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

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Experimental techniques

33 1

The numerator in Eq. 3.10 represents photon counting errors (see Eq. 3.8)

and the denominator corresponds to the total number of counts, both

summed over the total number of the measured data points,

For example

the two diffraction patterns shown in

Figure

3.47 are characterized by Robs

3.59% and 30.9% for the higher and the lower quality data, respectively. The

range shown in this figure includes a total of 1201 data points and the total

number of accumulated counts varies by nearly two orders of magnitude:

-6.4~10' and -8xlo3, for the high and poor quality data, respectively. The

Robs figure of merit reflects the overall quality of the powder diffraction

pattern and is comparable with other residuals (R-factors) commonly

calculated to represent the quality of fitting in the Rietveld method (see Eqs.

6.18 to 6.22 in section 6.7): a low Robs characterizes high quality data, while

a high Robs corresponds to low quality diffraction patterns.

3.7.2

Factors affecting resolution

The intrinsic one-dimensionality of the powder diffraction experiment

implies that every attempt should be made to collect diffraction data with the

highest possible resolution. As we established in section 3.6, many factors

affect the resolution of the experimental data, the most important being the

radius of the goniometer, the wavelength of the used radiation, the receiving

slit aperture and the step size or sampling step.

The goniometer radius is generally fixed. However, since the angular

resolution remains constant for a given condition of the specimen, linear

resolution of the goniometer is inversely proportional to its radius.

Therefore, when especially high resolution is an issue, x-ray diffraction data

should be collected using the instrument with the largest available radius. It

is important to remember that a large goniometer radius usually translates

into a decreased diffracted intensity and the powder diffraction experiment

will take a longer time to achieve an identical

Robs, when compared to that

using the goniometer with a smaller radius.

Similarly, using lower energy x-rays

(i.e. x-rays with higher

wavelengths) will normally result in the improved resolution of the powder

diffraction pattern but not

KallKa2

doublets (see

Figure

3.46).

The caveat in

using long wavelengths is the reduction of the volume of the "visible"

reciprocal lattice when compared to that using short wavelengths because

only those reciprocal lattice points which have

d*hkl

2lh (e.g. see Chapter

Figure

2.28) can be positioned on the surface of the Ewald's sphere.l

An additional reduction of the number of experimentally measurable reciprocal lattice

points occurs due to a physical limit of any goniometer, which makes it impossible to

collect data at Bragg angles exceeding

-150"

20.

332

Chapter

Furthermore, as discussed above (see section 3.6), the easy selection of the

wavelength is only feasible when using a synchrotron x-ray source.

Unlike the goniometer radius and radiation wavelength, both the

diffracted beam aperture and the step size (or sampling step), are easily

controlled by the experimentalist. Their effects on the resolution of the

powder diffraction pattern were examined in section 3.6, which should be

consulted for proper selection of beam apertures.

Experimental techniques

3.8

Additional reading

R. Jenkins and R.L. Snyder, Introduction to x-ray powder diffractometry,

John Wiley and Sons, New York (1 996).

Parrish and J.I. Langford, Powder and related techniques, in: X-ray

techniques, International Tables for Crystallography, vol. C, Second

edition, A.J.C. Wilson and E. Prince, Eds., Kluwer Academic Publishers,

Boston/ Dordrecht/London (1999) p. 42.

3. Jens Als-Nielsen and Des McMorrow, Elements of modem x-ray

physics, John Wiley and Sons, Ltd., New York (2001).

4. D. Louk, Laboratory x-ray powder diffraction, in: Structure

determination from powder diffraction data. IUCr monographs on

Crystallography 13, W.I.F. David, K. Shankland, L.B. McCusker, and

Ch. Baerlocher, Eds., Oxford University Press, Oxford, New York

(2002).

5. P.W. Stephens, D. E. Cox, and A.

Fitch, Synchrotron radiation

powder diffraction, in: Structure determination from powder diffraction

data. IUCr monographs on Crystallography 13, W.I.F. David,

Shankland, L.B. McCusker, and Ch. Baerlocher, Eds., Oxford University

Press, Oxford, New York (2002).

6. R. J. Hill and I. C.

Madsen, Sample preparation, instrument selection and

data, in:

Structure determination from powder diffraction data. rUCr

monographs on Crystallography 13, W.I.F. David,

Shankland, L.B.

McCusker, and Ch. Baerlocher, Eds., Oxford University Press, Oxford,

New York (2002).

R. M. Ibberson and W. I. F. David, Neutron powder diffraction, in:

Structure determination from powder diffraction data. IUCr monographs

on Crystallography 13, W.I.F. David,

Shankland, L.B. McCusker, and

Ch. Baerlocher, Eds., Oxford University Press, Oxford, New York

(2002).

R.J. Hill, Data collection strategies: fitting the experiment to the need, in:

The Rietveld method. IUCr monographs on crystallography 5, R.A.

Young, Ed., Oxford University Press, Oxford, New York (1993).

334

Chapter

9. D.K. Bowen and B.K. Tanner. High resolution x-ray diffractometry and

topography.

T.J.

International Ltd, Padstow, United Kingdom (1998).

10.Modern powder diffraction. Reviews in mineralogy, v. 20, D.L. Bish and

J.E. Post, Eds., Mineralogical Society of America, Washington, DC

(1989).

1 1.

L.V.

Azaroff and M.J. Buerger, The powder method in x-ray

crystallography, McGrow-Hill, New York (1 958).

12. H. Lipson and H. Steeple, Interpretation of x-ray powder diffraction

patterns, Macmillan,

LondonISt. Martin's Press, New York (1970).

13. H.P. Klug and L.E. Alexander, X-ray diffraction procedures for

polycrystalline and amorphous materials,

2"d ed. John Wiley, New York

(1 974).

14. B.D. Cullity, Elements of x-ray diffraction, 2nd ed. Addison-Wesley,

Reading, MA (1 978).

15. D.C. Creagh and S. Martinez-Camera, Precautions against radiation

injury, in: International Tables for Crystallography, vol. C, Second

edition, A.J.C. Wilson and E. Prince, Eds., vol. C, Kluwer Academic

Publishers, Boston1 DordrechtILondon (1999) p. 949.

16. Accuracy in powder diffraction

11.

Proceedings of the International

Conference, May 26-29, 1992. E. Prince and J.K. Stalick, Eds., National

Institute of Standards and Technology, Washington, DC (1992).

17. Accuracy in powder diffraction. Proceedings of a Symposium on

Accuracy in Powder Diffraction, June 1 1-1 5, 1979. S. Block and C.R.

Hubbard, Eds., National Bureau of Standards, Washington, DC (1980).

18. A laboratory manual for x-ray powder diffraction (computer file on a

CD), L.J. Poppe, Ed., U.S. Geological Survey, Woods Hole, MA (2001).

19. A. J.C. Wilson, Mathematical theory of x-ray powder diffractometry,

Gordon and Breach, New York (1963).

Experimental techniques

335

3.9

Problems

Answers to all problems listed below are located in the file Chapter-3-

Problems-Solutions.pdf on the CD accompanying this book.

True or false? Brief exposure to the incident x-ray beam is harmless,

since the tissue is exposed to high energy photons only for a short

moment of time.

2. True or false? It is okay to be in a room without wearing an individual

dosimeter when the source of x-rays is energized as long you are not

involved in the experiment.

3. True or false? As low as is reasonably achievable (ALARA) with respect

to individual exposure to ionizing radiation is a fundamental principle of

current radiation safety practice.

Assume that you are to collect powder diffraction data from a powder

with the purpose to establish and refine its crystal structure. Previously

you have used the following powder diffractometer:

(A) Sealed x-ray tube source, curved position sensitive detector, the

radius of its goniometer is 150 mm and diffraction data are collected

in the transmission mode using cylindrical specimens. You employed

this equipment to characterize the phase purity of your materials.

addition to this device, other departments at your university have

the following powder diffractometer systems:

(B) Sealed x-ray tube source, Bragg-Brentano goniometer, radius 185

mm, scintillation detector.

scintillation detector.

(D) Sealed x-ray tube source, Bragg-Brentano goniometer, radius 250

mm, cooled solid-state detector.

Establish the order in which you would call people in charge of the

diffractometers to arrange for data collection and explain why.

Now assume that you have 20 different samples to characterize with

respect to their phase composition. Furthermore, each department charges

$50.00 per hour for the use of their equipment (the money goes to a

special account, which pays for a service contract and routine

maintenance). Assuming the availability of the same diffractometers as in

problem 4, what would be the order on your calling list and why?

336

Chapter

When you called the person in charge of the diffi-actometer D (see

problem 4) she told you that the goniometer axis is horizontal and that

the x-ray source arm is stationary. Do you need to worry about mixing

your powder with a binder?

7. As part of the preparation for your experiment described in problem 4

you used a mortar and pestle to grind the sample. You were completely

satisfied with the result since the powder appeared fine and homogeneous

to your eyes, but when you discussed the process with your thesis

advisor, he asked you to sift the powder through a 25 pm sieve. This

discussion happened just before you were about to take off across

campus since your allotted time on the powder diffractometer starts in 10

minutes. Describe your course of action and explain why?

You made a flat sample in the preparation for a highly precise powder

diffraction experiment using diffractometer D (see problem 4). The

sample completely fills a cylindrical opening 25 mm in diameter. The

lowest Bragg angle during the experiment will be

10". The set of

divergence slits available on this instrument includes the following

apertures: 0.05, 0.1, 0.25, 0.5, 0.75, 1,2, and 5 mm. Knowing that the

distance between the focus of the x-ray tube and the divergence slit is 60

mm, select the most appropriate divergence slit to be used in your

experiment and explain why?

You are having difficulties with making a flat sample for a high precision

powder diffraction experiment

the powder just will not spread evenly in

a cylindrical hole -20 mm diameter and

deep. You are considering

the following options:

a) compact the powder by pressing against a glass slide;

b) make a suspension in petroleum ether and pour it into the hole;

c) compact the powder by pressing against a glass slide with a strip of

rough sand paper glued to one of its surfaces;

d) use a different sample holder and dust the powder on top.

Arrange these options in the order which should result in the best

quality specimen for powder diffi-action and explain why.

10. You were able to arrange time on the powder diffractometer

(see

problem

4).

Several sample holders are available:

a) powder fills a cylindrical hole 25

diameter and

mm deep;

b) powder fills a cylindrical hole 20

diameter and 1 mm deep;

c) powder fills a square hole 30x30~1 mm3;

d) powder is dusted on top of a round rough spot 25

in diameter.

Experimental techniques

337

Arrange these sample holders in the order from the most to the least

suitable for a high precision powder diffraction experiment, and explain

why. Assume that you are working with a molecular compound.

1 1.

Answer problem 10 if you are working with an intermetallic compound

containing a lanthanide element.

12. You are using Fe

radiation to collect powder diffraction data

employing powder diffractometer

(see problem 4). After several quick

scans, you established that the receiving slit with the aperture of 0.03'

results in both acceptable resolution and intensity. Bragg peaks appear to

have a full width at half maximum between 0.4 and

0.5'

28.

What is the

largest allowable step during data collection and why?

13. You are about to perform a quick scan from 5 to 70'

to verify the

crystallinity of your material and to check whether or not the receiving

slit is of adequate width. You are planning for the experiment which will

take 10 to 15 min overall. Which scanning mode (step scan or continuous

scan) is the best to accomplish the task and why?

14.

the preparation for an overnight experiment, you performed a quick

scan, which resulted in the powder diffraction pattern shown in

Figure

3.48.

The receiving slit was narrow,

0.01'. You are planning to

double the receiving slit aperture and your goal is to measure the highest

intensity data point with

or better error at the 99.9% confidence level

during a step scan. Estimate the counting time parameter for the

upcoming overnight experiment.

15. Assume that the overnight experiment described in problem 14 is to be

performed using the continuous scanning mode from

60'

with a

sampling step

A28

0.005". Estimate both the scanning rate and the time

it will take to finish the experiment.

Chapter

10 15 20 25 30 35 40 45 50

Bragg

angle,

(deg.)

Figure

3.48.

The result of a quick scan experiment using a narrow receiving slit to determine

correct data collection parameters for

future overnight experiment.

Chapter

PRELIMINARY DATA PROCESSING AND

PHASE ANALYSIS

4.1

Introduction

the first three chapters of this book, we considered the fundamentals of

crystallographic symmetry, the phenomenon of diffraction from a crystal

lattice, and the basics of a powder diffraction experiment. Familiarity with

these broad subjects is essential in understanding how waves are scattered by

crystalline matter, how structural information is encoded into a three-

dimensional distribution of discrete intensity maxima, and how it is

convoluted with numerous instrumental and specimen-dependent functions

when projected along one direction and measured as the scattered intensity

versus the Bragg angle

28.

We already learned that this knowledge can be

applied to the structural characterization of materials as it gives us the ability

to decode a one-dimensional snapshot of a reciprocal lattice and therefore, to

reconstruct a three-dimensional distribution of atoms in an infinite crystal

lattice by means of a forward Fourier transformation.

Our experience with applications of the powder method in diffraction

analysis was for the most part, conceptual, and in the remainder of this book,

we will discuss key issues that arise during the processing and interpretation

of powder diffraction data. Despite the apparent simplicity of one-

dimensional diffraction patterns, which are observed as series of constructive

interference peaks (both resolved and partially or completely overlapped),

created by elastically scattered waves and placed on top of a nonlinear

background noise, the complexity of their interpretation originates from the

complexity of events involved in converting the underlying structure into the

experimentally observed data. Thus, nearly every component of data

processing in powder diffraction is computationally intense.

340

Chapter

The presence of symmetry (Chapter

coupled with well-defined

analytical relationships determining both the directions and intensities of

scattered beams (Chapter

2),

in addition to known properties of both the

specimen and instrument employed to obtain a powder diffraction pattern

(Chapter 3), makes it possible to develop both the general methodology and

algorithm(s) suitable for automation. Given the amount of numerical data

collected in a typical powder diffraction experiment,' their interpretation and

processing usually involves a broad use of computers.

We have no intent to comprehensively cover and/or evaluate any specific

product among a large variety of available software packages or stand-alone

programs (both

freeware and commercial). Instead, we will illustrate

multifaceted aspects of data processing using just a few computer codes,

while paying attention to both the capabilities and limitations of the powder

diffraction method, in addition to showing many examples of how the

analysis of the powder diffraction pattern may be accomplished in practice.

4.2

Interpretation of powder diffraction data

Given the nature of the powder diffraction method, the resultant

experimental data can be employed to obtain and/or confirm the following

information:

Phase composition of a material, including both qualitative and

quantitative analyses coupled with searches of various databases.

Indices of Bragg reflections, observed integrated intensities, and precise

lattice parameters.

Distribution of atoms in the unit cell, i.e. the crystal structure, either to

verify that the material has one of the already known types of crystal

structures or to solve it from first

principle^.^

Precise structural details including equilibrium positions of atoms in the

unit cell, individual atomic displacement and population parameters by

employing the Rietveld method.

Various microscopic structural characteristics of the specimen.

The first four items in this list represent the most common goals that are

usually achieved during characterization of polycrystalline materials using

For example, the range 10

90' scanned with a step A28

0.02' results in 4001

measured data points.

In the context of this book, structure solution from first principles (also referred to as the

initio

structure determination) means that all crystallographic data, including lattice

parameters and symmetry, and the distribution of atoms in the unit cell, are inferred from

the analysis of the scattered intensity as a function of Bragg angle, collected during a

powder diffraction experiment. Additional information, such as the gravimetric density of

a material, its chemical composition, basic physical and chemical properties, may be used

as well, when available.