More Päivi's suggestions implemented.
Former-commit-id: f391fdb73b126a241e3900d5282841f03abd5fb0
This commit is contained in:
Marek Nečada
parent
8942753a13
commit
0e45ae0d05
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@ -76,6 +76,14 @@
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series = {Artech {{House Antennas}} and {{Propagation Library}}}
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}
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@book{condon_theory_1935,
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title = {The {{Theory}} of {{Atomic Spectra}}},
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author = {Condon, E. U. and Shortley, G. H.},
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year = {1935},
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publisher = {{Cambridge University Press}},
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isbn = {978-0-521-09209-8}
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}
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@article{dellnitz_locating_2002,
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title = {Locating All the Zeros of an Analytic Function in One Complex Variable},
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author = {Dellnitz, Michael and Sch{\"u}tze, Oliver and Zheng, Qinghua},
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@ -945,6 +953,13 @@ matrix method for multilayer calculations.},
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number = {4}
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}
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@misc{SCUFF/MMN,
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title = {{{SCUFF}}-{{EM}}},
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author = {Reid, Homer},
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year = {2018},
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note = {https://github.com/texnokrates/scuff-em}
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}
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@misc{SCUFF2,
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title = {{{SCUFF}}-{{EM}}},
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author = {Reid, Homer},
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@ -1147,6 +1162,16 @@ matrix method for multilayer calculations.},
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number = {8}
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}
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@book{wigner_group_1959,
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title = {Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra},
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author = {Wigner, Eugene P. and Griffin, J. J.},
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year = {1959},
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edition = {Revised},
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publisher = {{Academic Press}},
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file = {/home/mmn/.zotero/zotero/w4aj0ekp.default/zotero/storage/8T5VQVHL/Group theory and its application to the quantum mechanics of atomic spectra by Eugene P. Wigner, J. J. Griffin (z-lib.org).djvu},
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isbn = {978-0-12-750550-3}
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}
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@article{xu_calculation_1996,
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title = {Calculation of the {{Addition Coefficients}} in {{Electromagnetic Multisphere}}-{{Scattering Theory}}},
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author = {Xu, Yu-lin},
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@ -2205,9 +2205,9 @@ noprefix "false"
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, taking the sums over scatterers inside one unit cell, to get the extinction
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and absorption cross sections per unit cell.
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From these, quantities such as absorption, extinction coefficients are
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obtained using suitable normalisation by unit cell size, depending on lattice
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dimensionality.
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From these, quantities such as absorption, extinction and scattering coefficien
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ts are obtained using suitable normalisation by unit cell size, depending
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on lattice dimensionality.
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\end_layout
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\begin_layout Standard
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@ -207,11 +207,11 @@ TODO Zkontrolovat všechny vzorečky zde!!!
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In order to make use of the point group symmetries, we first need to know
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how they affect our basis functions, i.e.
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the VSWFs.
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\end_layout
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\begin_inset space \space{}
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\end_inset
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\begin_layout Standard
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Let
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the VSWFs.
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Let
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\begin_inset Formula $g$
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\end_inset
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@ -220,8 +220,10 @@ Let
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\end_inset
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, i.e.
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a 3D point rotation or reflection operation that transforms vectors in
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\begin_inset space \space{}
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\end_inset
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a 3D point rotation or reflection operation that transforms vectors in
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\begin_inset Formula $\reals^{3}$
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\end_inset
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@ -281,8 +283,8 @@ Spherical harmonics
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, transform as
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\begin_inset CommandInset citation
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LatexCommand cite
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after "???"
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key "dresselhaus_group_2008"
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after "Chapter 15"
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key "wigner_group_1959"
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literal "false"
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\end_inset
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@ -429,11 +431,11 @@ noprefix "false"
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\end_inset
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(and analogously for the regular waves
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and analogously for the regular waves
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\begin_inset Formula $\vswfrtlm{\tau}lm$
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\end_inset
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).
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.
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\begin_inset Note Note
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status open
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@ -767,7 +769,7 @@ With these transformation properties in hand, we can proceed to the effects
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& =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(\kappa\left(\vect r-\vect r_{\pi_{g}p}\right)\right)\right.\\
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& \quad+\left.\outcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\kappa\left(\vect r-\vect r_{\pi_{g}p}\right)\right)\right)\\
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& =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm{\pi_{g}^{-1}q}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(\kappa\left(\vect r-\vect r_{q}\right)\right)\right.\\
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& \quad+\left.\outcoeffptlm{\pi_{g}^{-1}q}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\kappa\left(\vect r-\vect r_{q}\right)\right)\right)
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& \quad+\left.\outcoeffptlm{\pi_{g}^{-1}q}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\kappa\left(\vect r-\vect r_{q}\right)\right)\right).
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\end{align*}
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\end_inset
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@ -799,7 +801,11 @@ For a given particle
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\begin_inset Formula $p$
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\end_inset
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, i.e.
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,
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\begin_inset space \space{}
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\end_inset
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i.e.
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the set
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\begin_inset Formula $\left\{ \pi_{g}\left(p\right);g\in G\right\} $
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\end_inset
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@ -853,7 +859,17 @@ noprefix "false"
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\end_inset
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(TODO avoid notation clash here in a more consistent and readable way!)
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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(TODO avoid notation clash here in a more consistent and readable way!)
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\end_layout
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\end_inset
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\begin_inset Formula
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\begin{align}
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\rcoeffp p & \overset{g}{\longmapsto}\tilde{J}\left(g\right)\rcoeffp{\pi_{g}^{-1}(p)},\nonumber \\
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@ -942,7 +958,7 @@ ing problem matrix
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\begin_inset Formula $G$
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\end_inset
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consisting of matrices
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consisting of matrices
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\begin_inset Formula $D^{\Gamma_{n}}\left(g\right)$
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\end_inset
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@ -1050,7 +1066,7 @@ literal "false"
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or
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\begin_inset CommandInset citation
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LatexCommand cite
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after "???"
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after "Chapter 2"
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key "bradley_mathematical_1972"
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literal "false"
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@ -1187,7 +1203,10 @@ Also for periodic systems,
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\end_inset
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from the left hand side of eqs.
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\begin_inset space \space{}
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\end_inset
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Multiple-scattering problem unit cell block form"
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@ -1230,7 +1249,7 @@ s happens unless
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lies somewhere in the high-symmetry parts of the Brillouin zone.
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However, the high-symmetry points are usually the ones of the highest physical
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interest, for it is where the band edges are typically located.
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The subsection does not aim for an exhaustive treatment of the topic of
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This subsection does not aim for an exhaustive treatment of the topic of
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space groups in physics (which can be found elsewhere
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\begin_inset CommandInset citation
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LatexCommand cite
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@ -1414,7 +1433,10 @@ The transformation to the symmetry adapted basis
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: this can happen if the point group symmetry maps some of the scatterers
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from the reference unit cell to scatterers belonging to other unit cells.
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This is illustrated in Fig.
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\begin_inset space \space{}
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\end_inset
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "Phase factor illustration"
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@ -1426,7 +1448,10 @@ noprefix "false"
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.
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Fig.
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\begin_inset space \space{}
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\end_inset
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "Phase factor illustration"
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@ -1449,21 +1474,21 @@ a shows a hexagonal periodic array with
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\end_inset
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.
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If we delimit our representative unit cell as the Wigner-Seitz cell with
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origin in a
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We delimit our representative unit cell as the Wigner-Seitz cell with origin
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in a
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\begin_inset Formula $D_{6}$
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\end_inset
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point group symmetry center (there is one per each unit cell).
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Per unit cell, there are five different particles placed on the unit cell
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boundary, and we need to make a choice to which unit cell the particles
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on the boundary belong; in our case, we choose that a unit cell includes
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the particles on the left as denoted by different colors.
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point group symmetry center (there is one per each unit cell); per unit
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cell, there are five different particles placed on the unit cell boundary,
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and we need to make a choice to which unit cell the particles on the boundary
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belong; in our case, we choose that a unit cell includes the particles
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on the left as denoted by different colors.
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If the Bloch vector is at the upper
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\begin_inset Formula $M$
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\end_inset
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point,
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point,
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\begin_inset Formula $\vect k=\vect M_{1}=\left(0,2\pi/\sqrt{3}a\right)$
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\end_inset
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@ -1503,7 +1528,10 @@ horizontal
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\end_inset
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as in eq.
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\begin_inset space \space{}
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\end_inset
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:excitation coefficient under symmetry operation"
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@ -1577,7 +1605,7 @@ If we set instead
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\begin_inset Formula $\vect k=\vect K=\left(4\pi/3a,0\right),$
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\end_inset
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the original
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the original
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\begin_inset Formula $D_{6}$
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\end_inset
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