Madelung/quadrupoles/README.txt

This directory contains a set of quadrupolar crystals that can be used
to test electrostatic energies for quadrupole-quadrupole interactions.
The quadrupolar analogues to the structural Madelung constants for
ionic crystals were first worked out by Nagai and Nakamura who
computed the energies of certain selected quadrupole arrays and
obtained a number of these constants.[1] Their work is based on
earlier work on dipolar crystals that was done by Luttinger & Tisza
[2].  We have generated only the lowest energy configurations for the
linear quadrupoles 

 Lattice    Quadrupole Direction     Energy constants   Notes
 -------    --------------------     ----------------   ------
   SC             111                     -8.30         Fig 6a
   BCC            011                    -21.7          Fig 7b
   FCC            111                    -80.5          Fig 8
             
The electrostatic energy for one of these dipolar arrays is

  E = C (3/4) N^2 Q^2

where C is the energy constant above, N is the number of quadrupoles
per unit volume, and Q is the strength of the dipole.

In the units used by OpenMD, with quadrupole moments of 1
Debye-angstrom, lengths in angstroms, and energies reported in
kcal / mol, the electrostatic energies are:

  E = 14.39325 (3/4) C N^2 Q^2

E here is an energy density, so it must be multiplied by the total
volume of the box.


---------------------------------------------------------------------
[1] O. Nagai and T. Nakamura, "Quadrupole Interaction in Crystals,"
    Progress of Theoretical Physics 24 (2), 432-454 (1960) 
    doi: 10.1143/PTP.24.432,   also see Errata: Progress of
    Theoretical Physics 30 (3), 412 (1963).  doi: 10.1143/PTP.30.412a

[2] J. M. Luttinger and L. Tisza, "Theory of Dipole Interaction in
    Crystals," Phys. Rev. 70, 954–964 (1946) doi: 10.1103/PhysRev.70.954
Revision:	1921
Committed:	Thu Aug 1 18:23:07 2013 UTC (12 years, 5 months ago) by gezelter
Content type:	text/plain
File size:	1851 byte(s)
Log Message:	Fixed a few DumpWriter / FlucQ bugs. Still working on Ewald, updated Quadrupolar test structures.
#	User	Rev	Content
1	gezelter	1916	This directory contains a set of quadrupolar crystals that can be used
2			to test electrostatic energies for quadrupole-quadrupole interactions.
3			The quadrupolar analogues to the structural Madelung constants for
4			ionic crystals were first worked out by Nagai and Nakamura who
5			computed the energies of certain selected quadrupole arrays and
6			obtained a number of these constants.[1] Their work is based on
7			earlier work on dipolar crystals that was done by Luttinger & Tisza
8			[2]. We have generated only the lowest energy configurations for the
9			linear quadrupoles
10
11			Lattice Quadrupole Direction Energy constants Notes
12			------- -------------------- ---------------- ------
13			SC 111 -8.30 Fig 6a
14			BCC 011 -21.7 Fig 7b
15			FCC 111 -80.5 Fig 8
16
17			The electrostatic energy for one of these dipolar arrays is
18
19			E = C (3/4) N^2 Q^2
20
21			where C is the energy constant above, N is the number of quadrupoles
22			per unit volume, and Q is the strength of the dipole.
23
24	gezelter	1921	In the units used by OpenMD, with quadrupole moments of 1
25	gezelter	1916	Debye-angstrom, lengths in angstroms, and energies reported in
26			kcal / mol, the electrostatic energies are:
27
28			E = 14.39325 (3/4) C N^2 Q^2
29
30			E here is an energy density, so it must be multiplied by the total
31			volume of the box.
32
33
34			---------------------------------------------------------------------
35			[1] O. Nagai and T. Nakamura, "Quadrupole Interaction in Crystals,"
36			Progress of Theoretical Physics 24 (2), 432-454 (1960)
37	gezelter	1921	doi: 10.1143/PTP.24.432, also see Errata: Progress of
38			Theoretical Physics 30 (3), 412 (1963). doi: 10.1143/PTP.30.412a
39	gezelter	1916
40			[2] J. M. Luttinger and L. Tisza, "Theory of Dipole Interaction in
41			Crystals," Phys. Rev. 70, 954–964 (1946) doi: 10.1103/PhysRev.70.954