Harold T. Stokes and Dorian M. Hatch, Department of Physics and Astronomy, Brigham Young University, Provo, Utah, 84602, USA, branton_campbell@byu.edu
This program finds possible paths taken by atoms in a reconstructive phase transition. This is accomplished by finding maximal common subgroups of the two phases.
Version 2.0 contains an improved algorithm which contains major revisions. (Phys. Rev. B 65, 144114-1-12 (2002) describes the algorithm for versions 1.x.) Differences between version 1.x and 2.0 are described at the end of this document.
Be aware that the on-line version of COMSUBS has a run-time limit of one hour. After one hour, a "timeout" message will be sent to the output and the program will be terminated. If you need more then one hour to generate output, use the version for linux which can be downloaded as part of the ISOTROPY software package.
Below, crystal 1 and crystal 2 refer to the structures in the two phases.
COMSUBS uses the following settings of space groups in International
Tables of Crystallography:
monoclinic: unique axis b, cell choice 1
rhombohedral: hexagonal axes
origin choice 2 (point of inversion at origin)
(1) title line. This line is copied to the output, but is otherwise ignored by the program.
(2) space group symmetry of crystal 1 (1-230)
(3) lattice parameters of crystal 1: a,b,c,alpha,beta,gamma, where a,b,c are the lengths of the lattice vectors defining the conventional unit cell, alpha is the angle between b and c, beta is the angle between a and c, and gamma is the angle between a and b. Angles are in degrees. Units for lengths are arbitrary.
(4) number of Wyckoff positions occupied by atoms in crystal 1.
(5) information about each Wyckoff position (each on a separate line). This information can be given in two different forms: (a) the symbol for the atom and the x,y,z dimensionless coordinates of one of the atoms in that position. (b) the symbol for the atom, the symbol for the Wyckoff position (a,b,c,etc.), and the values for the x,y,z structural parameters (as given for that Wyckoff position in the International Tables for Crystallography). Note that, for example, if only a value for y is needed, enter a zero for the value of x.
(6) space group symmetry of crystal 2 (1-230)
(7) lattice parameters of crystal 2: a,b,c,alpha,beta,gamma.
(8) number of Wyckoff positions occupied by atoms in crystal 2.
(9) information about each Wyckoff position
(10) Each of the remaining lines contains a keyword followed by data if needed. The possible keywords are:
size: the minimum and maximum size of the primitive unit cell of the subgroup with respect to the primitive unit cell of crystal 1. If only one value is entered, it will be interpreted to be the maximum size. The default values for the minimum and maximum size are both 1.
strain: the minimum and maximum values allowed for principal values of the strain tensor. The default values are 0.66 and 1.5.
shuffle: the maximum atomic displacement allowed. This is measured with respect to a "center of mass" of the crystal.
bonds: preserve bonds of length less than the given length d, i.e., all neighbors closer than the distance d in crystal 1 will also be closer than d in crystal 2.
subgroup: the minimum and maximum space group symmetry (1-230) of the subgroup to be considered. The default values are 1 and 230.
neighbor: the minimum distance allowed between nearest neighbors along a linear path between the two crystals. The default distance is zero.
lattice: a particular choice of subgroup lattice to be considered. This is given by (1) the space-group symmetry of the subgroup, (2-4) the lattice vectors a,b,c of the subgroup each in terms of the lattice vectors a,b,c of crystal 1, and (5-7) the lattice vectors a,b,c of the subgroup each in terms of the lattice vectors a,b,c of crystal 2. All seven items must be separated by spaces. If the space-groups symmetry is not known, enter any space group which specifies the lattice centering intended. Each lattice vector of the subgroup is given by three rational dimensionless numbers separated by commas (no spaces). The specification of the space-group symmetry of the subgroup does not restrict the considerations of those symmetry. It merely communicates to the program the setting of the lattice vectors in the subgroup. For example:
lattice 166 1/2,0,1/2 0,1/2,-1/2 -1,1,1 0,1,1 -1,-1,0, 1,-1,1
Note that rational numbers are expressed in terms of the ratio of two integers. By default, all possible lattice vectors of the subgroup are considered, within the constraints imposed by the length and strain keywords (or their default values). When the lattice of the subgroup is selected, the constraints imposed by the length, strain, and neighbor keywords (or their default values) are ignored.
madelung: relative madelung energy/atom of the structure half-way between the the two crystals. This energy is given relative to the average madelung energy/atom in the two crystals. On the lines following this keyword, enter the symbol for each type of atom followed by the net charge of that atom. Each type of atom must be entered on a separate line. If the relative madelung energy is high, the path will be energetically unfavorable due to distribution of charges. Such paths can be eliminated from further investigation.
(1) title, as read from the standard input
(2) space group symmetry of crystal 1, as read from the standard input
(3) lattice parameters of crystal 1, as read from the standard input
(4) number of Wyckoff positions in crystal 1, as read from the standard input
(5) each Wyckoff position, as read from the standard input along the the letter symbol for the Wyckoff position. This allows the user to be sure that the program has correctly interpreted the atomic position given in the standard input.
(6) space group symmetry of crystal 2, as read from the standard input
(7) lattice parameters of crystal 2, as read from the standard input
(8) number of Wyckoff positions in crystal 2, as read from the standard input
(9) each Wyckoff position, as read from the standard input along the the letter symbol for the Wyckoff position.
(10) acknowledgement of each keyword encountered
(11) number of atoms in the primitive unit cell of each crystal. The user should check these values to be sure that the data about the structures of the crystals have been correctly entered.
(12) volume/atom in the two crystals. For real data, these two values should be approximately equal. Otherwise, the user should suspect that the input data may have been incorrectly entered.
(13) nearest-neighbor distance in the two crystals. Unreasonable values here should alert the user that the input data may have been incorrectly entered.
(14) each subgroup found under the imposed constraints. For each subgroup, the following information is given:
(a) numbering of subgroups
(b) principal values of the strain tensor for the path from crystal 1 to crystal 2. Pure rotations are removed from the strain tensor.
(c) nearest-neighbor distance between any pair of atoms along a linear path between crystal 1 and crystal 2.
(d) space-group symmetry of the common subgroup
(e) subgroup in the setting of crystal 1:
(f) same information for the subgroup in the setting of crystal 2.
(g) subgroup with a structure half-way between that of crystals 1 and 2.
Version 2.0 contains some major revisions over previous versions.
(1) All superlattices for each size of unit cell of the subgroup are now considered.
(2) For each common superlattice which obeys the strain constraint, all possible mappings of atoms from crystal 1 to crystal 2 are generated, without any consideration given to possible space-group symmetries of the common subgroup.
(3) Space-group symmetry of the common subgroup is obtained only after a mapping has been found which obeys the constraints of minimum allowed nearest-neighbor distances and maximum allowed atomic shuffle.
(4) The following are differences in the action of keywords (for more details, see the description of the keywords above):
(a) length: this keyword is no longer implemented. Instead, all lattices of each allowed size are considered.
(b) subgroup: the default subgroups considered now include the triclinic subgroup #1.
(c) neighbor: sets a minimum value for nearest neighbor distance along the entire linear path between the two crystals.
(d) size: may also indicate a minimum size of the unit cells of the subgroup.
(e) shuffle: a new keyword which sets maximum atomic displacements allowed in the transition.