||||
\end_inset
|
||||
|
||||
-dimensional lattice.
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
\begin_inset Foot
|
||||
status open
|
||||
|
||||
|
|
@ -318,11 +347,17 @@ Note that
|
|||
|
||||
\end_inset
|
||||
|
||||
In electrostatics, one can solve this problem with Ewald summation.
|
||||
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
In electrostatics, this problem can be solved with Ewald summation [TODO
|
||||
REF].
|
||||
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.
|
||||
We use the same idea here, but the technical details are more complicated
|
||||
than in electrostatics.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
|
@ -520,7 +555,7 @@ reference "eq:W definition"
|
|||
|
||||
\end_inset
|
||||
|
||||
and legendre
|
||||
and
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:W sum in reciprocal space"
|
||||
|
|
|
|||
Loading…
Reference in New Issue