PSEUDO
a program for a pseudosymmetry search
E. Kroumova,
J. M. Igartua, S. Ivantchev. M. I. Aroyo, J. M. Pérez-Mato
Departamentos de Física de la Materia Condensada
y Física Aplicada II, Universidad del País Vasco,
Apdo 644, 48080 Bilbao, Spain
The problem
The existence of pseudosymmetry in a crystal structure is indicative of a
slightly distorted structure of higher symmetry. If the distortion is small
enough, it can be expected that the crystal acquires this more symmetric
configuration at a higher temperature after a phase transition.
The aim of the program is to search for a pseudosymmetry in a given structure
among the minimal supergroups of the structure's space group, thus giving the
possibility to predict possible continuous phase transitions for that structure.
The method
Let the initial structure be S0 with symmetry given by the space
group H.
We will consider the case when the pseudosymmetry corresponds to a supergroup G of
H.
The method(&) used for the search of pseudosymmetry can be divided in two main
parts:
- Calculation of all minimal supergroups of H.
- Calculation of the displacements necessary to obtain a structure with
symmetry given by each one of the minimal supergroups.
Minimal supergroups
First all of the minimal supergroups should be found. Each supergroup is
characterized by the additional operations it contains with respect to the group
H, e.g. the coset representatives in the decomposition of G with respect to H.
The search for a pseudosymmetry is in fact a test for the existence of these
additional operations in the initial structure.
All of the minimal supergroups, considering also the different supergroups of
the same type, are obtained using the program
MINSUP.
Displacements
Once all of the minimal supergroups are calculated, the displacements necessary to
obtain a structure with symmetry given by each one of this supergroups should be
calculated.
To do that first each one of the additional operations for the supergroups is
applied to the initial structure giving as a result the transformed structure:
St = g S0. Then
comparing the positions of the atoms of the same type in the initial and
the transformed structure/s are calculated the displacements necessary to make
both structure coincident.
Special Cases
The general procedure described above should be completed with some additional
steps in two special cases:
- Change of the symmetry from polar to non polar one, when there is an
infinite number of supergroups of the same type corresponding to the infinite
number of the possibilities to choose the position of the origin in the polar
group.
- Structures with monoclinic or triclinic symmetry for which the number
of the different
supergroups is also infinite and the search for a pseudosymmetry have to be
restricted depending on the pseudosymmetry of the lattice.
Non polar pseudosymmetry for a polar structure When the initial
structure S0 is a polar one, there is no fixed
position for the origin. If the supergroup G for which the pseudosymmetry should
be checked is a non polar group than the location of the additional symmetry element(s)
corresponding to this group can be anyone with respect to the origin chosen for
the description of the structure in the polar group, which means that there
will be an infinite number of supergroups corresponding to the different
possibilities to choose the origin in the initial structure. In more formal
terms, the additional operation g=(W|w+p) has a continuous parameter p in the
translational part. The value of this parameter should be optimized using some
criteria, which will permit to select only one from the infinite number of
supergroups for which the procedure for the search of pseudosymmetry described
above will be applied.
In our method the value of this continuous parameter is chosen so that the
maximal displacement necessary to obtain the high symmetry structure has the
minimal possible value.
Monoclinic and triclinic symmetry(#)
The procedure used for the calculation of all of the
minimal supergroups of the same type is based on the affine normalizers of the
space groups(*). The problem that arises with the monoclinic and triclinic space
groups is that their affine normalizers are not space groups.
In this case an additional considerations related with the symmetry of the lattice will
be used to obtain the subgroup normalizer ``suitable'' for the given structure
and this new normalizer will be used when the procedure for the search of pseudosymmetry
is applied.
NOTE: By now is used the Euclidean normalizers for the monoclinic and the triclinic groups.
(&) For more information about the procedure, see the References.
(#) Not available by now.
The program
The program PSEUDO uses the method described above and permits to search
for a pseudosymmetry among the minimal supergroups of the structure's space
group. The data that the program needs is the space group, the cell constants
and the atoms in the asymmetric unit of the structure, and the result contains
all of the pseudosymmetries found among the minimal supergroups. You can use
conventional or non conventional description of the structure, providing the
matrix that relates the non conventional basis with the conventional one in the
latter case. All of the result is given with respect to the basis of the initial
group H.
Input Data
The input data is given in three steps: the first two of them are related with
the structure data and the third is the choice of the supergroups.
- First the formulae, the space group number and the cell constants for
the initial structure should be given. Also, it is obligatory to give
the number of the atoms in the asymmetric unit.
- The next step is to give the data for the atoms in the asymmetric unit.
For each one of these atoms you should give:
- a label of the atom site which contains the chemical element and a number;
- the multiplicity and the letter of the Wyckoff position
- the (x,y,z) coordinates for the atom (in relative units);
- the site occupation factor.
- When all of the structure data is given, the minimal supergroups for
which the pseudosymmetry will be checked have to be chosen. To do that you a are
provided with the table of the minimal supergroups.
From this table you can chose one, more, or all of the supergroups.
At this step also the value for the maximal "permitted" displacement of the
atoms from their high symmetry positions should be given.
This is an empirical value, and from the
calculations we have made by now(*) using this program it seems that a reasonable
value for this displacement is normally 0.75Å.
The choice of the minimal supergroups is the last step of the input. After that
the program checks the structure for a pseudosymmetry for each one of the
selected supergroups and gives the result from this search.
(*) We have used this program in the search for pseudosymmetry
among the compounds with symmetry P212121,
Pnma, Pba2 and Pmc21 (see the References).
Output data
The result from the search for pseudosymmetry is divided in three parts:
- The full data for the initial structure - space group, cell constants and
the atoms in the unit cell, given in the conventional setting.
- The summary of the result from the search which contains the supergroups
for which a pseudosymmetry has been found as well as the maximal, minimal and
relative displacements for each one of these supergroups.
- Finally, a full report for the result is given which contains for
each one of the supergroups for which a pseudosymmetry has been found
the displacements of the atoms necessary to obtain the
structure with symmetry given by the supergroup and also the unit cell of
this high symmetry structure.
All of the data is given with respect to the
conventional basis of the space group of the initial structure.
References
1) J. M. Igartua, M. I. Aroyo, J. M. Pérez-Mato,
Phys. Rev. B54 18 (1996) 12744-12752.
2) J. M. Igartua, M. I. Aroyo, E. Kroumova, J. M. Pérez-Mato,
Acta Cryst. B55 (1999) 177-185.
(1996) 12744-12752.
[PSEUDO]
PSEUDO