Faculty of Physics and Nuclear Techniques, AGH University of Science
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
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.
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:
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)
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 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
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).
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.
Two algorithm corrections:
The basis vector calculations have been incorrect for a special case when coincidence of the following conditions took place:
Accuracy control has been introduced near special points in the k-vector space. If the combination of the selected k-vector version and the selected parameter values leads to a k-vector which is located in the near vicinity (distance less than 0.001) of a special (high-symmetry) point the input procedure suggests a choice of another k-vector version, which leads to better accuracy and more adequate symmetry description
Some errors has been corrected which affected the display of input data, but had no influence on the calculation process and the final results:
Special accuracy control has been introduced with regard to special atomic positions. The user provided values of atomic coordinates are treated as special if they coincide with one of the special values (n/24) up to the third decimal place. If this is the case an exact 8-digit value is assumed, the other values are rounded to the third decimal place. The internal comparison of atomic coordinates is carried out with 4 digit accuracy, in order to avoid the effects of rounding errors.
For the groups of the monoclinic system, exhibiting centered lattices (IT 5,8,9,12,15) the calculation procedure has been redesigned and the calculation results are now presented in coordinate systems compatible with the IT(1983) conventions (eg. unique axis c, cell choice 2 or "B 112/n" for the IT 15 group). In the older versions the results have been presented in consistency with the IT (1965) ( i.e. "B 112/b" for the IT 15 group).
Further developement of the graphic possibilities inludes: