Implement remaining minor Päivi's comments, comment on Γ branches.
Former-commit-id: 985cf66a7fde1b8b66807f82d4e9dc2942419e60
This commit is contained in:
Marek Nečada
parent
3b6dedf4a2
commit
b56c9f8ee3
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@ -2171,7 +2171,7 @@ reference "eq:absorption CS single"
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-th particle reads according to
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:regular vswf translation"
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reference "eq:reqular vswf coefficient vector translation"
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plural "false"
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caps "false"
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noprefix "false"
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@ -2194,7 +2194,7 @@ whereas the contributions of fields scattered from each particle expanded
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is, according to
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:singular vswf translation"
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reference "eq:singular to regular vswf coefficient vector translation"
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plural "false"
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caps "false"
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noprefix "false"
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@ -121,8 +121,8 @@ Although large finite systems are where MSTMM excels the most, there are
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Other methods might be already fast enough, but MSTMM will be faster in
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most cases in which there is enough spacing between the neighboring particles.
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MSTMM works well with any space group symmetry the system might have (as
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opposed to, for example, FDTD with cubic mesh applied to a honeycomb lattice),
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which makes e.g.
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opposed to, for example, FDTD with a cubic mesh applied to a honeycomb
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lattice), which makes e.g.
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application of group theory in mode analysis quite easy.
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\begin_inset Note Note
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status open
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@ -134,7 +134,7 @@ Topology anoyne?
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\end_inset
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And finally, having a method that handles well both infinite and large
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finite system gives a possibility to study finite-size effects in periodic
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finite systems gives a possibility to study finite-size effects in periodic
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scatterer arrays.
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\end_layout
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@ -171,7 +171,7 @@ noprefix "false"
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\begin_inset Formula $d$
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\end_inset
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-dimensional integar multiindex
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-dimensional integer multi-index
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\begin_inset Formula $\vect n\in\ints^{d}$
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\end_inset
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@ -473,7 +473,7 @@ noprefix "false"
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\end_inset
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is close enough to zero.
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However, this approach is quite expensive, for
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However, this approach is quite expensive, since
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\begin_inset Formula $W\left(\omega,\vect k\right)$
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\end_inset
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@ -540,7 +540,7 @@ TODO write this in a clean way
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).
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A somehow challenging step is to distinguish the different bands that can
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all be very close to the empty lattice approximation, especially if the
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particles in the systems are small.
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particles in the system are small.
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In high-symmetry points of the Brilloin zone, this can be solved by factorising
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\begin_inset Formula $M\left(\omega,\vect k\right)$
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@ -586,7 +586,7 @@ literal "false"
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\end_layout
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\begin_layout Subsection
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Computing the Fourier sum of the translation operator
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Computing the lattice sum of the translation operator
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\begin_inset CommandInset label
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LatexCommand label
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name "subsec:W operator evaluation"
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@ -597,7 +597,7 @@ name "subsec:W operator evaluation"
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\end_layout
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\begin_layout Standard
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The problem evaluating
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The problem in evaluating
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:W definition"
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@ -608,11 +608,16 @@ reference "eq:W definition"
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\begin_inset Formula $\tropsp{\vect 0,\alpha}{\vect m,\beta}\sim\left|\vect R_{\vect m}\right|^{-1}e^{i\kappa\left|\vect R_{\vect m}\right|}$
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\end_inset
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that does not in the strict sense converge for any
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, so that its lattice sum does not in the strict sense converge for any
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\begin_inset Formula $d>1$
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\end_inset
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-dimensional lattice.
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-dimensional lattice unless
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\begin_inset Formula $\Im\kappa>0$
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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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@ -647,14 +652,13 @@ literal "false"
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\end_inset
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.
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Its basic idea is to decompose the divide the lattice-summed function in
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two parts: a short-range part that decays fast and can be summed directly,
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and a long-range part which decays poorly but is fairly smooth everywhere,
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so that its Fourier transform decays fast enough, and to deal with the
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long range part by Poisson summation over the reciprocal lattice.
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The same idea can be used also in this case case of linear electrodynamic
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problems, just the technical details are more complicated than in electrostatic
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s.
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Its basic idea is to decomposethe lattice-summed function in two parts:
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a short-range part that decays fast and can be summed directly, and a long-rang
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e part which decays poorly but is fairly smooth everywhere, so that its
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Fourier transform decays fast enough, and to deal with the long range part
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by Poisson summation over the reciprocal lattice.
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The same idea can be used also in the case of linear electrodynamic problems,
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just the technical details are more complicated than in electrostatics.
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\end_layout
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\begin_layout Standard
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@ -695,11 +699,20 @@ literal "false"
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and can be applied to our case.
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If we formally label
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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 Marginal
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status open
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\begin_layout Plain Layout
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Check signs.
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FP: Check signs.
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\end_layout
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\end_inset
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\end_layout
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\end_inset
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@ -820,7 +833,7 @@ In all three dimensionality cases, the lattice sums are divided into short-range
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status open
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\begin_layout Plain Layout
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Check sign of s
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FP: Check sign of s
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\end_layout
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\end_inset
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@ -831,26 +844,41 @@ Check sign of s
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\begin_layout Standard
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\begin_inset Formula
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\begin{multline}
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\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=-\frac{2^{l+1}i}{\kappa^{l+1}\sqrt{\pi}}\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},-\vect s}\right)\left|\vect{R_{n}+\vect s}\right|^{l}\ush lm\left(\vect{R_{n}+\vect s}\right)e^{i\vect k\cdot\left(\vect{R_{n}+\vect s}\right)}\int_{\eta}^{\infty}e^{-\left|\vect{R_{n}+\vect s}\right|^{2}\xi^{2}}e^{-\kappa/4\xi^{2}}\xi^{2l}\ud\xi\\
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+\frac{\delta_{l0}\delta_{m0}}{\sqrt{4\pi}}\Gamma\left(-\frac{1}{2},-\frac{\kappa}{4\eta^{2}}\right)\ush lm\left(\vect{R_{n}+\vect s}\right),\label{eq:Ewald in 3D short-range part}
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\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=-\frac{2^{l+1}i}{\kappa^{l+1}\sqrt{\pi}}\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},-\vect s}\right)\left|\vect{R_{n}+\vect s}\right|^{l}\ush lm\left(\vect{R_{n}+\vect s}\right)e^{i\vect k\cdot\left(\vect{R_{n}+\vect s}\right)}\\
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\times\int_{\eta}^{\infty}e^{-\left|\vect{R_{n}+\vect s}\right|^{2}\xi^{2}}e^{-\kappa/4\xi^{2}}\xi^{2l}\ud\xi\\
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+\delta_{\vect{R_{n}},-\vect s}\frac{\delta_{l0}\delta_{m0}}{\sqrt{4\pi}}\Gamma\left(-\frac{1}{2},-\frac{\kappa}{4\eta^{2}}\right)\ush lm\left(\vect{R_{n}+\vect s}\right).\label{eq:Ewald in 3D short-range part}
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\end{multline}
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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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NEPATŘÍ TAM NĚJAKÁ DELTA FUNKCE K PŮVODNÍMU
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\begin_inset Formula $\sigma_{n}^{m(0)}$
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Here
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\begin_inset Formula $\Gamma(a,z)$
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\end_inset
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?
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\end_layout
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is the incomplete Gamma function.
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The last (
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\begin_inset Quotes eld
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\end_inset
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self-interaction
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\begin_inset Quotes erd
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\end_inset
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) term in
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Ewald in 3D short-range part"
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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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, which appears only when the displacement vector
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\begin_inset Formula $\vect s$
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\end_inset
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coincides with a lattice point, is often noted separately in the literature.
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\begin_inset Note Note
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status open
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@ -870,7 +898,7 @@ The long-range part for cases
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status open
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\begin_layout Plain Layout
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check sign of
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FP: check sign of
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\begin_inset Formula $\vect k$
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\end_inset
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@ -882,8 +910,8 @@ check sign of
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\begin_inset Formula
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\begin{multline}
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\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=-\frac{i^{l+1}}{\kappa^{d}\mathcal{A}}\pi^{2+\left(3-d\right)/2}2\left(\left(l-m\right)/2\right)!\left(\left(l+m\right)/2\right)!\times\\
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\times\sum_{\vect K\in\Lambda^{*}}\ush lm\left(\vect k+\vect K\right)\sum_{j=0}^{\left[\left(l-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\left|\vect k+\vect K\right|/\kappa\right)^{l-2j}\Gamma\left(-j,\frac{k^{2}\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)}{4\eta^{2}}\right)}{j!\left(\frac{1}{2}\left(l-m\right)-j\right)!\left(\frac{1}{2}\left(l+m\right)-j\right)!}\left(\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)\right)^{2j+3-d}\label{eq:Ewald in 3D long-range part 1D 2D}
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\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)=-\frac{i^{l+1}}{\kappa^{d}\mathcal{A}}\pi^{2+\left(3-d\right)/2}2\left(\left(l-m\right)/2\right)!\left(\left(l+m\right)/2\right)!\times\\
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\times\sum_{\vect K\in\Lambda^{*}}\ush lm\left(\vect k+\vect K\right)\sum_{j=0}^{\left[\left(l-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\left|\vect k+\vect K\right|/\kappa\right)^{l-2j}\Gamma\left(\frac{d-1}{2}-j,\frac{\kappa^{2}\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)}{4\eta^{2}}\right)}{j!\left(\frac{1}{2}\left(l-m\right)-j\right)!\left(\frac{1}{2}\left(l+m\right)-j\right)!}\left(\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)\right)^{2j+3-d}\label{eq:Ewald in 3D long-range part 1D 2D}
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\end{multline}
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\end_inset
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@ -895,7 +923,7 @@ and for
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\begin_inset Formula
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\begin{equation}
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\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=\frac{4\pi i^{l+1}}{\kappa\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}\frac{\left(\left|\vect k+\vect K\right|/\kappa\right)^{l}}{\kappa^{2}-\left|\vect k+\vect K\right|^{2}}e^{\left(\kappa^{2}-\left|\vect k+\vect K\right|^{2}\right)/4\eta^{2}}\ush lm\left(\vect k+\vect K\right).\label{eq:Ewald in 3D long-range part 3D}
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\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)=\frac{4\pi i^{l+1}}{\kappa\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}\frac{\left(\left|\vect k+\vect K\right|/\kappa\right)^{l}}{\kappa^{2}-\left|\vect k+\vect K\right|^{2}}e^{\left(\kappa^{2}-\left|\vect k+\vect K\right|^{2}\right)/4\eta^{2}}\ush lm\left(\vect k+\vect K\right).\label{eq:Ewald in 3D long-range part 3D}
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\end{equation}
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\end_inset
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@ -969,27 +997,184 @@ The Ewald parameter
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\end_inset
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.
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For one-dimensional, square, and cubic lattices, the optimal choice is
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\begin_inset Formula $\eta=\sqrt{\pi}/p$
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\end_inset
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where
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\begin_inset Formula $p$
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\end_inset
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is the direct lattice period
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\begin_inset CommandInset citation
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LatexCommand cite
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key "linton_lattice_2010"
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literal "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 Marginal
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status open
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\begin_layout Plain Layout
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What would be a good choice?
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Whatabout different geometries?
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\end_layout
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\end_inset
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\end_layout
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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 Marginal
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status open
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\begin_layout Plain Layout
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I have some error estimates derived in my notes.
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FP: I have some error estimates derived in my notes.
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Should I include them?
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\end_layout
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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For a two-dimensional lattice, the incomplete
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\begin_inset Formula $\Gamma$
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\end_inset
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-function
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\begin_inset Formula $\Gamma\left(\frac{1}{2}-j,z\right)$
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\end_inset
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in the long-range part has a branch point at
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\begin_inset Formula $z=0$
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\end_inset
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and special care has to be taken when choosing the appropriate branch.
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If the wavenumber of the medium has a positive imaginary part,
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\begin_inset Formula $\Im\kappa>0$
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\end_inset
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, then the translation operator elements
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\begin_inset Formula $\trops_{\tau lm;\tau'l'm}\left(\kappa\vect r\right)$
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\end_inset
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decay exponentially as
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\begin_inset Formula $\left|\vect r\right|\to\infty$
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\end_inset
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and the lattice sum in
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:W definition"
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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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converges absolutely even in the direct space, and it is equal to the Ewald
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sum with the principal value of the incomplete
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\begin_inset Formula $\Gamma$
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\end_inset
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function being used in
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Ewald in 3D long-range part 1D 2D"
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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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For other values of
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\begin_inset Formula $\kappa$
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\end_inset
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, the branch choice is made in such way that
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\begin_inset Formula $W_{\alpha\beta}\left(\vect k\right)$
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\end_inset
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is analytically continued even when the wavenumber's imaginary part crosses
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the real axis.
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The principal value of
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\begin_inset Formula $\Gamma\left(\frac{1}{2}-j,z\right)$
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\end_inset
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has a branch cut at the negative real half-axis, which, considering the
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lattice sum as a function of
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\begin_inset Formula $\kappa$
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\end_inset
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, translates into branch cuts starting at
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\begin_inset Formula $\kappa=\left|\vect k+\vect K\right|$
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\end_inset
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and continuing in straight lines towards
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\begin_inset Formula $+\infty$
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\end_inset
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.
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Therefore, in the quadrant
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\begin_inset Formula $\Re z<0,\Im z\ge0$
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\end_inset
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we use the continuation of the principal value from
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\begin_inset Formula $\Re z<0,\Im z<0$
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\end_inset
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instead of the principal branch
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\begin_inset CommandInset citation
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LatexCommand cite
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after "8.2.9"
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key "NIST:DLMF"
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literal "false"
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\end_inset
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, moving the branch cut in the
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\begin_inset Formula $z$
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\end_inset
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variable to the positive imaginary half-axis.
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This moves the branch cuts w.r.t.
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\begin_inset Formula $\kappa$
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\end_inset
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away from the real axis.
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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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TODO Figure.
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\end_layout
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\end_inset
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Detailed physical interpretation of the remaining branch cuts is an open
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question.
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\begin_inset Note Note
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status open
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