Computer-Aided Crystallography 13

13.1 Introduction

This chapter has been designed to further the knowledge gained from a study of the earlier chapters of

this book. The computer programs that are supplied as theWeb Program Packages and described here

are complementary to that work, and enable the reader to gain practical experience of concepts and

methods germane to X-ray structure analysis. While these stand-alone programs are provided for

quick and easy access for problem solving within the context of this book, we emphasize that the

serious structure analyst must also refer to the other important program systems readily available.

13.1.1 Collaborative Computational Projects

The Collaborative Computational Projects Number 4, for Macromolecular Crystallography (CCP4),

and Number 14 for Powder and Small-Molecule Single-Crystal Diffraction (CCP14), aim to collect

and support the best and most commonly used programs for crystal structure determination by single-

crystal and powder techniques. The software is located on web sites [1, 2] and is freely available to

workers in the academic and research communities, and the wide dissemination of new ideas,

techniques, and practice is encouraged.

Computing is an essential feature in any modern X-ray crystallographic investigation. Here, we can

provide only a flavor of what is available, but enough, we hope, to demonstrate the great importance of

an intelligent application of computational methods to this subject. The program suite provided here

has the following functions:

• To study the derivation of point groups;

• To carry out systematic point-group recognition using crystal or molecular models;

• To simulate the procedures and calculations involved in the determination of crystal structures by

X-ray diffraction data from both single-crystal and powder specimens.

• To carry out various other related computations.

On the basis of a familiarity with these programs, it should be only a small step to proceed, when

needed, with the more comprehensive and detailed crystallographic software to which we have

referred here and in earlier chapters.

The Web Program Packages can be accessed under the web site reference http://extras.springer.com. The programs are very straightforward to use, and it is recommended that the complete suite of

M. Ladd and R. Palmer, Structure Determination by X-ray Crystallography:Analysis by X-rays and Neutrons, DOI 10.1007/978-1-4614-3954-7_13,# Springer Science+Business Media New York 2013

635

programs, together with the data files supplied, should be downloaded to a folder in the PC, according

to instructions given on the web site by the publisher (the ISBN number of the book will be needed).

The programs may be amended from time to time, as improvements and additions are applied to

them. The first set with this fifth edition will be dated 1 January 2012 (Version 5.1), so that any date

after that implies a revision or addition to those programs, and will be so notified on the web site.

However, the programs ITO12, ESPOIR, and LEPAGE will not be subject to such changes unless

they are first revised by their own authors.

13.1.2 Structure of the Web Program Packages

The companion program suite comprises four folders: XRSYST contains the programs for the single-

crystal techniques, PDSYST contains the programs for powder techniques, POWDER contains a single

program system (ESPOIR) also for powder data, and GNSYST contains all other general programs

referred to in the text and problem sections of the book; all the folders contain data as necessary. There

are also six GINO files present, concerned with plotting Fourier maps: they must remain unaltered.The structure of the program suite is illustrated in Fig. 13.1. In general, the programs, which are

provided as IBM-compatible .EXE files, are opened by a double click on the program name or icon. In

some cases, it can be helpful to open the programs in a Command Prompt window and enlarge the

screen. We now describe the different programs and their uses in detail, with the aid of examples.

13.2 Derivation of Point Groups (EULR)

In Sect. 1.4ff we discussed the symmetry operations R and �R (R ¼ 1, 2, 3, 4, 6), taken singly or in

combinations, and gave stereogram representations of them in Fig. 1.32 of that chapter. The first

nontrivial combinations of symmetry operations follow from combining R with �1, and it is easy to

show, with the aid of stereograms, that R and �1 together lead to R/m for R ¼ 2, 4, and 6; for R ¼ 1

↓↓ ↓ ↓ ↓

↓ ↓ ↓ ↓

PROGRAMSUITE

XRSYST PDSYST POWDER GNSYS

XRAYa

(AND GINO FILES)MAKDAT

+Data Sets

ITO12LEPAGEQ-VALSRECIP

+Data Sets

ESPOIRRASMOLb

+

Data Sets

EULRFOUR1DFOUR2DINTXYZLSLIMATOPSMOLGOMSQUARESYMMTRANS1ZONE

+

Data Sets

Fig. 13.1 Structure of the

program packages on web

site http://extras.springer.

com (the IBSN number of

the book will be needed

here). aThere are also six

files (five .DLL and one .

CON) present concerned

with plotting that must

remain unaltered. bThe

RASMOL file is concerned

with drawing. There are

also two other data files in

POWDER for each

substance that must remain

unaltered: one with crystal

data (.DAT) and one with

reflection data (.HKL)

636 13 Computer-Aided Crystallography

and 3, point groups �R already include �1 as an operation in the group. Thus, we have quickly derived 13crystallographic point groups: 1, 2, 3, 4, 6, �1, m ( � �2), �3, �4, �6, 2/m, 4/m, and 6/m.

The point groups that contain more than one symmetry operator display the essence of Euler’s

theorem on the combination of rotations. We have used this theorem implicitly in Sect. 1.4.2,

explicitly in Sect. 2.8ff, and we may state it formally as

R2R1 � R3 (13.1)

which means that, from a given situation, symmetry operation R1 followed by operation R2 is

equivalent to operation R3 applied to the original situation, and we know from the definition of

point group that the three symmetry operations have at least one point in common. We saw this

theorem in operation in the examples of point groups mm2 and 4mm, Figs. 1.28 and 1.29, and it is thebasis of a procedure for determining the remainder of the 32 crystallographic point groups. In general,

the order of carrying out the symmetry operations is important, although the result is not affected with

symmetry operations of degree 2 or less.

We next combine the operations R with another symmetry, operation, say 2, and we need to know

immediately the relative orientations of the rotation axes R and 2 that we use symbolically to

represent these operations. Are they perpendicular to each other or even coincident, and are there

other possibilities to consider?

The program EULR, opened by a double click on the file name, has been devised to follow the

steps of the derivation of point groups described elsewhere, for example, in the references that appear

on the monitor screen when this program is opened. This program is not interactive, but it shows how

Euler’s theorem can be used with the combinations of operations 2 and the permitted values of R to

develop six sets of orientations of rotation axes.

Then, independently of the program, consider replacing two of the rotation axes in each of the six

sets by inversion axes. Why not just one of the rotation axes, or all three of them?

As a final step, we must consider if any new point groups are obtained by incorporating a center of

symmetry into any point group where one is not already present. An extension of the program caters

for certain non-crystallographic point groups that are encountered in studying the symmetry of simple

molecules.

13.3 Point-Group Recognition (SYMM)

There are several ways in which one can approach systematically the recognition of the point group of

a crystal or a molecular model. In the method used here [3], molecules and crystals are divided into

four symmetry types, Table 13.1, dependent upon the presence of a center of symmetry and one

mirror plane or more, or a center of symmetry alone, or one mirror plane or more but no center of

Table 13.1 Crystallographic point groups typed by m and/or �1 or neither

Neither m nor �1 Only m Only �1 Both m and �1

1, 2, 3, 4, �4, 6 m, mm2, 3m �1, �3 2

m, mmm, �3m

222, 32, 422 4mm, �42m 4

m,4

mmm

622, 23, 432 �6, 6mm 6

m,6

mmm

�6m2, �43m m�3, m�3m

13.3 Point-Group Recognition (SYMM) 637

symmetry, or neither of these symmetry elements; hence, the first step in the scheme is a search for

these elements.

In order to demonstrate the presence of a center of symmetry, place the given model in any

orientation on a flat surface; then, if either the plane through the uppermost atoms (for a chemical

species), or the uppermost face (for a crystal), is parallel to the plane surface supporting the model

and the two planes in question are both equivalent and inverted across the center of the model, then a

center of symmetry is present.

If a mirror plane is present, it divides the model into enantiomorphic (right-hand/left-hand) halves.

A correct identification of these symmetry elements at this stage places the model into one of the four

types listed in Table 13.1.

The reader may care to examine a cube or a model of the SF6 molecule, which shows both a center

of symmetry and mirror planes, and a tetrahedron or a model of the CH4 molecule, which shows

mirror planes but no center of symmetry. Models of a cube and a tetrahedron may be constructed

easily:

Cube

Draw a square of side, say 40 mm, on a thin card. On each side of this square draw another identical

square. Lightly score the edges of the first square and fold the other four to form five faces of a cube,

and fasten with “Selotape.” There is an advantage in leaving the sixth face of the cube open, as we

shall see, but we shall imagine its presence when needed.

Tetrahedron

On similar card, draw an equilateral triangle of side ca. 39ffiffiffi2

pmm. On each side of the triangle, draw

another identical triangle. Lightly score the edges of the first triangle, fold the other three triangles in

the same sense to meet at an apex, and fasten with “Selotape.”

Note that on placing the tetrahedron inside the cube, it will be found that an edge of the tetrahedron

is a face diagonal of the cube, thus aligning the symmetry elements common to both models.

If these models are to be used with the point-group recognition program, allocate model numbers

7 and 19 for the cube (or SF6) and tetrahedron (or CH4), respectively. The identification of the point

group of a model then proceeds along the lines indicated by the block diagram of Fig. 13.2, on which

the program SYMM is based [4].

After assigning the model to one of the four “types,” the principal rotation axis, the rotation axis of

highest degree, is identified, together with the number of such axes if more than one, the presence and

orientations of mirror planes, twofold rotation axes, and so on.

The program SYMM is interactive and the directions on the monitor screen should be followed.

If an incorrect response is given during a path through the program, the user will be returned to that

question in the program where the error occurred, for an alternative response to be made. Two such

returns are allowed before the program rejects that particular examination for a further preliminary

appraisal.

It is necessary that the crystal and molecular models to be used are allocated a model number

appropriate to their symmetry, and Table 13.2 provides the necessary key, based on Krantz wood

crystal models, together with molecular examples or possible molecular examples of the point

groups; a set of solid crystal models, appropriately numbered, is equally satisfactory.

It will be realized that some of the molecules listed are not rigid bodies, and will show the

required symmetry only if their functional groups are orientated correctly. The program responds to

the non-crystallographic point groups 1m and 1=m only. The 1 symbol is replaced by the word

infinity in the program description itself, but zero is used to input 1 to the program when asked for

the point-group symbol.

638 13 Computer-Aided Crystallography

Fig.13.2

Flowdiagram

forthepoint-grouprecognition

13.4 Structure Determination Simulation (XRAY)

The purpose of the XRAY program package is to facilitate an understanding of the practical

applications of techniques of structure determination by single-crystal X-ray diffraction that we

have discussed in the earlier chapters, albeit here in two dimensions. Example structures have been

selected that give, in projection, results that are readily interpretable in terms of chemical structures.

It is all too easy, we believe, to use modern, sophisticated structure-solving packages without really

understanding the nature of the calculations taking place within them. In particular, for reasons that we

have discussed in Sect. 11.7.1, very subtle difficulties are sometimes encountered.

A subsidiary program, MAKDAT, enables sets of primary data to be constructed in the precise

format required by XRAY. Alternatively, one may choose to prepare a data set independently, in which

case the layout of data shown by the example sets provided should be followed exactly.

It is stressed that the program system does not teach the subject of structure determination. Rather,

it provides the basic concepts with a medium for their exploration, and so relates closely to techniques

Table 13.2 Point groups and model numbers for the program SYMM with molecular examples

or possible examples

Point group Model number/s Example or possible example

Crystallographic point groups1 91 CHBrClF, monochlorofluoromethane2 77 H2O2, hydrogen peroxide3 84, 93 H3PO4, phosphoric acid4 85, 94 (CH3)4C4, tetramethylcyclobutadiene6 88, 97 C6ðCH3Þ�6, hexamethylcyclohexadienyl�1 78, 79 C6H5CH2CH2C6H5, dibenzyl�3 48 [Ni(NO2)6]

4�, hexanitronickel(II) ion�4 86, 95 [H2+PO4]

�, dihydrogen phosphate ion�6 89, 98 C3H3N3(N3)3, 2,4,6-triazidotriazine2/m 68–70, 72–75, 80 CHCl¼CHCl, trans-1,2-dichloroethene4/m 56 [Ni(CN)4]

2�, tetracyanonickel(II) ion6/m 37 C6(CH3)6, hexamethylbenzenem ð�2Þ 83, 92, 99 C6H5Cl3, 1,2,4-trichlorobenzenemm2 16, 64, 71, 76 C6H5Cl, chlorobenzene3m 42 CHCl3, trichloromethane4mm 87, 96, 100 [SbF5]

2�, pentafluoroantimony(III) ion6mm 81 C6(CH2Cl)6, hexa (chloromethyl) benzene222 67 C8H12,cycloocta-1,5-diene32 43, 47 [S2O6]

2�, dithionate ion422 55 Co(H2O)4Cl2, tetraaquodichlorocobalt622 36 C6(NH2)6, hexaminobenzenemmm 59–63, 65, 66 C6H4Cl2, 1,4-dichlorobenzene�6m2 44–46, 90 [CO3]

2�, carbonate ion4/m mm 49–54 [AuBr4]

�, tetrabromogold(III) ion6/m mm 29–35 C6H6, benzene�42m 57, 58 ThBr4, thorium tetrabromide�3m 38–41 C6H12, chair-cyclohexane23 27 C(CH3)4, 2,2-dimethylpropanem�3 22–25 [Co(NO2)6]

3�, hexanitrocobalt(III) ion�43m 17–21, 28 CH4, methane432 26 C8(CH3)8, octamethylcubanem�3m 1–15 SF6, sulfur hexafluoride

Non-crystallographic point groups�82m – S8, sulfur5 – C5(CH3Þ�5, pentamethylcyclopentadienyl5m – C5H5NiNO, nitrosylcyclopentadienylnickel�1�0m2 – (C5H5)2Re, bis-cyclopentadienylruthenium�5m – (C5H5)2Fe, bis-cyclopentadienyliron1m 101 HCl, hydrogen chloride1=m 102 CO2, carbon dioxide

640 13 Computer-Aided Crystallography

that are in use today. The program is interactive, and the messages on the monitor screen indicate the

steps to be carried out by the user. Nevertheless, it will be useful to discuss here some of the features

and capabilities of the system.

Each data set contains information about the crystal unit cell, crystal symmetry in the appropriate

projection, wavelength of radiation used, a set of Fo(hk0) data, and other information required for the

calculation of electron density maps, structure factors, and jEj values. As the procedures are two-

dimensional, the symbols a, b, g, h, k, x, y, and so on are employed. Where the projection is other than

that on (001), appropriate adjustments are made in setting up the primary data file; thus, true y and z

become x and y to the program.

The following basic procedures may be carried out within the XRAY program package:

• Patterson function, normal and sharpened

• Superposition (minimum) function

• Structure factor calculation

• Least-squares refinement

• Electron density function

• Direct methods: calculation of jEj values• Direct methods: calculation of E map

• Distances and angles calculation

• Scale and temperature factors by Wilson’s method

• jEj values calculated from the structure

The program is executed in the usual manner. Several PAUSE situations occur throughout the

routines in the system, so that material on the screen may be read. Continuation is effected by just

depressing the Enter key. All primary data file names must be four-letter words plus the suffix .TXT,

for example, NIOP.TXT, corresponding to a nickel o-phenanthroline complex, although only the first

four letters of the name, NIOP, are used to call the data set; the program checks for the suffix .TXT. A

name for the output results file, say NOUT, is entered at the keyboard. In some routines, such as the

calculation of jEj values, other output files are organized, with appropriate messages to the screen.

Certain other files pertaining to coordinates are created at an appropriate stage, and their significance

will become clear later. The particular calculations available are then listed on the monitor screen,

and we shall give a brief description of each routine in turn.

13.4.1 Patterson Function

The Patterson function P(uv) may be calculated using either F2o or a “sharpened” mode, in which the

coefficients used are ðjEj2 � 1Þ values, thus providing a sharpened, origin-removed Patterson map.

For the sharpened Patterson, jEj values must be first calculated, and the program so directs. Then, the

system returns to the Patterson routine. Sharpening generally introduces a small number of spurious

maxima, and it is always useful to compare sharpened and unsharpened maps.

The Patterson and electron density maps may be viewed as contoured maps on the screen, which

enhances the interactive nature of the program system. It is possible, for example, to determine the

coordinates of the heavy atom directly from the map on the screen, then go to the structure factor

routine and input the heavy atom x, y coordinates, so as to obtain partial phase information, and

thence calculate a first electron density map. It should be possible then to recognize other atom

peaks from this first Fourier map, so beginning the process of structure determination by successive

Fourier synthesis.

At some stage it may be helpful to print and contour a Patterson or electron density figure field, so

as to get a clearer picture of the projection, particularly where contours below the relative level 10 are

13.4 Structure Determination Simulation (XRAY) 641

involved. All Fourier maps are scaled to a maximum of 100, and 10 is the lowest level that is

contoured on the screen.

It should be noted that, if the WINDOWS “screen saver” comes into operation while a plot is on

the screen, then owing to some delay, the cross-wires might become fixed in position. On reactivating

the plot, a second, mobile cross-wires may appear. To avoid these events, the screen saver may be

deactivated.

13.4.2 Superposition Function

This routine calculates a minimum function M(x, y) at each grid point of the projection, Sect. 7.4.6:

Mðx; yÞ ¼ Minimum of fPðxþ Du1; yþDv1Þ;Pðxþ Du2; yþ Dv2Þ; . . . ;Pðxþ Dun; yþ DvnÞg where

Du1;Dv1;Du2;Dv2; :::Dun;Dvn represent n displacement vectors, which may indicate either atom

positions or a set of vectors, including 0; 0, obtained from a partial interpretation of a Patterson map;

symmetry-related positions should be entered. The grid points x and y are determined by the values

set in the primary data. The minimum function, if successful, should indicate atomic positions that

can then be used as discussed above.

13.4.3 Structure Factor Calculation

Each atom contributing to the structure factor calculation requires the following data:

1. Atom type identity number: a list is given at the start of the routine.

2. Fractional x and y coordinates of the atoms.

3. Population parameter: 1, unless the atom is in a special position.

4. An overall (isotropic) temperature factor is given initially from the primary data; it may be altered

by routine 10. In either case, the value will be allocated to each atom, in preparation for subsequent

refinement.

In the calculation of structure factors, the coordinates may be entered either from a file or at the

keyboard. If entered at the keyboard, the final line must be END. If entered from a file, the file must be

named XYS.TXT and the first line must be the number of atoms to follow. The output from this

routine is self-explanatory, and after an jFcj calculation and least-squares refinement, the current

coordinates are retained in the file COORDS.TXT. This file may be invoked in a subsequent

calculation, or edited as desired, before being used as input data. The file XYS.TXT remains

unaltered by the program.

13.4.4 Least-Squares Refinement

The routine uses the atoms already located. It may be just the heavy atoms, or a number of other atoms

may be included as well. The routine uses the diagonal least-squares approximation to the ideal full-

matrix procedure: it is fast, and satisfactory for emphasizing the principles involved. The x and ycoordinates and the isotropic temperature factor B (initially the same for each atom, from the primary

data) are refined.

The changes dx, dy, and dB are determined and applied to each atom, and the R factor listed

together with other parameters. The cycle can be repeated until no further improvement is obtained,

as judged by near-constancy in the R factor, or by very small changes in the dx and dy shifts. The scale

642 13 Computer-Aided Crystallography

factor is determined as SjFcj/SFo and may be applied to the data at the end of any cycle of refinement.

In this diagonal approximation, 60% of the calculated shifts are applied by the program. The value

may be altered at any cycle, as desired. At each cycle, the new coordinates are stored in the file

COORDS.TXT (see Solution 13.1 for the results).

13.4.5 Electron Density Maps

When some phase information becomes available, an electron density summation with the phases

allocated to the corresponding Fo data should reveal a portion of the structure, if the information from

the partial structure is correct. Then, more atom positions can be interpreted from the map and built

into the next structure factor calculation, and so on.

The same program routine will also calculate a difference electron density map, using Fo � jFcj ascoefficients, provided that the R factor is less than 0.3. In the difference synthesis, atoms placed

incorrectly appear as low or negative density regions whereas unallocated atomic positions will show

positive density, both relative to the general level of the figure field. It follows that when all atom

positions have been determined correctly, the difference map figures should show a nearly level,

ideally zero, figure field. Thus, a difference map may be used to make appropriate adjustments to

atomic positions.

Sometimes a fairly well-refined structure may show a significant positive region on the difference

map, which may indicate the presence of solvent of crystallization. We note also that incorrect

temperature factors can lead to variations in the level of the figure field, although this effect would be

expected to be small when individual temperature factors are applied, as from a least-squares

refinement.

13.4.6 Direct Methods: Calculation of |E| Values

We have shown that jEj values may be calculated from the equation

jEj2 ¼ K2F2o

ePj

f 2j expð�Bl�2 sin2 yÞ( ) (13.2)

where the symbols have the meanings as described elsewhere, and the values of B and K may be

obtained from a Wilson plot. However, a single isotropic B factor may not be representative of the

structure; consequently, a Wilson plot may deviate from linearity. In an alternative procedure [5], we

write

jEj2 ¼ KðsÞF2o;corr

ePj

f 2j

( ) (13.3)

where K(s) is a factor that includes adjustments for the scaling of Fo and the temperature effect on fj.

In implementing this method, a number n of ranges is set up in equal increments of s2, where

s ¼ (sin y)/l. For each range, K(s) is computed asP

es2P

F2o

�, where each value of Fo is given

13.4 Structure Determination Simulation (XRAY) 643

its appropriate reflection multiplicity, according to the space group. The K(s) values, then as a

function of s, are interpolated [6] so as to derive F2o;corr for each reflection. Then, jEj2 is given by

jEj2 ¼ jFo;corrj2es2

(13.4)

It is desirable to enter only the s and K(s) values corresponding to the extreme ends of the data

range, because extrapolation, particularly at the low y end of a K-curve, can be uncertain. Data are

output to the monitor that enable these values to be estimated. A facility is provided for printing in the

main output file any reflections excluded by the procedure. There should be none, but if there are any,

it means that the values entered for the extreme ends of the K-curve have not been chosen

satisfactorily.

The jEj values are written to a file EVALS, together with some statistics of the jEj distribution; jEjvalues greater than or equal to a chosen limit, ELIM, are written to the main output file, and a

P2

listing is set up in the file SIG2, in descending order of jEj magnitude. The file EDATA contains the

jEj values greater than or equal to ELIM arranged in parity groups.

The program comes to a halt at this stage, so that theP

2 listing can be printed and signs for the

jEj values developed along the lines already discussed in the text; see Sect. 8.2.2. Data sets are

provided only for centrosymmetric structures, but similar procedures are used at the beginning of a

determination of a non-centrosymmetric crystal structure. In practice, it may be necessary to re-run

the jEj values link with a slightly lower value of the limit ELIM, if insufficient data are produced for a

successful sign determination process.

After a set of signs has been determined, the next program, Routine, 7, is used to prepare an E map.

Alternatively, where the plane group is p2gg, the program FOUR2D, Sect. 13.6.2, can be used.

13.4.7 Calculation of E Maps

An jEj map is an electron density map calculated, in the case of a centrosymmetric structure,

with �jEj values. This routine in the program provides for a straightforward transfer of the jEjvalues, with their signs as determined through the

P2 routine, to a Fourier calculation. As jEj values

are sharpened coefficients, a few spurious peaks may be anticipated. Thus, chemical knowledge has to

be brought to bear on the extraction of a sensible chemical structure or fragment. Once this has been

done, and shown to be satisfactory, electron density calculations may be carried out with normal Fo

coefficients.

This link of the program, however, permits modifications of signs, for further E maps, without

restarting the system from scratch. A second call to routine 7 lists the current set of data as h, k, and s.A value of zero for the sign s indicates an unsigned reflection which does not contribute to the E map.

To the question “Do you want to retain some of the current values of cos(phi)?” the answer no implies a

re-input of new signs. The answer yes implies that some changes are desired, and the opportunity to do

so follows. The changes are then made, and a further E map calculated.

From the E map, atomic positions should be found that can be entered into the structure factor

calculation, with or without a least-squares refinement, and an Fo Fourier map then calculated as

described above.

644 13 Computer-Aided Crystallography

13.4.8 Bond Lengths and Bond Angles

This routine calculates bond lengths and bond angles, and distances between nonbonded atoms in the

structures. The amount of information extracted depends on the distance limits input to the routine.

Generally, there is no need to set a limit greater than the van der Waals contact distances, typical

values of which are listed in Table 13.3. In interpreting the results from this routine, it must be

remembered that, in working in two dimensions, some variations from standard numerical values are

to be expected, because of the distortion of the molecule in projection. In order to minimize this

effect, however, structures have been selected in which the plane of the molecule lies nearly in the

plane of the projection.

13.4.9 Scale and Temperature Factors by Wilson’s Method

The general procedure of Wilson’s method has been discussed elsewhere, and is implemented in this

routine. The data output contains a breakdown of the individual parts of the calculation. In particular, it

lists the data for plotting the Wilson line, and thus checking on the linearity of the plot.

For reliable statistics, the portion of reciprocal space under consideration should include all

reflections other than systematic absences. Accidental absences, that is, reflections with intensities

too low to be recorded, should be included at values 0.55 of the localized minimum Fo, that is, the

minimum Fo in a given range, for a centric distribution, and 0.66 of the localized minimum Fo for an

acentric distribution. Some of the data sets do not have the accidental absences included. The NO2G

data is, however, complete in this respect. A check on this aspect of a data set is given by inspecting

the number of reflections in each of the ranges of the Wilson plot routine; they should be approxi-

mately equal.

13.4.10 |E| Values Calculated from the Structure

This link has been incorporated to show how jEj values may be calculated from a crystal structure. It

follows that all atoms, preferably including hydrogen atoms, must be present in their correct

locations. Then, jE(hk)j is calculated from

jEðhkÞj ¼ 1ffiffiffiffiffiffiffiffiffiffiehks2

p ðA2Z þ B2

ZÞ1=2 (13.5)

Table 13.3 Van der Waals radii for some common species

Atom Radius (A) Atom Radius (A)

H 1.20 C 1.85

N 1.50 O 1.40

F 1.35 Si 2.10

P 1.90 S 1.83

Cl 1.80 As 2.00

Se 2.00 Br 1.95

Sb 2.20 Te 2.20

I 2.15 –CH3 2.00

>CH2 2.00 –C6H5 1.85a

aHalf-thickness of the phenyl ring

13.4 Structure Determination Simulation (XRAY) 645

where ehk is the epsilon factor for the hk reflection, s2 is given by

s2 ¼Xj

Z2j (13.6)

and AZ and BZ are given by

AZ ¼Xj

Zj cos 2pðhxj þ kyjÞ; BZ ¼Xj

Zj sin 2pðhxj þ kyjÞ (13.7)

and the sums are taken over the N atoms in a complete unit cell. It may be noted that the temperature

factor is not involved in the definition of jEj, because for a “point atom” fj ¼ Zj for all h, k. The phase

associated with jEj is given generally by tan–1(BZ/AZ), having due regard to the sign of both AZ and BZ,

but will be 0 or p for centrosymmetric structures. It follows that the term E(00)1 is given by

ðPj ZjÞ=ðP

j Z2jÞ1=2 which, for identical atoms, is

ffiffiffiffiN

p; the value of E(00) is listed, with jEj statistics

in the results files EVALS and ECALC.

13.5 Crystal Structure Analysis Problems

These problems have been devised in conjunction with the XRAY program system. The different data

sets may not all operate equally well with all methods of structure solving provided by the system.

Hence, although the operations available are indicated on the monitor screen during execution of the

program, we suggest here those procedures by which satisfactory results may be obtained for each

data set provided. Organic species mostly have been chosen because it is not difficult to find examples

that show well resolved and interpretable projections.

There are certain features associated with working in two dimensions that we should remember:

• Because of the relatively small amount of data and a certain degree of inclination of the molecule

to the plane of projection, some bond lengths and angles will not calculate to typical values.

• Fourier maps will not necessarily be true to scale, and will not present the b angle in oblique

projections, but they will be satisfactory insofar as they give good practice with the structure

determining methods, and enable atoms to be located. When the axis of projection is not

perpendicular to the plane of projection, the true axes of the projection should be modified by

an angular term. For example, for a monoclinic unit cell projected on to (100), the axes are b and

c sin b. The sinb term may be important where the coordinates are measured from a map and the

b-angle is very different from 90�. For the projection of a triclinic cell on to (001), the axes are

a sin b and b sin a. However, since coordinates are almost always refined by least squares, the

correction may often be ignored.

• It is rare to be able to locate hydrogen atoms in structures determined from projections, but they

may be positioned by geometrical considerations.

The XRAY program is entered as already described. Generally, it will help to enlarge the screen.

Solutions are provided for the structure determinations in Sects. 13.5.1 and 13.5.6 but, with other

1 E(00) ¼ E(000).

646 13 Computer-Aided Crystallography

examples, the correctness of the results should be judged according to the criteria of correctness

already discussed.

13.5.1 Ni o-Phenanthroline Complex (NIOP)

Enter the data file name NIOP and then a name for the general output file, say NOUT. The compound

crystallizes in space group P212121, the plane group of the projection is p2gg and Z ¼ 4. Open the

menu, select the Patterson link, carry out Patterson and sharpened Patterson syntheses, and plot the

maps on the screen. Print at least one of the maps, so as to make for easy comparison with the other. In

p2gg, the general equivalent positions are

� x; y; 12þ x; 1

2� y

� �so that interatomic vectors will have the Patterson coordinates

� 2x; 2y; 12; 12� 2y; 1

2� 2x; 1

2Þ�

On the screen, along the lines x ¼ 1

2and y ¼ 1

2there are two large peaks that may be taken as

Ni–Ni vectors; they are double-weight (why?). From them we obtain the atomic coordinates as ca.

0.24, 0.18; the peak corresponding to the 2x, 2y vector is not well resolved in this projection. Other

peaks indicate possible Ni–S vectors, but the results may not be completely satisfactory. It may be

useful to keep copies of the Patterson maps for later reference.

Use the coordinates of the nickel atom in the asymmetric unit to calculate structure factors and

then an electron density map. This map shows the Ni atom positions and two other strong peaks to

which the S atoms can be allocated. Repeat the structure factor and electron density calculations with

these atoms (three per asymmetric unit), or first apply a least-squares refinement. The electron density

map may not be obvious to interpret in terms of all carbon atoms. If necessary, print the asymmetric

unit of the Fourier map and contour it carefully; the lowest contour on these plots is 10 and the

maximum is 100.

It may help in this example to contour the figure field at level 5. Search for peaks that would make

up the picture of the phenanthroline complex, Fig. 13.3. It may not be possible to find all the

remaining atomic positions at this stage, but enough will be located to enable a better electron density

map to be calculated.

When all 21 atoms, excluding hydrogen, have been found, several cycles of least-squares refine-

ment, with scale factor adjustment, should converge with an R-factor of about 9.8%, which is

probably the best result that can be obtained with this data set. The difference electron density map

at this stage will be almost featureless. A small, negative peak near the location of the nickel atom

may indicate that the isotropic temperature factor is not a completely satisfactory approximation for

this species, or that there are insufficient terms for true convergence of the Fourier series, Sects. 6.2.1

and 6.9.1.

Bond lengths and angles may be calculated. Because the c dimension is only 4.77 A, the molecule

is quite well resolved in this projection, and the lengths and angles should have fairly sensible

chemical values. The results for the nickel and sulfur atoms are Ni–S(1) ¼ 1.953 A, Ni–S

(2) ¼ 1.917 A, and S(1)–Ni–S(2) ¼ 95.08�; small variations from these values may reflect the

state of the refinement.

13.5 Crystal Structure Analysis Problems 647

13.5.2 2-Amino-4,6-dichloropyrimidine (CL1P)

The title compound, C4H3N3Cl2, crystallizes in space group P21/awith Z ¼ 4. The plane group of the

projection on (010) is p2, doubled along the x-axis because of the translation of the a-glide plane. The

data for this structure and for CL2P produce satisfactory Wilson plots.

There are two chlorine atoms in the asymmetric unit, not related by symmetry. Hence, these two

chlorine atoms together with the two related by the twofold symmetry will give rise to eight non-

origin Cl–Cl vectors:

Type 1, single weight for each: � ð2x1; 2y1Þ; �ð2x2; 2y2ÞType 2, double weight for each: � ðx2 � x1; y2 � y1Þ; �ðx2 þ x1; y2 þ y1Þ where the single-weight

vectors terminate at the corners of a parallelogram, and the double-weight vectors terminate at the

mid-points of its sides.

Solve the Patterson projection for positions of the chlorine atoms, and then complete the structure

determination for the non-hydrogen atoms. It will be helpful to print more than one copy of the

Patterson map, and then to join them such that the origin is at the center of the composite. (Hint: the

coordinates of one of the chlorine atoms are ca. 0.16, 0.16.)

The data for this projection will refine to ca. 12.1%. The bond lengths and angles from this

projection indicate a tilt of the molecule out of the plane of projection.

13.5.3 2-Amino-4-methyl-6-chloropyrimidine (CL2P)

Consider the unit cell data for this compound, C4H6N2Cl, and that for the dichloropyrimidine

just studied:

2-Amino-4-methyl-6-chloropyrimidine 2-Amino-4,6-dichloropyrimidine

a (A) 16.426 16.447

b (A) 4.000 3.845

c (A) 10.313 10.283

b (�) 109.13 107.97

Z 4 4

Space group P21/a P21/a

Fig. 13.3 The molecular structure of C12H14N6S2Ni

648 13 Computer-Aided Crystallography

The two sets of crystal data are sufficiently similar for the two pyrimidine derivatives to be treated

as isomorphous. Hence, it should be possible to allocate trial atomic coordinates from the structure of

the dichloropyrimidine; one of the chlorine atoms in the dichloro-compound has been replaced here

by a methyl group.

Calculate Patterson maps for this compound and, by comparison with the previous structure,

obtain atomic coordinates for a trial structure of this compound. Then refine the trial structure by

successive Fourier syntheses and least squares. Alternatively, use the coordinates from CL1P mutatis

mutandis and calculate a first electron density map.

This data set refines to an R value of approximately 12.3%, but some of the bond lengths differ

from the accepted values for this compound because of the inclination of the molecule to the plane of

projection.

13.5.4 m-Tolidine Dihydrochloride (MTOL)

m-Tolidine dihydrochloride, crystallizes in space group I2, a non standard setting of C2, with Z ¼ 2.

The molecules occupy special positions on twofold axes. The plane group is p2 in the projection on to(010). Thus, the chlorine atoms are related by twofold symmetry to give a Patterson vector at 2x, 2z.

The projection can be solved by the heavy-atom method, and refined to ca. 22% with the given

data. The atoms are well resolved, albeit with some distortion and the bond lengths are at variance

with standard values, because no account can be taken of the third dimension in their calculation from

this projection.

13.5.5 Nitroguanidine (NO2G)

Nitroguanidine, C(NH2)2NNO2, crystallizes in space group Fdd2, with the unit cell dimensions

a ¼ 17.639 A, b ¼ 24.873 A, c ¼ 3.5903 A, and Z ¼ 16. The small c dimension, approximately

equal to the van der Waals nonbonded distance between carbon atoms, means that good resolution

will arise in the projection on (001). The plane group of this projection is p2gg, with the a and bdimensions halved and four molecules in the transformed unit cell.

This structure is suitable for the direct methods procedure. In two dimensions, two reflections

suffice to fix the origin provided they are chosen one from any two of the parity groups h even/kodd, h odd/k even, h odd/k odd; h even k even is a structure seminvariant and cannot be used to restrict

the origin.

In p2gg, the sign relationships in reciprocal space may be summarized as sðhkÞ ¼ sðh kÞ ¼ð�1Þhþksð�hkÞ, so that both positive and negative signs will be generated by the

P2 equation.

One or more letter symbols may be used as necessary, in order to aid the sign allocation process.

In some cases, the signs attaching to such letters evolve during the procedure; otherwise, trial

E maps must be constructed with permuted values for the letter signs. This structure will refine to an

R value of approximately 5%.

As an alternative procedure at the E map stage, the data can be assembled as lines of h, k, jEj, s(E),and used in conjunction with the program FOUR2D, which has been written for plane group p2gg.

13.5 Crystal Structure Analysis Problems 649

13.5.6 Bis(6-sulfanyloxy-1,3,5-triazin-2(1H)-one) (BSTO)

This compound, (C3H2N3O2S)2, crystallizes in space group P21/m, with Z ¼ 2, so that the molecules

occupy special positions on m-planes. In projection on (010), the plane group is p2 and the molecules

occupy general positions in the plane group.

The Patterson maps indicate more than one peak of similar height in the asymmetric unit, so that it

may be necessary to investigate both of them in order to find a good trial structure. This structure

responds well to the superposition technique, so that the solution is not as difficult as might have been

expected. Refinement to ca. 15.5% can be achieved.

13.5.7 2-S-methylthiouracil (SMTX and SMTY)

2-S-methylthiouracil, C4H6N2OS, is triclinic, with space group P�1 and Z ¼ 2. Data for the (100) and

(010) projections are supplied, for which the plane group is p2 in each case. This structure may

present more difficulty than that in the previous example, because several peaks of similar height

occur in the Patterson maps.

The correct choice refines here to ca. 13.0% (R ¼ 7.3% with three-dimensional data has been

reported in the literature). The molecule is mostly well resolved, but not all atoms, particularly the

carbon attached to sulfur, are clearly resolved in this projection. Note that, for the (100) projection, the

axes marked x and y on the plot are, strictly, y sin g and z sin b, respectively.The second data set for this compound relates to the (010) projection. By solving it, a three-

dimensional model for the compound can be built up.

The given selection of problems provides good practice in current, basic structure-solving meth-

ods. Other problems can be built up as desired; data for suitable structures can be found in the early

volumes of Acta Crystallographica. If you do this, remember to adopt the correct format.

13.6 General Crystal Structure and Other Programs

13.6.1 One-Dimensional Fourier Summation (FOUR1D)

This program calculates a one-dimensional Fourier summation, r(x). The data comprise lines of index

h and coefficients A(h) and B(h),which must be available in a file named ABDAT.TXT (example

given). If r(x) is centrosymmetric, B(h) should be entered as zero for each data line. The figure field

for r(x) is established in the file RHOX.TXT from whence it can be plotted; it is normalized to a

maximum value of 50. The file FUNCTN.TXT contains the true values of the data in a form suitable

for the Fourier transform program TRANS1. The desired interval of subdivision N is entered at the

keyboard; its maximum value in the program is 100.

13.6.2 Two-Dimensional Fourier Summation (FOUR2D)

This program computes a two-dimensional Fourier summation for plane group p2gg. The datamust be in

a file named TWODAT.TXT (example given) as lines of h, k, Fo, and s; s is the sign (�1) that multiplies

Fo. The interval of subdivision is 40 along both the x and y axes, and the output from the file RHOXY.

650 13 Computer-Aided Crystallography

TXT may be joined along the duplicated lines y ¼ 20. Four molecules are contained within the unit

cell of the plot, which is normalized to a maximum of 100. Contour the plot in steps of 10 units.

The source code FOUR2D.F90 for this program, written in FORTRAN 90, is also supplied.

Those conversant with FORTRAN 90 may wish to modify this program for other plane groups.

The procedure is straightforward [7], making use of the electron density equations given in the

International Tables for X-ray Crystallography, vol I [8]. Consider, for example, plane groups p2:we can write r(xy) as

rðxyÞ ¼ KðCT2FC1 � ST2GS1Þ (13.8)

where K is a constant involving the normalization of the output results, CT2 and ST2 are the transposes

of matrices of cos(2phX) and sin(2phnY), of order hmax � nX (nX ¼ nY ¼ 0/40, 1/40, 2/40,. . ., as setcurrently in the program), and C1 and S1 are matrices of cos(2pknY/b) and sin(2pknY/b), of orderkmax � nY. In plane group p2, F and G are matrices of order hmax � kmax, with elements

½FoðhkÞ þ FoðhkÞ� and ½FoðhkÞ� FoðhkÞ�, respectively. A step to form these elements could be

inserted into the program.

13.6.3 One-Dimensional Fourier Transform (TRANS1)

This program calculates the Fourier transform of a one-dimensional function f(x). The function is

divided into an even number of intervals, up to a maximum of 100, and contained, one datum to a line,

in the file FUNCTN.TXT; only the values of the function are used as data. The number of data (the

interval N of subdivision of the function) is entered at the keyboard, followed by the maximum

frequency hmax for the output coefficients.

Because of sampling conditions, Sect. 6.6.7, if N is chosen as 30, hmax could be conveniently

10–15. The output in the file ABDAT.TXT can then be used with FOUR1D to recreate the original

function, f(x). Note: Because FOUR1D writes a file named FUNCTN.TXT, the original values of this

function will be lost unless saved in another file.

13.6.4 Reciprocal Unit Cell (RECIP)

This program determines the parameters of the reciprocal unit cell from those of the corresponding

direct space unit cell (or vice versa) and the volumes of both cells. The input consists of the reciprocal

constant K (chosen as unity here) and the parameters a, b, c, a, b, and y, say, 5.0 A, 6.0 A, 7.0 A, 90.0�,105.0�, and 90.0�. The output is self-explanatory.

13.6.5 Molecular Geometry (MOLGOM)

This program calculates bond lengths, bond angles, and torsion angles. It requires the following data

input from a file named MOLDAT.TXT (example given):

Unit-cell parameters: 5.0, 6.0, 7.0, 90.0, 105.0, 90.0

Number of atoms (e.g.,)

Atom number and x, y, z coordinates

13.6 General Crystal Structure and Other Programs 651

As prompted, the atoms forming a torsion angle are entered at the keyboard. The convention

relating to the sign of the torsion angle has been discussed, Sect. 8.5.2 and Appendix C. The results

are listed in the file GEOM.TXT. Note that a program error at the torsion angle stage means that either

the data is incorrect or that a torsion angle cannot be defined by the order of the atoms given. If the

coordinates are in absolute measure, as with the test data, then a, b and c are each entered as unity.

13.6.6 Internal and Cartesian Coordinates (INTXYZ)

This program converts the geometry of a molecule in terms of its internal coordinates, that is, bond

lengths, bond angles, and torsion angles, to a set of Cartesian coordinates for the molecule. The data

must be supplied from a file named CART.TXT (example given), and take the form of lines of atom

code number, bond angle, torsion angle, bond length, as indicated on the monitor screen (with results)

after the program is opened; the convention for torsion angles given in Appendix C applies. The first

entry is always

0 0:0 0:0 0:0

In subsequent lines, the code number of the current atom is the atom number of a previous atom towhich the current atom is linked. As an example of input data, consider the molecular fragment shown

here in Fig. 13.4, with its internal coordinates, or geometry, as given. Then the input data has the

following format:

Atom code number

(not entered) Input data for file CART.TXT

1 0 0.0 0.0 0.0

2 1 0.0 0.0 1.49

3 2 109.0 0.0 1.50

4 3 110.0 0.0 1.54

5 4 107.0 180.0 1.51

6 4 105.0 �30.0 1.52

Atom number 1 is at the origin, and the fragment 1–2–3–4–5 is planar. Does the 4–6 bond lie

above or below this plane? The Cartesian coordinates given by the program are in the file

METRIC.TXT as:

Atom X Y Z

1 0.0000 0.0000 0.0000

2 �1.4900 0.0000 0.0000

3 �1.9784 1.4183 0.0000

4 �0.7815 2.3874 0.0000

5 �1.3472 3.7875 0.0000

6 0.32437 1.64686 0.7341

652 13 Computer-Aided Crystallography

13.6.7 Linear Least Squares (LSLI)

This program determines the best-fit straight line to a series of data points that must number at least 3

and, in this program, cannot exceed 100. Data must be entered from the keyboard or from a file named

LSSQ.TXT (example given). The first two lines of data, title, and number of data pairs, must alwaysbe entered at the keyboard. Then the remainder of the data follow in lines of xi, yi; unit weights for

each observation are assumed in this program.

It is implicit that errors in x are significantly smaller than those in y. The goodness-of-fit is reflectedby the values of s(a), s(b), and Pearsons’s r coefficient. If the errors in the parameters a and b are to

be propagated to another quantity z, then they follow the law given in Sect. 8.6.

13.6.8 Matrix Operations (MATOPS)

This program accepts an input of two 3 � 3 matricesA and B, and forms A + B,A � B,A � B,AT,

BT, Trace(A), Trace(B), Det(A), Det(B), Cofactor(A), Cofactor(B), A�1, and B�1. If results are

required on only one matrix, A, then B is set by the program as the unit matrix

1 0 0

0 1 0

0 0 1

0@

1A

13.6.9 Q Values (QVALS)

This program is useful in conjunction with work on indexing powder diffraction patterns. Given the

unit cell parameters a, b, and c in A, and a, b, and g in degrees, the program produces a set of values of

104Q; maximum values for h, k, and l are also entered at the keyboard. Provision is made for the

indices k and l to take negative values. Thus, in any symmetry higher than triclinic, duplicate values

of Q will be generated and may be discarded as required. The results are in the file INDEX.

Fig. 13.4 Hypothetical fragment C6, for input to program INTXYZ

13.6 General Crystal Structure and Other Programs 653

13.6.10 Le Page Unit-Cell Reduction (LEPAGE)

This program was written by A.L. Spek of the University of Utrecht and kindly made available by him

to the academic community. For reduction, choose the D-option when prompted, and enter the unit

cell parameters as indicated, one to a line. It may be desirable to vary the “2-axis criterion” by means

of the C-option.

The program reports inter alia the input unit cell, the reduced unit cell, the conventional

crystallographic unit cell (which may be the same as the reduced cell), and the transformation

matrices for a, b, c and x, y, z. Other options are provided by the program, but they need not concern

us here. Note that this program refers to the triclinic system by the symbol a: the alternative name for

the triclinic system is anorthic.

A useful mnemonic for using any transformation matrix M and its inverse has been given in Sect.

2.5.5 and the scheme of Fig. 2.16 in the same chapter.

13.6.11 Zone symbols/Miller indices (ZONE)

This program calculates the Miller indices of a plane from the input of two zone symbols, or the

symbol of a zone from the input of the Miller indices for two planes.

13.7 Automatic Powder Indexing: ITO12

This program has been made available to the academic community by courtesy of Dr. J. Vissser,

Technisch Physische Dienst, Delft [9]. The format of the input data is very specific and must be

followed. The data file must be named ITOINP.DAT, and set out as follows; the parentheses indicate

FORTRAN formats:

Line 1: Title, up to 80 alphanumeric characters (A80).

Line 2: Leave blank; it is related to a number of parameters that take default values in the program,

and which we need not discuss here.

Line 3: Four parameters: a zeroshift, 0.0 is recommended; a print controlminimum value ofM20 for a

lattice to be printed, 4.0 is recommended; a print control minimum value of lines indexed for a

lattice to be printed, 14.0 is recommended; the number of data, between 20 and 40. These four

parameters are entered as real numbers, and terminate at character numbers 10, 20, 30, and 40,

respectively, in the line (4F10.5).

Lines 4: The data, 104Q or sin2 y or 2y values (in ascending order), or d (in descending order), in nlines each containing eight such values, the data ending at character numbers 10, 20,. . .,80 in each

line (8F10.5).

Line 5: Leave blank.Line 6: The word END as its first three characters (A3).

An example input file ITOINP.DAT, which relates to Problem 9.6a, is provided with the Program

Suite POWDER folder. One output file is generated by the program, and provides an echo of the input

data file. A second output file contains the results of the indexing; it is mostly self-explanatory.

However, the following column heads have meanings as follow:

654 13 Computer-Aided Crystallography

The best results follow based on the first 20 indexed lines, and then those based on all lines in the

data set, up to 40; finally the best, refined unit cells are printed. From a study of the observed

reflections, deductions may be made about the space group of the crystal. The program and example

data set are together in a folder named ITO12.

13.8 Automatic Powder Structure Solving: ESPOIR

This program has been made available to the academic community by courtesy of Professor A Le

Bail, Laboratoire Fluorures, Universite du Main, Le Mans, and we discussed it in Sect. 12.11. Only

the essentials needed to run a data set are included here, and the reader is referred to an original

reference [10] for a fuller exposition of the features of this technique.

We follow through a sequence of instructions for using ESPOIR. Because a large number of files

can be generated in using this program, we have placed the program and data in a separate directory,

named POWDER.

13.8.1 Aragonite

As a first example, we consider the aragonite form of calcium carbonate, CaCO3. The space group is

Pmcn, a nonstandard setting of Pnma. Since there are four formula entities per unit cell, the Ca and C

atoms lie on special positions, but the oxygen atoms could occupy one set of general positions plus

one set of special positions or three sets of special positions. In practice, both arrangements may need

to be tried. We report here the successful choice, that is, with occupancies of1

2for each of Ca, C, and

O1, and unit occupancy for O2. The following procedure is typical:

Open Espoir.exe.

From the Espoir window: File ! Open File and Open Arag.dat

From the Espoir window: Run ! EspoirA number (10) of tests follow; the end is signified by the Run Terminate box

From the Espoir window: View ! Open .spf for the chosen, best run. Open .imp for the run data and

the 10 test results.

The chosen set of x, y, z coordinates is shown below, together with general equivalent position

appropriate to space group Pmcn; the parameters can be refined by least squares.

NEWNR New sequence number for the zone after evaluation (cp. OLDNR)

A Provisional value of QA for the zone

B Provisional value of QB

FMAAS Provisional value of QF

QUALITY Measure of fit for the zone based on probability theory

OLDNR Old sequence number

CNTR Flag, equal to 0 for primitive zone or 1 for a centered zone

NOBS & NCALC Number of observed and calculated lines used in evaluating QUALITY

ZERSHFT Estimate of 2y zero error for the zone

13.8 Automatic Powder Structure Solving: ESPOIR 655

The known parameters for the Aragonite structure are given below, showing that there is good

agreement with the structure determined, taking into account symmetry-related atoms:

x y z

Ca 14

0.4151 0.2405

C 14

0.7621 0.0852

O114

0.9222 0.0956

O2 0.4735 0.6807 0.0873

13.8.2 a-Alumina (Corundum)

In the POWDER folder there are three pairs of files for a-alumina, labeled Al2O3A, Al2O3B, and

Al2O3C. For each of A, B, and C, one file type (.dat) contains crystal data and program settings, and

the other (.hkl) contains the Fo(hkl) data. We consider just Al2O3A, in which the structure is treated

from scratch, applying distance constraints for Al–Al of 3.0, Al–O of 1.6, and O–O of 1.6 A,

respectively. We know also the space group, R�3c, and that Z ¼ 6. This data contains more output

options, as we shall see.

Open Espoir.exe.

From the Espoir window: File ! Open File and Open Al2O3A.dat.

From Espoir window: Run ! Espoir.

Ten test trials are now performed, followed by Run Terminate.

From the Espoir window: Open ! View. The .spf and .imp files are present, as with previous run. In

addition, several other flies are presented; of particular interest are Profile and Structure.

From the Espoir window: Open ! Profile. The red plot shows the pattern of agreement between

calculated and observed jFj values, whereas the blue plot is the difference pattern, which highlightstheir discrepancies. Close Profile.

From the Espoir window: View ! Structure. The RASMOL program is now invoked.

From the RasWin window: Open ! Display.Under Display, Sticks, Spacefill and Ball & Sticks are probably the most useful. Select Ball &

Spokes.

From the RasWin window: Open ! Colours. Select ! CPK (normal).

From the RasWin window: Open! Options. Select Specular. This mode enhances the appearance of

the model. Under Option, the links Labels is useful, but can be best seen withMonochrome (under

Colours). A Stereo link also exists under Options. A stereo viewer will be needed, and it may be

necessary to decrease the horizontal distance between the stereo pairs.

From the RasWin window: Open! Export, and write the diagram in the .BMP mode for subsequent

printing.

On returning to the original folder (POWDER), a number of new files relating to Al2O3A will

be found. One of the files, Al2O3Astru (a .dat file), contains structural data about the crystal and

x y z x y z

Ca 0.7552 0.9150 0.2602 �x; 12þ y; 1

2� z ! 0.2448 0.4150 0.2498

C 0.2498 0.7624 0.0859 x; y; z ! 0.2498 0.7624 0.0959

O1 0.7553 0.0781 0.9074 �x; �y; �z ! 0.2447 0.9219 0.0926

O2 0.5289 0.3188 0.9123 �x; �y; �z ! 0.4711 0.6812 0.0877

656 13 Computer-Aided Crystallography

the view, together with a list of x, y, z coordinates found from that run; the results for two such

runs were:

x y z

A1 Al 0.6667 0.3333 0.9812

O 0.0000 0.6667 0.7500

A2 Al 0.3274 0.6725 0.8146

O 0.6887 0.9995 0.2498

The two runs do not show apparently the same coordinates because of the random nature of the

process. We shall discuss these results shortly, but first we consider the other two data sets.

Next, carry out the procedure with Al2O3B. This data set is arranged to fit to the Fo data rather than a

regenerated pattern, and makes use of chosen values for the occupation numbers; the execution time

is much shorter. One set of coordinates, the best fit, will be produced.

From the View window: Select Al2O3B.spf.Alternatively, return to the POWDER folder: Select A12O3B.spf. The results of two runs are

listed below.

x y z

B1 Al 0.3422 0.6702 0.8146

O 0.6417 0.6716 0.9165

B2 Al 0.3414 0.6668 0.8146

O 0.3424 0.9772 0.9159

Finally, repeat the second procedure now with Al2O3C. In this example, the constraints of the

special positions of the type 0, 0, z for Al and x, 0,1

4for O have been added to the data set.We obtain

the coordinates and a plot of the best-fit structure. Two such runs are shown below:

x y z

C1 A1 0.0000 0.0000 0.6470

O 0.6940 0.0000 0.2500

C2 A1 0.0000 0.0000 0.6479

O 0.3060 0.0000 0.2500

In order to interpret the totality of these results, we list the special positions for space group R�3c

with Z ¼ 6, and the centers of symmetry in the unit cell:

ð0; 0; 0; 13; 23; 23; 23; 13; 13Þþ

12 Al at� ð0; 0; z; 0; 0; 12þ zÞ 18O at � ðx; 0; 1

4; 0; x; 1

4; �x; �x; 1

4Þ

�1 at ð0; 0; 0; 0; 0; 12; 0; 1

2; 0; 1

2; 0; 0; 0; 1

2; 12; 1

2; 0; 1

2; 1

2; 12; 0; 1

2; 12; 12Þ

The program does not necessarily select all atoms from one and the same asymmetric unit, so that

we have to consider the full implication of the space group symmetry and choice of origin. For

example, we take result C2 as a norm. Then, if O in set Cl is moved across the center of symmetry at

1,0,0, that set then agrees with C2. In sets B, we add the translations 23; 13; 13in each case, which leads to

0, 0, 0.1479 (�0.6479); 0, 0, 0.1479 (�0.6479) for B1: and 0, 0, 0.1479 and 0, 0,

13.8 Automatic Powder Structure Solving: ESPOIR 657

0.2492 for B2, both of which agree reasonably with set C. In a similar way, set A can be transformed

to 0, 0, 0.6479 and 0, 0.3333, 0.2500, and 0, 0, 0.1479 (�0.6479) and 0.3113, 0,0.2498. Except

for Al2O3C, where the constraints are strong, we would not always expect to get the same numerical

values exactly in subsequent trials, because of the random nature of the movement of the atoms.

Recently reported parameters for the a-alumina structure are:

13.9 Problems

The problems for this chapter arise throughout the text itself, Sect. 13.5.1ff. Solutions are provided

for two of them.

References

1. http://www.ccp4.ac.uk/

2. http://www.ccp14.ac.uk/

3. Ladd MFC, Int J (1976) Math Educ Sci Technol 7:395

4. Bibliography, Ladd (1989)

5. Karle J et al (1958) Acta Crystallogr 11:757

6. Ladd MFC (1978) Z Kristallogr 147:279

7. Ladd MFC, Davies M (1968) Z Kristallogr 126:210

8. Loc. cit., Chapter 1

9. http://www.ccp14.ac.uk/tutorial/crys/program/ito12.htm

10. Mileur M, Le Bail A. http://www.cristal.org/sdpd/espoir

x y z

Al 0 0 0.6477

O 0.3064 0 14

658 13 Computer-Aided Crystallography