7Lattice constants
 
The lattice constants of a crystal structure define the unit cell metric i.e. the translation lattice. The length of the basis vectors a, b, c must be given in Å, the angles a, b, g  in degrees. The agreement of the given lattice constants with the defined crystal system derived from the space-group number and the used setting will be checked automatically to the greatest possible extent by PowderCell. It should be taken into consideration, that at first the correct space group or setting must be edited or changed, because the program uses the so defined symmetry to set the invariants of the lattice constants. Sometimes it will be more complicated, if the setting of the former space group doesn't exist in the edited space group.  In this case the setting will be corrected automatically to 1. 
example: 	For a cubic structure only the input of the length of the basis vector a is necessary.
However, PowderCell doesn't check departures from the recommended rules for a general definition of basis vectors and angles between them. So e.g. the monoclinic angle should be selected higher than 90° but smaller than 120°. Otherwise it will be recommended to transform the basis vectors. For the definition of the orthorhombic basis vectors it will be recommended to define a < b < c. But very often this is impossible, especially if standard settings should be used. Therefore PowderCell don't try to fit this automatically.

remark: It is not correct to believe that the given lattice constants define automatically the crystal system. But it is exactly the opposite: The crystal symmetry defines the relation between the lattice constants! That means, if the length of all three basis vectors are equal and the angles are 90° than it is an important hint that the present structure could be described by the cubic symmetry. But that's no proof! On the other hand, if you know the symmetry of the structure you are able to define the relation between the basis vectors.
It follows that the metric of the unit cell is a necessary condition but no sufficient. As example some Cryolithes can be given [26]. There only the monoclinic angle shows a small deviation from 90°, but the structure symmetry is clearly monoclinic and not orthorhombic (Na3AlF6  - b=90°...90.3°, Na3FeF6  - b=90.4°, Na3NiF6  - b=90.1°, Na3VF6  - b=90.5°).

Substitution and occupation
 
No seldom it is possible to observe in nature, that one and the same geometrically described position has been occupied by a lot of atoms or ions (e.g. mixed crystals, alloys etc.). In this case one assumes that this substitution (or replacement) can be described by the average of the scattering factors of atoms affected. This is exactly the way used in PowderCell. The weight of the individual atoms will be defined using the so-called substitution or replacement factor. This must be seen as probability. The values must be given between 0 and 1. The 1 defines a complete substitution (100%) of this position by the atom or ion, respectively.
On the other hand in some cases all symmetry-equivalent positions cannot be occupied what is caused e.g. by the stoichiometry of the phase. For this situation the so-called occupation factor has been defined. This factor must be seen as probability, too. 
Per default the substitution as well as the occupation will be set as 1. With regard to the calculation of the structure factor both the substitution and the occupation have the same influence on the result. Therefore PowderCell uses always their product (see input data) — the so-called multiplied substitution and occupation factor (SOF). This value can be in maximum 1. But the user must check it himself that the sum of all SOF for one and the same atomic position shouldn't be higher than 1.
The input of SOF is arranged behind the z-coordinate within the table of the dialog box structure data. The standard value is 1.
remark: If you input a SOF as 0, the program calculates the scattering parts also for this atomic positions, but they have no influence on the result because their weight ist 0.
If you would like to replace in NaCl the Na-ion  by 10% K, at first an additional atom/ion must be inserted (e.g. behind Na+). It is enought to input K+ in the column ion. PowderCell uses a look-up table to check the existence of such an atom/ion in the file asfpcw.dat and completes the entry by the atomic number Z. It's not allowed to input an own identifier for this substitution. Now only the SOF must be entered. In principle, it is possible to input the relative atomic coordinates, but it is not necessary if this line follows directly. 

 
Only if the inserted substitution hasn't the same position as the atom in the column before the relative atomic coordinates or the Wyckoff letter (if it's clear) must be entered.

 
 If you press OK PowderCell arranges the atomic positions in the right order. If you open the worksheet structure data onesmore you will find the following table:

 
Within the program one atomic position can be substitute in maximum by 10 different types of atom.

Wyckoff positions
 
In contrast to the former DOS version PowderCell for Windows is able to distinguish between the several Wyckoff positions. This new feature allows the investigation of the Wyckoff splitting which is interesting with regard to the use of group-subgroup relations. Furthermore it simplifies the input and inspection of the relative atomic coordinates describing the asymmetric unit.
Additionally, by the use of identification of Wyckoff-position types it is possible to optimize the generation of atomic positions considering their specific local symmetry. In contrast to the DOS version using all symmetry operations in form of the general positions the Windows version uses only the generators selected (see International Tables for Crystallography). Therefore the implemented procedures for the generation of atomic positions of both the DOS and the Windows version are quite different. 
The combination of generators produces all symmetry operations. In dependence on the local symmetry, for all special positions specific generators will be combined, respectively. This prevents a multiple generation of special positions and represents the optimized Wyckoff-specific generation of atomic positions. Furthermore, the new data structure of symmetry information in the symmetry file pcwspgr.dat is more compact, because you need in maximum 5 generators in comparision to 48 general positions used for the DOS version. On the one hand this decreases typing errors, on the other hand it is more easy to locate these, because 1/2 or 2/3 of all generated positions must be wrong using the new generation procedure. Additionally, the new symmetry-data structure in pcwspgr.dat simplified its extension by non-conventional settings for orthorhombic and monoclinic space-group types.

Relative atomic coordinates
 
The relative atomic coordinates of a structure always correspond to a definition of the unit cell, respectively, i.e. they are specified by the setting of the given space-group type. The coordinates are free of choice and not limited on the interval between 0 and 1. PowderCell automatically translates all atoms within the unit cell, but in principle that's really not necessary for the calculation of the structure factor. The necessity of this translation into the unit cell is caused by the procedure for identification of the Wyckoff positions and the generation of all symmetry-equivalent atomic positions using a specific generator sequence. In connection with the identification of the Wyckoff position it is necessary that the coordinates of a given atom must be brought into the line with the defined Wyckoff-position types. 

example:	In the space-group type P bca (61) the Wyckoff-position type 4a corresponds to the coordinates (0,0,0). The symmmetry-equivalent positions are (½,0,½), (0,½,½) and (½,½,0). One can use all of them as input in the worksheet structure data but PowderCell will transform these always in (0,0,0), automatically.
 
Caused by the enlargement using several group-subgroup relations, PowderCell fits a given atomic position to a special position of this space-group type, if the deviation is smaller than 0.0005. 
The atomic coordinates can be entered in structure data as decimal numbers as well as decimal fractions.

example:	0.3333
		1.037
		1/3
		-2.3783
 
As shown in the examples we recommend to input a four-figure number or better, a decimal fraction. 
Within the procedure of identification of the Wyckoff position the program brings the coordinates into the line with the characteristic of the Wyckoff-position type (e.g. for (x,x,½) only the x-coordinate can be varied, whereas the y-coordinate is set automatically as copy of x). This makes the input of atomic coordinates more comfortable. 
But you can also input only the Wyckoff letters. In the special case of no degree of freedom in the atomic position it is enough to input the letter. PowderCell completes the entry by the multiplicity of this position and by all coordinates. In this moment when you input the letter in the (correct) column in the table the characteristic of the Wyckoff-position type will be inserted. That means that it is not necessary to know this for the given space-group type. If the entered Wyckoff letter doesn't exist in the defined space-group type a short hint will be inserted, where the latest Wyckoff letter will be given. 

Indexing used
 
For the identification of lattice lines, lattice points, net planes or directions in reciprocal space or reflections different stiles (e.g. usually different brackets) have been defined. Generally, the Miller indexing for lattice planes or lines will be used. For the indexing of lines and planes in a hexagonal coordinate system the so-called  Miller-Bravais indexing has been introduced to emphasize the additional symmetry. However, the Miller-Bravais indexing contains no additional information. PowderCell is able to use both for the indexing of reflections (even though for this the Laue indexing is used).
 
lattice lines
 
Symmetry-equivalent lattice lines will be described by a set of integers áuvwñ with no common divisor  — the general indexing for lattice lines. If you use this style, i.e. this brackets, you don't distinguish between all symmetry-equivalent lines and the given statement is made for all of them. 
In contrast to that for an individual line the style [uvw] will be used, generally. 
According to their definition, the lattice lines áuvwñ have the same origin, but there are exist an infinite number of it caused by the translation symmetry. All [uvw] show a parallel arrangement of an infinite number of lines with the condition for each of them, that at least one lattice point (and therefore an infinite number) is located on the line.

remark: In agreement with their definition it is possible to define lattice lines as normals to planes in the reciprocal lattice.
Furthermore it seems to be interesting that in contrast to the ordinary geometry a crystallographic line is characterized by it's direction too. In non-centrosymmetric point groups you must even distinguish between the lines [uvw] and [-u-v-w] but these are parallel to each other!
 
lattice points
 
Using ·UVW· an individual lattice point will be described. The full set of all symmetry-equivalent lattice points is given by :UVW:. 
 
lattice planes
 
The Miller indices (hkl) of a lattice plane are integers and have no common divisor. The h, k and l describe the relative reciprocal distances from the origin to the point of intersection of the lattice plane with the basis vectors, respectively. {hkl} defines the totality of all symmetry-equivalent lattice planes. Additionally, the {hkl} are lines in reciprocal space. Usually, {hkl} will be defined as normal vector to a lattice plane. Then the inverse value of this reciprocal lattice vector describes the interplanar distance between parallel planes. With the exception of the cubic system a line [uvw] in direct space and a line (hkl) in reciprocal space (with u=h, v=k and w=l) don't characterize the same direction, generally, i.e. these are not parallel to eatch other!
 
# interferences
 
Unfortunately the indexing of reflections is not defined consequently. Very often you will find in papers the same style (brackets) used for the diffracted lattice planes. But this is sometimes dangerous. The most important disadvantage is, that no difference is declared between the interference measured on a single crystal and in powder diffraction, where all symmetry-equivalent lattice planes contribute to the detected intensity. Therefore we use the following definition in PowderCell:

*HKL*	interference from (hkl)
  HKL	interferences from {hkl}

 

Bragg equation
 
One of the most important equation in the diffraction theory is the Bragg equation. It presents the correlation between the wavelength l, the diffraction angle q and the interplanar distance dhkl. Often this equation will be used in two different forms without any detailed comment:

			or

Therefore we shall try to clarify the difference between these two equations.
 
Normally for the interpretation of reflections (indexing, determination of lattice constants etc.) equation (2) will be used. PowderCell uses the Laue indexing for interferences: HKL. Generally the relation between HKL and the corresponding  {hkl} can be described by:

 
i.e.  H = n · h or  K = n · k  etc.
q, r and n are integers. They describe the order of interference q and the order of lattice planes r summarized to the higher order n. Within the program equation (2) will be used.
Therefore dHKL is no interplanar distance but a fraction of this caused by the higher order.

Bravais lattices
 
For simplification of the mathematical description of a crystal structure, in former days so-called crystal lattices have been introduced. On this way the description of a three-dimensional periodic arrangement of atoms or molecules can be substitute by a mathematical point lattice. Every lattice point represents one and the same part of the crystal structure. The content of a lattice point is given by the asymmetric unit multiplied by the generators selected (exclusively all pure translations).
For the description of any three-dimensional periodic ordered structure only a very limited number of lattices are necessary. These are well known as the 14 Bravais lattices, which are given in the following table:
 
	a)	triclinic	primitive lattice	P		b)	monoclinic	primitive lattice	P		c)		face centered lattice	A,B,C		d)	orthorhombic	primitive lattice	P		e)		face centered lattice	A,B,C		f)		body centered lattice	I		g)		all-face centered lattice	F		h)	hexagonal	primitive lattice	P		i)	rhombohedral	primitive lattice	R		k)	tetragonal	primitive lattice	P		l)		body centered lattice	I		m)	cubic	primitive lattice	P		n)		body centered lattice	I		o)		all-face centered lattice	F

For the description of trigonal crystal structures the coordinate system of the hexagonal lattice will be used, generally. Therefore you don't find it separately. Also for rhombohedral crystals the hexagonal lattice will be used beside the listed primitive. However, the additional centerings must be corrected by the introduction of reflection conditions (systematic extinctions).

Space-group type
 
Any crystal can be described by different characteristics: 

1.	metric: The lattice constants define the translation symmetry of the point lattice. 
2.	space-group type: It defines the arrangement of the symmetry elements within the unit cell (inclusively derived properties, e.g. local symmetries, Wyckoff letters etc.). 
3.	crystal structure: The individual atoms of the structure don't occupy all possible Wyckoff positions, i.e. they present only a part of all local symmetries. 
Therefore it is possible to postulate, that any structure crystallizes in it's own space group, because it has specific lattice constants and/or specific atomic positions. 
So one can say that two structures can be described in P 432 , i.e. in the same space-group type, but not in the same space group. It is even conceivable that one and the same chemical substance crystallizes in two different modifications of the same space-group type (e.g. MnF3 · 3H2O [26]). 
In the International Tables for Crystallography, Vol.A only the space-group types are described in detail  [5,16]. In PowderCell they are defined by the number of space-group type and the number of the setting. These are one of the necessary inputs in the dialog box structure data. 

Number of space-group type
 
In PowderCell the number of space-group type will be used to identify the space-group symmetry. By this number the program gets all necessary information reading the encoded data in the symmetry file  pcwspgr.dat.  The numbering of the space-group type corresponds to the IT [16]. In the dialog box structure data the Hermann-Mauguin symbol is shown. So an incorrect evaluation of the number of space-group type is nearly impossible.
Please notice, that even for monoclinic and orthorhombic space-group types a lot of different settings exist which must be given separately. In these cases the number of space-group types is always the same.

Standard and non-conventional settings
 
The publishing of the International Tables for Crystallography (IT), e.g. Vol A [16] supports a general accepted classification of crystal structures. On the base of group theory there you will find a systematic description of the two- and three-dimensional arrangement of symmetry elements. The space-group types can be described as combination of symmetry elements under consideration of certain conventions (Bravais lattices etc.). Even in the monoclinic crystal system one can define the structures using different basis vectors, i.e. unit cell settings. In these cases the absolute arrangement of symmetry elements will be changed, too. But also in other crystal systems (e.g. trigonal, tetragonal, cubic) different origins or basis vectors are described. Often they will be caused by the defined conventions which sometimes don't enables a clear description of space-group symmetry. However, especially the change of origin is unproblematically because these settings are documented additionally in detail in IT.
In contrast to that, for orthorhombic space-group types only one setting is given, respectively. Only for the monoclinic symmetry different cell choices are quoted, but only one setting will be described in detail whereas some other will be given in a shorter form. The consequence of a simple permutation of the basis vectors is described in the introduction part of the IT (Vol A). Considering this permutation and the different cell choices in the monoclinic system in maximum 18 different settings must be distinguished. Especially because of the implementation of group-subgroup relations it prooves to be necessary to consider all these different settings listed in Tab. 4.3.1 (Vol A) in IT. Therefore diffractograms can be calculated for more than 740 different settings implemented in the symmetry file pcwspgr.dat.
Within PowderCell only these settings will be regarded as standard settings which are described in IT in detail (more than 230!). This includes the settings differed by a shifted origin, but not the different settings and cell choices given for the monoclinic space-group types. 
In contrast to the numbering of settings in the former DOS version the numbering here is based on the order of Tab. 4.3.1 in the IT. This seemed to be consequently and simplified the automatism of the transformation procedure within the program.
 
# monoclinic settings
 
In the monoclinic crystal system either 6 or 18 settings can be derived in dependence of the space-group type, respectively. These settings described in detail in IT correspond to the settings given in the column 1 and 3 of the following table. Because of the consideration of all settings incompatibillities result to the numbering of settings of the former DOS version. The assignments of the setting numbers between DOS and Windows version is changing as follows: 
 
1 will be 1, 2 transforms to 7, 3 to 13, 4 will be changed to 3, 5 to 9 and the 6. setting will be described by setting number 15. PowderCell converts the structure files of the DOS version automatically. However, a message will be given as hint which must be confirmed that the conversion is really necessary.

 
 
For new constructed parameter files it is urgent to use the following numbers of settings! 
These will be given in the following table as number of row or in several cases as number in brackets behind the Hermann-Mauguin symbol.
#  
setting	     1	    2	    3	    4	    5	    6
No of space-
group type	
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
3	P  1 2 1	P  1 2 1	P  1 1 2	P  1 1 2	P  2 1 1	P  2 1 1
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
4	P  1 21 1	P  1 21 1	P  1 1 21	P  1 1 21	P  21 1 1	P  211 1
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
5	C  1 2 1	A  1 2 1	A  1 1 2	B  1 1 2	B  2 1 1	C  2 1 1
	A  1 2 1 (7)	C  1 2 1 (8)	B  1 1 2 (9)	A  1 1 2 (10)	C  2 1 1 (11)	B  2 1 1 (12)
	I  1 2 1 (13)	I  1 2 1 (14)	I  1 1 2 (15)	I  1 1 2 (16)	I  2 1 1 (17)	I  2 1 1 (18)
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
6	  1 m 1	P  1 m 1	P  1 1 m	P  1 1 m	P  m 1 1	P  m 1 1
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
7	P  1 c 1	P  1 a 1	P  1 1 a	P  1 1 b	P  b 1 1	P  c 1 1
	P  1 n 1 (7)	P  1 n 1 (8)	P  1 1 n (9)	P  1 1 n (10)	P  n 1 1 (11)	P  n 1 1 (12)
	P  1 a 1 (13)	P  1 c 1 (14)	P  1 1 b (15)	P  1 1 a (16)	P  c 1 1 (17)	P  b 1 1 (18)
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
8	C  1 m 1	A  1 m 1	A  1 1 m	B  1 1 m	B  m 1 1	C  m 1 1
	A  1 m 1 (7)	C  1 m 1 (8)	B  1 1 m (9)	A  1 1 m (10)	C  m 1 1 (11)	B  m 1 1 (12)
 	I  1 m 1 (13)	I  1 m 1 (14)	I  1 1 m (15)	I  1 1 m (16)	I  m 1 1 (17)	I  m 1 1 (18)
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
9	C  1 c 1	A  1 a 1	A  1 1 a	B  1 1 b	B  b 1 1	C  c 1 1
	A  1 n 1 (7)	C  1 n 1 (8)	B  1 1 n (9)	A  1 1 n (10)	C  n 1 1 (11)	B  n 1 1 (12)
	I  1 a 1  (13)	I  1 c 1 (14)	I  1 1 b (15)	I  1 1 a (16)	I  c 1 1 (17)	I  b 1 1 (18)
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
10	P  1 2/m 1	P  1 2/m 1	P  1 1 2/m	P  1 1 2/m	P  2/m 1 1	P  2/m 1 1
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
11	P  1 21 /m 1	P  1 21 /m 1	P  1 1 21 /m	P  1 1 21 /m	P  21 /m 1 1	P  21 /m 1 1
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
12	C  1 2/m 1	A  1 2/m 1	A  1 1 2/m	B  1 1 2/m	B  2/m 1 1	C  2/m 1 1
	A  1 2/m 1 (7)	C  1 2/m 1 (8)	B  1 1 2/m (9)	A  1 1 2/m (10)	C  2/m 1 1 (11)	B  2/m 1 1 (12)
	I  1 2/m 1 (13)	I  1 2/m 1 (14)	I  1 1 2/m (15)	I  1 1 2/m (16)	I  2/m 1 1 (17)	I  2/m 1 1 (18)
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
13	P  1 2/c 1	P  1 2/a 1	P  1 1 2/a	P  1 1 2/b	P  2/b 1 1	P  2/c 1 1
	P  1 2/n 1 (7)	P  1 2/n 1 (8)	P  1 1 2/n (9)	P  1 1 2/n (10)	P  2/n 1 1 (11)	P  2/n 1 1 (12)
	P  1 2/a 1 (13)	P  1 2/c 1 (14)	P  1 1 2/b (15)	P  1 1 2/a (16)	P  2/c 1 1 (17)	P  2/b 1 1 (18)
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
14	P  1 21 /c 1	P  1 21 /a 1	P  1 1 21 /a	P  1 1 21 /b	P  21 /b 1 1	P  21 /c 1 1
	P  1 21 /n 1 (7)	P  1 21 /n 1 (8)	P  1 1 21 /n (9)	P  1 1 21 /n 10	P  21 /n 1 1 11	P  21 /n 1 1 (12)
	P  1 21 /a 1 13	P  1 21 /c 1 14	P  1 1 21 /b 15	P  1 1 21 /a 16	P  21 /c 1 1 17	P  21 /b 1 1 (18)
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
15	C  1 2/c 1	A  1 2/a 1	A  1 1 2/a	B  1 1 2/b	B  2/b 1 1	C  2/c 1 1
	A  1 2/n 1 (7)	C  1 2/n 1 (8)	B  1 1 2/n (9)	A  1 1 2/n (10)	C  2/n 1 1 (11)	B  2/n 1 1 (12)
	I  1 2/a 1 (13)	I  1 2/c 1 (14)	I  1 1 2/b (15)	I  1 1 2/a (16)	I  2/c 1 1 (17)	I  2/b 1 1 (18)

 
orthorhombic settings
 
Please notice the specific settings of the space-group types 48, 50, 59 68 and 70! There you must distinguish between 12 different settings resulting from the different choice of origins (look Tab. 4.3.1 in IT ).
 
For new constructed parameter files it is urgent to use the following numbers of settings! 
These will be given in the following table as number of column or in several cases as number in brackets behind the Hermann-Mauguin symbol.
#  
setting	     1	    2	    3	    4	    5	    6
 No of space-
group type
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
16	P 2 2 2	P 2 2 2	P 2 2 2	P 2 2 2	P 2 2 2	P 2 2 2
17	P 2 2 21	P 2 2 21	P 21  2 2	P 21  2 2	P 2 21  2	P 2 21  2
18	P 21  21  2	P 21  21  2	P 2 21  21	P 2 21  21	P 21  2 21	P 21  2 21
19	P 21  21  21	P 21  21  21	P 21  21  21	P 21  21  21	P 21  21  21	P 21  21  21
20	C 2 2 21	C 2 2 21	A 21  2 2	A 21  2 2	B 2 21  2	B 2 21  2
21	C 222	C 2 2 2	A 2 2 2	A 2 2 2	B 2 2 2	B 2 2 2
22	F 222	F 2 2 2	F 2 2 2	F 2 2 2	F 2 2 2	F 2 2 2
23	I 2 2 2	I 2 2 2	I 2 2 2	I 2 2 2	I 2 2 2	I 2 2 2
24	I 21  21  21	I 21  21  21	I 21  21  21	I 21  21  21	I 21  21  21	I 21  21  21
25	P mm2	P mm2	P 2mm	P 2mm	P m2m	P m2m
26	P mc21	P cm21	P 21 ma	P 21 am	P b21 m	P m21 b
27	P cc2	P cc2	P 2aa	P 2aa	P b2b	P b2b
28	P ma2	P bm2	P 2mb	P 2cm	P c2m	P m2a
29	P ca21	P bc21	P 21 ab	P 21 ca	P c21 b	P b21 a
30	P nc2	P cn2	P 2na	P 2an	P b2n	P n2b
31	P mn21	P nm21	P 21 mn	P 21 nm	P n21 m	P m21 n
32	P ba2	P ba2	P 2cb	P 2cb	P c2a	P c2a
33	P na21	P bn21	P 21 nb	P 21 cn	P c21 n	P n21 a
34	P nn2	P nn2	P 2nn	P 2nn	P n2n	P n2n
35	C mm2	C mm2	A 2mm	A 2mm	B m2m	B m2m
36	C mc21	C cm21	A 21 ma	A 21 am	B b21 m	B m21 b
37	C cc2	C cc2	A 2aa	A 2aa	B b2b	B b2b
38	A mm2	B mm2	B 2mm	C 2mm	C m2m	A m2m
39	A bm2	B ma2	B 2cm	C 2mb	C m2a	A c2m
40	A ma2	B bm2	B 2mb	C 2cm	C c2m	A m2a
41	A ba2	B ba2	B 2cb	C 2cb	C c2a	A c2a
42	F mm2	F mm2	F 2mm	F 2mm	F m2m	F m2m
43	F dd2	F dd2	F 2dd	F 2dd	F d2d	F d2d
44	I mm2	I mm2	I 2mm	I 2mm	I m2m	I m2m
45	I ba2	I ba2	I 2cb	I 2cb	I c2a	I c2a
46	I ma2	I bm2	I 2mb	I 2cm	I c2m	I m2a
47	P mmm	P mmm	P mmm	P mmm	P mmm	P mmm
48	P nnn (1)	P nnn (3)	P nnn (5)	P nnn (7)	P nnn (9)	P nnn (11)
	P nnn (2)	P nnn (4)	P nnn (6)	P nnn (8)	P nnn (10)	P nnn (12)
49	P ccm	P ccm	P maa	P maa	P bmb	P bmb
50	P ban (1)	P ban (3)	P ncb (5)	P ncb (7)	P cna (9)	P cna (11)
	P ban (2)	P ban (4)	P ncb (6)	P ncb (8)	P cna (10)	P cna (12)
51	P mma	P mmb	P bmm	P cmm	P mcm	P mam
52	P nna	P nnb	P bnn	P cnn	P ncn	P nan
53	P mna	P nmb	P bmn	P cnm	P ncm	P man
54	P cca	P ccb	P baa	P caa	P bcb	P bab
55	P bam	P bam	P mcb	P mcb	P cma	P cma
56	P ccn	P ccn	P naa	P naa	P bnb	P bnb
57	P bcm	P cam	P mca	P mab	P bma	P cmb
58	P nnm	P nnm	P mnn	P mnn	P nmn	P nmn
59	P mmn (1)	P mmn (3)	P nmm (5)	P nmm (7)	P mnm (9)	P mnm (11)
	P mmn (2)	P mmn (4)	P nmm (6)	P nmm (8)	P mnm (10)	P mnm (12)
60	P bcn	P can	P nca	P nab	P bna	P cnb
61	P bca	P cab	P bca	P cab	P bca	P cab
62	P nma	P mnb	P bnm	P cmn	P mcn	P nam
63	C mcm	C cmm	A mma	A mam	B bmm	B mmb
64	C mca	C cmb	A bma	A cam	B bcm	B mab
65	C mmm	C mmm	A mmm	A mmm	B mmm	B mmm
66	C ccm	C ccm	A maa	A maa	B bmb	B bmb
67	C mma	C mmb	A bmm	A cmm	B mcm	B mam
68	C cca (1)	C ccb (3)	A baa (5)	A caa (7)	B bcb (9)	B bab (11)
	C cca (2)	C ccb (4)	A baa (6)	A caa (8)	B bcb (10)	B bab (12)
69	F mmm	F mmm	F mmm	F mmm	F mmm	F mmm
70	F ddd (1)	F ddd (3)	F ddd (5)	F ddd (7)	F ddd (9)	F ddd (11)
	F ddd (2)	F ddd (4)	F ddd (6)	F ddd (8)	F ddd (10)	F ddd (12)
71	I mmm	I mmm	I mmm	I mmm	I mmm	I mmm
72	I bam	I bam	I mcb	I mcb	I cma	I cma
73	I bca	I cab	I bca	I cab	I bca	I cab
74	I mma	I mmb	I bmm	I cmm	I mcm	I mam	

Laue group 
 
Generally interference phenomenons will assumed as centrosymmetrically, i.e. with regard to the X-ray diffraction the X-ray beam isn't able to distinguish between direction and opposite direction, i.e. the detected intensity of HKL and -H-K-L will be identically. This theorem is the so-called Friedel's law and reduces the 32 point groups to 11, which really can be distinguish experimentally. These 11 centrosymmetric point groups are called Laue groups.
However, a special case occurs in the trigonal crystal system in the Laue group  –3m. On the one hand there it's possible to locate the mirror plane perpendicular to the basis vector a, on the other hand you are able to locate the mirror plane perpendicular to [210]. Regarding the translation lattice the Laue group –3m can be described by two clearly distinguishable isomorphic groups  –3 1 m and –3 m 1. Therefore in practice 12 Laue groups will be occured. The same you will find for the point groups too, so that in practice the number of point groups will be increased from 32 to 35 (siehe z.B. [26]).

     crystal system	Laue group	non-centrosymm. 
		point group
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾ 
     triclinic	–1	1
     monoclinic	2/m	2, m
     othorhombic	mmm	222, mm2
     tetragonal	4/m	4, –4
	4/m mm	4mm, –4m2, 422
     trigonal	–3	3
	–3m1	3m1, 321
	–31m	31m, 312
     hexagonal	6/m	6, –6
	6/m mm	6mm, –6m2, 622
     cubic	m3	23
	m–3m	–43m, 432
 
The Friedel's law is canceled if the investigated substance contains an element emitting a radiation of which energy of the characteristic component is insignificantly smaller than the energy of the characteristic radiation used in experiment. This effect is called anomaleous dispersion and is relevant for non-centrosymmetric crystal structures.

The classification of data used in PowderCell is based on IZUMI [6]. He puts his data file at free disposal. There beside the reflection conditions also the encoded Laue group will be given. The Laue group will be used in PowderCell for different problems, e.g. to identify the monoclinic axis or to separate the setting for rhombohedral space-group types. 
 
The symmetry of the Laue group only then will be used, if the anomaleous dispersion in the dialog box powder diffraction / experiment is disabled. 

Symmetry or inversion centre 
 
The existence of an inversion centre can be used to simplifications due the calculation of the structure factor. Usually, one assumes that the so-called phase angle in centrosymmetric structures becomes zero. That's correct, but only if the inversion centre is located in the origin of the lattice. In contrast to this, in the first setting of some tetragonal and cubic space-group types the inversion centre has been defined outside the origin. In this cases the phase angle doesn't become zero but 45°. 
A well-known example is the structure of silicon (F 41–3m, No 227). The representation shown in papers is usually the following (silicon atom on Wyckoff position 8a):
 
	     structure of silicon (1. setting) 
 
It will be described by the first setting locating the inversion centre in (1/8,1/8,1/8). If one uses the 2. setting (inversion centre in the origin of the lattice; Si again on 8a) then the phase angle is 0°, but the structure representation of one unit cell is strange.
 
 	     structur of silicon (2. setting) 
 
If you are surprised, please compare the two calculated powder pattern! The input data you can create by yourself and with help of PowderCell. The lattice constants are not important (you can input an arbitrary value)  and the atomic position and the number of space-group type was described above (8a).

Generators
 
In PowderCell for the generation of all atomic position so-called generators will be used. The initial data of the crystal structure must be described in the asymmetric unit. The generators have been extracted manually from the IT [1996]. 
The generators of a space-group type include the identity , the centering vectors for non-primitive lattices and  a special selection of symmetry elements (the so-called generators selected), which combination determines all other symmetry elements. The derived symmetry elements define the commonly used general positions, if any will be multiplied with the atomic coordinate (x,y,z). The most important advantage of the new generation (in contrast to the former DOS version) is the small size of necessary information for each space-group type. This decreases spelling mistakes too, but if such exist a detection is very easy because one half or two third of all generated positions must be wrong.
To describe all space-group types and the different settings it's enough to define 15 generator types. These correlate with the following symmetry elements: -1, 2x, 2y, 2z, 2110, 21-10, mx, my, mz, mxxz, mx-xz, 3, 3111, 4, -4. The generator itself consist of generator type and an additional translation part defined by the translation vector of the symmetry element and their shift from the origin. The generators can be used to create all special positions, too (see: generation of atomic positions). Using the Wyckoff notation a multiple generation will be prevented. There only some generators will be used in dependence of the local symmetry, respectively. The order of these or which must be selected is declared in the symmetry file  pcwspgr.dat. There only a simple encoding has been developed. In maximum 5 generators gi must be given (exclusively the lattice centering vectors and the identity). The equation for the calculation of the code number is:
 
	  .
 
If one doesn't need a generator (except the identity and lattice centering vectors) the encoded number is  2^0=1. In case of selection of the first and the third generator the encoding results

	 .

Wyckoff position types
 
In PowderCell a procedure for an identification of special positions — the so-called Wyckoff positions — has been installed. This supports the user to check imported data or to understand the interaction between space-group and local symmetry: On the other hand the program uses the Wyckoff notation for an optimized generation of atomic positions. This is very useful, especially for a structure manipulation — the aim of this program. It is planned to fix the local symmetry for selected atoms/ions or molecules. Currently you must be careful of this by yourself.
In the description of all space-group types and their settings (more than 745) a total of 208 different types of special positions have been extracted. They will be used to encode the special-position types in the symmetry file pcwspgr.dat. There each special position is characterized by two numbers: the encoded Wyckoff-position type and its generator sequence.
However, the extracted types of special positions classify the first coordinate only, but not all symmetry-equivalents. This is the reason that the number of different types is relative small. 
The disadvantage of this classification is, that all atomic coordinates must be transformed to the condition of the identified Wyckoff-position type. Using the generators selected, only with the transformed coordinates the generation of atomic positions will be carried out correctly.
 
The procedure can be described shortly in the following manner: At first PowderCell creates the atomic positions using all generators. The comparision between all locations generated and the possible Wyckoff-position types for the actual space-group number supplies the Wyckoff letters wanted. In this part of procedure a multiple generation is allowed and in general necessary. But if the program has been allocated the Wyckoff-position types the following structure manipulation or generation of atomic positions will be used the Wyckoff-specific generation.
 
For a better classification it is recommendable to split the position types into a group with no, one or two degrees of freedom. The following tables show all types of special positions used in PowderCell. With the help of the given code number it is possible to translate the Wyckoff-position data from the symmetry file pcwspgr.dat.
 
#  
	no degree of freedom
 position	type	  position	type	  position	type	
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
0 , 0 , 0	    1	1/4 , 1/4 , 0	  32	1/2 , 3/4 , 0	  63	
0 , 0 , 1/4	    2	1/4 , 1/4 , 1/4	  33	1/2 , 3/4 , 1/4	  64	
0 , 0 , 1/2	    3	1/4 , 1/4 , 1/2	  34	1/2 , 3/4 , 1/2	  65	
0 , 0 , 3/4	    4	1/4 , 1/4 , 3/4	  35	1/2 , 3/4 , 3/4	  66	
0 , 1/4 , 0	    5	1/4 , 1/2 , 0	  36	5/8 , 0 , 1/4	  67	
0 , 1/4 , 1/8	    6	1/4 , 1/2 , 1/4	  37	5/8 , 1/8 , 1/8	  68	
0 , 1/4 , 1/4	    7	1/4 , 1/2 , 1/2	  38	5/8 , 3/8 , 5/8	  69	
0 , 1/4 , 3/8	    8	1/4 , 1/2 , 3/4	  39	5/8 , 5/8 , 3/8	  70	
0 , 1/4 , 1/2	    9	1/4 , 3/4 , 0	  40	5/8 , 5/8 , 5/8	  71	
0 , 1/4 , 5/8	  10	1/4 , 3/4 , 1/4	  41	2/3 , 1/3 , 0	  72	
0 , 1/4 , 3/4	  11	1/4 , 3/4 , 1/2	  42	2/3 , 1/3 , 1/4	  73	
0 , 1/2 , 0	  12	1/4 , 3/4 , 3/4	  43	2/3 , 1/3 , 1/2	  74	
0 , 1/2 , 1/4	  13	1/3 , 2/3 , 0	  44	3/4 , 0 , 0	  75	
0 , 1/2 , 1/2	  14	1/3 , 2/3 , 1/4	  45	3/4 , 0 , 1/4	  76	
0 , 1/2 , 3/4	  15	1/3 , 2/3 , 1/2	  46	3/4 , 0 , 1/2	  77	
0 , 3/4 , 0	  16	1/3 , 2/3 , 3/4	  47	3/4 , 1/4 , 0	  78	
0 , 3/4 , 1/8	  17	3/8 , 0 , 1/4	  48	3/4 , 1/4 , 1/4	  79	
0 , 3/4 , 1/4	  18	3/8 , 1/8 , 1/8	  49	3/4 , 1/4 , 1/2	  80	
0 , 3/4 , 1/2	  19	3/8 , 3/8 , 3/8	  50	3/4 , 1/4 , 3/4	  81	
1/8 , 0 , 1/4	  20	1/2 , 0 , 0	  51	3/4 , 1/2 , 0	  82	
1/8 , 1/8 , 1/8	  21	1/2 , 0 , 1/4	  52	3/4 , 1/2 , 1/4	  83	
1/8 , 1/8 , 3/8	  22	1/2 , 0 , 1/2	  53	3/4 , 1/2 , 1/2	  84	
1/8 , 1/8 , 5/8	  23	1/2 , 0 , 3/4	  54	3/4 , 1/2 , 3/4	  85	
1/8 , 1/8 , 7/8	  24	1/2 , 1/4 , 0	  55	3/4 , 3/4 , 1/4	  86	
1/8 , 3/8 , 1/8	  25	1/2 , 1/4 , 1/4	  56	3/4 , 3/4 , 1/2	  87	
1/8 , 5/8 , 1/8	  26	1/2 , 1/4 , 1/2	  57	3/4 , 3/4 , 3/4	  88	
1/8 , 7/8 , 1/8	  27	1/2 , 1/4 , 3/4	  58	7/8 , 0 , 1/4	  89	
1/4 , 0 , 0	  28	1/2 , 1/2 , 0	  59	7/8 , 1/8 , 1/8	  90	
1/4 , 0 , 1/4	  29	1/2 , 1/2 , 1/4	  60	7/8 , 7/8 , 7/8	  91	
1/4 , 0 , 1/2	  30	1/2 , 1/2 , 1/2	  61			
1/4 , 0 , 3/4	  31	1/2 , 1/2 , 3/4	  62			

#  
	one degree of freedom
 position	type	 position	type	position	type	
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
0 , 0 , z	100	1/4 , y , 1/2	134	x , 1/2 , 1/4	168	
0 , 1/4 , z	101	1/4 , y , 3/4	135	x , 1/2 , 1/2	169	
0 , 1/2 , z	102	1/4 , y , y	136	x , 1/2 , 3/4	170	
0 , 3/4 , z	103	1/4 , y , -y	137	x , 3/4 , 0	171	
1/8 , 1/8 , z	104	1/4 , y , y+1/2	138	x , 3/4 , 1/4	172	
1/8 , 7/8 , z	105	1/4 , y , -y+1/2	139	x , 3/4 , 1/2	173	
1/4 , 0 , z	106	1/2 , y , 0	140	x , 3/4 , 3/4	174	
1/4 , 1/4 , z	107	1/2 , y , 1/4	141	x , 7/8 , 1/8	175	
1/4 , 1/2 , z	108	1/2 , y , 1/2	142	-x , x , 0	176	
1/4 , 3/4 , z	109	1/2 , y , 3/4	143	x , x , 0	177	
1/3 , 2/3 , z	110	1/2 , y , y	144	x , x , 1/4	178	
1/2 , 0 , z	111	1/2 , y , -y	145	x , x , 3/8	179	
1/2 , 1/4 , z	112	1/2 , y , y+1/2	146	x , x , 1/2	180	
1/2 , 1/2 , z	113	3/4 , y , 0	147	x , x , 5/8	181	
1/2 , 3/4 , z	114	3/4 , y , 1/4	148	x , x , 3/4	182	
2/3 , 1/3 , z	115	3/4 , y , 1/2	149	x , x , x	183	
3/4 , 0 , z	116	3/4 , y , 3/4	150	x , -x , 0	184	
3/4 , 1/4 , z	117	7/8 , y , 1/8	151	x , -x , 1/6	185	
3/4 , 1/2 , z	118	x , 0 , 0	152	x , -x , 1/4	186	
3/4 , 3/4 , z	119	x , 0 , 1/6	153	x , -x , 1/3	187	
7/8 , 1/8 , z	120	x , 0 , 1/4	154	x , -x , 1/2	188	
0 , y , 0	121	x , 0 , 1/3	155	x , -x , 2/3	189	
0 , y , 1/4	122	x , 0 , 1/2	156	x , -x , 5/6	190	
0 , y , 1/2	123	x , 0 , 2/3	157	x , x+1/4 , 1/8	191	
0 , y , 3/4	124	x , 0 , 3/4	158	x , x+1/4 , 7/8	192	
0 , y , y	125	x , 0 , 5/6	159	x , x+1/2 , 0	193	
0 , y , -y	126	x , 1/8 , 1/8	160	x , x+1/2 , 1/4	194	
1/8 , y , 1/8	127	x , 1/8 , 7/8	161	x , x+1/2 , 1/2	195	
1/8 , y , 7/8	128	x , 1/4 , 0	162	x , x+1/2 , 3/4	196	
1/8 , y , y+1/4	129	x , 1/4 , 1/8	163	x , -x+1/2 , 1/4	197	
1/8 , y , -y+1/4	130	x , 1/4 , 1/4	164	x , 2x , 0	198	
1/4 , y , 0	131	x , 1/4 , 1/2	165	x , 2x , 1/4	199	
1/4 , y , 1/8	132	x , 1/4 , 3/4	166	x , 2x , 1/2	200	
1/4 , y , 1/4	133	x , 1/2 , 0	167	x , 2x , 3/4	201	
 
#  
	two degrees of freedom	
position	type	position	type	position	type	
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
0 , y , z	202	x , 1/2 , z	208	x , y , 0	214	
1/4 , y , z	203	x , 3/4 , z	209	x , y , 1/4	215	
1/2 , y , z	204	x , x , z	210	x , y , 1/2	216	
3/4 , y , z	205	x , -x , z	211	x , y , 3/4	217	
x , 0 , z	206	x , 2x , z	212			
x , 1/4 , z	207	x , x+1/2 , z	213			

pcwspgr.dat
 
The symmetry file pcwspgr.dat contains all information about the space-group types and different settings used in PowderCell. It is possible to edit this ASCII file but mostly the data are encoded (in a simple manner). The structur of the space-group records is based on IZUMI. He developed this for the use in his  Rietveld program RIETAN [6]. In the last years a set of additional information have been inserted and some bugs could be eliminated. 
The individual record for an arbitrary space-group type contains:
  
  the number of space-group type used in IT generally,  
  the number of setting,
  the corresponding Laue group, 
  the existence of a inversion centre in origin (0...no, 1...yes), 
  the number of general positions (without consideration of an inversion centre located in origin and lattice centering vectors producing additional general positions too),  
  the number of special positions describing the existing local symmetries,  
  the number of generators selected,  
  the full Hermann-Mauguin symbol,  
  the reflection conditions,  
  the Wyckoff-position types and the corresponding local symmetry (described by the generator sequence) and  
  the generators itself, defined by the generator type an an additional translation part.  
 
All date have been written in free format except the reflection conditions. The line containing the reflection conditions is probably based on former FORTRAN programs used by IZUMI. Respectively two characters define the reflection condition of a reflection type connected with this position in the line.  
 
example:
spgr 227  1 15  0 48  8  5   F 4_1/d -3 2/m
1917 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

PowderCell uses the key-word spgr as code for the beginning of a space-group record . It follows the number of space-group type (227) and the setting (1.) The third number (15) represents the Laue group, which is important if the anomaleous dispersion is deactivated. The 0 represents a flag symbolizing the absence of an inversion centre in the origin (but there exists an inversion centre, as you can see from the last generator described by type1). Using the listed generators 48 general positions will be created. Besides the position of general symmetry 8 special positions exist. To generate all 48 different symmetry elements 5 generators are given and must be feeded into the program. At the end of the first line of the record the full Hermann-Mauguin symbol  (F 4_1/d -3 2/m) will be presented. 
In the second line the reflection conditions will be given in dependence on the defined reflection type. So e.g. the intensity of 00L will be calculated only than, if the condition 19 (i.e. L=4n) will be meeted. For the reflection type 0K0 the condition 17 is determined. This corresponds to the condition K=4n. 

The condition of reflection type 7 has a high importance for PowderCell. It will be used to extract the lattice type. In our example the condition Nr 7 defines an all-face centered lattice (F).

The third line contains all information about the existing Wyckoff positions: the type and the generator sequence. The types don't shall be described here in detail. However, the encoding of the generator sequences shall be represented in a short manner.
For each Wyckoff position two values will be presented in the record. The second of this describes the generator sequence. In the line you will find the same order of special positions as listed in IT. The order starts with that position describing by the Wyckoff letter a but doesn't consider the general position, i.e. if no special position exists no value is given in this line of the record. In our example some of the 8 Wyckoff-position types will be described by the same generator sequence, e.g. for a and b will be used code 16 or c and d are described by code 6. That means that both positions will be characterized by the identical local symmetry. However, the Wyckoff positions e,f,g,h will be generated by a different generator sequence — 22, 26, 30, 46. The encoding is very simple using a bit structure. In our example the generator type 1 will be presented by bit 0, type 5 by bit 1, type 13 by bit 2, type 3 by bit 3 and type 4 by bit 4. In dependence of the generator sequence the bits are activated or not. If within the unit cell only one symmetry-equivalent position exits, no generator must be used and therefore no bit is activated. The result is code 1.

Generation of atomic positions
 
For the description of the generation of atomic position 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/m
1917 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 has been 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 2^4=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 (press the right mouse key over the respective generator type in the example record!) correspond to the symmetry elements -1 und 2110. 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, 2110 and 2y (8 positions). The used identifier is 32e. From 26 can be followed that the bits 0, 2 and 3 are activated (-1, 3111 and 2y), so that the position can be described as 48f. The 30 contains the activated bits 0, 1, 2 and 3 (-1, 2110, 3111 and 2y). Therefore it is called 96g. The 46 corresponds to the activated bits 0,1,2,4 (-1, 2110, 3111 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.

remark: It should not be astonished that for non-primitive lattices the positions given in IT and that 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.

Hermann-Mauguin symbol (HMS)
 
For description of space-group symmetry PowderCell uses the number of space-group type and their settings. In some procedures the Hermann-Mauguin symbol will be used to select the setting. Therefore, PowderCell shows the Hermann-Mauguin symbol on request. Schoenflies or Hall symbols are not yet supported by the program.
The advantage of the Hermann-Mauguin symbols is that you will be informed about the location of symmetry elements, immediately. However, not always definit! Therefore the Hall symbolic has been introduced which describes the arrangement of symmetry element unambiguously. 

The Hermann-Mauguin symbol contains the most important symmetry elements which enables the derivation of the full space-group symmetry. Within the nomenclature mirror and glide planes get a higher priority than rotation or screw axis. The complete arrangement of symmetry elements can be created by combination of that given in the Hermann-Mauguin symbol. To evaluate the symbol correctly the corresponding lattice directions must be known. 

		position of symmetry elements in HMS
	crystal system	   1.	   2.	   3.	
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
	triclinic	   --	   -- 	   --
	monoclinic axes   b 	   --	[010]	   --
	                               c	   --	   --	[001]
	                               a	[100]	   --	   --
	orthorhombic	[100]	[010]	[001]
	tetragonal	[001]	á100ñ	á110ñ
	hexagonal	[001]	á100ñ	á1-10ñ
	trigonal
	   hexagonal axes	[001]	á100ñ	
	   rhombohedral axes	[111]	á1-10ñ
	cubic	á001ñ	á111ñ	á110ñ
 
The symmetry elements will be given in dependence of the position characterized by an individual lattice direction. One distinguishes a short and a full symbol, e.g. P mmm and the same as full symbol: P 2/m 2/m 2/m. But also the full symbol don't contain all symmetry elements, e.g. in the given example the inversion centre is absent. 

remark: Please notice the differences for some trigonal space-group types! There sometimes the same symmetry occurs in different settings. But it must be distinguished e.g. between P 3 1 2 and P 3 2 1. In dependence of the corresponding setting of axes unfortunately different procedures must be used, e.g. the generation of HKL. The use of the IT is unproblematically because these settings have been described by an additional space-group type (see:Laue group).

Reflection condition
 
To decrease the processing time the laws of extinction (absent reflections) will be considered in PowderCell. These are contained in the symmetry file pcwspgr.dat as  reflection condition. That means that a powder pattern can be calculated for that structures exclusively, described by a corresponding setting in pcwspgr.dat. However, the group-subgroup relations will be given for standard settings only. This includes the settings with different origins, but  not the different settings or cell choices of the monoclinic and orthorhombic space-group types.
Systematic extinctions — the aim of this help screen — occurs then, if centered lattices (integral extinctions), glide planes (zonal extinctions) or screw axis (serial extinctions)
have been defined as generators. Lattice centerings constitute only the result of a mathematical model. Therefore actual physical extinctions don't occur. They result from the comfortable definition of the translation lattice. If you describe the same structure using a primitive lattice you don't will find additional or absent reflections. Only the indexing of the calculated reflections will be changed. However, translation parts of symmetry elements always create systematic extinctions. 

remark: Only one well-known exception exist: The 31 in cubic space-group types contains an ineffective translation part because it's the same as this caused by projection of the translation lattice on direction [111].
 
Accidental extinctions will be observed whenever atomic positions are located in an apparently higher symmetry. Excellently this can be shown with the first step of the automatical group-subgroups transition in PowderCell. There in practice the symmetry will be decreased and the same structure will be described in this lower symmetry once more. In the calculated powder patter some additional reflections occur but with no intensity, i.e. they are extinguished coincidentally. If you shift only one of the atoms outside a virtual (non-existing) symmetry element you will observe an intensity variation. The intensity of the extinguished reflections will increase.
 
remark: Another kind of accidental extinctions one observe for KCl. This crystallizes in structure type of NaCl. Both the K and the Cl ion interact with the X-ray beam, but it is practically not able to distinguish between their because of the identical number of electrons. The consequence is that some reflections will be extinguished accidentally. During the evaluation of the powder pattern one would derive a cubic primitive lattice instead of an all-face centered. The lattice constant a would be the half of the correct value. One lattice point would contain only a average atom described by 18 electrons instead of a K ion and a Cl ion, respectively.
 
Generally PowderCell is optimized in consideration of reflection conditions given in pcwspgr.dat. Nevertheless it is conceivable that systematically extinguished reflections will be calculated or usually observed are absent . If you find such case don't hesitate to contact us.
 
In the following tables all generator types characterized by additional translation parts are included. Furthermore the reflection conditions are given, respectively.
 
#  
lattice type
 
symbol	observable reflections	extinguished reflections
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
      P	HKL arbitrary	non
      I	H+K+L = 2n	H+K+L = 2n+1
      F	H , K , L all even or	H+K = 2n+1 or K+L = 2n+1
	               all odd	or H+L = 2n+1
      A	K+L = 2n	K+L = 2n+1
      B	H+L = 2n	H+L = 2n+1
      C	H+K = 2n	H+K = 2n+1
      R	(only for hexagonal axes)
	-H+K+L = 3n or
	H-K+L		
 
 #  
glide plane
		concerned		reflection conditions
  symbol	orientation	reflections	(observable reflections)
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
	a	     (010)	    H0L		H = 2n
		     (001)	    HK0		H = 2n
	b	     (100)	    0KL		K = 2n
		     (001)	    HK0		K = 2n
	c	     (100)	    0KL		L = 2n
		     (010)	    H0L		L = 2n
		     (110)	    HHL		L = 2n
		     (1-100)	    HH.L		L = 2n
		     (11-20)	    HH.L		L = 2n
	d	     (100)	    0KL		K+L = 4n   (K , L = 2n)
		     (010)	    H0L		H+L = 4n   (H , L = 2n)
		     (001)	    HK0		H+K = 4n   (H , K = 2n)
		     (110)	    HHL		2H+L = 4n
	n	     (100)	    0KL		K+L = 2n
		     (010)	    H0L		H+L = 2n
		     (001)	    HK0		H+K = 2n
 
 #  
screw axis	
		concerned		reflection conditions
   symbol	orientation	reflections	(observable reflections)
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
	21	     [100]	    H00		H = 2n
		     [010]	    0K0		K = 2n
		     [001]	    00L		L = 2n
	41 , 43	          [100]	    H00		H = 4n
		     [010]	    0K0		K = 4n
		     [001]	    00L		L = 4n
	42	     [100]	    H00		H = 2n
		     [010]	    0K0		K = 2n
		     [001]	    00L		L = 2n
	31 , 32	     [00.1]	    00.L		L = 3n
	61 , 65	     [00.1]	    00.L		L = 6n
	62 , 64	     [00.1]	    00L		L = 3n
	63	     [00.1]	    00L		L = 2n

 

Asymmetric unit
 
PowderCell uses the data of the asymmetric unit to generate all atomic positions. Using the coordinates given in the asymmetric unit and the space-group symmetry defined in pcwspgr.dat the positions of atoms will be created. The atomic numbers listed in the structure file (e.g. import of *.cel-files) or the defined ionizations determine the coefficients for the calculation of the atomic scattering factors. The SOF (substitution and occupation will be multiplied to the Site Occupation Factor) represents the weight of the scattering power of the corresponding atomic position. The values given in the asymmetric unit represent the basis of each structure processed in PowderCell. For one and the same structure the number of atoms that must be entered is dependent on the point-group symmetry (see e.g. subgroups).
 
For each atomic position the following data must be known to carry out calculations within PowderCell:

( i )	atomic number or ionization state (ion in the table of the dialog box structure data)
( ii)	relative atomic coordinates or alternatively the Wyckoff position (i.e. often it is enough to input the letter)
(iii)	substitution and/or occupation 
(iv)	isotropic temperature factor or equivalent temperature factor (optional, per default 0).
 
But these data can be used only if either the space-group number and the setting or the Hermann-Mauguin symbol is known to derive the space-group type.
If the symmetry is unknown but all atomic positions within the unit cell are available it is possible to calculate the powder pattern using the space-group type P1. Unfortunately, a lot of extinctions can be occured corresponding to the unknown symmetry. The shown diagram is always correct.
 
It is possible to display beside the full structure the entered atomic positions of the asymmetric unit, too.For this purpose you must press the following botton:
 
 	The icon on this switch doesn't will be changed in dependence of the representation mode.

PDF (ASTM, JCPDS, ICDD)
 
Beside the typical crystal structure data bases also other data bases exist which have been created for an identification of polycrystalline mixtures. Additionally to the determined lattice constants, the reference, some details to the experimental conditions etc. the interplanar distances, the corresponding relative intensities and the derived indexing will be given. A generally used powder diffraction data base is the PDF (Powder Diffraction File). In 1940 the support of this data base was undertaken as common task of the ASTM (American Society for Testing Materials) and the predecessor of the ACA (American Crystallographic Association). Distributed as ASTM for a long time it has been renamed in JCPDS (Joint Commitee on Powder Diifraction Standards). In these days you can order this data base under the name ICDD (International Centre for Diffraction Data). Currently it contains approximately 85000 entries and is distributed on CD-ROM. One of the famous news is, that the PDF contains now also the powder pattern of all crystal structures collected in the ICSD. From this foolows, that for a lot of substance (e.g. corundum) more than one PDF entries exist. Especially for the search procedure this occurs to a multiple listing of one and the same hit.

Very often the intensities given in the respective entries have essential disadvantages. On the one hand it is possible that the measurements have been carried out on texturized samples. On the other hand very often the peak maximum instead of the peak shape is given as relative intensity. This makes it more difficult to compare different powder patterns.
Therefore, the calculation of diffractograms using LAZY PULVERIX or PowderCell is more reliable, usually. This is especially important for the semiquantitative phase analysis, although even the newest entries of the ICDD have been completed by I/Ic-values.

Crystal structure data
 
On the one hand crystal structure data can be loaded using the structure files of the former DOS version of PowderCell (*.cel). On the other hand you are able to import structure data from ICSD or in the SHELX-format. But PowderCell also contains a worksheet where you can input or edit the data manually.
It was one of the most important intention of us to minimize the necessary data for a complete generation of the unit cell so far as possible, so that also non-crystallographers are able to use the program. Therefore, for the processing of a structure only the following data must be entered in the worksheet structure data:

 	the lattice constants: the length of the basis vectors a, b, c and the angles a, b, g ,
 	the space-group number and the setting number, 
 	all atoms describing the asymmetric unit (identifier, atomic number, „ionization" state, Wyckoff position (optional; will be determined automatically by PowderCell),  atomic coordinates, substitution and occupation (optional; per default 1), temperature factor (optional; per default 0)).
 
As you can see from the last item, you may consider isotropic temperture factors as well as fractional occupations or substitutions within PowderCell. 

Import of structure data
 
Besides the loading of own simple structure files (*.cel-files used in the former DOS version of PowderCell) the program is able to import data exported by the Inorganic Crystal Structure Database ICSD. Additionally, it has been recommended to be able to import SHELX-data because even the program SHELX is used very intensively in the structure analysis of single crystals. Until now the import of CIF formated structure data is not supported yet.

The import of the structure data is very easy using File Þ Load in the main menue. In PowderCell an additional import function doesn't exist. In the Load dialog only the correct file format must be selected to call the respective import procedure.

Import of *.cel files

PowderCell is able to import the structure files of the former DOS version. These are characterized by a simple format and can be created or edited in an usual ASCII editor. Because of the simplicity of the data structure it shall be described in the following. Sometimes it is easier to create structure file on this way than in the worksheet of the dialog box structure data. However, at first you should try to do this there because it is very comfortable e.g. by the consideration of Wyckoff letters. 

As example of an ASCII-formatted structure file (DOS) that of corundum is shown:
 
cell	4.76095	4.76095	12.9962	90	90	120	
Al	13	0	0	0.35215	1	0.14	
O	8	0.6937	0	0.25	1	0.22	
rgnr	167	
corundum, hexagonal axes
 
fundamental rules for editing *.cel-files

1.	The number of blanks between the several data is indefinitely, i.e. the files are free formatted. Only one exception exists: Please notice that the identifier is defined by a string given by the first 4 characters of a line. That means that the atomic number must be given later. But the 4 characters must be characters and not tabulars, i.e. the second atom in the given example is oxygene described by an „O“ followed by 3 or more spaces. This identifier which is longer than 4 characters will be ignored with the beginning of the fifth character.
2.	The comma as decimal point or as separation of the several data is not allowed.
3.	Empty lines are not allowed, only if they are inserted in the comment part after the last data line with the key-word rgnr.
4.	It will be recommended to define decimal numbers up to the fourth digit. Irrational numbers (e.g. 1/3 or 1/6) should be rounded-off (0.3333 or 0.1667). The input as fraction hasn't been supported. Only in the worksheet structure data this is possible but also there the entered data will be converted during the data storage into decimal numbers.
5.	The structure-data file must have the extension „.cel“.
 
Some lines of the structure file are commented using keywords. This simplifies the orientation within the file. However, for the atoms identifier will be used. This enables a simple description of all symmetry-equivalent atoms in the structure representation or supports the measurements of atomic distances and bonding or torsion angles. 
The first line starts with the keyword cell. This points out, that the following data are the lattice constants.  The length of basis vectors must be given in Å and the angles in degrees. In the next lines the atomic data will be given.  For every atom an own line will be defined. Each line starts with the identifier. Then the atomic number, the relative atomic coordinates, the SOF (multiplied substitution and occupation factor) and the isotropic temperature factor follow. If one find no hint in the structure report regarding substitution or occupation, but isotropic temperature factors are given you must set the default for the SOF (i.e. 1). Then you can enter the temperature factors (see the given example). Usually both are absent and you will find in the *.cel-files only the atomic coordinates as last data in each line.
If you must consider substitutions the first atom defines the identifier and the atomic coordinates for all substitutional atoms, equally. The substitutional atoms are described only by the atomic number and the degree of substitution. PowderCell recognizes an substitution if the identifier contains of 4 blanks. All substituents must be declared one after another. That means that these lines will be characterized by only three quantities in maximum: the atomic number, the substitution (possibly multiplied with a occupation of the given position) and a possible isotropic temperature factor. 
In the last column the space-group number and the setting number  must be defined. It starts with the keyword rgnr. The setting number is optional, if the first will be described. 
Please notice: There are exist differences in the definition of the monoclinic settings between the former DOS and the present Windows version ! (see: settings)
 
The further lines can be used for comments or remarks.

Import of ICSD files
 
PowderCell enables the import of structure data exported by the ICSD (Inorganic Crystal Structure Database). The ICSD is distributed together with a DOS program (Retrieve) for a comfortable use of the data base. This program is able to export the originary binary records as ASCII data. On the one hand it is possible to export the records into a file with the extension *.cry. Until now this is not yet readable for PowderCell. On the other hand Retrieve is able to "print" into a file. Per default the extension is defined as *.txt. For this data structure a import filter has been programmed and implemented into PowderCell. 
In the *.txt-files you will find a lot of keywords analyzed by the program. Because of the variation of data structure enabled by Retrieve also for the export, it is very possible that in rare cases the file is not readable. Please check in such cases the data structure of the exported file.
In contrast to the *.cel-files you can export the atomic coordinates as fractions or decimal numbers.
For the determination of the space-group symmetry the space-group number will be analyzed (the number in brackets behind the Hermann-Mauguin symbol). If more than one setting are defined for this space-group type the Hermann-Mauguin symbol will be analyzed additionally. 

#remark: Despite of the variety of settings considered it is very possible that a structure description uses another symmetry arrangement. This can't be processed in PowderCell directly without changes. To give you the chance to edit the unusual structure description it will be imported with some restrictions. If there the setting number can't be assigned correctly using the data file icsdkonv.dat, it will be defined as first setting. However, this has the disadvantage that the monoclinic angle will be defined as b. So it is conceivable that you must enter the correct monoclinic angle after the adaption of the space-group symmetry, manually.
Furthermore we found a lot of records where wrong space-group numbers will be given. As example gypsum can be cited. In former versions of the ICSD e.g. the space group I 1 2/c 1 will be described by the space-group number 13 but in IT this will be defined as 15 (of course PowderCell uses 15, too (setting number is 14)). Therefore, of course also the setting can't be assigned correctly but in the dialog box structure data both the space-group and the setting number can be corrected manually without problems. For the assignment of Hermann-Mauguin symbol and setting number used in PowderCell you will find the necessary correlation in the help screen standard and non-conventional settings:monoclinic; orthorhombic.
By the way, in the cited ICSD edition 345 structures will be found during the search for the space-group number 13 representing a primitive lattice. However, there will be given approximately 80 (!) structure descriptions using a face or body centred lattice but usually these are defined by space-group number 15. 
We found out that in the present version this mistake described above has been corrected except of a single structure description. 
 
However, we detected some problems during the detailed use of exported ICSD data. 
Especially in the german Windows version the decimal point is per default defined as comma. Therefore in former days we recommended to change this setting in the Windows control panel. In the meantime we adapted the import filter in this manner that it is able to read both the point and the comma.
 
The second problem detected is caused by Retrieve itselfs. If the standard deviation is activated, sometimes the output string is to long for the defined column length. That means it will be broken and continued in the next line. This disapproves PowderCell and it will be shown a message, that it isn't possible to import the present file. Therefore we recommend to deactivate the output of the standard deviation, especially because the program can't use it. In principal it is always possible to edit the exported data manually. Please notice, that all data corresponding to one and the same atomic position must be given in a single line!
 
For structures described by relative huge lattice constants (length of basis vectors) sometimes the atomic coordinates will be given up to five digits. In this case you will find only one space between the coordinates of a position. If there signs orrur PowderCell is not able to split the coordinates, respectively and shows a message again, that it isn't possible to import the present file. Therefore please think for the necessary space between the separate data which used by the program. Again it is possible to edit this data manually using an arbitrary ASCII editor.

Import of SHELX data
 
The import of SHELX files is more complicate than that of ICSD data. The reason is that SHELX describes the crystal structures using a listing of general positions. However, there is no relation to a space-group symmetry never mind a space-group number or at least the Hermann-Mauguin symbol. The advantage is that on this way the definition of an arbitrary symmetry arrangement is possible. But possibly this is also a disadvantage because the relation to the IT may be lost.
Therefore, for a data conversion one needs all general positions of all settings used in the program. These must be compared with that listed in the SHELX file. For the comparison PowderCell uses the file slx.dat containing these extensive data. However, there a further problem results, shortly described in the end of the help screen generation of atomic positions: not all general positions described by their coordinates in the IT will be generated identically in PowderCell. There no rules exist which position represents which lattice point. Therefore, for centred lattices the positions of all lattice points must be considered during the comparison of data. Caused by the SHELX data structure, however, it is not necessary to consider that position resulting by the generation of an inversion center located in origin.

Please notice, that different versions of SHELX produce sometimes different file entries. So we got some hints from users, that apparently the program doesn't import the SHELX data correctly. The reasons of this problems were always, that we support not only all used keywords, because we don't know the complete syntax. If you observe such problems please contact us.

Export of structure data
 
To exchange structure data with other programs it has been recommended to export the structure data into a format readable by these programs. It would be very nice e.g. to be able to export SHELX files, but there are requested additional information. Unfortunately, we don't have a description of the full format. On the other hand, it seems to be very "flexible". Therefore we don't support the SHELX export, yet. 
In contrast to that, PowderCell is able to export the imported ICSD files as *.cel files.

Regarding to the crystal structure determination from powder data we use the full pattern refinement program BGMN. This is a special kind of Rietveld program with innovative formulations for the description of reflection profiles and for the refinement itself.
Because PowderCell has been developed especially for these problems with wich we are confronted in BAM, the export possibillities are directed to the software used in our laboratories. If you have suggestions for an implementation of further export formats, we need at least an exact description of this.

Unit cell volumen
 
The unit-cell volume will be used in PowderCell e.g. to determine „absolute“ and comparable intensities referring to different chemical phases or the X-ray density. The value itself will be shown in the dialog box structure data because the calculation of the volume refers to the lattice constants which can be edited there. 
For the theoretical astimation of the real diffraction power of a given crystal structure the calculated structure factor will be divided by the volume of the unit cell. For the different crystal symmetries special formulation can be given but for reasons of simplification PowderCell uses a general equation :

 

PowderCell 
programmed by Werner Kraus & Gert Nolze (BAM Berlin)
© subgroups derived by Ulrich Müller (Gh Kassel)

General remarks
 	aim of the program, used indexing etc.
 	crystal structure generation and manipulation tools
 	reflection intensity 
 	used refinement procedures
  
How to use
 	installation of the program
 	input of structure data
 	crystal structure representation 
 	powder pattern simulation 
 	refinement

 introduction of the program 
 used indexing 

 structure generation
 used crystal symmetry file
 Wyckhoff position identification 
 substitution and occupation

 atomic scattering factor 
 anomaleous dispersion
 temperature factor
 preferred orientation 
 background 
 integral intensity

 installation
 hard- and software requirements

 switch ON/OFF
 edit representation options (radii, bonds etc.)
 structure info (lengths and angles)
 structure projection 
 show more than a single unit cell
 choose transformation or set subgroups 

 load experiment
 edit refinement parameters 
 start refinement
 see refinement results

 switch ON/OFF 
 edit experimental conditions 
 see refinement parameters
 view reflection listing

Dipl.-Krist. Dr. Gert Nolze 
 
Federal Institute for Materials Research and Testing 
Lab. BAM-V.13:  X-Ray Diffraction and Microanalysis
Unter den Eichen 87
D — 12205 Berlin
Telephone:  +49 - 30 - 8104 3109
Fax:             +49 - 30 - 8104 3187
 
e-mail:   gert.nolze@bam.de 

Dipl.-Ing. Werner Kraus
 
Federal Institute for Materials Research and Testing
Lab. BAM-I.33:  X-Ray Structure and Phase Analysis
Rudower Chaussee 5
D — 12489 Berlin
Telephone: +49 - 30 - 6392 5845
Fax:            +49 - 30 - 6392 5787
 
e-mail:   w.kraus@bam.de 

Prof. Dr. Ulrich Müller
 
University Kassel
Department 19, Biology/Chemistry
Heinrich-Plett-Straße 40
D — 34132 Kassel
Telephone:  +49 - 561 - 804 4425
Fax:                +49 - 561 - 804 4466 

System requirements
 
PowderCell has been developed under Windows 3.1 using Borlands Delphi 1. However, the 16-bit program also runs under Windows 95, Windows NT or OS/2. Unfortunately the graphic procedures of Borlands PASCAL compiler are not very efficient. Therefore we recommend a recent powerful computer, i.e. the CPU should be at least a 486 DX. But only with a recent Pentium you will have fun to work with PowderCell. We checked the program on computers only which had more than 8 MByte as main memory. So we are not able to say anything about computers with lower main memory.  
Because of the fact that the transformation of the source code into Delphi2 (generates 32 bit code) was more complicate than we thought, we distribute only a 16 bit version, currently. However, it is planned to develop a 32 bit version, especially because the graphic procedures used in Delphi2 shall be up to 10 times faster.
 
known hardware and software problems:
 
1.	On some computers we were confronted with problems in the use of graphic boards. Almost certainly they are caused by the graphic drivers made by the manufacturer. We located bugs in the Windows-own colour dialog. The selected colours don't will be activated in the structure drawing. Unfortunately, we couldn't delimit this problem.Caused by the same reason, sometimes the chosen colour will be given wrong in the table of atoms and ions (representation options) but in structure representation they are shown correctly.
 
2.	The conversion of ICSD data are sometimes problematically. A more detailed description you will find in the help screen ICSD import *.txt.
 
3.	Sometimes problems occur regarding the ASCII export of data (e.g.POV-Ray, powder pattern data). The reason is, that as standard setting in Windows the comma will be used as decimal point. We recommend to change this in the Windows system panel, permanently.

Installation
 
The installation of PowderCell is'n t a really one. You must only copy the following files in one and the same directory:

  1.	pcw.exe   (program file)
  2.	pcwspgr.dat    (space group symmetry)
  3.	asfpcw.dat   (coefficients for the calculation of the atomic scattering factors, radii, relative mass etc.)
  4.	pcaac.dat   (atomic attenuation coefficients)
  5. 	sub_i.dat, sub_iia.dat, sub_iib.dat, sub_iic.dat   (subgroup data)
  6.	sup_i.dat, sup_iia.dat, sup_iib.dat, sup_iic.dat   (supergroup data, not complete)
  7.	icsdkonv.dat, slx.dat    (ICSD and SHELX import data)
  8.	pcwg.hlp, nacl.avi, batio3_k.bmp   (english help)
  9.	pcw.cfg    (configuration file; optional; will be created if the program starts)
10.	orient.dat    (preferred orientation file; optional; will be created if the program needs it)
11.	bitmap files (icons) for the switches

Usually, you will get the program in a compressed form, e.g. pcw.zip or as selfextracted file pcwbeta.exe. 
If you got a zipped file you must decompress it using the Norton Commander or the program pkunzip.exe. How you must handle the NortonCommander to unzip a compressed file, you can read in the manual or in the help file of the program. If you want to use pkunzip you must enter the following command:

	pkunzip pcw.zip

In this case the files pkunzip.exe and pcw.zip should be copied in the wished directory before. However, there are other possibilities, too. About this you will be informed if you enter pkunzip without any additional information.
In principal we recommend to use the point as decimal point but not the comma. This setting you can vary in the Windows system panel. This is important if you like to export data (PovRay files etc.).

References

# [1]	Allmann, R.: Röntgen-Pulverdiffraktometrie. Clausthaler Tektonische Hefte 29, Köln: Verlag Sven von Loga, 1994
# [2]	Caglioti, G., Paoletti, A. & Ricci, F.P.: Choice of collimators for a crystal spectrometer for neutron diffraction. Nucl. Instrum. 3(1958), 223-228  
# [3]	Cromer, D.T.: Calculation of anomalous scattering factors at arbitrary wavelength. J.Appl.Cryst. 16(1983), 437
# [4]	Dollase, W.A.: Correction of intensities for preferred orientation of the March model. J.Appl.Cryst. 19(1986), 267-272 
# [5]	Hahn, T.: International Tables for X-ray Crystallography, Vol. A. Kynoch Press Birmingham, 1983 
# [6]	Izumi, F.: User's manual: Rietveld Analysis System. publ. by: Science and TechnologyAgency, National Institute for Research in Inorganic Materials,1987, 1-1 Namiki, Tsukuba, Ibaraki 305, Japan
# [7]	March, A.: Mathematische Theorie der Regelung nach der Korngestalt bei affiner Deformation. Z. Kristallogr. 81(1932), 285-297
# [8]	Reck, G., Kraus, W. & Nolze, G.:  ECM-15, 1994, Dresden, Germany, München: R.Oldenburg Verlag
# [9]	Sasa, Y. & Uda, M.: Structure of stoichiometric USi2. J.Solid State Chem. 18(1976), 63-68 
# [10]	Toraya, H. & Marumo, F.: Preferred orientation correction in powder pattern-fitting. Mineral. J. 10(1981), 211-221
# [11]	Davis, B.L. & Smith, D.K.: Tables of Experimental Reference Intensity Ratios. Powder Diff. 3(1988),205-208
# [12]	Davis, B.L., Smith, D.K. & Holomany, M.A.: Tables of Experimental Reference Intensity Ratios - Tables No.2, December, 1989. Powder Diff. 4(1989),No.4,201-205
# [13]	Davis, B.L., Kath, R. & Spilde, M.: The Reference Intensity Ratio: Its Measurement and Significance. Powder Diff. 5(1990),No.2,76-78
# [14]	Hubbard, C.R. & Snyder, R.L.: RIR - Measurement and Use in Quantitative XRD. Powder Diff. 3(1988),74-77	
# [15]	Cromer, D.T. & Mann, J.B.: X-ray scattering factors computed from numerical Hartree-Fock wave functions.Acta.Cryst. A24 (1968),321-324
# [16]	Hahn, T.: International Tables for Crystallography, Vol. A- Space-group symmetry. Kluver Academic Publishers, Dordrecht/Boston/London, 1995
# [17]	Haidu, F.: Revised parameters of the analytic fits for coherent and incoherent scattered X-ray intensities of the first 36 atoms. Acta Cryst. A28(1972),250-252
# [18]	Hill, R.J.: Rietveld refinement round robin. I. Analysis of standard X-ray and neutron data for PbSO4. J. Appl. Cryst. 25(1992),589-610
# [19]	Kath, R.L., Spilde, M.N., Davis, B.L. & Smith, D.K.: Reference intensity ratio and mass absorption measurements of eleven biotites. Powder Diff. 6(1991), No.4,183-186
# [20]	Kleber, W.: Einführung in die Krsiatllographie. Verlag Technik, Berlin, 1978
# [21]	Kostov, I.: Kristallografija. Mir, Moskva, 1965
# [22]	Paufler, P. & Leuschner, D.: Kristallographische Grundbegriffe der Festkörperphysik. Reihe Mathematik und Physik, Akademie.Verlag, Berlin, 1975
# [23]	Rietveld, H.M.: A profile refinement method for nuclear and magnetic structures. J. Appl. Cryst. 2(1969),65-71
# [24]	Williams, J.H.: Phsy.Rev. 44(1933),146
# [25]	Yvon, K., Jeitschko, W. & Parthe, E.: LAZY PULVERIX - a program to calculate theoretical X-ray and neutron diffraction patterns. Beschreibung, 1.12.1977
# [26]	Massa, W.: Kristallstrukturbestimmung. Teubner Studienbücher Chemie, B. G. Teubner, Stuttgart, 1996
# [27]	Bergmann: persönlich Mitteilung, 1996
# [28]	Hahn, T. & Wondratschek, H.: Symmetry of Crystals - Introduction to International Tables for Crystallography, Vol.A. Lecture Notes provided to the summmer school at Gjuletchitsa, Bulgaria, June 1994 
# [29]	Le Page, Y.: LEPAGE - program for lattice symmetry determination. NRC of Canada, Ottawa
# [30]	Visser, J.W. & de Wolff, P.M.: Absolute intensities. (1964),Report 641.109, Technisch Physische Dienst, Delft, NL
# [31]	H. Schneiderhöhn, Einführung in die Kristallographie, Verlag Karl Alber Freiburg, 1949

Introduction
 
1. The aim of the program 

The basic idea of the program is the specific use of crystallographic know-how for an intuitive generation of structure models. Very often the success of the so-called Rietveld programs — e.g. DBW3.2S (WILES al.), LHPM8 (HILL), EDINP/ALLHKL (PAWLEY), GSAS (LARSON et al.), PREP/PROF (RIETVELD) or RIETAN (IZUMI [6]) — depends on the initial structure model. Exactly this characterizes the aim of the program. The user shall be able to manipulate in an easy way known crystal structures (translation and rotation of atoms or molecules; change, delete and insert atoms or molecules, symmetry reduction etc.) or to create a new in a relative short time only by the use of crystallographic and crystal chemical knowledge. On this way  PowderCell tries to support the structure determination. 
The criterion for the quality of the created or modified model is the calculated powder pattern. That means that one of the most important requirements is an excellent X-ray or neutron measurement on a powder of the interesting substance. On the one hand the deviation of the experimental and theoretical diffractogram has their reason in an inacceptable adaption of the model on the real strcuture. On the other hand it is very possible that the measurements includes texture components. However, the comparison with a theoretical diffractogram requires its exact calculation. Therefore the simulation must allow the variation of essential experimental parameters, e.g. used radiation, consideration of anomaleous dispersion, diffraction geometry, fixed or variable slits, different intensity corrections, consideration of doublet, background etc. 
However, not only in research it is useful. Also for the solution of problems connecting with phase identification, phase mixtures, unknown preferred orientations, the simulation of a powder pattern of a phase described exclusively by single crystal data, one can apply the different features of PowderCell. On the other hand it is very possible to employ the program within the teaching on universities or colleges. It is very simple e.g. to demonstrate the different diffraction conditions onto the resulting powder pattern or the symmetry relations between groups and subgroups.
 
2. data import
 
For the generation of structures PowderCell uses only information which are absolutely necessary. These one will find in structure reports or different data bases. The own structure-data format is quite simple and corresponds to that of the DOS version. Beside this own format also ICSD data can be imported. Because of the fact that a lot of crystallographers uses SHELX for the structure determination on single crystals PowderCell is able to import these data, too.
Generally, PowderCell enables the consideration of crystal structures described by the conventions of the International Tables for Crystallography [16]. However, not only the standard settings will consider but also the different settings existing in the monoclinic ond orthorhombic crystal system. These are caused by the definition of different cell choices or permutation of the basis vectors in the several space-group types. On the one hand this is very important for the import ICSD records, on the other hand this is the base for the generation of group-subgroup relations.
 
3. general and special features
 
The possibility to transform the different settings within the monoclinic or orthorhombic space-group types automatically simplifies the processing of structures, especially for non-crystallographers, if the data have been given in a non-conventional settings.
Of course PowderCell enables the use of statistical substitution and occupation, an important criterion for the solution of mineralogical or technical problems (alloys, mixed crystals). Additionally, the influence of the anomaleous dispersion on the powder pattern can be simulated for the considered radiations, respectively. But also the possibility to study a given preferred orientation or to use this effect for a better fit of theory and experiment can be useful sometimes.
During the development of the Windows version we were able to implement the derivation of sub- and supergroups using data from U.Müller (University Kassel, Germany). From that PowderCell is one of the first programs generating group-subgroup relations in a simple way. With its help anybody should be able to use group-subgroup relations e.g. for the derivation of structure similarities. But also the investigation of phase transitions or the decrease of  symmetry by superposition with anisotropic properties (Neumann's principle) seem to be an application field for PowderCell. The uniqueness of this feature could be made an impression on scientists who havn't an interest on powder diffraction.  
Simultaneously a Wyckoff-position identification has been introduced. During the structure manipulation this will allow the fixing of the local symmetry. On this way a definite variation of a structure can be created e.g. without a change of the stoichiometry of the substance. Unfortunately this isn't implemented yet.
In former days a lot of user asked us for the extension of the program regarding diffraction patterns of mixtures. In this way it should be possible to create reference patterns very easy. Therefore PowderCell enables to consider up to 10 structures simultaneously. However, please notice the limitations of such assumptions. Using the current version you are able to simulate mixtures using mass as well as volume fraction in %. Even the comparison with experimental powder pattern can be carried out. However, difference plot as well as R values can be shown supporting the fit of simulation and experiment.
 
4. graphic output
 
PowderCell hasn't been developed for a graphical processing or a photorealistic representation of crystal structures, urgently. However, basic tools have been implemented, e.g. rotation, translation, clipping  and zooming. It is also possible to generate more than one unit cell. A useful add-on is the description of atoms and bonds. But additionally one can use a former tool to show arbitrary interatomic distances, binding or torsion angles etc. Furthermore, additional bonds can be defined. Even on this way you are able to create a description of the structure arrangement as polyhedrons. 
The program supports sufficient export formats for the graphics output. We recommend the use of the vector format *.wmf.  For a photorealistic representation of crystal structures PowderCell is able to export POV-Ray files. This is a powerful freeware raytracing program. Please notice that the program creates PostScript files but only for the representation of powder pattern. The PostScript export has been developed in cooperation with S.Weber.
 
Beside the storage in several formats it is of course possible to use Windows clip board for a comfortable copy and paste of graphics into  reports or papers. Then also the WMF will be used.

      		 Index
#       A
absent reflections 
absorption
angular range 
anomaleous dispersion 
switch on / off 
asfpcw.dat
ASTM 
asymmetric function 
asymmetric unit 
representation of 
transformation of 
atom 
coordinates of 
identifier 
ionisation state and bonds 
number in the asymmetric unit 
number in the unit cell 
radius 
atomic scattering factors 
effective 
number 
selection of 
 
#       B 
background 
basis vectors of the lattice 
Bi-Gaussian  
bond 
angle 
edit 
length 
set colors 
set invisible 
set line type 
special 
bitmap-graphic export 
Bragg equation 
Bragg-Brentano focusing 
absorption correction
Bravais lattice 
 
#       C 
calculation after structure manipulation
clipboard 
coordinate system 
comparison between 
diffractograms 
structures
crystal 
lattice 
structure 
manipulation
symmetry 
encoding in pcwspgr.dat 
system 
hexagonal 
monoclinic 
orthorhombic 
rhombohedral 
trigonal 
    
#       D 
 
Debye-Waller factor 
anisotropic 
isotropic 
diffractogram 
enable and disable 
import
increment 
menue 
scan function 
start and end angle 
distribution functions 
doublet 
switch on / off 
 
#       E
edit the structure data 
export 
diffraction data
structure data 
extinctions 
serial 
zonal 
integral 

#       F 
 
Friedel's Law 
fundamentals 
triclinic 
FWHM 

#       G 

Gaussian profile
generator 
generation 
atomic positions 
interferences HKL
loops
geometry factor 
graphic export 
Guinier focusing 
 
#       H 
hexagonal axes 
HKL list 
Hermann-Mauguin symbol 
definition of symmetry directions 
 
#       I
I / Ic 
ICDD 
ICSD 
import 
ICSD 
SHELX 
indexing 
Miller 
Miller-Bravais 
reflections 
initial data
input data 
edit in worksheet edit initial data 
installation 
integral intensity 
convolution 
line intensity correction 
intensity 
absolute 
relative 
International Tables for Crystallography 
non-conventional settings 
interplanar spacings 
inversion centre 
ionic radii 
 
#       J
JCPDS 
 
#       K
 kinematic theory

#       L 
LaTeX 
lattice 
centred 
constants 
plane 
distances 
indexing 
representation 
points 
reciprocal 
Laue group 
Laue indexing 
laws of extinctions 
Lorentz factor 
Lorentzian profile  
  
#       M 
 manipulation
mass 
absolute 
attenuation coefficient
fraction in % 
relative
menue bar 
diffractogram 
structur 
Miller indexing 
Miller-Bravais indexing 
monochromator
multiplicity 
effective 
of atomic positions 
of planes 
of reflections 
 
#       N 
neutron diffraction 
Niggli matrix 
normalization 
of convoluted intensity
of the integral intensity 
 
#       O 
occupation factor 
 
#       P 
pcaac.dat 
pcwspgr.dat 
PDF 
phase analysis 
semiquantitative 
use 
phase angle
PLG correction 
polarization factor 
PostScript™ export 
POV-Ray™ export 
preferred orientation 
print 
content of activated windows 
problems 
profile function 
asymmetric 
symmetric 
projection 
on (hkl) 
Pseudo-Voigt function 
       
#       Q 
quantitative phase analysis 
 
#       R 
radiation 
density
selection 
radius 
atomic
ionic
reduced cell 
references 
refinement  
parameters
results
start 
reflection 
conditions 
indexing 
list 
maximum number 
maximal  H, K and L
non-symmetry-equivalent
position
type
weight 
replacement factor 
representation 
asymmetric unit 
   	basis vector directions (coordinate system) 
diffractogram 
structure 
automatic scaling
changing the degree of central projection 
cross or parallel 
mono picture 
options 
perspective 
standard projection 
stereo picture 
cell edges 
more than one unit cell 
reciprocal lattice 
RIR 
rotation angle 
calculation 
for projection on (hkl) 
rotation matrix 
multiplication of
rotation of structure 
reset 
using matrix pc.rot 

#       S 
scattering geometry 
select 
screen copy 
selection 
atoms 
bonds
molecules 
all atoms in a range 
settings 
select 
SHELX import 
slit system 
variabel or fixed 
slx.dat 
space group 
number 
rhombohedral 
settings 
type 
Save 
all rotations in pc.rot 
asymmetric unit 
diffractogram 
data
graphic 
report file
structure 
as *.wmf (metafile) 
as PostScript file 
in space group P1
subgroups 
selection 
 substitution factor 
structure 
 amplitude 
data import 
ICSD 
SHELX93 
data 
edit  
load 
save 
factor 
representation options 
supergroups
symmetry 
encoding in pcwspgr.dat 
centre 
system requirements 
 
#       T 
temperatur factor 
thermal diffuse scattering (TDS)
texture 
tool bar 
torsion angle 
transformation 
 
#       U 
unit cell 
additional space 
monoclinic axes 
number of atoms 
relative masse
rhobohedral 
save in space group P1
settings 
volume 

#       V 
vector graphics 
volumen 
fraction in % 
unit cell 

#       W 
wavelength 
Wyckoff position 
type 
 
#       X 
X-ray density 
 
#       Y
 
#       Z

Intensity — general remarks
 
Since the discovery of X-ray interferences in the beginning of this century the correct calculation of the reflection intensities has been an important task. In practice the quantitative phase analysis is one of the famous problems. Certainly, there the structure data must be known. In contrast to the large group of scientists using e.g. this standard method only a small number attend to the determination of crystal structure data.  
Both commen interest groups use the same fundamentals for a correct calculation of reflection intensities. Nevertheless the methods are quite different because of the unknowns, respectively. For the structure analysis only the relative intensities of a maximum of considerable reflections are necessary. If it would be possible to leave asside texture effects, in the quantitative phase analysis only a single reflection for each phase would be sufficed to determine the share, respectively. But from the view of a better reliability this wouldn't be acceptable. It will be shown that even single parameter as e.g. texture (better: preferred orientation) can have an essential influence on the result of a quantitative phase analaysis. 
Actually, the basic problem is to recognize which information are necessary. From the view of structure analysis only the intensity ratios must be known (and the more are available for a deconvolution the better). For this case the well-known quadratic relation between intensity and structure amplitude can be described sufficiently using the simple equation for the structure factor without further corrections. Therefore one finds the mathematical sign for proportionality between intensity and structure amplitude.
However, it should be possible to calculate powder patterns of mixtures under consideration of certain assumtions (e.g. comparable grain-size distribution, degree of imperfections, homogenization, preferred orientation etc.). Even in case of identical experimental conditions comparisons of the diffraction power of different substances are practicable, although die quantity and the character of the unknown will be underestimated by the researcher no rarely. 
For the calculation of the absolute intensity it is necessary to localize the several unknown or disturbance variables and to estimate their influence on the result. The use of an inner standard can be carried to a more reliable result, but this don't must be in each case. Also this method has its advantages, but there are limitations too.
At long last for the correction of the ideal intensity only the mass attenuation (only then problematical, if substitutions or occupations occur) and the unit-cell volume must be considered. In contrast to that the consideration of textures (preferred orientations) is more complicate.
However, please notice, that each Rietveld program must solute the same problems. The user of these programs must believe that the implemented procedures are sensible enough to seperate the different influences.
Within PowderCell you are able to process mixtures containing up to 10 different phases, with all advantages and disadvantages described above. We tried to offer you a simple tool with a common automatism in refinement. 

Powder pattern simulation
 
This concerns data which don't will be defined by external data files. They must be changed within the program. There belong:

a)	radiation (anode material, neutrons, consideration of doublet);
b)	start and end angle of the simulated powder pattern in (2q);
c) 	increment (step size in D2q);
d) 	width of reflection calculated in ;
e) 	diffraction geometry (monochromators) ;
f) 	variable or fixed slits ;
g) 	anomaleous dispersion ;
 
Some of these parameter/data must be activated using a special switch. Other data must be edited manually using the keyboard. In each case you must use the dialog box powder diffraction / experiment.

Structure factor
 
The structure factor will be calculated within PowderCell by the following equation:

in which the isotropic temperature factor will be described by 
 
	 . 
 
There k is the occupation factor, g is the substitution and f describes the atomic scattering factor. At first it will be summarized about all atoms of one and the same position (x,y,z) which substitute one another. In this case beside the part j of the substitution only the atomic scattering factor and the temperature factor vary. The sum of the product of g and f can be assumed as effective atomic scattering factor describing the position i in the structure. It is also possible to seperate this general part of the equation to minimize the processing time:

 
k is the occupation factor which describes the general occupation of all symmetry-equivalent positions of the i-th atom in the asymmetric unit. PowderCell doesn't use this equation in the given optimized form. The reason is that the input is organized as product of k and g. This is listed in structure data as SOF. However, the program correcs the atomic scattering factore directly by the temperature factor. (In the following equations this hasn't been considered, separately.)
In several cases it is useful to consider the anomaleous dispersion. However, this complicates the expression of the equation and so its calculation:

 
centrosymmetric crystals: The advantage of centrosymmetric structures is connected with the location of the symmetry center in the origin of the unit cell. Then the calculation of the structure factore will be simplified. On the one hand all sinus terms disappear for the two centrosymmetric positions because these are characterized by opposite values, on the other hand the cosinus terms have identical values. Thus it is not necessary to calculate these twice. Therefore the resulting equation has the following simple form:

 
where the upper limit of the sum is described by the half of atomic positions of the unit cell. This requires that the correct atomic positions will be choosen, i.e. only these are included which are not connected by the existing symmetry centre.
In this case the structure factor is a real number. This reduces the processing time noticeably. Pleace notice, that even in the cubic and tetragonal crystal system the second setting of an centrosymmetric structure description is defined by a symmetry centre located in the origin of the unit cell. 

remark: This simplification don't should be confused with Friedel's Law, which enables the use of the Laue groups instead of the point groups. This only occurs to a reduction of the interferences which must be considered!

I / Ic and RIR
 
For a better estimation of the scattering power of the individual structures the International Centre for Diffraction Data (ICDD) has been tried to introduce standardizeable data within the powder data bases for a long time. On this way it should be possible to carry out a semiquantitative phase analysis. This idea follows a recommendation of VISSER and DE WOLFF [30] in 1964. Because the reference is defined as the most intensive reflection of corundum the value has been called as  I / Ic. Using this, the ratio between the most intensive reflections of the interesting substance and that of the corundum shall be described. To permit also other standards the so-called RIR value (Reference Intensity Ratio) has been introduced. In practice, this is defined by an 1:1 mixture of both phases where the RIR is given by the intensity ratio of the most intensive reflections. However, there are important requirements on the sample preparation [13,14]:
 
a)	the preparation should be free of preferred orientation and micro extinction
b)	within the full 2q range the illuminated volume should be equivalent 
c)	correction of polarization effects of a used monochromator
d)	use of integral intensities.

In reacent years listings of RIR have been published [11,12] irregularly. This shall clarify the importance of these values regarding to a semiquantitative phase analysis (e.g. [13,14]). However, the reliability of the RIR determined by experimental data is relative small which shall be demonstrated using the following table. There different RIR for natural or synthetic minerals are confronted each other [12]:

mineral	place	formula	HKL	RIR
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
Quartz	Hot Springs, AR	SiO2 n	101	2.7i	
Quartz	Hot Springs, AR	SiO2 n	101	3.0p	
Quartz	Black Hills Peg.	SiO2 n	101	5.1i	
Quartz	Hot Springs, AR	SiO2 n	101	4.7i	
Quartz	synthetic	SiO2 n	101	3.6i	
Quartz	Dolni Bory, CSSR	SiO2 n	101	4.5i	
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
Halite	synthetic	NaCl n	200	4.4i	
Halite	Timbalier Bay, LA	NaCl n	200	3.4i
¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
Calcite	synthetic	CaCO3 n	014	4.2i	
Calcite	Black Hills, SD	CaCO3 n	014	4.0i	
Calcite	Cherokee Co, KS	CaCO3 n	014	2.6i	
Calcite	Creel, Chihua, Mex	CaCO3 n	014	2.2p	
Calcite	synthetic	CaCO3 n	014	2.0i	
Calcite	Bitola, Yugoslav.	CaCO3 n	014	2.4i	
 
Even synthetic Calcit has been characterized by significantly different RIR, although in both cases the RIR have been determined using the integral intensities (i) but not the peak maximum (p).
 
Therefore the large interest for a comfortable calculation of the RIR is understandable, especially then if experimental data don't exist. Thus, some programs really calculate I / Ic values (eg MICRO-POWDER [1]). For quartz an I / Ic=4.2 will be given, whereas for Calcite a value of 3.2 will be calculated. However, it is clear that the user should be considered the limitations of such a formulation. 

asfpcw.dat 
 
In this file all constants are contained which are necessary for the calculation of the atomic scattering factors and their anomaleous dispersion correction.

 

Atomic scattering factor
 
X-ray diffraction: Atoms cannot be seen as mathematical points because their size is comparable with the wavelength used. Furthermore, even the electrons can exceed the assumed atomic or ionic radii with the result, that they must be described by different scattered phases. This difference in scattered phase occurs to a partial interference and a real loss regarding to the scattered amplitude of the respective atom. This results to the well-known dependence of efficience of the scattering  from the Bragg angle. It will described in dependece on sinq/l.
The efficience between the scattered atoms will be expressed by the atomic scattering factor. It can be understood as ratio between the scattered amplitude of a stationary atom  to that of a single electron. Therefore, in case of q = 0° the atomic scattering factor f corresponds to the periodic number and decreases very fast with increasing sinq/l. Thus, for typical light atom structures (eg organic compounds) it is not worthwhile to measure a diffractogram at higher angles 2q (Mo Ka: up to 50°, Cu Ka: up to 120°), usually . 
If thermal motions of the atoms will be suspected caused by thermal influences, the atomic scattering factor must be corrected by the so-called temperature factor. Therefore, the  effective atomic scattering factor depends not only from the temperature but in turn of the respective atom itself.
For the calculation of the atomic scattering factor an approximation formula will be used. It is recommended within the IT. There also corresponding values for special sinq/l will be given. However, for the analytic form the necessary coefficient are listed which enables the calculation of an arbitrary sinq/l. This equation has been implemented in PowderCell. Though the coefficients havn't been taken from the IT, but kindly have been calculated by Dr. Wendschuh-Josties using FPRIME [3]. Thus the newest level of knowledge could be considered. The atomic scattering factor f will be calculated by the following analytical expansion:
	  
There a, b and c represent element specific coefficients.

remark:	Please notice, that the structure amplitude of a given reflection doesn't vary by the change of the radiation (anode materials for X-ray generation), used. This corresponds with the Bragg equation, where sinq/l=1/2d (but this is not connected with the atomic scattering factor, causally). However, if one considers the anomaleous dispersion the structure amplitudes of non-centrosymmetrical reflections show a small different shift in correspondence to the used radiation.
 
Because of the fact that X-rays interact with the electron shell, several ionic states of the elements will be considered within PowderCell. Since the sceleton electrons within the lower electron shells show a higher electron density than the valence electrons, in practice a difference in the scattering power for medium scattering angles cannot be detected. That means, that with a good approximation it is sufficient to use always the atomic scattering factors of the neutral atoms [26]. Within PowderCell the use of the coefficients of the atomic scattering factor is connected with the chosen radii. From this it follows, that only for the defined radii the corresponding coefficients have been implemented.
 
neutron diffraction: In comparison to the X-rays, for the elastic coherent neutron diffraction no dependence in sinq/l appears. This simplifies the calculation of the reflection intensities, considerably. Merely if magnetic moments would be evaluated a sinq/l-dependence would be resulted. Currently this hasn't been considered yet, on the one hand because of a non-existing interest of the user, on the other hand because of our missing knowledge.

Polarization factor
 
Due to the interaction between the X-radiation and the atoms their electrons will be stimulated to an oscillation caused by the electro-magnetic alternating field. The oscillations are located within a plane perpendicular to the beam direction. However, the electrons don't scatter in their oscillation direction. So it follows, that the share of the scattering vector within the plane defined by the incident and diffracted beam disappears, if 2q comes closer to 90°. For non-polarized X-radiation the polarization factor P is defined as follows [1,21,26]: 

It is independent from the equipment.
If one uses a graphite monochromator the incident radiation is prepolarized by this effect, insignificantly. Usually, this influence will be corrected by an experimantally determined  factor K:

Because of the insignificant degree of polarization very often this effect can be ignored (for Mo Ka radiation the deviation will be given smaller than 1% [26]). Therefore it hasn't been considered within PowderCell.

Lorentz factor
 
In places the Lorentz factor is defined variously. Therefore, from case to case the given dependences must be analysed exactly.
ALLMANN defines no additional geometry factor, but includes this in the Lorentz factor [1]. In contrast to that, KLEBER extracts this and defines the rest as Lorentz factor. PowderCell follows this recommendation [21]. 
In correspondence to the Bragg equation, within the Lorentz factor it will be considered that an interference does not appear on an exact angle position, but have a certain blurredness (no reciprocal lattice point). Using the Ewald construction it means, that in dependence on the Bragg angle the beam runs more slowly or faster through the respective reciprocal lattice points. In other words, a short scattering vector passes the reflection position in a shorter time than a longer, which penetrates the Ewald sphere nearly tangentially. This can be corrected using the  following term:

@LOR_FACT.BMP@ 

Geometry factor
 
For a powder a further correction must be carried out. There the normals of these lattice planes, which are suitable for an interference, form a circular cone arround the incident direction of the X-radiation. The half aperture angle is defined by 90°-qHKL. The scattered X-rays describe a circular cone arround the incident direction, too, but the half aperture angle is now 180°-2q. The sections of the cones with the unit sphere represent circles, usually described by different diameters. The bigger the circle characterized by the normal direction of the lattice plane (identically to a small q), the bigger is the probability that a grain is located in reflection condition. This presupposes a homogeneous orientation distribution of the grains (coherent region) within the powder. The probability is proportional to cosq. Since the intensity will be measured only on one selected position of the scattered cone the detected intensity is invers proportional to the size of the intersected circles, i.e. proportional to 1/sin(180°-2q)=1/sin2q.
Therefore, the geometry factor is described by [21]:

 

PLG correction
 
Usually the polarization, Lorentz and geometry factors will be combined to a so-called PLG correction (sometimes it is described as LPG or as LP correction [1]). Therefore, very often complete PLG correction termes are listed in dependence of several focusing conditions or different constructions of diffractometers [25]. In PowderCell the identical corrections as in LAZY PULVERIX will be used:
 
Debye-Scherrer and Bragg-Brentano:

Guinier (kinematic theory):

Guinier (dynamic theory):

where b is the angle between the normal direction of the sample surface and the incident beam (within the program fixed on 45°) and a is the diffraction angle of the monochromator used (within the program the use of quartz reflection 011 has been assumed). The fixed d001=3.343Å and the eligible radiation determines a. The used interference, .ie. really a, as well as b can be varied. However, the corresponding data must be edited manually within the configuration file pcw.cfg using an arbitrary ASCII editor.
Generally, a and b depend from the special geometry of the Guinier camera used as well as from the interplanar spacings of the diffracted lattice planes of the monochromator crystal. The given equations are valid for Guinier cameras after Guinier—de Wolf or Guinier—Haegg and for the use of quartz as monochromator crystal [25].

Temperature factor
 
Very often for a correct intensity calculation of a reflection it is indispensible to relativize uncertainly determined locations of atoms using a so-called temperature factor (Debye-Waller factor). This describes a blurr in location (oscillation or vibration) caused by an energy supply (e.g. heat), whereas the given coordinates suggest an exact position within the unit cell. The temperature factors depend from the atomic number (i.e. really from the mass) because there will be assumed silently that for a low atomic mass and a constant energy supply  higher oscillation amplitudes result. Therefore, the given coordinates always represent the equilibrium positions corresponding to that at the absolute zero mark (if further phase transition will be ignored). With increasing temperature the intensity of reflections should be decrease. Due to the derivation of the temperature factor an additional phase shift i.e. intensity attenuation occurs caused by increasing average oscillation amplitudes u and by decreasing interplanar distances dhkl. The atomic scattering factors will be attenuated by thermal oszillation, additionally to their usual reduction caused by the dependence of the scattered angle [26].
 
In practice different formulations for an estimation of the temperature influence will be used. For a lot of materials characterized by a melting point which is significantly higher than room temperature and where hardly any light elements exist, thermal oscillation don't will be considered during a structure refinement. In this case the correction term will be ignored by PowderCell. This represents the standard setting (coefficient is fixed as zero). 
In former days usually isotropic temperature factors have been determined, i.e. one assumes a spherical deviation of the given location. 

 
# The square of the oscillation amplitude u defines the isotropic temperature factor U  so that it is possible to write:

		 
# Within the Debye-Waller factor B the factor 8p² will be considered, additionally. Thus the equation can be given in the following well-known form:

 
This expression will be considered in PowderCell. 
 
# Usually, current structure refinements use anisotropic temperature factors Uij. These represents a tensor generally described by 6 independent components Uij. The anisotropic oscillation will be clarified by an ellipsoide, where the main axes will be defined as U1, U2 and U3. Using orthogonal coordinate system these correspond to the main diagonal elements U11, U22 and U33. However, the mixed Uij characterize the location to the reciprocal basis vectors. Therefore, in oblique-angled coordinate systems the Uij contain contributions to the length of the main axes. The Uij will be given in Å² and have a range of valueas from 0.005 up to 0.02 for heavy atoms in inorganic crystal structures. For organic compounds values between 0.02 and 0.06 are acceptable. In rare cases also values between 0.1...0.2 have been described.
Sometimes it is possible to find structure date where bij values instead of Uij will be given. Beside the factor 2p² the bij contain the lentgh of the reciprocal axes, additionally. The disadvantage is that such data are structure dependent so that the temperature factors of different structures cannot be confronted. b11 is defined as:

		 
 
The consideration of anisotropic temperature factors hasn't been implemented directly.  
# However, in several journals so-called equivalent isotropic temperature factors Ueq will be given derived arithmetically from the anisotropic temperature factors. PowderCell substitutes the isotropic temperature factor by Ueq automatically, if anisotropic coefficients will be imported by ICSD data.

Some hints for mistakes in temperature factors [26]:

1.	One-sided orientation of the anisotropy without any correspondence to the existing binding conditions indicates an inadequate extinction correction of a anisotropic crystal design (habitus) and a high value of attenuation coefficient.
2.	Individual temperature factors are to small or negative. Perhaps on this position a heavy atom is located.
3.	Individual temperature factores values are to high. Either there is a lighter or no atom or the position is only partially occupied.
4.	The oscillation ellipsoide is presented by a physically nonsensical form. 

Thermal Diffuse Scattering (TDS)
 
The X-ray scattering will be influenced by the thermal motion of the atoms. This will be corrected using the so-called temperature factor. However, it is known that also an additional term caused by long-range correlated oscillation (lattice oscillations). The result of this oscillations is a diffuse background characterized on the reflection position by a small-intensity, broad, but pointed maxima. Although the TDS at high Bragg angles makes up to 25% of the netto intensity (but there is no influence on the atomic position - only on the size of the temperature factors) it will be neglected. 

Radiations
 
PowderCell supports the most important anode materials used in X-ray tubes. Therefore the necessary coefficients for the calculation of the corresponding scattering factors has been provided. However, PowderCell enables the consideration of corrections, except in some cases. The following table shows an overview.

				                        mass      anomaleous
       anode	     Ka1	     Ka2	Ia2/Ia1       attenuation   dispersion
 ¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
	Cr	2.28970 (2)	2.293606 (3)	  0.515	yes	yes
	Mn	2.101820 (9)	2.10578 (2)	  0.519	yes	yes
	Fe	1.936042 (9)	1.939980 (9)	  0.500	yes	yes
	Co	1.788965 (9)	1.792850 (9)	  0.497	yes	yes
	Ni	1.657910 (8)	1.661747 (8)	  0.495	yes	yes
	Cu	1.540562 (2)	1.544390 (2)	  0.497	yes	yes
	Zr	0.78593 (1)	0.79015 (1)	  0.502	no	yes
	Nb	0.74620 (1)	0.75044 (1)	  0.498	no	yes
	Mo	0.709300 (1)	0.713590 (6)	  0.499	yes	yes
	Ag	0.5594075 (6)	0.563798 (4)	  0.499	yes	yes
 ¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
		     La1	     La2	Ia2/Ia1
 ¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾
	W	   1.4767	    1.4877	  0.115	no	yes
 
In this process it is important to know, that only the scattering factors may be calculated for atoms up to the atomic number of 104. The necessary data have been edited in the file asfpcw.dat. Please notice, that the used data are not identical with that given in IT. Rather they have been calculated by Dr. Wendschuh-Josties using the program FPRIME [3,15]. 
Regarding the mass attenuation some restrictions exist. One the one hand only atoms with the atomic number up to 96 will be considered, although in most cases this will be sufficient. On the other hand these data are not available for all X-radiations supplied in PowderCell. 
For the calculation of the mass attenuation the total photon interaction cross section (given in [barns/atom] / 10, IT, Vol.C, Tab. 4.2.4.2.) will be used. The file pcaac.dat, which originally contained the necessary data for the atomic numbers up to 84, has been adopted also from RIETAN [6] initially. The data have been extended up to 96 using the values given in IT. Therefore the data differ for elements with atomic number up to 84, regarding to the IT. 
 
Initially, only the consideration of X-ray diffraction has been intended. Recently also the calculation of neutron patterns is available. There a fixed wave length will be used derived from the condition for the given correction values. Thus the velocity of the neutrons is given as 2200 m/s. So a wave length of 1.7982 Å results.

Diffraction geometry
 
PowderCell only considers focusing powder diffraction methods. Because of their special arrangement these methods direct a divergent beam in such way, that in ideal case all rays meet in a point or a vertical line within the detector plane. The advantage of the use of a divergent beam is the significantly higher intensity gain at acceptable reflection width in a powder pattern.
The optimale line sharpness affects positively on the evaluation of the diffractograms.
The influence of these geometrical relations (diffractometer construction etc.) on the observed intensity of the diffraction pattern will be described by the geometry factor G.
In PowderCell the corresponding correction term used additionally includes the Lorentz- and polarization correction and will be called PLG correction. For the different diffraction geometries the corresponding correction term can be chosen, respectively. Furthermore it is possible to neglect any correction, so that an external processing can be attached using own correction models.
Altogether the following diffraction geometries have been considered till now:

a)	Bragg-Brentano (identical with Debye-Scherrer),
b)	Guinier (in which the diffraction will be described kinematically),
c)	Guinier (in which the diffraction will be described dynamically) and
d)	none.
 
Using the Guinier correction the diffraction angle of the monochromator can be varied, respectively.
 
For neutron diffraction a standard function has been implemented. 
 
The diffraction geometry can be chosen in the window powder diffraction / experiment.

Anomaleous dispersion
 
The consideration of the anomaleous dispersion on the diffraction pattern is really meaningful in relative rare cases. Usually the influence is very small. Therefore it should be neglected. Otherwise the running time can be doubled approximately, because for the generation of the HKL not the centro-symmetrical  Laue group but perhaps the non-centro-symmetrical point group must be considered (see also structure factor calculation). Only if the comparison of experimental results and simulated diffraction pattern shows significant deviations it can be suspected that Friedel's law is invalid in first approximation. In such cases we recommend to consider the anomaleous dispersion. The anomaleos dispersion influences the intensity, if the energy of the used X-radiation is nearby the absorption edge of an element containing in the processed crystal structure.
 
The additional scattering contributions become complex and will be described by a real (Df') and an imaginary (Df'') component. These will be summarized to the atomic scattering factor. Df' becomes positive but also negative, whereas Df'' is always positive, i.e. using the anomaleous scattering always a small phase angle will be added.

 
For an optimal correction of the anomaleous dispersion within PowderCell the constants Df' and Df'' have been calculated by Dr. Wendschuh-Josties using the program FPRIME [3,15]. He put it at our disposal. For all within the program per default considered radiations they are fixed. However, if you want to use own radiations the consideration of an anomaleous dispersion is impossible. This becomes important, if also infinitissimal deviations of considered radiations will be entered because this is also a "strange" wavelength. 

Symmetrical profil functions
 
PowderCell is able to convolute the calculated line intensities using different profile functions. The shape of these curves is always 1. The used functions are well-known in principle. They will be used in a lot of different programs (e.g. Rietveld refinement etc.). Usually there the symmetrical profile functions will be mixed by an asymmetric function. This presupposes a certain experience and intuition, because in a lot of cases one observes an overlapping of reflections which must be weighted subjectively. The software requires certain initial data for the refinement procedure. These values must be determined using the experimental data.
 
The following symmetrical profile functions have been implemented in PowderCell and may be selected very easy:
 
Gaussian 

where h describes the maximum intensity (line or integral intensity) and b is connected with the full width at half maximum FWHM: b = 2.772 / FWHM²
 
Lorentzian 

Pseudo-Voigt

Cell transformation
 
In PowderCell a lot of non-conventional settings are considered, what is mainly caused by the idea for an implementation of subgroups and supergroups. However, to use not only the maximal sub- or supergroups, what would be inacceptable from practical point of view, it should be possible to transform non-conventional settings into standard settings. Furthermore, the different data bases offer a lot of crystal structures, not always described in a standard setting. Therefore PowderCell shall be able to interprete this settings correctly.
Thus, an automatism has been implemented, which enables at least for monoclinic, orthorhombic and rhombohedral space-group types the derivation of non-conventional settings in each other or in standard settings. 
The setting choice can be selected using the corresponding button in the tool bar or the main menue: Structure Þ Transform. There a selection menue will be opend, where the Hermann-Mauguin symbols or the transformation instructions will be shown as orientation.

 
For a certain number of tetragonal or cubic space-group types two different settings will be given in IT. The difference between their is a shift of the origin. Till now PowderCell is not able to transform between these settings. However, the transformation is very simple and everyone should be able to do this manually by a shift of the origin and change of the setting number. To this end a translation vector has been implemented in the dialog box structure data. To all atomic positions of the asymmetric unit this vector will be added. This is equivalent to a shift of the origin in opposite direction by the same amount. 
The same is hold for a number other space-group types, e.g. C -1 (typical representive:Albit, ICSD 63191). The additional centering vector must be considered manually. An automatic transformation is not implemented. In contrast to that the diffractogram can be calculated for C -1, because the symmetry record for this setting is included in pcwspgr.dat.

In correspondence to the chosen setting the atomic positions of the asymmetric unit and the metric of the unit cell (lattice constants) will be transformed by PowderCell. The transformation itself uses program-internal, fixed algorithms. An adaption to the recommendations of the IT  or the use of the reduced cell has not been considered yet. 

In [26] an interesting special case will be given, which is not considered in PowderCell too. It shows the importance of the understanding of the nature of space-group symmetry:

„A special case is the cubic space group P a-3 (Nr. 205, complete symbol P 21/a-3): In contrast to the cubic symmetry it is not allowed to change the axis arbitrarily in this centro-symmetrical space group. The reason is that the directions of the glide planes will be changed, respectively. If one like to use the symmetry codes of the International Tables (and the corresponding reflection conditions) for the setting P a-3, it is necessary to set the axis choice (orientation) in identical manner as in IT. Therefore, perpendiculary to a must be define a b-glide plane, perpendiculary to b must be given a c-glide plane and perpendiculary to c an a-glide plane. This can be checked using the reflection conditions. If one finds glide planes in all three directions, but with incorrect glide vectors, the axis must be transformed, respectively. Otherwise the symmetry operation for the non-conventional settings P b-3 or P c-3 must be used. 
 
# Furthermore, rhombohedral space-group types can be described using rhombohedral or hexagonal axis, alternatively. Since only one alternative setting exists, an additional selection menue is not necessary. The program transforms into the alternative setting, immediately. Even there problems occur regarding the calculation accuracy. Usually, the transformed angles in the rhombohedral axis systems will not be idendically or the angle g in the hexagonal system results to a value, which is not exactly 120°. As information a message will be shown hinting on the differences. The calculated values will be shown, addidionally. If there are further doubt, you should compare the two corresponding powder pattern. There should be no difference, neither in the additional generated reflections nor in the shape of the intensity curve.

Subgroups
 
The purpose of PowderCell is an intuitive variation or derivation of crystal structures with the aim to produce a better theoretical description of experimental powder patterns. Even there, subgroup and supergroup relations become important because they impressively demonstrate the interaction between symmetry variation and the change of scattered intensity.
Subgroups can be clearly defined as decrease of symmetry density within a crystal structure. Inevitably, this does not mean a decrease of the space-group number, what apparently will be assumed sometimes. Not seldom even the opposite will be observed.
The transition to a subgroup relatively often ends in a quite unusual setting of the unit cell, the so-called non-conventional settings. However, these can be transformed into standard settings. But sometimes this is neither desirable nor necessary [26]. Therefore PowderCell supports the calculation of the powder pattern for non-conventional settings as well as  transformation into the standard setting including the following calculation of the diffractogram. For the generation of so-called master trees of symmetry, i.e. the full set of all very possible subgroup transitions (not only maximal), always the standard settings must be used.
 
In practice the subgroups are very useful for the understanding of structural relationships e.g. for the analysis of a possible twinning. Especially for the description of crystallographic phase transitions of 2. order they play an important part, because such transition requires a corresponding group-subgroup relation.
In dependence of the character of the decrease of symmetry density it is possible to classify the subgroups. Thus one distinguishes between 3 classes of transitions indicating the reduction of symmetry [26]:

1.	translationengleiche subgroups. The translation lattice do not vary. However, individual symmetry elements will be removed. Thus the result is a new space-group symmetry with lower symmetry density but also with a lower space-group number. In IT these will be called as subgroup type I.
2.	klassengleiche subgroups. There the crystal class (i.e. the crystal system) do not vary. However, the translation lattice can be changed. On the one hand existing centering vectors will be discontinued (type IIa), on the other hand individual basis vectors becomes longer by a simple multiplication (type IIb).
3.	isomorphic subgroups. In principle these are klassengleiche subgroups. However, a special case of it, because the space-group type must be conserved (type IIc).
 
The data for the derivation of the maximal subgroups have been put us at our disposal from Prof. U. Müller. 
For the generation of subgroups PowderCell merely needs the transformation instructions for the new basis vectors and the shift of the origin of the derived unit cell. The basis vectors and the atomic positions will be transformed internally. The necessary matrices will be generated by the program, automatically. For enlarged unit cells the transformed atomic positions must be multiplied using the corresponding translations. By comparison of the calculated  X-ray density of the old and the new cell it is possible to check that all positions within the unit cell have been occupied. 
Only just the real "metamorphosis" of the structure starts by the determination of the Wyckoff-position types regarding to the new space-group type. There all atoms will be analysed exactly which have been derived from one and the same position of the asymmetric unit of the former unit cell. If one Wyckoff-position type has been found all other symmetry-equivalent atoms will be created using the generation instruction of the new space-group symmetry. Perhaps there are some individual atoms which must be analysed onesmore by the same procedure. The identification is only finished, if all atoms of the former unit cell will be described explicitely by the use of the symmetry of the new cell. In the new asymmetric unit only these atoms will be taken which correspond to the Wyckoff-position types. For every location of the old asymmetric unit this procedure will be repeated. Thus, the so-called Wyckoff splitting can be analysed on a simple manner. 

remark: So PowderCell has been used to proof the Wyckoff splitting carried out by U.Müller, manually. By the way, this was also a wonderful check of the several routines used in the program. The check includes all space-group types, given in standard setting. As reliability criterion the calculated powder pattern has been used. It is clear that independently of the describing symmetry or chosen lattice, the resulting powder pattern must be exactly the same. Of course, in dependence of the chosen symmetry several coincidental extinctions will appear. However, their intensities result to zero as long as no changes in lattice constants or atomic positions have been carried out. 

Within the program the subgroup transitions  can be processed easily. 

Supergroups
 
A space group is then a supergroup of another, if this is a subgroup of the space group. There the classification „translationengleich“ and „klassengleich“ will be preserved, so that it can be seen as a simple inversion. However, the derivation of supergroups is not so easy as for subgroups. 
HAHN & WONDRATSCHEK wrote about that [28]:
 
"Supergroups of a group G are much more difficult to survey than subgroups of G. The reason is easy to comprehend: The group G is a closed set of well-defined elements; if G is a space group these elements are its symmetry operations. The subgroups of G are space groups, plane groups, layer groups, rod groups, line groups, and finite groups. Therefore, in principle it is possible to determine all subgroups of the space group even if their number is infinite. The problem to determine all supergroups of G is much less clear. It is undefined at all as long as the set is not specified from which the elements of the supergroups shall be drawn. Such sets may the set E of all isometries, the set A of all invertible affine mappings, or even sets of more general mappings.
Even if the set is specified to which the supergroups belong the problem is much more difficult than the search for subgroups of a space group G. The supergroup of a space group in the group E may be space groups, continuous groups, or even groups which belong to neither of these kinds but are of some intermediate character. An example for such a group is the symmetry of a crystal structure projected along a non-rational direction onto a plane. The translation group of this projection is neither discrete nor continuous. Very little is known about these groups, and no classification of them seems to exist.
Therefore, the supergroup data of IT A (1983) only serve as a guide to the reader when looking for subgroups of space groups. For a space group H the entries under the heading "Minimal non-isomorphic supergroups" provide a list of those space groups G for which H is a maximal non-isomorphic subgroup of G."
 
This simple inversion has been carried out in PowderCell. In analogy to the subgroups, currently only the transition from standard settings into supergroups is possible. Of course, the supergroups will be describe exclusively in standard settings. However, in one of the next versions this procedure will be extended on arbitrary settings for the starting space group. But also then the transition describes exclusively standard settings. Unfortunately this procedure has not been finished yet.
 
For further explanations to the supergroups, see Section 2.15 of IT A (1983). 

Full Width at Half Maximum

Asymmetric profil function
 
This function describing the asymmetric intensity distribution of an interference consists of a symmetrical profile function (Gaussian, Lorentzian etc.) corrected by a special correction funktion. The superposition of both results in an asymmetrical profile function related to the well-defined Bragg angel. The mathematical description is the following: 

The use of such asymmetrical term is dependent of the experimental arrangement, the measured angle range etc. If one use a manual refinement we recommend to neglect this correction. 
Currently the program doesn't support this function. We are not dedicated of the quality of such correction because of the resulting negative intensity in one of the reflection flanks. Therefore we plan to inplement another way for the creation of a reflection asymmetry. One idea is to split the reflection profile in two seperate parts.

Preferred orientation - Texture
 
PowderCell uses multiplicities of lattice planes for an optimized calculation of diffraction pattern. The aim is the decrease of the processing time. To this end only the absolute necessary  part of reciprocal space will be considered during the generation of the reflection. The size of the considered space is defined by the point-group or Laue-group symmetry of the given crystal structure. This requires a homogeneous orientation distribution. However, usually this is not so. Very often there are compact samples or powders which cannot be prepared without texture or it cannot be neglected absolutely. 

remark: The term texture is characterized by an inhomogeneous distribution of crystal orientations of a lot of grains or crystallites (mostly more than 100000 in the considered volume). That means, that with regard to the normal direction of the sample some directions occour overproportionally. One typical example is the well-known fibre texture, where the fibre axis corresponds to a small-indexed lattice direction, mostly. Perpendicular to this direction no further preferred orientations exist. This special and simple texture will be observed on crystals with fibrous (or plated - but than the direction is described by a small-indexed reciprocal lattice vector) habit which can be directed more or less during the preparation.
 
However, an observed texture can be much more complicate than usually assumes, especially for low-symmetry crystals. Therefore, for a mathematical description one needs as much as possible complete distributions of multiplicities for different lattice planes, which must be derived from measured spatial intensity distributions. 
If one speaks about preferred orientations, one assumes a very simple texture for the given sample. This will be described by only one texture component, where its axis or normal direction corresponds to the sample normal. Usually, it is nothing known about any distribution of multiplicity, because only a single powder pattern has been measured. In maximum only the sample has been rotated around its normal for a better statistic. Then, only the existence of singular deviations between theoretical and experimental reflection intensities let suppose an occurence of texture components, which must be considered during the comparison of theoretical and experimental curve. Till now PowderCell corrects the influence of a preferred orientation on the reflection intensities by the use of an additional factor, which corresponds in a way with an effective multiplicity of lattice planes.
The derivation of the effective multiplicity is based on a simple trick. The generation of the HKL will be carried out as before, i.e. only the really necessary spatial part will be considered. However, no multiplicity factors will be used for the calculation of the real reflection intensities but the angles between the reference direction of the preferred orientation and the individual symmetry-equivalent lattice-plane normals. Therefore these directions must be determined using the generators (or symmetry matrices). Unfortunately, these matrices have been defined for the transformation of space lattice vectors, so that the reciprocal vectors must be transformed before. After the multiplication with the respective symmetry matrix the vector must be retransformatted into the reciprocal space. On this way a symmetry-equivalent reciprocal lattice vector will be generated. Using this the necessary angle to the reference direction (texture component) can be calculated. Its value characterizes the probability of the existence of this individual lattice plane parallel to the sample normal (or perpendicular to the diffraction vector). This is equivalent to the intensity weight of this plane and influences the summarized intensity for the single interference HKL. If the normal of the individual lattice plane (hkl) — derived from {hkl} — includes a small angle with the assumed direction of the preferred orientation, than its influence is comparable big. If the described angle goes to 90° the probability to find this plane parallel to the sample surface is very small. In dependence of the model used the sharpness of the preferred orientation will be described by a special factor which must be optimized. This can be done manually or automatically using the refinement capabilities of the program. 
Currently, PowderCell uses two different models for the simulation of a preferred orientation.
 
1. model
 
The first model has been used e.g. in RIETAN and was programmed by RIETVELD in a similar form too. It simulates the influence of a preferred orientation on the powder pattern using two parameters in an empirically determined gaussian deviation. The quality of this function have been analyzed extensively by SASA & UDA [9] and TORAYA & MARUMO [10].

 
The use of the calculated angle Y  between the reference direction and the normal of the individual lattice plane is dependent on the diffraction geometry.

Bragg-Brentano focusing:

 
cylindrical chamber:

 
where Y  is defined as the acute angle between the normal of the diffracting lattice plane (hkl) and the vector of the preferred orientation. The direction of the vector of the preferred orientation is equivalent to the normal H of the cleavage plane or the grown plane of a plately crystal, or corresponds to the growing direction R of a needle-shaped crystal.
On this way the used multiplicity MU can be easily substitute by an effective multiplicity O, which will be defined as sum about all N symmetry-equivalent reciprocal lattice directions {hkl} and will be considered during the intensity calculation. Unfortunately, with regard to a semi-quantitative phase analysis an important restriction exists. In principle, for all reflections a general attenuation occurs caused by angle-dependent consideration of the preferred orientation. Thus, the calculated intensities between different crystal structures and/or dergrees of texture are not comparable. During the determination of absolute intensities it must be considered, that reflections favourabled by preferred orientation become a higher intensity as in normal state. Of course, the intensities of the majority of the reflections decrease. Therefore it has been assumed, that the total multiplicity related to the existence of isometrical crystallites will be preserved. So a factor of normalization will be introduced, which considers the ratio of the accumulated multiplicities of all reflections to the sum of the effective multiplicities:

 
2. model
 
This is an equation which has been developed by MARCH in 1932 and will be used in actual Rietveld programs, too. Mainly it is useful for the description of the preferred orientations of axial crystals [7].

 
There MU is the multiplicity without any preferred orientation.
This equation has the advantage to contain the increase as well as the decrease of intensity caused by the defined preferred orientation. Therefore it is in general use in Rietveld programs to determine the fractions of a multiphase aggregate (mixture). The most important disadvantages are the limited validity for crystals with a lower symmetry and the fact that also for crystals with higher symmetry (e.g. the axial crystal systems) a general description about the full 2q range is nearly impossible regarding a constant quality.
 
The coefficients can be vary manually in the dialog box powder diffraction / phase options.  

Normalization or corrections of line intensities
 
Using the normalization (set in powder diffraction / phase options) one is able to analyze or simulate the diffraction patterns of phase mixtures — as far it is actually possible. The so determined values can be used e.g. for a calculation of the material specific constant  I / Ic. This is useful for a first estimation of the scattering power of different materials and therefore the basis for the successful use of internal or external standards. However, the identifier "normalization" is a little misleaded because in most cases of the given possibilities it is more the question of intensity corrections which are necessary for any quantitative phase analysis.
 
Beside the change of the kind of the correction it is possible to activate and set a line maximum value. However, this only will be used in the reflection listing HKL-list. So one may compare the intensities with other tables, e.g. the JCPDF. On the diffraction pattern this value has no influence. In contrast to that the scaling factor represents the proportional factor regarding to the absolute intensity. Using this the scaling of the diffraction pattern will be changed but not the values of the HKL-list.
The different corrections allow to calculate the volume or mass fraction in a mixture. PowderCell supports three different kinds of corrections. 
The first divides the intensity by the square of the unit cell volume — I / V². This enables the output of volume fractions for each phase. 
The second considers the X-ray density. The product of I, 1/r and 1/V² results to I / Vm. Using this the output of mass fractions is possible.
The last one considers beside the unit cell volume the linear attenuation coefficient — I / 2µV². Of course, the factor 2 is not really necessary but the program uses it in this form. This normalization supports the comparision of diffraction power of different but single phases, i.e. this is no valid for mixtures!

Multiplicity of lattice planes
 
The multiplicity H of a lattice plane {hkl} defines the number of all symmetry-equivalent lattice planes. On the one hand it depends from the indexing of the lattice plane itselfs, on the other side the given quantities of all lattice planes are characterical for the point-group symmetry, respectively. Thus, the indexing {hkl} and H are connected very closely because the general {hkl} shows a typical face (habit), where H gives the numbery of the contained symmetry-equivalent lattice planes.

example:	point group 23 (cubic): {111} forms a tetrahedron; consequently H is 4. Therefore it exists a further tetrahedron with the indexing {11-1}, which is not symmetry-equivalent. For both faces the structure factore can be extremly different from each to another.
Usually, the multiplicity will be used to increase the processing time, because for all symmetry-equivalent lattice planes (or interferences) the characteristical properties (in our case the diffracted intensity) must be identically. Therefore, it is not necessary to recalculate the structure factor over and over again. For each interference the total intensity must be combined with H, only. If the probability for the occurence for each individual (hkl) of {hkl} is not identically (see preferred orientation) this must be considered additionally using a respective correction factor. The structure factor itselfs is preserved and independent of such effects.

Reflection intensity
 
Since X-rays only interact with the electrons of atoms or ions the scattering power of a crystalline phase depends above all on its electron density. Usually, the result of this interaction between X-radiation and electrons of a crystal structure will be expressed by the intensity of the scattered radiation. The analytical formulation is:

where A considers the attenuation, PLG characterizes the polarization, Lorenz  and geometry correction, FHKL defines the structure factor, OHKL represents the influence of a possible preferred orientation and — if no texture exists — H arranges the multiplicity of the given lattice plane {hkl}. The factor K is a constant and enables the normalization of the relative intensities regarding to the unit cell volume and/or the mass (density). On this way it seems to be possible to analyse mixtures or to compare the scattering power of different materials. Usually, for a crystalline phase only relative intensities will be determined and K and A can be neglected. However, for an analysis of the scattering power of a single phase only A will be considered during the calculation.
 
If one try to compare the scattering power of different materials it is necessary to counterbalance the influences of the electrons against each other. Thus, it seems to be plausible, that the absolute value of the structure amplitude must be divided by the volume of the unit cell VEZ in order to create a normalized intensity per Å³  as comparable value between different materials. Since the absolute value of the structure amplitude will be considered quadratically within the calculation it must also be divided by the square of the volume.

remark: That is very easy to show. Let us describe e.g. a face-centered cubic lattice as a primitive rhombohedral. The relation between the cubic (ac) and rhombohedral (ar und a) lattice constants can be given as follows:
			  
For the rhombohedral unit cell definition, structure amplitudes result which are only 1/4 of the absolute value compared with that for the cubic face-centered cell. In correspondence, the volume of the rhombohedral cell is also 1/4 of that of the cubic cell. Since the intensity is independent from the used setting and the structure factor will be considered quadratically the calculated intensity must be weighted by the factor 1/V².
 
Since, in spite of the same structure type different crystalline phases show different volumina of interactions (i.e. a different number of unit cells interact with the X-ray beam) it is necessary to consider the mass attenuation m/r in form of the linear absorption coefficient if the scattering power of pure materials (in the meaning of an external standard) shall be compared. The linear absorption coefficient describes the penetration depth. In case of the Bragg-Brentano focusing the attenuation is identically for the incident beam as well as for the diffracted radiation. Therefore an additional factor 2 results. For other diffraction geometries it is impossible to compare the scattering power of different phases using PowderCell because their influence have not been considered. The linear absorption coefficient m can be extracted from the atomic absorption coefficients ma and the unit cell volume VEZ using the following equation:

 
If one likes to analyze a mixture (e.g. internal standard) the mass attenuation must be neglected during the calculation. For a mixture the penetration depth is identically for each phases. Of course, it would be more reliable to consider a averaged linear absorption coefficient. However, in this case it is only a proportional factor which is equal for all phases. Therefore, it can be neglected. In dependence of the question it must be chosen between the different correction terms. The following expressions have been implemented in PowderCell which considers different basis conditions:
 
pure phase without absorption:

 
pure phase with absorption:	 (only Bragg-Brentano geometry)

 
phase of a mixture:

 
 
The used factors 100 and 1000 have been chosen arbitrarily as scaling factors in order to give handable intensity values.

Phase mixtures
 
During the quantitative phase analysis one distinguishes between to different ways:

a)	phase analysis using an external standard and
b) 	phase analysis using an internal standard.

Both cases can be simulated by PowderCell. However, as in practice two different methods must be used.
 
external standard
 
This method enables the comparison of the different scattering power of arbitrary materials. To this end an intensity correction will be carried out using the consideration of the linear absorption coefficient µ. For some radiations used in the program the set of data has been maked up in the data file pcaac.dat. Since the dependence of the scattering power has been implemented for the Bragg-Brentano focusing, exclusively, this method can be used only for the simulation of this diffraction geometry. We recommend to activate the consideration of anomaleous dispersion in this cases only then, if one assumes an influence on the result. Of course, for other radiations which are not considered in the program the anomaleous dispersion correction is impossible, because the atomic scattering coefficients are unknown. 
 
internal standard and phase mixtures
 
In phase mixtures a correction of the extinction is not necessary. The reason is that from all extinction coefficients calculated for each phase, an average value results which must be used for all phases, equally. Therefore, we neglect this. 
In correspondence to the remarks regarding the external standard, the same is valid for the anomaleous dispersion. We recommend to activate it only in case of doubt. Please notice, that mixtures will be given in mass%, usually. That means that the reflection intensities must be corrected additionally by a division with the density. To this end PowderCell uses the calculated X-ray density.
 
The theoretical description of mixtures using diffraction pattern has also some disadvantages or possible mistakes. In spite of the exact knowledge of the unit cell volume sometimes some atomic coordinates don't will be given in the structure reports, e.g. contained crystal water or non-considered structure elements (adsorbed water in clays etc.). Of course, all these neglected absorber falsify the calculated X-rax density. Additionally, by the use of the inserted equations a lot of requirements will be assumed, e.g. the scattered volume is identically over the range of the diffraction angle, a constant radiation density within the beam, a constant grain size, an ideal homogenization, a comparable degree of imperfectness (crystal lattice defects), no or at least a well-known preferred orientation etc. However, this is nearly impossible to carry out so that the calculated reflection intensities represent a first approximation. Nevertheless, in a lot of cases the determined results correspond very good with the expected values.
 
PowderCell contains three different kinds of intensity corrections (in principle these are not normalizations). The use of this is dependents on the problem, respectively. 
To simulate a mixture, the switcher should be set on  I=I / V² or I=I / mV. This means that the calculated intensity will be devided by the square of the unit cell volume or — in the second case — by the volume VEZ and the absolute mass Ma of the unit cell.

remark: Properly, the second representation is really not ideal because usually the intensity will be devided always by V². According to that it will be devided by the X-ray density r. Since this is equivalent to the product of mass and volume we used this simplified representation.
The division by V² results during the derivation of the structure factor from the electron density deviation function, where the volume represents only a proportional term and will be neglected. Since the structure factor will be considerd within the intensity calculation in form of the quadratic of the structure amplidude, also the unit cell volume must be considered in a quadratic form. The X-ray density correction enables only the use of mass% inspite of volume%. Even for selfmade mixtures the weight fraction is better known than the volume fraction of the several phases. 

To compare the scattering power of pure phases it is necessary to set the switcher on  I=I / 2µV². Then the phase-specific extinction will be considered because in this case the volume must be considered is not identically.
 
The description  „normalization ON maximum“ enables the comparison with the results of other programs or with the JCPDF data base. Especially during the generation of subgroups  this setting should not be used. The reason is the possible splitting of reflections combined with its ideal overlapping. Thus the reflections will be summarized to different intensity levels.
 
The setting must be carried out in the screen powder diffraction / phase options.

X-ray density
 
For the transformation of mass fraction into volume fraction and vice versa PowderCell uses the X-ray density of the corresponding phase.
However, the calculates X-ray density is approximately equal to the real density, if the structure has been described completely.
For the calculation of the X-ray density the relative mass, the absolute mass and the unit cell volume will be used. The equation for the determination of the relative mass of the unit cell has the following form if all existing occupations (k) and substitutions (g) will be considered:

 
The product kp·gq corellates with the initial value of the substitution and occupation (SOF) within the worksheet structure data. s defines the number of atoms which substitute the atom p. t gives the number of generated positions within the unit cell. Usually, the calculation of MEZ can be carried out more elegant if the known Wyckoff position will be used. Then it is not necessary to summarize over all atoms within the unit cell but only of the asymmetric unit. n shall be the number of atoms in the asymmetric unit whereas z is the multiplicity of the given Wyckoff, respectively:

 
However, PowderCell doesn't use this simplification because it is not a time-critical procedure.

The absolute mass will be calculated using the following equation

 
Thus, the X-ray density follows (in g/cm³) from the relation:

Asymmetric function
 
For a better fit between theory and experiment it is also possible to adapt the given symmetrical profile functions within PowderCell using an implemented asymmetrical function. However, this adaption has been originally developed from RIETVELD for the refinement of neutron powder pattern (i.e. exclusively for gaussian) also eigentlich nur für Gaußprofile) and from our point of view it is not really necessary if one remembers the  aim of PowderCell. But even for the completeness of the powder diffraction part of the program we believe that some users like to have such correction possibillity. Thus, with the help of the following described equation an asymmetrical profile function will be calculated. It describes the superposition of symmetrical and asymetrical part. The asymmetrical term is defined as follows:

 
There A is the asymmetry parameter and q describes the Bragg angle of the interference HKL. The most important disadvantage of this model function is, that the asymmetry produces in dependence of sign(Dq) negative intensities. Therefore, the powder pattern shows sometimes intensity edges at the end of the calculated reflection interval (only for a respective enlargement). Thus we recommend to use the asymmetric correction in individual cases only.
 
The used expression represents only one idea to adapt experimental diffractograms empirically. In [1] further models will be given, but this cannot be used in PowderCell. There e.g. a correction function will be given especially for the convolution with a Pseudo-Voigt profile. So a superposition of two single profiles L and P will be generated where the maximum of both functions is shifted by a small d. The resulting intensity I(2q) will be calculated using the following equation

 
Here, additionally the parameter w must be fitted, which corresponds to the weight of the single profiles.
But, there are exist also further ideas where more than one parameters must be adapted, e.g.

 
Here for each reflection the Ai must be optimized. From our point of view this is not acceptable within a program like PowderCell. However, for Rietveld programs such procedure will be used.
 
An alternative method to describe the asymmetry of reflections has been developed by BERGMANN in his Rietveld-Programm BGMN [27]. Neiher he uses a function to consider the variation of FWHM nor for the description of the asymmetry. He uses a so-called geometry function, where only experimental parameters will be considered (wavelength-characeristic of the used radiation, tube focus, slit system, radii of the focusing circle etc. Of course, that seems to be very complicate and the calculation needs approximatly 20-30 min. However, the results are very good. 

Background
 
From the beginning the simulation of background in PowderCell was one of the most complicate problem because no analytical description is known. Thus, also in commercial programs polynoms will be used.
In former days only a constant background could be defined. Using the refinement option in the program also polynoms up to the 14th degree can be used to describe the experimental background. The parameters can be seen on the result screen. There they are given at the end of the refined parameter. The single constants cannot be edited regarding a further refinement. Also limitations on single coefficients are impossible currently because only the output on screen or printer has been created.

Reduced cell
 
Sometimes the choice of the basis vectors of a unit cell do not correspond with the conventional setting described in the IT. Therefore and also for the comparison of apperant different structures a reduced primitive cell has been defined. This follows the condition a £ b £ c and all angels either £90° or ³90°. For each crystal exist exactly one cell which fulfills this conditions.
Using this procedure one is able to determine if a structure has been described in another setting. A further advantage is that it is possible to derive the really conventional unit cell using the so-called Niggli-Matrix.
Niggli-Matrix:    

For the determination of the reduced cell special programs has been developed, e.g. LEPAGE [29].

Hexagonal coordinate system
 
For lattice planes in the hexagonal or trigonal crystal systems the Miller-Bravais indexing often will be used. There the description works with 4 indexes: {hkil}. The additional index i corresponds to the [-1-10] direction which is symmetry-equivalent to the basis vectors a and b. Because this vectors are linear dependent the index i can be calculated from the h and k on a simple way:
 
		i = - (h+k)

This gives a very simple way to determine all symmetry-equivalent reflections using a cyclic permutation of H, K and i: HKiL ® KiHL ® iHKL.
 
PowderCell enables the use of the Miller-Bravais indexing. 

Coordinate systems for rhombohedral crystals
 
For description of rhombohedral crystals to equivalent coordinate systems can be used. On the one hand a primitive lattice allows the definition of a non-centered cell (the so-called R lattice: all length of basis vectors but also all angles between there are identically). There the three-fold axis correspond to [111].
On the other hand a R lattice can also described using the hexagonal basis vectors (a=b, a=b=90°, g=120°). However, additional lattice points must be inserted: 2/3, 1/3, 1/3 und 1/3, 2/3, 2/3. This occurs to the usual systematical extinctions. Within the hexagonal system the three-fold axis is located parallel to the [001].

remark: Beside the obverse setting of a rhombohedral unit cell characterizing the conventional setting also an alternative setting exists. One produces this reverse setting by 60° rotation of the basis vectors a and b arround c. Thus the centering vectors (or location of additional lattice points) will be changed: 2/3,1/3,2/3 and 1/3,2/3,1/3. In PowderCell exclusively the observe setting will be supported. Please notice that also an internal transformation of the cellis impossible.
 
Of course, in dependence of the basis vectors (hexagonal or rhombohedral axis) also the coordinates af the atomic positions will be changed. Therefore the exact definition of the setting is very important for the result. However, PowderCell allows the automatical transformation of the cell metric and the coordinates into another.
 
For trigonal structures of the point groups 32, 3m und -3m two different cases of allocation of symmetry elements 2 and m will be observed, but both has been considerd in PowderCell. The reason is that additionally to the three-fold axis parallel to c only symmetry elements perpendicular to this are allowed which are located parallel to the direct or reciprocal base vectors. So one considers between e.g. 3 1 m and 3 m 1. Unfortunately on this difference will be noticed very rarely.

Structure representation in stereo mode
 
Usually in PowderCell the crystal structure will be shown as mono picture. However there is also implemented to show the structure as stereo picture. In this case two single pictures will be generated and drawn side by side. The single pictures are tilted by 6° against each other. Similar pictures one find also in IT (Vol. A) for the cubic space-group types. So it is more easy to interprete and understand the complete set of the symmetry-equivalent locations of a general position. Besides this in a lot of older crystallography textbooks or papers of crystal structure analysis very often stereo pictures have been used [31].
A little practice provided it is very helpful to use this 3-dimensional impression of a atomic arrangement. Oner consideres between cross and parallel lookers. However there are simple tools, e.g. special tipping mirrors which allows the view of the stereo pictures.

     Pressing this button the program shows the active crystal structure in two single pictures. Commonly single pictures will be represented using parallel or central perspective. Here exclusively the central perspectiv will be used. 
Repeating the press on this button the stereo representation is deactivated and the crystal structure will be shown as single picture. During this operation the icon of the button doesn't change. 
 
As described above the view of the stereo representation allows to choose between the parallel or cross looking. Therefore the corresponding switcher is visible only in the stereo mode and substitutes the button for the definition of the wished kind of perspective.
 
      The stereo view changes from cross to parallel looking.
      Momently the crystal structure representation is shown in parallel mode. To change to the cross mode this button must be pressed.
 
The suitable representation mode (parallel or cross) the user must find out for himself. 
We recommend to public pictures using the parallel mode exclusively.

Lattice base vectors
 
For a better overview relating the orientation of the structure during its rotation the directions of the base vectors in form of a small cross can be faded in. The length of the shown lines do not coorespond with the lattice parameters. These are identically (normalized).
Using the central perspective the direction of the basis vectors are not parallel to the edges of the unit cell. The small tilt is caused by the factor perspective — a factor, which describes the distance to a vanishing point — and the differences in the lattice parameters. The reason the shift of the origin used for the crystal structure and the coordinate cross which is relateted to a position (½,½,½). After the rotation the cross is shifted on the left down corner of the window. 
If you don't believe it please use the parallel prespective. The the direction of edges and the cross is identically.
To switch on/off use the following button:
 
 	The use of this button fade in or fade out a cross describes the directions of the base vectors. The icon of the switcher does not change. 

Perspective in structure representation
 
Besides the parallel perspective the program offers also the central perspective for the representation of crystal structures. Both, the central as well as the parallel perspective have advantages but also disadvantages. Thus, the actual problem will defines the best mode.
Even for the use of a crystallographic projection of the structure on a chosen lattice plane, i.e. the projection vector corresponds to a reciprocal lattice vector) of course the parallel perspective is recommended. Only in this mode all atoms located on this vector will be drawn one upon the other. On the other hand the use of parallel perspective gives a very small impression of the 3-dimensionality (see also stereo pictures) Then the use of the central perspectiv can be recommended where the degree of perspective can be adapted using the factore perspective.
 
 	If you press this button you change the central perspective to the parallel perspective. 
 	This button allows to switch from parallel to central perspective. The vanashing point can be varied in the window structure representation options using the factor perspective. 

The coordinate system only shows the directions
of the base vectors. The lengths are normalized!

Kinematic interference theory
 
Natural crystal are characterized by a so-called mosaic structure. This causes the higher divergence and the lower coherency of the reflected beam regarding the incident beam. The result is that the reflected beam will be emmited by the crystal without a further diffraction. Thus the mosiac structure represents the ideal case of a defect crystal [26]. On this condition the intensity of an interference is proportional to the square of the structure factor:

If the crystal is more similar to a perfect crystal i.e. there is approximately no mosaic structure the diffracted beams can be functioned as "primary" beams. One observes a lost of intensity which is called primary extinction. For a really perfect crystal the reflection intensity is equivalent to the absolute value of the structure factor — the structure amplitude. 
 
For the description of diffraction effects in perfect crystals the dynamic interference theory has been developed. However, usually powders are not perfect crystals so that the more easy kinematic assumtions are sufficient for the calculation of reflection intensities. Only for semiconductors dynamic formulations must be considered. On the other hand, if you use single crystals as monochromator you should be know the degree of perfectness.

combination: during a crystallographic combination all 
generated positions describe a point location i.e. both 
symmetry elements really exists in this space group.

Please select a subset of this!

Graphic export of structures and diffractogram
 
Some of the most important functions of programs like PowderCell is the possibility to export and import data but also to output graphs or structure pictures on printer devices or save it in typical graphic formats. Within the DOS version we supported the export as WPG  (WordPerfect Graphic) and GEM metafile format. Based on fundamental source code of S. Weber we developed an adapted export filter for PostScript. Especially the colour fading in the atom representation is an idea of S. Weber.
Since the development of the Windows version also the generation of Windows Metafiles has been carried out. Beside the use of the clip board this presents the most recommendable way for a graphic transfer between different Windows applications. 
The used formats produces excellent graphics based on the vector oriented description of the pictures. However, all these are not comparable with a "realistic" representation, which can be found on workstations or Mac's created by expansive commercial programs. To get such photorealistic structure pictures an export filter for POVRAY has been programmed which enables to edit the ASCII source code. 
At least the possibility exists to export the structure picture and the diffractogram as bitmap.
 
The graphic export can be called using the tool bar (see red bordered button in the following bitmap):

 
Then a common dialog window will be shown where it is necessary to select the wished format. In dependence to the active window (structure or diffractogram) the respective crystal structure or the powder pattern will be exported. 
Especially for the structure representation different variants will be supported: atoms as hollow circles, filled by pattern, colorized or with changing colors etc. Unfortunately not each of the supported formats realizes all these variation. 

Use of the clip board 
 
The structure picture, the diffractogram but also the HKL listing or text windows like the comment in the worksheet structure data can be copied into the clip board. The content of this can be used in other Windows programs i.e. in office applications. It is necessary only to use the function "Paste" of the respective program. Sometimes it is recommended to export the graphics in one of the supported vector formates even in such cases if the graphics shall be edited in other programs like CorelDraw. 

POV-Ray output
 
The aim of PowderCell is not the creation of mostly photo-realistic pictures or what we imagine about it (e.g. ball-stick models). However we like to support the user in generation of nice structure pictures. Because of the existence of freeware raytracing programs, we assumed that it is more easy to use such excellent sofware which is comparable with commercial products. Therefore we export input files for POV-Ray (Persistence Of Vision Raytracer). 
The generated POV-Ray scenes are ASCII files based on the standard of POV-Ray 3.0. PowderCell exports only 3 or 4 grouped textures which can be edit very easy. The most important help for POV-Ray user is the description of all atomic positions given as cartesian coordinates which has been collected into groups with identical properties (atomic number, color etc.).

remark: Using POV-Ray single pictures as well as small animations can be produced by individual variation of surface (object) properties or light sources. The animation mostly results by the movment of the camera what enables e.g. the virtual flight through a crystal structure. For each camera position a single picture will be generated (usually *.tga files). Using a special viewing program all pictures will be fit together. The produced film has often a size of more than 10 Mbyte in dependence of the number of individual pictures, the resolution and the color depth. The calculation for a single pictures can take a lot of time, even in dependence of the number of objects and light sources but also of the used computer performence. However, the calculation of structure pictures takes only some minutes.

PostScript
 
PostScript is a device and resolution independent high-level programming language It enables to work with loops, procedures and operators. PostScript supports a lot of data types (real numbers, boolean, arrays, strings etc.). 
Device and resolution independence means that the same file can be printed using a common 300dpi laser printer but also professional equipment with a higher resolution than 1000dpi. Usually the data will be interpreted within the used device, i.e. the printer must be able to do this.
But also for TeX or LaTeX user the output of PostScript files represents an important feature because the include of these files using dvitops is relative comfortable.
If you haven't a PostScript printer you can use PostScript files too. As freeware the programs GhostScript and GhostView allow the output of these files on an arbitrary Windows controlled printer.

Vector formats
 
Also for the DOS version some exportable graphic formats had been created for the comfortable import into documents — WPG (WordPerfect Graphic) and GEM (a relative old format which can be imported in CorelDraw without any problems). The export format of WordPerfect Graphic is based on standard commands of version 1.0 and has been programmed for a direct import into WordPerfect and WinWord. In principle also now it is possible to use these formats. However not all changes and extensions in graphical output will be revised e.g. the labeling of atoms or bonds etc. Therefore the export of the graphics using these two format cannot be recommended.
Even during the conversion from DOS to Windows the creation of Windows Metafiles (WMF) has been implemented which represents beside the use of the clip board the most comfortable and recommendable method for the graphic transfer between Windows applications. Since the output on screen, on printer, the use of the clip board and the generation of WMF are very closed, the output possibilities always will be harmonized.
On the other hand other "vector-oriented" formats exist generated by PowderCell (PostScript, POV-Ray). However there each additional graphical detail must be edited manually which causes small incompatibilities in the graphic output on screen and on print.

Unit cell edges
 
Usually the cell edges defines the smallest volume of a geometrical unit describing a complete crystal by the use of a three-dimensional translation symmetry exclusively. The visualization of the edges is profitable either if no atom is located there or if the represented volume contains more than a single unit cell. Therefore it is recommended to fade out the edges only then if e.g. one like to visualize a single molecule.
 
   If you press this button the unit cell edges will fade out. If you do this onesmore the edges will be faded in.
Please notice that the bitmap don't changes during the operation.

1_SE.BMP

 

3_111_SE.BMP

3_SE.BMP

MXXZ_SE.BMP

MXXZA_SE.BMP

MZ_SE.BMP

MY_SE.BMP

MX_SE.BMP

21-10_SE.BMP

2110_SE.BMP

1-_SE.BMP

2Z_SE.BMP

2Y_SE.BMP

2X_SE.BMP

code	Laue group
    1	triclinic (-1)
    2	monoclinic, monoclinic axes a (2/m)
    3	monoclinic, monoclinic axes b (2/m)
    4	monoclinic, monoclinic axes c (2/m)
    5	orthorhombic (mmm)
    6	tetragonal (4/m)
    7	tetragonal (4/m mm)
    8	trigonal, rhombohedral coordinate system (-3)
    9	trigonal, hexagonal coordinate system (-3)
  10	trigonal, rhombohedral coordinate system (-3m)
  11	trigonal, hexagonal coordinate system (-3m)
  12	hexagonal (6/m)
  13	hexagonal (6/m mm)
  14	cubic (m-3)
  15	cubic (m-3m) 

lattice	reflection	 additional
 type	condition	translations
    P	       0	       ---
    A	       4	0 , 1/2 , 1/2
    B	       5	1/2 , 0 , 1/2
    C	       6	1/2 , 1/2 , 0
    I	     13	1/2 , 1/2 , 1/2
    F	       7	0 , 1/2 , 1/2
      		1/2 , 0 , 1/2
    		1/2 , 1/2 , 0
    Rh	     14	2/3 , 1/3 , 1/3
    		1/3 , 2/3 , 2/3
    Rr	     21	         --- 

code	reflection condition	code	reflection condition
    1	H = 2n	  17	K = 4n
    2	K = 2n	  18	L = 3n
    3	L = 2n	  19	L = 4n
    4	K+L = 2n	  20	L = 6n
    5	H+L = 2n	  21	|H| >= |K| >= |L|
    6	H+K = 2n	  22	2H+K = 2n
    7	H,K,L odd / even	  23	2H+K = 4n
    8	K+L = 4n	  24	H+2K = 2n
    9 	H+L = 4n	  25	H+2K = 4n
  10	H+K = 4n	  26	H = 2n , K = 2n
  11	2H+l = 2n	  27	K = 2n , L = 2n
  12	2H+L = 4n	  28	H = 2n , L = 2n
  13	H+K+L = 2n	  29	K+L = 4n , K = 2n , L = 2n
  14	-H+K+L = 3n	  30	H+L = 4n , H = 2n , L = 2n
  15	H-K+L = 3n	  31	H+K = 4n , H = 2n , K = 2n
  16	H = 4n		 

position	reflection 	position	reflection
	     type 		    type
      1	00L	8	0KK
      2	0K0	9	HH0
      3	0KL	10	HHL
      4	H00	11	H0H
      5	H0L	12	HKK
      6	HK0	13	HKH
      7	HKL	14	HHH

reflection conditions

	Atomic radii of the major period elements
H	0.78													He															1.22Li	1.57	Be	1.13	B	0.95	C	0.86	N	0.8	O		F		Ne	1.52		1.10												1.60		T	1.07	T	0.89	T	0.77	T	0.70	T	0.66	T	0.64Na	1.92	Mg	1.60	Al	1.43	Si	1.34	P	1.30	S		Cl		Ar	1.86		1.55		1.39										1.92		T	1.40	T	1.26	T	1.17	T	1.10	T	1.04	T	0.99		P	1.40	P	1.32	P	1.17	P	1.10	P	1.04K	2.36	Ca	1.97	Ga	1.39	Ge	1.39	As	1.40	Se	1.60	Br		Kr		2.29		1.91				1.34		1.35						1.98				T	1.26	T	1.22	T	1.18	T	1.14	T	1.11				P	1.31	P	1.22	P	1.17	P	1.14	P	1.20 Rb	2.53	Sr	2.16	In	1.57	Sn	1.58	Sb	1.61	Te	1.70	I		Xe	2.46		2.10		1.52		1.53		1.56				T	1.44	T	1.40	T	1.36	T	1.32	T	1.28	T	2.18				P	1.48	P	1.40	P	1.35	P	1.28	P	1.13Cs	2.74	Ba	2.25	Tl	1.71	Pb	1.74	Bi	1.82	Po		At		Rn	2.46		2.18		1.66		1.60		1.77				T	1.47	T	1.46	T	1.46				P	1.54
in each case: first line 12-fold coordinated, second line 8-fold coordinated, 
T...tetraedrical coordinated, P...covalent radius

	Atomic radii of the minor period elements
Sc	1.65	Ti	1.45	V	1.36	Cr	1.28	Mn	1.31	Fe	1.27	Co	1.26	Ni	1.24	Cu	1.28	Zn	1.37			1.41		1.32		1.24		1.26		1.24		1.22		1.21		1.24		1.34																T	1.34	T	1.31																P	1.36	P	1.39												Q	1.23	Q	1.22	Q	1.21Y	1.81	Zr	1.60	Nb	1.47	Mo	1.40	Tc	1.34	Ru	1.32	Rh	1.34	Pd	1.37	Ag	1.44	Cd	1.52			1.55				1.36				1.28		1.30		1.33		1.40		1.47																T	1.52	T	1.48																P	1.55	P	1.56												Q	1.33	Q	1.32	Q	1.33La	1.86	Hf	1.59	Ta	1.46	W	1.41	Re	1.37	Os	1.34	Ir	1.35	Pt	1.38	Au	1.44	Hg	1.55			1.53		1.42		1.37		1.34		1.30		1.31		1.34		1.40		1.50																T	1.50	T	1.48																		P	1.54												Q	1.33	Q	1.32	Q	1.31		 
in each case: first line 12-fold coordinated, second line 8-fold coordinated, 
T...tetraedrical coordinated, P...covalent radius, Q...quadratic coordinated

	Ionic radii of the major period elements
H														He	1.22 –	1.54															2.08	1.36Li		Be		B		C		N		O		F		Ne	1.60 +	0.78	2+	0.34	3+	0.20	4+	0.2	5+	0.15	6+	0.09	7+	0.07	0.6		0.31		0.24		0.15		0.11	2 –	1.32	 –	1.33	0.68		0.39				0.19	3 –	1.71		1.40		1.36											1.40		1.33Na		Mg		Al		Si		P		S		Cl		Ar	1.92 +	0.98	2+	0.78	3+	0.57	4+	0.39	5+	0.35	6+	0.34	7+	0.26			0.95		0.65		0.50		0.41		0.34		0.29	 – 	1.81	0.98		0.71		0.53		0.44	3 –	2.12	2 –	1.74		1.81						4 –	2.71				1.85		1.81K		Ca		Ga		Ge		As		Se		Br		Kr	1.98 +	1.33	2+	1.06	3+	0.62	4+	0.44	5+	0.47	6+	0.35	7+	0.39			1.33		0.99		0.62		0.53	3 –	2.22		0.41	 –	1.96	1.33		0.98			4 –	2.72			2 –	1.91		1.95											1.98		1.96											1.99 Rb		Sr		In		Sn		Sb		Te		I		Xe	2.18 +	1.49	2+	1.27	3+	0.92	4+	0.74	5+	0.62	6+	0.56	7+	0.50	1.48		1.13		0.81		0.71	3+	2.45	4+	0.89	 –	2.20			1.48		1.15			4 –	2.94				0.81		2.16										2 –	2.11		2.19											2.21											2.18Cs		Ba		Tl		Pb		Bi		Po		At		Rn +	1.65	2+	1.43	3+	1.05	4+	0.84	5+	0.74	1.69		1.35		0.95		0.84			1.67		1.31	 +	1.49					1.44
radii of Goldschmidt, Pauling and Zachariasen

	Ionic radii of minor period elements
Sc		Ti		V		Cr		Mn		Fe		Co		Ni		Cu		Zn	3+	0.83	4+	0.64	5+	0.4	6+	0.35	7+	0.46	3+	0.67	2+	0.82	2+	0.78	 +	0.63	2+	0.83	0.81		0.68		0.59		0.52	4+	0.52	2+	0.83		0.72		0.69		0.96		0.74	0.78		0.62	4+	0.61	3+	0.64		0.50		0.75					2+	0.51				3+	0.69		0.69			3+	0.70										2+	0.76	3+	0.65			2+	0.91				2+	0.72				0.80Y		Zr		Nb		Mo		Tc		Ru		Rh		Pd		Ag		Cd	3+	1.06	4+	0.87	5+	0.69	6+	0.62	7+	0.56	4+	0.65	3+	0.68	2+	0.72	 +	1.31	2+	1.03	0.93		0.80		0.70	4+	0.68				0.63						1.26		0.97	0.93		0.79	4+	0.69		0.66																0.67							La		Hf		Ta		W		Re		Os		Ir		Pt		Au		Hg	3+	1.22	4+	1.02	5+	0.68	4+	0.68	6+	0.55	4+	0.67	4+	0.66	2+	0.87	 +	1.37	2+	1.12	1.15		1.01								0.65		0.64						1.10	1.06		0.89										0.65										1.09 Ac		Th		Pa		U		4+	1.09			2+	1.05						4+	1.14
radii of Goldschmidt, Pauling and Zachariasen or experimental data

Basic formulaes for the triclinic crystal lattice
 
absolute value of a lattice vector:

 
unit cell volume:

 
interplanar distance:

 

ICSD import (*.txt)
COL	Collection Code 30285 (recorded Jan 1, 1980)
 NAME	Aluminium silicate hydroxide *
 MINR	Kaolinite - from Brooklyn, New York
 FORM	Al2 Si2 O5 (O H)4
 TITL	The Crystal Structure of Kaolinite
 REF	ZEKGA 83 (1932) 75-88
 AUT	Gruner W J
 SYM	x, y, z
 SYM	x, -y, 0.500+z
 CELL	a=5.140 b=8.900 c=14.510 a=90.0 b=100.2 g=90.0
      	V=653.3 Z=4
 SGR	C 1 c 1       (9) - monoclinic
 ANX	A2B2X9
 PARM	Atom__No      OxStat    Wyck   -----X-----  -----Y-----  -----Z-----  -SOF-
   	Al	1	 3.000	4a  	1/4    	-0.167	0
      	Al	2	 3.000	4a  	1/4     	0.167	0	
 	Si	1	 4.000	4a  	0.008	1/2    	0.188
      	Si	2	 4.000	4a  	0.008	0.167	0.188
      	O	1	-2.000	4a 	-0.046	1/2    	0.076
      	O	2	-2.000	4a 	-0.046	0.167	0.076
      	OH1	3	-2.000	4a 	-0.046	-0.167	0.076
      	O	4	-2.000	4a  	0.033	0.333	0.229
      	O	5	-2.000	4a  	0.283	-0.417	0.229
      	O	6	-2.000	4a  	0.283	0.083	0.229
      	OH1	7	-2.000	4a  	0.046	0       	0.424
      	OH1	8	-2.000	4a 	-0.454	0.167	0.424
      	OH1	9	-2.000	4a  	0.046	0.333	0.424
      	H	1	 1.000	16 Atoms not located in Unit Cell
 REM	XDP (X-ray diffraction from a powder)
 TEST	No R value given in the paper. (Code 51)
 TEST	At least one temperature factor missing in the paper. (Code 53)

ICSD import (*.cry)
0000   30285  100   Aluminium silicate hydroxide *  300   Kaolinite - from Brooklyn, New York  400   Al2 Si2 O5 (O H)4  500   The Crystal Structure of Kaolinite  600   32ZEKGA     83     75     88     0  700   Gruner W J  800   5.14000  -1.E+0  8.90000  -1.E+0 14.51000  -1.E+0 90.000  -1.E+0  810   100.200  -1.E+0 90.000  -1.E+0  4-1.000  -1.E+0 6.53E+0002  900   C1C1       1200   Al  1 3.000  4a 0.25000  -1.E+0-0.16700  -1.E+0 0.00000  -1.E+0 1210          0.E+0    0.E+0 1.0000  -1.E+000 1200   Al  2 3.000  4a 0.25000  -1.E+0 0.16700  -1.E+0 0.00000  -1.E+0 1210          0.E+0    0.E+0 1.0000  -1.E+000 1200   Si  1 4.000  4a 0.00800  -1.E+0 0.50000  -1.E+0 0.18800  -1.E+0 1210          0.E+0    0.E+0 1.0000  -1.E+000 1200   Si  2 4.000  4a 0.00800  -1.E+0 0.16700  -1.E+0 0.18800  -1.E+0 1210          0.E+0    0.E+0 1.0000  -1.E+000 1200   O   1-2.000  4a-0.04600  -1.E+0 0.50000  -1.E+0 0.07600  -1.E+0 1210          0.E+0    0.E+0 1.0000  -1.E+000 1200   O   2-2.000  4a-0.04600  -1.E+0 0.16700  -1.E+0 0.07600  -1.E+0 1210          0.E+0    0.E+0 1.0000  -1.E+000 1200   O   3-2.000  4a-0.04600  -1.E+0-0.16700  -1.E+0 0.07600  -1.E+0 1210          0.E+0    0.E+0 1.0000  -1.E+01H 1200   O   4-2.000  4a 0.03300  -1.E+0 0.33300  -1.E+0 0.22900  -1.E+0 1210          0.E+0    0.E+0 1.0000  -1.E+000 1200   O   5-2.000  4a 0.28300  -1.E+0-0.41700  -1.E+0 0.22900  -1.E+0 1210          0.E+0    0.E+0 1.0000  -1.E+000 1200   O   6-2.000  4a 0.28300  -1.E+0 0.08300  -1.E+0 0.22900  -1.E+0 1210          0.E+0    0.E+0 1.0000  -1.E+000 1200   O   7-2.000  4a 0.04600  -1.E+0 0.00000  -1.E+0 0.42400  -1.E+0 1210          0.E+0    0.E+0 1.0000  -1.E+01H 1200   O   8-2.000  4a-0.45400  -1.E+0 0.16700  -1.E+0 0.42400  -1.E+0 1210          0.E+0    0.E+0 1.0000  -1.E+01H 1200   O   9-2.000  4a 0.04600  -1.E+0 0.33300  -1.E+0 0.42400  -1.E+0 1210          0.E+0    0.E+0 1.0000  -1.E+01H 1200   H   1 1.000  4a-1.00000  -1.E+0-1.00000  -1.E+0-1.00000  -1.E+0 1210          0.E+0    0.E+0 4.0000  -1.E+000 1600   XDP 9900    30285

Tool bar to influence the structure representation
 
Even from the beginning of the development of PowderCell the visualization of structure data played an important role. Therefore it is not surprising that the program offers some useful tools to vary the structure representation in a comparatively easy manner. If a structure screen is activated a special menue bar will be faded in on the right side on the screen to influence the visualization.

 
It allows the manipulation of the representation which is only applied on the selected structure, in contrast to the representation options given in the tool bar on the bottom of the screen which defines global properties applied on all structures. Each setting is valid exclusively for the respectively active structure window (except the rotation angle, which represents a value only and always can be changed).
Besides the realization of rotation and zooming functions all rotations can be saved and loaded additionally using a rotational matrix. The visualization of a crystal structure will be carried out as mono picture using  parallel or central perspective but also as stereo picture for parallel or cross looker. In all these modes the icons on the buttons will be adapted on the representation type respectively.
Sometimes it could be very interesting for different reasons to look at the asymmetric unit — i.e. the content of the initial structure data. However, also small additional features may be very useful like the possibility to fade in the cell edges or the crystallographic basis vectors as hint for their directions. Of course a reset of all rotations has been implemented.
Since PowderCell visualizes bonds also then if the user don't like this a switcher allows to fade out it. 

If additional bonds shall be generated or existing shall be deleted the program contains a bond editor. In contrast to the described functions above this feature has been included in the tool bar, because it represents a structure manipulation tool. 
 
Additionally to the known functions a labelling of the atoms and the bonding lengths has been included. For the atoms the name of this position defined in the worksheet structure data will be used. The lengths of the bonds will be given in Å.
Furthermore the possibility exists to show all atoms or all selected but also all unselected positions. The calculation of the powder pattern is independent on this variation of structur representation. The same is valid for the demonstration of the asymmetric unit.

Rotation of the crystal structure
 
For the understanding of the crystal structure the spatial imagination about the arrangement of atoms or molecules within the unit cell is essentially. Even for PowderCell this represents the starting point for the following structure modelling. However, also for other applications it is very often an advantage to visualize the initial data for the calculation of the powder pattern on the screen because not very rare errors occur during the generation of structure files.
To estimate the complexity of a structure it is recommended to study the atomic arrangement very intensively. Therefore additional rotations have benn defined which allows the observation of the structure from an arbitrary perspective. This is possible using well defined but free rotation angles, the rotation matrix or other  projection tools contained in PowderCell. The rotation directions have been defined using 6 different buttons. Unfortunately the use of left and right mouse key like in the DOS version canot be taken over. The used mouse driver don't support such function. Furthermore, in contrast to the DOS version each rotation must be started by a separate click on the left mouse key over the respective button. Thus for complicate structures and slow PC's it is possible that the visualization of the rotated structure is not finished in the background and copied in the video memory yet before the carry out of the next click. In such case no screen output will be shown. Only then if the program has finished a rotation and no further movement has been produced the structure will be represented on the screen after all movements. However all rotations will be carried out correctly. The reason is the small velocity of the drawing rotines used in Delphi or Windows. 
The crystal structure always will be rescaled during this procedure based on the atomic positions. Until now the used radii dont will be considered.

Zoom in structure representation
 
Escpecially during the use of the stereo mode of structure visualization it is recommended to scale down the single pictures. Sometimes also the effect troubles if the automatic scaling draws the single pictures bigger than the window dimension has been defined. In such cases it is recommended to scale down the representation to have a good survey. This is very important regarding the use of the clip board or grafical export because there really only this will be considered what is visible in the window respectively. 
In contrast to the DOS version a horizontal and vertical shift of the structure picture has not been implemented. This will be realized using Windows basic tools. 
The degree of scaling up and down is fixed and cannot be adapted manually. 

Angle for structure rotation
 
The input value in degrees is valid likewise for all rotation axis and must be changed manually if it is sensible. A countineous rotation around a selected axis is impossible. Because of the low drawing velocity it seems to be absurd to try this. The limitation is based on the used Delphi compiler (16 bit code).

Using this a former saved rotation matrix 
can be loaded. So arbitrary structures 
can be adjusted identically.

Using this button all used rotations
can be saved into a rotation matrix.

By the use of Reset the structure will be
projected on the lattice plane (001).

By the use of this buttons a rotation
around the given axis will be carried 
out. The used angle is shown below
whereas the arrows define the 
direction of rotation.

With this button the structure representation can
be changed between stereo or mono mode.
The icon on the button do not changes.

Press this button to fade in the 
length of all given bonds in Å.

Using this button it is possible to
scale up and down the structure.

With this the drawing of bonds
can be prevented. This does not
means that the bonds are deleted!

The rotation angle can be chosen free. 
Each rotation will be started by pressing 
the left mouse key.

This button enables the visualization of grouped
atoms or ions selected before. It is possible to
draw all selected, all unselected but also  the 
complete number of generated positions.

Besides the labeling of bonds also the identifiers
of all atoms shown on the picture can be faded 
in. The labeling is valid also for the different
graphical export of pictures.

To change between the representation of 
asymmetric unit and unit cell this button has
been implemented.

To fade in and out the small crystallographic 
coordinate system this button can be used.

Using this the cell edges can be feded in 
and out. This could be very useful for the 
representation of single molecules.

Here it is possible to change between
central -and parallel perspective.

Save and load the rotation matrix
 
If crystal structures are very similar (especially if the crystal system is equivalent) it would be very useful to confront this structures directly for a detailed comparison. Therefore it would be very nice to repeat the same rotations of the second structure. Often this seems to be very difficult because it takes a long time to find a suitable projection for the crystal structure. To repeat this PowderCell always allows to save a transformation matrix containig all rotations carried out. If one loads this matrix an arbitrary structure can be located on same manner. The used file pcw.rot containig the matrix components is located in the directory of PowderCell. This file has always the same name i.e. that it will be overwritten permanently. The use of more than one matrix has not been provided. 

Menue for the powder pattern representation
 
If one activates the powder pattern window on the right side of the screen a pop-up menue containig several buttons will be faded in. Fig. A shows this menue bar.
 
Fig. A:	  		Fig. B:	  
 
It allows to strech or upset the diffractogram by the use of fixed steps which cannot be edited in the program. Furthermore all single patterns can be faded in as well as the sum of all. Very important is the import of measured data and the calculation of the difference between experiment and calculated pattern. The existing reflections can be indexed using different qualities (see Fig. B). Besides the common description using the Miller indexing HKL for hexagonal crystals the Miller-Bravais-Indexing will be supported. However, also the interplanar distances or the detektor angle 2q (q describes the Bragg angle) can be used as identification of reflections. In special cases the structure amplitudes, the line or the corresponding profile intensity can be used for the respective reflection. 

Reset of all movements.

Using this button it is possible to index the reflections.
If there are many reflections only the most intensive 
will be indexed. During the scale up more and more 
reflection will be indexed.

By a direct comparison with a measured
diffractogram the difference plot can be 
faded in pressing this button.

If more than one structure will be considered
the single patterns can be faded out but also in.

Especially in mixtures the sum of all single pattern is 
much more interesting than the single curves itselfs. 
This simplifies the comparison of theory and experiment.
By the way, the given maximum intensity are related 
on the sum so that the single phases cannot be 
resolved very often.

To fade in the refined background
polynom please press this button. 

By the use of this buttons the angle area
can be scaled up and down.

Using this buttons the intensity of the 
diffractogram representation can be 
scaled up and down.

If this switcher is active the press on it
fade in or out an imported experimental 
diffraction pattern.

After the manually scaling of intensity it 
enables to Reset this by press on this button.

To reset (delete) the refined background
you must press this button. Then all poly-
nomial parameters will be set to 0.

The program allows to choose the shown range
in the diffractogram (angle and/or intensity) very
easy, only by selection with the mouse button. 
To reset this you can use this switcher. 

After the scale up the full angle area can be visualized
by this simple shift using a fixed step width which will 
be calculated automatically.

Select atoms and molecules 
 
One of the most interesting features in PowderCell is the possibility to catty out special manipulations like a structure modelling. However, all atoms or molecules which shall be considered during the manipulation must selected before. There no button exists exclusively for this function. It is only possible to use the selection function using the main menue of the program. If one choose this a submenue will be displayed which contains the different tools to select or unselect atoms or molecules:

 
Thus one is able to select single atoms but also the complete structure. All variants seem to be useful in dependence of the given problem.
In principle one differs between a structure controlled selection (single atom, range or even the complete unit cell represented by the asymmetric unit or Select all) and a bond controlled selection (described by the synonym: molecule). Especially the last one can be very useful if for relative complex crystal structures, e.g. Ceolithes, only defined "molecules" shall be translated or rotated. The critical criterion is that the molecule will be defined by the generated bonds. Since the bonds in PowderCell will be recognized in dependence of the used radii of the respective atoms or ions. So it is sometimes possible that the resulting molecule is a little bit bigger than the chemical pendant. In such case it is possible to combine the different selection tools for a deselection of single atoms or whole ranges. Please notice: The given tools can be combined to select or unselect.
 
For the practical use of the tools it is necessary to understand their characteristical properties. The program works definitively conform with the space group definition in the IT. That means that the space group symmetry cannot be switch off during rotational or translational operations. On the one hand this simplifies the practice. On the other hand from the crystallographic point of view it is surely reasonable to do this so. One reason is that during the manipulation only the atoms of the asymmetric unit will be used. On the one hand this has the advantage that the manipulation is more easy because the number of atoms is decreased as far as possible. On the other hand some difficulties result where some shall be given in the follow.
 
1.	Molecules or selected ranges which are not described by one and the same Wyckoff position cannot be moved in an arbitrary direction without destroying the metric of the molecule. Therefore the subgroups have been implemented to enables a selective decrease of local symmetry.
 
2.	Unfortunately the selected areas of a molecule must not describe the metric. The reason is that on the one hand the input file is not molecule related i.e. PowderCell doesn't requires a desription of the molecule in a bonding form. Even though the Wyckoff position adaption in the program it is very possible that bonds of a molecule will be cutted. In these cases separate manipulations must be carried out. We assume that this should be very rare but we cannot exclude such problems following from the generation procedure used. Besides the multiple step manipulation sometimes it is recommended to proof the possibility of redefinition of the asymmetric unit. 
example:	The last may be used e.g. for gypsum. There the sulfur atom is sittig on the position: (0,y,1/4). y is reqired as 0.9218. The corresponding oxygen atoms will defined by a comparable small y. Scanning the structure or use the labelling all symmetry-equivalent positions nearly the given S can be can be investigated and the corresponding coordinates can be used in the structure data file. 
 
3.	Since the manipulation will be carried out on the atoms of the asymmetric unit exclusively the connection to the whole structure sometimes will be lost. Therefore the display of the complete content of the unit cell is able. Unfortunately the selection within the unit cell is sometimes problematically because it is not easy to know the content of the asymmetric unit. Usually the program gives some hints, but it supports the transfere into the asymetric unit.

Tool bar
 
In order to make the handling of PowderCell so easy as possible the most important functions can be called using special buttons on the upper tool bar. This enables a fast operation without the use of the main menues. Please press about the buttons of the following given tool bar to get information about the task of this.

However, also during the operation of PowderCell the buttons will be described by a small hint, if the mouse pointer touches the range of the button (without pressing the mouse key). 
In principle the upper tool bar has been subdivided into four ranges. The first two buttons are charactered by basic functions: load, import, create and edit structur files. The second range contains the most important functions related to the crystal structure representation.

The third one contain the very important export possibilities of PowderCell. However also it shall partition off the structure representation buttons from this part of the menue applying oneself to the diffraction calculation or refinement.

Load structure data
 
The basic of the work with PowderCell are created *.cel-files or impartable structure data. Of course, at first  these must be loaded or imported in PowderCell. This will carry out by pressing the „Open“ button in the tool bar which is selected in the following graphic by a red frame.

	
The small yellow hint of PowderCell explains it correctly by load structure.

If one select this button or the corresponding entry in the main menue File Þ Load (please consider also the short cut: Ctr+L) the File open dialog will be faded in and a structure file can be loaded or imported. The required import filter must be selected in the corresponding dialog box (file format). 

 
File open is a standard Windows dialog. So the shown graphic is only an example for the german Windows version.
 
If there exist incompatibilities regarding the setting number of monoclinic or orthorhombic space groups during the data import a message will be given. In this case the first setting will be used automatically and the user must try to solve this problem manually, e.g. by the use of an additional translation of the origin or the edit of the real setting number. Sometimes it could be possible that the data must be transformed into another setting because the given data cannot be interpreted by the program. If you have problems to do this don`t hesitate to contact the authors. 

Edit structure data

The structure data can be displayed, edited and changed in a special dialog window. To call this window the corresponding button in the tool bar must be selected using the mouse pointer. The button is marked in the following graphic by a red frame:

 
If one touchs the button with the mouse pointer and don't press it a small hint with the description edit initial data will be faded in. Of course, the dialog box edit structure data can also be activated be the use of the main menue Structure Þ Edit initial data.
 
If no structure file exists the structure data can also be created by the use of the editor. At first the editor must be called. As you can see some data are inserted as dummy values. Now the structure data must be inserted. Usually it is recommented to start with the space group number because then PowderCell knows the relation between the lattice vectors and e.g. for a cubic crystal only one lattice constant must be edited. Please notice, that the inserted data only then qwill be activated if the windows will be closed by OK. Only then the structure will be generated using the defined data. Be sure that you saved the inserted structure data e.g. by the use of the main menue option Structure Þ Save. 
 
The structure data of all structures considered in this session can be listed in this and only this window. There is no possibility to compare the data directly by the use of a second or third editor box.  
The dialog box shows in the upper part beside the lattice constants, the space-group type, the setting also derived data e.g. the # number of generated atoms (also the occupation factors will be considered in a second number), the unit cell volume, the X-ray density, the relative mass or the extended Hermann-Mauguin symbol. Only this data can and must be edited which are written in black. All gray numbers cannot be changed and will be adapted by PowderCell.
   example:	In the following graphic an example window is shown where only the lattice constant a can be varied because of the cubic symmetry of perowskite. As explained before the defined symmetry (space-group number and setting) are the reason of the fixed relation of the lattice constants. In contrast to other programs PowderCell is not able to interprete the Hermann-Mauguin symbol. However it will be displayed.

 
The shown table contains the relative atomic coordinates, the site occupation factor (SOF) multiplied by the substitution factor, but also the isotropic temperature factor for each atom of the asymmetric unit. The exact Wyckoff position in form of the Wyckoff letter must not be inserted. In principle PowderCell checks all positions after the transfer by OK and inserts the Wyckoff letter automatically, e.g. all existing letters will be overwritten. 
 
Although the use of the Wyckoff letter is very interesting even then if a structures is given by special positions. So it is only necessary to insert the Wyckoff letter and PowderCell completes immediately the coordinates and describes the characteristic of degree of freedom. If there no degree of freedom no further data must be inputed. In the given upper example this is applied to all three positions defining the asymmetric unit of perowskite. However, also for positions with one or two degrees of freedom the editing will be easier than only this feature. In this case the letter will be inserted and the undefined coordinate will be completed. This is very comfortable and works also for positions like x,x,x or x,2x,1/6. However, then the Wyckoff letter must be inserted onesmore after the input of the quantity. The program creates the missing combinations automatically.
 
The oxydation state shall hint to the different number of valence electrons (ionicity). This value is also optionally. If no value has been inserted, PowderCell inserts the atom in the table which will be identified by the atomic number Z.
On the other side it is possible to define the atomic number only by the use of the corresponding ion. During the edit of the values PowderCell interpretes simultaneously, if such an ion can be considered in the program. This is dependent from the availablility of the necessary constants for the calculation of the atomic scattering factors or the correction of anomaleous dispersion. E.g. we have no data for O2-, so that in this case usually the atom (O) will be used. In order to learn which ions will be supported actually, the dialog screen structure representation options should be called. There all ions will be listed, combined with the shown periodic table. If one click there an atom the corresponding element and all ions will be shown in the table. Unfortunately, a direct taking over of selected data into the edit structure data screen is not possible.
 
Additional atons can be inserted by the creation of new line using the button + atom. However, also superflouos atoms can be deleted be use of the button - atom. Cancel deletes all variations and the original state will bee activeted
Also imported data will be inserted immediately in the table. If there are exist incompatibilities  regarding monoclinic or orthorhombic settings a message like the following will be given:

 

The crystal structure representation 
 
In principle, in PowderCell the structure representation is only a useful resource. From the first it wasn't the intention of this program to create photorealistic pictures like Crystalmaker or Diamond. The program uses the structure representation only as intuitive ability for structure modification or as visualiser for the check of used structure data. But sometimes it is not necessary to display the crystal structure (e.g. during refinement of powder pattern or if one is sure that the structure data are correct). To this end the tool bar contains a special button which enables to dactivate the crystal structure representation. It is marked in the following bitmap by an red frame and shall represent a LED:

 
Only if the LED is On, the loaded structure data will be transformed into a picture of the structure. Of course, the necessary data will be generated, too. The advantage is only that no additional windows are necessary which simplifies the operation of the program. 
The structure previewer can be selected/deselected using the main menue: Structure Þ Representation ON. If the structure doesn't will displayed the corresponding submenues in the main menue are no activated.

As visible in the upper graphic the structure representation options can be selected also then if the representation is active. In case of the upper figure no structure file has been imported. Therefore all other buttons are not active, except the option menue. On this way one is able to vary the different colors or line styles of bonds or cell line before the first structure will be imported.
Of course it is possible to activate the structure representation during the work with the program. In this case the powder pattern representation will be transformed into a small window whereas all imported structures will be generates as picture.

Basic structure representation options 
 
It was recommended to implement option for an easy differentiation of atoms, bonds, cell lines etc in order to support the intuitive character of the program handling.
In order to open the window containing the settings of basic structure representation options the button with the painting palette on the tool bar must be selected using the mouse pointer. It is marked in the following bitmap by an red frame:

 
However, you can call the screen structure representation options also by the use of the main menue: Options Þ Structure representation. 

 
Essentially, this screen contains a periodic law. Using this it is very easy to locate the supported ionic states of the different elements. If one select an element the table below it lists all considered ionic states immediately. Furthermore the radius and the color used in the representation can be adapted manually. The column radius shows as hint the value defined PowderCell as standard. 
Using the button size=r for all all values in used size will be reset to the standard values. The standard values have been defined in the data file asfpcw.dat.

In the DOS version it was possible to represent also ions where no atomic scattering coefficients exist. Sometimes this was very confusing. Therefore we believe that it is better to use only this ions where we have the corresponding coefficients. A separate file containing all radii doesn't exist now. In contrast to the DOS version the asfpcw.dat contains beside the constants for the calciulation of the atomic scattering factors also the atomic mass and the radii. From this the disadvantage follows that Ions like O²¯ will not be listed because we havn't the corresponding data. However, the user is able to define the radii as he like it.
# Þ as help please look atomic radii (major- and minor period elements) and ionic radii (main- and minor period elements).
 
In analogy to the representation options of the atoms or ions also the bond and cell lines can be adapted. Additionally to the color the line character can be changed (thin or thick, dotted etc.). Unfortunately some problems with different graphic devices have been observed. However the reason of this effect could not located until now.
Furthermore two additional factors can be varied: the perspective factor and the size factor. The perspective factor characterizes the degree of central perspective. If perspective will be set greater than 1(standard value), the distance between viewer and projection point will decrease, i.e. the central perspective seems to be stronger. For values smaller then 1 it is vice versa.
The size factor gives a proportional value which will be multiplied with all atomic or ionic radii in the crystal structure representation. Even for closed packing structures this enables to see anything. Of course, this also would be possible if all radii would be edited manually. However then the drawing of bonds doesn't work correctly. Usually size factor should be not bigger than 1. If this factor will be set to 0, no atoms or ions will be drawn. Only the bonds between these are visible. Using this possibility the program allows to create coordination polyhedra etc.

Structure information:   atom — bond — angle
 
Beside basic knowledge in geometric crystallography which is necessary for a successful structure-analytical use of PowderCell also crystal-chemical or molecular-chemical knowledge is very profitable. Nevertheless in the centre of interest are molecules where sometimes the stereology of the molecule or of similar groups are known since a lot of years. Thus, usually it would be very nice to get information about bond length or to use the knowledge about bonding or torsion angles during the structure simulation. This has been implemented in the program.
To use this function the structure must be created, i.e. it must be displayed on screen. In the tool bar a button (red framed in the following graphic) must be selected using the mouse pointer:

	 
There is no entry in the main menue to do this.
In the left bottom of the sreen a window will be faded in, called atom information by click. There different data ranges will be shown but no values are given. Also the number of data ranges in each column are different. To get information about the structure at first the window containing the interesting structure must be selected. To this end the mouse pointer must be pressed about this window. Now only the respective atoms of the structure must be selected using the left mouse button. 

Sometimes this selection procedure can be very complicate in dependece of the complexity of the structure. In principle on must notice only, that the wished atom must be hit as far as possible in its centre. The reason is that PowderCell takes the next atom of the selected position in the projection (marked by the mouse pointer) as selected. From this effective algorithm a nother problem results: hidden atoms. In this case the structure should be rotated befor the atoms will be selected. On the other hand it is possible to rotate also during the selection procedure. Additionally it is recommended to set the size factor not greater than 0.3. Then one has a good general view of the structure. If the drawn bonds derange the program enables to fade out some or all of it. 
 
At first the selected atom will be given by its relative atomic coordinates. In the first column the identifier of the atom will be introduced. Please notice that this identifier is valid for all symmetry-equivalent atoms in the structure. The identifier is defined in the content of the asymmetric unit edited in the worksheet structure data. There no counter has been introduced which enables to differ the different symmetry-equivalent atoms (also if this seems to be so in the following example). 
If one selects a further atom the identifier and the atomic coordinates will be given onesmore. However, now the distance of this positions will be displayed in Å. Selecting a third atom the distance to the second will be given and the angle (in °) between the last three selected atoms will be calculated. If one select bonded atoms (the first to the second and the secon to the third) the given angle describes the bonding angle.
By the selection of the fourth atom a more or less relevant torsion angle (in °) will be displayed. It is defined as angle between that planes described by the first three and the last three atoms (of the selected four). The second and the third atom characterize the sectional line of the two planes but also the rotational axis which anables to transform the two planes into another using the calculated torsion angle.
 
During the selection mode always the last four will be considered. However in the structure data window all distances and bonding angles will be listed.  

In the following picture the above described window is shown. The distances and angles corresponds to that resulting from the SO4-tetraedron of baryte.

 
At first three O atoms lying on one and the same triangle have been selected (at first O3, then O2 and at last O1). At the end the central sulfur atom has been chosen. As expected the two  O—O distances are nearly identically, wheras the O—S distance (bond length between O and S) is claerly smaller. The angle between the 3 O atoms corresponds with  60.039° in practice with the ideal angle of a equilateral triangle. The given angle of 33.481° characterizes the angle between a tetrahedron edges and a O—S bond. For an ideal tetrahedron an angle of 35.264° would be resulted (angle between [110] and [111]). The reason of the difference is the lower symmetry of the baryte structure (P nma). Or other said: The lower symmetry explains the distortion of the tetrahedron. The torsion angle of 35.398° does not differ so much from the ideal value 35.264° (angle between (111) and (110)). In this special case the distorsion of the tetrahedron cannot be detected so good.

Edit bonds
 
The generation of bonds has been carried out using a simple procedure. The radii of the atoms or ions assumed as spheres will be added and then compared with the distance of the atomic positions in the structure. If the deviation of the sum of the radii is in a defined reliable interval (+10%) a bond will be drawn. 
This procedure has a lot of disadvantages and there is no consideration of chemical knowledge but on the other hand this is not the aim of this program. In spite of this argumentation we believed that it should be possible to create some tools to be able to edit bonds manually. To this end a small tool has been implemented. The corresponding button is marked by a red frame on the following tool bar.

 
If one press this button or chooses the submenue Structure Þ Edit bond a dialog box will be opened. Now you must select two atoms in the structure representation. This will be inserted automatically in this box. If there no bond exist one is able to create a so-called special bond. For additional bonds the color and the style can be defined. From this follows that if one likes to change the color of a part of existing bondsthese must be deleted before. Then they don't exist in the array of created bonds displayed by the same color. Now they can create onesmore whith an own color and style. In the following example two O atoms will be given, describing a part of a SO4 tetrahedron. Between this atoms a "bond" is drawn in the program.

 
Let us assume that this "bond" shall be deleted. Of course, you must press the button delete bonds. Using reset bonds all bonds will be redrawn. However, it is also possible to draw between all symmetry-equivalent positions of the atoms selected only this bonds which are in a range given by the minimal and maximal distance.

Structure manipulation: delete, shift and rotate
 
Within PowderCell it is possible to shift or rotate atoms, groups of atoms or complete molecules under keeping of the defined space-group symmetry. This function can be called  using the red framed button in the  tool bar (see the next graphic).

 
Of course, it is possible to use the main menue: Select Þ Move sel. atoms:

 
Please notice, that at first atoms must be selected! This is valid also for the delete function what is a way of structure manipulation too. The difference to shift and rotate is only, that also all unselected atoms can be deleted which is sometimes more easy to handle.
The manipulation interacts with the calculation of the powder pattern. But be careful that in correspondence to the considered spacegroup possibly more then the primary selected atoms will be deleted. Usually it is advisable to delete this position in the dialog box structure data, where all atoms of the asymmetric unit will be listed.
If one has selected the wished part of structure the manipulation dialog can be called.

 
The dialog is separated into a rotation and translation range. The rotational axis corresponds to that used during the structure representation, i.e. they don't be equal with the crystallographic basis vectors. In contrast to that the translation direction correspond to the direction of the crystallographic basis yectors. x and -x represent translations along a of the lattice, y and -y correspond to a shift along b and z or -z is a translation parallel to c. Beside this translations the programm allows to define translation directions fixed by two atoms. Thus it is possible to shift along bonds. The step rate can be given in relative or absolute units.
In analogy to this it is possible too to carry out rotations around a direction defined by two atoms (e.g. a bond). In this case the angle a describes the used rotation angle for each step. Therefore the two other angles are fixed and cannot be changed (as shown in the upper examplary dialog box).

More than one unit cell
 
In principle the content of the unit cell completely describes the crystal structure. However, In a lot of cases molecules will be broken and even their positions cannot be estimated. Also it would be very nice to diplay e.g. special closed packed planes. Very often this is only then possible if more than one unit cell can be displayed. The program enables to do this pressing the red framed butten in the tool bar:

 
There is no entry in the main menue to do this.
The dialog box Set additional space for unit cell will be displayed where it is possible to edit the necessary values. Even for complex crystal structures characterized by a relative high number of occupied positions the limit of 2000 atoms will be attained.

 
The separation of + and - indicate the direction in which the cell dimension shall be changed. But there not only positive values can be edit but each positive, rational number. The values represent relative koordinates, i.e. 1 means that in this direction a complete unit cell will be generated. 
Currently it is only possible to generate along the basis vectors (especially for low-symmetry crystal system this procedure is not optimal). During the generation the translation symmetry of the lattice will used, exclusively, i.e. all relative coordinates will be reproduced by addition with an integer. A spherical generation of atomic positions, i.e. all atoms located in a sphere around a unit cell, has not been carried out. 
Using the button x,y,z=0 all values can be reset to 0.
 
Please notice that the representation of  multiple unit cells don't influence the calculation of the  structure factor.

Projection of the structure on a lattice plane
 
Sometimes it is very interesting to look at the structure from a special direction, e.g. to study existing symmetry elements or differnces in packing density of atomic layers. Even an inhomogeneous distribution of electron density occurs to an increase of rfelection intensity for the corresponding interferences. Therefore it is useful to visualize such effects. To this end the projection of the structure on a lattice plane (parallel to the screen plane) has been developed in PowderCell . Here the projection vector will be defined by the reciprocal lattice vector (hkl). The function can be called using the red framed button on the tool bar:

 
There is no entry in the main menue to do this.
After pressing the button a small dialog box will be displayed where the Miller indices must be intruduced. In the following graphic the projection vector (111) has been arranged.

 
Within the program the projection will be carried out by the use of two rotational angles calculated from the given projection vector. The start is always the standard projection (001). However, no additional information to the used reference direction (horizontal of the screen plane) will be given. Even for cubic crystals this would be useful. 
To estimate the packing density of atomic layers as analogon to the electron density the projection plan must be tilted by 90° around the horizontal or vertical axis. Only then the different layers can be seen. It is recommended to enlarge the structure representation. Use more than one unit cell! 

		  (111) in GaAs 

The upper Picture shows the polarity of Ga and As layers in the semiconductor GaAs. Along the [111] direction on a As layer (brown) follows with a short distance a Ga layer (blue). For the inverse direction the polarity changes. This is the reason for the difference in the etch behaviour of semiconductors like GaAs. 

Rotation angle for used for the projection
 
To be able to use an arbitrary projection plane for the crystal structure — except the standard projection plane (001) — it is only necessary to introduce two rotations which locate the reciprocal lattice vector H characterized by the indices (hkl) parallel to the screen axis zB. In the used procedure H will rotate around zB by an angle jz to localize H in the plane perpendicular to yB. After that H will rotate around  yB by jy to remove H to zB. jy represents the angle between the lattice planes (hkl) and (001). Therefore, for the determination the well known crystallographic equations can be used. In contrast to that the angle jz shall be derived now.
The basis vectors a, b and c shall be described by the orthonormal coordinate system e1, e2, e3, which is identically with the coordinate system of the screen:

 
For the corresponding components the following variables shall be introduced:

 
According to the definition of the reciprocal lattice, H can be represented regarding to the orthonornal coordinate system as:

 
jz describes the angle between the vectors e1 and H' — the projection of H in the e1 - e2 plane. To get the coordinates of H', only the e3 component of H must be neglected.

 
The angle jz results from the following equation:

 

Subgroup selection
 
For a chosen space-group type only then subgroups can be selected if the program contains any information about it. The information about the availability and the existing type is stored in the data file subavai.dat. The individual subgroup data are given in the four data files sub_i.dat, sub_iia.dat, sub_iib.dat and sub_iic.dat.
The subgroup selection takes place using the tool bar by pressing of the left mouse button about the corresponding switcher — red framed in the following graphic.

 
However, one can also use the main menue: Structure Þ Subgroups. There the four different types I, IIa, IIb and IIc will be offered. Only these types for which really subgroup translations are defined can be called (black written). All others are are gray written and cannot be chosen.
For the space-group type P 63/m mc the dialog box for the subgroups of type I (translationengleiche) is identically to the following graphic:

 
Now one only must select the wished subgroup using the mouse or the keyboard. The transformation of all necessary data will be carried out automatically.

remark: The group-subgroup relations have been defined exclusively for standard settings. If you use non-conventional settings you are forced to transform it into the standard setting.
 
The dialog box contains beside the space-group number, the setting and the Hermann-Mauguin symbol only information about the basis vector transformation. An additional origin shift is not given, but of course defined in the corresponding subgroup data file. Using these data the transformation matrix for the atomic coordinates will be derived within the program.
In the dialog box structure data the derived content of asymmetric unit and lattice constants etc. can be studied. To save the new structure file you can use the main menue: File Þ Save.

The data for the calculation of the powder pattern (angle range, radiation used etc.) will be copied from the initial structure.

Selection of space-group setting
 
PowderCell is able to transform non-conventional settings into standard setting, automatically. On the one hand this is required for the use of group-subgroup relations, on the other hand it is very useful for structure data related to old space-group descriptions or non-conventional settings.
Because of the similar data structure of subgroups and transformation, but also caused by the direct connection with the subgroups (available only for standard settings) no additional button has been considered in the tool bar. Therefore this function will be called by the same button:

 
The „transformation“function can be also found using the main menue Structure Þ Transform. This submenue can be chosen only then if PowderCell is able to do this.
Then a dialog will be faded in. The content is of course different in dependence of the chosen space-group type and the setting used. For the space-group number 38 the following graphic shows the dialog if the first setting shall be transformed into another. This is sometimes very useful if different structures described in other symmetry shall be compared (structure similarities).

 
Also here only the setting must be selected.
Generally, in the transformation procedure using in the program no origin shift has been considered. Therefore the data for the basis vector transformation are sufficient for the definite description. The transformation matrix for the atomic position will be derived by the program.
 
For the calculation of the powder pattern the used data will be copied from the initial seting. 

Print the actual screen
 
The graphical output of programs like PowderCell seems to be us very important because of the lot of scripts and papers must be maked out nowadays. Nevertheless, now and then it is desirable to be able to print out the crystal structure or the calculated diffraction representation using the print dialog of Windows directly. This can be called using a button of the tool bar (red framed in the following graphic):

 
However, also data can be printed out, e.g. the list of reflections, the result of refinement etc. That means that always the content of the actual screen will be printed. Please be careful that even the wished window has been activated before pressing the print dialog. 

The shown print dialog depends on the specific dialog used on the actual computer. It doesn't created by the program. However, the configuration or print quality can be changed as usual.

Export current window 

In PowderCell the export of the graphics as well as listings etc. is possible. Generally, the used export procedure is dependent from the active window or box. If the powder pattern or the structure representation is active the graphic export will be called. If e.g. the HKL-listing or the generated bonds have been activated an ASCII export will be occured. For each case the red framed button must be pressed to call this procedures:

	 	

Graphic export 

If the export button has been pressed the export dialog box will be called. It depends from the active window (crystal structure or powder pattern). In principle it is the same dialog box, however the supported formats are different for the respective pictures.

Whereas for the powder pattern only the Windows Meta File (WMF) will be supported (beside the output as bitmap (BMP)) the export of structure pictures will be supported by additional formats: WMF, PostScript, PovRay, WordPerfect Graphic (WPG) and GEM. The last two represent relicts from the DOS version. Therefore some descriptions (e.g. labeling of bonds and atoms) are not implemented. Thus, we recommend to use WMF or PostScript. One possibility to create photorealistc pictures is the program Pov-Ray. The scripts generated by PowderCell can be used by Pov-Ray directly. Sometimes small variations of the camera distance are necessary.

ASCII data export 

This export considers only some special data.  The user should check the possibilities by himself because we always extend this export. Currently the structure data, the generated positions, the bond length, the HKL-listing and the refinement results can be exported using the red framed button. In contrast to that the phase options or used experimental setting simulating the powder diffraction cannot be exported.
As example the follwing graphic shows the message faded if the HKL-list shall be exported.

 remark: The export of powder pattern data is supported by an additional procedure.

 Copy to clipboard

One of the most important advantage of Microsoft's Windows os the implementation of the clipboard. Using this different programs are able to communicate on a simple way. However, this communication must be supported from the respective programs. PowderCell is able to copy the content of the active window in the clipboard using the red-framed button: 

In dependence of the active window (structure or powder pattern) this feature can be called by the main menue: Diffraction Þ Copy to clipboard or Structure Þ Copy to clipboard. Additionally, there a further possibility exists. Press the right mouse button over the active window and the following menues will be faded in:

	for the structure:   	for the powder pattern:    

In both cases Windows Meta Files will be copied to the clipboard. Unfortunately, some programs don't like to support all properties of this format, e.g. the vertical description of reflections. Sometimes it is also recommended to distinguish between the "Paste" and "Paste content" of other programs. If this difference exist you should use the last one.
On the other hand PowderCell allows to export the graphics in several formats. This represents another possibility to implement graphic generated with PowderCell into reports or articles.

Fade in/out the powder pattern
 
As required the calculation of the powder pattern can be switched off (e.g. complicate structures of low symmetry shall be manipulate, low calculation power etc.). To this end the corresponding button must be pressed on the tool bar. It is represented by the blue LED (red framed in the following two graphics). In the first (a) the LED is on i.e. the diffraction pattern will be calculated. Please notice, that only than the last two buttons are active if a powder pattern has been loaded. Thus, the tool-bar part in (a) hints that a powder pattern has been imported before. In the second (b) the LED is off and no powder patter is given:
 
	a)   	b)  
 
Of course, the calculation can be also switched on or off using the main menue: e.g. Diffraction Þ Diffraction OFF (see the following graphic — the last one):

 
The activation of the diffraction calculation will be carried out on the same way. However then the menue has the entry Diffraction ON.
The bitmap on the button shall represent the state of radiation like on X-ray diffractometers. Only if the LED is highlighted the further diffractomtry options can be chosen (variation of experimental conditions, multiple phase generation, reflection listing).

 Variation of the experimental settings
 
For the calculation of the powder pattern some "experimental" condition must be predefined which influence the diffraction more or less intensive. The settings can be varied using the main menue: Diffraction Þ Experiment . However, also the tool bar contains a button which calls the experimental condition dialog. The corresponding button is marked in the following graphic by a red frame. The bitmap shall represent a working X-ray tube.

Pressing this button the following dialog will be opened. There a lot of different settings will be shown describing the condition of a virtual experiment. The defined conditions are valid for all considered phases. More information you get by the selection of the respective ranges on the following graphic. Please try it!

 
Information about the reflection listing and the phase options are described in another place.
Using the automatic unit cell transformation or group-subgroup generation the defined properties will be copied  

Phase mixtures
 
PowderCell is able to simulate phase mixtures. To define the phase specific settings please use the button in the following graphic. It is marked on the representation of the tool bar by a red frame:

Of course, also the main menue can be used: Diffraction Þ Phase options. In both cases the following dialog box will be faded in. The following graphic contains the possibillity to get detailed information by a simple click using the mouse.

 
In this dialog box of the window powder diffraction the most important values regarding the intensity normalization are fixed. Additionally the representation colors and styles of the individual curves can be set. Please notice, that the color of the current phase corresponds to the chosen phase. But more interesting are the values concerning the normalization of the line (or integral) intensity or the used profile function (see the basic remarks to the intensity). Furthermore, here it is possible to consider different models for a description of a preferred orientation. 
The chosen values can be fixed as standard values. This is very useful if one always carries out qualitative or quantitative investigations. In the crystal structure analysis they have no eminent influence. If there arise obvious problems during the calculation of the powder pattern it is sometimes useful to delete the configuration file pcw.cfg. PowderCell creates after an error message (no file exists!) a standard file. Of course, own pre-settings (e.g. the chosen colors) are lost. 

Reflection listing
 
One of the most interesting information source within PowderCell represents the reflection list. Beside the critical inspection of the calculated values, e.g. the check of used extinction laws, the relative or absolute reflection intensites, the given structure amplitude as amount of the structure factor (e.g. to compare with other program applications), the list of used multiplicity MU or the variation of the FWHM (full width at half maximum) in dependence of the Bragg angle, also a connection between the listed reflections HKL and the graphical powder pattern representation has been carried out. Thus, the active reflection in the listing is simultaneously marked by a vertical, colored line in the diffractogram. 
 
The table can be called using the tool bar. The respective button is selected in the next graphic by a red frame.

Once again the dialog box can also be called using the main menue: Diffraction Þ HKL - List . The short key Alt-D + Alt-H represents the fastest way to open the reflection list.

If more than one structure have been considered, be careful that the right listing will be studied. Usually the reflection list of the at least activated structure picture will be shown. However, it is easy to change the listings using the choose of structure file.

Usually, the program has an successful algorithm to consider the systematic extinction laws. However, please notice that even for derived subgroups arbitrary extinction may occur (in dependence of the decrease of symmetry). If you change special positions to general positions using the manipulation tools the resulted zero-intensities should disappear. 

remark: In contrast to the DOS version the Windows version does not allow to summarize all reflections with very similar d-values. On the other hand it is possible to define intensity limits to reduce the number of listed reflections. However, in contrast to the DOS version, all reflections will be considered during the convolution of the integral intensity. This limitation influences exclusively the reflection table.
 
For the estimation of the number of reflections considered in the chosen angle range, this will be shown below the table. Additionally, the number of activated reflection in the table will be given.
Please notice, that sometimes blanks may occur in the column showing the intensity. The reason is, that if reflections have the same Bragg angle, the intensity will summarize. Especially for non-centrosymmetrical structures this follows automatically. 
It is easy to change to the other dialogs (see also experiment and phase options).

Refinement parameters

Refinement results

Refinement 

The requirements on a refinement process are the knowledge about the involved phases and their crystal structure data and, of course, a set of experimental data. At first the structure data must be imported and the diffraction pattern must be calculated. Only then it is possible to import an experimental powder pattern.

Here it is posible to choose between X-radiation and 
neutrons as radiation source. By the use of X-radiation
the anode material must be selected additionally. The
wave length heve been defined within the program.

By the use of X-radiation here the anode material can 
be selected. Additionally, it is possible to enter another 
radieation (of course: wave length). To do this the 
arrow in the last line must be chosen. However, some 
corrections cannot be considered during the calculation.

Using this switcher the consideration of the 
Ka2-component of the chosen radiation can
be activated or deactivated.

Here the anomaleous dispersion can be switched 
ON or OFF. Friedel's law shows that in most cases 
it is not necessary to consider it.

Here the really used wave length are given.  It is 
possible to change this, however no permanently. 
A restart of the program sets the old values.

PowderCell supports different diffraction geometries. These can be 
selected here. From the choice of a geometry special automatical 
intensity corrections result. If you measuring arrangement has not 
been considerd, you may coorect the intensities outside of the 
program. Please use None to define no correction.

Some diffractometers use a variable slit system. The advantage is,
that the measured square of the sample is identically. However, this 
influences the detected intensity very strong.PowderCell is able to 
correct the simuklated pattern using a simple sinQ-dependence.

Here the diffraction angle of the used
monochromator will be given. It depends 
on the used crystal as well as interference. 

Here a constant background component can be
inserted. This value gives also the constant
part of the refined background polynome. 

For the powder pattern calculation the start 
angle 2q must be defined. Please notice that 
the start angle must be smaller than the end 
angle. The given d-value below is only an 
orientation for the user to prevent the input 
of absurd small Bragg angles.

Also the end angle of the range must be given. 
During the start of the program these values will 
be imported using the configuration file pcw.cfg.
Of course, the end angle should be smaller than 180°  :-)).

This number defines the width of a
calculated reflection in FWHM units.
For profiles with a high Lorentzian
part the number should be enlarged.
So edges in the diffractogram may 
be prevented.

The step size gives the distance between the detector positions. 
If one likes to import experimental data the step size is defined by 
the experiment.
It can be changed if the imported powder pattern has been removed. 
If multiple phases shall be considered a step size valid for all phases 
must be defined (in analogy to the radiation).

To change the line style or the color
of the powder pattern, please use
this part of the dialog box.

For the HKL-listing a maximal intensity value
can be defined. Therefore, here the wished
number must be inserted. 

The integral intensity must be convoluted
to create a comparable intensity shape.
The respective profile function must be 
selected here.

If you consider more than one structure
(or phase) you must choose it here to 
change the phase specific values.

The scaling factor represents an proportional
number for the fit to an experimental powder
pattern. With regard to phase mixtures the ratio 
of the scaling factors will be used to determine
the share of each phase.

The variation of the reflection width in dependence
of the Bragg angle can be described analytically.
The used dependences can be selected here.

The calculated reflection intensities may
be corrected by the unit cell volume, the
mass attenuation coefficient or the X-ray
density. The different normalizations can
chosen here.

To consider a preferred orientation the 
wished model can be selected here.

The use of this filter allows to separate 
groups of reflections. The reflection 
type must be selected here.

The given numbers represent the lower limit of
intensity or structure amplitude. Only reflections
with higher values will be listed.  

The given d-values are calculated using the start
and end angles. They cannot be changed directly.
They are shown as additional information.

Spezielle Funktionen
 
Speziell zum Testen des Programmes ist im Hauptmenü ein Punkt aufgeführt, der hauptsächlich für G. Nolze eingerichtet wurde und dem Testen der Untergruppen und Transformationen (monoklin, orthorhombisch, rhomboedrisch) dient. Bei Anwahl der entsprechenden Funktion werden alle verfügbaren maximalen Untergruppen oder Aufstellungen der aktuellen Struktur erzeugt und in ein File geschrieben. Der Name des Files entspricht dem der Struktur. Als Datei-Extension wird *.sgr verwendet.
Anwender von PowderCell können bei eventuellen Zweifeln diese Funktion nutzen, um sich von der richtigen Generierung der Untergruppen bzw. Transformatinen zu überzeugen. Allerdings muß die Berechnung des Diffraktogramms eingeschaltet und der Schalter der Intensitäts-Korrektur auf OFF I'=I/V² , OFF I'=I/2µV² oder OFF I'=I/mV gesetzt sein.