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

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Fundamentals of diffraction

waves behave as particles (e.g. photons), and particles (e.g. neutrons and

electrons) behave as waves with wavelength

determined by the de Broglie

equation:

where

is Planck's constant

6.626~10"~ Jas),

is the particle's rest

mass, and v is the particle's velocity (mv

particle momentum).

For example, a neutron (rest mass,

1.6749~10-~~ kg) moving at a

constant velocity v

3000 m/s will also behave as a wave with

1.3

Furthermore, charged particles, e.g. electrons, can be focused using magnetic

lenses. Thus, modem high-resolution electron microscopes allow direct

imaging of atomic structures (for the most part in two dimensions on a

surface) with the resolution sufficient to distinguish individual atoms. Direct

imaging methods, however, require sophisticated equipment and the

accuracy in determining atomic positions is substantially lower than that

possible by means of diffraction techniques.' Hence, direct visualization of a

structure with atomic resolution is invaluable in certain applications but the

three-dimensional crystal structures are determined exclusively from

diffraction data.

The process of transforming diffraction patterns in order to reinstate the

underlying crystal structures in the three-dimensional direct space is

governed by the theory of diffraction. The latter rests on several basic

assumptions, yet it is accurate and practical. We have no intent to cover the

comprehensive derivation of the x-ray diffraction theory since it is mainly of

interest to experts, and can be found in many excellent books and

review^.^

Therefore, in this chapter we will discuss the nature and sources of radiation

that are in common use today and consider the principles and fundamental

laws of diffraction in general. We will also consider diffraction from

crystalline matter

specifically from polycrystalline materials

and describe

diffraction pattern as a function of crystal symmetry, atomic structure and

conditions of the experiment.

Despite recent progress in the three-dimensional x-ray holography [e.g. see M. Tegze,

Faigel, S. Marchesini,

Belakhovsky, and A. I. Chumakov, Three-dimensional

imaging of atoms with isotropic 0.5

resolution, Phys. Rev. Lett.

82,4847

(1999)], which

in principle enables visualization of the atomic structure in three dimensions, its accuracy

in determining coordinates of atoms and interatomic distances is much lower than possible

by employing conventional diffraction methods.

For example, see the International Tables for Crystallography, vol.

Second edition,

Shmueli, Ed. (2001) and vol. C, Second edition, A.J.C. Wilson and E. Prince, Eds. (1999),

Published for the International Union for Crystallography by Kluwer Academic

Publishers,

Boston/Dordrecht/London,

and references therein.

Chapter

2.2

Properties and sources of radiation

Nearly immediately after their discovery, x-rays were put to use to study

the internal

structure of objects that are opaque to visible light but

transparent to x-rays,

e.g. parts of a human body using radiography, which

takes advantage of varying absorption: bones absorb x-rays stronger than

surrounding tissues. It is interesting to note that the lack of understanding of

their nature, which did not occur until 19 12, did not prevent the introduction

of x-rays into medicine and engineering. Today the nature and the properties

of x-rays and other types of radiation are well understood and they are

briefly considered in this section.

2.2.1

Nature and properties of x-rays

Electromagnetic radiation is generated every time when electric charge

accelerates or decelerates. It consists of transverse waves where electric

(E)

and magnetic

(H)

vectors are perpendicular to one another and to the

propagation vector of the wave

(k),

see

Figure

2.2, top. The x-rays have

wavelengths from -0.1 to -100 A, which are located between y-radiation and

ultraviolet rays as also shown in

Figure

2.2, bottom. The wavelengths, most

commonly used in crystallography, range between -0.5 and -2.5

since

they are of the same order of magnitude as the shortest interatomic distances

observed in both organic and inorganic materials. Furthermore, these

wavelengths can be easily produced in almost every research laboratory.

When x-rays propagate through a substance, the occurrence of the

following processes should be considered in the phenomenon of diffraction:

1. Coherent scattering (section

2.5), which produces beams with the same

wavelength as the incident (primary) beam. In other words, the energy of

the photons in a coherently scattered beam remains unchanged when

compared to that in the primary beam.

2. Incoherent (or Compton) scattering, in which the wavelength of the

scattered beam increases due to partial loss of photon energy in collisions

with core electrons (the Compton effect).

Absorption of the x-rays, see section 2.3.2.1, in which some photons are

dissipated in random directions due to scattering, and some photons lose

their energy by ejecting electron(s) from an atom (i.e. ionization) andlor

due to the photoelectric effect (i.e. x-ray fluorescence).

Incoherent scattering is not essential when the interaction of x-rays with

crystal lattices is of concern, and it is generally neglected. When absorption

becomes significant, it is usually taken into account as a separate effect.

Thus, in the first approximation only coherent scattering results in the

diffraction from periodic lattices and will be considered in this chapter.

Fundamentals of diffraction

10.12 10-11 10.10 10.8 10-8

10-7

10-8 10.5 i0.4

lo-3

lo-2

10.1 100

Wavelength,

(m)

Figure

2.2.

Top

the schematic of the transverse electromagnetic wave in which electric

(E)

and magnetic

(H)

vectors are mutually perpendicular, and both are perpendicular to the

direction of the propagation vector of the wave,

The wavelength,

is the distance between

the two neighboring wave crests. Bottom

the spectrum of the electromagnetic waves. The

range of typical x-ray wavelengths is shaded. The boundaries between different types of

electromagnetic waves are diffuse.

Generally, the interaction of x-rays (or any other type of radiation with

the proper wavelength) with a crystal is multifaceted and complex, and there

are two different levels of approximation

kinematical and dynamical

theories of diffraction.

In the kinematical diffraction, a beam scattered once

is not allowed to be scattered for the second, third and so on times. Thus, the

kinematical theory of diffraction is based on the assumption that the

interaction of the diffracted beam with the crystal is negligibly small. This

requires the following postulations: i) a crystal consists of individual mosaic

blocks

crystallites'

which are slightly misaligned with respect to one

another; ii) the size of the crystallites is small, and iii) the misalignment of

the crystallites is large enough, so that the interaction of x-rays with matter

at the length scale exceeding the size of mosaic blocks is negligible.

On the contrary, the theory of the dynamical

diffvaction accounts for

scattering of the diffracted beam and other interactions of waves inside the

Crystallite usually means a tiny single crystal (microcrystal). Each particle in a

polycrystalline material usually consists of multiple crystallites that join together in

different orientations.

small powder particle can be a single crystallite as well.

104

Chapter

crystal, and thus the mathematical apparatus of the theory is quite complex.

Dynamical effects become significant and the use of the theory of dynamical

diffraction is justified only when the crystals are nearly perfect or when there

is an exceptionally strong interaction of the radiation with the material.

the majority of crystalline materials, however, dynamical effects are small

and they are usually noticeable only when precise single crystal experiments

are conducted. Even then, numerous dynamical effects

(e.g. primary and/or

secondary extinction, simultaneous diffraction, thermal diffuse scattering,

and others) are usually applied as corrections to the kinematical diffraction

model.

The kinematical approach is simple, and adequately and accurately

describes the diffraction of x-rays from mosaic crystals. This is especially

true for polycrystalline materials where the size of crystallites is relatively

small. Hence, the kinematical theory of diffraction is used in this chapter and

throughout this book.

2.2.2

Production of

x-rays

The x-rays are usually generated using two different methods or sources.

The first is a device, which is called an x-ray tube, where electromagnetic

waves are generated from impacts of high-energy electrons with a metal

target. These are the simplest and the most commonly used sources of x-rays

that are available in a laboratory of any size, and thus, an x-ray tube is

known as a laboratory or a conventional x-ray source. Conventional x-ray

sources usually have low efficiency and their brightness' is fundamentally

limited by the thermal properties of the target material. The latter must be

continuously cooled because nearly all kinetic energy of the accelerated

electrons is converted into heat when they decelerate rapidly (and sometimes

instantly) during the impacts with a metal target.

The second is a much more advanced source of x-ray radiation

the

synchrotron, where high energy electrons are confined in a storage ring.

When they move in a circular orbit, electrons accelerate towards the center

of the ring, thus emitting electromagnetic radiation. The synchrotron sources

are extremely bright (or brilliant2) since thermal losses are minimized and

Brightness is measured as photon flux

a number of photons per second per unit area

where the area is expressed in terms of the corresponding solid angle in the divergent

beam. Brightness is different from intensity of the beam, which is the total number of

photons leaving the target, because intensity can be easily increased by increasing the area

of the target irradiated by electrons without increasing brightness.

The quality of synchrotron beams is usually characterized by brilliance, which is defined

as brightness divided by the product of the source area (in

rnrn2)

and a fraction of a useful

photon energy, i.e. bandwidth [see, for example, J. Als-Nielsen and

McMorrow,

Elements of modern x-ray physics, John Wiley

Sons, New York

(2001)l.

Fundamentals

diffraction

105

there is no target to cool. Their brightness is only limited by the flux of

electrons in the high energy beam. Today, the so-called third generation of

synchrotrons is in operation and their brilliance exceeds that of the

conventional x-ray tube by nearly ten orders of magnitude.

Obviously, given the cost of both the construction and maintenance of a

synchrotron source, owning one would be prohibitively expensive and

inefficient for an average crystallographic laboratory. All synchrotron

sources are multiple user facilities, which are constructed and maintained

using governmental support

(e.g. they are supported by the United States

Department of Energy in the US and by similar agencies in Europe and

Japan), which is fully justified by the unique structural information that can

be obtained using brilliant, variable energy x-ray beams.

general, there is no principal difference in the diffraction phenomena

using the synchrotron and conventional x-ray sources, except for the

presence of several highly intense peaks with fixed wavelengths in the

conventionally obtained x-ray spectrum and their absence,

i.e. the

continuous distribution of photon energies when using synchrotron sources.

Here and throughout the book, the x-rays from conventional sources are of

concern unless noted otherwise.

2.2.3 Conventional sealed x-ray sources

As noted above, the x-ray tube is a conventional laboratory source of x-

rays. The two types of x-ray tubes in common use today are the sealed tube

and the rotating anode tube. The sealed tube consists of a stationary anode

coupled with a cathode, and both are placed inside a metallglass or a

metallceramic container sealed under high vacuum as shown in

Figure

2.3.

The x-ray tube assembly is

simple and maintenance-free device.

However, the overall efficiency of an x-ray tube is very low

approximately

1% or less. Most of the energy supplied to the tube is converted into heat,

and therefore, the anode must be continuously cooled with chilled water to

avoid target meltdown. The input power to the sealed x-ray tube (-0.5 to

kW) is therefore, limited by the tube's ability to dissipate heat, but the

resultant energy of the usable x-ray beam is much lower than 1% of the input

power because only a small fraction of the generated photons exits through

each window. Additional losses occur during the monochromatization and

collimation of the beam (see section

2.3).

the x-ray tube, electrons are emitted by the cathode, usually

electrically heated tungsten filament, and they are accelerated towards the

anode by a high electrostatic potential (30 to 60 kV) maintained between the

cathode and the anode. The typical current in a sealed tube is between 10 and

mA.

The x-rays are generated by the impacts of high-energy electrons

106

Chapter

with the metal target of a water-cooled anode, and they exit the tube through

beryllium (Be) windows, as shown in

Figure

2.3

and

Figure

2.4.

standard sealed tube has four Be windows located 90" apart around the

circumference of the cylindrical body. One pair of the opposite windows

corresponds to a point-focused beam, which is mostly used in single crystal

diffraction, while the second pair of windows results in a line-focused beam,

which is normally used in powder diffraction applications, see

Figure

2.4.

Given the geometry of the x-ray tube, the intensities of both the point-

and line-focused beams are nearly identical, but their brightness is different:

the point focus is brighter than the linear one. The use of the linear focus in

powder diffraction is justified by the need to maintain as many particles in

the irradiated volume of the specimen as possible. The line of focus (is. the

projection of the cathode visible through beryllium windows) is typically 0.1

to 0.2 mm wide' and

to 12 mm long. Similarly, point focus is employed in

single crystal diffraction because a typical size of the specimen is small

(0.1 to 1 mm). Thus, high brightness of a point-focused beam enables one to

achieve high scattered intensity in a single crystal diffraction experiment.

X-rays

window

Cathode

Figure

2.3.

The schematic (left) and the photograph (right) of the sealed x-ray tube. The

bottom part of the tube is metallic and it contains the anode (high purity copper, which may

be coated with a layer of a different metal, e.g. Cr, Fe, Mo, etc., to produce a target other than

copper), the windows (beryllium foil), and the cooling system. The top part of the tube

contains the cathode (tungsten filament) and it is manufactured from glass or ceramics,

welded shut to the metal canister in order to maintain high vacuum inside the tube. The view

of two windows (a total of four) and the "Water out" outlet is obscured by the body of the

tube (right). High voltage is supplied by a cable through a coupling located in the glass (or

ceramic) part of the tube. Both the metallic can and the anode are grounded.

The projection of the cathode on the anode surface is wider,

mm,

see

Figure

2.9.

Fundamentals of diffraction

107

Cathode Wmdows,

Cathode

projection

lme focus

projection

Figure

2.4. The schematic explaining the appearance of two different geometries of the x-ray

focus in a conventional sealed x-ray tube (left) and the disassembled tube (right). The photo

on the right shows the metallic can with four beryllium windows, two of which correspond to

line- and two to point-foci. The surface of the anode with the cathode projection is seen inside

the can (bottom, right). What appears as a scratch on the surface of the anode is the damage

from the high intensity electron beam and a thin layer deposit of the cathode material

(W),

which occurred during the lifetime of the tube. The cathode assembly is shown on top, right.

2.2.4

Continuous and characteristic x-ray spectra

The x-ray spectrum, generated in a typical x-ray tube, is shown

schematically in

Figure

2.5.

It consists of several intense peaks, the so-

called characteristic spectral lines, superimposed over a continuous

background, known as the white radiation. The continuous part of the

spectrum is generated by electrons decelerating rapidly and unpredictably

some instantaneously, other gradually

and the distribution of the

wavelengths depends on the accelerating voltage but not on the nature of the

anode material. White radiation, also known as

bremsstrahlung

(German for

"braking radiation"), is generally highly undesirable in x-ray diffraction

analysis applications.'

While it is difficult to establish the exact distribution of the wavelengths

in the white spectrum analytically, it is possible to establish the shortest

wavelength that will appear in the continuous spectrum as a function of the

accelerating voltage. Photons with the highest energy (i.e. rays with the

shortest wavelength) will be emitted by the electrons, which are stopped

instantaneously by the target.

In this case, the electron may transfer all of its

kinetic energy

One exception is the so-called Laue technique, in which white radiation is employed to

produce diffraction patterns from stationary single crystals.

to a photon with the energy

Chapter

(2.2)

where

is the rest mass,

is the velocity and

is the charge of the electron

(1.602~ lo-'' coulomb),

is the accelerating voltage,

is the speed of light in

vacuum (2.998~10~ mls),

is Planck's constant (6.626~10"~ J.s),

is the

frequency and

is the wavelength of the wave associated with the energy of

the photon.

After combining the right hand parts of Eqs. 2.2 and 2.3- and solving with

respect to A, it is easy to obtain the equation relating the shortest possible

wavelength (Asw in A) and the accelerating voltage (in

V).

Wavelength,

Figure

2.5.

The schematic of a typical x-ray emission spectrum, for clarity indicating only the

presence of continuous background and three characteristic wavelengths:

Ka,, Ka2,

and

KP,

which have high intensities. The relative intensities of the three characteristic spectral lines

are approximately to scale, however, the intensity of the continuous spectrum and the

separation of the

KallKa2

doublet are exaggerated. Fine structure of the

K(3

spectral line is

not shown for clarity. The vertical arrow indicates the shortest possible wavelength of white

radiation,

hsw,

as determined by

Eq.

2.4.

Fundamentals

diffraction 109

The three characteristic lines are quite intense and they result from the

transitions of upper level electrons in the atom core to vacant lower energy

levels, from which an electron was ejected by the impact with an electron

accelerated in the x-ray tube. The energy differences between various energy

levels in an atom are element-specific and therefore, each chemical element

emits x-rays with a constant,

i.e, characteristic, distribution of wavelengths

that appear due to excitations of core electrons by high energy electrons

bombarding the target, see Table

2.1.

Obviously, before core electrons can

be excited from their lower energy levels, the bombarding electrons must

have energy, which is equal to or exceeds that of the energy difference

between the two nearest lying levels of the target material.

Table

2.1.

Characteristic wavelengths of five common anode materials and the K absorption

edges of suitable P-filter materials.'

Wavelength (A)

K-

absorption

Metal

Kaa K~I Ka2 KP filter edge, (A)

Cr 2.29105 2.28975(3) 2.293652(2)

2.08491(3) V 2.26921 (2)

Fe 1.93739 l.93608(1) 1.94002(1)

1.75664(3) Mn l.896459(6)

Co 1.79030 1.789OO(l) l.79289(1) l.62082(3) Fe 1.7436 U(5)

.%I187 1.5405929(5) 1.54441(2) l.39225(1) Ni l.488140(4)

Mo 0.71075 0.7093171(4) 0.71361(1) 0.63230(1) Nb 0.653134(1)

Zr 0.688959(3)

The weighted average value, calculated as ha,,,,

(2hK,,+hK,,)/3.

The transitions from

and

shells to the

shell, i.e.

and

M+

are designated as Ka and

radiatioq2 respectively. Here

corresponds

to the shell with principal quantum number n

to n

2 and

to n

The Ka component consists of two characteristic wavelengths designated as

Kal and Ka2, which correspond to 2pl12

Isll2 and 2p3l2

Isln transitions,

respectively, where

and

refer to the corresponding orbitals. The

subscripts 112 and 312 are equal to the total angular momentum quantum

number,

j.3

The KP component also consists of several spectral lines, the

The wavelengths are taken from the International Tables for Crystallography, vol. C,

Second edition, A.J.C. Wilson and

Prince, Eds., Kluwer Academic Publishers, Boston1

DordrechtILondon (1999). For details on absorption and filtering, see section 2.3.2.1.

According to IUPAC

[R.

Jenkins,

Manne, J. Robin, C. Cenemaud, Nomenclature,

symbols, units and their usage in spectrochemical analysis. VIII Nomenclature system for

x-ray energy and polarization, Pure Appl. Chem.

63,

735 (1991)l the old notations, e.g. Cu

Kal and Cu KP should be substituted by the initial and final levels separated by a hyphen,

e.g. Cu K-L3 and Cu K-M3, respectively. However, since the old notations remain in

common use, they will be retained throughout this book.

s when

0 and

112 when

0, where

is the orbital, and s is the spin

quantum numbers. Since

adopts values 0, 1, 2,

..,

n-1, which correspond to

orbitals and s

112,

is equal to 112 for

orbitals, 112 or 312 forp orbitals, and so on.

110 Chapter

strongest being KPI and KP3, which are so close to one another that they are

practically indistinguishable in the x-ray spectra of many anode materials.

There are more characteristic lines in the emission spectrum

(e.g. La-y and

Ma-c), however, their intensities are substantially lower and their

wavelengths are greater that those of Ka and KP. Therefore, they are not

used in x-ray diffraction analysis and will not be considered here.'

addition to their wavelengths, the strongest characteristic spectral lines

have different intensities: the intensity of Ka, exceeds that of Ka2 by a

factor of two, and the intensity of

Ka, is approximately five times that of the

intensity of the strongest KP line, although the latter ratio varies

considerably with the atomic number. Spectral purity, i.e. the availability of

a single intense wavelength, is critical in most diffraction applications and

therefore, various monochromatization methods are used to eliminate

multiple wavelengths (and hence multiple Bragg peaks from the identical

sets of crystallographic planes), which is discussed in section 2.3. Although

the continuous x-ray emission spectrum does not result in distinct diffraction

peaks from polycrystals, its presence increases the adverse background noise

and therefore, white radiation should be minimized.

Typical anode materials that are used in x-ray tubes (Table

2.1)

produce

characteristic wavelengths between -0.5 and -2.3 A. However, only two of

them are in the most common use. These are Cu in powder and Mo in

single-

crystal diffractometry. Other anode materials can be used in special

applications, for example Ag anode

Ka,

0.5594218 A) can be used to

increase the resolution of the atomic structure since using shorter wavelength

broadens the range of

sineI3L in which diffraction peaks can be measured.

Bragg peaks, however, are observed closer to each other and the resolution

of the diffraction pattern may deteriorate. On the other hand, Cr, Fe or Co

anodes may be used instead of a Cu anode in powder diffraction (or Cu

anode instead of Mo anode in single crystal diffractometry) to increase the

resolution of the diffraction pattern (Bragg peaks are observed further apart)

but the resolution of the atomic structure decreases.

2.2.5

Rotating anode

x-ray

sources

The low thermal efficiency of the sealed x-ray tube can be substantially

improved by using a rotating anode x-ray s~urce,~ which is shown in Figure

Except for one experimental artifact shown later in

Figure

2.18,

where two components

present in the

characteristic spectrum of

(filament material contaminating Cu anode

of a relatively old x-ray tube) are clearly recognizable in the diffraction pattern collected

from the oriented single crystalline silicon wafer.

For more details on rotating anode x-ray sources see

W.C.

Phillips, X-ray sources,

Methods Enzymol.

114,300

(1985)

and references therein.