387 lines
8.2 KiB
Plaintext
387 lines
8.2 KiB
Plaintext
#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\lyxformat 584
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\begin_header
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\use_package amsmath 1
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\use_package amssymb 1
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\use_package cancel 1
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\use_package esint 1
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\use_package mathdots 1
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\use_package mathtools 1
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\use_package mhchem 1
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\use_package stackrel 1
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\index Index
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\shortcut idx
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\color #008000
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\end_index
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\secnumdepth 3
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\tocdepth 3
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\paragraph_indentation default
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\end_header
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\begin_body
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\begin_layout Standard
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\lang english
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\begin_inset FormulaMacro
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\newcommand{\vect}[1]{\mathbf{#1}}
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\end_inset
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\lang finnish
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\begin_inset FormulaMacro
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\newcommand{\Kambe}[1]{#1^{\mathrm{K}}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\Linton}[1]{#1^{\mathrm{L}}}
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\end_inset
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\end_layout
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\begin_layout Standard
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Here and in Kambe's papers,
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\begin_inset Formula $\kappa$
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\end_inset
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is the wavenumber (
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\begin_inset Formula $k$
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\end_inset
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in Linton).
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Here
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\begin_inset Formula $\vect K_{p}$
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\end_inset
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is a point of the reciprocal lattice (
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\begin_inset Formula $\vect K_{p}=\Kambe{\vect K_{pt}}=\Linton{\vect{\beta}_{\mu}}$
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\end_inset
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)
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\end_layout
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\begin_layout Section
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\begin_inset Quotes eld
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\end_inset
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Gammas
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\begin_inset Quotes erd
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\end_inset
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\end_layout
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\begin_layout Standard
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For
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\begin_inset Formula $\kappa$
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\end_inset
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positive,
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\Kambe{\Gamma_{p}}\equiv\begin{cases}
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\sqrt{\kappa^{2}-\left|\vect K_{p}\right|^{2}} & \kappa^{2}-\left|\vect K_{p}\right|^{2}>0\\
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i\sqrt{\left|\vect K_{p}\right|^{2}-\kappa^{2}} & \kappa^{2}-\left|\vect K_{p}\right|^{2}<0
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\end{cases}
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\]
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\end_inset
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\begin_inset Formula
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\[
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\Linton{\gamma_{\mu}}\equiv\begin{cases}
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\sqrt{\left(\frac{\vect K_{p}}{\kappa}\right)^{2}-1} & \kappa-\left|\vect K_{p}\right|\le0\\
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-i\sqrt{1-\left(\frac{\vect K_{p}}{\kappa}\right)^{2}} & \kappa-\left|\vect K_{p}\right|>0
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\end{cases}
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\]
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\end_inset
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hence
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\begin_inset Formula
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\[
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\Kambe{\Gamma_{p}}=-i\kappa\Linton{\gamma_{\mu}},
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\Linton{\gamma_{\mu}}=i\frac{\Kambe{\Gamma_{p}}}{\kappa}.
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\]
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\end_inset
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\end_layout
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\begin_layout Section
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D vs sigma
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\end_layout
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\begin_layout Standard
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In-plane sums [Linton 2009, (4.5)], replacing
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\begin_inset Formula $n,m\rightarrow L,M$
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\end_inset
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,
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\begin_inset Formula $k\rightarrow\kappa$
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\end_inset
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\end_layout
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\begin_layout Standard
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\lang english
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\begin_inset Formula
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\begin{eqnarray*}
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\sigma_{L}^{M(1)} & = & -\frac{i^{L+1}}{2\kappa^{2}\mathscr{A}}\left(-1\right)^{\left(L+M\right)/2}\sqrt{\left(2L+1\right)\left(L-M\right)!\left(L+M\right)!}\times\\
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& & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}\sum_{j=0}^{\left[\left(L-\left|M\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/2\kappa\right)^{L-2j}e^{iM\phi_{\vect{\beta}_{pq}}}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(L-M\right)-j\right)!\left(\frac{1}{2}\left(L+M\right)-j\right)!}\left(\frac{\gamma_{pq}}{2}\right)^{2j-1}
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\end{eqnarray*}
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\end_inset
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[Kambe II, (3.17)], replacing
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\lang finnish
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\begin_inset Formula $n\rightarrow j$
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\end_inset
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\lang english
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,
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\lang finnish
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\begin_inset Formula $A\rightarrow\mathscr{A}$
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\end_inset
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,
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\begin_inset Formula $\vect K_{pt}\to\vect K_{p}$
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\end_inset
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,
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\begin_inset Formula $\Gamma\left(\frac{1}{2}-j,e^{-i\pi}\Gamma_{p}^{2}\omega/2\right)\to\Gamma_{j,p}$
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\end_inset
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and performing little typographic modifications
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\lang english
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\begin_inset Formula
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\begin{align*}
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D_{LM} & =-\frac{1}{\mathscr{A}\kappa}i^{\left|M\right|+1}2^{-L}\sqrt{\left(2L+1\right)\left(L+\left|M\right|\right)!\left(L-\left|M\right|\right)!}\times\\
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& \quad\times\sum_{p}e^{i\vect K_{p}\cdot\vect c_{ijt}}e^{-iM\phi_{K_{p}}}\sum_{j=0}^{\left(L-\left|M\right|\right)/2}\frac{\left(\Gamma_{p}/\kappa\right)^{2j-1}\left(K_{p}/\kappa\right)^{L-2j}\Gamma_{j,p}}{j!\left(\frac{1}{2}\left(L-\left|M\right|\right)-j\right)!\left(\frac{1}{2}\left(L+\left|M\right|\right)-j\right)!}
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\end{align*}
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\end_inset
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Using the relations between
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\begin_inset Formula $\Kambe{\Gamma_{p}}=-i\kappa\Linton{\gamma_{\mu}}$
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\end_inset
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, we have (also, we replace the
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\begin_inset Formula $\mu$
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\end_inset
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index with
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\begin_inset Formula $p$
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\end_inset
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)
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\begin_inset Formula
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\begin{align*}
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D_{LM} & =-\frac{1}{\mathscr{A}\kappa}i^{\left|M\right|+1}2^{-L}\sqrt{\left(2L+1\right)\left(L+\left|M\right|\right)!\left(L-\left|M\right|\right)!}\times\\
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& \quad\times\sum_{p}e^{i\vect K_{p}\cdot\vect c_{ijt}}e^{-iM\phi_{K_{p}}}\sum_{j=0}^{\left(L-\left|M\right|\right)/2}\frac{\left(-i\gamma_{p}\right)^{2j-1}\left(K_{p}/\kappa\right)^{L-2j}\Gamma_{j,p}}{j!\left(\frac{1}{2}\left(L-\left|M\right|\right)-j\right)!\left(\frac{1}{2}\left(L+\left|M\right|\right)-j\right)!}
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\end{align*}
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\end_inset
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and now, trying to make the exponents look the same as in Linton,
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\begin_inset Formula $2^{-1}2^{2j-L}2^{1-2j}=2^{-L}$
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\end_inset
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(OK),
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\begin_inset Formula $K_{p}^{L-2j}=K_{p}^{L-2j}$
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\end_inset
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(OK),
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\begin_inset Formula
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\begin{align*}
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D_{LM} & =-\frac{1}{2\kappa\mathscr{A}}i^{\left|M\right|+1}\sqrt{\left(2L+1\right)\left(L+\left|M\right|\right)!\left(L-\left|M\right|\right)!}\times\\
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& \quad\times\sum_{p}e^{i\vect K_{p}\cdot\vect c_{ij}}e^{-iM\phi_{K_{p}}}\sum_{j=0}^{\left(L-\left|M\right|\right)/2}\frac{\left(-i\right)^{2j-1}\left(K_{p}/2\kappa\right)^{L-2j}\Gamma_{j,p}}{j!\left(\frac{1}{2}\left(L-\left|M\right|\right)-j\right)!\left(\frac{1}{2}\left(L+\left|M\right|\right)-j\right)!}\left(\frac{\gamma_{p}}{2}\right)^{2j-1}
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\end{align*}
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\end_inset
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There are now these differences left:
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\end_layout
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\begin_layout Itemize
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\lang english
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Additional
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\begin_inset Formula $\kappa$
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\end_inset
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factor in
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\begin_inset Formula $D_{LM}$
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\end_inset
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\end_layout
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\begin_layout Itemize
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\lang english
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\begin_inset Formula $i^{L+1}\left(-1\right)^{\left(L+M\right)/2}\left(-1\right)^{j}$
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\end_inset
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vs.
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\begin_inset Formula $i^{\left|M\right|+1}\left(-i\right)^{2j-1}$
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\end_inset
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\end_layout
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\begin_layout Itemize
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\lang english
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Opposite phase in the angular part.
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\end_layout
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\begin_layout Itemize
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\lang english
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Plane wave factor in
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\begin_inset Formula $D_{LM}$
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\end_inset
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\end_layout
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\begin_layout Standard
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\lang english
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Let's look at the
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\begin_inset Formula $i,-1$
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\end_inset
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factors (note that
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\begin_inset Formula $L+M$
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\end_inset
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is odd):
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\begin_inset Formula $\left(-i\right)^{2j}=\left(-1\right)^{j},$
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\end_inset
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leaving
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\begin_inset Formula $i^{L+1}\left(-1\right)^{\left(L+M\right)/2}$
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\end_inset
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vs.
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\begin_inset Formula $i^{\left|M\right|+1}i$
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\end_inset
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.
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So there is might be a phase difference due to different conventions, but
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it does not depend on
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\begin_inset Formula $j$
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\end_inset
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, so one should be able to transplant the
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\begin_inset Formula $z\ne0$
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\end_inset
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sum from Kambe without major problems.
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\end_layout
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\begin_layout Section
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Ewald parameter (integration limits)
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\Linton{\eta}=\sqrt{\frac{1}{2\Kambe{\omega}}}
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\]
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\end_inset
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(Based on comparison of some function arguments, not checked.)
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\end_layout
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\end_body
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\end_document
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