Pecharsky V.K., Zavalij P.Y. Fundamentals of Powder Diffraction and Structural Characterization of Materials

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Experimental techniques 301

3.5.6

Effects of sample preparation on powder diffraction data

To summarize this section, a high quality specimen for powder

diffraction may be difficult to prepare and it is not as simple as it seems. The

task requires both experience and creativity. The adverse effects of an

improperly prepared sample can be illustrated by the following three figures,

shown in Rietveld format, where the experimental powder diffraction data

were collected in the Bragg-Brentano geometry from two different

specimens prepared from the same material, which was intentionally left in

the form of course powder.

the first example (Figure 3.29), the sample holder, which had a

cylindrical hole, 25 mm in diameter and

mm deep to hold the powder, was

filled completely with the powder using the technique shown in Figure 3.22.

The aperture of the incident beam was selected in a way so that the length

12, see Figure 3.24 and Figure 3.25) of the irradiated area was

-20 mm at 28

20". Assuming that the calculated diffraction pattern in

Figure 3.29 represents correct distribution of relative intensities, it is easy to

see that some of the observed diffraction peaks at random Bragg angles are

much stronger than expected. These anomalies are associated with the non-

ideal specimen, which contained course grains. As a result, the total number

of particles was far from infinite, andlor their orientations were not random.

50 60

Bragg angle,

(deg.)

Figure 3.29.

Poorly ground powder, suitable incident beam aperture. Experimental data are

shown using small circles; the calculated diffraction pattern is shown using solid lines. The

difference

Yobs

Ycalc

is shown at the bottom of the plot and the calculated positions of Bragg

peaks are marked using vertical bars. The data were collected without spinning the sample.

Chapter

To illustrate the origin of these random intensity spikes it is worthwhile to

recall the spottiness of the

Debye rings seen in the film in

Figure

3.1

and

modelled for the first four strongest peaks from this film in

Figure

3.30.

Assume that a course-grained LuAu specimen is under examination using a

powder diffractometer. Contrary to the film data, a much smaller fraction of

the Debye ring is registered by the detector in a conventional powder

diffractometer. Two possible traces of the receiving slit traversing

Debye

rings during data collection are indicated by the two bars in

Figure

3.30. The

sequence of high and low intensity spots varies from one Debye ring to

another along these two different directions. This will result in random,

considerable and unpredictable variations in the registered intensity as

established by a specific distribution of particles in the sample.

We note that the actual path of the detector slit is always the same, e.g.

that shown by the solid elongated rectangle in

Figure

3.30. When the

particles in the specimen are rearranged, the distribution of spots along each

Debye ring will change, thus effectively changing the path of the receiving

slit of the detector,

e.g. to that shown by the dashed elongated rectangle.

conventional powder diffractometry, it is all but impossible to recognize

these random and severe intensity spikes or dips from the visual analysis of

the collected data, but the same can be done easily from a simple visual

analysis of the film or area detector data.

Figure

3.30.

The model showing the spottiness of the first four distinct Debye rings in the x-

ray film with the powder diffraction pattern of LuAu from

Figure

3.1.

The relative intensities

of Bragg peaks vary considerably in the two directions shown as elongated rectangles (each

direction represents the possible trace of the receiving slit traversing the

Debye rings during

data collection). The Debye rings on the x-ray film are elliptic because the Debye-Scherrer

camera is cylindrical rather than spherical. The character notations in the figure characterize

the expected measured intensity of Bragg peaks as follows: VVS

very very strong, VS

very strong, S

strong, W

weak and VW

very weak. RS is the receiving slit.

Experimental techniques 303

the second example (Figure 3.31), the same course powder was placed

in a 0.2 mm deep sample holder and the powder covered a smaller area

(10x5mm2). The aperture of the diffractometer was left identical to the first

experiment. After comparison of Figure 3.31 with Figure 3.29 it is easy to

see that the relative intensities of peaks at low Bragg angles are clearly

reduced. This happened because the projection of the incident x-ray beam at

low Bragg angles (-20 mm) exceeded the sample length (10 mm), and since

only a fraction of the incident beam energy was scattered by the sample, the

diffracted intensity at low Bragg angles was correspondingly reduced.

Furthermore, the reduction of the number of particles in the irradiated

sample volume (smaller depth) further exaggerated intensity spikes at

random Bragg angles. Clearly, the presence of fewer particles worsens the

randomness in the distribution of their orientations.

On the other hand, when the specimen is nearly ideal

(i.e., when the

particles are small and nearly spherical) and when the aperture of the

diffracted beam is compatible with the sample size, highly accurate powder

diffraction data can be obtained as shown in Figure 3.32. Assuming that the

calculated diffraction pattern here also represents the proper diffracted

intensity, the difference between the observed and calculated diffracted

intensity is minimal over the whole range of Bragg angles and there are no

spikes or dips at random Bragg angles as was seen in both Figure 3.29 and

Figure 3.31.

Figure

3.31.

Poorly ground powder and an improper incident beam aperture. The data were

collected without spinning the sample. Notations are identical to

Figure

3.29.

304

Chapter

The particles in the specimen that was used to collect the data illustrated

Figure

3.32 consisted of multiple cellular grains with an average diameter

the result of rapid solidification. The powder was subjected to a

brief homogenization heat treatment at 950•‹C (5 min), which was sufficient

to even out the distribution of Ni and Sn in the crystal lattice of the major

phase. The powder was sifted through a 10 pm sieve after the heat treatment

and only particles smaller than -10 pm in diameter were used to prepare a

flat specimen for data collection employing the Bragg-Brentano focusing

geometry. The data were collected using a scintillation detector with

monochromatization of the diffracted beam by a curved graphite single

crystal monochromator. A variable divergence slit was also employed to

minimize the background. The peak-to-background ratio for the strongest

Bragg reflection at 28

42" was

-1

80: 1.

20 30 40 50 60 70 80

Bragg angle,

(deg.)

Figure

3.32.

High quality, nearly spherical powder prepared by high-pressure gas atomization

from the melt and proper sample length,

The x-ray powder diffraction data were collected

from a continuously spinning sample

(20

rnrn

diameter and 1

deep) prepared as shown in

Figure

3.22.

Notations are the same as in

Figure

3.29.

The powder contains a small fraction

of a second phase, which is identified by the series of vertical bars shifted downwards. The

inset shows the scanning electron microscopy image of the powder morphology. (Powder

courtesy of

Dr.

I.E.

Anderson.)

Experimental techniques

3.6

Data acquisition

When the sample for a powder diffraction experiment has been properly

prepared, the next step, i.e. acquiring experimental diffraction data is also

exceedingly important in obtaining reliable diffraction data. Needless to say,

many of the data acquisition variables can be adjusted to produce a better or

worse quality data set. These include instrumental parameters, i.e., the

wavelength (energy) of the x-rays, their monochromatization, apertures of

goniometer optics, power settings, and data collection parameters,

i.e.,

scanning mode, scan range, step in data collection, and counting time.

3.6.1

Wavelength selection

Usually the wavelength of the x-rays can be freely selected only when

using synchrotron radiation.

conventional laboratory conditions when the

wavelength of the x-rays needs to be changed, this normally means that the

x-ray tube has to be replaced, which should be followed by tube ageing and

realignment of the goniometer. These operations should be carried out by

trained personnel and they are rarely performed in practice to collect data

from a single sample. Therefore, the selection of the x-ray tube (and the

anode) type is usually done based on practical considerations,

i.e., based on

which type of materials are customarily studied, and what is the purpose of

the powder diffraction examination of the majority of samples.

The most typical wavelength selection in powder diffiactometry

with a

copper (Cu) anode, although other anode materials may be used. These

include chromium (Cr), iron (Fe), cobalt (Co) and molybdenum (Mo)

anodes. The corresponding characteristic wavelengths for these anode

materials are listed in

Table

2.1,

Chapter

Long wavelengths (Cr, Fe, Co)

are preferred when the accuracy of lattice parameters is of greatest concern

since it is possible to measure intensity scattered at high Bragg angles.' Short

wavelength (Mo) can be used to examine a large volume of the reciprocal

lattice. However, since powder diffraction data are one-dimensional, the

resolution may be too low for complex crystal structures andlor materials

with low crystallinity when excessive Bragg peak overlap makes the

diffraction pattern extremely difficult to analyze and proces~.~

See Chapter 5, section 5.13 for details about various factors affecting precision of the unit

cell dimensions.

Short wavelength radiation may adversely affect resolution of the data collected using

goniometers with small radii (usually under -200 mm). When a large radius goniometer

(usually 250

mrn

or greater) is employed, the reduction of the wavelength from Cu- to Mo-

anode has little effect on pattern resolution because the widths of Bragg peaks decrease

with the decreasing wavelength. Regardless of the goniometer radius, the use of short

wavelengths requires a more precise alignment when compared to long wavelengths.

306

Chapter

Another important consideration in selecting x-ray wavelengths for a

powder diffraction experiment is whether or not the studied material

contains chemical elements with their K-absorption edges located just above

the used characteristic wavelength. For example, the K-absorption edge of

Co is -1.61

A. The strongest KallKa2 spectral lines of copper have

wavelengths -1.54 A. Hence, nearly all KallKa2 characteristic Cu radiation

will be absorbed by Co, and this type of x-ray tube is hardly suitable for x-

ray powder diffraction analysis of Co-based materials. The same is true for

Fe-based materials because the K-absorption edge of Fe is -1.74 A. On the

other hand, Co and Fe anodes are well-suited for collecting powder

diffraction data from Co- and Fe-containing substances using Co KallKa2

and Fe KallKa2 doublets.

Certain combinations of characteristic wavelengths and chemical

elements may cause considerable x-ray fluorescence (e.g. see

Figure

3.16).

This phenomenon is similar to the fluorescence in the visible spectrum and it

originates from the fact that electrons in an atom can be excited and removed

from their ground states by energy transfer from photons of sufficient

energy. Electrons from higher energy levels produce fluorescent x-rays when

they lower their energy by occupying the formed vacancies. Fluorescent

radiation is dissipated in all directions and it usually results in an increase of

the background. Chemical elements with strong true absorption will

generally produce strong x-ray fluorescent background.

3.6.2

Monochromatization

Either the diffracted or incident beam should be monochromatized during

data collection. When conventional x-ray sources are employed, additional

peaks due to the presence of weaker spectral lines with various energies in

the characteristic spectrum will "contaminate" the diffraction picture and

exaggerate peak overlap. The strongest unwanted wavelength for any anode

material is

KP1 (see

Table

2.1

in Chapter

2).

When a continuous

(synchrotron or neutron) spectrum is available, it would be impossible to see

discrete Bragg reflections as a function of Bragg angle without

monochromatization, as directly follows from Bragg's equation.

The simplest monochromatization tool that can be used in powder

diffractometry is a

P-filter (also see Chapter 2, section 2.3.2.1). Filter

materials have their K-absorption edges just above the wavelength of the

strongest

KP spectral line of the anode of the x-ray tube, and their

performance is based on how completely they absorb the characteristic

impurity wavelengths and how well they transmit the desired parts of the

characteristic x-ray spectrum. For example, to eliminate (absorb) nearly all

the

P-component of a copper anode

1.39 A) but to transmit most of the

Experimental techniques 307

a-component (h

1.54

the filter should be made from Ni (the K

absorption edge of Ni is -1.494. This results in nearly an eight-fold

difference in the linear absorption coefficients of Ni for

Kp and Ka parts of

the characteristic copper spectrum.

Various p-filters are most often used to monochromatize the diffracted

beam

(e.g. see Figure 3.6, left), but sometimes they are used to eliminate

Kp radiation from the incident beam in conventional x-ray sources. The

advantages of p-filters are in their simplicity and low cost. The

disadvantages include: (1) incomplete monochromatization because a small

fraction of Kp spectral line intensity always remains in the x-ray beam;

(2)

the intensity of the Ka spectral line is reduced by a factor of two or more,

and (3) the effectiveness of a p-filter is low for white x-rays above Ka and it

rapidly decreases below Kp. Therefore, p-filters are nearly helpless in

eliminating the background, especially when the latter is enhanced by x-ray

fluorescence (the filter itself fluoresces due to the true Kp absorption).

The most common monochromatization option used in modern powder

diffractometry is by means of crystal monochromators (see Figure 3.6, right

and Chapter 2, section 2.3.2.2). Monochromators transmit only specific,

narrowly selected wavelengths. As follows from the Bragg equation (nh

2dmsinOm), for a constant interplanar distance, dm, only one wavelength, h,

will be transmitted at a given monochromator angle, Om, assuming that n

A great variety of crystal monochromators are used in practice, but the best

results are usually obtained using curved crystal monochromators, as shown

in Figure

3.33

since they achieve the most precise focusing of the x-ray

beam and therefore, lower the intensity losses.

A well aligned monochromator usually leaves only Ka, and Ka2

characteristic wavelengths and considerably reduces background when it is

used to monochromatize the diffracted beam.

fact, a diffracted beam

monochromator is very effective in nearly complete elimination of even

severe fluorescence (see Figure 3.16).

some instances, high quality curved

monochromators can be used to eliminate the Ka2 component in

combination with the relatively large focusing distance and narrow

monochromator slit. Spectral purity of x-rays is a definite advantage of this

monochromatization approach. Furthermore, if necessary, the

monochromator angle, Om, can be selected to eliminate the Kalla2 doublet

and to leave only the Kp part, which is truly monochromatic in light anode

materials, including the most commonly used Cu anode.

On a downside, high quality crystal monochromators are relatively

expensive, and they also reduce the intensity of characteristic x-rays by a

factor of two to three. It is worth noting that when the monochromator is

made from a low quality single crystal or when it is improperly aligned, the

resulting reduction of the transmitted intensity may be severe.

Chapter

Figure

3.33.

The schematic of monochromatization of the diffracted beam using a curved

crystal monochromator.

receiving slit,

curved monochromator,

monochromator scatter slit,

-detector. The dash-dotted arc represents the goniometer

circle. The dashed arc shows the focusing circle of the monochromator.

A third monochromatization option, i.e. energy dispersive solid-state

detectors, became available quite recently.

adjusting the energy window,

it is possible to make such a detector sensitive to only the specific energy of

x-ray photons, and therefore, the monochromatization is achieved

electronically (see Chapter 2, sections 2.3.2.3 and 2.4.3). The most important

advantage of this approach is in the virtual absence of the loss of intensity.

On the downside, continuous cooling of the detector is required.

Furthermore, if the powder diffractometer is used in special applications, for

example to examine diffraction patterns from large single crystals, very

strong diffracted intensity with different wavelengths could be accidentally

registered.'

All monochromatization options discussed above have been used

successfully in powder diffractometry. With point detectors,

i.e. with those

detectors, which register, diffracted intensity at a specific angle, one point at

a time, either or both the incident and diffracted beam can be

monochromatized. When position sensitive or image plate detectors are

used, the only feasible option is to use a

P-filter or a crystal monochromator

to achieve monochromatization of the incident beam. As shown in

Figure

3.16, for some materials the background becomes too high, which makes

diffraction data nearly useless in the determination of the atomic parameters

of the material.

An example or registering

and

wavelengths is found in

Figure

2.18

Chapter

Experimental techniques

309

3.6.3

Incident beam aperture

The aperture of the incident beam should be selected to match the

diffraction geometry and sample size. For the commonly used Bragg-

Brentano focusing geometry, the most important requirement is that at any

Bragg angle the length of the projection of the incident beam does not

exceed the length of the sample (see

Figure

3.24,

Figure

3.25 and Eq. 3.1).

This is achieved by a proper selection of the divergence slit (see

Figure

3.6).

When the slits are calibrated in degrees, the use of Eq. 3.1 to determine the

proper size of the divergence slit is straightforward.

the majority of

commercial diffractometers, however, slits are calibrated in mm and their

angular divergence can be estimated (also see Eq. 2.5 in Chapter 2) using the

following simple relationship, which assumes the infinitesimal size of the x-

ray tube focus.'

Eq. 3.5,

is the angular divergence of the slit in degrees,

is the slit

opening in mm, and

is the distance from the x-ray tube focus to the

divergence slit in mm. It may be useful to check that the correct aperture of

the incident beam has been selected by mounting a fluorescent screen on the

sample holder at the lowest Bragg angle that will be examined during the

experiment. Alternatively, it is always possible to select a divergence slit

narrower than the acceptable maximum to ensure that the projection of the

incident beam fits within the sample length at the minimum Bragg angle.

Unnecessary reduction of the aperture of the incident beam, however, results

in the proportional reduction in the diffracted intensity, see

Figure

3.35,

below.

the transmission geometry the requirements are different. When a flat

transmission sample is used, the aperture of the incident beam is defined by

the largest Bragg angle of interest, since at

0 the sample is perpendicular

to the incident beam (and not parallel, as in the Bragg Brentano geometry).

Equation 3.1 then becomes as follows (where the notation are the same as in

Eq. 3.1)

As established in Chapter

(see

Figure

2.9), this is a valid approximation because the

typical projection of the I

wide line focus of the x-ray tube, visible at a small take-off

angle (usually

6'),

results in the source size on the order of

0.1

mm.

Chapter

The opposite constraint on the relationships between the incident beam

size and sample size should be followed when a cylindrical transmission

sample is under examination: the projection of the beam should be large

enough to irradiate the whole sample at any position it may occupy during

spinning. This includes small precession around the goniometer axis. If this

requirement is not followed, the inhomogeneities in powder packing density

may cause sporadic changes in diffracted intensities.'

The effects of the incident beam aperture on both the intensity of the

diffracted beam and the resolution of the goniometer in the Bragg-Brentano

focusing geometry are shown in

Figure

3.34

and

Figure

3.35.

Bragg peak

intensities increase rapidly and nearly linearly as a function of the incident

beam aperture as long as the opening of the divergence slit keeps the

incident beam in check and the lengths of its projection

(L,

Eq.

3.1) remains

shorter than the diameter of the specimen (20 mm), as shown in

Figure

3.36.

34.6 34.8 35.0 35.2 35.4 35.6 35.8 36.0

Bragg angle,

(deg)

Figure

3.34.

The set of x-ray powder diffraction patterns collected from the nearly spherical

LaNi4,85Sno.ls powder (see

Figure

3.32,

inset) on a Rigaku TTRAX rotating anode powder

diffractometer using Mo

radiation. Goniometer radius

285

mm;

receiving slit RS

0.03"; flat specimen diameter d

mrn.

Incident beam apertures were 0.05, 0.17,0.25,0.38,

0.5,0.75, 1, 1.5,2" and completely opened (-5O), respectively. An automatic variable scatter

slit was used to reduce the background. The data were collected with a fixed step A20

O.OlO,

and the sample was continuously spun during the data collection.

The homogeneity of an incident beam may become an issue when a cylindrical specimen

experiences large amplitude oscillations around the goniometer axis during spinning.