|by A. Filipponi (1995), revised May 1999, Aug 2000|
The program identifies all the distinct two, three, and four-body local configurations around a photoabsorber atom. These atomic configurations will be often referred to as ``peaks'' of the two-body (g2), three-body (g3), and four-body (g4) distribution functions respectively. The nomenclature is borrowed from the theory of the liquid structure. In a crystalline or molecular case the n-body distribution functions (gn) are the sum of thermally broadened Gaussian peaks centered on particular geometries. In the liquid case the gn functions oscillate around the value of 1, they show peaks corresponding to highly probable configurations and are 0 in excluded volume regions. Each configuration generates several single and multiple scattering contributions to the x-ray absorption cross-section. The analysis is performed on a cluster, specified by an input formatted file denominated ABC.ato (ABC is a three character string), that contains basic information on the atoms including type, position, and neighbours. The file can be prepared automatically by the program crymol (see crymol documentation for details). The cluster can be either representative of the whole structure (as in the case of a molecule) or of a portion of it up to a distance D from the central photoabsorber atom, as in the case of a crystal. In this latter case the sub cluster analysis can be performed up to maximum distance RCUT £ D for which the counting of the atomic configurations is still correct in the cluster. If the user requires RCUT > D a new crymol run with the appropriate input should be executed. For each photoabsorber atom the program selects all possible different arrangements of 1, 2 and 3 neighbouring atoms. The output is a table of peaks of the pair, triplet and quadruplet distribution functions g2, g3 and g4 with the constraint that one of the atoms must be of the photoabsorber type. Possible equivalences of peaks after a permutation of the photoabsorber are considered. This can reduce considerably the number of different structures that have to be considered in a case in which many atoms of the photoabsorber type exist in the structure.
|>> > in case the number of atoms exceeds 100 an additional card is needed << <|
ABC is the same (A3) string of the input file, the extensions are given by the program automatically. The open qualifier is STATUS=`UNKNOWN', so that previous version are overwritten under UNIX operative systems. The file ABC.gnp contains a list of the g2, g3, and g4 peaks found in the cluster below the given cutoff, the format is useful for a rapid inspection of the peak analysis performed by gnpeak. The file ABC.gnx contains the same information of the .gnp file, but in numerical form directly usable as part of an input for the gnxas program which actually performs the signal computations (see the gnxas.doc for details). The file ABC.chi contains a list of the main cn paths related to the gn peaks. First the c2 and c4 related to a g2 distribution are listed, then all the c3, and c4 related to a g3, and finally all the c4 related to g4 peaks are listed. The association of cn contribution to gn peaks actually follows a physical intuition, since all the paths generated by the same geometrical arrangement of 2, 3 or 4 bodies are in this way listed together. The use of this file is mainly to provide a link with previous approaches where a wide use of the multiple scattering series was performed. Finally the file ABC.dbw is generated only in the case when the cluster is disordered and it has been analysed with a large EPS tolerance so that several configurations are included into the same peak. It contains the structural covariance matrices of the geometrical parameters for the various g2 and g3 peaks. Covariance matrices are not calculated for g4 peaks.
There is however a different ``local'' perspective for the configuration counting which considers what is seen from a given atom in the structure. This is exactly what is required for calculating the x-ray absorption signal due to a photoabsorption process involving that atomic site. This counting involves the conditional n-body distribution functions that describe the probability that n atoms are placed in a given arrangement provided that one of them is of a specified type and is in a given position. The two configuration counting schemes are equivalent. Coming back to the previous example, in the f.c.c. lattice is well known that the number of first neighbours at the distance R is 12. This is the so called COORDINATION NUMBER corresponding to the ``first peak'' of the ``conditional two-body distribution function''. Thus the two-body configuration with a degeneracy of 6 generates a shell of neighbours with coordination number of 12. The doubling from 6 to 12 arises because both of the atoms in the two-body configuration are identical and can act as the photoabsorber. More complex relationships occur in the case of three and four body peaks.
Let us now establish what are the parameters sufficient to describe a n-body peak, starting with the two-body ones.
|TWO BODY PEAKS|
A two-body peak, that can be schematically represented as:
R12, TY1, TY2
Only those peaks in which at least one of the two atoms is of the photoabsorber type atoms are interesting for the EXAFS calculation. Let A2 be the degeneracy of the peak. Then there are two possible cases for the counting in the local perspective. Either TY1 is different from TY2 and then the contribution to the number of configuration (conf #) NC2 (the so called ``coordination number'') is NC2=A2 or TY1=TY2 and NC2=2×A2 (as in the previous example).
|THREE BODY PEAKS|
A three-body peak can be schematically represented as:
There are three possible symmetry conditions on the triangle which can be checked with a permutation of the atoms: in the most general case when the three sides, or the atoms, are not equivalent the triangle is scalene ``S''. If two atoms are equivalent as well as the two bond connecting them to the third atom, then the triangle is isosceles ``I''. Finally if both three atoms and sides are the same (q = 60°) the triangle is equilateral ``E''. Now it is important to consider the various possible positions of the photoabsorber since each different position will generate a different structural signal. Similarly to the neighbour coordination number, each of these conditional three-body arrangements will be associated with a coordination number, counting the number of configurations of this kind seen from a given photoabsorber. Clearly each of the three possible positions of the photoabsorber will generate A3 contribution to a coordination number. According to the symmetry properties sometimes the number of configurations (conf #) NC3 can be 2×A3 or 3×A3 (equilateral case).
|FOUR BODY PEAKS|
In a four-body peak there are six inter-atomic distances, the geometry is completely described by six roto-translational invariants and the atomic types in positions 1, 2, 3, and 4. Schematic diagrams for four-body peaks are reported below:
||In light of the chemical applications of the theory, where
the most stable and well defined distances are likely to be the
shortest ones among the six sides of the tetragon and the
chemically meaningful bond angle be the angle between these
sides, the following convention will be used:
and the atom types are
TY1, TY2, TY3, TY4
Notice that position 1 is the central position in the star, position 2 is at the end of the first distance R12, position 3 is at the end of the second distance R13, and position 4 is at the end of the third distance R14. The three angles q[^213] q[^214] and q[^314] are three angles around the vertex 1.
and the atom types are
TY1, TY2, TY3, TY4
Notice that position 1 is the first end of the chain, position 2 is at the other end of the first distance R12, position 3 is at the end of the second distance R23, and position 4 is at the other end of the chain with respect to position 1, at the end of the third distance R14. The angle q[^123] is the one between the first two bonds, the angle q[^234] is the one between the second and third bonds of the chain. The spatial orientation of the three bonds requires a further angle which is the Dih edral a ngle f among the three bonds.
The degeneracy of the peak will be indicated by
Notice that similarly to the g3 case the same four-body peaks can generate several (up to four) different conditional peaks with the photoabsorber placed in different positions. And each permutation of the photoabsorber can be either singly degenerate or two-fold, three-fold and also four-fold degenerate. This degeneracy times A4 is indicated by the number of configurations (conf #) NC4.
|>> > Notice! << <|
associated c2 and c4 paths ® I,Xn,PATH,SYP,RP,DEG,NP
I progressive number 1,2,3 ... of the c2 or c4 contributions cn is c2 or c4 according to the symbol
I progressive number 1,2,3 ... of the c3 or c4 contributions cn is c3 or c4 according to the symbol
I progressive number 1,2,3 ... of the c4 contributions cn always c4 in this case
File translated from TEX by TTH, version 2.78.
On 26 Sep 2000, 09:11.