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# Magnetic Refinements

Topas v5 will perform magnetic refinements. There are various options for handling the magnetic symmetry which are explored in a tutorial: Magnetic Tutorial.

Two common pitfalls are comparing moments between different pieces of software (there are several different conventions in use) and scaling moments appropriately when the magnetic cell size is different to the nuclear. Scaling is discussed in the tutorial. There are notes on moment conventions on Branton's IUCr commission website (IUCr commission website). I've reproduced the June 2012 version of these below, though they may change.

# Topas Coordinate System

The topas coordinate system is in 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).

The refined moments can be displayed in alternate systems using macros:

```    MM_CrystalAxis_Display( 0.00033`, 0.00000, 3.08356`)
MM_Cartesian_Display(-0.96490`, 0.00000`, 2.92859`)```

There are also macros which let you refine in these coordinate systems. For example, the macro MM_Cartesian_Refine(@, 0, @, 0, @, 0) corresponds to the following:

`    Let X correspond to cross product.`
```    Reciprocal lattice is:

a* = b X c
b* = c X a
c* = a X b```

In MM_Cartesian_Refine, the lattice is in Cartesian coordinates, ie.

```    ax i
bx i + by j
cx i + cy j + cx k```

where i, j, k are unit Cartesian vectors, Or,

```    i || a
j || a X (a X b) = a X c*
k || a X b = c*```

### Crystal coordinate systems for defining components of magnetic moments

N.B. This is a frozen summary. Go to IUCr commission website for up to date information.

(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^(-1).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.

### Usage

FULLPROF: Systems (2) and (5) can both be used for refinements. The implementation of system (5) requires that the a and c axes be orthogonal, excluding triclinic cases.

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

GSAS: Systems (4) is used for refinements; but systems (1) and (3) can be displayed as output.

TOPAS: The parameters of system (1) are explicitly defined (in an unreleased version), though built-in macros for systems (2) and (4) are available for refinements.

(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 arises naturally when performing representational analyses; it is the only coordinate system that properly isolates magnetic modes from lattice-strain modes by definition, which permits the rigorous study of magneto-elastic effects. It is also computationally convenient because 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 will often be the best for publishing a magnetic structure.

(3) This system does not appear to be in common use.

(4) Orthonormal coordinates are advantageous because of the ease with which the components of the moment are related to its magnitude. But they are not very natural when the crystal axes are not orthogonal. If an orthonormal coordinate system is needed, this one is the most widely used.