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

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Chapter

1.21.2

Incommensurate modulation

As noted above, when

is irrational, the incommensurate modulation

occurs, and this is shown schematically in

Figure

1.53.

The exact description

of incommensurately modulated structure is impossible using only

conventional crystallographic symmetry in the unit cell of any size smaller

than the crystal. The periodicity of the structure can only be restored by

using two different periodic functions. The first function is the conventional

crystallographic translation, and the second one is the sinusoidal modulation

function with certain period, which is incommensurate with the

corresponding translation, and amplitude.

addition to one-dimensional modulations considered above, both two-

and three-dimensional modulations are possible. Furthermore, atomic

parameters affected by modulations may be one or several of the following:

positional (as shown in

Figure

1.52

and

Figure

1.53),

occupancy, thermal

displacement, and orientation of magnetic moments. The latter, i.e.

commensurately or incommensurately modulated orientations of magnetic

moments are quite common in various magnetically ordered structures

(e.g.

pure lanthanide metals such as Er and Ho), and both the value of the

modulation vector and the amplitude of the modulation function often vary

with temperature.

Symmetry of modulated structures is represented algebraically by using

rotation matrices and translational vectors in an N-dimensional superspace,

where N

Additional dimensions are needed to describe the symmetry of

Figure

1.53.

The ideally periodic one-dimensional structure, the corresponding modulation

function with the period

llq,

which is incommensurate with

and the amplitude

(top),

and the resulting incommensurately modulated structure (bottom).

Fundamentals

crystalline state

the modulation functions. Thus, one-dimensional modulation is described in

a four-dimensional superspace using

4x4

rotation matrices and

1 vectors

dimensions for a normal space plus one for the modulation function),

while two- and three-dimensional modulations require

5x5

and

6x6

rotation

matrices and

1 and

vectors, respectively.

1.21.3

Quasicrystals

The symmetry of quasicrystals can be represented by introducing a

different perturbation function, which is based on the Fibonacci1 numbers.

infinite Fibonacci sequence is derived from two numbers,

and 1, and is

formed according to the following rule:

This results in the series of numbers

Assume that we have a sequence of words containing letters

(for long

distance or fragment) and S (for short distance or fragment), which are

constructed by replacing each letter in the previous word using the following

substitution rule: letter S is replaced by letter L, while letter

is replaced by

the word LS. Starting from

as the first word, the infinite sequence of

words is obtained, and the first six members of this sequence are shown in

Figure 1.54.

The fi-equency of occurrence of letters L and

in this sequence is

represented in Table 1.21, and it is easy to recognize that they are identical

to the consecutive members

(Fn+, and F,) of the Fibonacci series. The

corresponding limit when the number of words, n, approaches infinity is the

golden mean,

Leonardo Pisano Fibonacci (1 170

1250). Medieval Italian mathematician who in 1202

wrote

Liber

abaci

"Book of the abacus"

in which he formulated the problem leading to

the sequence of numbers 1, 1, 2,

13,

21,

34,

55,

.. .

(without the first term, i.e.

without 0): "How many pairs of rabbits can be produced in a year from one pair of rabbits

assuming that every month each pair produces one new pair of rabbits, which becomes

productive one month after birth?"

Chapter

Figure

1.54.

The sequence of words containing quasiperiodic sequences of letters

and

based on the following substitution rule:

and

LS.

The golden mean can also be represented as a continuous fraction, which

contains only one number, 1, and therefore, it is sometimes referred to as the

"most irrational" number.

Table

1.21.

The frequency of occurrence of letters

and

in the infinite series of words

based on the substitution rule

and

LS.

n Word f~ fs

LSL

3 LSLLS 3

LSLLSLSL

LSLLSLSLLSLLS

LSLLSLSLLSLLSLSLLSLSL 13

LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS

21 13

The perturbation (modulation) function used in the description of

aperiodic structures is obtained by associating interatomic distances (or

larger fragments in the crystal structure) with length ratio

to 1 to letters

Fundamentals

crystalline state

and S and the resulting modulation function is no longer a sinusoidal wave,

but is saw-tooth-like. It is worth noting that the periodicity in this simple

one-dimensional case

(Figure

1.54

and

Table

1.21)

is absent but the order is

perfect: as soon as the law has been established, the "structure" of the series,

i.e. the location of S and

can be predicted at any point starting from the

origin or any other known location.

Similar to modulated structures, symmetry of quasicrystals also can be

described in space with more than three dimensions. For example, two-

dimensional and three-dimensional quasiperiodicity can be described in

five-

and six-dimensional space, respectively, using

5x5

and

6x6

rotation

matrices. Yet, the properties of these matrices (Eqs. 1.44 to

1.47)

remain the

same for both modulated structures and quasicrystals.

The more detailed description of the non-conventional symmetry goes

beyond the scope of this book' as it has little use in powder diffraction,

because even the three-dimensional diffraction from aperiodic crystals is

quite complex. When the diffraction picture is projected along one

dimension, its treatment becomes too complicated and the crystal structure

of aperiodic crystals is rarely, if ever, completely studied by means of

powder diffraction techniques beyond simple phase identification.

Nevertheless, this section has been included here for completeness, and to

give the reader a flavor of recent developments in

~rystallography.~

A more complete description of de Wolffs approach to treatment of various types of

aperiodic crystals can be found in the International Tables for Crystallography, vol. B,

Second edition,

Shmueli, Ed., Published for the International Union of Crystallography

by Kluwer Academic Publishers, Boston/Dordrecht/London (2001).

The discovery of five-fold symmetry prompted the ad-interim Commission on Aperiodic

Crystals of the International Union of Crystallography to change the definition of a crystal

periodic three-dimensional arrangement of identical unit cells to the following: "...by

'crystal' we mean any solid having an essentially discrete diffraction diagram, and by

'aperiodic crystal' we mean any crystal in which three-dimensional lattice periodicity can

be considered to be absent". International Union of Crystallography. Report of the

Executive Committee for 1991,

Acta Cryst. A48,922

946 (1992).

Chapter

1.22 Additional reading

1. C. Giacovazzo, H.L. Monaco, D. Viterbo, F. Scordari, G. Gilli, G.

Zanotti, and M. Catti, Fundamentals of crystallography. IUCr texts on

crystallography 7, Second Edition, Oxford University Press, Oxford,

New York (2002).

Schwarzenbach, Crystallography, John Wiley

Sons, New York

(1996).

M.B. Boisen and G.V. Gibbs. Mathematical crystallography

introduction to the mathematical foundations of crystallography. Reviews

in mineralogy, Vol. 15 (revised). Washington, DC: The Mineralogical

Society of America (1992).

Farmer, Groups and symmetry, Amer. Math. Soc., Providence,

(1995).

5. C. Hammond, The basics of crystallography and diffraction. IUCr texts

on crystallography

Oxford University Press, Oxford, New York

(1997).

E. Prince, Mathematical techniques in crystallography and materials

science. Second edition, Springer-Verlag, Berlin, Heidelberg, Germany

(1 992).

C. Janot, Quasicrystals.

primer, Clarendon Press, Oxford (1992).

IUCr Teaching Pamphlets:

http://www.iucr.org/iucr-top/comm/cteachlpamphlets.html

9. International Tables for Crystallography, vol. A, Fifth revised edition,

Theo Hahn, Ed., Published for the International Union of Crystallography

by Kluwer Academic Publishers,

Boston/Dordrecht/London,

(2002).

10. International Tables for Crystallography. Brief teaching edition of

volume A, Fifth revised edition. Theo Han, Ed., Kluwer Academic

Publishers, Boston/Dordrecht/London (2002).

Fundamentals of crystalline state

1.23 Problems

Answers to all problems are located in the file Chapter-l-Problems-

Solutions.pdf on the CD accompanying this book.

Consider a two-dimensional lattice shown in

Figure

1.55, left, which was

discussed earlier in section 1.3. One half of the molecules in this lattice

have been modified in a regular way so that their atoms now have sad

faces, as shown in

Figure

1.55, right. This may be a schematic

illustrating the formation of a magnetically ordered structure, where

happy and sad faces represent opposite directions of magnetic moments.

Suggest the most probable unit cell in this new lattice assuming that the

correct unit cell in the original lattice is shown by a shaded

parallelogram.

Consider two mirror planes that intersect at

90'. Using geometrical

representation of two planes establish which symmetry element(s) appear

as the result of this combination of mirror planes. What

is(are) the

location(s) of new symmetry element(s)? Name point group symmetry

formed by this combination of symmetry elements.

Consider two mirror planes that intersect at

45'.

Using geometrical

representation of two planes establish which symmetry element(s) appear

as the result of this combination of mirror planes. What

is(are) the

location(s) of new symmetry element(s)? Name point group symmetry

formed by this combination of symmetry elements.

Figure

1.55.

The original lattice containing indistinguishable molecules in which a proper unit

cell is shaded (left) and a new lattice derived

switching

112

of atoms from happy to sad

faces in a regular fashion.

Chapter

4. Consider the following sequence of numbers: 1, 112, 113, 114,

1/N,

Is this a group assuming that the combination law is multiplication,

division, addition or subtraction? If yes, identify the combination law in

this group and establish whether this group is finite or infinite.

Consider the group created by three non-coplanar translations (vectors)

using the combination law defined by Eq. 1.1. Which geometrical form

can be chosen to illustrate this group? Is the group finite?

Determine both the crystal system and point group symmetry of a brick,

which is shown schematically in

Figure

1.56

and in which

c and

a=~=~=900?

Determine both the crystal system and point group symmetry of benzene

molecule, C6&?

Determine the point group symmetry of the octahedron. How many and

which symmetry elements are present in this point group symmetry?

9. The following relationships between lattice parameters:

90 or 120•‹, and y

90" potentially define a "diclinic" crystal system

(two angles

90"). Is this an eighth crystal system? Explain your answer.

10. The relationships

90•‹, and

90"

point to a

monoclinic crystal system, except that

What is the reduced

(standard) Bravais lattice in this case? Provide equations that reduce this

lattice to one of the 14 standard Bravais types.

11. Consider space group symmetry Fdd2 and without using the International

Tables for Crystallography establish the following: (a) the crystal system;

(b) the corresponding point group symmetry; (c) the corresponding Laue

class; (d) the relationships between the unit cell dimensions; and (e)

explain the space group symbol.

Figure

1.56.

Illustration of a brick (parallelepiped) in which three independent edges have

different lengths.

Fundamentals

crystalline state

12. Consider independent atoms with the following coordinates in the space

group symmetry C2lm: Atoml:

0.15,

0.0,

0.33; Atom2:

.5,

y=0.11,~=0.5;andAtom3:x=0.25,y=0.25,z=0.25.Usingthe

International Tables for Crystallography carry out the following tasks:

a) Apply the coordinates and centering vectors listed for the general

equivalent position to generate all symmetrically equivalent atoms from

the three listed independent atoms (the total in each case should be the

same as the multiplicity of the general position).

b) Find atoms with equal coordinate triplets (remember that the difference

by a full translation in one, two or three directions refers to the same

atom) and cross them out. The total number of atoms left is the

multiplicity of the corresponding special position.

c) Establish both the multiplicity and the Wyckoff notation of special

position for each of the three listed independent atoms.

d) To which symmetry element(s), if any, do the independent atoms belong?

e) Which of the three original independent atoms occupies the general

equivalent position?

13. The crystal structure of a material is described in space group symmetry

P63/mmc with the following atomic coordinates

Atom

Bal

0 0 0.25

Ba2

0.3333 0.6667

0.91 10

0 0 0

0.3333 0.6667 0.1510

0.4816 -0.0368 0.25

0.1685 0.3370 0.4169

Using the International Tables for Crystallography describe every atom

in terms of the multiplicities and Wyckoff letters of their site positions

and establish the content of the unit cell, the simplest chemical formula

and the number of formula units'

(Z)

per unit cell.

14. Imagine that there is an "edge-centered" lattice (for example unit cell

edges along

contain lattice points at 112~). If this were true, the

following lattice translation is present:

(0,

0, 112). Convert this lattice to

one of the standard lattices.

Usually, a formula unit corresponds to the simplest chemical formula or to the

stoichiometry of the molecule of a material.

Chapter 2

FUNDAMENTALS OF DIFFRACTION

2.1

Introduction

the previous chapter, we introduced basic concepts of symmetry and

discussed the structure of crystals in terms of three-dimensional periodic

arrays of atoms and/or molecules, sometimes perturbed by various

modulation functions.

doing so, we implicitly assumed that this is indeed

reality. Therefore, it is time to think about the problem from a different point

of view: how atoms or molecules can be observed

either directly or

indirectly

and thus, how is it possible to determine the crystal structure of a

material and verify the concepts of crystallographic symmetry?

To begin answering this question, consider the following mental

experiment: imagine yourself in a dark room next to this book

(Figure

2.1,

left). Since human eyes are sensitive to visible light, you will not be able to

see the book, nor will you be able to read these words in total darkness. Only

Figure

2.1.

The illustration of an observer placed in the absolutely dark room with a book

(left) and the same room with the light source producing visible rays of light (right).

100

Chapter

when you turn on the light, the book becomes visible and the information

stored here becomes accessible

(Figure

2.1,

right). The fundamental

outcome of our experiment is that the book and its content can be observed

by means of a visible light after it has been scattered by the object (the book)

and detected by eyes.

general, a source of rays and a suitable detector (such as the light bulb

and the eye, respectively) are required to observe common objects. Atoms,

however, are too small to be discerned using any visible light source because

atomic radii' range from a few tenths of an angstrom to a few angstroms, and

they are smaller than 111000 of the wavelengths present in visible light (from

-4000 to -7000

A).

A suitable wavelength to observe individual atoms is

that of x-rays. The latter are short-wave electromagnetic radiation discovered

by W.C. Roentgen,* and they have the wavelengths that are commensurate

with both the atomic sizes and shortest interatomic distances.

Unfortunately, the index of refraction of x-rays is

-1

for all materials and

they cannot be focused by a lens in order to observe such small objects as

atoms are, as it is done by glass lenses in a visible light microscope or by

magnetic lenses in an electron microscope. Thus, in general, x-rays cannot

be used to image individual atoms

dire~tly.~ However, as was first shown by

Max von Laue in 1912 using a single crystal of hydrated copper sulphate

(CuS04.5H20), the periodicity of the crystal lattice allows atoms in a crystal

to be observed with exceptionally high resolution and precision by means of

x-ray diffraction. As we will see later, the diffraction pattern of a crystal is a

transformation of an ordered atomic structure into reciprocal space rather

than a direct image of the former, and the three-dimensional distribution of

atoms in a lattice can be restored only after the diffraction pattern has been

transformed back into direct space.

Particles in motion, such as neutrons and electrons, may be used as an

alternative to x-rays. They produce images of crystal structures in reciprocal

space because of their dual nature: as follows from quantum mechanics,

Atomic radius may be calculated self-consistently or it may be determined from

experimental structural data. Effective size of an atom varies

a function of its

environment and nature of chemical bonding. Several different scales

covalent, ionic,

metallic, and Van der Waals radii

are commonly used in crystallography.

Wilhelm Conrad Roentgen (1845-1923). German physicist who on November 8, 1895

discovered x-rays and was awarded the first ever Nobel Prize in Physics in 1901 "in

recognition of the extraordinary services he has rendered by the discovery of the

remarkable rays subsequently named after him". For more information about

W.C.

Roentgen see

http://www.nobel.se/physics/laureates/l90l/index.html

on the Web.

Direct imaging of atoms is feasible using x-ray holography, in which the wave after

passing through a sample is mixed with a reference wave to recover phase information and

produce three-dimensional interference patterns. For more information see R. Fitzgerald,

X-ray and y-ray holography improve views of atoms in solids, Phys. Today

54,21 (2001).