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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 Madelung constants for ionic crystals were |
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first worked out by Sauer who computed the energies of certain |
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selected dipole arrays (ordered arrays of zero magnetization) and |
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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] who |
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tabulated energy constants for the Sauer arrays (and other periodic |
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structures). We have repeated the Luttinger & Tisza series summations |
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to much higher order and obtained the following energy 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 -2.676788684 |
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A BCC 001 0 |
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A BCC 111 -1.770078733 |
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A FCC 001 2.166932835 |
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A FCC 011 -1.083466417 |
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|
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* BCC minimum -1.985920929 |
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|
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B SC 001 -2.676788684 |
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B BCC 001 -1.338394342 |
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B BCC 111 -1.770078733 |
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B FCC 001 -1.083466417 |
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B FCC 011 -1.807573634 |
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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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There's also an additional minimum energy structure for the BCC |
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lattice that was found by Luttinger & Tisza. All of the energy |
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constants can be recomputed to a very high degree of accuracy using |
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the "fieldValues" python script contained in this directory. Dipolar |
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arrays matching these configurations which are suitable for use with |
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OpenMD can be generated with the "buildDipolarArray" program. |
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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, dipoles are measured in 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 (1 Debye each) |
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in a (40 angstrom)^3 box with a lattice spacing of 2 angstroms, so the |
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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 * (-2.676788684) * N^2 * mu^2 |
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= 64000 * (-38.52769) * (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 |
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Also note the errata contained in: Phys. Rev. 72, 257 (1947) |
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doi: 10.1103/PhysRev.72.257 |
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