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This directory contains a set of dipolar crystals that can be used to |
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test electrostatic energies for dipole-dipole interactions. The |
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dipolar analogues to the structural Madelung constants for ionic |
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crystals were first worked out by Sauer who computed the energies of |
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certain selected dipole arrays (ordered arrays of zero magnetization) |
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and obtained a number of these constants.[1] |
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|
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This theory was developed more completely by Luttinger & Tisza [2] and |
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they tabulated energy constants for the Sauer arrays (and other |
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periodic structures). We have repeated the Luttinger & Tisza series |
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summations to much higher order and obtained the following energy |
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constants: |
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|
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Array Type Lattice Dipole Direction Energy constants |
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---------- ------- ---------------- ---------------- |
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A SC 001 -2.67678868438 |
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A BCC 001 0 |
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A BCC 111 -1.770 |
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A FCC 001 2.16693283503 |
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A FCC 011 -1.08346641751 |
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|
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B SC 001 -2.67678868438 |
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B BCC 001 -1.33839434219 |
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B BCC 111 -1.770 |
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B FCC 001 -1.08346641751 |
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B FCC 011 -1.80757363405 |
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|
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Type "A" arrays have nearest neighbor strings of antiparallel dipoles. |
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|
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Type "B" arrays have nearest neighbor strings of antiparallel dipoles |
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if the dipoles are contained in a plane perpendicular to the dipole |
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direction that passes through the dipole. |
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|
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Note that these arrays are not necessarily the minimum energy |
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structures, and those interested in this problem should consult the |
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Luttinger & Tisza paper for more details. |
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|
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The electrostatic energy for one of these dipolar arrays is |
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|
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E = C N^2 mu^2 |
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|
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where C is the energy constant above, N is the number of dipoles per |
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unit volume, and mu is the strength of the dipole. |
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|
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In the units used by OpenMD, with dipoles of 1 Debye, lengths in |
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angstroms, and energies reported in kcal / mol, the electrostatic |
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energies are: |
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|
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E = 14.39325 C N^2 mu^2 |
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|
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E here is an energy density, so it must be multiplied by the total |
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volume of the box. |
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|
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For example, the A_sc_001.md sample has a 8000 dipoles in a (40 angstrom)^3 box |
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with a lattice spacing of 2 angstroms, so the resulting energy is: |
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|
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E = Total volume * Energy per unit volume |
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= 40 * 40 * 40 * 14.39325 * (-2.67678868438) * N^2 * mu^2 |
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= 64000 * (-38.527688) * (1/8)^2 * 1^2 |
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= 38527.69 KCal/Mol |
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|
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Here we've used the definition of N as the number of dipoles per unit |
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volume: N = 1/a^3 = 1/(2^3) |
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|
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--------------------------------------------------------------------- |
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[1] J. A. Sauer, "Magnetic Energy Constants of Dipolar Lattices," |
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Phys. Rev. 57, 142–146 (1940) doi: 10.1103/PhysRev.57.142 |
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|
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[2] J. M. Luttinger and L. Tisza, "Theory of Dipole Interaction in |
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Crystals," Phys. Rev. 70, 954–964 (1946) doi: 10.1103/PhysRev.70.954 |
