notes/ewald_23_z_nonzero.lyx

   1 #LyX 2.4 created this file. For more info see https://www.lyx.org/
   2 \lyxformat 584
   3 \begin_document
   4 \begin_header
   5 \save_transient_properties true
   6 \origin unavailable
   7 \textclass article
   8 \use_default_options true
   9 \maintain_unincluded_children false
  10 \language finnish
  11 \language_package default
  12 \inputencoding utf8
  13 \fontencoding auto
  14 \font_roman "default" "default"
  15 \font_sans "default" "default"
  16 \font_typewriter "default" "default"
  17 \font_math "auto" "auto"
  18 \font_default_family default
  19 \use_non_tex_fonts false
  20 \font_sc false
  21 \font_roman_osf false
  22 \font_sans_osf false
  23 \font_typewriter_osf false
  24 \font_sf_scale 100 100
  25 \font_tt_scale 100 100
  26 \use_microtype false
  27 \use_dash_ligatures true
  28 \graphics default
  29 \default_output_format default
  30 \output_sync 0
  31 \bibtex_command default
  32 \index_command default
  33 \paperfontsize default
  34 \use_hyperref false
  35 \papersize default
  36 \use_geometry false
  37 \use_package amsmath 1
  38 \use_package amssymb 1
  39 \use_package cancel 1
  40 \use_package esint 1
  41 \use_package mathdots 1
  42 \use_package mathtools 1
  43 \use_package mhchem 1
  44 \use_package stackrel 1
  45 \use_package stmaryrd 1
  46 \use_package undertilde 1
  47 \cite_engine basic
  48 \cite_engine_type default
  49 \use_bibtopic false
  50 \use_indices false
  51 \paperorientation portrait
  52 \suppress_date false
  53 \justification true
  54 \use_refstyle 1
  55 \use_minted 0
  56 \use_lineno 0
  57 \index Index
  58 \shortcut idx
  59 \color #008000
  60 \end_index
  61 \secnumdepth 3
  62 \tocdepth 3
  63 \paragraph_separation indent
  64 \paragraph_indentation default
  65 \is_math_indent 0
  66 \math_numbering_side default
  67 \quotes_style english
  68 \dynamic_quotes 0
  69 \papercolumns 1
  70 \papersides 1
  71 \paperpagestyle default
  72 \tablestyle default
  73 \tracking_changes false
  74 \output_changes false
  75 \html_math_output 0
  76 \html_css_as_file 0
  77 \html_be_strict false
  78 \end_header
  79
  80 \begin_body
  81
  82 \begin_layout Standard
  83
  84 \lang english
  85 \begin_inset FormulaMacro
  86 \newcommand{\uoft}[1]{\mathfrak{F}#1}
  87 \end_inset
  88
  89
  90 \begin_inset FormulaMacro
  91 \newcommand{\uaft}[1]{\mathfrak{\mathbb{F}}#1}
  92 \end_inset
  93
  94
  95 \begin_inset FormulaMacro
  96 \newcommand{\usht}[2]{\mathbb{S}_{#1}#2}
  97 \end_inset
  98
  99
 100 \begin_inset FormulaMacro
 101 \newcommand{\bsht}[2]{\mathrm{S}_{#1}#2}
 102 \end_inset
 103
 104
 105 \begin_inset FormulaMacro
 106 \newcommand{\sgn}{\operatorname{sgn}}
 107 {\mathrm{sgn}}
 108 \end_inset
 109
 110
 111 \begin_inset FormulaMacro
 112 \newcommand{\pht}[2]{\mathfrak{\mathbb{H}}_{#1}#2}
 113 \end_inset
 114
 115
 116 \begin_inset FormulaMacro
 117 \newcommand{\vect}[1]{\mathbf{#1}}
 118 \end_inset
 119
 120
 121 \begin_inset FormulaMacro
 122 \newcommand{\ud}{\mathrm{d}}
 123 \end_inset
 124
 125
 126 \begin_inset FormulaMacro
 127 \newcommand{\basis}[1]{\mathfrak{#1}}
 128 \end_inset
 129
 130
 131 \begin_inset FormulaMacro
 132 \newcommand{\dc}[1]{Ш_{#1}}
 133 \end_inset
 134
 135
 136 \begin_inset FormulaMacro
 137 \newcommand{\rec}[1]{#1^{-1}}
 138 \end_inset
 139
 140
 141 \begin_inset FormulaMacro
 142 \newcommand{\recb}[1]{#1^{\widehat{-1}}}
 143 \end_inset
 144
 145
 146 \begin_inset FormulaMacro
 147 \newcommand{\ints}{\mathbb{Z}}
 148 \end_inset
 149
 150
 151 \begin_inset FormulaMacro
 152 \newcommand{\nats}{\mathbb{N}}
 153 \end_inset
 154
 155
 156 \begin_inset FormulaMacro
 157 \newcommand{\reals}{\mathbb{R}}
 158 \end_inset
 159
 160
 161 \begin_inset FormulaMacro
 162 \newcommand{\ush}[2]{Y_{#1,#2}}
 163 \end_inset
 164
 165
 166 \begin_inset FormulaMacro
 167 \newcommand{\hgfr}{\mathbf{F}}
 168 \end_inset
 169
 170
 171 \begin_inset FormulaMacro
 172 \newcommand{\hgf}{F}
 173 \end_inset
 174
 175
 176 \begin_inset FormulaMacro
 177 \newcommand{\ghgf}[2]{\mbox{}_{#1}F_{#2}}
 178 \end_inset
 179
 180
 181 \begin_inset FormulaMacro
 182 \newcommand{\ghgfr}[2]{\mbox{}_{#1}\mathbf{F}_{#2}}
 183 \end_inset
 184
 185
 186 \begin_inset FormulaMacro
 187 \newcommand{\ph}{\mathrm{ph}}
 188 \end_inset
 189
 190
 191 \begin_inset FormulaMacro
 192 \newcommand{\kor}[1]{\underline{#1}}
 193 \end_inset
 194
 195
 196 \begin_inset FormulaMacro
 197 \newcommand{\koru}[1]{\utilde{#1}}
 198 \end_inset
 199
 200
 201 \begin_inset FormulaMacro
 202 \newcommand{\swv}{\mathscr{H}}
 203 \end_inset
 204
 205
 206 \begin_inset FormulaMacro
 207 \newcommand{\expint}{\mathrm{E}}
 208 \end_inset
 209
 210
 211 \end_layout
 212
 213 \begin_layout Standard
 214
 215 \lang english
 216 \begin_inset Formula
 217 \begin{eqnarray}
 218 \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 \\
 219  &  & \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 \\
 220  & = & -\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 \\
 221  &  & \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 \\
 222  & = & -\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 \\
 223  &  & \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}
 224 \end{eqnarray}
 225
 226 \end_inset
 227
 228 For
 229 \begin_inset Formula $z\ne0$
 230 \end_inset
 231
 232
 233 \begin_inset Formula
 234 \begin{align*}
 235  & =-\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)!\\
 236  & \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)!}\\
 237  & =-\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)!\\
 238  & \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)!}
 239 \end{align*}
 240
 241 \end_inset
 242
 243
 244 \end_layout
 245
 246 \begin_layout Section
 247
 248 \lang english
 249 Ewald long range integral
 250 \end_layout
 251
 252 \begin_layout Standard
 253
 254 \lang english
 255 Linton has (2.24):
 256 \begin_inset Formula
 257 \[
 258 G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{2\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}e^{i\vect K_{m}\cdot\vect r}\int_{1/\eta}^{\infty\exp\left(i\pi/4\right)}e^{-\kappa^{2}\gamma_{m}^{2}\zeta^{2}/4}e^{-\left|\vect r_{\bot}\right|^{2}/\zeta^{2}}\zeta^{1-d_{c}}\ud\zeta
 259 \]
 260
 261 \end_inset
 262
 263 Try substitution
 264 \begin_inset Formula $t=\zeta^{2}$
 265 \end_inset
 266
 267 : then
 268 \begin_inset Formula $\ud t=2\zeta\,\ud\zeta$
 269 \end_inset
 270
 271  (
 272 \begin_inset Formula $\ud\zeta=\ud t/2t^{1/2}$
 273 \end_inset
 274
 275 ) and
 276 \begin_inset Formula
 277 \[
 278 G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{4\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}e^{i\vect K_{m}\cdot\vect r}\int_{1/\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\kappa^{2}\gamma_{m}^{2}t/4}e^{-\left|\vect r_{\bot}\right|^{2}/t}t^{\frac{-d_{c}}{2}}\ud t
 279 \]
 280
 281 \end_inset
 282
 283 Try subst.
 284
 285 \begin_inset Formula $\tau=k^{2}\gamma_{m}^{2}/4$
 286 \end_inset
 287
 288
 289 \end_layout
 290
 291 \begin_layout Standard
 292
 293 \lang english
 294 \begin_inset Formula
 295 \[
 296 G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{4\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}e^{i\vect K_{m}\cdot\vect r}\left(\frac{\kappa\gamma_{m}}{2}\right)^{d_{c}}\int_{\kappa^{2}\gamma_{m}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{m}^{2}/4\tau}\tau^{\frac{-d_{c}}{2}}\ud\tau
 297 \]
 298
 299 \end_inset
 300
 301
 302 \end_layout
 303
 304 \end_body
 305 \end_document