Rework periodic part.
Former-commit-id: 4b15d1e9342e7c6b049c5bd29fd98c9766bf38a4
This commit is contained in:
Marek Nečada
parent
eb26821d11
commit
3ce1210d8a
|
|
@ -1063,11 +1063,28 @@ FP: Check signs.
|
|||
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\sigma_{l,m}\left(\vect k,\vect s\right)=\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},\vect s}\right)e^{i\vect{\vect k}\cdot\left(\vect R_{\vect n}-\vect s\right)}\sswfoutlm lm\left(\vect{R_{n}}-\vect s\right),\label{eq:sigma lattice sums}
|
||||
\sigma_{l,m}\left(\vect k,\vect s\right)=\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},\vect s}\right)e^{i\vect{\vect k}\cdot\vect R_{\vect n}}\sswfoutlm lm\left(\kappa\left(\vect s+\vect{R_{n}}\right)\right),\label{eq:sigma lattice sums}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\sigma_{l,m}\left(\vect k,\vect s\right)=\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},\vect s}\right)e^{i\vect{\vect k}\cdot\left(\vect R_{\vect n}-\vect s\right)}\sswfoutlm lm\left(\vect{R_{n}}-\vect s\right),\label{eq:sigma lattice sums-1}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
we see from eqs.
|
||||
|
||||
\begin_inset CommandInset ref
|
||||
|
|
@ -1129,6 +1146,10 @@ W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k) & =\sum_{\lambda=\left|l-l'\right|+1
|
|||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
\begin_inset Formula
|
||||
\[
|
||||
W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k)=\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r_{\beta}-\vect r_{\alpha}\right),\quad\tau'\ne\tau,
|
||||
|
|
@ -1137,6 +1158,19 @@ W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k)=\sum_{\lambda=\left|l-l'\right|+\lef
|
|||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Formula
|
||||
\[
|
||||
W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k)=\sum_{\lambda=\left|l-l'\right|+\left|\tau-\tau'\right|}^{l+l'}\tropcoeff_{\tau lm;\tau'l'm'}^{\lambda}\sigma_{\lambda,m-m'}\left(-\vect k,\vect r_{\alpha}-\vect r_{\beta}\right),\quad\tau'\ne\tau,
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
|
|
@ -1223,14 +1257,19 @@ FP: Check sign of s
|
|||
\begin_layout Standard
|
||||
\begin_inset Formula
|
||||
\begin{multline}
|
||||
\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=-\frac{2^{l+1}i}{\kappa^{l+1}\sqrt{\pi}}\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},-\vect s}\right)\left|\vect{R_{n}+\vect s}\right|^{l}\ush lm\left(\vect{R_{n}+\vect s}\right)e^{i\vect k\cdot\left(\vect{R_{n}+\vect s}\right)}\\
|
||||
\times\int_{\eta}^{\infty}e^{-\left|\vect{R_{n}+\vect s}\right|^{2}\xi^{2}}e^{-\kappa^{2}/4\xi^{2}}\xi^{2l}\ud\xi\\
|
||||
+\delta_{\vect{R_{n}},-\vect s}\frac{\delta_{l0}\delta_{m0}}{\sqrt{4\pi}}\Gamma\left(-\frac{1}{2},-\frac{\kappa^{2}}{4\eta^{2}}\right)\ush lm\left(\vect{R_{n}+\vect s}\right).\label{eq:Ewald in 3D short-range part}
|
||||
\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=-\frac{2^{l+1}i}{\kappa^{l+1}\sqrt{\pi}}\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},-\vect s}\right)\left|\vect s_{\vect n}\right|^{l}\ush lm\left(\uvec s_{\vect n}\right)e^{i\vect k\cdot\vect{R_{n}}}\\
|
||||
\times\int_{\eta}^{\infty}e^{-\left|\vect s_{\vect n}\right|^{2}\xi^{2}}e^{-\kappa^{2}/4\xi^{2}}\xi^{2l}\ud\xi\\
|
||||
+\delta_{\vect{R_{n}},-\vect s}\frac{\delta_{l0}\delta_{m0}}{\sqrt{4\pi}}\Gamma\left(-\frac{1}{2},-\frac{\kappa^{2}}{4\eta^{2}}\right)\ush lm\left(\uvec s_{\vect n}\right),\label{eq:Ewald in 3D short-range part}
|
||||
\end{multline}
|
||||
|
||||
\end_inset
|
||||
|
||||
The formal
|
||||
where we labeled
|
||||
\begin_inset Formula $\vect s_{\vect n}\equiv\vect s+\vect R_{\vect n}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
The formal
|
||||
\begin_inset Formula $\left(1-\delta_{\vect{R_{n}},-\vect s}\right)$
|
||||
\end_inset
|
||||
|
||||
|
|
@ -1355,7 +1394,19 @@ The explicit form of the long-range part of the lattice sum depends on the
|
|||
\end_inset
|
||||
|
||||
.
|
||||
|
||||
In the following, let us label
|
||||
\begin_inset Formula $\vect k_{\vect K}\equiv\vect k+\vect K$
|
||||
\end_inset
|
||||
|
||||
, where
|
||||
\begin_inset Formula $\vect K$
|
||||
\end_inset
|
||||
|
||||
is a point in the reciprocal lattice, and let
|
||||
\begin_inset Formula $\mathcal{A}$
|
||||
\end_inset
|
||||
|
||||
be the lattice unit cell volume (or area/length in the 2D/1D cases).
|
||||
\end_layout
|
||||
|
||||
\begin_layout Paragraph
|
||||
|
|
@ -1369,7 +1420,7 @@ Case
|
|||
\begin_layout Standard
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)=\frac{4\pi i^{l+1}}{\kappa\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}\underbrace{e^{i\vect K\cdot\vect s}}_{\text{nemá tu být \ensuremath{\vect{k\cdot s}?}}}\frac{\left(\left|\vect k+\vect K\right|/\kappa\right)^{l}}{\kappa^{2}-\left|\vect k+\vect K\right|^{2}}e^{\left(\kappa^{2}-\left|\vect k+\vect K\right|^{2}\right)/4\eta^{2}}\ush lm\left(\vect k+\vect K\right)\label{eq:Ewald in 3D long-range part 3D}
|
||||
\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)=\frac{4\pi i^{l+1}}{\kappa\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{-i\vect k_{\vect K}\cdot\vect s}\frac{\left(\left|\vect k_{\vect K}\right|/\kappa\right)^{l}}{\kappa^{2}-\left|\vect k_{\vect K}\right|^{2}}e^{\left(\kappa^{2}-\left|\vect k_{\vect K}\right|^{2}\right)/4\eta^{2}}\ush lm\left(\uvec k_{\vect K}\right)\label{eq:Ewald in 3D long-range part 3D}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
|
@ -1384,14 +1435,115 @@ regardless of chosen coordinate axes.
|
|||
\end_layout
|
||||
|
||||
\begin_layout Paragraph
|
||||
Case
|
||||
\begin_inset Formula $d=2$
|
||||
Cases
|
||||
\begin_inset Formula $d=1,2$
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
In the quasiperiodic cases, we decompose vectors into parallel and orthogonal
|
||||
parts with respect to the linear subspace in which the Bravais lattice
|
||||
lies (the reciprocal lattice lies in the same subspace),
|
||||
\begin_inset Formula $\vect v=\vect v_{\perp}+\vect v_{\parallel}$
|
||||
\end_inset
|
||||
|
||||
, and we label
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\gamma_{\vect k_{\vect K}}\equiv\gamma_{\vect k_{\vect K}}\left(\kappa\right)\equiv\left(\left|\vect k_{\vect K}\right|^{2}-\kappa^{2}\right)^{\frac{1}{2}}/\kappa,\label{eq:lilgamma}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\Delta_{d;j}\left(x,z\right)\equiv\int_{x}^{\infty}t^{-\frac{d_{c}}{2}-n}\exp\left(-t+\frac{z^{2}}{4t}\right)\ud t,\label{eq:Delta_j}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
where
|
||||
\begin_inset Formula $d_{c}=3-d$
|
||||
\end_inset
|
||||
|
||||
is the complementary dimension of the lattice.
|
||||
Then
|
||||
\begin_inset Formula
|
||||
\begin{multline}
|
||||
\sigma_{l}^{m}\left(\vect k,\vect s\right)=\frac{-i}{2\pi^{d_{c}/2}\mathcal{A}\kappa}\frac{\left(2l+1\right)!!}{\kappa^{l}}\sum_{\vect K\in\Lambda^{*}}e^{-i\vect k_{\vect K}\cdot\vect s}\times\\
|
||||
\times\sum_{j=0}^{l}\frac{\left(-1\right)^{j}}{j!}\left(\frac{\kappa\gamma_{\vect k_{\vect K}}}{2}\right)^{2j}\Delta_{d;j}\left(\frac{\kappa^{2}\gamma_{\vect k_{\vect K}}^{2}}{4\eta^{2}},-i\kappa\gamma_{\vect k_{\vect K}}\left|\vect s_{\perp}\right|\right)\times\\
|
||||
\times\sum_{l'=\max\left(0,l-2j\right)}^{l-j}4\pi i^{l'}\left(2\left|\vect s_{\bot}\right|\right)^{2j-l+l'}\frac{\left|\vect k_{\vect K}\right|^{l'}}{\left(2l'+1\right)!!}\sum_{m'=-l'}^{l'}\ush{l'}{m'}\left(\uvec k_{\vect K}\right)\times\\
|
||||
\times\int\ud\Omega_{\vect r}\,\ush lm\left(\uvec r\right)\ushD{l'}{m'}\left(\uvec r\right)\left(\frac{\left|\vect r_{\perp}\right|}{\left|\vect r\right|}\right)^{l-l}\left(\frac{-\vect r_{\perp}\cdot\vect s_{\perp}}{\left|\vect r_{\perp}\right|\left|\vect s_{\perp}\right|}\right)^{2j-l+l'}.\label{eq:Ewald in 3D long-range part 1D 2D}
|
||||
\end{multline}
|
||||
|
||||
\end_inset
|
||||
|
||||
The angular integral on the last line of
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:Ewald in 3D long-range part 1D 2D"
|
||||
plural "false"
|
||||
caps "false"
|
||||
noprefix "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
gives a set of constant coefficients characteristic to a chosen convention
|
||||
for spherical harmonics and coordinate axes; relatively simple closed-form
|
||||
expressions are obtained for 2D periodicity if we choose the lattice to
|
||||
lie in the
|
||||
\begin_inset Formula $xy$
|
||||
\end_inset
|
||||
|
||||
plane, so that both
|
||||
\begin_inset Formula $\vect r_{\perp},\vect s_{\perp}$
|
||||
\end_inset
|
||||
|
||||
are parallel to the
|
||||
\begin_inset Formula $z$
|
||||
\end_inset
|
||||
|
||||
axis, as done in
|
||||
\begin_inset CommandInset citation
|
||||
LatexCommand cite
|
||||
key "kambe_theory_1968"
|
||||
literal "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
, see also Supplementary Material.
|
||||
In the special case
|
||||
\begin_inset Formula $\vect s_{\perp}=0$
|
||||
\end_inset
|
||||
|
||||
the expressions can be considerably simplified as most of the terms vanish
|
||||
and
|
||||
\begin_inset Formula $\Delta_{d;j}\left(x,0\right)=\Gamma\left(1-d_{c}/2-j,x\right)$
|
||||
\end_inset
|
||||
|
||||
, but the general case is needed for evaluating the fields in space (see
|
||||
Section
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "subsec:Periodic scattering and fields"
|
||||
plural "false"
|
||||
caps "false"
|
||||
noprefix "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
) or if there is an offset between two particles in a unitcell that is not
|
||||
parallel to the lattice subspace.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
Reasonable explicit forms assume that the lattice lies inside the
|
||||
\begin_inset Formula $xy$
|
||||
\end_inset
|
||||
|
|
@ -1462,12 +1614,17 @@ FP: check sign of
|
|||
j\le s\le\min\left(2j,l-\left|m\right|\right)\\
|
||||
l-n+\left|m\right|\,\mathrm{even}
|
||||
}
|
||||
}\frac{1}{\left(2j-s\right)!\left(s-j\right)!}\frac{\left(-\kappa s_{\perp}\right)^{2j-s}\left(\left|\vect k+\vect K\right|/\kappa\right)^{l-s}}{\left(\frac{1}{2}\left(l-m-s\right)\right)!\left(\frac{1}{2}\left(l+m-s\right)\right)!}\label{eq:Ewald in 3D long-range part 1D 2D}
|
||||
}\frac{1}{\left(2j-s\right)!\left(s-j\right)!}\frac{\left(-\kappa s_{\perp}\right)^{2j-s}\left(\left|\vect k+\vect K\right|/\kappa\right)^{l-s}}{\left(\frac{1}{2}\left(l-m-s\right)\right)!\left(\frac{1}{2}\left(l+m-s\right)\right)!}\label{eq:Ewald in 3D long-range part 1D 2D-1}
|
||||
\end{multline}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
|
|
@ -1486,23 +1643,12 @@ status open
|
|||
|
||||
\end_inset
|
||||
|
||||
where
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\gamma\left(z\right)=\left(z^{2}-1\right)^{\frac{1}{2}},\label{eq:lilgamma}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\Delta_{j}\left(x,z\right)=\int_{x}^{\infty}t^{\frac{-1}{2}-n}\exp\left(-t+\frac{z^{2}}{4t}\right)\ud t.\label{eq:Delta_j}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
If the normal component
|
||||
\begin_layout Plain Layout
|
||||
where If the normal component
|
||||
\begin_inset Formula $s_{\bot}$
|
||||
\end_inset
|
||||
|
||||
|
|
@ -1538,12 +1684,20 @@ noprefix "false"
|
|||
\end_inset
|
||||
|
||||
.
|
||||
If
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
If
|
||||
\begin_inset Formula $s_{\bot}\ne0$
|
||||
\end_inset
|
||||
|
||||
, the integral
|
||||
\begin_inset Formula $\Delta_{j}\left(x,z\right)$
|
||||
\begin_inset Formula $\Delta_{d;j}\left(x,0\right)$
|
||||
\end_inset
|
||||
|
||||
can be evaluated e.g.
|
||||
|
|
@ -1555,7 +1709,7 @@ using the Taylor series
|
|||
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\Delta_{j}\left(x,z\right)=\sum_{k=0}^{\infty}\Gamma\left(\frac{1}{2}-j-k,x\right)\frac{\left(z/2\right)^{2k}}{k!}
|
||||
\Delta_{d;j}\left(x,z\right)=\sum_{k=0}^{\infty}\Gamma\left(1-\frac{d_{c}}{2}-j-k,x\right)\frac{\left(z/2\right)^{2k}}{k!}
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
|
@ -1586,27 +1740,19 @@ noprefix "false"
|
|||
|
||||
\end_inset
|
||||
|
||||
by parts (note that the signs are wrong in
|
||||
\begin_inset CommandInset citation
|
||||
LatexCommand cite
|
||||
key "kambe_theory_1968"
|
||||
literal "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
)
|
||||
by parts (with signs corrected here):
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
\Delta_{j+1}\left(x,z\right)=\frac{4}{z^{2}}\left(\left(\frac{1}{2}-j\right)\Delta_{j}\left(x,z\right)-\Delta_{j-1}\left(x,z\right)+x^{\frac{1}{2}-j}e^{-x+\frac{z^{2}}{4x}}\right)\label{eq:Delta_j recurrent}
|
||||
\Delta_{d;j+1}\left(x,z\right)=\frac{4}{z^{2}}\left(\left(\frac{1}{2}-j\right)\Delta_{d;j}\left(x,z\right)-\Delta_{d;j-1}\left(x,z\right)+x^{\frac{d_{c}}{2}-j}e^{-x+\frac{z^{2}}{4x}}\right)\label{eq:Delta_j recurrent}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
with the first two terms
|
||||
with the first two terms for 2D periodicity
|
||||
\begin_inset Formula
|
||||
\begin{align*}
|
||||
\Delta_{0}\left(x,z\right) & =\frac{\sqrt{\pi}}{2}e^{-x^{2}+\frac{z^{2}}{4x}}\left(w\left(-\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)+w\left(\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)\right),\\
|
||||
\Delta_{1}\left(x,z\right) & =i\frac{\sqrt{\pi}}{z}e^{-x^{2}+\frac{z^{2}}{4x}}\left(w\left(-\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)-w\left(\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)\right),
|
||||
\Delta_{2;0}\left(x,z\right) & =\frac{\sqrt{\pi}}{2}e^{-x^{2}+\frac{z^{2}}{4x}}\left(w\left(-\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)+w\left(\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)\right),\\
|
||||
\Delta_{2;1}\left(x,z\right) & =i\frac{\sqrt{\pi}}{z}e^{-x^{2}+\frac{z^{2}}{4x}}\left(w\left(-\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)-w\left(\frac{z}{2\sqrt{x}}+i\sqrt{x}\right)\right),
|
||||
\end{align*}
|
||||
|
||||
\end_inset
|
||||
|
|
@ -1717,9 +1863,8 @@ FP: I have some error estimates derived in my notes.
|
|||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
One pecularity of the two-dimensional case is the two-branchedness of the
|
||||
function
|
||||
\begin_inset Formula $\gamma\left(z\right)$
|
||||
One pecularity of the two-dimensional case is the two-branchedness of
|
||||
\begin_inset Formula $\gamma_{\vect k_{\vect K}}\left(\kappa\right)$
|
||||
\end_inset
|
||||
|
||||
and the incomplete
|
||||
|
|
@ -1735,7 +1880,7 @@ One pecularity of the two-dimensional case is the two-branchedness of the
|
|||
\end_inset
|
||||
|
||||
the function
|
||||
\begin_inset Formula $\gamma\left(z\right)$
|
||||
\begin_inset Formula $\gamma_{\vect k_{\vect K}}\left(\kappa\right)$
|
||||
\end_inset
|
||||
|
||||
appears with even powers, and
|
||||
|
|
@ -1898,21 +2043,6 @@ Detailed physical interpretation of the remaining branch cuts is an open
|
|||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
Generally, a good choice for
|
||||
\begin_inset Formula $\eta$
|
||||
\end_inset
|
||||
|
||||
is TODO; in order to achieve accuracy TODO, one has to evaluate the terms
|
||||
on TODO lattice points.
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
|
@ -1927,23 +2057,6 @@ status open
|
|||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Paragraph
|
||||
Case
|
||||
\begin_inset Formula $d=1$
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
For one-dimensional chains, the easiest choice is to align the lattice with
|
||||
the
|
||||
\begin_inset Formula $z$
|
||||
\end_inset
|
||||
|
||||
axis.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Subsubsection
|
||||
Choice of Ewald parameter and high-frequency breakdown
|
||||
\end_layout
|
||||
|
|
@ -2016,7 +2129,10 @@ Whatabout different geometries?
|
|||
|
||||
However, in floating point arithmetics, the magnitude of the summands must
|
||||
be taken into account as well in order to maintain accuracy.
|
||||
There is a particular problem with the
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
There is a particular problem with the
|
||||
\begin_inset Quotes eld
|
||||
\end_inset
|
||||
|
||||
|
|
@ -2026,7 +2142,7 @@ central
|
|||
|
||||
reciprocal lattice points in the long-range sums for which the real part
|
||||
of
|
||||
\begin_inset Formula $\left|\vect k+\vect K\right|^{2}-\kappa^{2}$
|
||||
\begin_inset Formula $\left|\vect k_{\vect K}\right|^{2}-\kappa^{2}$
|
||||
\end_inset
|
||||
|
||||
is negative: the incomplete
|
||||
|
|
@ -2077,7 +2193,7 @@ central
|
|||
\end_inset
|
||||
|
||||
needs to be adjusted in a way that keeps the value of
|
||||
\begin_inset Formula $\Gamma\left(a,\frac{\left|\vect k+\vect K\right|^{2}-\kappa^{2}}{4\eta^{2}}\right)$
|
||||
\begin_inset Formula $\Gamma\left(a,\left(\left|\vect k_{\vect K}\right|^{2}-\kappa^{2}\right)/4\eta^{2}\right)$
|
||||
\end_inset
|
||||
|
||||
within reasonable bounds.
|
||||
|
|
@ -2197,6 +2313,13 @@ left out for the time being
|
|||
|
||||
\begin_layout Subsection
|
||||
Scattering cross sections and field intensities in periodic system
|
||||
\begin_inset CommandInset label
|
||||
LatexCommand label
|
||||
name "subsec:Periodic scattering and fields"
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
|
@ -2391,14 +2514,36 @@ TODO fix signs and exponential phase factors
|
|||
|
||||
|
||||
\begin_inset Formula
|
||||
\begin{align*}
|
||||
\vect E_{\mathrm{scat}}\left(\vect r\right) & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect n,\alpha}{\tau}lm\vect u_{\tau lm}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\\
|
||||
& =\sum_{\alpha\in\mathcal{P}_{1}}e^{-i\vect k\cdot\left(\vect r-\vect r_{\alpha}\right)}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r-\vect r_{\alpha}\right).
|
||||
\end{align*}
|
||||
\begin{align}
|
||||
\vect E_{\mathrm{scat}}\left(\vect r\right) & =\sum_{\left(\vect n,\alpha\right)\in\mathcal{P}}\sum_{\tau lm}\outcoeffptlm{\vect n,\alpha}{\tau}lm\vect u_{\tau lm}\left(\kappa\left(\vect r-\text{\vect R_{\vect n}-\vect r_{\alpha}}\right)\right)\nonumber \\
|
||||
& =\sum_{\alpha\in\mathcal{P}_{1}}\sum_{\tau lm}\outcoeffptlm{\vect 0,\alpha}{\tau}lm\sum_{m'=-1}^{1}\vswfrtlm 21{m'}\left(0\right)\sum_{\lambda=\left|l-1\right|+\left|\tau-2\right|}^{l+1}\tropcoeff_{\tau lm;21m'}^{\lambda}\sigma_{\lambda,m-m'}\left(-\vect k,\vect r-\vect r_{\alpha}\right).\label{eq:Scattered fields in periodic systems}
|
||||
\end{align}
|
||||
|
||||
\end_inset
|
||||
|
||||
In the scattering problem, the total field intensity is obtained by adding
|
||||
the incident field to
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:Scattered fields in periodic systems"
|
||||
plural "false"
|
||||
caps "false"
|
||||
noprefix "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
; whereas in the lattice mode problem the total field is directly given
|
||||
by
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:Scattered fields in periodic systems"
|
||||
plural "false"
|
||||
caps "false"
|
||||
noprefix "false"
|
||||
|
||||
\end_inset
|
||||
|
||||
.
|
||||
\end_layout
|
||||
|
||||
\end_body
|
||||
|
|
|
|||
Loading…
Reference in New Issue