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aip,jcp, |
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amsmath,amssymb, |
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preprint,% |
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< |
% reprint,% |
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> |
%reprint,% |
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%author-year,% |
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%author-numerical,% |
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jcp]{revtex4-1} |
| 1410 |
|
constant), $N$ is the number of dipoles per unit volume, and $\mu$ is |
| 1411 |
|
the strength of the dipole. Energy constants (converged to 1 part in |
| 1412 |
|
$10^9$) are given in the supplemental information. |
| 1413 |
+ |
\begin{figure}[!htbp] |
| 1414 |
|
|
| 1415 |
+ |
\includegraphics[width=3.5in]{Dipoles_rcutConvergence-crop.pdf} |
| 1416 |
+ |
|
| 1417 |
+ |
\includegraphics[width=3.5in]{Quadrupoles_rcutConvergence-crop.pdf} |
| 1418 |
+ |
\caption{Convergence to the analytic energy constants as a function of |
| 1419 |
+ |
cutoff radius (normalized by the lattice constant) for the different |
| 1420 |
+ |
real-space methods for (a) dipolar and (b) quadrupolar crystals.The energy constants for hard, SP, GSF, TSF and analytic methods are represented by black sold-circle, red solid-square,green solid-diamond and grey dashed line respectively. |
| 1421 |
+ |
The left panel shows results for the undamped kernel ($1/r$), while the damped |
| 1422 |
+ |
error function kernel, $B_0(r)$ was used in the right panel. } |
| 1423 |
+ |
\label{fig:Dipoles_rcutCovergence-crop.pdf} |
| 1424 |
+ |
\label{fig:QuadrupolesrcutCovergence-crop.pdf} |
| 1425 |
+ |
\end{figure} |
| 1426 |
|
For the purposes of testing the energy expressions and the |
| 1427 |
|
self-neutralization schemes, the primary quantity of interest is the |
| 1428 |
|
analytic energy constant for the perfect arrays. Convergence to these |
| 1434 |
|
potential-shifting and self-interaction term, but does not shift the |
| 1435 |
|
forces and torques smoothly at the cutoff radius. The SP method is |
| 1436 |
|
essentially an extension of the original Wolf method for multipoles. |
| 1425 |
– |
|
| 1437 |
|
\begin{figure}[!htbp] |
| 1438 |
< |
\includegraphics[width=4.5in]{energyConstVsCutoff} |
| 1438 |
> |
|
| 1439 |
> |
\includegraphics[width=3.5in]{Dipoles_alphaConvergence-crop.pdf} |
| 1440 |
> |
|
| 1441 |
> |
\includegraphics[width=3.5in]{Quadrupoles_alphaConvergence-crop.pdf} |
| 1442 |
|
\caption{Convergence to the analytic energy constants as a function of |
| 1443 |
< |
cutoff radius (normalized by the lattice constant) for the different |
| 1444 |
< |
real-space methods. The two crystals shown here are the ``B'' array |
| 1445 |
< |
for bcc crystals with the dipoles along the 001 direction (upper), |
| 1432 |
< |
as well as the minimum energy bcc lattice (lower). The analytic |
| 1433 |
< |
energy constants are shown as a grey dashed line. The left panel |
| 1434 |
< |
shows results for the undamped kernel ($1/r$), while the damped |
| 1443 |
> |
cutoff damping alpha for the different |
| 1444 |
> |
real-space methods for (a) dipolar and (b) quadrupolar crystals.The energy constants for hard, SP, GSF, TSF, and analytic methods are represented by the black sold-circle, red solid-square, green solid-diamond, and grey dashed line respectively. |
| 1445 |
> |
The left panel shows results for the undamped kernel ($1/r$), while the damped |
| 1446 |
|
error function kernel, $B_0(r)$ was used in the right panel. } |
| 1447 |
< |
\label{fig:energyConstVsCutoff} |
| 1447 |
> |
\label{fig:Dipoles_alphaCovergence-crop.pdf} |
| 1448 |
> |
\label{fig:Quadrupoles_alphaCovergence-crop.pdf} |
| 1449 |
|
\end{figure} |
| 1438 |
– |
|
| 1450 |
|
The Hard cutoff exhibits oscillations around the analytic energy |
| 1451 |
|
constants, and converges to incorrect energies when the complementary |
| 1452 |
|
error function damping kernel is used. The shifted potential (SP) and |
| 1456 |
|
for obtaining accurate energies. The Taylor-shifted force (TSF) |
| 1457 |
|
approximation appears to perturb the potential too much inside the |
| 1458 |
|
cutoff region to provide accurate measures of the energy constants. |
| 1448 |
– |
|
| 1459 |
|
{\it Quadrupolar} analogues to the Madelung constants were first |
| 1460 |
|
worked out by Nagai and Nakamura who computed the energies of selected |
| 1461 |
|
quadrupole arrays based on extensions to the Luttinger and Tisza |
| 1678 |
|
``Bare Coulomb.'' Equations \ref{eq:b9} to \ref{eq:b13} are still |
| 1679 |
|
correct for GSF electrostatics if the subscript $n$ is eliminated. |
| 1680 |
|
|
| 1671 |
– |
\newpage |
| 1672 |
– |
\bibliographystyle{aiprev} |
| 1681 |
|
\bibliography{multipole} |
| 1682 |
|
|
| 1683 |
|
\end{document} |
