Paper Info
Title
The Charge-Group Summation Method For Electrostatics Of Periodic Crystals
Abstract
The electrostatic fields and potentials associated with neutral periodic crystals are defined by sums that are not absolutely convergent. The sums depend on the order of summation. The mean-zero periodic solution phi of Poisson's equation provides a natural potential and electric field. This sum is inconsistent with the electric field in ferroelectric materials. We introduce summation methods based on the concept of neutral charge groups, a notion common in computational chemistry. If a charge group has moments of order < kappa vanishing, then the sum for partial derivative(alpha)phi defined by summing first by charge groups is absolutely convergent for vertical bar alpha vertical bar >= 3 - kappa. In the borderline case vertical bar alpha vertical bar = 2 - kappa, consider charge groups coming from primitive unit cells. For a macroscopic shape Omega, the unit-cell R Omega algorithm sums over unit cells in R Omega and then takes the limit R -> infinity. This yields an answer for partial derivative(alpha)phi that depends on Omega and the unit cell. The limits differ from partial derivative(alpha)phi by a constant given as an easily approximable integral. For quadrupolar primitive unit cells, kappa = 2. The unit-cell R Omega algorithm gives a potential phi that differs from phi by a constant. This constant enters additively in the energy content per unit volume. If one could grow or cut crystals respecting these unit cells, this shape-dependent energy effect could be verified experimentally. For dipolar primitive unit cells, kappa = 1. Given a shape Omega and unit cell, the algorithm yields a well-defined electric field that differs from the mean-zero periodic field by a constant vector. If one could grow or cut crystals respecting these dipolar unit cells, this shape-dependent ferroelectric effect [C. Kittel, Introduction to Solid State Physics, 8th ed., Wiley, New York, 2004] could be verified experimentally.
Year
DOI
Venue
2021
10.1137/20M1353654
SIAM JOURNAL ON APPLIED MATHEMATICS
Keywords
DocType
Volume
Madelung constant, electrostatics, electric field, charge group, potential, ferroelectric
Journal
81
Issue
ISSN
Citations 
2
0036-1399
0
PageRank 
References 
Authors
0.34
0
2
Name
Order
Citations
PageRank
Jeffrey Rauch122.54
L. Ridgway Scott2153.63