Blake A.J.(ed.) Crystal Structure Analysis

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292 Introduction to twinning

energetic material α-NTO(Bolotina et al., 2005). The unit cell dimensions

of this compound are a =5.12, b = 10.31, c = 17.99Å, α = 106.6,β = 97.8,

γ = 90.1

◦

, and the space group is P1. Symmetry checking shows that the

triclinic unit cell can be transformed to a metrically nearly orthorhom-

bic unit cell with dimensions a = 5.12, b = 10.31, c = 31.14 Å by the

matrix

⎛

⎝

100

010

112

⎞

⎠

Table 18.5. Coset decom-

position of mmm with

respect to

1 (calculated

using TWINLAWS). The

notation used is similar to

that in Table 18.3.

1 −1

[100]

[010]

[001]

The lattice effectively has mmm symmetry (order 8), but the space

group belongs to point group

1 (order 2). There are therefore three

twin laws (8/2 − 1 = 3) to consider. A unique set of twin laws can

be obtained by decomposing mmm into cosets with

1 (Table 18.5); we

shall use the two-fold rotationsabout the [100], [010] and [001] directions

of the orthorhombic cell as the twin laws. These operations need to be

expressed with respect to the triclinic axes, and this is achieved by form-

ing triple matrix products as in examples 3 and 4; in each case the three

matrices are (from right to left) the triclinic to orthorhombic transfor-

mation, a two-fold axis in the orthorhombic cell and the orthorhombic

to triclinic transformation:

⎛

⎝

100

010

−0.5 −0.5 0.5

⎞

⎠

⎛

⎝

10 0

0 −10

00−1

⎞

⎠

⎛

⎝

100

010

112

⎞

⎠

⎛

⎝

100

0 −10

−10−1

⎞

⎠

⎛

⎝

100

010

−0.5 −0.5 0.5

⎞

⎠

⎛

⎝

−10 0

010

00−1

⎞

⎠

⎛

⎝

100

010

112

⎞

⎠

⎛

⎝

−10 0

010

0 −1 −1

⎞

⎠

⎛

⎝

100

010

−0.5 −0.5 0.5

⎞

⎠

⎛

⎝

−100

0 −10

001

⎞

⎠

⎛

⎝

100

010

112

⎞

⎠

⎛

⎝

−100

0 −10

111

⎞

⎠

All

four domains (i.e. the reference domain and the three generated by

the three twin laws above) were found to be signiﬁcantly populated,

though the populations were found to be different in different crystals.

R converged to ∼0.04. Further details of this structure determination are

given in Bolotina et al. (2005) and in Schwarzenbach et al. (2006), where

the twinning is interpreted in terms of layer stacking faults.

Example 8. The following exempliﬁes analysis of a non-merohedrally

twinned crystal. Trimethyltin hydride (Me

SnH) is a gas under ambient

conditions, and its melting point is ∼160 K. A sample was crystallized

Examples 293

in situ in a capillary. The diffraction pattern failed to index using routine

procedures, but was indexed using the twin-indexing package DIRAX.

Indexing problems are the most common sign of non-merohedral twin-

ning, but it is often also clear from features such as split peaks in the

diffraction pattern itself that a crystal is not single.

The unit cell chosen for data collection (on a four-circle instrument

with a point detector) was a metrically monoclinic C-centred cell with

dimensions a = 6.255(2), b = 12.113(4), c = 15.963(6) Å, β = 91.66(6)

◦

although it was noted that γ was signiﬁcantly different from 90

◦

90.10(3)

◦

. After data collection it was clear from the Laue symmetry of

the data set that the true crystal system was triclinic. These data imply



= 4, which is high. In addition, the dataset showed strong pseudo-

translational symmetry of the form (h + k + 2l) = 4n, this information

being readily available in the output of SIR97 (Altomare et al., 1999), and

in a Patterson map that showed a very large peak at (

2).

Non-merohedral twinning occurs when a metrically higher-

symmetry supercell exists. Sometimes (though quite rarely in the

author’s experience) this supercell, rather than the true cell, is identiﬁed

on indexing, and this is what occurred here. Strong pseudo-translational

effects and a high implied Z



usually indicate that this has occurred, and

a Patterson synthesis is a useful tool to identify the correct cell.

The unit cell was transformed with the matrix

⎛

⎝

100

0.5 0.5 0

0.25 0.25 0.5

⎞

⎠

and re-reﬁned to give the triclinic setting a = 6.262, b = 6.822, c =

8.640 Å, α = 67.41

◦

, β = 80.92

◦

, γ = 62.62

◦

. The structure solved easily

by Patterson methods, and the carbon atoms were located in a sub-

sequent difference map. Isotropic reﬁnement converged to R = 0.068,

= 0.081 with unit weights, anisotropic reﬁnement led one C atom to

become non-positive-deﬁnite. The difference map showed a very large

peak (+6.42 eÅ

−3

) in a chemically unreasonable position.Application of

ROTAX identiﬁed a two-fold axis along the [−1 2 0] direct lattice direc-

tion (or alternatively the (021) reciprocal lattice direction) as a potential

twin law. This is described by the matrix

⎛

⎝

−100

−1 −10

−0.5 1 −1

⎞

⎠

This matrix can also be derived by recognising that twinning may

occur by two-fold rotation about the b-axis of the monoclinic supercell.

In terms of the triclinic axis system this symmetry operation is given by

a triple matrix product consisting of the transformation from the triclinic

to the monoclinic cell, the two-fold about the monoclinic b axis, and the

294 Introduction to twinning

monoclinic to triclinic transformation:

⎛

⎝

100

0.5 0.5 0

0.25 0.25 0.5

⎞

⎠

⎛

⎝

−10 0

010

00−1

⎞

⎠

⎛

⎝

100

−120

0 −12

⎞

⎠

Incorporation of the twin law into the model gave an R factor of 0.031

and even allowed H atoms to be located in a difference map.

The dataset used for this example was collected using a point detector,

though this is unusual nowadays. Although twinning can be applied

to a dataset collected with an area detector and integrated as though

the crystal were single, it is almost always better to take twinning into

accountduringintegration,usingmore than one orientationmatrix.This

feature is available in modern integration packages such as SAINT v7,

EVALCCD and TWINSOLVE.

An extensive database of papers describing twinning has been assem-

bled by Spek and Lutz (Utrecht University, The Netherlands), and is

available on the internet at http://www.cryst.chem.uu.nl/lutz/twin/

gen_twin.html.

Worked examples for several twinning problems have been assem-

bled by Herbst-Irmer, and are available from http://shelx.uni-

ac.gwdg.de/∼rherbst/twin.html. Further examples of non-merohedral

twinning problems are given by Dauter (2003), Choe et al. (2000),

Colombo et al. (2000), Gaudin et al. (2000), Guelylah et al. (2001),

Cooper et al. (2002) and Tang et al. (2001). A worked example

(Herbst-Irmer and Sheldrick, 1998) is available from http://shelx.uni-

ac.gwdg.de/∼rherbst/twin.html.

References

Beurskens, P. T., Beurskens, G., Bosman, W. P., de Gelder, R., Garcia-

Granda, S., Gould, R. O., Israel, R. and Smits, J. M. M. (1996). The

DIRDIF96 Program System, Technical Report of the Crystallography

Laboratory, University of Nijmegen, The Netherlands.

Bolotina, N., Kirschblaum, K. and Pinkerton, A. A. (2005). Acta Crystal-

logr. B61, 577–584.

Boyle, P. D. (2007) COSET. A program for deriving potential merohe-

dral and pseudomerohedral twin laws by coset decomposition. North

Carolina State University, Raleigh, NC, USA.

Britton, D. (1972). Acta Crystallogr. A28, 296–297.

Bruker-Nonius (2002). SAINT, version 7. Bruker-Nonius, Madison,

Wisconsin, USA.

Catti, M. and Ferraris, G. (1976). Acta Crystallogr. A32, 163–165.

Choe, W. V., Pecharsky, K., Pecharsky, A. O., Gschneidner, K. A., Young,

V. G. and Miller, G. J. (2000). Phys. Rev. Lett. 84, 4617–4620.

Churakov, A. V. and Howard, J. A. K. (2004). Acta Crystallogr. C60,

o557–o558.

References 295

Colombo, D. G., Young, V. G. and Gladfelter, W. L. (2000). Inorg. Chem.

39, 4621–4624.

Cooper, R. I., Gould, R. O., Parsons, S. and Watkin, D. J. (2002). J. Appl.

Crystallogr. 35, 168–174.

Dauter, Z. (2003). Acta Crystallogr. D59, 2004–2016.

Duisenberg, A. J. M. (1992). J. Appl. Crystallogr. 25, 92–96.

Flack, H. D. (1983). Acta Crystallogr. A39, 876–881.

Flack, H. D. (1987). Acta Crystallogr. A43, 564–568.

Gaudin, E., Petricek, V., Boucher, F., Taulelle, F. and Evain, M. (2000).

Acta Crystallogr. B56, 972–979.

Giacovazzo, C. (1992). (ed.) Fundamentals of crystallography. Oxford

University Press, Oxford, UK.

Guelylah, A., Madariaga, G., Petricek, V., Breczewski, T., Aroyo, M. I.

and Bocanegra, E. H. (2001). Acta Crystallogr. B57, 221–230.

Hall, S. R., Flack, H. and Stewart, R. F. (1992). Xtal3.2, University of

Western Australia.

Herbst-Irmer, R. and Sheldrick, G. M. (1998). Acta Crystallogr. B54,

443–449.

Herbst-Irmer, R. and Sheldrick, G. M. (2002). Acta Crystallogr. B58,

477–481.

Jameson, G. B. (1982). Acta Crystallogr. A38, 817–820.

Kahlenberg, V. (1999). Acta Crystallogr. B55, 745–751.

Le Page, Y. (1982). J. Appl. Crystallogr. 15, 255–259.

Le Page, Y. (1999). Acta Crystallogr. A55, Supplement, Abstract

M12.CC.001; this refers to a lecture given by Le Page at the IUCr

Conference in Glasgow, 1999.

Pratt, C. S., Coyle, B.A. and Ibers, J.A. (1971). J. Chem. Soc. pp. 2146–2151.

Rees, D. C. (1980). Acta Crystallogr. A36, 578–581.

Schlessman, J. and Litvin, D. B. (1995). Acta Crystallogr. A51, 947–949.

Schwarzenbach, D., Kirschblaum, K. and Pinkerton, A. A. (2006). Acta

Crystallogr. B62, 944–948.

Sheldrick, G. M. (2001). XPREP. Bruker AXS Inc., Madison, Wisconsin,

USA.

Sheldrick, G. M. (2002). TWINABS. Bruker AXS Inc., Madison, Wiscon-

sin, USA.

Sheldrick, G. M. (2005). CELL_NOW. Bruker AXS Inc., Madison, Wis-

consin, USA.

Sheldrick, G. M. (2008). Acta Crystallogr. A64, 112–122.

Spek, A. L. (2003). J. Appl. Crystallogr. 36, 7–13

Tang, C. Y., Coxall, R. A., Downs, A. J., Greene, T. M. and Parsons, S.

(2001). J. Chem. Soc. Dalton Trans. pp.

2141–2147.

van der Sluis, P. (1989). Thesis, University of Utrecht, The Netherlands.

van Scheltinga, A. T., Valegard, K., Haidu, J. and Andersson, I. (2003).

Acta Crystallogr. D59, 2017–2022.

Yeates, T. O. (1988). Acta Crystallogr. A44, 142–144.

296 Introduction to twinning

Exercises

1. To which point groups do the following space groups

belong?

1, P2

/c, P2

, Cmca, I4, P3

21, R3m, P6

/mmc, Pa3.

2. Explain why it is often stated that a low value for

|E

− 1| can indicate twinning. What values of this

parameter are expected for untwinned structures, and

what values might be expected for a twinned struc-

ture? Under what circumstances might this parameter

be misleading?

3. Suggest twin laws that might arise from structures with

the following unit cells. In each case state which reﬂec-

tions would be affected and what features would help

diagnose the twinning.

(a) Monoclinic, with β ∼ 90

◦

(b) Monoclinic P with a ∼ c.

equal.

4. Consider a triclinic crystal structure with a unit cell with

approximately orthorhombic metric symmetry.

(a) How many domains are possible if the crystal

forms a twin and the space group is P

(b) What twin laws are possible if the space group

is P

group is P1?

5. In Example 6 a mirror perpendicular to [100] was used

to model twinning. Write down in matrix form the

twin laws corresponding to 6

[001]

and m

[110]

that are

equivalent to this operation.

6. Which reﬂections would be affected in the presence of

the following twin laws?

⎛

⎝

−100

0 −10

001

⎞

⎠

⎛

⎝

−10−0.33

0 −10

00 1

⎞

⎠

7. Suggest twin laws that might arise from structures

with the following unit cells. In each case state which

reﬂections would be affected and what features would

help diagnose the twinning.

(i) Orthorhombic P , a = 4.49, b = 16.74, c = 9.01 Å.

(ii) Monoclinic P, a = 5.50, b = 11.49, c = 6.34 Å,

β = 98.3

◦

8. Diffraction data were collected on the low-temperature

phase of oxalyl chloride, (COCl)

. A frame from the

diffraction pattern is shown in Fig. 18.5.

(a) Comment on the appearance of this diffraction

pattern.

(b) Discuss strategies that might be used to index this

pattern.

orthorhombic unit cell a = 5.342(4), b = 7.270(5), c =

16.676(11) Å. The following (next page) was found

assuming orthorhombic symmetry using XPREP.

Show that these data are consistent with the cor-

rect space group P2

/c with a = 16.67, b = 5.34, c

= 7.26 Å, β = 90

◦

Fig. 18.5 A frame of diffraction from oxalyl chloride.

Exercises 297

b-- c-- n-- 21-- -c- -a- -n- -21- --a --b --n --21

N 347 317 316 7 240 235 239 12 85 97 94 26

NI>3s 4 70 68 0 79 72 73 0 28 26 22 8

<I> 0.4 115.8 116.2 0.2 204.6 335.6 275.4 0.2 59.9 40.3 78.9 347.8

<I/s>0.5 1.9 1.8 0.2 2.6 2.6 2.5 0.4 2.3 2.1 2.0 2.8

Identical indices and Friedel opposites combined before calculating

R(sym)

No acceptable space group - change tolerances or unset chiral flag

or possibly change input lattice type, then recheck cell using H-option

Mean |E*E-1| = 1.327 [expected .968 centrosym and .736 non-centrosym]

(d) Calculate Z



and comment on the mean value

|E

− 1| = 1.327.

(e) A Patterson map calculated using the second cell

given in part (c) showed a very strong non-origin

peak at [

3 0

3]. Suggest a transformation to a

smaller unit cell.

(f) What are the dimensions of this smaller cell?

(g) The structure of oxalyl chloride was successfully

modelled as a twin. What is the likely twin law?

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The presentation of

results

Alexander Blake

19.1 Introduction

The ﬁnal stage of a crystal-structure analysis is its presentation. This can

occur within your own research group, as a conference poster or oral

contribution, on the internet, or as a refereed article in a journal. In each

case the requirements are different and you must tailor the presentation

tothe medium used.Aswith all communicationskillsthepresentationof

crystallographic results improves with practice. The reporting of struc-

tural results in crystallographic or chemical journals is usually guided

by the relevant Notes, Instructions or Guidance for Authors published in

these journals. Some journals accept only electronic submissions via a

web interface (or possibly by e-mail, FTP or on a disk) in a speciﬁc

computer-readable format. Since April 1996 Section C of Acta Crystallo-

graphica has accepted submissions only as Crystallographic Information

Files (CIF); since its launch in 2001 Section E of Acta Crystallographica has

required CIF submissions, as does the New Crystal Structures section of

Zeitschrift für Kristallographie.

Even among crystallographic journals there has been a marked trend

away from publishing primary data (co-ordinates and displacement

parameters). Greater selectivity in the choice of molecular geometry

parameters to be published is also being encouraged. Even when jour-

nals did publish extensive information it did not always convey the

structural information effectively, but now it is even more important to

include effective graphical representations of your structures. With the

spread of graphical abstracts in the contents pages of journals, a picture

that is clear and attractive can be effective in attracting the attention of

a reader browsing a hardcopy journal or an index on the web.

This section will deal ﬁrst with molecular graphics, then with the

productionof tables,and ﬁnally with thedifferent methods of delivering

your results.Archiving will also be brieﬂy mentioned. The use of the CIF

will be referred to here as necessary but will be covered in more detail

in the following chapter.

299

300 The presentation of results

19.2 Graphics

Althoughmostobviouslyassociatedwith the production ofhigh-quality

views of the ﬁnal structure, molecular graphics are also used as an

aid in initial structure determination, in the interpretation of difference

electron density maps and to investigate disorder and other situations

requiring modelling. Here, we are concerned only with the ﬁrst of these,

and the main consideration is the quality of the resulting illustration in

terms of its clarity, effectiveness and information content. Early graph-

ics programs [e.g. ORTEP (Johnson, 1965, 1976)] were not interactive

and the program had to be re-run every time a new view was required.

Happily, modern programs are interactive [e.g. XP (Sheldrick, 2001),

CAMERON (Pearce et al., 1996), Mercury (Macrae et al., 2006)], allowing

continuous or stepped rotation of the molecule.

19.3 Graphics programs

The range of graphics programs is vast and it is not practical to offer a

comprehensivesurveyhere:anexcellentsourceofinformationon poten-

tially useful programs is the CCP14 website (http://www.ccp14.ac.uk).

The majority are free (at least to academic users) or cost very little. A

major factor in your choice of program is its range of features and how

these match your needs. Are atomic displacement ellipsoids required?

Are polyhedral representations important? Do you want to display a

ball-and-stick drawing of a molecule within its van der Waals enve-

lope? A further point concerns the ability of the program to read and

write data in certain formats. For example, if you regularly need to rep-

resent the data in ﬁles from the Cambridge Structural Database it is

desirable that your program can do this without manual editing of the

input ﬁle. More generally, a program’s ability to generate plot ﬁles in

standard formats such as HPGL, PostScript, TIFF or JPEG makes it pos-

sible to incorporate these into documents or transmit them by e-mail or

FTP, or to a networked printer or plotter.

There are also commercially available programs, usually supplied

by diffractometer manufacturers as a complete package for structure

analysis, but in some cases these can be purchased separately from

the instrumentation. Such packages have the advantage of integra-

tion: for example, the solution and reﬁnement programs communicate

directly with the graphics module and the problems that can arise due

to incompatible data formats are avoided. Integration is also available

in freely available software [e.g. WinGX (Farrugia, 1999)] that provides

a graphical interface linking various programs.

The increasing power of desktop computers and the availability

of cheap laser printers with 600 dpi or higher resolution means that

publication-quality illustrations can be produced with what is now

very standard and affordable hardware. If you are considering the pur-

chase of a system for structure analysis the advice is the same as for

19.4 Underlying concepts 301

any computer-related purchase. First choose the software you need,

then select hardware you know will run it. Buy the fastest processor

(for structure solution and reﬁnement), the best screen (for viewing the

structure), and the best printer (for hardcopy output) you can afford.

Of course, if you plan to submit only electronic versions of your illus-

trations, then a printer with adequate features to allow proof checking

is all you will need.

19.4 Underlying concepts

The positions of the atoms in a structure are derived from their frac-

tional co-ordinates (x, y, z) on the crystal unit cell axes. Any graphics

program needs to read these, along with the cell parameters required

to convert to the orthogonal co-ordinate system in which the necessary

calculations will be performed. The actual orthogonal axis set (x

, y

, z

)

is arbitrary and is not important. Provided that the program accepts and

can use symmetry operators, it is necessary to read in only those atoms

comprising the crystallographic asymmetric unit: symmetry-equivalent

parts of the structure, whether for a molecule straddling a special posi-

tion or for a packing diagram, can then be generated by the program. If

the program cannot handle symmetry then all the atoms required for a

particular drawing must be generated before being input. (One reason

for doing this could be to exploit a particular drawing style not avail-

able in the graphics routine you normally use.) For the drawing itself

the program uses a separate co-ordinate system (x

, y

, z

) in which the

axes are deﬁned relative to the drawing medium (usually a screen) and

co-ordinates are generated only in order to produce the plot. The axes

of this co-ordinate system are variously deﬁned in different programs

and this represents a minor source of possible confusion if you use a

number of these.

The rotation (or view) matrix transforms the initial arbitrary view

deﬁned by the orthogonal co-ordinates (x

, y

, z

) into the plotting co-

ordinates (x

, y

, z

) corresponding to the viewing direction required.

Much of the ease of use of a program is associated with the ﬂexibility

and simplicity with which this can be done. The following options may

be available:

• direct input of the nine elements of a rotation matrix – of limited

interest;

• along cell axes or other crystallographic directions;

• with respect to molecular features such as

(i) the direction perpendicular to the mean plane through

selected (or all) atoms;

(ii) along the vector between two atoms (which need not be

bonded together).

A good general approach is to start by looking along the direction

perpendicular to the mean plane through all the non-H atoms, then