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

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Experimental techniques

Figure

3.20.

The two critical cases of non-random particle orientation distributions due to

distinctly anisotropic particle shapes: platelet-like (left) and needle-like (right) particles. The

arrows indicate the directions around which the particles may rotate freely.

When particles are in the form of thin needles

(Figure

3.20, right), the

orientations of the axes of the needles may be nearly random in the plane of

the sample. Furthermore, each needle has an additional rotational degree of

freedom, i.e. rotation about its longest axis, and such samples are expected to

have an in-plane preferred orientation. Even more complex preferred

orientation effects may occur when elongated particles are flat, i.e. when

they are ribbon-like. Then ribbons may align in the plane of the sample, and

because they are flat, their surfaces may arrange parallel to the sample

surface as well. This is the case of two combined preferred orientation axes:

one is along the axis of the ribbon and the second is perpendicular to the flat

face.

Regardless of which type of preferred orientation is present in the

sample, it will in its own and systematic way affect diffracted intensities.'

severe cases, nothing more than lattice parameters (if any) can be determined

from highly textured powder diffraction data since it is impossible to

precisely account for the changes in the diffracted intensity caused by the

exceedingly non-random distribution of particle orientations.

Depending on the diffraction geometry, proper sample preparation

techniques are different. We will begin with flat samples used in

combination with the Bragg-Brentano focusing geometry and horizontal

goniometer axis, since it is the most common geometry employed today. The

two types of sample holders frequently used in the Bragg-Brentano method

are shown schematically in

Figure

3.21. They can be made of metal, plastic

or glass. The difference between the two is that the holder on the left has a

See Chapter 2, section 2.10.6 for a description of mathematical models, which may be

used to account for the effects of preferred orientation on the scattered intensity.

292

Chapter 3

shallow cylindrical or rectangular hole (usually

114

mm deep) to

accommodate the powder. The container on the right has a rough spot on its

surface. Rough surface is usually created using a sand blaster with hard

particles

pm in diameter. When the roughness of the sample surface

is of the same order as particle size, this helps to diminish the preferred

orientation effects.

When a relatively large quantity of powder is available, the sample for

ray diffraction can be prepared by filling the volume of a cavity (Figure

3.21, left) with the dry powder. The excess powder should be removed from

the surface of the sample holder by a single sweep with a razor blade (Figure

3.22) or by an edge of a glass slide. Under normal circumstances, the powder

should never be compacted inside the hole using a smooth flat surface, e.g.

the flat side of a glass slide. If the powder is packed by pressing against a

smooth flat surface, this will inevitably rearrange particle orientations in the

specimen causing strong preferred orientation (see Figure 3.20). Compacting

is only forgivable if particles in the powder are nearly spherical, which is

quite rare. The powder inside the hole may be compressed, if necessary (e.g.

when the material is lightweight), by gently pressing against a rough surface,

which has the roughness commensurate with the average particle size.'

figure

3.21.

Exanlples of flat sample holders used in powder d~ffractometers with reflection

geometry. The holder on the left has a cavity, which is filled with the powder. The holder on

the right has a rough area, which accommodates a thin layer of the powder.

Figure

3.22.

Excess powder is removed from the top of the sample holder with a single sweep

by an edge of a razor blade or a glass slide. The direction of the sweep depends on the

physical properties of the powder and is usually established by trial-and-error.

Flat surfaces with varying roughness may be created by sandblasting a set of glass slides

using particles of different diameter.

quick solution may be achieved by gluing a piece

of sand paper with appropriate roughness on a glass slide.

Experimental techniques

293

Though filling the specimen holder with powder produces the best

quality flat samples for powder diffraction experiments, the procedure

requires some experience, and a uniform distribution of particle orientations

may be difficult to achieve, especially when working with light, fluffy

powders. It may be helpful, therefore, to prepare a viscous suspension of

powder using chemically inert, low boiling temperature liquid, which does

not dissolve the material and then pour the suspension into the hole. Excess

mixture is then removed by a single cut with a razor blade, and the

remaining solvent should be evaporated before installing the sample on a

di ffractometer.

Another method of reducing preferred orientation while mounting the

powder into the sample holder is the so-called side or back filling. It requires

a special or modified sample holder with an opening on the side or on the

back. The surface of the holder is covered with a glass slide during packing

and the slide surface facing the powder should have nearly the same

roughness as the average particle size. Using this technique the powder can

be packed better and with lower preferred orientation effects especially on

the surface of the sample, which is the most critical part of the specimen in

x-ray powder diffraction due to the limited penetration depth of x-rays.

An effective way of avoiding preferred orientation is spraying the fine

powder suspended in a quick drying polymer solution. Small droplets

spheroidise before the solution dries in-flight, and the tiny solid spheres that

form usually contain only a few particles embedded in each droplet.' This

method removes preferred orientation nearly completely because the

resulting particles are spherical and thus maintain random orientations

during mounting. It is, however, complex and introduces a substantial

amount of a polymer, which increases background noise thus reducing the

overall quality of the resultant powder diffraction pattern.

Good quality specimens with minimal preferred orientation effects can be

prepared by dusting the ground powders through a sieve directly on a sample

holder thus covering the rough spot. This is the only feasible option when

using sample holders without the hole to accommodate the powder. It is best

to cover the sample holder with a specially made mask, which is removed

when the dusting is completed

(Figure

3.23).

Excellent powdered samples (see

Figure

3.32),

can be prepared by using the high-pressure

gas atomization (HPGA) technique. HPGA involves melting a material of interest and then

spraying the melt through a nozzle employing a high-pressure non-reactive gas (e.g.

nitrogen, argon or helium). Liquid droplets (usually between -10 and -100 pm in

diameter) spheroidise and then rapidly solidify in-flight maintaining nearly spherical form.

The resulting powders may require brief homogenization

andlor recrystallization heat

treatment before they may be employed to collect powder diffraction data. HPGA-

prepared powders are not embedded into polymer shells, however, this technique requires

large amount of a starting material and is cost-ineffective in routine diffraction studies.

Dusted powder

Mask

Chapter

Figure

3.23.

Sample for powder diffraction prepared

dusting ground powder on the sample

holder through

mask.

Usually it takes several passes of dusting the powder on top of the sample

holder surface to ensure complete and uniform coverage of the area, which

will be irradiated during the x-ray diffraction experiment. It is also possible

to apply a thin layer of oil, grease, or slow drying glue to the appropriate

spot on a sample holder surface, and then dust the powder on top. When the

dusting is complete, the excess powder is easily removed by flipping the

sample holder

andlor by gently blowing air over the sample surface. The

downside of using fluid to hold the powder on the surface is that the powder

will remain contaminated with the fluid after the experiment is finished.

Another fact to keep in mind when preparing samples by dusting, is that it is

nearly impossible to create a smooth surface of the specimen: pressing a

sample holder against a flat surface is not an option as it usually induces

strong preferred orientation effects.

The most common approach to prepare flat samples for transmission

geometry is by dusting the powder on an x-ray transparent film covered with

a thin layer of slow drying varnish or glue and letting it dry before installing

the sample on the goniometer. Obviously, both the film and dry

varnishlglue

should not be crystalline.

Cylindrical specimens are usually prepared by sinking a thin glass

capillary into a liquid binder (e.g., purified petroleum jelly or liquid varnish)

and then by dipping the capillary into a loose powder. Alternatively, powder

can be mixed with oil to a consistency of thick slurry and then the capillary

is simply dipped into the mixture.

both cases, the capillary may need to be

exposed to the powder several times to ensure complete and uniform

coverage of its surface.

some instances, especially when the studied powder is air or moisture

sensitive, it can be placed inside a low absorbing glass capillary (e.g. a

borosilicate glass), after which the capillary is sealed. Filling capillaries with

powders is usually a tedious process and it requires larger diameter

capillaries than those usually used for surface coverage. Furthermore, it may

Experimental techniques

295

be difficult to avoid preferred orientation in packed capillaries, even though

cylindrical samples are the least susceptible to preferred orientation.

3.5.3

Sample

size

Several additional items must be considered when preparing samples for

x-ray powder diffraction experiment. One is the length of a flat sample,

along the optical axis of the goniometer in the Bragg-Brentano geometry. It

should be large enough so that at any Bragg angle during data collection, the

projection of the x-ray beam on the sample surface does not exceed the

length of the specimen. Referring to

Figure

3.24

and assuming that the

angular divergence of the beam is cp, it is easy to derive the relationship for

the varying irradiated length,

as a function of cp, goniometer radius,

and

Bragg angle,

Figure

3.24.

The length of the projection of the incident beam,

on the surface of the flat

sample in Bragg-Brentano geometry.

focal point of the x-ray source,

-divergence slit,

goniometer radius,

angular divergence of the incident beam,

Bragg angle. The

location of the goniometer axis is indicated by the open circle.

296

Chapter

Eq.

3.1,

is the angular divergence of the incident beam in degrees

and

is the goniometer radius in mm. When the angular divergence of the

incident beam is small

-lo) and

-5O, the approximation, also

shown in

Eq.

3.1, may be used, where

is the angular divergence of the

incident beam in radians, and

is in

rnm.

Based on

Eq.

3.1, the distance

12,

becomes critical at low Bragg angles,

wide divergence slit apertures and large goniometer radii (Figure

3.25).

Practical sample sizes in powder

diffraction

are usually kept below 25 mm in

length, and when low Bragg angle data are desired, the appropriate aperture

of the incident beam should be selected to avoid the situation when the size

of the x-ray spot exceeds the size of the sample. If it does, the measured

intensities will be underestimated at low Bragg angles because the sample

will be illuminated by only a fraction of the incident beam when compared

to that at high Bragg angles. Analytical accounting of this effect is unfeasible

due to inhomogeneous distribution of intensity (photon flux) in the cross

section of the incident beam.

100

Bragg angle,

(deg.)

Figure

3.25.

Irradiated lengths,

and

12,

of the flat specimen in the Bragg-Brentano geometry

as functions of Bragg angle calculated using

Eq.

3.1

for different angular divergences of the

incident beam assuming goniometer radius,

285

mm.

Experimental techniques

297

3.5.4

Sample thickness and uniformity

The second important factor is the absorption of x-rays by the sample. In

the Bragg-Brentano geometry the sample should be completely opaque to x-

rays. Assuming that the absorption of 99.9

of the incident beam intensity

represents complete opacity, then the beam intensity should be reduced by a

factor of

1000

and the following equation can be written (also see Eq. 2.8 in

Chapter 2):

where:

and

are intensities of the incident and transmitted beams,

respectively,

peff

is effective linear absorption coefficient (in mm-'), which is specific

for each sample and it should also account for the porosity of the powder,

is beam path (in mm) through the sample and it is related to the sample

thickness, t, as

thine.

After solving Eq.

3.2

with respect to t, the minimum sample thickness (in

mm) can be estimated from

3.45

sin

€I,,,

eff

Eq. 3.3,

Om,

represents the maximum Bragg angle to be measured

during the experiment. Hence, it is usually easy to prepare a satisfactory

Bragg-Brentano specimen from dense metallic alloys containing elements

heavier than A1 since they have large linear absorption coefficients, but it

may be difficult to prepare a specimen of sufficient thickness Erom materials

with low density andlor containing light atoms

(C,

and H), e.g. organic

compounds.

Conversely, transmission geometry requires that the sample is minimally

absorbing. This is usually not a problem if the studied specimen is a

molecular substance. However, when the material is a dense alloy or

intermetallic compound containing heavy elements, the preparation of a high

quality specimen for transmission powder diffraction may be problematic.

With flat transmission samples the best approach is to try to arrange no more

than a single layer of particles mounted on the film. When cylindrical

specimens are employed, the radius of the capillary should be reduced to a

practical minimum. Unfortunately, these measures usually reduce the

298

Chapter

number of particles in the irradiated volume and the quality of the resulting

diffraction pattern deteriorates.

The third important issue, which is generic to any type of specimen for

powder diffraction except cylindrical, is the uniformity of the sample. A

portion of the specimen that scatters the incident beam changes as a function

of Bragg angle

(e.g. see

Figure

3.25 and Eq. 3.1 above, and Eq. 3.6 below).

Thus, when the packing density of the powder is not uniform, this will result

in random changes of the number of particles in the irradiated volume and in

the measured diffracted intensity that are nearly impossible to account for. It

is difficult to achieve a uniform packing density because compacting the

powder to improve uniformity will generally result in increased preferred

orientation with different but also deleterious effects on the resulting relative

intensities. Effects of sample non-uniformity are best reduced by rapidly

spinning the sample during data collection. As mentioned above, every effort

should be made to collect powder diffraction data from a spinning sample

since it also considerably improves both the number of particles and

randomness of their orientations in the irradiated volume.

3.5.5

Positioning the sample with respect to the goniometer axis

No matter how much time has been spent on the sample preparation and

how good the resulting specimen is, it always needs to be properly

positioned on the goniometer. Consider, for example,

Figure

3.26, which

shows the effect of sample displacement in the Bragg-Brentano geometry.

When the sample is properly aligned

(i.e. when its surface coincides with

the goniometer axis as shown by the dashed image of the flat sample), the

correct Bragg angle 0 is measured, provided the sample is completely

opaque (see Chapter 2, section 2.8.2 and Eq. 2.43). However, when the

sample is displaced by the distance

from the goniometer axis, this

displacement results in a different measured Bragg angle, O,, even though

both the incident and diffracted beams form the same angle

with the

sample surface. Furthermore, if sample displacement is severe, the focusing

of the diffracted beam is no longer precise and this will result in the loss of

the resolution of the diffractometer. The latter issue becomes particularly

important when using goniometers with small radii.

Similar errors in the measured Bragg angle may occur in the transmission

geometry (see

Figure

3.27). Moreover, when a cylindrical sample

continuously deviates from the goniometer axis in a circular fashion during

spinning, considerable Bragg peak broadening, 60, may be observed as also

indicated in

Figure

3.2

Experimental techniques

Figure

3.26.

The effect of sample displacement by distances from the goniometer axis on the

measured Bragg angle,

O,,

in the Bragg-Brentano geometry. The goniometer axis is indicated

by the small open circle in the center of the drawing. The optical axis is shown as the dash-

dotted line. The ideal location of the sample is shown as the light-shaded dashed rectangle.

The axis around which the sample is usually spinning during data collection is indicated by

the vertical arrow.

Figure

3.27.

Effect of sample displacement by distance

from the goniometer axis on the

measured Bragg angle,

8,,

in the transmission geometry. When the misaligned cylindrical

sample spins around the goniometer axis as shown by a thick broken arrow, peak broadening,

60,

results. The goniometer axis is indicated by the small open circle in the center of the

drawing. The optical axis is shown as the dash-dotted line. The ideal location of the sample is

indicted as the light-shaded dashed circle in the center of the drawing.

300 Chapter

Errors in the registered Bragg angles associated with the non-ideal

positioning of the sample are usually not as severe when compared to those

observed in intensity measurements due to improper sample preparation.

They can be nearly completely eliminated by maintaining the goniometer

properly aligned. Furthermore, sample positioning errors in Bragg angles

are systematic, and they can be accounted for analytically, based on the

known geometry of a powder

diffi-actometer. For example, sample

displacement error in the Bragg-Brentano geometry (Figure

3.26)

expressed as

cos

-€I=-

Eq. 3.4,

is sample displacement,

is goniometer radius and the

resulting difference in Bragg angles,

is in radians (compare Eq. 3.4

with Eqs. 2.40 and 2.44 in Chapter 2). This error is commonly observed due

to the varying sample thickness (especially when the sample was prepared

by dusting) and the varying sample transparency. It is worth noting that the

displacement, s, can be refined together with lattice parameters, provided

low Bragg angle peaks are included in the refinement since they are the most

sensitive to s, as shown in Figure

3.28.

Bragg angle,

(deg.)

Figure

3.28.

The effect of sample displacement,

on the observed Bragg angles calculated

from

Eq.

3.4

assuming Bragg-Brentano geometry and goniometer radius

285

mm.

the observed Bragg angle,

is the Bragg angle in the absence of sample displacement.