356 lines
6.8 KiB
Plaintext
356 lines
6.8 KiB
Plaintext
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#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\pdf_title "Sähköpajan päiväkirja"
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\pdf_author "Marek Nečada"
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\end_header
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\begin_body
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\begin_layout Title
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Symmetry-adapted basis functions for honeycomb lattice at
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\begin_inset Formula $K$
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\end_inset
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-point
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\end_layout
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\begin_layout Section
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Generation theorem
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\end_layout
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\begin_layout Standard
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Let
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\begin_inset Formula $\mathbf{G}$
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\end_inset
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be a group and
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\begin_inset Formula $\Gamma^{i}\left\{ R\to\mathbf{D}^{i}\left(R\right)\right\} $
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\end_inset
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some
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\begin_inset Formula $d_{i}$
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\end_inset
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-dimensional rep of
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\begin_inset Formula $\mathbf{G}$
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\end_inset
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.
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Let the group ring (corresponding to the given rep indexed by
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\begin_inset Formula $i$
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\end_inset
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) elements be defined as [Bradley&Cracknell (2.2.2)]
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\begin_inset Formula
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\[
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W_{ts}^{i}=\frac{d_{i}}{\left|\mathbf{G}\right|}\sum_{R\in\mathbf{G}}\mathbf{D}^{i}\left(R\right)_{ts}^{*}R.
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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From [Bradley&Cracknell, theorem 2.2.1]:
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\end_layout
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\begin_layout Standard
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If
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\begin_inset Formula $\phi$
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\end_inset
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is an arbitrary function of
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\begin_inset Formula $V$
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\end_inset
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(a linear space in which the realisation of the group operation act) such
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that
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\begin_inset Formula $W_{ss}^{i}\phi\ne0$
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\end_inset
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(
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\begin_inset Formula $s$
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\end_inset
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is fixed and is a number in the range 1 to
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\begin_inset Formula $d_{i}$
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\end_inset
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;
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\begin_inset Formula $i$
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\end_inset
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is idx of the rep) then the funs
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\begin_inset Formula $W_{ts}^{i}\phi=\phi_{t}^{i}$
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\end_inset
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,
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\begin_inset Formula $t=1$
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\end_inset
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to
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\begin_inset Formula $d_{i}$
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\end_inset
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, form a basis for the rep
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\begin_inset Formula $\Gamma^{i}$
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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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Particle-centered transformations
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\end_layout
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\begin_layout Standard
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Now let's see what are the point group actions on SVWF in the origin [Schulz]:
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\end_layout
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\begin_layout Standard
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\begin_inset Tabular
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<lyxtabular version="3" rows="4" columns="3">
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<features rotate="0" tabularvalignment="middle">
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<column alignment="center" valignment="top">
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<column alignment="center" valignment="top">
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<row>
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\begin_inset Text
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\begin_layout Plain Layout
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\begin_inset Formula $Z$
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\end_inset
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-axis rotation by
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\begin_inset Formula $2\pi/N$
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\end_inset
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\end_layout
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\end_inset
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</cell>
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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\begin_inset Formula $C_{N}M_{l}^{m}=e^{\pm?i2\pi m/N}M_{l}^{m}$
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\end_inset
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\end_layout
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\end_inset
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</cell>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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\begin_inset Formula $C_{N}N_{l}^{m}=e^{\pm?i2\pi m/N}N_{l}^{m}$
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\end_inset
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\end_layout
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\end_inset
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</cell>
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</row>
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<row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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Horizontal (xy) reflection
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\end_layout
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\end_inset
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</cell>
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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\begin_inset Formula $\sigma_{h}M_{l}^{m}=\left(-1\right)^{m+l+1}M_{l}^{m}$
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\end_inset
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\end_layout
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\end_inset
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</cell>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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\begin_inset Formula $\sigma_{h}N_{l}^{m}=\left(-1\right)^{m+l}N_{l}^{m}$
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\end_inset
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\end_layout
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\end_inset
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</cell>
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</row>
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<row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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Vertical (yz) reflection
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\end_layout
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\end_inset
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</cell>
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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\begin_inset Formula $\sigma_{yz}M_{l}^{m}=-M_{l}^{-m}$
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\end_inset
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\end_layout
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\end_inset
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</cell>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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\begin_inset Formula $\sigma_{yz}N_{l}^{m}=N_{l}^{-m}$
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\end_inset
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\end_layout
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\end_inset
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</cell>
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</row>
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<row>
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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Vertical (xz) reflection
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\end_layout
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\end_inset
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</cell>
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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\begin_inset Formula $\sigma_{xz}M_{l}^{m}=\left(-1\right)^{m+1}M_{l}^{-m}$
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\end_inset
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\end_layout
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\end_inset
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</cell>
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
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\begin_inset Text
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\begin_layout Plain Layout
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\begin_inset Formula $\sigma_{xz}N_{l}^{m}=\left(-1\right)^{m}N_{l}^{-m}$
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\end_inset
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\end_layout
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\end_inset
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</cell>
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</row>
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</lyxtabular>
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\end_inset
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\end_layout
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\begin_layout Section
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Transformations in a lattice
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\end_layout
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\begin_layout Standard
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\end_layout
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\end_body
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\end_document
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