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

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262 Powder diffraction

Table 17.1. Parameters commonly reﬁned during Rietveld analysis.

Sample-related Instrument-related

scale factor EITHER sample height error

unit cell parameters OR zero-point error

fractional co-ordinates wavelength?

atomic displacement parameters

sample contribution to peak shape instrument contribution to peak shape

preferred orientation correction

parameters, and w

a weighting factor.







(

obs

)

− y

(

calc

)







(

obs

)





(17.1)

exp





n − p





i=1

(obs)



(17.2)

exp

(17.3)

Bragg



hkl



hkl





obs





− I

hkl

(

calc

)





hkl

(



obs



)

(17.4)

hasthe advantage thatitrepresentsdirectlythe quantityminimized.

It is, however, inﬂuenced by, e.g., background points and a high back-

groundcan give a misleadingly low R

value – background-subtracted

values are more useful. R

Bragg

is most closely related to single-

crystal R-factors. It is, however, biased by the structural model, which is

usedto partition intensities of overlappingreﬂectionstoobtainI

hkl

(‘obs’)

values. The best indication of the quality of a structural reﬁnement is

often a visual inspection of the agreement of observed and calculated

patterns and the difference proﬁle.

An example of a Rietveld reﬁnement of an inclusion compound is

given in Fig. 17.6 (Evans et al., 2001). A number of commercial and

academic software packages are available for Rietveld reﬁnement. The

most widely used are GSAS, Fullprof, descendants of the DBWS code

and Topas (Bruker, 2000; Larson and von Dreele, 1994; Wiles and

Young, 1981; Rodriguez-Carvajal, 1990). Many of these packages offer

the opportunity to reﬁne a model simultaneously with both X-ray and

neutron data, allowing one to utilize the often complementary infor-

mation from the two techniques. They also allow ﬁtting of multiple

phases to each dataset, which can allow quantitative analysis or allow

17.5 Rietveld reﬁnement 263

Intensity (Arbitrary units)

15 25 35

2-theta (degrees)

45 55

Fig. 17.6 A Rietveld reﬁnement of an NLO-active layered inclusion compound

DAZOP[MnCr(ox)

]·0.6CH

CN [DAZOP = 4-(4-dimethylamino-phenylazo)-1-methyl-

pyridinium]. Observed data are shown as small crosses, calculated data as a solid line

and the difference as the lower solid line. Small vertical tick marks show 2θ values where

reﬂections are predicted.

for the presence of minor impurities during reﬁnement of a phase of

interest.

The practicalities of Rietveld reﬁnement are beyond the scope of this

text. Typically, though, in the early stages of a reﬁnement one would

want to adjust manually or reﬁne the scale parameter and factors such

as the detector zero-point error or sample height until predicted peak

positions match those observed. Parameters describing the peak shape

should then be adjusted until observed and calculated shapes match.

Finally, factors affecting intensities (atomic co-ordinates and displace-

ment parameters) should be reﬁned. As the reﬁnement improves (or

with good software) one should be able to reﬁne many of these param-

eters simultaneously. Details of parameters typically used in a Rietveld

reﬁnement are given in Table 17.1.

Rietveld reﬁnement does contain many traps for the uninitiated.

Reﬁnements are far more likely to diverge compared to single-crystal

reﬁnements; one is far more likely to ﬁnd false minima; it is much eas-

ier with powder work for a wrong model to ﬁt the data well; many

parameters are highly (sometimes completely) correlated so should not

be reﬁned together; there are many ‘fudge factors’ contained in soft-

ware packages that may improve the quality of ﬁt but have no physical

meaning; there are many more reﬁnement options (‘buttons to click’)

than in single-crystal packages. Never reﬁne any parameter unless you

know exactly what it is doing! Finally, it is worth re-emphasizing that

the information content in a powder pattern is almost always lower

than in a single-crystal experiment. In all but the simplest systems

one will not have the 10 observations per parameter that one would

264 Powder diffraction

like in single-crystal work. If you have a choice, do the single-crystal

experiment!

17.6 Structure solution from powder

diffraction data

It should be emphasized that the Rietveld technique is a reﬁnement

method – one needs an approximate set of starting co-ordinates from

which to begin reﬁnement. Until the late 1990s this meant that Rietveld

reﬁnement was largely conﬁned to extended metal oxides and chalco-

genides where starting models could be inferred from other known

materials. In recent years there have, however, been dramatic break-

throughs in solving structures ab initio from powder data (Harris et al.,

2001, 2002; David et al., 2006).

The process of structure solution of an unknown material from pow-

der data can be divided into several steps. Firstly, one must record the

highest-quality data possible on an (ideally) pure sample. One must

then determine the unit cell parameters from observed peak positions.

For simple systems indexing can be performed by hand (see problems at

the end of the chapter). In most cases indexing is performed by software

packages.Some operate alongsimilarlines tothoseused in single-crystal

work, others perform exhaustive searches of real space. Indexing pow-

der data is by no means a trivial task and is often the bottleneck to

structure solution. To maximize the chances of success peak positions

must be determined with a high degree of accuracy (not just preci-

sion), for which the use of internal standards and careful peak ﬁtting

is recommended.

Next, the space group symmetry must be determined from systematic

absences. This is again non-trivial, as peak overlap at high 2θ typically

means that it is hard to tell if a particular reﬂection class is present or not.

It may be easy to decide that the (010) reﬂection is absent and the (020)

present, but gaining deﬁnitive information on higher-order reﬂections

is hard. There are software packages that use intensity statistics from

whole-pattern ﬁtting to help with this (Markvardsen et al., 2001).

Once the unit cell and space group are known it is often sensible to

perform a Pawley (or Le Bail) reﬁnement (Pawley, 1981; Le Bail et al.,

1988). These reﬁnement methods are similar to Rietveld reﬁnements but

are performed without a structural model. They are essentially peak-

ﬁtting routines with allowed peak positions constrained by the unit cell

size/shape (which is reﬁned) and symmetry and a single 2θ -dependent

peak shape for the whole pattern. If peaks are not ﬁtted well during a

Pawley reﬁnement then either the cell or space group is wrong or impu-

rities are present. The Pawley reﬁnement also gives an indication of the

best ﬁt that will be achievable in the model-dependent Rietveld reﬁne-

ment. If the ﬁnal Rietveld agreement factors are signiﬁcantly higher

than those for the Pawley reﬁnement, or the ﬁts visually worse, then the

model should be examined carefully.

17.7 Non-ambient studies 265

At this stage one of a number of different routes can be followed. One

possibility is to extract integrated peak intensities from the pattern and

use techniques similar to those employed in single-crystal studies such

as direct methods or Patterson synthesis to solve the structure. Paw-

ley/Le Bail reﬁnements are the best way of obtaining these intensities.

Various software packages exist for structure solution, some dedicated

to overcoming the inherent uncertainties in intensities due to peak over-

lap and data shortage from a powder pattern. Very recently, so-called

‘charge-ﬂipping’ algorithms have been applied to powder data with

some success (Baerlocher et al., 2007; Oszlanyi and Suto, 2004). Struc-

ture completion can then be performed via a series of Fourier difference

maps and Rietveld reﬁnement. For example the 33-atom structure of

γ -ZrW

was solved in this way (Fig. 17.7) (Evans et al., 1997).

Fig. 17.7 The 33-atom structure of γ -

ZrW

solved by direct methods and

Fourier maps.

Alternatively (and perhaps more powerfully for molecular species

where the connectivity of the molecule, or a signiﬁcant part of it, is

known), one can utilize information about the cell contents in the form

of known molecular fragments and their geometric degrees of freedom

and attempt direct-space structure solution. From the molecular cell

contents one generates a trial structural model and compares its calcu-

lated diffraction pattern to the experimental data. Using a Monte-Carlo

or simulated annealing approach one then adjusts the structural model

in a random fashion and examines again the agreement between the

observed and calculated patterns. The move is then accepted or rejected

based on user-deﬁnable criteria and the process is repeated until the

best agreement between model and experiment is obtained. With efﬁ-

cient algorithms many hundreds of thousands of trial structures can be

generated and tested relatively rapidly even on desktop computers to

produce a structural model that can be improved by Rietveld methods.

Many variations on this general methodology and alternative genetic

and differential evolution algorithms have been developed. Materials

of remarkable complexity have been solved in this way. The 62-atom

structure of the NLO-active inclusion compound of Fig. 17.6, for which

single crystals could not be grown, was solved by a related technique

(Evans et al., 2001).

It should be noted that each stage of the structure solution pathway

from indexing to reﬁnement is complex and potentially insoluble. The

range of techniques for overcoming each barrier is, however, expanding

rapidly in what is a fast-moving research area.

17.7 Non-ambient studies

Experiments beyond a simple room-temperature diffraction pattern

can yield considerable insight into the properties of materials. One of

the most readily accessible thermodynamic variables experimentally

is temperature. Commercial attachments are available for most diffrac-

tometers to cool/heat samples from liquid He temperatures to ∼1600

◦

Variable-temperature experiments allow one to follow, inter alia, phase

266 Powder diffraction

transitions in materials, study temperature-dependent polymorphism,

follow hydration/dehydration pathways, and follow the synthesis of

materials in real time (Evans and Evans, 2004).

The possibility of performing a powder diffraction experiment dur-

ing more complex chemical reactions should not be overlooked. Various

workershaveshownthat,usinghigh-energy(and therefore highly pene-

trating) X-ray beams at synchrotrons, one can literally monitor reactions

occurring in test tubes (or more sophisticated reactors!) in real time by

powder diffraction methods (Evans et al., 1998; Francis and O’Hare,

1998). By way of an example, O’Hare and co-workers have shown that

one can record diffraction patterns of ∼100 mg of a suspension of solid

SnS

in toluene in as little as 5 s. One can then introduce a molecule

such as cobaltocene and monitor in real time the structural changes and

kinetics of the host-guest intercalation reaction.

The behaviour of materials under applied pressure is also readily

studied by powder diffraction methods. Using small-volume diamond

anvil cells most suitable for X-ray work, pressures of several hun-

dred GPa at temperatures up to several thousand K can be achieved

(Paszkowicz, 2002). Larger volumes of samples can be studied by neu-

tron techniques using gas pressure cells (up to ∼1 GPa on several cm

of sample) or more sophisticated designs such as the Paris–Edinburgh

design cell (up to ∼30 GPa/370 K on 30 mm

). Such studies have

revealed a wealth of important structural chemistry in a variety of

molecular systems.

References

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268 Powder diffraction

Exercises

1. Graphite is a layered material that undergoes inter-

calation chemistry with alkali metals. The ﬁrst two

reﬂections in the powder diffractionpatterns of graphite

and a K intercalation compound were observed

at 26.58/54.76

◦

and 16.56/33.47

◦

2θ, respectively.

Calculate d-spacings for each reﬂection and suggest

hkl indices. Why are only certain classes of hkl reﬂec-

tions typically seen in powder diffraction patterns of

these materials? How might you try to observe other

reﬂections? (λ = 1.54 Å).

2. Figure 17.8 shows powder diffraction patterns of two

inorganic materials recorded with λ = 1.54 Å. Index

Intensity

2-theta

10 20 30 40 50 60 70 80 90

d=3.83027

d=2.71812

d=2.22507

d=1.92216

d=1.71996

d=1.57263

d=1.36174

d=1.28355

d=2.33633

d=2.02402

d=1.16938

d=1.21910

d=1.11232

Intensity

2-theta

310 20 30 40 50 60

d=5.31987

d=4.33397

d=2.83559

d=2.65118

d=2.49925

d=2.37211

d=2.26187

d=2.16430

d=2.08010

d=1.93578

d=1.87430

d=1.81853

d=1.76782

d=1.71987

d=1.67641

d=1.63636

d=1.59860

d=3.06201

d=1.56336

Fig. 17.8 Diffraction data recorded with λ = 1.54 Å for two materials. d-spacings are given in Å.

Exercises 269

Intensity

2-theta

8.1 9 10 11 12 13 14 15 16 17

9.102

9.094

9.124

9.433

9.465

9.481

Fig. 17.9 Diffraction data recorded at T = 237, 248, 260, 271, 282 and 294 K for [FeL

](BF

)

(L = 2,6-di(pyrazol-1-yl)pyridine (Money

et al., 2003). Lowest temperature at the bottom of the ﬁgure. d-spacings are given in Å for one peak.

each and comment on their symmetry. Comment on any

reﬂections you cannot index.

3. For the second example of exercise 2 calculate the cell

parameter from each reﬂection indexed. Which data

should be used to obtain precise cell parameters? Why?

4. What experimental factors can cause systematic errors

in cell parameter determination? How would one obtain

the most precise and most accurate cell parameters

possible?

5. Figure 17.9 shows diffraction data recorded for an

octahedral Fe

complex (Fig. 17.10) at six different

temperatures. Comment on these data.

6. Use the Scherrer formula (Section 17.4.3) to obtain a

crude estimate of the size of the crystalline domains in

Fig. 17.5(a) and (b).

Fig. 17.10 The structure of [FeL

](BF

)

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Introduction to twinning

Simon Parsons

18.1 Introduction

Twinning is not an uncommon effect in crystallography, although it has

long been considered to be one of the most serious potential obstacles

to structure determination. Computer software has now been devel-

oped to such an extent that previously intractable twinning problems

have yielded results of comparable precision to those obtained with

untwinned samples. Structure determinations from twinned crystals are

therefore now quite common, and the aim of this chapter is to present

an introduction to the phenomenon of twinning.

18.2 A simple model for twinning

Twinning may occur when a unit cell (or a supercell) has higher symme-

try than implied by the space group of the crystal structure.An example

of a system that might be susceptible to twinning is a monoclinic crystal

structure where the unique angle, β, is equal, or very close, to 90

◦

.In

this case the crystal structure has point group 2/m, but the lattice has

point group mmm. The elements of these point groups are:

2/m :1,m⊥b,2//b,

mmm :1,m⊥a,2//a, m⊥b,2//b, m⊥c,2//c,

The important issue is that mmm contains symmetry elements that do

not occur in 2/m. Under these conditions ‘mistakes’ can occur during

crystal growth such that different regions of the crystal (domains) have

their unit cells relatedby symmetryoperations that are elements of mmm

but not of 2/m – a two-fold rotation axis about the a-axis direction for

example.

Thisideacanbe illustrated by building up a stack of bricks. The overall

shape or outline of a brick has symmetry mmm, but if we consider the

‘dent’(bricklayers call this the ‘frog’)on one sideplus the words ‘London

Brick’ the point symmetry is only 2. The most obvious way to build a

stack of bricks is to place all the bricks in the same orientation, such

as in Fig. 18.1ii: notice that the bricks are related to each other by the

two-fold axes perpendicular to the page or simple translation – both are

271