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

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Fundamentals of diffraction

20 1

The magnitude of the preferred orientation can be evaluated using the

following function:

which is unity in the case of random orientation, otherwise

1. When all

grains are perfectly aligned (single crystal) the function 2.84 becomes

infinity.

Only even orders are taken into account in Eqs. 2.83 and 2.84 due to the

presence of the inversion center in the diffraction pattern. The number of

harmonic coefficients

and terms

k(h)

varies depending on lattice symmetry

and desired harmonic order

The low symmetry results in multiple terms

(triclinic has 5 terms for

2) and therefore, low orders 2 or 4 are usually

sufficient. High symmetry requires fewer terms (e.g. cubic has only 1 term

for

4), so higher orders may be required to adequately describe preferred

orientation. The spherical harmonics approach is realized in

GSAS.'

example of the preferred orientation modeled using 2nd and 4th order

spherical harmonics in the orthorhombic crystal system [space group Cmc2,,

co2

0.17(1),

c~~

1.65(1),

co4

0.17(2),

c~~

0.04(1),

c~~

0.56(2)] is

shown in

Figure 2.50.

Figure

2.50.

The illustration of the complex distribution of reciprocal lattice vectors modeled

using a spherical harmonic preferred orientation function for the (100) reflection.

B. Von Dreele, Quantitative texture analysis by Rietveld refinement,

Appl. Cryst.

30,

517 (1997).

202

Chapter

Here the surface represents the probability to find the reciprocal lattice

point (100) in the diffractometer coordinate system assuming Bragg-

Brentano focusing geometry. The Z-axis is perpendicular to the sample, and

X-

and Y-axes are located in the plane of the sample.

To date, the spherical harmonics approach is the most comprehensive

method developed to account for the preferred orientation effects but in

routine experiments it should be used with great care. The order of

expansion should be increased gradually and only as long as improvements

are obvious and the results make sense.

unnecessarily large number of

harmonic coefficients may give excellent agreement between the observed

and calculated diffraction patterns but incorrect structural, especially thermal

displacement parameters may result.

its full form, Eq. 2.82 may be used in

complex texture analysis, where powder diffraction data have been collected

not only as a function of Bragg angle 28 but also at different orientations

along

Debye rings and with tilting the sample.

2.10.7

Extinction factor

Extinction effects, which are dynamical in nature, may be noticeable in

diffraction from nearly perfect andlor large mosaic crystals. Two types of

extinction are generally recognized: primary, which occurs within the same

crystallite, and secondary, which originates from multiple crystallites.

Primary extinction is caused by back-reflection of the scattered wave into the

crystal and it decreases the measured scattered intensity

(Figure

2.51,

left).

Furthermore, the re-reflected wave is usually out of phase with the incident

wave and thus, the intensity of the latter is lowered due to destructive

interference. Therefore, primary extinction lowers the observed intensity of

very strong reflections from perfect crystals. Especially in powder

diffraction, primary extinction effects are often smaller than experimental

errors; however, when necessary they may be included in Eq. 2.65 as:'

E,,

sin2

cos2

where

and

are Bragg (28

7c)

and Laue

(28

0) components, both

defined as various functions of the extinction parameter,

which is

normally

refined variable:

T.M.

Sabine,

R.B.

Von Dreele, and J.E. Jorgensen, Extinction in time-of-flight neutron

powder diffractometry, Acta Cryst.

A44,

374

(1988);

T.M.

Sabine, A reconciliation of

extinction theories, Acta Cryst.

A44,368 (1988).

Fundamentals of diffraction 203

Figure

2.51.

The illustration of primary (left) and secondary (right) extinction effects, which

reduce intensity of strong reflections from perfect crystals and ideally mosaic crystals,

respectively. The solid lines indicate actual reflections paths. The dashed lines indicate the

expected paths, which are partially suppressed by dynamical effects. The shaded rectangles on

the right indicate two different blocks of mosaic with identical orientations.

Eq. 2.86,

is the shape factor (it is unity for a cube of edge

314

for a sphere of diameter

and

813n for a cylinder of diameter

D),

the wavelength,

Fhkl

is the calculated structure amplitude and

is the

number of unit cells per unit volume.

Secondary extinction (Figure

2.51,

right) occurs in a mosaic crystal when

the beam, reflected from a crystallite, is re-reflected by a different block of

the mosaic, which happens to be in the diffracting position with respect to

the scattered beam. This dynamical effect is observed in relatively large,

nearly perfect mosaic crystals; it reduces measured intensities of strong

Bragg reflections, similar to the primary extinction. It is not detected in

diffkaction from polycrystalline materials and therefore, is always neglected.

2.11

Structure factor

So far we discussed all prefactors in Eq. 2.65, which were dependent on

multiple parameters except for the crystal structure of the material. The only

remaining term here is the structure factor,

IF^^^^^,

which is the square of the

absolute value of the so-called structure amplitude, Fhkl. It is this factor that

includes multiple contributions, which are determined by the distribution of

atoms in the unit cell and other structural features.

2.1

1.1

Structure amplitude

When the unit cell contains only one atom, the resulting diffracted

intensity is only a function of the scattering ability of this atom (see section

2.5.2). However, when the unit cell contains many atoms and they have

different scattering ability, the amplitude of the scattered wave is given by a

complex function, which is called the structure amplitude:

204 Chapter

F(h)

gJtJ

(s)

(s) exp(27cih

)

j=1

where:

P(h)

is the structure amplitude of a Bragg reflection with indices hkl,

which are represented as vector

in three dimensions. The structure

amplitude itself is often shown in a vector form since it is a complex

number;

is the total number of atoms in the unit cell and

includes all

symmetrically equivalent atoms;

is sineh&;

is the population (or occupation) factor of the

jth

atom

(g'

1 for a fully

occupied site);

is the temperature factor, which describes thermal motions of the

jth

atom;

fJ(.s) is the atomic scattering factor describing interaction of the incident

wave with a specific type of an atom as a function of sinelh for x-rays or

electrons, and it is simply

(i.e. is independent of sinelh) for neutrons;

i=fi;

h-x'

is a scalar product of two vectors:

(h, k,

and vector xj

(2,

y',

2).

The latter represents fractional coordinates of the

jt"

atom in the unit

cell:

Taking into account Eq. 2.88, the previous Eq. 2.87 can be written in the

expanded form as:

Fhkl

g't'

(s)

exp[2ni(hn'

lz'

(2.89)

j=1

2.1 1.2

Population factor

Considering a randomly chosen unit cell, each available position with a

specific coordinate triplet (2,

y',

may be only occupied by one atom

(fractional population parameter

1) or it may be left unoccupied

(g'

0).

On the other hand, even very small crystals contain a nearly infinite number

of unit cells (e.g. a crystal in the form of a cube with 1 pm side will contain

Fundamentals of diffraction 205

lo9 cubic unit cells with a

and diffraction is observed fkom all of the

unit cells simultaneously. Hence, the resulting structure amplitude is

normalized to a certain mean unit cell, which represents the distribution of

atoms averaged over the entire volume of the studied sample.

the majority

of compounds, each crystallite is fully ordered and the content of every unit

cell may be assumed identical throughout the whole crystal, so the

population factor remains unity for every atom. Occasionally, population

factor(s) may be lower than one but greater than zero, and some of the

common reasons of why this occurs are briefly discussed below.

It is possible that in some of the unit cells atoms are missing. Thus,

instead of a complete occupancy, the corresponding site in an average unit

cell will contain only a fraction

1) of the

jth

atom.

cases like that,

it is said that the lattice has defects and

smaller than 1 reflects a fraction of

the unit cells where a specific position is occupied. Obviously, a fraction of

the unit cells where the same position is empty complements

to unity and

it is equal to 1

g'.

different situation arises when in an average unit cell atom

is located

close to a finite symmetry element, e.g. mirror plane.

mirror plane

produces atom j', symmetrically equivalent to j, so that the distance between

and

becomes unreasonably small and core electrons of the two atoms

overlap but in reality they cannot be located at such close distance (Figure

2.52, left). This usually means that in some of the unit cells atom

is located

on one side of the mirror plane, while in the others it is positioned on the

opposite side of the same plane. If this is the case, then in the absence of

"conventional" defects, the population factors

and

are related as

21, and

because of mirror symmetry.

general,

=lln, where n is the multiplicity of the symmetry element

which causes the overlap of the corresponding atoms. When the culprits are

a mirror plane, a two fold rotation axis or a center of inversion,

and

0.5. For a three fold rotation axis n

and

113, and so on (Figure

2.52, middle and right). When defects are present in addition to the overlap,

is no longer equal to

and two or more population factors may become

independent.

The condition illustrated in Figure 2.52 usually happens when the real

symmetry of the unit cell (and the lattice) is lower than the symmetry of the

average unit cell as detected from

diffraction data. Hence, the following

should be taken into the consideration:

1. Distribution of atoms

and j' in the lattice is random, so the crystal is

partially disordered.

this case, the average cell may still be used to

describe the average crystal structure even when it has a higher symmetry

than the real unit cell.

Chapter

Figure

2.52.

The illustration of forbidden overlaps as a result of an atom being too close to a

finite symmetry element: mirror plane (left), three-fold rotation axis (middle), and four-fold

rotation axis (right). Assuming that there are no defects in a crystal lattice, these distributions

require

112,

113

and

114,

respectively.

Atoms

and

are distributed in the lattice in an ordered fashion. This

usually means that the incorrect space group symmetry was used to

describe the crystal structure. The symmetry therefore, should be lowered

by modifying or removing those elements that cause atoms to overlap.'

Yet another possibility is the so-called statistical mixing of various atoms

in the same site. It may be observed when different unit cells contain two or

more types of atoms in the same position.'

the most general case, the

occupational disorder can be expressed as:

where

represents the unoccupied fraction of the unit cells (i.e. defects) and

is the number of different types of atoms that occupy given site position in

different unit cells.

all cases, considered in this section, population factor(s) could be

refined even though some of them may be constrained by symmetry or other

relationships. For example when

0 and

Eq.

2.90, the following

constraint should be in effect:

g2.

Given the sensitivity of the

For example, overlaps due to a mirror plane may be avoided after doubling the unit cell

dimension along the mirror plane and substituting it with a glide plane. When certain

symmetry

operation(s) are excluded from the space group symmetry, this may sometimes

result in a substantial change, including switching to a lower symmetry crystal system.

Strictly speaking, it should be referred to as "practically the same position", since different

atoms would interact differently with their surroundings, thus causing locally different

environments unless their volumes and electronic properties are quite similar

(e.g. Si and

Ge; Fe3+ and co3+, and other). Consequently, in the majority of cases their positions

(coordinates) will be at least slightly different. On the other hand, the differences in the

positions of these atoms can rarely be detected from diffraction experiment and therefore,

are neglected.

Fundamentals

diffraction

207

population parameters to the potentially overestimated symmetry, it is

important to analyze their values, especially if they converge into simple

factions, e.g. 112, 113, and so on (see

Figure

2.52).

2.11.3

Temperature factor

At any temperature higher than absolute zero, certain frequencies in a

phonon spectrum of the crystal lattice are excited and as a result, atoms are

in a continuous oscillating motion about their equilibrium positions, which

are determined by coordinate triplets (x, y,

z).

To account for these

vibrations, the so-called temperature factor is introduced into the general

equation (Eq. 2.87) of the structure amplitude.

It is worth noting that according to a recommendation issued by the

International Union of Crystallography

(IUCr), the corresponding

parameters representing the temperature factor should be referred as "atomic

displacement parameters" instead of the commonly used "temperature

parameters" or "thermal parameters". This suggestion is based on the fact

that these parameters, when determined from x-ray diffraction experiment,

represent the combined total of several effects in addition to displacements

caused by thermal motion. They include deformation of the electron density

around the atom due to chemical bonding, improperly or not accounted

absorption, preferred orientation, porosity, and so on, even if they influence

the structure factor to a much lesser degree than thermal motion.

Oscillatory motions of atoms may be quite complex and as a result,

several different levels of approximations in the expression of the

temperature factor can be used.

the simplest form, the temperature factor

of the

jth

atom is represented as:

where

is the displacement parameter of the

jt"

atom,

is the Bragg angle

at which a specific reflection

hkl

is observed and

is the wavelength. This is

the so-called isotropic approximation, which assumes equal probability of an

atom to deviate in any direction regardless of its environment,

i.e. atoms are

considered as diffuse spheres.

Eq. 2.91

where (U2)i is the root mean square deviation of the

jth

atom from its

equilibrium position (x, y,

in A2.

208

Chapter

Considering Eq. 2.92, the isotropic displacement parameters are only

physical when they are positive (also see section

2.1

1.4).

Depending on the

nature of the material, they usually vary within relatively narrow ranges at

room temperature. For inorganic ionic crystals and intermetallic compounds

the typical range of

B's is -0.5 to -1 A2; for other inorganic and many

coordination compounds B varies from -1 to -3 A2, while for organic and

organometallic compounds and for solvent or other intercalated non-bonded

molecules or atoms this range extends from -3 to -10

or higher. As can

be seen in

Figure

2.53, the high value of the atomic displacement parameter

results in a rapid decrease of the structure amplitude when the Bragg angle

increases (also see Eq. 2.87).

As mentioned at the beginning of this section, the temperature factor

absorbs other unaccounted or incorrectly accounted effects. The most critical

are absorption, porosity and other instrumental or sample effects (see

Chapter 3), which in a systematic way modify the diffracted intensity as a

function of the Bragg angle. As a result, the

parameters of all atoms may

become negative. If this is the case, then the absorption correction should be

re-evaluated and re-refined or the experiment should be repeated to

minimize the deleterious instrumental influence on the distribution of

intensities of Bragg peaks.

The next level of approximation accounts for the anisotropy of thermal

motions in a harmonic approximation and describes atoms as ellipsoids in

one of the three following forms, which are, in fact, equivalent to one

another:

where

pi,

,..pi,

B,',

.B;~

and

u/,

. .

.u:,

are the anisotropic atomic

displacement parameters (six parameters' per atom).

This is true for atoms located in the general site position (point symmetry

I),

where all six

parameters are independent from one another. In special positions, some or all of the

Fundamentals of diffraction 209

Figure

2.53.

Temperature factor as a function of sinelk for several different atomic

displacement parameters:

0.5,

1.0

and

4.0

A'.

As follows from Eqs.

2.94

and 2.95, the relationships between

and

are identical to that given in Eq. 2.92 and both are measured in

A2.

The

parameters in Eq. 2.93 are dimensionless but may be easily converted into

Bij

uy.

Very high quality powder diffraction data are needed to obtain

dependable anisotropic displacement parameters and even then, they may be

reliable only for those atoms that have large scattering factors (see next

section).' On the other hand, the refinement of anisotropic displacement

parameters is essential for those crystal structures, where strongly scattering

atoms are distinctly anisotropic.

The anisotropic displacement parameters can also be represented in a

format of a tensor,

Tij, i.e. a square matrix symmetrical with respect to its

principal diagonal. For

it is given as

and for other types of anisotropic displacement parameters (Eqs. 2.93 and

2.95) the matrices are identical except that the corresponding elements are

and

Ui,.,

respectively. The diagonal elements of the tensor, Tii (i

2, 3),

anisotropic displacement parameters will be constrained by symmetry. For example, BI3

(PI3, UI3) and

BZ3

(P23, U23) for an atom located on a mirror plane perpendicular to Z-axis

are constrained to

More often than not, the anisotropic atomic displacement parameters determined from

powder diffraction data affected by preferred orientation will be incorrect (unphysical).

210

Chapter

describe atomic displacements along three mutually perpendicular axes of

the ellipsoid. Thus, similar to the isotropic displacement parameter, they may

not be negative and should have reasonable values at room temperature as

established by the nature of the material. Generally, the ratios between

w.i,

should not exceed

unless the large anisotropy can be explained.

All nine elements of the tensor

Tij

establish the orientation of the

ellipsoid in the coordinate basis of the crystal lattice. Hence, any or all non-

diagonal elements may be positive or negative but certain relationships

between them and the diagonal parameters should be observed as shown in

Eq.

2.97

for

Bij.

If any of these three relationships between the anisotropic

displacement parameters is violated, then the set of parameters has no

physical meaning.

The anisotropic displacement parameters can be visualized as ellipsoids

(Figure

2.54)

that delineate the volume where atoms are located most of the

time, typically at the 50

probability level. The magnitude of the

anisotropy and the orientations of the ellipsoids may be used to validate the

model of the crystal structure and the quality of refinement by comparing

"thermal" motions of atoms with their bonding states. Because of this, when

new structural data are published, the ellipsoid plot is usually required when

the results are based on single crystal

diffkaction data.

Figure

2.54.

The atomic displacement ellipsoids of carbon and nitrogen atoms shown at the

probability level for the hexamethylenetetramine molecule as determined from powder

diffraction data (see problem

in Chapter

7).

Hydrogen atoms were refined in the isotropic

approximation

(Eq.

2.91)

and are shown as small diffuse spheres.