MAGNEXT - Systematic Absences for Magnetic Space Groups
MAGNEXT Option A is a database that contains the systematic absences that can be observed in non-polarised neutron magnetic diffraction experiments, for a given Magnetic Space Group in one the two settings currently in use in the literature (see below). The user has to introduce the label of the required space group or to choose it from a table.
MAGNEXT tool provides for the given magnetic space group one systematic absence for each one of the subspaces of the reciprocal space for which systematic absences appear, treating separately the systematic absences for general and special reflections. The program also indicates, for the cases where that information is relevant or illustrative, a list of values of h, k, l and their corresponding symmetry-adapted structure factor form.
In addition, a complete list of the structure factor forms for any relevant (i. e., not completely forbidden by general absences) diffraction vector type is accesible from the output interface. The systematic absences and subsequent information can be obtained in a non conventional setting by clicking on the link [Show systematic absences in a different setting] and giving the change of basis (Important: only change of basis matrix with determinant equal to 1 are accepted).
MAGNEXT includes three alternative input choice apart from the Option A. A different interface for any of these input choices is provided.
Option B is used to obtain the systematic absences for a magnetic space group in any setting given by means of the symmetry cards of a set of generators.
Option C is used to get a list of Magnetic Space Groups compatible with a given set of systematic absences in standard settings and optionally with a non-magnetic space group or a crystal class.
Finally, although the program works mainly for commensurate magnetic structures, MAGNEXT includes another input choice which consist of a simple tool for incommensurate structures that provides the systematic extinction rules for magnetic superspace groups given by means of the symmetry cards of a set of generators.
Important: Lists of systematic absences are given in a non-redundant way, i. e., those systematic absences contained in a more general ones are not explicitly indicated. This is valid in general for any input option of MAGNEXT and the resulting outputs.
BNS and OG settings
There are two settings or notations currently in use for magnetic space groups. These two settings are different because they come from the way some Magnetic Space Groups are derived, and affect only the Groups Type IV. Not only the Generators, General Positions and Wyckoff Positions but also the Space Group and even the systematic absences vary depending on the chosen setting.
The OG setting refers to the Opechowski-Guccione notation (1). In this setting Space Groups labels, symmetry operations, etc are referred to a unit cell which the crystal has if the magnetic order is neglected. This implies that when magnetic ordering is considered the OG unit cell doesn't reproduce the crystal periodicity (two neighbor unit cells are in general different).
Magnetic Space Groups in OG setting are derived from two Fedorov Groups G and H, being H a k-subgroup of index 2 of G. The group is built calculating G' = H + (G-H)1'
The BNS setting refers to the Belov-Neronova-Smirnova notation (2). In this setting, the unit cell is in general different from the paramagnetic one, and takes into account the magnetic ordering from the beginning. BNS unit cell describes reliably the periodicity of the crystal.
Magnetic Space Groups in BNS setting are derived from only one Fedorov Group G, adding a "primed sublattice" generated by an operation that combines time-inversion with a fractional traslation. The group is built calculating G' = G + Gt1', where t is the mentioned translation.
1. Opechowski and Guccione (1965). Magnetism, edited by G. T. Rado and H. Suhl, Vol. II, Part A, pp. 105-165. New York: Academic Press
2. Belov et al. (1957). Sov. Phys. Crystallogr. 2, 311-322
"Generalized" BNS and OG settings
In Option B a "generalized" BNS or OG setting is required. This means that the magnetic space group generators given by the user must be in any setting in which the basic vectors (1|1,0,0), (1|0,1,0) and (1|0,0,1) are lattice traslations of the group ("generalized" BNS setting) or either lattice traslations or lattice traslations associated to a time-inversion ("generalized" OG setting).
Non-magnetic space group
In Option C a non-magnetic space group can be introduced. This group is just the commonly called "space group" of the crystal and is the space group the crystal has when the time-reversal symmetry is neglected. Giving a non-magnetic space group only those magnetic space groups that corresponds to the given non-magnetic space group will be provided.
Propagation vector k
MAGNEXT provides the propagation vector for Magnetic Superspace Groups Type IV in OG setting. This vector, which is defined with respect to the OG unit cell, does not always coincide with the actual propagation vector in the experiment (this only happens if the lattice defined by the OG unit cell is not the same as the one of the paramagnetic phase). In this cases, the real propagation vector can be derived from the one provided by MAGNEXT applying the trasnformation matrix between the paramagnetic and OG unit cells.
Magnetic Superspace Groups
For the systematic absences in incommensurate structures, MAGNEXT works also with Magnetic Superspace Groups. The Magnetic Superspace Group must be specified introducing in the text area some generators of the group in symmetry cards notation (see below). The propagation vector associated to the Magnetic Superspace Group, k, must be introduced as well, using the letters a, b and c to denote an incommensurate component (for example: k = (a,-a,0), k = (1/2,1/3,c), etc).
Symmetry cards notation
Option B input mode requires the specification of symmetry elements that belong to a Magnetic Space or Superspace Group. For Magnetic Space Group case, the symmetry elements must be indicated in a way similar to these examples:
where the last number indicates if a time inversion is associated to the symmetry element (last number equal to -1), or not (equal to 1).
For Magnetic Superspace Group case, the way to introduce symmetry elements is similar, but using x1, x2, x3 and x4 coordinates instead of x, y and z. For example:
-x3,-x1,-x2+1/2,x1+x4,-1. For some cases, in addition to the symmetry cards the transformation of the magnetic moment coordinates mx, my and mz is given.