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

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252 Powder diffraction

become essential when phenomena such as phase transitions cause sin-

gle crystals to shatter under working conditions. Recent advances in the

speed of powder diffraction studies (whole patterns being collected in

a matter of minutes, seconds or less) using advanced sources/detectors

mean that phenomena such as host-guest inclusion reactions, chemical

transformations in the solid state and crystallization can now be fol-

lowed in real time, allowing valuable kinetic and mechanistic insight

into the process of chemical transformations (Evans and Evans, 2004).

Despite powder diffraction being seen as a ‘poor cousin’ to single-crystal

techniques by many, it is a key member of the family of analyti-

cal methods that can be brought to bear on understanding structural

problems. This chapter highlights areas of potential interest to the small-

molecule community; for more in-depth and mathematical descriptions

of powder diffraction the reader should look elsewhere (Klug and

Alexander, 1974; Jenkins and Snyder, 1996; Cullity and Stock, 2001;

Pecharsky and Zavalij, 2003; Dinnebier and Billinge, 2008). Specialist

schools on structural and magnetic Rietveld reﬁnement are organised

biennially by the Physical Crystallography Group of the BCA (see

www.crystallography.org.uk).

17.2 Powder versus single-crystal diffraction

In a conventional single-crystal experiment a beam of monochromatic

X-rays/neutrons is incident on a suitably mounted and oriented sin-

gle crystal. The phenomenon of diffraction leads to diffracted beams

being produced in certain directions in space (see earlier chapters). The

positions and intensities of these beams are recorded by ﬁlm, point-

detector or (most commonly nowadays) area-detector methods. After

crystal selection and data collection the analysis is usually broken down

into four essentially separate stages that are described in detail in other

chapters:

1. indexing to ﬁnd the unit cell;

2. integration of raw images to produce a single data ﬁle listing

intensities and hkl values for each reﬂection;

3. structure solution (typically by direct methods or Patterson

synthesis);

4. structure completion and reﬁnement.

In a powder experiment (Fig. 17.1), instead of a single crystal one has a

collection of randomly oriented polycrystallites exposed to the beam.

Each of these polycrystallites can be thought of as giving rise to its

own diffraction pattern, and individual ‘spots’ on a ﬁlm become spread

out into rings of diffracted intensity (these rings are the intersections

of cones of diffracted intensity with the ﬁlm). The intensity of these

rings can be recorded using ﬁlm/area-detector methods, but are most

commonly measured by scanning a point detector or 1D line detec-

tor across a narrow strip of the rings. In either case one can represent

17.2 Powder versus single-crystal diffraction 253

(a) (b)

2-theta

Fig. 17.1 (a) shows diffraction from an oriented single crystal, (b) from a collection of 4 crystals at different orientations with respect

to the incident beam and (c) from a polycrystalline material. (d) shows the resulting I versus 2θ plot obtained by scanning across the

outlined rectangle of (c).

the diffraction data as a plot of total diffracted intensity against the

diffraction angle 2θ .

Figure 17.1 immediately shows one of the inherent problems of

powder diffraction. The 3D intensity distribution of a single-crystal

experiment is compressed into the one dimension of 2θ space, lead-

ing to a vast loss of information due to peak overlap. In a metrically

cubic crystal, for example, interplanar spacings are given by d

hkl

a/(h

)

1/2

. The (221) and (300) reﬂections(which will in general be

of different intensity) will occur at identical values of 2θ and only infor-

mation on their summed intensity is available from a powder diffraction

experiment. For cells of lower symmetry one may get accidental over-

lap (partial or complete) of different hkl reﬂections. For a triclinic cell of

the complexity that would be routine for modern single-crystal meth-

ods (3400 Å

), using a typical laboratory powder diffractometer with

λ = 1.54 Å there would be ∼5500 reﬂections predicted between 0 and

◦

2θ(d

min

= 1.09 Å). Even for a highly crystalline material this will

lead to a considerable degree of peak overlap (Sivia, 2000; David, 1999).

In order to minimize the effects of peak overlap it is important to choose

an experimental setup (see Section 17.3) that gives peak widths that are

as sharp as possible. In powder diffraction this is referred to as a ‘high-

resolution’ experiment. Note that this is a different meaning from that

typically implied in single-crystal studies (data recorded to high sin θ/λ

to allow high resolution in Fourier maps).

254 Powder diffraction

There are a number of methods that attempt to alleviate overlap

problems. In one approach some independent information regarding

overlapping peaks can be retrieved from intensity variations around the

Debye–Scherrer rings of deliberately textured samples (Wessels et al.,

1999); in another one can make use of anisotropic thermal expansion

to try to resolve different families of reﬂections at different tempera-

tures. In the absence of such methods, overlap can be minimized only

by recording the highest-resolution data attainable for a given sample.

The problems of peak overlap and the compression of 3D data into 1D

are at the heart of the differences in data analysis between single-crystal

and powder methods. Typically stages 2–4, and often stage 1, listed

abovearemergedinto oneandoneworks with the whole experimentally

recorded dataset throughout. The need for high resolution for many

experiments also means that CCD detectors are not widely used, and

pointdetectors or speciallydesigned1D/2D position-sensitive detectors

are employed.

17.3 Experimental methods

There are a host of experimental methods available for recording

powder diffraction data, each with its own inherent advantages and

disadvantages. The two most commonly used geometries for obtain-

ing X-ray diffraction data in home laboratories are shown in Fig. 17.2.

In the simplest ‘reﬂection’ or Bragg–Brentano setup (Fig. 17.2a), one

has an X-ray line source at 3. A ﬂat-plate sample is mounted at 4 and

a point detector at 6. The sample is scanned through an angle θ as

the detector is moved through 2θ . The most common laboratory X-

ray source is a sealed Cu tube. To produce monochromatic radiation

(CuKα

λ = 1.540596 Å) one can place the line source of the tube at posi-

tion 1 and a curved focusing Johannsen monochromator (e.g. Ge 111)

at position 2 to produce an effective line source at position 3. Either

a scintillation counter or a linear position-sensitive detector is com-

monly placed at position 6. This arrangement gives high resolution,

but can suffer from high backgrounds for samples that ﬂuoresce under

Cu irradiation (e.g. Co-containing materials). Fluorescence effects can

be reduced by instead placing a monochromator between the sample

and detector. Perhaps the most common laboratory setup uses a post-

sample pyrolytic graphite monochromator at 5 and a point detector,

giving an approximately 2:1 mixture of CuKα

(λ = 1.540596 Å) and

Kα

(λ = 1.544493 Å) radiation.

It is also possible to use an energy-dispersive detector at 6 and

dispense with the monochromator altogether. Commercially available

detectors can eliminate Kβ radiation, leaving an α

/α

mix. The advan-

tage of this is that a typical monochromator is only ∼25–35% efﬁcient,

so its omission leads to dramatic gains in intensity. In the latest gener-

ation of laboratory instruments it is common to use silicon-strip-based

linear detectors. This means that a post-sample monochromator is no

17.3 Experimental methods 255

X-ray tube

Soller

slit

Divergence

slit

Sample

Receiving

slit

Soller

slit

Detector

slit

Detector

(a) (b)

Antiscatter

slit

Secondary

monochromator

Fig. 17.2 (a) and (b) Typical laboratory powder diffraction setups (see text for details).

Diagrams are not to scale; typical distances 1–3, 3–4 and 4–6 would be ∼200 mm, a sample

area of ∼100 mm

would be illuminated. Below: schematic 3D view of a traditional ﬂat-

plate diffraction setup (reproduced from Philips publicity material).

longer employed and an Ni ﬁlter is placed in front of the detector to

remove most of the Kβ radiation. It should be noted that, with the high

count rates achievable, signiﬁcant discontinuities in background can

be observed around strong reﬂections due to the absorption edge of

the ﬁlter.

There are a host of other optical components present in a typical lab-

oratory setup. Between the source and sample one uses a divergence

slit to control the area of sample illumination. To obtain quantitatively

useful intensities it is crucial that the beam remains smaller than the

sample at all angles. To help achieve this, modern instruments may

use a divergence slit that changes size with diffraction angle. This will

lead to systematic changes in peak intensity with 2θ, which must be

corrected in quantitative work. A similar antiscatter slit is often placed

between the sample and detector. To reduce the effects of axial diver-

gence, which can lead to signiﬁcant peak asymmetry, Soller slits are

256 Powder diffraction

used. These are a series of thin metal plates placed in the beam paral-

lel to the plane of Fig. 17.2. For a point detector one must also select a

suitable detector slit. Each of the components in the system will inﬂu-

ence the ﬁnal peak shape in the diffraction pattern (see below), with

ﬁner Sollers or a smaller detector slit giving a better instrumental res-

olution. Each additional component will, however, lead to a signiﬁcant

loss in intensity. With a 0.05-mm detector slit one will get only

4 of

the count rate obtainable with a 0.2 mm slit. The optimal experimental

setup will be dependent on the sample, the instrument and the informa-

tion required. A typical ‘quick’ data collection covering 5−90

◦

2θ on a

conventional laboratory instrument might take 30 min, a higher-quality

scan for Rietveld reﬁnement 12 h or more depending on the instrument

conﬁguration. Line/area detectors may reduce these times by a factor

of 10–100.

Flat-plate samples can be prepared in a number of ways, either

as bulk powders pressed into a recessed holder or sprinkled on an

amorphous surface such as glass or (preferably) a ‘zero-background’

sample holder such as a 511-cut Si wafer. Flat-plate methods are,

however, prone to problems due to preferred orientation, whereby a

non-random arrangement of crystallites is presented to the beam. This

can severely skew diffraction intensities – in extreme cases making

experimental patterns appear completely different from calculated data

or database standards. Several methods for reducing preferred orien-

tation have been described in the literature (Klug and Alexander, 1974;

www.mluri.sari.ac.uk/commercialservices/spraydrykit.html).

The positions and intensities of reﬂections are also inﬂuenced by

factors such as the sample surface roughness and sample absorption

properties. For organic samples low absorption can lead to a signiﬁcant

portion of the diffracted intensity occurring from below the ideal sample

surface, leading to peak shifts and broadening. Surface roughness leads

to peaks being artiﬁcially strong at high 2θ. This method of data col-

lection is therefore perhaps best suited to relatively strongly absorbing

samples or ‘quick’ qualitative measurements.

The transmission setup of Fig. 17.2b is particularly well suited for

studies on low-absorbing organic/molecular materials. Here, the sam-

ple is placed at position 2, usually mounted in a thin-walled glass

capillary of 0.2–1.0 mm internal diameter and spun in the plane of the

page (Fig. 17.3). Samples can also be mounted on thin mylar sheets.

The use of capillaries signiﬁcantly reduces preferred orientation effects,

though sample mounting is slightly more time consuming. For highly

absorbing samples unusual peak shapes may also be observed, but these

can now be calculated/modelled during reﬁnement.

As with any piece of scientiﬁc equipment, the performance of a

powder diffractometer should be regularly checked. Various standard

materials areavailable tocheck the alignment of and intensitiesrecorded

by the system (www.nist.gov). There are several commercial suppliers

of powder diffractometers, with many of the modern designs allowing

a number of different experimental conﬁgurations on the same basic

17.3 Experimental methods 257

Detector

Sample

Tube

Divergence

slits/

Sollers

Mono-

chromator

Receiving

slit

Antiscatter

slits/

Sollers

Fig. 17.3 Top: a typical laboratory instrument corresponding to the ﬂat-plate setup of

Fig. 17.2. Bottom: ﬂat plate and capillary holders.

instrument. The introduction of new optical devices such as X-ray mir-

rors to replace monochromators gives further ﬂexibility in experimental

design.

Signiﬁcantly higher ﬂuxes and higher resolution is available at a syn-

chrotron source. Diffractometers such as ID31 at the ESRF and I11 at

Diamond receive a useful ﬂux several orders of magnitude greater than

a typical laboratory instrument and can give very high-resolution data.

The range of energies emitted by a synchrotron source means that wave-

lengths can be selected for speciﬁc experiments. By selecting a short

wavelength one canobtain data tohigher values of sin θ/λwith a shorter

scan range; by choosing a longer wavelength the diffraction pattern is

spread out in 2θ, potentially allowing better resolution of overlapping

peaks. One can also select a wavelength close to an absorption edge to

tune scattering factors (resonant or near-edge experiments). It is also

possible to use the entire spectrum of radiation produced and perform

energy-dispersive diffraction. In Bragg’s law (λ = 2d

hkl

sin θ) one is then

measuring different d

hkl

values by varying λ at ﬁxed θ rather than vary-

ing θ at ﬁxed λ. This can allow complex experimental setups to be used,

but with current detectors gives lower-resolution data, which can also

be harder to analyze quantitatively.

Powder neutron diffraction offers signiﬁcant potential advantages

over X-ray methods in some situations. In particular, since scattering

occurs from the nucleus rather than electrons, one can detect light atoms

in the presence of heavy atoms (e.g. O/H in the presence of metals). The

penetrating nature of neutrons also gives more conﬁdence that one is

258 Powder diffraction

studying the bulk of a sample rather than a thin surface layer and allows

the use of more complex sample environment equipment. Neutrons are

also scattered by magnetic moments in a material, giving the possibility

of magnetic structure determination. Neutron diffraction can be per-

formed either at a reactor (generally using constant λ neutrons) or at a

spallation source (usually by the time-of-ﬂight method that takes advan-

tage of the full range of neutron wavelengths produced by the source).

Perhaps the major drawback of this technique for the molecular chemist

is the fact that H scatters neutrons incoherently. To avoid unreasonably

high backgrounds it is often necessary to deuterate samples. However,

with the high ﬂuxes available for instruments such as GEM at ISIS and

D20 at ILL (and becoming available on HRPD and at other facilities)

studies on normal hydrogenated materials are becoming increasingly

feasible.

17.4 Information contained in a

powder pattern

The powder diffraction pattern of any material (or mixture of materi-

als) contains information ‘stored’ in three distinct places. Peak positions

are determined by the size, shape and symmetry of the unit cell. Peak

intensities are determined by the arrangement of scattering density (i.e.

atomic co-ordinates) within the unit cell. The peak shape is determined

by a convolution of instrumental parameters (source, optics and detec-

tor contributions) and important information about the microstructure

(domain size, strain) of the sample. This latter information is not usually

considered in small-molecule crystallographic work, but is more notice-

able in powder analysis as peak shapes are immediately apparent when

one visualizes a dataset, and must be considered during many forms of

data analysis.

One feature that distinguishes powder diffraction from single-crystal

work is that structural analysis (i.e. the determination of fractional

co-ordinates of the atoms in the material) is not always, indeed not

normally, the goal of the experiment. The sections below describe

some of the different applications of powder diffraction. They are

arranged approximately in order of increasing complexity of anal-

ysis, and are the applications most likely to be of interest to the

small-molecule/chemistry community.

17.4.1 Phase identiﬁcation

Each (crystalline) phase present in a bulk sample will give rise to a

characteristic set of peaks in a powder diffraction pattern. These can

be compared to a database of known diffraction patterns or compared

against patterns calculated from single-crystal diffraction data. Many

powder diffractometer manufacturers supply search/match software

to compare experimental datasets against the powder diffraction ﬁle

17.4 Information contained in a powder pattern 259

(PDF-2), a collection of around 186 000 (February 2007) datasets main-

tained by the International Centre for DiffractionData (www.icdd.com).

Very recently a large percentage of the Cambridge Structural Database

(Allen, 2002) (approximately 400 000 entries in February 2007) have been

made commercially available as calculated powder patterns in a for-

mat suitable for automated search/match algorithms. This so-called

PDF-4/Organics contained 312 000 entries in February 2007. Many

single-crystal reﬁnement packages provide a facility for simulating a

diffractionpattern fromeither a reﬁnedstructuralmodel or directlyfrom

experimental single-crystal data. These simulations can be compared

with experimental data. Many other resources for calculating powder

patterns are available via the web.

Perhaps the most important application of phase identiﬁcation to the

small-molecule crystallographer is in conﬁrming whether the powder

pattern of a bulk sample corresponds to a structure determined from a

single crystal obtained during the same synthesis – there are innumer-

able examples where the few single crystals produced in a synthesis are

due to minor products from side reactions or impurities. The presence

of crystalline co-products (e.g. KCl from a salt-elimination reaction) can

also be readily identiﬁed. A relatively quick powder diffraction experi-

ment can often shed considerable light on otherwise conﬂicting pieces

of analytical data.

With regard to synthesis, especially for solid-state syntheses of

extended materials, powder diffraction provides a straightforward way

of monitoring the course of reaction. Peaks due to starting materials

and other impurities can be readily identiﬁed, allowing the progress

of a reaction to be followed. In many ways powder diffraction is the

solid-state chemist’s equivalent of solution-state NMR.

17.4.2 Quantitative analysis

It is also possible to obtain quantitative information about the compo-

sition of a multiphase sample from powder diffraction data. Various

techniques have been developed based on the analysis of intensities of

individual peaks due to different phases contributing to the pattern, on

whole-pattern intensity analysis, or on multiphase Rietveld reﬁnement

(see below). Speciﬁc details are beyond the scope of this chapter and the

reader is referred elsewhere (Dinnebier and Billinge, 2008).

It is worth noting that extreme care should be taken when deter-

mining/interpreting quantitative composition. Results can be severely

inﬂuenced by methods of sample preparation (see above), data col-

lection and analysis. Careful calibration experiments on the system of

interest are essential. It is also worth noting that it is possible to estimate

the quantity of amorphous material in a sample by powder diffraction

measurements (amorphous materials generally give rise to a gradually

oscillating contribution to the background of the diffraction pattern and

can easily be overlooked), bycarefulquantitative dilutionof a powdered

sample with an additional crystalline phase.

260 Powder diffraction

17.4.3 Peak-shape information

Whilst the position and intensity of peaks in a powder pattern are deter-

mined by the unit cell size and contents, their shape and width are

determined by both instrumental effects (which can be corrected for or

modelled) and sample properties such as the size and strain of crys-

tallites and stacking faults (Fig. 17.4) (Klug and Alexander, 1974). The

simplest expression for peak broadening due to sample size (the Scher-

rer formula) predicts that peak width and particle size are related by

fwhm = Kλ/(size × cos θ), where K is a shape factor (often 0.9), fwhm

the peak full width at half-maximum in radians, and λ the wavelength;

absolute numbers from this expression should be treated with caution.

Sample strain leads to a peak-width dependence on tan θ. Note that,

although size and strain both cause peaks to broaden with increasing

2θ, one can distinguish between these effects from their different 2θ

dependence (1/ cos θ and tan θ, respectively). This does, however, need

high-quality data recorded over a wide 2θ range. Practical guidance on

determination of sample size and strain is given in a recent IUCr Round

Robin (Balzar et al., 2004).

Fig. 17.4 Size-strain.

Figure 17.5 illustrates the effects of sample size on peak shapes.

Figure 17.5a shows the diffraction pattern of an FePt alloy that con-

tains ∼2.2 nm nanoparticles. Figure 17.5b shows a material with ∼8nm

domains. It is worth remembering that there are many different ways

of deﬁning the ‘size’ of a material, and that diffraction methods report

the volume-weighted mean column height of the crystallites present.

The apparent ‘size’ of the sample is therefore dependent on the shape

of the domains (only for h00 reﬂections of a perfect cube is the volume-

weightedcolumnheightdirectlyrelatedto crystallite size). The apparent

size determined will also be dependent on the size distribution (often

log normal) present. If precise size information is required then support-

ing evidence from TEM/SEM is very important. In more sophisticated

treatmentshkl-dependent peak widthscan be usedto obtaininformation

Intensity

2-theta

10 20 30 40 50 60 70 80 90

(a)

(b)

Fig. 17.5 Diffraction data from (a) ∼2 nm FePt particles and (b) ∼8 nm particles.

17.5 Rietveld reﬁnement 261

on the anisotropies of size and strain in a sample. More details on the

interpretation of peak shapes are given elsewhere (Scardi and Leoni,

2002; Warren, 1969).

17.4.4 Intensity information

The intensities of peaks in a diffraction pattern contain information

about atomic co-ordinates and displacement parameters, just as in a

single-crystal experiment. Early structural work using powder diffrac-

tion data analyzed extracted intensities (e.g. by weighing carefully

cut-out peaks in the early days!) and reﬁnement methods essentially

identical to those used in single-crystal work. Nowadays, it is more

common to employ whole-pattern ﬁtting methods to extract structural

information – principally the Rietveld method discussed in Section 17.5.

For any quantitative work involving powder diffraction intensities it is

essential to consider how aspects of the experimental setup (use of vari-

able slits, Lorentz-polarization factors, etc.) inﬂuence intensities. If the

data are scaled in any way (e.g. to correct for variable slits) it is important

to propagate the standard uncertainty of the intensity in an appropriate

manner.

17.5 Rietveld reﬁnement

One of the main factors that has driven the explosion of powder diffrac-

tionmethodsinrecentyears is the popularizationoftheRietveldmethod

(Rietveld, 1969; Young, 1995; McCusker et al., 1999). In this method a

powder pattern is expressed in terms of y

obs

, the intensity observed

at a given value of 2θ . One can use a structural model (equivalent

to that used in a single-crystal reﬁnement), a model to describe how

experimental peak shapes vary as a function of 2θ, and a model for the

background, to determine the calculated intensity, y

calc

, at each exper-

imental value of 2θ. The most commonly used function for describing

powder peaks is a pseudo-Voigt (a mixture of Gaussian and Lorentzian

contributions), though more sophisticated approaches can model peak-

shape contributions fromthe experimental setup and sample size/strain

directly.

One then typically uses a least-squares method to adjust struc-

tural parameters such as unit cell dimensions, fractional atomic co-

ordinates and displacement parameters, and instrument/experiment-

related parameters to minimize the difference between y

obs

and y

calc

over the wholeexperimental pattern (Table. 17.1). Thequality of a reﬁne-

ment can be monitored in terms of agreement factors R

or R

Bragg

goodness-of-ﬁt/χ

(which compare the R

value to the statistically

expected value R

exp

). Standard expressions for agreement factors are

given in (17.1)–(17.4); n is the number of observations, p the number of