qpms/notes/ewald_23_z_nonzero.lyx

306 lines
8.0 KiB
Plaintext
Raw Normal View History

#LyX 2.4 created this file. For more info see https://www.lyx.org/
\lyxformat 584
\begin_document
\begin_header
\save_transient_properties true
\origin unavailable
\textclass article
\use_default_options true
\maintain_unincluded_children false
\language finnish
\language_package default
\inputencoding utf8
\fontencoding auto
\font_roman "default" "default"
\font_sans "default" "default"
\font_typewriter "default" "default"
\font_math "auto" "auto"
\font_default_family default
\use_non_tex_fonts false
\font_sc false
\font_roman_osf false
\font_sans_osf false
\font_typewriter_osf false
\font_sf_scale 100 100
\font_tt_scale 100 100
\use_microtype false
\use_dash_ligatures true
\graphics default
\default_output_format default
\output_sync 0
\bibtex_command default
\index_command default
\paperfontsize default
\use_hyperref false
\papersize default
\use_geometry false
\use_package amsmath 1
\use_package amssymb 1
\use_package cancel 1
\use_package esint 1
\use_package mathdots 1
\use_package mathtools 1
\use_package mhchem 1
\use_package stackrel 1
\use_package stmaryrd 1
\use_package undertilde 1
\cite_engine basic
\cite_engine_type default
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\justification true
\use_refstyle 1
\use_minted 0
\use_lineno 0
\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\is_math_indent 0
\math_numbering_side default
\quotes_style english
\dynamic_quotes 0
\papercolumns 1
\papersides 1
\paperpagestyle default
\tablestyle default
\tracking_changes false
\output_changes false
\html_math_output 0
\html_css_as_file 0
\html_be_strict false
\end_header
\begin_body
\begin_layout Standard
\lang english
\begin_inset FormulaMacro
\newcommand{\uoft}[1]{\mathfrak{F}#1}
\end_inset
\begin_inset FormulaMacro
\newcommand{\uaft}[1]{\mathfrak{\mathbb{F}}#1}
\end_inset
\begin_inset FormulaMacro
\newcommand{\usht}[2]{\mathbb{S}_{#1}#2}
\end_inset
\begin_inset FormulaMacro
\newcommand{\bsht}[2]{\mathrm{S}_{#1}#2}
\end_inset
\begin_inset FormulaMacro
\newcommand{\sgn}{\operatorname{sgn}}
{\mathrm{sgn}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\pht}[2]{\mathfrak{\mathbb{H}}_{#1}#2}
\end_inset
\begin_inset FormulaMacro
\newcommand{\vect}[1]{\mathbf{#1}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\ud}{\mathrm{d}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\basis}[1]{\mathfrak{#1}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\dc}[1]{Ш_{#1}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\rec}[1]{#1^{-1}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\recb}[1]{#1^{\widehat{-1}}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\ints}{\mathbb{Z}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\nats}{\mathbb{N}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\reals}{\mathbb{R}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\ush}[2]{Y_{#1,#2}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\hgfr}{\mathbf{F}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\hgf}{F}
\end_inset
\begin_inset FormulaMacro
\newcommand{\ghgf}[2]{\mbox{}_{#1}F_{#2}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\ghgfr}[2]{\mbox{}_{#1}\mathbf{F}_{#2}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\ph}{\mathrm{ph}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\kor}[1]{\underline{#1}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\koru}[1]{\utilde{#1}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\swv}{\mathscr{H}}
\end_inset
\begin_inset FormulaMacro
\newcommand{\expint}{\mathrm{E}}
\end_inset
\end_layout
\begin_layout Standard
\lang english
\begin_inset Formula
\begin{eqnarray}
\sigma_{n}^{m(1)} & = & -\frac{i^{n+1}}{2k^{2}\mathscr{A}}\left(-1\right)^{\left(n+m\right)/2}\sqrt{\left(2n+1\right)\left(n-m\right)!\left(n+m\right)!}\times\nonumber \\
& & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/2k\right)^{n-2j}e^{im\phi_{\vect{\beta}_{pq}}}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(n-m\right)-j\right)!\left(\frac{1}{2}\left(n+m\right)-j\right)!}\left(\frac{\gamma_{pq}}{2}\right)^{2j-1}\nonumber \\
& = & -\frac{i^{n+1}}{2k^{2}\mathscr{A}}\sqrt{\pi}2^{n+1}\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\times\nonumber \\
& & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}Y_{n}^{m}\left(\frac{\pi}{2},\phi_{\vect{\beta}_{pq}}\right)\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/2k\right)^{n-2j}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(n-m\right)-j\right)!\left(\frac{1}{2}\left(n+m\right)-j\right)!}\left(\frac{\gamma_{pq}}{2}\right)^{2j-1}\nonumber \\
& = & -\frac{i^{n+1}}{k^{2}\mathscr{A}}\sqrt{\pi}2\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\times\nonumber \\
& & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}Y_{n}^{m}\left(\frac{\pi}{2},\phi_{\vect{\beta}_{pq}}\right)\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/k\right)^{n-2j}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(n-m\right)-j\right)!\left(\frac{1}{2}\left(n+m\right)-j\right)!}\left(\gamma_{pq}\right)^{2j-1}\label{eq:2D Ewald in 3D long-range part}
\end{eqnarray}
\end_inset
For
\begin_inset Formula $z\ne0$
\end_inset
\begin_inset Formula
\begin{align*}
& =-\frac{i^{n+1}}{k^{2}\mathscr{A}}\sqrt{\pi}2\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\\
& \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}Y_{n}^{m}\left(\frac{\pi}{2},\phi_{\vect{\beta}_{pq}}\right)\sum_{j=0}^{n-\left|m\right|}\frac{\Delta_{npq}}{j!}\left(-1\right)^{j}\left(\gamma_{pq}\right)^{2j-1}\sum_{s\overset{*}{=}j}^{\min(2j,n-\left|m\right|)}\binom{j}{2j-s}\frac{\left(-\kappa z\right)^{2j-s}\left(\beta_{pq}/k\right)^{n-s}}{\left(\frac{1}{2}\left(n-m-s\right)\right)!\left(\frac{1}{2}\left(n+m-s\right)\right)!}\\
& =-\frac{i^{n+1}}{k^{2}\mathscr{A}}\sqrt{\pi}2\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\\
& \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}Y_{n}^{m}\left(\frac{\pi}{2},\phi_{\vect{\beta}_{pq}}\right)\sum_{j=0}^{n-\left|m\right|}\Delta_{npq}\left(\gamma_{pq}\right)^{2j-1}\sum_{s\overset{*}{=}j}^{\min(2j,n-\left|m\right|)}\frac{\left(-1\right)^{j}}{\left(2j-s\right)!\left(s-j\right)!}\frac{\left(-\kappa z\right)^{2j-s}\left(\beta_{pq}/k\right)^{n-s}}{\left(\frac{1}{2}\left(n-m-s\right)\right)!\left(\frac{1}{2}\left(n+m-s\right)\right)!}
\end{align*}
\end_inset
\end_layout
\begin_layout Section
\lang english
Ewald long range integral
\end_layout
\begin_layout Standard
\lang english
Linton has (2.24):
\begin_inset Formula
\[
G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{2\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}e^{i\vect K_{m}\cdot\vect r}\int_{1/\eta}^{\infty\exp\left(i\pi/4\right)}e^{-\kappa^{2}\gamma_{m}^{2}\zeta^{2}/4}e^{-\left|\vect r_{\bot}\right|^{2}/\zeta^{2}}\zeta^{1-d_{c}}\ud\zeta
\]
\end_inset
Try substitution
\begin_inset Formula $t=\zeta^{2}$
\end_inset
: then
\begin_inset Formula $\ud t=2\zeta\,\ud\zeta$
\end_inset
(
\begin_inset Formula $\ud\zeta=\ud t/2t^{1/2}$
\end_inset
) and
\begin_inset Formula
\[
G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{4\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}e^{i\vect K_{m}\cdot\vect r}\int_{1/\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\kappa^{2}\gamma_{m}^{2}t/4}e^{-\left|\vect r_{\bot}\right|^{2}/t}t^{\frac{-d_{c}}{2}}\ud t
\]
\end_inset
Try subst.
\begin_inset Formula $\tau=k^{2}\gamma_{m}^{2}/4$
\end_inset
\end_layout
\begin_layout Standard
\lang english
\begin_inset Formula
\[
G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{4\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}e^{i\vect K_{m}\cdot\vect r}\left(\frac{\kappa\gamma_{m}}{2}\right)^{d_{c}}\int_{\kappa^{2}\gamma_{m}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{m}^{2}/4\tau}\tau^{\frac{-d_{c}}{2}}\ud\tau
\]
\end_inset
\end_layout
\end_body
\end_document