Generation of Basis Vectors for Magnetic Structures and Displacement Modes

Z L Davies,A S Wills

doi:10.1155/2016/3960145

Z L Davies, A S Wills

Open Access

https://doi.org/10.1155/2016/3960145

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Abstract

Increasing attention is being focused on the use of symmetry-adapted functions to describe magnetic structures, structural distortions, and incommensurate crystallography. Though the calculation of such functions is well developed, significant difficulties can arise such as the generation of too many or too few basis functions to minimally span the linear vector space. We present an elegant solution to these difficulties using the concept of basis sets and discuss previous work in this area using this concept. Further, we highlight the significance of unitary irreducible representations in this method and provide the first validation that the irreducible representations of the crystallographic space groups tabulated by Kovalev are unitary.

Highlights

The use of symmetry-adapted functions is well established in many fields, such as electronic structure calculations and vibrational mode analysis
Analysis using these functions has been applied to several neutron scattering techniques, including spherical neutron polarimetry [10], single crystal diffraction [11, 12], and powder diffraction [13,14,15]
In magnetism and crystallography these functions are usually referred to as basis vectors (BVs) and they define the order parameters of some property resulting from a phase transition; the coefficients of the BVs define the state of a system with respect to a reference

Summary

Introduction

The use of symmetry-adapted functions is well established in many fields, such as electronic structure calculations and vibrational mode analysis. Analysis using these functions has been applied to several neutron scattering techniques, including spherical neutron polarimetry [10], single crystal diffraction [11, 12], and powder diffraction [13,14,15]. The foundation of this method is the derivation of symmetry-adapted functions using the techniques of representation theory. A representation of some system property under its symmetry operations is constructed as the direct product of two component representations Γ = ΓV ⊗ ΓPerm. We begin by defining what constitutes an appropriate set of solutions and how they are derived

Properties of Basis Sets

Presentation of the Operators

Overgeneration

Undergeneration

Unitary Check

Conclusions