notes/hexlattice_kpoint_projections.lyx

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  30 \pdf_title "Sähköpajan päiväkirja"
  31 \pdf_author "Marek Nečada"
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  62 \index Index
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  80
  81 \begin_body
  82
  83 \begin_layout Title
  84 Symmetry-adapted basis functions for honeycomb lattice at
  85 \begin_inset Formula $K$
  86 \end_inset
  87
  88 -point
  89 \end_layout
  90
  91 \begin_layout Section
  92 Generation theorem
  93 \end_layout
  94
  95 \begin_layout Standard
  96 Let
  97 \begin_inset Formula $\mathbf{G}$
  98 \end_inset
  99
 100  be a group and
 101 \begin_inset Formula $\Gamma^{i}\left\{ R\to\mathbf{D}^{i}\left(R\right)\right\} $
 102 \end_inset
 103
 104  some
 105 \begin_inset Formula $d_{i}$
 106 \end_inset
 107
 108 -dimensional rep of
 109 \begin_inset Formula $\mathbf{G}$
 110 \end_inset
 111
 112 .
 113  Let the group ring (corresponding to the given rep indexed by
 114 \begin_inset Formula $i$
 115 \end_inset
 116
 117 ) elements be defined as [Bradley&Cracknell (2.2.2)]
 118 \begin_inset Formula
 119 \[
 120 W_{ts}^{i}=\frac{d_{i}}{\left|\mathbf{G}\right|}\sum_{R\in\mathbf{G}}\mathbf{D}^{i}\left(R\right)_{ts}^{*}R.
 121 \]
 122
 123 \end_inset
 124
 125
 126 \end_layout
 127
 128 \begin_layout Standard
 129 From [Bradley&Cracknell, theorem 2.2.1]:
 130 \end_layout
 131
 132 \begin_layout Standard
 133 If
 134 \begin_inset Formula $\phi$
 135 \end_inset
 136
 137  is an arbitrary function of
 138 \begin_inset Formula $V$
 139 \end_inset
 140
 141  (a linear space in which the realisation of the group operation act) such
 142  that
 143 \begin_inset Formula $W_{ss}^{i}\phi\ne0$
 144 \end_inset
 145
 146  (
 147 \begin_inset Formula $s$
 148 \end_inset
 149
 150  is fixed and is a number in the range 1 to
 151 \begin_inset Formula $d_{i}$
 152 \end_inset
 153
 154 ;
 155 \begin_inset Formula $i$
 156 \end_inset
 157
 158  is idx of the rep) then the funs
 159 \begin_inset Formula $W_{ts}^{i}\phi=\phi_{t}^{i}$
 160 \end_inset
 161
 162 ,
 163 \begin_inset Formula $t=1$
 164 \end_inset
 165
 166  to
 167 \begin_inset Formula $d_{i}$
 168 \end_inset
 169
 170 , form a basis for the rep
 171 \begin_inset Formula $\Gamma^{i}$
 172 \end_inset
 173
 174 .
 175 \end_layout
 176
 177 \begin_layout Section
 178 Particle-centered transformations
 179 \end_layout
 180
 181 \begin_layout Standard
 182 Now let's see what are the point group actions on SVWF in the origin [Schulz]:
 183 \end_layout
 184
 185 \begin_layout Standard
 186 \begin_inset Tabular
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 192 <row>
 193 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 194 \begin_inset Text
 195
 196 \begin_layout Plain Layout
 197 \begin_inset Formula $Z$
 198 \end_inset
 199
 200 -axis rotation by
 201 \begin_inset Formula $2\pi/N$
 202 \end_inset
 203
 204
 205 \end_layout
 206
 207 \end_inset
 208 </cell>
 209 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
 210 \begin_inset Text
 211
 212 \begin_layout Plain Layout
 213 \begin_inset Formula $C_{N}M_{l}^{m}=e^{\pm?i2\pi m/N}M_{l}^{m}$
 214 \end_inset
 215
 216
 217 \end_layout
 218
 219 \end_inset
 220 </cell>
 221 <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
 222 \begin_inset Text
 223
 224 \begin_layout Plain Layout
 225 \begin_inset Formula $C_{N}N_{l}^{m}=e^{\pm?i2\pi m/N}N_{l}^{m}$
 226 \end_inset
 227
 228
 229 \end_layout
 230
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 236 \begin_inset Text
 237
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 245 \begin_inset Text
 246
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 248 \begin_inset Formula $\sigma_{h}M_{l}^{m}=\left(-1\right)^{m+l+1}M_{l}^{m}$
 249 \end_inset
 250
 251
 252 \end_layout
 253
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 255 </cell>
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 257 \begin_inset Text
 258
 259 \begin_layout Plain Layout
 260 \begin_inset Formula $\sigma_{h}N_{l}^{m}=\left(-1\right)^{m+l}N_{l}^{m}$
 261 \end_inset
 262
 263
 264 \end_layout
 265
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 267 </cell>
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 271 \begin_inset Text
 272
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 280 \begin_inset Text
 281
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 283 \begin_inset Formula $\sigma_{yz}M_{l}^{m}=-M_{l}^{-m}$
 284 \end_inset
 285
 286
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 288
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 290 </cell>
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 292 \begin_inset Text
 293
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 295 \begin_inset Formula $\sigma_{yz}N_{l}^{m}=N_{l}^{-m}$
 296 \end_inset
 297
 298
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 300
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 302 </cell>
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 306 \begin_inset Text
 307
 308 \begin_layout Plain Layout
 309 Vertical (xz) reflection
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 311
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 313 </cell>
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 315 \begin_inset Text
 316
 317 \begin_layout Plain Layout
 318 \begin_inset Formula $\sigma_{xz}M_{l}^{m}=\left(-1\right)^{m+1}M_{l}^{-m}$
 319 \end_inset
 320
 321
 322 \end_layout
 323
 324 \end_inset
 325 </cell>
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 327 \begin_inset Text
 328
 329 \begin_layout Plain Layout
 330 \begin_inset Formula $\sigma_{xz}N_{l}^{m}=\left(-1\right)^{m}N_{l}^{-m}$
 331 \end_inset
 332
 333
 334 \end_layout
 335
 336 \end_inset
 337 </cell>
 338 </row>
 339 </lyxtabular>
 340
 341 \end_inset
 342
 343
 344 \end_layout
 345
 346 \begin_layout Section
 347 Transformations in a lattice
 348 \end_layout
 349
 350 \begin_layout Standard
 351
 352 \end_layout
 353
 354 \end_body
 355 \end_document