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

Подождите немного. Документ загружается.

Preliminary data processing and phase analysis

391

12.H.M. Berrnan, T. Battistuz, T.N. Bhat, W.F. Bluhm, P.E. Bourne,

Burkhardt,

Feng, G.L. Gilliland,

Iype,

Jain, P. Fagan, J. Marvin,

D. Padilla,

Ravichandran, B. Schneider,

Thanki, H. Weissig, J.D.

Westbrook, and C. Zardecki, The Protein Data Bank, Acta Cryst. D58,

899

(2002).

13.H.M. Berman, J.D. Westbrook, Z. Feng,

Iype, B. Schneider, and C.

Zardecki, The Nucleic Acid Database, Acta Cryst. D58, 889 (2002).

392

Chapter

4.6

Problems

Answers to all problems listed below are located in the file Chapter-4-

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

Consider three powder diffraction patterns, which are shown in

Figure

4.25

Figure

4.27.

For each pattern select all applicable processing

steps and explain your reasoning assuming that the goal is to produce

digitized (reduced) powder patterns for phase identification.

a) Smooth the data (Yes/No/Probably),

b) Eliminate background (Yes/No/Probably),

c) Strip

Ka2

contributions (Yes/No/Probably).

A powder diffraction pattern collected from a metallic alloy was

processed into two digitized patterns. The background was eliminated

first, as illustrated in

Figure

4.28

and second, as shown in

Figure

4.29.

Assume that in both cases preliminary processing was continued as

follows:

Ka2

components were stripped and Bragg peak positions and

intensities were determined using an automatic peak search. Compare the

reliability of thus obtained digitized patterns and explain your reasoning.

40 50

Bragg angle,

(deg.)

Figure

4.25.

Powder diffraction pattern collected using a conventional x-ray source.

Preliminary data processing and phase analysis

393

Bragg angle,

(deg.)

Figure

4.26.

Powder diffraction pattern collected using a synchrotron x-ray source. (Data

courtesy of Dr.

M.J.

Kramer.)

Figure

4.27.

Powder diffraction pattern collected using a reactor-based neutron source. (Data

courtesy of Dr.

W.B.

Yelon.)

Chapter

~etailic alloy,

KCL

20 30 40 50 60

Bragg angle, 20 (deg.)

Figure

4.28.

Example of the automatically determined background (thick line at the bottom of

the plot shown using the scale identical to the experimental data).

30 40 50 60

Bragg angle, 20 (deg.)

Figure

4.29.

Example of the background represented by a polynomial (thick line at the bottom

of the plot shown using the scale identical to the experimental data).

Preliminary data processing and phase analysis

395

Consider the powder diffraction pattern shown in

Figure

4.30

and

answering Yes/No/Maybe: Is this pattern suitable for phase

identification? Is the material suitable for crystal structure determination

using powder diffraction? Explain your reasoning. What other

conclusions (if any) can be made from a visual analysis of this pattern?

Diffraction data'

(Table

4.4)

were collected from a white ceramic plate.

Using the Mineral Database2 and

strongest of the

observed peaks

identify the material.

Table

4.4.

Digitized pattern representing data collected from a white ceramic plate.

111,

20,"

d,A I/Io

20,

I/I0

20,

57 25.556

3.4827 99

57.465 1.6023 9

77.202 1.2346

35.125 2.5528

8 61.268

1.5117 7

80.648 1.1903

39 37.747

2.3812 38

66.481 1.4052 8

88.949 1.0995

100 43.324

2.0867 57

68.165 1.3745 10

91.139 1.0787

49 52.514

1.7412

76.834 1.2396

95.203 1.0431

Bragg angle,

(deg.)

Figure

4.30.

Powder diffraction pattern collected from an organo metallic compound on a

Rigaku TTRAX powder diffractometer using Mo

radiation. The data were collected in a

continuous scanning mode: scan rate was

deg/min, sampling step

0.01".

In problems

the data were collected on a powder diffractometer with Bragg-

Brentano geometry using Cu Ka radiation. Errors in d-spacing should not exceed

0.02

ford

otherwise they should be less than

0.01

This is a freely accessible database available at

http://webmineral.corn/X-Ray.shtm1.

396

Chapter

Diffi-action data

(Table

4.5)

were collected from a light-blue colored

powder. Using the Mineral Database and 3 strongest of the

observed

peaks identify the material.

Table

4.5.

Digitized pattern representing data collected from a light-blue colored powder.

111,

20,

111,

20,

111,

20,

100 12.778

6.9223 4

41.846

2.1570 6

53.388 1.7147

36 25.724

3.4603 10

43.508

2.0784 3

57.955 1.5900

10 33.514

2.6717 3

49.088 1.8543

2 58.613 1.5737

11 36.438

2.4637 3

51.238 1.7815 2 62.381 1.4873

6 39.753 2.2656 3 52.856 1.7307 3 62.764 1.4792

6. Diffraction data

(Table

4.6)

were collected from a powder containing

fluorine. Using the Mineral Database and

3 strongest of the 15 observed

peaks identify the material. Additional information about the powder: no

weight loss has been detected during a thermogravimetric experiment

carried out between

-25

and -500•‹C.

Table

4.6.

Digitized pattern representing data collected from a powder containing fluorine,

which is stable between room temperature and

-500•‹C.

111,

20,"

d,8,

IlI,

20,

IlI,

20,

25.850 3.4438

68 33.080

2.7057

37 49.555

1.8380

Diffraction data

(Table

4.7)

were collected from a powder containing

manganese. Using the Mineral Database and 3 strongest of the 15

observed peaks identify the material.

Table

4.7.

Digitized pattern representing data collected from a manganese-containing powder.

Illo

20,

IlI,

20,

111,

20,

d, 8,

100 28.630 3.1 153 66 56.560 1.6258 18 72.249 1.3066

52 37.296 2.4090 16 59.275 1.5577 7 86.457 1.1246

13 40.947 2.2022 7 64.736 1.4388 9 93.578 1.0569

15 42.744 2.1 137 13 67.136 1.3931 20 100.604 1.0011

5 46.026 1.9703 15 72.125 1.3085 5 102.886 0.9850

Diffraction data

(Table

4.8)

were collected from a two-phase powder

containing Mn, A1 and

Results of mass spectroscopic analysis with

respect to all known chemical elements show that there are no other

elements present in concentration exceeding 100 parts per million by

weight. Using the Mineral Database and

6 strongest of the

observed

peaks identify both compounds that are present in the mixture.

Preliminary data processing and phase analysis

Table

4.8.

Digitized x-ray diffraction pattern representing data collected from a two-phase

powder containing Al, Mn and

ZIZo

20,

ZIZo

29,

IIZo

20,

33 25.460 3.4956 13 40.947 2.2022 66 56.560 1.6258

Chapter

UNIT CELL DETERMINATION AND

REFINEMENT

5.1

Introduction

As we established in Chapter 1, crystal lattices, used to portray periodic

three-dimensional crystal structures of materials, are constructed by

translating an identical elementary parallelepiped

the unit cell of a lattice

in three dimensions. Even when a crystal structure is aperiodic, it may still

be represented by a three-dimensional unit cell in a lattice that occupies a

superspace with more than three dimensions.

the latter case, conventional

translations are perturbed by one or more modulation functions with

different periodicity.

Given the fact that the unit cell remains unchanged throughout the

infinite lattice, the crystal structure of a material may be considered solved

when both the shape and the content of the unit cell of its lattice, including

the spatial distribution of atoms in the unit cell, have been established.

Unavoidably, the determination of any crystal structure starts from finding

the shape and the symmetry of the unit cell together with its dimensions,

i.e.

the lengths of the three unit cell edges

(a,

and

and the values of the three

angles

(a,

and

between the pairs of the corresponding unit cell vectors,

e.g. see

Figure

1.4

in Chapter 1.

5.2

The

indexing problem

powder diffraction, the very first step in solving the crystal structure,

i.e. finding the true unit cell, may present considerable difficulties because

the experimental data are a one-dimensional projection of the three-

dimensional reciprocal lattice recorded as a function of a single independent

400

Chapter

variable

the Bragg angle. Thus, the directions are lost and only the lengths

of the reciprocal lattice vectors are measurable in a powder diffraction

experiment. This is quite different from scattering by a single crystal where

both the length and direction of each vector in reciprocal space are

preserved, provided the intensity of a corresponding Bragg peak exceeds the

background and is measurable.

The loss of directions is illustrated in

Figure

5.1

for a two-dimensional

case. It is easy to see that when different vectors from the two-dimensional

reciprocal lattice are projected on the lld axis (which may be chosen

arbitrarily as long as it intersects the origin of the reciprocal lattice), they all

have the same direction and are distinguishable only by their lengths. It is

worth reminding one's self that the distribution of points along the

lld axis

Figure

5.1

determines the Bragg angles at which scattered intensity

maxima can be observed in a powder diffraction experiment, as directly

follows from the

Braggs' law: d*

lld

2sinOlnI.

lld

Figure

5.1.

An illustration of a two-dimensional reciprocal lattice (top) and its one-

dimensional projection on the

lld

axis (bottom). The scales in the two parts of the drawing are

identical because

lld

d*.

The

lld

axis is shifted downwards from the origin of the reciprocal

lattice for clarity. The reciprocal lattice point

(32)

is shown as a filled black circle both in the

lattice and in its projection together with the corresponding reciprocal vector

d*(32).

Unit cell determination and refinement

Regardless of the nature of the diffraction experiment, finding the unit

cell in a conforming lattice is a matter of selecting the smallest

parallelepiped in reciprocal space, which completely describes the array of

the experimentally registered Bragg peaks. Obviously, the selection of both

the lattice and the unit cell should be consistent with crystallographic

conventions (see section 1.12, Chapter

I), which impose certain constraints

on the relationships between unit cell symmetry and dimensions.

Since each point in reciprocal space represents a series of

crystallographic planes, the description of diffraction data by means of any

lattice may therefore, be reduced to assigning triplets of Miller indices to

every observed Bragg peak based on the selected unit cell. Recalling the

definitions of both the direct and reciprocal lattices (Eqs. 1.1 and 1.16,

respectively) and considering Figure

5.1

(or Figure

2.40

in Chapter

which

illustrates a three-dimensional case), the assignment of indices in a periodic

lattice is based on Eq. 5.1.' The latter establishes relationships between the

unit vectors

a*,

b* and

and the corresponding reciprocal vectors d*(jko in

terms of triplets of integers

and I.

d*,,

ha*

kb*

lc*

(5.1)

This process is commonly known as indexing of diffraction data and in

three dimensions it usually has a unique and easy solution when both the

lengths and directions of reciprocal vectors,

d*jkl, are available. On the

contrary, when only the lengths,

d*hkl, of the vectors in the reciprocal space

are known, the task may become extremely complicated, especially if there

is no additional information about the crystal structure other than the array of

numbers representing the observed l/djkl

d*hkl] values.

The difficulty and reliability of indexing are closely related to the

absolute accuracy of the array of d*jkl values, i.e. to the absolute accuracy

with which positions of Bragg reflections have been determined. For

Conventional lattices may be perturbed by functions with different periodicity, e.g. by

sinusoidal or saw-tooth-like modulations, see section 1.21 in Chapter

In the simplest

case (one-dimensional modulation), Eq.

5.1

becomes

d*hkr

ha*

kb*

lc*

assuming that the perturbation function is periodic and has the modulation vector

In a

case of three-dimensional modulation, a total of six indices

(h, k,

m, n,

and

are

required to identify every point observed in reciprocal space:

d*hkl

ha*

kb*

lc*

mq,

nq2

pq,,

where

q,, q2

and

are the modulation vectors of the corresponding

perturbation functions. If this is the case, vectors

q,, q2

and

should be established in

addition to

a*,

and

before assignment of indices can be performed. Since even the

three-dimensional diffraction pattern of a modulated structure is a projection of four- to

six-dimensional superspace, indexing of single crystal diffraction data is quite complex. It

is rarely, if ever, successful from first principles when only powder diffraction data are

available.