MODY for Windows - ver 1.30a

by: W.Sikora, F.Białas(*), L.Pytlik, J.Malinowski

Faculty of Physics and Nuclear Techniques, AGH University of Science and Technology,
al.Mickiewicza 30, 30-059 Krakow, Poland

(*)WSB - National Louis University, ul.Zielona 27, 33-300 Nowy Sacz, Poland

More detailed description of the program algorithms and functions can be found in our publication, which should be also referenced, when any results produced using our program are presented:

W.Sikora, F.Białas and L.Pytlik - MODY: a program for calculation of symmetry-adapted functions for ordered structures in crystals,
Journal of Applied Crystallography, 37(2004), 1015-1019


Short introduction to symmetry adapted description

In the experimental investigations of crystalline solid state compounds many various physical quantities are encountered. The basic nature of the given physical quantity determines its mathematical description. There are things that are definitely described by scalars, namely electron charge density, probability of site occupation and others. For such things one number (scalar) is enough to specify all what is needed for a given crystalline site. There are other quantities which definitely require a vector description. For that class of quantities there is an additional division with respect to the behavior under space inversion and time reversal. The first subclass, in which the vector changes its sign after space inversion is called the polar vectors class and it includes quantities like atom displacement, dipolar moment etc. The second subclass comprises quantities for which the vector does not change its sign under space inversion transformation and that subclass is called the axial vectors class. It includes quantities like angular momentum, magnetic moment, spin etc. There is also a class which requires a tensor description i.e. the whole tensor (matrix) has to be defined to specify the quantity completely for a given crystal site. That class includes e.g. moments of inertia, quadrupolar moments, local susceptibilities and others.

Whatever is the quantity being studied in the experiment the simplest possible description usually consist of specification of all individual values of a given quantity on all the crystal sites in every crystal unit cell. The basic symmetry which usually leads to a more convenient description is the translation symmetry of the crystal. By making use of that symmetry physicist are able to express the values in each cell by the values on sites of the "basic" (zero) crystal cell and the wave vector describing the particular type of ordering, observed for a given quantity in the experiment. However the full symmetry of the crystal (described by its space group) contains also information about the local point symmetry of crystal sites (symmetry axes, reflection planes, screw axes etc), which is exactly the symmetry used by chemists in the description of molecules. Therefore the optimal symmetry description of the local crystal quantities should employ the full symmetry of the crystal, i.e. the symmetry determined by the elements contained in its space group. In order to make use of that symmetry one has to know all the transformation rules of a given quantity under the action of all symmetry operations allowed for a given space group.

Once the rules are known the quantity can be described by special form of functions called symmetry adapted functions. Because their properties origin from the transformation rules in a given space group no wonder they are closely related to the so called irreducible representations (IR's) of the space group i.e. set of matrices which strictly "mimic" the transformation rules under the action of symmetry operations. These symmetry adapted functions mentioned above are usually called basis functions (basis vectors - BV's) attributed to a given IR, because under the action of symmetry operations they transform exactly according the given irreducible representation. The basic advantage of using the symmetry adapted functions is a significant reduction of the number of parameters required for a full specification of a given quantity in all the crystal sites. The simplest analogy seems to be the use of vibrational eigenmodes, when it is enough to specify the amplitude of the given eigenmode instead of individual atomic displacements.

What does the program do ?

In order to calculate the symmetry adapted functions (basis functions) the program uses the so called projection operators method. The projection operators are constructed for individual irreducible representations, and are special by the fact that after action of such operator on almost arbitrary initial function the resulting function (the projection result) transforms according to the irreducible representation used for the construction of the operator.

In the construction process all the details required are the of space group symmetry of the crystal itself and additionally description of the so called wave vector group , responsible for the symmetry related to the wave vector of the ordering observed. In particular the set of wave vectors equivalent in the reciprocal space of the crystal, called the wave vector star, and one selected vector called the arm of the wave vector star are required. The wave vector group decides, which irreducible representation are present in the decomposition of the characteristics full representation (e.g. mechanical or magnetic representation) for a given crystal symmetry

The detailed description of projection operators and the mathematical procedure are described in:

  1. Cotton F.A. Chemical application of Group Theory Second Edition Wiley-Interscience a Division of John Wiley and Sons, Inc. New York London Sydney Toronto 1963, 1971
  2. Birman J. L. Theory of Crystal Space Groups and Infra-Red and Raman Lattice Processes of Insulating Crystals The City College of the University of New York, Springer-Verlag, Berlin Heidelberg New York, 1974
  3. Izyumov Yu. A. and Syromyatnikov V. N. Phase Transitions and Crystal Symmetry, 1990 Dordrecht:Kluwer Academic Publishers

The program calculates the basis functions for sets of individual atomic sites in the cry stall cell. Such a set, consisting of a selected site and all the symmetry equivalents sites, is called an orbit in a given group. The orbit is specified by selection of the first atom in the orbit. It is worth mentioning that sometimes one orbit (set of symmetry equivalent sites) in the space group of the crystal splits into two (or more ) orbits for a given wave vector group. Therefor in order to obtain a complete symmetry adapted description of the crystal with a given type of ordering the user has to calculate the basis functions for all the first atoms choices, required for covering of all the sites in the unit cell, for which the ordering is observed (e.g. for magnetic ordering for all the sited on which magnetic moments is encountered)

What do I need to use the program ?

As described above the information required for basis functions calculation first of all includes the details of the space group. Therefore at the very beginning the user is asked to input the following data regarding the space group:

The other input data required are:

The user is also allowed to select (one or more) irreducible representations, for which the basis vector should be calculated. Once the inputs are specified the program is able to calculate all the required outputs. The calculations can be repeated as many times as possible after changing some of the input parameters e.g. the first atom, irreducible representations, orbits etc

What can I do with the results ?

The outputs produced by the program include:

The main output files (Modes) lists the complex components of the calculated basis functions for all the crystal sites contained in a given orbit of equivalent sites in the unit cell. The values are listed for all the irreducible representations selected by the user. In the header of every section information about dimension and multiplicity of the respective representation is shown. The results are presented for one or two wave vectors (the latter case if simultaneous calculation for k and -k has been requested).

The calculated basis vectors can be directly applied to construction of symmetry adapted ordering patterns (ordering modes), which can be very useful for all experimentalists, who want to interpret the experimental results by selecting (fitting) a theoretical ordering model which fits best to the measured data. The only problem encountered is the fact the ordering models describe actual physical quantities (magnetic moments, atomic displacements etc.) and have to be real not complex. The problem is solved by taking a linear combination of the calculated basis vectors over the all function belonging to the given irreducible representation, and over k and -k if required (e.g. modulated structures). It can be shown that by appropriate selection of the coefficients of the linear combination the results can be made real, and still some free parameters are left. Therefore the field of potential applications of the symmetry adapted models (thus the calculated basis vectors) seems quite vast and offering good prospects for futures applications (e.g. tensor type orderings).

Here you can find some Examples of application of Mody program results

The basis vectors calculated in the MODY program can be used for construction of model structures, which are very helpful as starting configurations in the diffraction profile refinement. The model structures are constructed as linear combinations of the calculated basis vectors. The condition that has to be imposed on the result is the following: all components of the resulting vector have to be purely real (no imaginary part can be present). The condition effectively acts as a set of relations between the coefficients of the linear combination, which guarantee the fulfillment of the reality condition. A program processing the respective set of linear equations is under development, but the whole process can be carried out manually. Three explicit examples of such calculations are given below.

Here's the MODY program History

You can download the MODY for Windows program by clicking HERE

or read more about the program by downloading it's help file HERE

Last updated on: March 19-th, 2007