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

Подождите немного. Документ загружается.

Experimental techniques

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

Divergence slit aperture, DS (deg) Divergence slit aperture, DS (deg)

Figure

3.35. The normalized integrated intensity of the strongest peak (left) and the average

full width at half maximum,

FWHM,

(right) of the three Bragg peaks shown in

Figure

3.34,

both as functions of the divergence slit aperture. The intensity of the strongest Bragg peak

was normalized with respect to its value when the divergence slit was completely opened (DS

So).

The corresponding values at the completely opened divergence slit are not shown to

clarify the behavior at low apertures.

Figure

3.36.The projections of the incident beam (white rectangles) on the sample surface

(filled circles) at

35"

modeled using

Eq.

3.1

DS),

goniometer radius

285

mm,

and a constant width,

mm.

Specimen diameter is

mm.

Chapter

Bragg peak intensity continues to increase so long as the aperture of the

incident beam is small enough and the projection of the incident beam is

fully contained within the sample boundaries. The background level

approximately follows the behavior of Bragg peak intensities, which is also

evident from

Figure

3.34. When the aperture of the divergence slit reaches

and exceeds 1.5", no further increase in the intensity of the diffracted beam

is observed. This is consistent with

Eq.

3.1,

Figure

3.25, and

Figure

3.36,

which indicate that for the used goniometer the projection of the incident

beam exceeds the sample (20 mm diameter) at

35" when DS aperture

(cp)

is between 1.0 and 1.5O.I Predictably, the linear behavior of the scattered

intensity when DS

0.5" extrapolates to

0 with DS

0.2

The varying incident beam aperture has minimal effect on the resolution

of the instrument due to excellent focusing. As shown in

Figure

3.35, right,

the average full width at half maximum (FWHM) increases from -0.073 to

-0,077" (i.e. only by -5%) when the divergence slit aperture increases from

0.05" to completely opened (i.e. by as much as -10,000%). The dependence

of the FWHM on the slit opening saturates at wide apertures, which is

consistent with the full illumination of the specimen when DS exceeds

1".

addition to controlling the aperture of the incident beam in the plane

perpendicular to the goniometer axis, its divergence in the plane parallel to

the goniometer axis should be controlled by Soller slits (see

Figure

3.6).

This is done to reduce the asymmetric broadening of Bragg peaks, which

may be very strong at low Bragg angles and small goniometer radii.

The effect of varying the axial divergence of the incident beam is shown

Figure

3.37, where one low- and one middle-Bragg angle peaks were

measured with different Soller slits. Obviously, large axial divergence results

in the appearance of a broad tail extending towards low Bragg angles and in

somewhat reduced resolution of the instrument due to the related change in

the peak shape caused by broadening of both the base and FWHM. As the

Bragg angle increases, the peak asymmetry becomes less obvious but is still

present.

The behavior of

as a function of

(cp)

is nearly linear for a constant

when

is small,

which is apparent from the second part of

Eq.

3.1.

The deviation from linearity observed in

Figure

3.35,

left, when

is between

0.5

and

lo,

i.e. when the projection of the incident

beam remains within the sample boundaries as seen in

Figure

3.36,

is associated with the

inhomogeneity of the incident beam. Its intensity as a function of

becomes nonlinear at

large divergence slit openings with the maximum in the center of the projection, gradually

and non-linearly decreasing towards its ends. The distribution of intensity in the incident

beam as a function of

is source-dependent and, if necessary, it may be measured

experimentally.

This is expected assuming the ideal homogeneity of both the incident beam and the sample

packing density. The former is true for small divergence slit openings, and the latter is true

for the used sample, which was prepared from the nearly spherical particles.

Experimental techniques

13.4 13.6 13.8 14.0 14.2

Bragg angle, 28 (deg.)

29.7 29.9 30.1 30.3 30.5

Bragg angle, 28 (deg.)

Figure

3.37.

Two individual Bragg peaks observed in the diffraction pattern of the

LaNi4,&,,, powder (see

Figure

3.32,

inset). The data were collected on a Rigaku

TTRAX

rotating anode powder diffractometer using Mo Ka radiation and two different sets of Soller

slits controlling the axial divergence of the incident beam:

2.5"

(solid lines) and

(dash-dotted lines). The Soller slit controlling the axial divergence of the diffracted beam was

constant,

2.5O.

The goniometer radius R

285 mm; receiving slit RS

0.03"; flat

specimen diameter d

20 mm; incident beam aperture DS

0.17". An automatic variable

scatter slit was employed to reduce the background. The data were collected with a fixed step

A20

0.01

and the sample was continuously spun during the data collection.

3.6.4

Diffracted beam aperture

The aperture of the receiving slit (Figure 3.38) is as important as that on

the incident beam side. The receiving slit should be selected as small as

reasonably possible to improve the resolution of the instrument. While the

size of the receiving slit does not affect the measurements at specific Bragg

angles (as does an improperly selected divergence slit), inevitably any

reduction in the receiving slit opening results in a proportional reduction of

the intensity registered by the detector at all Bragg angles. The balance

between the needed intensity and resolution may be difficult to find ab initio,

and it is best to perform several quick scans of a narrow Bragg angle range

with varying receiving slit and make a selection where both the resolution

and registered intensity are satisfactory. It is advisable to perform quick

scans in the range of Bragg angles that include the strongest Bragg peaks.

3 14

Chapter

The influence of the diffracted beam aperture on both the diffracted

intensity and the resolution of the goniometer in the Bragg-Brentano

geometry are shown in

Figure

3.38 and

Figure

3.39, respectively. Bragg

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

receiving slit opening when the receiving slit is narrow, then the increase

becomes slow and non-linear with an obvious tendency to saturation. The

background, however, increases steadily as a function of the receiving slit

aperture.

Unlike the incident beam aperture

(Figure

3.39, the varying receiving

slit opening has strong influence on the resolution of the instrument. When

the receiving slit aperture increases, the

Ka,

components of the two Bragg

peaks, 113 and 032, which are relatively well resolved when RS

0.04", are

observed as a main peak and a shoulder when RS

0.07". As shown in

Figure

3.39, this is due to the considerable change in the FWHM, which

increases from -0.073" for RS

0.01" to -0.104" when the receiving slit is

in a completely opened position, i.e. peak broadening exceeds 40%.

34.6 34.8 35.0 35.2 35.4 35.6 35.8 36.0

Bragg angle,

(deg)

Figure

3.38. The set of x-ray powder diffraction patterns collected from the LaNi,,85Sno,,5

powder (see the inset in

Figure

3.32)

on a Rigaku TTRAX powder diffractometer using Mo

radiation. Goniometer radius

285

mm;

Divergence slit DS

0.5'; flat specimen

diameter d

rnm.

Diffracted beam apertures were 0.01,0.02,0.03, 0.04,0.05, 0.06, 0.07,

0.08, 0.1, 0.12' and completely opened (-1

"),

respectively. An automatic variable scatter slit

was used to reduce the background. The data were collected with a fixed step A28

O.OlO,

and the sample was continuously spun during the data collection.

Experimental techniques

Integral

Receiving slit aperture,

(deg)

Receiving slit aperture,

(deg)

Figure

3.39.

The intensity of the strongest peak (left) and the average full width at half

maximum (right) of the three Bragg peaks shown in Figure

3.38,

as functions of the receiving

slit aperture. Both the integrated and peak intensities were normalized with respect to their

values at the completely opened receiving slit

(RS

lo).

The corresponding values at the

completely opened receiving slit are not shown to clarify the behavior at low apertures.

As noted above, a scatter slit can be used to reduce the background noise

before it reaches the detector. The aperture of the scatter slit should be

selected to enable unobstructed passage of the monochromatic diffracted

beam at any Bragg angle, see

Figure

3.40.

this example, the scatter slit

ScS is wide enough to transmit the beam without affecting its intensity. On

the contrary, the scatter slit ScS' is too narrow, and only a fraction of the

diffracted intensity will reach the detector.

The best practical way to select the aperture of the scatter slit manually is

to find a well-resolved and strong Bragg peak at low angles, and after setting

the detector at the peak maximum find the minimum opening of the scatter

slit that does not reduce the intensity of the peak without moving the detector

arm. Some goniometer designs enable continuous variation of the scatter slit

aperture, and if this is the case, it is advisable to expand the scatter slit gap

by a few percent after finding the minimum non-interfering opening.

Another option is to move (if possible) the slit by a few mm towards the

receiving slit. It is worth noting that it is always better to have a slightly

wider scatter slit aperture than a slightly narrower, because an improperly

selected scatter slit may result in the reduction of the measured intensity.

Chapter

Sample

Figure

3.40.

Examples of proper (ScS) and improper (ScS') selection of scatter slit aperture.

divergence slits, RS

receiving slits.

3.6.5

Variable aperture

Some of the modem powder diffractometers may be equipped with the

variable divergence, scatter and receiving slits (e.g., see

Figure

3.13). This

enables slit selection at the software level for any aperture, and in addition, it

is possible to vary the aperture of the incident beam continuously during data

collection. Thus, the variable slit option facilitates experiments where the

length of the irradiated area of the flat sample can be kept constant as shown

Figure

3.41.

This may be particularly useful when it is impossible to spin

the sample during data collection (e.g., when using high or low temperature,

or high pressure attachments) to improve particle orientations averaging.

Figure

3.41.

The schematic of goniometer optics during data collection employing variable

divergence and scatter slits apertures, which enables one to maintain the irradiated area of the

sample constant at any Bragg angle. DS

divergence slit, ScS

scatter slit, RS

receiving

slit.

Experimental techniques

The resultant diffraction pattern is then numerically processed to convert

the measured profile intensities to a constant incident beam divergence,

which is a standard in the majority of crystallographic software.

added

benefit when using variable slits is the availability of the software control

over the aperture of the scatter slit. Its opening may be varied automatically

and continuously thus providing the most effective background reduction at

all Bragg angles.

It is worth noting, however, that correcting the measured intensity for the

variable aperture of the incident beam introduces additional errors into the

experimental data, and if precise intensities are of greatest concern, using the

variable divergence slit option should be avoided. The errors associated with

the variable divergence slit may become especially severe if the

diffractometer is misaligned or when the incident beam is inhomogeneous.

On the other hand, maintaining a constant irradiated area eliminates errors

due to the inhomogeneity of sample packing,

e.g. when there is a difference

in packing density in the middle and on the sides of the flat specimen.

alternative to the continuously variable aperture methods is

measuring low and high Bragg angle regions separately with different but

fixed divergence slits. Thus, the low Bragg angle region is measured using

narrow, and the high Bragg angle region is measured using wide divergence

slits, respectively, to improve the intensity at high angles. This makes it

difficult to treat the whole powder diffraction pattern simultaneously in

many common applications, such as peak search and phase identification.

However, the majority of software programs designed for crystal structure

refinement enable processing of multiple diffraction patterns and the

multiple divergence slits approach is acceptable.'

3.6.6

Power

settings

Before making the selection of power settings, it is important to know the

power ratings of both the high voltage generator and x-ray tube. The two

adjustable instrumental parameters are accelerating voltage and tube current.

Their product (accelerating voltage in

kV and tube current in

mA)

establishes the output power of the generator and the input power of the

When relative diffracted intensities are of greatest concern, the multiple divergence slits

technique is preferred over the continuously varying divergence slit method. The two

largest contributions to intensity correction errors when slit opening varies arise from both

the imprecise mechanical control of the slit opening and from the inhomogeneity of the

incident beam. The latter becomes especially severe at large apertures (see

Figure

3.34

Figure

3.36

and relevant text). Neither of the errors can be easily accounted for. In the

former (multiple divergence slits) these

issues become irrelevant because multiple

diffraction patterns are employed in calculations with their own scale factors (see Chapter

Eq.

7.10 and section 7.3.8).

Chapter

ray source in Watts. It is usually the case that the maximum power is limited

by the x-ray tube power rating.

The accelerating voltage should normally be selected at or slightly above

the threshold of the most efficient generation of the characteristic emission

spectrum of the anode material. For example, the optimal voltage for the

excitation of characteristic radiation is

-45

kV for a Cu anode and

-80

for a Mo anode. However, most commercial high-voltage generators do not

operate above

kV, and the distribution of intensities between white and

characteristic radiation for a Mo anode is not at its optimum.

The second parameter, i.e. the tube current, should be selected as high as

possible without exceeding the allowable power rating of either the tube or

generator, since the intensity of the characteristic radiation in the incident

beam is proportional to the tube current. Finally, the standard lifetime of

most x-ray tubes (several thousand hours) can be extended considerably if

the tube is operated at

75%

or less of its rated power.'

3.6.7 Classification of powder diffraction experiments

Provided all instrumental parameters discussed above have been properly

selected, the next step in the acquisition of high quality experimental data is

to select the scanning mode, scan range, step of data collection, and counting

time. The scanning mode is applicable to both point detectors and short

linear or curved position sensitive detectors. When using long curved

position sensitive detectors

andlor image plates, the scanning mode generally

loses its meaning, since there is no need to move the detector and the entire

diffraction pattern is recorded at once, similar to using x-ray film for

recording the powder diffraction data. Most settings in data acquisition are

dependent on both the type of the powder diffraction experiment, and what

information is to be gathered from the acquired data. Based on the counting

time, powder diffraction experiments using point detectors can be broadly

classified as fast, overnight, and weekend experiments.

Fast experiments are usually conducted in the time frame from several

minutes to several hours depending on the brightness of the incident beam

(i.e. whether the x-ray source is a rotating anode or a sealed x-ray tube) and

the crystallinity of the material. Fast powder diffraction experiments give the

experimentalist a general idea about the complexity of a diffraction pattern

and data collected in a fast experiment are hardly ever useful beyond the

verification of the selected instrumental parameters for longer experiment@)

andlor for simple phase identification purposes. However, an hour-long

experiment collected from a nearly perfect sample may be suitable for

indexing and accurate refinement of the unit cell parameters.

For a properly aged x-ray tube; ageing procedure is described by the manufacturer.

Experimental techniques

319

Overnight experiments normally take from several hours to a whole day

or night. It makes sense to run these experiments overnight, since modern

powder diffractometers are completely automated.

Overnight powder

diffraction experiments usually provide good quality diffraction patterns,

which are satisfactory for further numerical processing that requires precise

peak positions and their intensities.

Weekend experiments normally take more than a day. They are

conducted when precise peak shapes are desired, for example, when

microstructural properties are analyzed based on the anomalies of Bragg

peak shapes, or when the crystal structure of the material is determined and

refined by using the Rietveld method (see Chapter

7).

3.6.8 Step

scan

The scanning mode defines how the detector and the x-ray source, or the

detector and the sample arms move during data collection (this depends on

the goniometer design, see

Figure

3.13

and

Figure

3.12, respectively): are

they in an intermittent (stepping) or in a continuous motion. When the

goniometer arms are in an intermittent motion, the diffracted intensity is

measured when both the x-ray source (sample) and the detector are at rest,'

and this is usually referred as the step scan mode.

A generic algorithm of data collection in the step scanning mode is

shown in the form of a flowchart in

Figure

3.42. It includes the following

sequence of events:

Movable goniometer arms advance to their initial (or next) positions.

Diffracted intensity is measured for a certain time (counting time) in this

configuration.

Current Bragg angle and intensity count are saved when counting time

expires.

Computer analyzes whether the last Bragg angle value has been reached:

if the answer is no, the loop is repeated by advancing goniometer arms to

the next position, which is calculated by adding a fixed step to the current

value of

28;

if the answer is yes, the data collection is finished.

The resulting powder diffraction pattern usually consists of intensity data

a numerical function of Bragg angle, as shown in

Figure

3.43.

In this context, "sample at rest" means that the angle between the sample surface and the

incident beam is constant. The specimen, however, normally continues to spin about an

axis, which is normal to its surface for better particle averaging and minimization of other

errors associated with preferred orientation and sample inhomogeneity.

320

Chapter

Figure

3.42.

The flow chart visualizing a generic step scan data acquisition algorithm. The

main loop is highlighted by using thick arrows.

Advance goniometer arms

Measure intensity for

Start

to the initial or next position

a specified time

The two most important parameters of the step scan experiment, defined

by the user, are the size of the step in terms of Bragg angle, A28, and

counting time,

The step size is constant throughout the entire experiment,

and it is usually selected between 0.01 and 0.05" of 28. Using Cu

radiation, the value A28

0.02' is quite standard. When using Mo

radiation, step size should be generally reduced to A20

0.01". High energy

synchrotron radiation may require steps as low as 0.001". Neutron powder

diffraction data are usually collected with 0.05 to

0.1' 28 steps.

When Bragg peaks are extremely broad, and when conducting fast

experiments, larger step sizes for analytical x-ray diffractometers are

acceptable. On the contrary, when the crystallinity of the examined material

is high and Bragg peaks are narrow, lower step sizes should be employed.

rule of thumb for the selection of the step size is that at least

to 12 data

points should be measured for well-resolved peaks within one full width at

half maximum. Even lower A28 values are used when precise Bragg peak

shape data are needed for structure refinement and microstructure

determination. When selecting A28 it is important to remember that small

step sizes improve resolution, which obviously cannot exceed the limits

imposed by the goniometer optics, while large step sizes reduce the duration

of the experiment.

Yes

Stop

Instrument specific header

Bragg angle 1, Number of accumulated counts 1[, Error 11

Bragg angle 2, Number of accumulated counts 2

Error 21

Bragg angle

Number of accumulated counts

Error

...,

...

Bragg angle N, Number of accumulated counts N[, Error N]

-------

Save

and current

photon count

Figure

3.43.

Example of powder diffraction data file format. Optional experimental errors

the measured intensity are shown in square brackets.