qpms/notes/kambe_linton_dict.lyx

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\index Index
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\begin_body

\begin_layout Standard

\lang english
\begin_inset FormulaMacro
\newcommand{\vect}[1]{\mathbf{#1}}
\end_inset


\lang finnish

\begin_inset FormulaMacro
\newcommand{\Kambe}[1]{#1^{\mathrm{K}}}
\end_inset


\begin_inset FormulaMacro
\newcommand{\Linton}[1]{#1^{\mathrm{L}}}
\end_inset


\end_layout

\begin_layout Standard
Here and in Kambe's papers, 
\begin_inset Formula $\kappa$
\end_inset

 is the wavenumber (
\begin_inset Formula $k$
\end_inset

 in Linton).
 Here 
\begin_inset Formula $\vect K_{p}$
\end_inset

 is a point of the reciprocal lattice (
\begin_inset Formula $\vect K_{p}=\Kambe{\vect K_{pt}}=\Linton{\vect{\beta}_{\mu}}$
\end_inset

)
\end_layout

\begin_layout Section
\begin_inset Quotes eld
\end_inset

Gammas
\begin_inset Quotes erd
\end_inset


\end_layout

\begin_layout Standard
For 
\begin_inset Formula $\kappa$
\end_inset

 positive,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\Kambe{\Gamma_{p}}\equiv\begin{cases}
\sqrt{\kappa^{2}-\left|\vect K_{p}\right|^{2}} & \kappa^{2}-\left|\vect K_{p}\right|^{2}>0\\
i\sqrt{\left|\vect K_{p}\right|^{2}-\kappa^{2}} & \kappa^{2}-\left|\vect K_{p}\right|^{2}<0
\end{cases}
\]

\end_inset


\begin_inset Formula 
\[
\Linton{\gamma_{\mu}}\equiv\begin{cases}
\sqrt{\left(\frac{\vect K_{p}}{\kappa}\right)^{2}-1} & \kappa-\left|\vect K_{p}\right|\le0\\
-i\sqrt{1-\left(\frac{\vect K_{p}}{\kappa}\right)^{2}} & \kappa-\left|\vect K_{p}\right|>0
\end{cases}
\]

\end_inset

hence 
\begin_inset Formula 
\[
\Kambe{\Gamma_{p}}=-i\kappa\Linton{\gamma_{\mu}},
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\Linton{\gamma_{\mu}}=i\frac{\Kambe{\Gamma_{p}}}{\kappa}.
\]

\end_inset


\end_layout

\begin_layout Section
D vs sigma
\end_layout

\begin_layout Standard
In-plane sums [Linton 2009, (4.5)], replacing 
\begin_inset Formula $n,m\rightarrow L,M$
\end_inset

, 
\begin_inset Formula $k\rightarrow\kappa$
\end_inset


\end_layout

\begin_layout Standard

\lang english
\begin_inset Formula 
\begin{eqnarray*}
\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\\
 &  & \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}
\end{eqnarray*}

\end_inset

[Kambe II, (3.17)], replacing 
\lang finnish

\begin_inset Formula $n\rightarrow j$
\end_inset


\lang english
, 
\lang finnish

\begin_inset Formula $A\rightarrow\mathscr{A}$
\end_inset

, 
\begin_inset Formula $\vect K_{pt}\to\vect K_{p}$
\end_inset

, 
\begin_inset Formula $\Gamma\left(\frac{1}{2}-j,e^{-i\pi}\Gamma_{p}^{2}\omega/2\right)\to\Gamma_{j,p}$
\end_inset

 and performing little typographic modifications
\lang english

\begin_inset Formula 
\begin{align*}
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\\
 & \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)!}
\end{align*}

\end_inset

Using the relations between 
\begin_inset Formula $\Kambe{\Gamma_{p}}=-i\kappa\Linton{\gamma_{\mu}}$
\end_inset

, we have (also, we replace the 
\begin_inset Formula $\mu$
\end_inset

 index with 
\begin_inset Formula $p$
\end_inset

)
\begin_inset Formula 
\begin{align*}
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\\
 & \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)!}
\end{align*}

\end_inset

and now, trying to make the exponents look the same as in Linton, 
\begin_inset Formula $2^{-1}2^{2j-L}2^{1-2j}=2^{-L}$
\end_inset

 (OK), 
\begin_inset Formula $K_{p}^{L-2j}=K_{p}^{L-2j}$
\end_inset

 (OK), 
\begin_inset Formula 
\begin{align*}
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\\
 & \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}
\end{align*}

\end_inset

There are now these differences left:
\end_layout

\begin_layout Itemize

\lang english
Additional 
\begin_inset Formula $\kappa$
\end_inset

 factor in 
\begin_inset Formula $D_{LM}$
\end_inset


\end_layout

\begin_layout Itemize

\lang english
\begin_inset Formula $i^{L+1}\left(-1\right)^{\left(L+M\right)/2}\left(-1\right)^{j}$
\end_inset

 vs.
 
\begin_inset Formula $i^{\left|M\right|+1}\left(-i\right)^{2j-1}$
\end_inset


\end_layout

\begin_layout Itemize

\lang english
Opposite phase in the angular part.
\end_layout

\begin_layout Itemize

\lang english
Plane wave factor in 
\begin_inset Formula $D_{LM}$
\end_inset


\end_layout

\begin_layout Standard

\lang english
Let's look at the 
\begin_inset Formula $i,-1$
\end_inset

 factors (note that 
\begin_inset Formula $L+M$
\end_inset

 is odd): 
\begin_inset Formula $\left(-i\right)^{2j}=\left(-1\right)^{j},$
\end_inset

 leaving 
\begin_inset Formula $i^{L+1}\left(-1\right)^{\left(L+M\right)/2}$
\end_inset

 vs.
 
\begin_inset Formula $i^{\left|M\right|+1}i$
\end_inset

.
 So there is might be a phase difference due to different conventions, but
 it does not depend on 
\begin_inset Formula $j$
\end_inset

, so one should be able to transplant the 
\begin_inset Formula $z\ne0$
\end_inset

 sum from Kambe without major problems.
\end_layout

\begin_layout Section
Ewald parameter (integration limits)
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\Linton{\eta}=\sqrt{\frac{1}{2\Kambe{\omega}}}
\]

\end_inset

(Based on comparison of some function arguments, not checked.)
\end_layout

\end_body
\end_document