# Magnetic Coordinates

### (for defining components of magnetic moments)

## Important coordinate systems

(1) Bohr Magneton/Angstrom units, with x||a, y||b and z||c

This is the reduced lattice coordinate system, where the magnetic metric tensor (M) is the same metric used for interatomic distances (G).

(2) Bohr Magneton units, with x||a, y||b and z||c

This is the crystal-axis coordinate system, where components of the moment are defined by their projections along the lattice basis vectors.

If we define L = {{a,0,0},{0,b,0},{0,0,c}}, then the magnetic metric tensor is M = L.G.L^(-1), which is unitless.

(3) Bohr Magneton units, with x||a, y||b* and z||(a x b*)

This is an orthonormal coordinate system defined by a and b*. The magnetic metric tensor is the identity matrix.

(4) Bohr Magneton units, with x||a, y||(c* x a) and z||c*

This is an orthonormal coordinate system defined by a and c*. The magnetic metric tensor is the identity matrix.

(5) Bohr Magneton units for the magnitude, plus two spherical-coordinate angles measured relative to the X and Z axes of system (4)

φ runs from 0 to 2π in the XY plane, and θ runs from 0 to π relative to Z.

(6) Bohr Magneton units in a Q-dependent orthonormal coordinate system, where x||Q, z is perpendicular to the scattering plane, and y lies within the scattering plane to yield right-handed axes

These Blume-Maleyev coordinates are used to describe the magnetic interaction vector rather than individual magnetic moment vectors.

## Usage

FULLPROF: Systems (2) is used by default for refinements, though systems (4) and (5) can also be used. System (3) is used by some of the modules of the CrysFML library.

JANA: System (1) is used internally. Systems (2) and (5) can both be used for refinements.

GSAS: System (4) is used for refinements. Systems (1) and (3) can be displayed as additional output.

TOPAS: System (1) has built-in refinable parameters. Built-in macros for refinements in systems (2) and (4) are also available.

## Comments

(1) For atomic positions, reduced lattice coordinates are unitless and independent of the cell parameters. For magnetic moments, on the other hand, reduced lattice-coordinates are less-intuitive because they are not unitless (Bohr magnetons per Angstrom); one must use the cell parameters to compute the magnitude of the moment. This system is computationally convenient in that it uses the same metric tensor used to compute interatomic distances. Reduced lattice-coordinate moments are not easily digestible by human readers, and are probably best kept out of sight.

(2) Crystal-axis coordinates have intuitive units (Bohr magnetons) and are easy communicate. The coordinate axes correspond to natural crystal lattice directions and require no additional conventions. However, when the crystal axes are not orthogonal, it does take some effort to relate the components of the moment to its magnitude. This system is preferred (most common and most natural) for publishing and otherwise disseminating magnetic structures to a broad audience.

(3) This system is less commonly used.

(4) This system was popularized by Busing and Levy [Acta Cryst 22, 457-464 (1967)] when they formalized the UB matrix. In an orthonormal coordinate system, the components of the moment and the magnitude of the moment have a very simple relationship. But orthonormal coordinates are not very natural when the crystal axes are not orthogonal. While orthonormal coordinates are not recommended for general dissemination of a magnetic structure (e.g. publications), this is the most widely used orthonormal coordinate system.

(5) The spherical coordinate system must be defined with respect to an underlying orthonormal coordinate system (e.g. 4). This system may be convenient when describing helical or helicoidal magnetic structures.

(6) The magnetic interaction vector (the projection of the magnetic structure factor onto the plane perpendicular to Q) also needs a vector component description. This is often done in systems (2), (3) or (4), though (6) is particularly useful in the context of single-crystal polarized-neutron measurements.

Non-orthogonal coordinate systems based on lattice basis vectors (e.g. systems 1 and 2) arise naturally when performing representational analyses of magnetic structures.

[Correspondence with John Evans regarding his efforts to reconcile refinements from multiple packages and to understand their various coordinate systems led to the creation of this page. On 21/04/15, this content was moved here from the old CMS Wiki site, which is no longer supported.]