International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk
International Tables for Crystallography (2018). Vol. H, ch. 2.10, pp. 200-218

Section 2.10.1. X-ray powder diffraction

P. S. Whitfield,a* A. Huqb and J. A. Kadukc,d,e

aEnergy, Mining and Environment Portfolio, National Research Council Canada, 1200 Montreal Road, Ottawa ON K1A 0R6, Canada,bChemical and Engineering Materials Division, Spallation Neutron Source, P.O. Box 2008, MS 6475, Oak Ridge, TN 37831, USA,cDepartment of Chemistry, Illinois Institute of Technology, 3101 South Dearborn Street, Chicago, IL 60616, USA,dDepartment of Physics, North Central College, 131 South Loomis Street, Naperville, IL 60540, USA, and ePoly Crystallography Inc., 423 East Chicago Avenue, Naperville, IL 60540, USA
Correspondence e-mail:  pamela.whitfield@psi.ch

2.10.1. X-ray powder diffraction

| top | pdf |

Most people first encounter X-ray powder diffraction with laboratory instrumentation, but concepts applying to laboratory systems are also applicable to synchrotron and neutron beamlines. The scattering mechanisms of X-rays in the laboratory and at a synchrotron are the same, and apparent differences are usually due to specifics of the wavelength used or beamline/instrument geometry. Differences and similarities between synchrotron and laboratory experiments will be highlighted where they are significant. For example, differences in the polarization factor result in different intensities from the same specimen on laboratory and synchrotron instruments (Fig. 2.10.1[link]). Issues specific to neutron powder diffraction will be covered in Section 2.10.2[link].

[Figure 2.10.1]

Figure 2.10.1 | top | pdf |

Calculated corundum powder patterns using the structure of Lewis et al. (1982[link]; PDF entry 04-004-2852) for laboratory and synchrotron instruments, using the default profile settings from the Powder Diffraction File. The synchrotron pattern is displaced by 200 intensity units for clarity.

A typical laboratory instrument has the Bragg–Brentano reflection geometry, either with θ–θ (fixed specimen) or θ–2θ (fixed tube) setup. Other laboratory configurations are possible, such as transmission (capillary or flat plate) and spot focus with area detectors. Some of the concepts described affect both reflection and transmission, but some will be more important for one geometry over the other. Synchrotron beamlines can also be operated in reflection or transmission geometry, but capillary transmission geometry is much more common than in the laboratory. Some of the apparent differences are due to the typically very small beam divergence, and the tunable wavelength can be very helpful in circumventing some problems.

It will become apparent that many problems relating to specimen preparation and data quality are directly and indirectly the result of samples being too coarse to produce a random powder. The word `powder' forms part of the name of the technique, but what makes a powder a powder?

2.10.1.1. Powders and particle statistics (granularity)

| top | pdf |

The question of when a powder is a `true' powder is not new. It was dealt with in Klug & Alexander (1954[link]) and Alexander & Klug (1948[link]), and more recently by Smith (Smith, 2001[link]; Buhrke et al., 1998[link]). The short answer is that at least 50 000 crystallites in the illuminated volume are necessary to obtain a random powder pattern.

The classic Debye rings of powder diffraction are formed by the random orientation of a large number of single crystallites, which are either physically separate or part of larger agglomerates. These rings used to be a common sight when film cameras were the norm, but can still be seen where two-dimensional (2D) or area detectors are used, most often on microdiffraction systems or synchrotron beamlines. Where there are sufficient crystallites diffracting, the spots from the crystallites merge into smooth rings. Problems with insufficient crystallites are often indicated by the presence of high-intensity spots in the Debye rings. When using 2D data sets, part or all of the intensity in the Debye rings may be integrated to produce an average 1D powder pattern.

More serious problems can arise in cases where 0D or 1D detectors are used. Most modern laboratory powder diffractometers use some form of 0D point detector (e.g. a scintillation counter) or 1D position-sensitive detector (PSD). When collecting data, these detectors pass through the Debye rings along a radius vector. Should the Debye ring be spotty, it is purely down to chance whether the detector will intersect with a spot of higher intensity or low intensity within the ring. An example of how spotty Debye rings can have an adverse effect on the integrated pattern can be seen in Fig. 2.10.2[link]. Unfortunately there is usually no indication of the problem in the resulting integrated 1D pattern. The uncertainty with regard to the intensity of the Bragg reflections is something that must be minimized should accurate relative intensities be required for an analysis. This reproducibility is the concern when the term `particle statistics' is used in relation to powder diffraction. The desirable smooth Debye rings shown in Fig. 2.10.3[link] were produced after reducing the crystallites to less than a few µm by milling. As shown in Fig. 2.10.4[link], rotating the coarse unmilled sample greatly improves the Debye rings compared with those seen in Fig. 2.10.2[link], but they are still not as uniform as those from the static milled sample in Fig. 2.10.3[link].

[Figure 2.10.2]

Figure 2.10.2 | top | pdf |

2D images of the spotty Debye rings of a coarse (~35 µm) cement powder using a Co Kα radiation 1 mm point source. Overlaid are SEMs of the sample material and integrated patterns from the thin slices indicated in the 2D patterns to illustrate what a point or 1D detector would see. Note: in these 2D data sets the low 2θ rings are on the right-hand side.

[Figure 2.10.3]

Figure 2.10.3 | top | pdf |

2D image from the same sample after reducing the crystallites down to a few µm, together with the properly averaged integrated data.

[Figure 2.10.4]

Figure 2.10.4 | top | pdf |

2D image showing the Debye rings when the unmilled sample from Fig. 2.10.2[link] is rotated. The slight spottiness shows that the quality is not as good as the milled sample, even when not rotated, as shown in Fig. 2.10.3[link].

When using a point detector, granularity often manifests itself in the presence of a sharp (instrumental width) peak at relatively high diffraction angle. After a sharp peak at ~68° 2θ was observed in the pattern of a railroad tank car corrosion deposit (Fig. 2.10.5[link]), examination of the specimen in an optical microscope indicated the presence of a single crystal grain of sand (quartz) on the surface. Re-grinding the specimen removed this artifact. Such sharp peaks tend to occur at relatively high diffraction angles, because at such angles the illuminated specimen area is smaller than at low angles, and the presence of a single crystal grain at the surface is relatively more important than when a larger area is illuminated.

[Figure 2.10.5]

Figure 2.10.5 | top | pdf |

The appearance of specimen granularity in a hand-ground specimen of a railroad tank car corrosion deposit. The pattern was measured using a point detector. The intense sharp peak at ~68° 2θ turned out to come from a single crystal grain of sand at the surface of the specimen. The grain was detected by examination (after the measurement) in an optical microscope.

An extreme example of granularity is provided by a hand-ground specimen of Scott's Moss Control Granules (Fig. 2.10.6[link]). The even spacing of the strong peaks suggested severe preferred orientation, but examination of the specimen in an optical microscope (Fig. 2.10.7[link]) revealed the presence of grains several tens of µm in size. Regrinding the sample in a McCrone micronizing mill reduced the crystallite size to a few µm (Fig. 2.10.7[link]), and resulted in random powder data which could be used successfully in a Rietveld refinement (Fig. 2.10.8[link]) and quantitative phase analysis.

[Figure 2.10.6]

Figure 2.10.6 | top | pdf |

An extreme example of granularity. The pattern is of a hand-ground specimen of Scott's Moss Control Granules. No preferred orientation model could fit the langbeinite peaks.

[Figure 2.10.7]

Figure 2.10.7 | top | pdf |

Optical microscope images of the surfaces of the hand-ground and micronized specimens of Scott's Moss Control Granules. The full length of the bar at the bottom is 100 µm. The hand-ground specimen contains grains much too large to yield a random powder pattern. Courtesy of B. J. Huggins, BP Analytical.

[Figure 2.10.8]

Figure 2.10.8 | top | pdf |

Rietveld refinement plot for micronized Scott's Moss Control Granules. No preferred orientation correction was necessary, and the specimen scattered as a random powder. For angles > 35° the vertical scale has been multipled by a factor of 5.

An example of granularity at a synchrotron beamline is provided by (Ba0.7Sr1.3)TiO4 (Fig. 2.10.9[link]). A Rietveld refinement using data collected from a static capillary specimen was unsuccessful. In an attempt to understand why, the diffractometer was driven to the 2θ angle of a strong peak, and a ϕ scan was carried out (rotating the capillary in steps). The intensity varied by a factor of five, as individual crystallites came into and out of diffracting position. Clearly, the intensities from such a measurement are not meaningful. When the capillary was rotated rapidly during a repeated ϕ scan, the intensity was constant and reliable.

[Figure 2.10.9]

Figure 2.10.9 | top | pdf |

A rocking curve (ϕ scan) of (Ba0.7Sr1.3)TiO4, with the detector fixed at 9.647° 2θ, the top of a strong peak in the synchrotron pattern. The jagged plot is from a static specimen, and shows individual grains moving in and out of diffracting position. The flat curve is from a rotating specimen, and indicates that a powder average was obtained.

Granularity can be encountered even in highly transparent organic specimens. A synchrotron pattern of 17α-estradiol showed that the sample was a mixture of the α-polymorph and an additional phase. Indexing the unknown peaks yielded the cell of the β-polymorph, the structure of which was unknown. The structure of the β-polymorph was solved using Monte Carlo simulated-annealing techniques, but the Rietveld refinement (Fig. 2.10.10[link]) was not nearly as good as a Le Bail fit using the same cell and profile. The errors were then clearly in the structural model and/or the data. Examination of the specimen under an optical microscope revealed the presence of needles ∼50 × 50 × 150–200 µm in size. Even the rapid rotation of the capillary specimen was not sufficient to obtain a powder average of such large crystallites.

[Figure 2.10.10]

Figure 2.10.10 | top | pdf |

Rietveld plot of a mixture of β-17α-estradiol hemihydrate and α-17α-estradiol. The largest errors occur at the peaks of the β phase. Examination of the sample with an optical microscope revealed the presence of large single crystals. The rapid specimen rotation at the synchrotron beamline could not yield a powder average from such a coarse sample.

Although granularity is normally considered to affect only the intensities of peaks, in extreme cases it can also affect the shapes. This is easily seen in a pattern from very coarse crystalline quartz in Fig. 2.10.11[link]. The strange looking 101 reflection at 26.6° contains contributions from individual single crystals. When a wider view is taken, the relative intensities are distorted from those expected, similar to that seen in Fig. 2.10.2[link]. Flat-plate data from highly-parallel-beam synchrotron beamlines are more (as opposed to less) susceptible, as shown by the comparison between flat-plate and capillary data of LaB6 from the Australian Synchrotron in Fig. 2.10.12[link]. Despite the use of ω-rocking and a Mythen position-sensitive detector, the flat-plate synchrotron data with 2–5 µm SRM660a LaB6 crystallites show worse splitting of the Bragg peaks than lower-resolution laboratory data with 100 µm quartz crystallites.

[Figure 2.10.11]

Figure 2.10.11 | top | pdf |

The main 101 reflection in data collected from a very coarse (~100 µm) highly crystalline quartz. The strange peak splitting is characteristic where there are very large crystallites present in the sample. The inset shows the diffraction pattern over a wider range and the strangely high intensity at 50° 2θ is caused by the detector intersecting a very intense diffraction spot similar to that seen in the lower part of Fig. 2.10.2[link].

[Figure 2.10.12]

Figure 2.10.12 | top | pdf |

Comparison between capillary (0.3 mm, 0.8265 Å) and rocking flat-plate (strip heater, 1.2386 Å, ω ±2°) data from the Australian Synchrotron. Data courtesy of Ian Madsen, CSIRO.

The quantitative effect of particle statistics on diffraction results can be seen in Table 2.10.1[link]. In the 15–50 µm sample the intensity varied from 4823 to 11 123 counts, which is a huge variation when trying to extract reliable intensities for analysis. Averaging over ten samples, the mean deviation was reduced from 18.2% to 1.2% when the smallest fraction of <5 µm was used. The absolute intensities of the largest fraction are significantly lower, which was attributed to extinction effects.

Table 2.10.1| top | pdf |
Intensity (counts) and mean deviation in intensity of the main quartz 101 reflection with a stationary sample of -325 mesh quartz powder

Data from Alexander et al. (1948[link]) and Klug & Alexander (1954[link]).

 Crystallite size
Data set15–50 µm5–50 µm5–15 µm<5 µm
1 7612 8688 10841 11055
2 8373 9040 11336 11040
3 8255 10232 11046 11386
4 9333 9333 11597 11212
5 4823 8530 11541 11460
6 11123 8617 11336 11260
7 11051 11598 11686 11241
8 5773 7818 11288 11428
9 8527 8021 11126 11406
10 10255 10190 10878 11444
Mean % deviation 18.2 10.1 2.1 1.2

The source of this huge variation in errors can be understood more clearly when the theoretical treatment for quartz from Smith (Smith, 2001[link]; Buhrke et al., 1998[link]) is considered, which followed principles first described by de Wolff (1958[link]). By considering the effects of crystallite size, illuminated volume and beam divergence, for a monodisperse 40 µm specimen Smith calculated that only 12 crystallites would be in the diffracting condition (assuming a point detector). Obviously this is nowhere near enough to create the desired smooth Debye rings. To obtain a standard error of less than 1% the number of diffracting crystallites should be over 52 900, which even the 1 µm sample fails to meet.

Why then does a powder diffraction experiment work? A number of factors may affect the effective number of crystallites, many of which will be mentioned in the following paragraphs. One that isn't is the multiplicity due to the crystal symmetry, meaning that there are always at least two equivalent orientations of each crystallite that would meet the diffraction condition, up to 48 for some cubic reflections. It is worth remembering that these figures relate to a single phase, so the impact on the errors in quantitative phase analysis can be considerable. For low-concentration phases, the number of crystallites is automatically smaller than those of the major phases, so the effects of granularity might be more pronounced.

When considering the granularity of a particular specimen there are a few things to consider. The number of diffracting crystallites depends on the illuminated volume (V), the size of the crystallites (s), the packing density (ρ) and the probability that a crystallite is in the correct orientation (P).

The illuminated volume V depends on a combination of instrument geometry, specimen geometry and the X-ray absorption of the sample. The footprint of the X-rays on a specimen in reflection depends on the beam width, beam length, the beam divergence (if any) and the diffraction angle. The effective beam width at the tube window with a typical long-fine-focus X-ray tube is 0.04 mm, with a length of 12 mm. The beam width may be increased up to 0.2 mm by using a broad-focus tube (Jenkins & Snyder, 1996[link]), but these are rarely used in modern powder diffractometers. With Bragg–Brentano geometry the beam divergence may be increased to cover the available specimen, but large divergence angles degrade the peak resolution. A parallel-beam primary optic produces negligible beam divergence. The beam width may be reduced using an exit slit, but the largest beam width attainable is dependent on the characteristics of the mirror.

Additionally, V also depends on how deeply the X-rays can penetrate into the sample. This depends on the linear absorption coefficient of the specimen for the particular radiation being used, and is given (Klug & Alexander, 1954[link]) by t = (3.2ρ sin θ)/(μρ′), in which μ is the linear absorption coefficient, ρ is the crystal density and ρ′ is the packing density. In the absence of other information, a reasonable assumption for the value of ρ′/ρ is 0.5. The penetration with Cu Kα radiation can range from >1 mm for an organic material to a few µm for heavily absorbing specimens. With Bragg–Brentano geometry this leads to a potential trade-off between improving particle statistics and the peak shifts resulting from sample transparency. As a rule, doubling the diffracting volume will reduce the errors in intensity by about 1.5 times (Zevin & Kimmel, 1995[link]), so is rarely sufficient on its own to solve problematic particle statistics.

Both the experimental data and theoretical treatment shown in Tables 2.10.1[link] and 2.10.2[link] show that with a typical laboratory setup, crystallites should ideally be in the range of a few µm in size to produce accurate intensity data. Reducing the crystallites to below 1 µm will improve the statistics further but may also induce crystallite-size and/or microstrain broadening depending on the instrument resolution. It is important to note that the crystallites must be uniformly small. Mineralogists often refer to `rocks in dust', where there are a small number of very large crystallites scattered among the sample. Scattering of X-rays is sensitive to statistics by volume. A few very large crystallites will dominate (and probably distort) the resulting pattern, so the `rocks in dust' scenario should be avoided whenever possible by correct specimen-preparation techniques.

Table 2.10.2| top | pdf |
Theoretical behaviour of different crystallite sizes of quartz in a volume of 20 mm3

Data from Smith (2001[link]).

Crystallite diameter (µm) 40 10 1
Crystallites per 20 mm3 5.97 × 105 3.82 × 107 3.82 × 1010
No. of diffracting crystallites 12 760 38 000

As we have seen previously, the granularity can be seen visually in a 2D data set. If the researcher has access to a 2D detector this is the quickest way to assess a sample. However, where such a system is not available, an alternative is to use ϕ scans. In simple terms this involves taking data sets of a static specimen but rotating the specimen by a particular angle between data sets, for example at 0, 90, 180 and 270° in ϕ. Ideally the patterns should overlap exactly, although in practice one is looking for reproducible relative intensities, as the absolute intensities may change slightly. The examples used here are the so-called `five fingers' of quartz. Although they are relatively weak reflections in the quartz pattern, three overlapping Kα1,2 doublets provide a conveniently compact example. The three data sets shown in Fig. 2.10.13[link] are -400-mesh quartz (<38 µm), a commercial quartz with a size less than 15 µm and a sample milled to less than 5 µm. Optical micrographs of the -400 mesh and milled quartz samples are shown in Fig. 2.10.14[link].

[Figure 2.10.13]

Figure 2.10.13 | top | pdf |

ϕ scans of the five fingers of quartz for (a) <38 µm, (b) <15 µm and (c) micronized samples.

[Figure 2.10.14]

Figure 2.10.14 | top | pdf |

Optical micrographs of (a) -400 mesh quartz at 100× magnification and (b) quartz milled in a McCrone micronizer for 15 min in isopropyl alcohol at 150× magnification.

The most obvious feature of the ϕ scans is that the reproducibility of the relative intensities is poor with the -400 mesh quartz sample. This has obvious consequences for any analytical technique relying on accurate peak intensities. All eight of the patterns from the micronized sample have practically identical relative peak intensities. It is worth comparing the similar results in the variability visible in Fig. 2.10.13[link] with the tabulated errors for the different methodology used for the data in Table 2.10.1[link].

The final approach to improving statistics is to increase the probability P that a crystallite is in the diffracting condition and visible to the detector. The latter is relevant today with 1D PSD detectors becoming more common, as the detector can simultaneously see multiple crystallite orientations at a particular incident beam angle, as shown in Fig. 2.10.15[link]. P also increases with beam divergence; although there are many advantages of parallel-beam geometry, improving particle statistics is not one of them.

[Figure 2.10.15]

Figure 2.10.15 | top | pdf |

Diagram showing the source of improved particle statistics in reflection geometry using a 1D position sensitive detector (PSD) versus a point detector.

P is much higher with capillary transmission geometry than for reflection geometry. By rotating the specimen about an axis normal to the beam the effective number of orientations `seen' by the detector increases greatly. This is the reason why a powder passing a 325-mesh sieve (<45 µm) almost always yields smooth Debye rings in a capillary (Klug & Alexander, 1954[link]), while the pattern would be granular in reflection. Specimen rotation has also been long employed in reflection geometry (de Wolff, 1958[link]; de Wolff et al., 1959[link]), and a sample spinner is now a standard attachment for commercial diffractometers. When properly applied, the use of a spinner can reduce the standard deviation of the integrated intensity by a factor of approximately 4–5 (∼7–8 for peak intensities) (de Wolff et al., 1959[link]), corresponding to a reduction in the effective crystallite size by a factor of 3 (Zevin & Kimmel, 1995[link]). However, depending on the sample, as seen in Table 2.10.2[link] this can be insufficient on its own as it rotates the specimen only in a single plane. Where a spinner is used in conjunction with a point counter, it is important that the spinner must complete at least one rotation during each step to maximize its effectiveness. In order to further improve the particle statistics it is possible to construct spinners that tilt back and forth along an axis normal to the beam (similar to the capillary concept) in addition to the normal axis of rotation. This is effective in improving particle statistics but adversely affects the para­focusing condition in Bragg–Brentano geometry. It is important to note that specimen rotation improves grain-sampling statistics, but does nothing to alter preferred orientation.

The term `micronized' is one that is frequently seen in papers on quantitative phase analysis. Potentially any kind of mill could be used to reduce the crystallites down to the desirable µm size range (such as shown in Fig. 2.10.14[link]a and b). However, most mills use high-energy percussion-like impacts between the grinding media and the sample, which tend to damage the crystal structure in softer materials and induce microstrain into the material. In extreme cases the sample can become completely amorphous. There is also the potential problem of modifying the polymorph with samples susceptible to such changes. The mill produced by McCrone (http://www.mccrone.com ) was designed specifically for the preparation of X-ray diffraction and X-ray fluorescence samples, and the shearing milling mechanism minimizes damage versus conventional impact milling. It is necessary to use wet milling to produce the best results, so it is up to the analyst to choose the best media compatible with both the sample and the polymer micronizing vials. Commonly used are ethanol, isopropyl alcohol, n-hexane and water; it is not advisable to use acetone, as this solvent dissolves the polymer jars supplied with this mill. A limitation of most forms of milling is the requirement for a relatively large amount of sample. In the McCrone mill a volume of >1 ml is usually required, although desperate scientists have been known to dilute the specimen with amorphous material, such as silica gel. The analyst should also be aware of the possible contamination of samples by degrading and eroding grinding elements (corundum or agate in the micronizing mill; possibly iron, WC, SiC etc. in other types of mill).

Obviously, a reduction of the crystallite size to the µm-sized region will produce size broadening if the instrument has sufficient resolution to detect it. It is worth bearing in mind that micronizing does not guarantee a problem-free sample. Micronized specimens almost always exhibit some microstrain broadening. In principle, this could be decreased by an annealing treatment, but this step is rarely practiced. When a mixture contains both very hard and very soft phases, the hard phases may not mill properly. This has been observed in mixtures containing organics and a minor quartz fraction. Despite milling for 30 min or more, the classical split 101 quartz reflection (such as seen in Fig. 2.10.11[link]) was still visible in some data sets, an indication of the `rocks in dust' phenomenon. Although the McCrone mill is designed to minimize microstructural damage to samples, damage can still occur with very soft materials, and ductile materials may weld as opposed to mill. With very soft and pliable materials a possible alternative could be to cryo-mill the samples, taking advantage of the increased brittleness of materials at low temperature.

2.10.1.2. Preferred orientation

| top | pdf |

Preferred orientation is usually undesirable in a powder diffraction pattern, although sometimes it is the information required, as in texture studies. One of the exceptions is the analysis of clays, where orientation is deliberately induced to identify related reflections. Preferred orientation manifests itself as continuous but non-uniform intensity in the Debye rings, and so is easily characterized with 2D detectors. Preferred orientation does not change the total diffracted intensity, but renormalizes some classes of reflections with respect to others.

Reference is commonly made to a preferred-orientation `correction'. Strictly speaking, what is done is `modelling' of the preferred orientation. The proper way to correct preferred orientation is through better specimen preparation.

Models for preferred orientation exist in many analysis packages, specifically the March–Dollase (Dollase, 1986[link]) and spherical-harmonics (Järvinen, 1993[link]) formalisms. Apparent severe preferred orientation may be a sign of large crystallites, which may result in one or more of the other problems outlined in this section.

Additional care must be taken where software corrections are used during quantitative phase analysis, where overlapping reflections can cause serious correlations and erroneous results. The March–Dollase correction is less prone to this, as an orientation direction must be supplied by the analyst. The spherical-harmonics correction has no such constraint. It behaves properly where peak overlap is not extensive, but negative peak intensities are not uncommon (especially when too high an order is used) when applying it without thought in complex mixtures. Negative peak intensities are obviously impossible, so the results of such an analysis must be viewed with great suspicion.

The presence of preferred orientation can be most easily discerned by comparing the observed pattern to a calculated pattern (random) of the same phase from the Powder Diffraction File or other source. The likelihood of preferred orientation can be assessed by calculating the Bravais–Friedel–Donnay–Harker (Bravais, 1866[link]; Friedel, 1907[link]; Donnay & Harker, 1937[link]) morphology from the crystal structure using Mercury (Sykes et al., 2011[link]) or other tools.

Orientation tends to occur in materials where the crystallites have either a needle or plate-like morphology. Plates are common in the analysis of mineral samples, such as the commercial phlogopite mica used here as an example. Conventional top-loading of such samples can result in very few reflections being visible because of almost perfect orientation of the plates during pressing, as seen in Fig. 2.10.16[link]. Where the aspect ratio of the crystallites is large, micronizing the sample does not reduce the preferred orientation significantly (Fig. 2.10.17[link]).

[Figure 2.10.16]

Figure 2.10.16 | top | pdf |

Top: diffraction pattern of top-loaded 400 mesh phlogopite mica. Bottom: calculated random pattern.

[Figure 2.10.17]

Figure 2.10.17 | top | pdf |

Diffraction pattern of top-loaded miconized phlogopite mica.

The most common approach to decrease preferred orientation of troublesome samples such as this mica is a technique known as back-loading. [Others are discussed in Buhrke et al. (1998[link]).] The concept is that the surface of the sample is not subjected to significant compression, yet remains flat. An example of a commercial back-loading holder is shown in Fig. 2.10.34[link]. These holders are filled while upside down with the back removed. The cavity is filled with sample using minimal pressing, the back of the holder is replaced, and then the whole assembly including specimen is flipped the right way up. Generally, the deeper the holder the lower the compressive force on the analysed surface, but the trade-off is the requirement for large amounts of sample. Many of the samples exhibiting plate-like morphology possess low-angle reflections (such as mica and illite) so the sample area cannot be reduced too much to reduce sample volume, or beam overspill may occur.

A variation of the back-loading sample holder is the side-loading sample holder. These are less common, although the sample is still loaded against some surface in the same fashion as the back-loading variant. As the name implies, the difference is that the sample is introduced from a hole in the side as opposed to the back, and the hole is then plugged after filling.

Simple back-loading of samples in itself is not always sufficient for very platy samples such as the high-aspect-ratio mica used here. Fig. 2.10.18[link] shows the result from back-loading a micronized sample of the mica onto a smooth surface.

[Figure 2.10.18]

Figure 2.10.18 | top | pdf |

Diffraction pattern of miconized phlogopite mica when back-loaded onto a smooth surface.

Although the result is improved, the specimen is still not a random powder. A useful approach in these circumstances is to make the surface of a back-loaded sample deliberately rough to break up the orientation of the plates. An easy way to achieve this is to load the sample onto the surface of sandpaper or a coarse ground glass slide. Sandpaper has the advantage of being disposable so avoiding cross-contamination among samples. Not all sandpaper has the desired jagged surface, so it may be necessary to experiment to find the best. The paper used for the data shown here was a 400-grit carborundum paper, the surface morphology of which is shown in Fig. 2.10.19[link]. The rough surface will cause some slight defocusing in a parafocusing setup and reduce the count rates somewhat, but in many cases the advantages outweigh the disadvantages.

[Figure 2.10.19]

Figure 2.10.19 | top | pdf |

20× optical micrograph of a cross section of the 400-grit carborundum paper used for back-loaded mica.

The result of back-loading the micronized mica onto the 400-grit carborundum paper is shown in Fig. 2.10.20[link]. The dominance of the 00l reflections is reduced even further than when mounted onto a smooth surface. The approach is simple enough that it is used routinely in at least one laboratory dealing with large numbers of mining and mineral samples (Raudsepp, 2012[link]). Back-loading samples is more time consuming than top-loading. Consequently, where high sample throughput is required, back-loading can be reserved for those samples where orientation is a problem.

[Figure 2.10.20]

Figure 2.10.20 | top | pdf |

Diffraction pattern of micronized phlogopite mica when back-loaded onto 400-grit carborundum paper.

Without resorting to transmission measurements, preferred orientation from platy samples may be almost, if not completely, eliminated by spray drying micronized samples (Hillier, 1999[link], 2002[link]; see Fig. 2.10.21[link]). This process produces spherical agglomerates (Fig. 2.10.22[link]) that have no tendency to orient if handled gently. The disadvantage is that a relatively large amount of sample is often required because of inefficient sample recovery. Equipment optimized to reduce sample loss for spray-dried XRD samples may be bought in kit form (http://www.claysandminerals.com/spraydrykit ), or constructed in house using a small air-brush and heated oven.

[Figure 2.10.21]

Figure 2.10.21 | top | pdf |

Diffraction pattern of top-loaded spray-dried phlogopite mica. The sample was not pressed; instead, a flat surface was produced by lightly scraping off excess material with a microspatula.

[Figure 2.10.22]

Figure 2.10.22 | top | pdf |

SEM micrograph of spray-dried micronized phlogopite mica (courtesy of M. Raudsepp, University of British Columbia).

One potential practical problem when using spray-dried material with θ−2θ geometry instruments is that the spherical particles can start to roll out of the specimen holder at higher 2θ angles (Raudsepp, 2012[link]). The effectiveness of spray drying can be seen as the relative intensities from the top-loaded spray-dried material are almost identical to those in data obtained from the capillary experiments. The spray-dried spheres are very delicate and pressing of the sample must be avoided where possible.

It is worth noting that the platy nature of this mica was so extreme that the micronized mica tended to orient slightly inside the capillary if too much energy was applied during the filling process (e.g. using ultrasonics). Arguably, a capillary measurement using a spray-dried material is the ultimate precaution against preferred orientation effects, and the excellent flow characteristics of the spheres mean that the agglomerates remain intact while filling the capillary (Fig. 2.10.23[link]). Fig. 2.10.24[link] gives a summary of the effectiveness of the different sample-preparation techniques for this particular mica sample in terms of the ratio of the integrated intensities of the 001 and 200 reflections. The spray-dried sample with careful top loading can produce a pattern practically equivalent to the capillary data set.

[Figure 2.10.23]

Figure 2.10.23 | top | pdf |

View through the alignment scope of the spherical spray-dried mica inside a 0.5 mm capillary.

[Figure 2.10.24]

Figure 2.10.24 | top | pdf |

Plot of the ratio of the integrated intensities of the 001/200 reflections of the mica using different sample-preparation techniques.

Plates are not the only problematic morphology. Needle-shaped crystallites such as those exhibited by wollastonite (Fig. 2.10.25[link]) and some organic compounds can also show significant problems when top-loaded. In fact, lath-like crystallites such as wollastonite can orient in two directions at the same time, so the behaviour can be more complicated than that of materials with plate-like morphology (see Figs 2.10.26[link], 2.10.27[link] and 2.10.28[link]).

[Figure 2.10.25]

Figure 2.10.25 | top | pdf |

SEM micrograph of wollastonite needles.

[Figure 2.10.26]

Figure 2.10.26 | top | pdf |

Effect of preferential orientation on data from top-loaded wollastonite compared with the calculated pattern from the literature wollastonite-1A structure (Ohashi, 1984[link]).

[Figure 2.10.27]

Figure 2.10.27 | top | pdf |

Rietveld refinement fit to the literature wollastonite-1A structure (Ohashi, 1984[link]) with data from a 0.3 mm capillary with no orientation corrections.

[Figure 2.10.28]

Figure 2.10.28 | top | pdf |

Rietveld refinement fit to the literature wollastonite-1A structure (Ohashi, 1984[link]) with data from a 0.2 mm capillary with no orientation corrections.

2.10.1.3. Absorption (surface roughness), microabsorption and extinction

| top | pdf |

Absorption, microabsorption and extinction effects all alter peak intensities, although particularly low absorption (e.g. from organics) can give rise to sample transparency in reflection geometry (as discussed in the section on the choice of sample mounting), where a peak shift and change in profiles can occur. Microabsorption and extinction solely affect the peak intensities.

Microabsorption (also known as absorption contrast) and extinction are effects that complicate quantitative phase analysis. They are both still related to size − particles in the case of microabsorption and crystallites in the case of extinction.

2.10.1.3.1. Absorption (surface roughness)

| top | pdf |

Absorption is an obvious issue when using capillaries in transmission (a convenient calculator is available on the 11-BM web site, http://11bm.xray.aps.anl.gov ), but absorption can also affect data obtained in reflection using Bragg–Brentano geometry through the mechanism commonly described as `surface roughness'. In essence, the increasing packing density with depth leads to lower intensities at low diffraction angles, leading to anomalously low or negative displacement parameters (much as absorption does in capillaries). There are two components to the effect (Fig. 2.10.29[link], Suortti, 1972[link]). The constant decrease in intensity is generally incorporated into the refined scale factor. The angle-dependent portion becomes more significant as the packing density is reduced.

[Figure 2.10.29]

Figure 2.10.29 | top | pdf |

The effect of surface roughness on the intensity compared to that of a bulk copper specimen. Data from Suortti et al. (1972[link]).

The effect is greatest with strongly absorbing materials analysed in reflection geometry, so care should be taken to produce a sample with a smooth surface and uniform density where possible. An example is provided by the patterns (Fig. 2.10.30[link]) of a commercial cobalt silicate (which turned out to consist of a mixture of phases). A pattern from a slurry deposited on a zero-background cell – a technique useful for small samples, but which produces a rough surface – yielded significantly lower peak and background intensities at low angles than a conventionally front-packed specimen. Refinement using the `rough' data yielded unreasonably negative displacement coefficients. Including a surface-roughness model in the refinement resulted in reasonable displacement coefficients, identical to those obtained using the `flat' data.

[Figure 2.10.30]

Figure 2.10.30 | top | pdf |

Powder patterns of a commercial cobalt silicate sample, measured from a (rough) slurry-mounted specimen (red) and from a (flat) conventional front-packed specimen (green). The surface roughness decreases the intensities of the low-angle peaks and background.

Models by Suortti (1972[link]) and Pitschke et al. (1993[link]) exist in most analysis software. They both yield similar results, but the Suortti correction is generally regarded as slightly more stable at 2θ angles below 20°.

2.10.1.3.2. Microabsorption

| top | pdf |

Microabsorption is widely regarded as the greatest impediment to the application of quantitative phase analysis with powder X-ray diffraction data. Misapplication of a microabsorption correction can degrade accuracy (Scarlett et al., 2002[link]). The infamous sample 4 in the Commission for Powder Diffraction quantitative phase analysis round robin (Scarlett et al., 2002[link]) was deliberately designed to be difficult to analyse accurately using any wavelengths available to a laboratory X-ray diffractometer because of microabsorption effects.

Brindley (1945[link]) published a theoretical correction (model) for microabsorption, but the range for an appropriate application can be easily exceeded. In the absence of such a correction, relative concentration errors can commonly be 20–30%. The extent of microabsorption for a phase was described by Brindley in terms of μD, where μ is the linear absorption coefficient and D is the particle (not crystallite) diameter. While a crystallite size can be estimated from the profile widths, determining the particle size requires additional information, such as laser light-scattering measurements. Where μD > 0.1 the Brindley correction may not be accurate, and ideally μD for every phase present should be less than 0.01, where microabsorption can safely be ignored. Brindley also suggested a rule-of-thumb for the maximum acceptable particle size for quantitative phase analysis, where Dmax = 1/(100μ).

The terms μ and D show that microabsorption can be affected by X-ray wavelength and particle size. The easiest approach for the analyst with a troublesome sample is to reduce the particle size by micronizing the sample. Some in-house laboratories may have multiple systems or the flexibility to change wavelengths. As a rule the linear absorption coefficient decreases with increasing energy, but users should beware of absorption edges that can create a serious discontinuity in this trend. Synchrotron beamlines have more flexibility for avoiding absorption edges and can achieve higher energies that are not practical in laboratory systems. Even when using this high-energy `sledgehammer' approach there is still a benefit to reducing the crystallite size to avoid some of the other effects mentioned in this chapter.

Microabsorption results from differences in linear absorption coefficients and particle sizes, and can sometimes arise in unexpected situations. Adding a NIST SRM 640b silicon internal standard to a micronized mullite sample (Kaduk, 2009[link]) in order to quantify the amorphous content resulted in a significant microabsorption effect, the result of differences in both particle size and absorption coefficients. The microabsorption could be overcome by micronizing the mullite/Si blend.

Even in cases where the absorption contrast is small, large differences in particle size can result in significant microabsorption effects. For anatase/rutile mixtures (μ = 489.4 and 534.2 cm−1, respectively) in which the anatase and rutile particle sizes were 3 and 150 µm, respectively, concentration errors of 20% relative were observed. The errors were corrected by micronizing the mixtures (Kaduk, 2013[link]). Similarly, mixtures of large-particle (23 µm) MFI zeolite (μ = 65.66 cm−1) and a quartz internal standard (μ = 96.39 cm−1, 10 µm) resulted in relative concentration errors of 5%; these errors were corrected by micronizing the mixtures.

Neutrons are absorbed much less than X-rays, which means that microabsorption is practically nonexistent in neutron diffraction data. The lack of microabsorption is why neutron diffraction is often regarded as the `gold standard' for quantitative phase analysis, although the beamline proposal process tends to make its application in conventional quantitative phase analysis uncommon. The application of quantitative phase analysis using neutron diffraction data is most often seen when studying phase evolution in an in situ experiment. Microabsorption is an issue specific to quantitative phase analysis and a more detailed discussion of the problem is given in Chapter 3.9[link] .

2.10.1.3.3. Extinction

| top | pdf |

Extinction effects are not common in X-ray powder diffraction but may be significant when using neutrons. Extinction is dependent on the size/shape of the coherently diffracting domains, and is a multiple-scattering phenomenon. Primary extinction occurs when a second diffracting event occurs within a single crystallite. Secondary extinction occurs in mosaic crystals and is not seen in powders. When primary extinction occurs, the re-diffracted beam will continue in the same direction as the incident beam but interfere destructively with it. Re-diffraction within a crystallite is not likely to occur where imperfections disrupt the ordering of a crystallite. Consequently, primary extinction is usually only seen in powders of highly ordered and crystalline materials. A classic example is powdered single-crystal silicon as used in studies on extinction such as that by Cline & Snyder (1987[link]). The most commonly encountered phase that can exhibit primary extinction is high-quality natural quartz.

Zachariasen (1945[link]) described an extinction correction (model) including terms relating to crystallite size, wavelength, structure factor and scattering angle. Extinction effects will be apparent with large crystallites and long wavelengths. Extinction effects are also greater for the more intense (low-angle) reflections, so extinction mimics the effects of small displacement parameters. In a single-phase system, unexpectedly low or even negative displacement parameters may be the only sign that extinction effects are present. In a multiphase system the effects of extinction will reduce the apparent phase fraction of the affected phase with respect to the rest of the sample. In fact, studying extinction experimentally is often done by using its effects on quantitative phase analysis to untangle the different effects (Cline & Snyder, 1987[link]). The frequently high quality of natural quartz makes the quantitative phase analysis of mineral samples the most likely scenario for the appearance of extinction in a practical laboratory setting.

The wide range of wavelengths and wide range of (sin θ)/λ used in time-of-flight (TOF) neutron diffraction makes extinction effects particularly pronounced. Consequently TOF data often require the application of an extinction correction (Sabine et al., 1988[link]). Constant-wavelength neutron diffraction frequently uses longer wavelengths than normally used in the laboratory or synchrotron beamlines, so the user must be aware of possible problems.

Despite the danger of `message fatigue', the dependence of primary extinction on crystallite size adds yet another reason to reduce the crystallite sizes to the order of 1 µm or so. Theoretically, single-crystal silicon will exhibit extinction with copper radiation with crystallite sizes of 5 µm.

2.10.1.4. Holders

| top | pdf |

2.10.1.4.1. Reflection sample holders

| top | pdf |

In a laboratory setting these are the most common type of holders – normally for use in a Bragg–Brentano instrument. A wide variety of sample holders for different applications are available. Several different holders and techniques will be described, but there are some issues common to all holders in reflection geometry, particularly with Bragg-–Brentano geometry.

In Bragg–Brentano parafocusing geometry care should be taken that the surface of the sample is flat. If the surface is not flat the parafocusing condition is violated and will degrade the peak resolution and positions; in addition, surface roughness can affect the intensities. Where there is a cavity it seems straightforward to make sure that the sample surface is level with the top surface of the holder. The peak positions obtained in Bragg–Brentano geometry are very sensitive to specimen displacement; a vertical displacement of 20 µm in a typical diffractometer will shift the peaks by approximately 0.01° 2θ. The derivation of the equation for the effect of displacement on peak position is given in Fig. 2.10.31[link]. The minus sign in the equation reflects the convention that the displacement is positive if it increases the radius of the diffracting circle, i.e. the sample is too low. Front-packed specimens are almost always too high, so the analyst needs to refine his/her technique to minimize the displacement errors.

[Figure 2.10.31]

Figure 2.10.31 | top | pdf |

Derivation of the equation relating peak displacement to sample displacement (s) in parafocusing geometry. R is the goniometer radius.

The sensitivity to specimen displacement is such that even dirt between the reference surface of the sample stage and the holder can produce a detectable peak shift. Dust accumulation inside a powder diffractometer is almost inevitable, so occasionally cleaning these surfaces is recommended.

Parallel-beam-geometry diffractometers have become popular in many laboratories because some of these problems are avoided. Although there are often some disadvantages in terms of peak resolution and grain sampling, they allow more flexibility in the mounting of specimens. For instance, rough sample surfaces and displacements do not cause the aberrations that are apparent in data from conventional parafocusing diffractometers when the same samples are analysed with a parallel-beam system.

Many different types of holders for reflection geometry are available commercially from the instrument vendors, but often home-made holders can be equally effective and customized for specific tasks. Most common are the different types of top-loading sample holders made from plastic or metal, often with a cavity to hold the sample. Commonly the cavities are larger or smaller than those offered by the vendors. The cavity may include some form of zero-background plate such as specially cut single-crystal silicon (Fig. 2.10.32[link]) or quartz, although this does add a significant cost. Some quartz plates may exhibit forbidden reflections or contain inclusions, so they should be tested before use in a sample spinner.

[Figure 2.10.32]

Figure 2.10.32 | top | pdf |

A home-made top-loading zero-background silicon holder with a 0.5 mm deep cavity.

In addition to the standard holders, more specialized holders may be bought or built, or indeed fabricated using a 3D printer. These include holders for air-sensitive samples (Fig. 2.10.33[link]), back-loading (Fig. 2.10.34[link]) and side-loading holders, holders for filter papers, clay samples etc. Any laboratory with a competent workshop can construct a wide variety of holders, including those for complex in situ work, which is discussed in Chapter 2.9[link] . One common theme is that any material in the X-ray beam path must be kept to a minimum to reduce attenuation. Ideally any such material (such as the polymer dome of the air-sensitive holder shown in Fig. 2.10.33[link]) should be as far away from the diffracting plane as possible. A secondary monochromator can be effective in stopping the parasitic scattering from reaching the detector, but with a PSD there is greater reliance on good design to reduce it as much as possible. A common approach with home-designed and -constructed sample holders for air- or moisture-sensitive samples is to cover the sample with a thin Kapton or Mylar film attached with a bead of silicone grease.

[Figure 2.10.33]

Figure 2.10.33 | top | pdf |

Commercial holder for air-sensitive samples. This particular holder for small samples has a flat silicon zero-background plate and a polymer dome which screws down against a rubber o-ring seal.

[Figure 2.10.34]

Figure 2.10.34 | top | pdf |

Filling a commercial back-loading sample holder. The holder is held against a base surface (sandpaper in this case) while filling, the back is replaced and then the holder is flipped over to reveal the sample surface once the clips are removed.

One of the most common questions asked by users of laboratory instruments is how to deal with small sample sizes. In an ideal world, a specialist microdiffraction system or capillary geometry could be used, but many laboratories do not have access to such equipment. How problematic such samples can be depends to some extent on the mass absorption coefficient of the sample. Conventional powder diffraction data relies on having a sample with an `infinite sample depth'. However, that depth can be very small for samples with very high absorption coefficients. In those cases, spreading the sample in a very thin layer can still yield reasonable relative intensities across a large range of 2θ angles. With low-absorbing samples such as organics the relative intensities will drop off at higher angles as the sample is no longer `infinitely thick', as shown in Fig. 2.10.35[link]. However, the peak positions will be more accurate than with deeper samples because of the lack of transparency effects in thin samples. Consequently, it is not uncommon to obtain two data sets from such samples: from a thin sample to obtain good peak positions, and from a deep one to obtain better relative intensities. The details of sample penetration are given in Chapter 5.4[link] .

[Figure 2.10.35]

Figure 2.10.35 | top | pdf |

Data from powdered sucrose on a Bragg–Brentano instrument, with the peak intensities normalized to the first reflection. The thin sample was prepared by sieving onto a low-background silicon plate made slightly tacky using hairspray. The inset shows that there is a slight peak shift between the two data sets as well as the predicted decay in relative intensities with the thin sample with 2θ angle.

Most modern holders are circular and the specimen is often loaded into a round cavity. As the beam `footprint' is rectangular, this is not the most efficient use of the material, as a significant portion will always remain outside the beam. Prior to the introduction of sample spinners, square and rectangular cavities were quite common. It is good practice to know the footprint of the beam at various diffraction angles by observing the illuminated area of a fluorescent specimen. Should the material be in particularly short supply and sample spinning is not absolutely necessary, the powder may be mounted in the minimum rectangular shape to be illuminated by the incident beam. Such an approach may be combined with the use of motorized divergence slits to maintain a constant beam length on the sample. Although most analysis software assumes constant divergence slits, the correction is well known and implemented in most commercial software.

In order to avoid background from the sample holder, thin specimens are usually mounted on flat zero-background plates. It is useful to have the surface of the plate lower than the reference surface (50 µm is a common value) to minimize specimen displacement effects. In practical terms, thin samples are historically referred to as smear mounts. Slurry mounting using ethanol or acetone often yields a self-adhesive specimen, but it is tricky to obtain the correct slurry rheology to produce a non-lumpy, thin and even layer across the surface; surface roughness is often apparent in the pattern. Loose, friable samples may be problematic with spinning specimens or the tilting specimens in θ−2θ geometry. A number of materials have been used over the years to adhere thin powder samples to flat plates; common ones are thin smears of Vaseline or grease, but analysts often have their own favourites. The particular material used to stick the sample to the surface is often the result of testing a large number of options to find the one with the lowest background and fewest non-Bragg reflections. An unusual alternative is hairspray, which produces a tacky surface when applied correctly whilst having a minimal effect on the resulting diffraction pattern. The medium chosen may also depend on whether the sample must be recovered intact, as contamination with grease might not be acceptable. The effect on the background of different adhesion materials can be seen in Fig. 2.10.36[link]. The Vaseline and vacuum grease smears add broad reflections at approx. 19 and 11° 2θ, respectively, with Cu Kα radiation. Where data collection starts above the main portion of the peak the effect may be hardly noticeable, but could be problematic when starting at low 2θ angles. Such broad patterns are straightforward to model with a Debye (diffuse scattering) function, and it is not necessary to subtract them from the raw data.

[Figure 2.10.36]

Figure 2.10.36 | top | pdf |

Diffraction pattern from a silicon-wafer zero-background holder, smears of Vaseline and Corning high-vacuum grease, and the surface treated with hairspray.

Should the instrument have parallel-beam geometry, an alternative approach is to use a fixed incident-beam angle, more commonly known as grazing-incidence geometry. In this way the volume of sample illuminated is constant with angle, so in the absence of secondary diffractometer optics the relative intensities will match those expected with conventional geometry. An unfortunate effect of conventional grazing-incidence geometry with long slits is that the peak widths degrade significantly at lower incident angles (Toraya & Yoshino, 1994[link]). It is possible to model the peak broadening in a Rietveld refinement (Rowles & Madsen, 2010[link]) but it is not straightforward. Use of an appropriate secondary optic can avoid the peak-broadening problem but introduces a complex, geometry-dependent intensity correction (Toraya et al., 1993[link]).

2.10.1.4.2. Transmission sample holders

| top | pdf |

Transmission geometry of any type is best suited to samples with low absorption such as organics and polymers, and is preferred for such samples when available. Transmission geometry has advantages when data are required at low diffracting angles. While the beam often has to be stopped-down in reflection geometry to avoid overspilling the sample, this undesired attenuation of the beam is not required for transmission geometry. Another advantage common to both the foil and capillary transmission techniques is that a small quantity of a powdered sample is usually sufficient. Samples small enough to be problematic with reflection geometry will often be perfectly adequate for transmission.

Data collection in transmission geometry is best done with either a parallel-beam or focusing geometry; the focus should be at the detector. Data can be collected using a divergent-beam setup, but the intensities obtained are very low and the resolution is usually poor. Parallel-beam geometry has the advantage that it is able to perform reflection and transmission measurements equally well.

2.10.1.4.2.1. Flat foils

| top | pdf |

Although less commonly used with modern diffractometers, the foil-type transmission sample mounting was quite common in some older-style X-ray cameras. Sprinkling powders onto single-sided Scotch tape was sometimes used with instrumentation such as Hägg–Guinier cameras, but care should be taken as the quality of the tapes as diffraction substrates can vary wildly; the crystallinity of the polymer can be high or low, and the adhesive sometimes contains mineral inclusions, such as talc. In the modern diffractometer, foil-type transmission data can sometimes be collected using the same rotating sample stage as for reflection measurements. Simply turning the stage by 90° and using a different holder can be sufficient if the optical configuration is suitable for both reflection and transmission. For solid organic samples such as polymers this foil transmission geometry has significant advantages because of the lack of transparency effects. It is worth noting, however, that the processing of polymers can induce significant texture, such that the data collected from a film in reflection geometry will not necessarily be identical to those collected in transmission. Should a reproducible pattern independent of geometry be required, then steps should be taken to reduce the sample to a true random powder and/or a 2D detector should be used.

With powder samples the technique requires the use of a transparent substrate, usually in the form of a thin polymer film or foil. In an analytical laboratory the easiest place to find such a substrate is the X-ray fluorescence laboratory, where very thin X-ray transparent polymer films are used for both sample supports and covers for liquid cells. Some of the materials used in these applications are familiar in the diffraction community as windows, i.e. Mylar and Kapton, but others such as polypropylene are not. The substrate will obviously add to the background, but a good substrate from a diffraction standpoint combines transparency with a lack of sharp features in the diffraction pattern. This makes fitting the background much easier. Any holder must be capable of stretching or holding the film flat across an opening for the X-ray beam. A commercial version of a foil-type holder is shown prior to assembly in Fig. 2.10.37[link]. Example data from three different XRF films are shown in Fig. 2.10.38[link], together with that from a thicker Kapton foil commonly used as window material. It is notable that, despite the two 7.6 µm Kapton films being almost twice as thick as the Mylar or polypropylene films, the scattering from them is almost identical. The lack of any distinctive, sharp features above 6° 2θ in the Kapton films makes them attractive in this region, but for low-angle data Mylar is probably the better choice. Although giving a generally higher background, the thicker 50 µm Kapton foils can be used very successfully (see Fig. 2.10.39[link]). Despite the greater attenuation they are much easier to handle, as their greater stiffness and weight makes them less susceptible to static electricity.

[Figure 2.10.37]

Figure 2.10.37 | top | pdf |

Parts prior to assembly of a transmission foil sample in the holder. In this instance, micronized quartz is held as a loose powder between two 50 µm Kapton foils while the upper foil is stretched into place by the black clip.

[Figure 2.10.38]

Figure 2.10.38 | top | pdf |

Transmission data from double layers (as used for powder samples) of different polymer substrate films. They include 3.6 µm Mylar, 4.0 µm polypropylene and 7.6 µm XRF films, and a thicker 50 µm Kapton foil.

[Figure 2.10.39]

Figure 2.10.39 | top | pdf |

Diffraction pattern from loose SRM640c powder between two 50 µm Kapton foils.

One advantage of transmission foil mounts is the small amount of sample required. In a similar way to producing smear mounts for reflection geometry, there are a number of ways to prepare the thin layer required. Loose powders may be trapped between two foils as in Fig. 2.10.39[link], or alternatively a slurry or smear mount may be used in a similar way to reflection geometry. Although the sample may adhere sufficiently such that a single foil can be used, it may be necessary to use a sandwich in the same way as a loose powder. For instance, slurries do not usually adhere well to Kapton foils, so it is often better to sacrifice a little intensity from the additional Kapton attenuation and ensure the sample does not fall away during data collection. Lack of adhesion could be regarded as an advantage with regards to recovery of valuable samples. Where an adhesive is used, the same considerations as with a smear mount in reflection still apply with regards to background etc.

Ideally the sample thickness should be perfectly uniform, but in practice this will rarely be achieved. Commonly a specimen in visible light transmission will appear something like that seen in Fig. 2.10.40[link]. Rotation is used to average out inhomogeneity in the specimen.

[Figure 2.10.40]

Figure 2.10.40 | top | pdf |

Transmitted light view of a micronized quartz sample through 50 µm Kapton foils.

Sedimentation during slurry mounting and compression of powders between two foils can lead to preferential orientation in foil transmission samples just as with flat-plate reflection specimens. Although the physical effect is the same for plate-like crystallites, it should be remembered that the crystallite orientation with respect to the beam is rotated by 90°, so the resulting diffraction patterns will not look the same. This becomes very apparent when comparing the foil transmission and reflection patterns from the micronized mica in Fig. 2.10.41[link].

[Figure 2.10.41]

Figure 2.10.41 | top | pdf |

Comparison of data from micronized 40S mica taken in reflection and transmission geometry, and spray-dried material in reflection geometry. For improved clarity the spray-dried and transmission data sets are translated by +1° and +2° 2θ respectively.

Foil transmission specimens are usually rotated in a similar fashion to a reflection sample, but the improvement in statistics falls short of that found in the capillary geometry described in the next section.

One thing worth considering is that there is an inherent angular intensity aberration due to the plate transmission geometry. Owing to geometrical considerations, the path length through the specimen (and support) increases with angle with a resulting increase in absorption. For refinement work, a 1/cos θ correction can be applied.

2.10.1.4.2.2. Capillaries

| top | pdf |

Capillaries are particularly suitable for small samples, air- and moisture-sensitive samples and organics where the absorption is low enough to cause transparency effects in reflection data. They are also commonly used for materials with platy morphologies such as clays to eliminate or greatly reduce preferred-orientation effects. They are less effective at reducing preferred orientation in materials with needle-like morphologies but are still useful, a possible analogy being that the crystallites pack into the capillary like a handful of pencils in a glass. The extent of the problems with needles depends on the aspect ratio of the needles and the diameter of the capillary used – smaller diameter capillaries usually being more problematic. Figs. 2.10.27[link] and 2.10.28[link] show the example of wollastonite powder mounted in 0.3 and 0.2 mm capillaries, respectively, where orientation effects become pronounced in the 0.2 mm capillary. Fortunately, needle-like morphology is observed more often in organic crystallites, where larger-diameter capillaries can be tolerated.

Glass and fused silica (`quartz') capillaries can be bought commercially in a range of diameters between 0.1 and 2 mm. Different compositions of glass are available that have varying absorption characteristics (Table 2.10.3[link]). The softer glass has a greater tendency to splinter but can be heat-sealed very easily by melting. Quartz tends to be stiffer and often breaks more cleanly when scored using a cutting stone, but requires a hydrogen flame for heat-sealing because of its high melting point. Alternative methods of sealing the open end of capillaries include using molten wax, epoxy and nail varnish. The choice may be restricted by the environment in which the capillary is being filled. In an argon-filled glove box the use of a flame or solvent-based method may not be feasible or desirable, whereas wax sealing with a heated filament is acceptable.

Table 2.10.3| top | pdf |
Absorption and physical characteristics of the capillaries whose data are shown in Fig. 2.10.46[link]

MaterialLinear absorption, Cu Kα (cm−1)Wall thickness (µm)Outside diameter (mm)
Quartz (Hampton Research) 76 10 0.50
Soda lime glass (Hampton Research) 126 10 0.50
PET (Advanced Polymers) 10 19 0.58
Polyimide (Cole-Palmer) 9 25 0.55

The small size and delicate nature of capillaries can make them extremely frustrating to fill, especially in environments such as glove boxes. Patience is an absolute must, especially with valuable or small samples where capillary breakage and sample loss are unacceptable. It is very important to make sure that the sample is fine enough to pass into the capillary without jamming. Even if it is fine enough, different powders can vary considerably in their tendency to aggregate. For example, NIST 640d silicon contains fine crystallites and flows extremely well, making it very easy to load into a capillary. However, some rutile powders can be very fine but don't flow well, making them difficult to load into smaller capillaries.

Once the small amount of material is in the capillary funnel (assuming it is a commercial capillary), it must be coaxed to drop to the bottom. This is usually done using some form of vibration. Anything from dedicated capillary-filling machines to ultrasonic baths, test-tube vibrators and nail files can be used. A common strategy is to drop the capillary down a vertical 50 cm glass tube, and allow the bouncing when the capillary hits the bottom to vibrate the sample. However, with very small and/or valuable samples the risk of the sample being vibrated out of the funnel may be too great to use automated techniques. In this case, very gently stroking the capillary using a fingernail to induce a low-frequency vibration may be the best option, changing the position at which the capillary is held to alter the vibration frequency as required. Agglomerates blocking a capillary can be very difficult to break up by vibrating the capillary manually, but an ultrasonic bath can often break up loosely bound agglomerates. Using a smaller-diameter quartz capillary or wire to tamp down a clog is possible, but riskier than using an ultrasonic bath.

The most commonly used capillaries range between 0.3 and 0.8 mm in diameter. Capillaries with a diameter less than 0.3 mm are extremely difficult to fill and very large ones can cause unwanted artifacts. For moisture-sensitive materials it is worth noting that significant moisture can adhere to the interior surface of commercial glass and quartz capillaries, so heating them in an oven prior to use is recommended.

The interplay between the sample absorption, radiation and optics can make the choice of capillary material and diameter a dynamic one. The capillary absorption is measured using the term μR, where μ is the effective linear absorption coefficient (taking account of the sample density) and R is the capillary radius. A convenient tool for estimating capillary absorption is available on the 11-BM web site (http://11bm.xray.aps.anl.gov ). Ideally, the value of μR should be less than 3 for the absorption corrections in most software packages to adequately cope with the effect of absorption. A recent analytical correction has been shown to be effective to μR = 10 (Lobanov & Alte de Viega, 1998[link]), but is not yet implemented in all current analysis software. A pre-analysis correction is always possible but not ideal. The effect of high capillary absorption can be seen visually by a reduction in peak intensity at lower angles, which correlates with the displacement parameters in a structure refinement. The easiest way to change μR is by changing the capillary diameter. More heavily absorbing samples usually require smaller capillaries, although using an alternative radiation such as Mo Kα to change the linear absorption coefficient is a possible alternative. Determining an accurate sample packing density experimentally can be tricky. There can be significant variability between supposedly identical capillaries, so ideally the empty portion of the actual capillary being used should be measured. The packing density generally ranges from 20–50% depending on the morphology of the crystallites and the amount of energy applied in vibrating the sample into the capillary (e.g. sonicating the sample will increase the packing density).

Where contaminating the sample is acceptable, another option is to dilute the sample with a material with very low absorption to reduce the overall sample absorption. There are two options here: either an amorphous material or a crystalline one. The addition of an amorphous material such as fumed silica (others could include amorphous boron, carbon black etc.) does not add any additional reflections to the pattern but will increase the background. Given that the backgrounds of capillaries using Cu Kα radiation are often quite high already, this may not be desirable. Alternatively, a material such as diamond powder can be used, which will add a small number of lines at high angles but does not add to the background. The closely defined crystallite sizes of diamond polishing powder can also improve the flow characteristics of materials that tend to agglomerate. The phase purity of polishing media is not relevant to their intended use, and some diamond polishing powders can contain some SiC, corundum or quartz. Check the phase purity of any diluting phase before use.

Fig. 2.10.42[link] shows the pattern from a 0.3 mm capillary of pure SnO2 (cassiterite) taken with Cu Kα radiation compared with that from reflection geometry. The linear absorption coefficient of SnO2 with Cu Kα radiation is ~1400 cm−1. Assuming a 50% packing density, μR with a 0.3 mm diameter capillary is 10.5, which is much higher than can be tolerated in any structural analysis. Absorption attenuates the lower-angle reflections as the X-rays cannot penetrate properly compared to the high angles. However, in addressing capillary absorption, less really can be more. Fig. 2.10.43[link] shows data sets from SnO2 diluted with 8000 grit diamond powder and with amorphous carbon black. As expected, the background is higher with the amorphous carbon but without the additional reflections from the diamond powder. Despite there being only approximately 10 vol% SnO2 in each of the sample mixtures, the raw low-angle intensities are much higher, and the relative intensities are comparable with those from the reflection data in Fig. 2.10.42[link]. Assuming a 50% packing density for the mixture, the value of μR with a 0.3 mm capillary would be approximately 2.3, which is in the acceptable range for structural analysis.

[Figure 2.10.42]

Figure 2.10.42 | top | pdf |

Comparison of the diffraction patterns of pure SnO2 from a 0.3 mm quartz capillary in transmission and reflection geometries with Cu Kα radiation. The very high absorption of SnO2 leads to severe attenuation of the lower-angle reflections in the transmission data.

[Figure 2.10.43]

Figure 2.10.43 | top | pdf |

Raw diffraction data from 0.3 mm capillaries of SnO2 diluted with 8000 grit diamond powder and carbon black. In each case the capillaries had approximately the same packing density of SnO2, so yielded almost identical intensities.

The relative intensities are such that a good-quality Rietveld refinement of a heavily absorbing compound such as SnO2 with Cu Kα laboratory data can be easily carried out. Fig. 2.10.44[link] shows the fit of the diamond-diluted sample to the literature cassiterite SnO2 structure. With very high dilution factors one should be careful not to compromise the particle statistics too much. Utilizing the full width of the detector with a full capillary will maximize the available statistics.

[Figure 2.10.44]

Figure 2.10.44 | top | pdf |

Rietveld refinement of the diamond-diluted data with the SnO2 cassiterite structure. The capillary background was subtracted prior to the fitting whilst maintaining the correct counting statistics. The Rwp value for this fit was 8.4%.

An alternative approach to dilution of heavily absorbing samples inside a capillary is to coat the outside (or inside) of a capillary. An appropriate absorption correction for annular samples does exist (Bowden & Ryan, 2010[link]), so this is not an impediment. However, it is not available in common software packages so may have to be applied to the raw data prior to a structural analysis. One requirement is that a known thickness of sample needs to be applied to the surface of the capillary as uniformly as possible. This can be difficult to achieve and may require the use of an adhesive to bond the sample sufficiently to the capillary while spinning. The additional effect of an adhesive on the background should be considered in the same way as for a smear mount. Similar results to dilution may be achieved if done with care, as shown in Fig. 2.10.45[link].

[Figure 2.10.45]

Figure 2.10.45 | top | pdf |

Comparison of data from SnO2 when diluted with diamond inside a 0.3 mm capillary and pure SnO2 coated on the outside of a 0.3 mm capillary.

Depending on the instrument geometry, a large diameter capillary can have an additional effect. Where an instrument does not have a focusing geometry (either primary or secondary), the peak resolution is degraded with increasing capillary diameter. With organic samples this can lead the analyst to use a smaller diameter capillary than optimal to retain reasonable resolution. Consequently, with organic samples where capillaries of 0.8 mm diameter are commonly used, it is highly recommended that an instrument with a primary focusing monochromator (or mirror) is used; the focus should be at the detector. Where the diffractometer is θ−θ geometry it is best to still collect capillary data as if it were a θ−2θ Debye–Scherrer instrument, simply by collecting `detector scans' or the equivalent in the data-collection software. This has no effect on the data in a perfect situation, but it means that the sample illumination is constant over all diffracting angles even if there is a misalignment of the primary beam with respect to the capillary axis (caused either by misaligned optics, a mis­aligned capillary stage, or both). In addition, a correction for capillary displacement can be applied to data collected in conventional Debye–Scherrer geometry (Klug & Alexander, 1954[link]) as the x and y displacements relative to the incident beam are constant over all 2θ angles.

Polymer capillaries are becoming increasingly common and are the standard at many synchrotron beamlines. They are easy to seal, but the lack of a funnel can make smaller sizes trickier to fill. A number of polymers can be used for capillaries, e.g. Mylar [poly(ethylene terephthalate) – PET] and Kapton [poly(oxydiphenylene pyromellitimide)]. The background from the capillary material itself is often more noticeable with a laboratory diffractometer than for higher-energy synchrotron instruments. A comparison of the background with a Cu Kα focusing mirror laboratory diffractometer from 0.5 mm quartz, soda lime glass, PET and polyimide capillaries is shown in Fig. 2.10.46[link]. A study of the different options for polymer capillaries in the laboratory environment was published by Reibenspies & Bhuvanesh (2006[link]), which highlighted the awkward reflection with polyimide visible just above 5° 2θ in Fig. 2.10.46[link]. It is also worth noting that the walls of polymer capillaries are not as stiff as those of quartz capillaries. If a low-temperature or other experiment might produce an internal vacuum (i.e. freezing a liquid sample), a polymer capillary can deform from a perfect cylinder, which may cause problems.

[Figure 2.10.46]

Figure 2.10.46 | top | pdf |

Comparison of the background from four different 0.5 mm-diameter capillaries. The quartz and glass capillaries are commercial capillaries for diffraction analysis. PET and Kapton capillary tubing are available from a number of different suppliers and are not made specifically for diffraction.

Mounting the filled capillary on the goniometer head can be achieved in different ways. Most commonly a hollow brass pin is used, but flat platforms are available (Fig. 2.10.47[link]). The various pins/platforms are a standard size, so they should fit no matter where they are sourced from. The flat platforms have a hole in the middle, but it is only suitable for inserting small-diameter capillaries. Large-diameter capillaries must be affixed to the platform surface with wax and are vulnerable to sagging with horizontal goniometers because of the lack of support. The brass pins will accept larger capillaries and are to be preferred with respect to improved support for the capillary where the capillary is held at both ends of the brass pin (Fig. 2.10.48[link]). Fixing the capillary onto the base is often done using wax or clay, although epoxy may be preferable if elevated temperatures are to be used. Coarse alignment is usually performed using a small desktop microscope before final alignment on the system. It is important to try to get the capillary rotating as straight as possible before mounting on the system, as removing tilt errors is much more difficult with the higher-magnification alignment scope mounted on the goniometer. Final alignment of a capillary is an exercise requiring patience. Never try to align out errors in two directions at once. Even if repeated attempts are necessary to stop the goniometer head in the correct position (Fig. 2.10.49[link]), only correct errors perpendicular to the view in the scope. Ideally, the final alignment should only require correction of a side-to-side movement rather than any wobble from tilt misalignment. However, it can still take some time. For systems where the goniometer spinning is controlled by computer software a wireless computer mouse is a very good investment, as it allows the person performing the alignment to stop the spinning capillary without taking their eyes off the sample.

[Figure 2.10.47]

Figure 2.10.47 | top | pdf |

Platform and pin mounts for capillary samples.

[Figure 2.10.48]

Figure 2.10.48 | top | pdf |

A 0.5mm capillary secured into a standard brass capillary pin using dental wax at both ends of the pin.

[Figure 2.10.49]

Figure 2.10.49 | top | pdf |

Goniometer head position in relation to the goniometer-mounted alignment scope.

References

Alexander, L. E. & Klug, H. P. (1948). Basic aspects of X-ray absorption in quantitative diffraction analysis of powder mixture. Anal. Chem. 20, 886–894.Google Scholar
Bowden, M. & Ryan, M. (2010). Absorption correction for cylindrical and annular specimens and their containers or supports. J. Appl. Cryst. 43, 693–698.Google Scholar
Bravais, A. (1866). Etudes Cristallographiques. Paris: Gauthier Villars.Google Scholar
Brindley, G. W. (1945). The effect of grain or particle size on X-ray reflections from mixed powders and alloys, considered in relation to the quantitative determination of crystalline substances by X-ray methods. London Edinb. Dubl. Philos. Mag. J. Sci. 36, 347–369.Google Scholar
Buhrke, V. E., Jenkins, R. & Smith, D. K. (1998). A Practical Guide for the Preparation of Specimens for X-ray Fluorescence and X-ray Diffraction Analysis. New York: Wiley-VCH.Google Scholar
Cline, J. P. & Snyder, R. L. (1987). The effects of extinction on X-ray powder diffraction intensities. Adv. X-ray Anal. 30, 447–456.Google Scholar
Dollase, W. A. (1986). Correction of intensities for preferred orientation in powder diffractometry: application of the March model. J. Appl. Cryst. 19, 267–272.Google Scholar
Donnay, J. D. H. & Harker, D. (1937). A new law of crystal morphology extending the law of Bravais. Am. Mineral. 22, 446–467.Google Scholar
Friedel, G. (1907). Etudes sur la loi de Bravais. Bull. Soc. Fr. Mineral. 30, 326–455.Google Scholar
Hillier, S. (1999). Use of an air brush to spray dry samples for X-ray powder diffraction. Clay Miner. 34, 127–135.Google Scholar
Hillier, S. (2002). Spray drying for X-ray powder diffraction specimen preparation. IUCr Commission on Powder Diffraction Newsletter, 27, 7–9. http://www.iucr.org/__data/assets/pdf_file/0017/21626/cpd27.pdf .Google Scholar
Järvinen, M. (1993). Application of symmetrized harmonics expansion to correction of the preferred orientation effect. J. Appl. Cryst. 26, 525–531.Google Scholar
Jenkins, R. & Snyder, R. (1996). Introduction to X-ray Powder Diffractometry. New York: Wiley-Interscience.Google Scholar
Kaduk, J. A. (2009). A Rietveld tutorial – mullite. Powder Diffr. 24, 351–361.Google Scholar
Kaduk, J. A. (2013). Personal communication.Google Scholar
Klug, H. P. & Alexander, L. E. (1954). X-ray Diffraction Procedures for Polycrystalline and Amorphous Materials. New York: Wiley-Interscience.Google Scholar
Lobanov, N. N. & Alte de Viega, L. (1998). Analytic absorption correction factors for cylinders to an accuracy of .5%. Abstract P 2-16, 6th European Powder Diffraction Conference, Budapest, Hungary, 22–25 August. Zurich: Trans Tech Publications.Google Scholar
Pitschke, W., Hermann, H. & Mattern, N. (1993). The influence of surface roughness on diffracted X-ray intensities in Bragg–Brentano geometry and its effect on the structure determination by means of Rietveld analysis. Powder Diffr. 8, 74–83.Google Scholar
Raudsepp, M. (2012). Personal communication.Google Scholar
Reibenspies, J. H. & Bhuvanesh, N. (2006). Capillaries prepared from thin-walled heat-shrink poly(ethylene terephthalate) (PET) tubing for X-ray powder diffraction analysis. Powder Diffr. 21, 323–325.Google Scholar
Rowles, M. R. & Madsen, I. C. (2010). Whole-pattern profile fitting of powder diffraction data collected in parallel-beam flat-plate asymmetric reflection geometry. J. Appl. Cryst. 43, 632–634.Google Scholar
Sabine, T. M., Von Dreele, R. B. & Jørgensen, J.-E. (1988). Extinction in time-of-flight neutron powder diffractometry. Acta Cryst. A44, 374–379.Google Scholar
Scarlett, N. V. Y., Madsen, I. C., Cranswick, L. M. D., Lwin, T., Groleau, E., Stephenson, G., Aylmore, M. & Agron-Olshina, N. (2002). Outcomes of the International Union of Crystallography Commission on Powder Diffraction Round Robin on Quantitative Phase Analysis: samples 2, 3, 4, synthetic bauxite, natural granodiorite and pharmaceuticals. J. Appl. Cryst. 35, 383–400.Google Scholar
Smith, D. K. (2001). Particle statistics and whole-pattern methods in quantitative X-ray powder diffraction analysis. Powder Diffr. 16, 186–191.Google Scholar
Suortti, P. (1972). Effects of porosity and surface roughness on the X-ray intensity reflected from a powder specimen. J. Appl. Cryst. 5, 325–331.Google Scholar
Sykes, R. A., McCabe, P., Allen, F. H., Battle, G. M., Bruno, I. J. & Wood, P. A. (2011). New software for statistical analysis of Cambridge Structural Database data. J. Appl. Cryst. 44, 882–886.Google Scholar
Toraya, H., Huang, T. C. & Wu, Y. (1993). Intensity enhancement in asymmetric diffraction with parallel-beam synchrotron radiation. J. Appl. Cryst. 26, 774–777.Google Scholar
Toraya, H. & Yoshino, J. (1994). Profiles in asymmetric diffraction with pseudo-parallel-beam geometry. J. Appl. Cryst. 27, 961–966.Google Scholar
Wolff, P. M. de (1958). Particle statistics in X-ray diffractometry. Appl. Sci. Res. 7, 102–112.Google Scholar
Wolff, P. M. de, Taylor, J. M. & Parrish, W. (1959). Experimental study of effect of crystallite size statistics on X-ray diffractometer intensities. J. Appl. Phys. 30, 63–69.Google Scholar
Zachariasen, W. H. (1945). Theory of X-ray Diffraction in Crystals. New York: Dover Publications Inc.Google Scholar
Zevin, L. S. & Kimmel, G. (1995). Quantitative X-ray Diffractometry. New York: Springer-Verlag.Google Scholar








































to end of page
to top of page