Generation of Atomic Positions

For the description of the generation of atomic positions in PowderCell let us take as example a record of the symmetry file pcwspgr.dat:

spgr 227  1 15  0 48  8  5   F 4_1/d -3 2/m1917 816 931 7 0 0 5 0 6 6 0
1 16  51 16  18 6  54 6  138 22  111 26  166 30  93 46
4   0 0.5 0.5
3   0.5 0.5 0
13  0 0 0
5   0.75 0.25 0.75
1   0.25 0.25 0.25

The meaning of the several data will be explained in the description of the record structure of pcwspgr.dat. From this it follows that the encoding of the special positions can be taken from the third line of the record. So the Wyckoff position a can be generated using the code 16. That means that only the bit 3 (the first bit is 0) is activated because 24=16. The multiplicity of this position is 2 because the bit 3 represents generator type 3 and this is defined as 2y. That's all! But don't forget the centering vectors of the F lattice! The additional three lattice points results a complete multiplicity of 8, i.e. you will find in IT the full Wyckoff symbol 8a.

The lattice typ can be extracted from the Hermann-Mauguin symbol but also from the reflection condition of the reflection type HKL. The second way is more elegant.

In analogy to the explanation before the second Wyckoff position has the symbol 8b.

The third and the fourth positions are described by code 6. This means that bit 0 and bit 1 is activated. The given generator types 1 and 5 correspond to the symmetry elements -1 und 2[110]. Using this symmetry operations 4 positions will be generated. The inversion centre doubles the positions and the two-fold axis doubles onesmore the new positions generated before. If you consider the lattice centerings it follows, that the complete identifier (multiplicity, Wyckoff letter) is 16c and 16d.

The analysis of code 22 shows that bits 0,1 and 3 are activated. These correspond to the generator types -1, 2[110] and 2y (8 positions). The used identifier is 16d. From 26 can be followed that the bits 0, 2 and 3 are activated (-1, 3[111] and 2y), so that the position can be described as 48f. The 30 contains the activated bits 0, 1, 2 and 3 (-1, 2[110], 3[111] and 2y). Therefore it is called 96g. The 46 corresponds to the activated bits 0,1,2,4 (-1, 2[110], 3[111] and 2z) and can be described by 96h. Therefore the general position is called 192i.

Until now only the number of symmetry-equivalent positions are known. The exact coordinates we havn't determined yet. Therefore not only the generator types must be used but the exact generators itself. In principle it's the same procedure .

Let us start with a general position (x,y,z). After the multiplication with the generator type 1 the position (-x,-y,-z) has been created. But as told before only the exact generator is interesting, i.e. also the translation part must be considered. Therefore the correct position is not (-x,-y,-z) but (-x+¼,-y+¼,-z+¼). The second generator is a superposition of the generator type 5 and the translation part (¾,¼,¾). On this way two additional positions have been generated:

 (y+¾,x+¼,-z+¾) (-y,-x+½,z+½)

To this moment we derived 4 symmetry-equivalent positions. The next generator is a three-fold axis and represents a special case. All other generators double the previously generated positions. In contrast to that the three-fold axis triples these. That means that the third generator will be created additional 8 positions. At first the generator will be multiply with the old 4 positions and in a second step these new 4 positions will be multiply onesmore with the same generator:

 (z,x,y) (y,z,x) (-z+¼,-x+¼,-y+¼) (-y+¼,-z+¼,-x+¼) (-z+¾, y+¾,x+¼) (x+¼,-z+¾,y+¾) (z+½,-y,-x+½) (-x+½,z+½,-y)

The fourth generator is the 2y. It addits 12 further positions to the 12 known:

 (-x+½,y+½,-z) (-z+½,x+½,-y) (-y+½,z+½,-x) (x+¼,-y+¾,z+¾) (z+¼,-x+¾,y+¾) (y+¼,-z+¾,x+¾) (-y+¾,x+¾,z+¼) (z+¾,y+¼,-x+¾) (-x+¼,-z+¼,-y+¼) (y+½,-x,z+½) (-z,-y+½,x+½) (x,z,y)

The last of the 5 generators is the 2z completed by a translation part of (0,½,½). This doubles all positions onesmore to a total of 48:

 (-x,-y+½,z+½) (-z,-x+½,y+½) (-y,-z+½,x+½) (x+¾,y+¼,-z+¾) (z+¾,x+¼,-y+¾) (y+¾,z+¼,-x+¾) (-y+¼,-x+¼,-z+¼) (z+¼,-y+¾,x+¾) (-x+¾,z+¾,y+¼) (y,x,z) (-z+½,y+½,-x) (x+½,-z,-y+½) (x+½,-y,-z+½) (z+½,-x,-y+½) (y+½,-z,-x+½) (-x+¾,y+¾,z+¼) (-z+¾,x+¾,y+¼) (-y+¾,z+¾,x+¼) (y+¼,-x+¾,z+¾) (-z+¼,-y+¼,-x+¼) (x+¾,z+¼,-y+¾) (-y+½,x+½,z) (z,y,x) (-x,-z+½,y+½)

To generate all symmetry-equivalent positions of a general position whithin the unit cell you must consider at least the lattice-centering vectors of the all-face centered cubic lattice: (½,½,0), (½,0,½) and (0,½,½). This quadruples the number of generated positions at least to a total of 192.

 It should not be astonished that for non-primitive lattices the positions given in IT and those derived in PowderCell seem to be not identically. The reason for this difference is the relative arbitrary assignment of the atomic positions to the lattice points. Let us take e.g. the fourth position derived in PowderCell — (-y,-x+½,z+½). This position is not listed explicitly in the IT. But you will find there an equivalent position with the coordinates (-y+½,-x,z+½). If you add the translation-centering vector (½,½,0) to this position you will get exactly the same coordinates derived by the program.

© Dr. Gert Nolze & Werner Kraus (1998)

Federal Institute for Materials Research and Testing
Unter den Eichen 87, D-12205 Berlin,
Germany