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

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Fundamentals of diffraction

2.6.

this design, a massive disk-shaped anode is continuously rotated at a

high speed while being cooled by a stream of chilled water. Both factors, i.e.

the anode mass (and therefore, the total area bombarded by high energy

electrons) and anode rotation, which constantly brings chilled metal into the

impact zone, enable a routine increase of the x-ray tube input power to -15

kW and in some reported instances to 50

kW, i.e. up to 20 times

greater when compared to a standard sealed x-ray tube.

The resultant brightness of the x-ray beam increases proportionally to the

input power, however, the lifetime of seals and bearings that operate in high

vacuum may be limited in some designs.' The considerable improvement in

the incident beam brightness yields much better diffraction patterns,

especially when diffraction data are collected in conditions other than the

ambient air (e.g. high or low temperature, high pressure and other), which

require additional shielding and windows for the x-rays to pass through, thus

resulting in added intensity losses.

Rotat~ng anode

Figure

2.6.

The schematic (left) and the photograph (right) of the direct drive rotating anode

assembly.

is point focus and

is line focus. The trace seen on the anode surface on the

right is surface damage caused by high-energy electrons bombarding the target and a thin

layer deposit of the filament material

(W),

which occurred during anode operation.

(Photograph courtesy of RigakuIMSC.)

In the laboratory of one of the authors

(VKP)

the direct drive rotating anode source

manufactured by RigakuIMSC has been in continuous operation (the anode is spinning and

the x-rays are on

hours per day, 7 days per week) for

months at the time of writing

this book and it is still operating without a breakdown.

112 Chapter 2

2.2.6 Synchrotron radiation

sources

Synchrotron radiation sources were developed and successfully brought

on line beginning in 1960's. They are the most powerful x-ray radiation

sources today. Both the brilliance of the beam and the coherence of the

generated electromagnetic waves are exceptionally high. The synchrotron

output power exceeds that of the conventional x-ray tube by many orders of

magnitude. Tremendous energies are stored in synchrotron rings (Figure

2.7), where beams of accelerated electrons or positrons are moving in a

circular orbit, controlled by a magnetic field, at relativistic velocities.

Electromagnetic radiation ranging from radiofrequency to

short-

wavelength x-ray region (Figure 2.8) is produced due to the acceleration of

charged particles towards the center of the ring. The x-ray beam is emitted in

the direction, tangential to the

electron/positron orbit (Figure

2.7).

Since there is no target to cool, the brilliance of the x-ray beam that can

be achieved in synchrotrons is four (first generation synchrotrons) to twelve

(third generation synchrotrons) orders of magnitude higher than that from a

conventional x-ray source.

Furthermore, given the size of the storage ring

(hundreds of meters in diameter), the average synchrotron beam consists of

weakly divergent beams that may be considered nearly parallel at distances

typically used in powder diffraction (generally less than 1 m). This feature

presents an additional advantage in powder diffraction applications since the

instrumental resolution is also increased.

Storage ring

Injection

point

Bending

magnet

Figure

2.7.

Schematic diagram of a synchrotron illustrating x-ray radiation output from

bending magnets. Electrons must be periodically injected into the ring to replenish losses that

occur during normal operation. Unlike in conventional x-ray sources, where both the long-

and short-term stability of the incident photon beam are controlled by the stability of the

power supply, the x-ray photon flux in

synchrotron changes with time: it decreases

gradually due to electron losses, and then periodically and sharply increases when electrons

are injected into the ring.

Fundamentals of diffraction

Wavelength,

Figure

2.8.

The schematic of a typical x-ray spectrum from a synchrotron.

Another important advantage of the synchrotron radiation sources, in

addition to extremely high brilliance of the x-ray beam, is in the distribution

of the beam intensity as a function of wavelength

(Figure

2.8).

The high

intensity, observed in a broad range of photon energies, allows for easy

selection of nearly any desired wavelength. Furthermore, the wavelength

may be changed when needed and energy dispersive experiments, during

which the diffraction angle remains constant but the wavelength varies, can

be conducted quite effectively.

Thus, synchrotron radiation finds more and more use today, although its

availability is restricted to the existing synchrotron sites. Some of the well-

known sites are the ALS

Advanced Light Source at Berkeley Lab, APS

Advanced Photon Source

Argonne National Laboratory, NSLS

National

Synchrotron Light Source at Brookhaven National Laboratory, SRS

Synchrotron Radiation Source at Daresbury Laboratory,

ESRF

European

Synchrotron Radiation Facility in Grenoble, and others.'

2.2.7

Other types of radiation

Other types of radiation that are commonly used in diffraction analysis

are neutrons and electrons. The properties of both are compared with those

of x-rays in

Table

2.2.

Neutrons are usually produced in nuclear reactors; they have variable

energy and therefore, a white spectrum. Maximum flux of neutrons is

usually obtained in an angstrom range of wavelengths. The main differences

when compared to x-rays are as follows: i) neutrons are scattered by nuclei,

which are much smaller than electron clouds, and the scattering occurs on

Web

links to worldwide synchrotron and neutron facilities can be found at

http:Nwww.iucr.org/cww-top/rad.index.html.

114

Chapter

points; ii) scattering factors of elements remain constant over the whole

range of Bragg angles; iii) scattering functions are not proportional to the

atomic number and they are different for different isotopes of the same

chemical element. Furthermore, since neutrons have spins, they interact with

the unpaired electron spins (magnetic moments) and thus neutron diffraction

data are commonly used to determine ordered magnetic structures. Other

differences between neutrons and x-rays are nonessential in general

diffraction theory.

Table

2.2.

Comparison of three types of radiation used in powder diffraction.

X-rays (conv./synch.) Neutrons Electrons

Nature wave particle particle

Medium atmosphere atmosphere high vacuum

Scattering by electron density nuclei and magnetic electrostatic

spins of electrons potential

Scattering function

As)

is constant at all

As)

Z 'I3,

Wavelength range, h 0.5-2.510.1-10

-1

0.02-0.05

Wavelength selection fixed,c Ka,Plvariable variable variable

Focusing

none

magnetic lenses

Lattice image

Direct structure image

Applicable theory of

diffraction

Use to determine

reciprocal

kinematical

direct, reciprocal

Yes

dynamical

relatively simple very complex

atomic structure

sinelh, Z- atomic number,

atomic scattering function.

If unknown, electron scattering factor

S,(s)

may be derived from x-ray scattering factor

f,(s)

asf,(s)

k[Z -f,(s)]/s2, where

is constant.'

According to Moseley's law, x-ray characteristic frequency is

c/h

D)~, while

for neutrons and electrons h

hlmv

h (2m~)-'/~, where

and

are constants, m is mass,

v is velocity, and

is kinetic energy of a particle.

One of the biggest disadvantages of the conventional (reactor-generated)

neutron sources is relatively low neutron fluxes at useful energies. This

problem is addressed in the new generation of highly intense pulsed

(spallation) neutron

source^.^

a spallation neutron source, bunches of

protons are accelerated to high energies, and then released bombarding a

heavy metal target in short but extremely potent pulses. The collision of each

proton with a heavy metal nucleus results in many expelled (spalled or

Electron diffraction techniques, Vol. 1,

M. Cowley, Ed., Oxford University Press,

(1992).

One of the brightest operational pulsed neutron sources (ISIS) is located in the

(http://www.isis.rl.ac.uk/).

In the US, the construction of the spallation source at the Oak

Ridge National Laboratory (http://www.sns.gov/) is scheduled for completion in 2006.

Fundamentals of diffraction

115

knocked out) neutrons at various energies. The resultant highly intense

(-1

times higher flux than in any conventional reactor) neutron beams have a

nearly continuous energy spectrum and they can be used in a variety of

diffraction studies, mostly in the so-called time-of-flight (TOF) experiments.

In the latter, the energy (and the wavelength) of the neutron that reaches the

detector is calculated from the time it takes for a neutron to fly from the

source to the specimen and from the specimen to the detector.

In addition to the direct imaging of crystal lattices

(e.g. in a high-

resolution transmission electron microscope), electrons may be used in

diffraction analysis. Despite the ease of the production of electrons by

heating a filament in vacuum, electron diffraction is not as broadly used as

x-ray diffraction. First, the experiments should be conducted in high

vacuum, which is inconvenient and may result in the decomposition of some

materials. Second, electrons strongly interact with materials. The latter

requires the use of the dynamical theory of diffraction, thus making structure

determination and refinement quite complex. Finally, the complexity and the

cost of a high-resolution electron microscope usually considerably exceed

those of a high-resolution powder diffractometer.

Neutron diffraction examples will be discussed when deemed necessary,

even though in this book we have no intent to cover diffraction of neutrons

(and electrons) at any significant depth. Interested readers can find more

information on electron and neutron diffraction in some of the references

provided at the end of this chapter.

2.3

Collimation

and

monochromatization

Both the polychromatic nature and the angular divergence of the primary

x-ray beam generated using either a sealed or rotating anode x-ray tube (see

Figure

2.3

Figure

2.6)

result in complex diffraction patterns when x-rays

are employed in the "as produced" condition. This occurs since i)

white

radiation causes a high background; ii)

the presence of the three intense

characteristic lines (Ka,, Ka2 and KP) in the spectrum results in three Bragg

peaks from each set of crystallographic planes (i.e. from each point in the

reciprocal space), and iii)

the angular divergence in all directions yields

broad and asymmetric Bragg peaks. Thus, the incident x-ray beam needs to

be modified (conditioned) in order to improve the quality of the powder

diffraction pattern.

Angular divergence (dispersion) can be reduced by collimation

the

process of selecting electromagnetic waves with parallel or nearly parallel

propagation vectors. The undesirable satellite wavelengths can be removed

by various monochromatization approaches

the processes that convert

polychromatic radiation into a single wavelength (Ka, or KP in conventional

116

Chapter

x-ray sources, or narrow bandwidth using a synchrotron source) or at least

into a double wavelength

(Ka,

plus

Ka2)

beam. The

Ka,

plus

Ka2

doublet

wavelength is, by and large, the only acceptable combination of

polychromatic x-rays that is in common use today.

The main problem in both collimation and monochromatization is not in

how to reduce both the angular and wavelength (energy) dispersions but how

to do so with the minimal loss of intensity (photon flux) of both the incident

and diffracted beams. Needless to say, different diffraction methods require

a different degree of collimation and monochromatization. Thus, high

resolution or low-angle scattering applications usually require parallel and

narrow beams containing a single wavelength, while routinely used powder

diffractometers have less strict requirements. For example, the commonly

used Bragg-Brentano focusing technique' was specifically developed to

work well with slightly divergent incident beams and this focusing method

will not produce the best results when coupled with a perfectly parallel

incident beam.

2.3.1

Angular divergence and collimation

The simplest collimation can be achieved by placing a slit between the x-

ray source and the sample, as shown in

Figure

2.9, top left. The angular

divergence of thus collimated beam is established by the dimensions of the

source, the size, and the placement of the slit. This slit is usually called the

divergence slit and in the majority of powder diffractometers, the placement

of the divergence slit is fixed at a certain distance from the x-ray tube focus.

Considering the geometry shown in

Figure

2.9 and assuming that the

distance between the slit and the tube focus is much larger than the slit

opening, the angular divergence of the collimated beam,

i.e. the angle

given as

where

is the divergence slit opening in mm,

is the width of the focus of

the x-ray tube (in mm) visible at a take-off angle

y~,

and

is the distance

between the tube focus and the slit in mm. Since

and

are usually fixed,

the variable beam divergence is customarily achieved by varying the slit

opening,

second divergence slit can be placed further on the way of the

beam to provide additional collimation as shown in

Figure

2.9, top right.

The so-called Bragg-Brentano focusing geometry is the most commonly used technique in

modern powder diffractometry. It is briefly discussed later in this chapter and in greater

depth in the next chapter.

Fundamentals of diffraction

Divergence

slit

Divergence Divergence

slit

Figure

2.9.

The schematic showing collimation of the incident x-ray beam by using a single

divergence slit (top, left) or coupled divergence slits (top, right). The schematic on the bottom

left illustrates the size of the source

(S)

when the projection of the cathode

(C)

is viewed at a

take-off angle,

v.'

Equation

2.5

is derived on the bottom, right.

Both collimation methods shown in

Figure

2.9

are commonly used in

powder diffractometers that are employed for routine powder diffraction

experiments. High resolution and low-angle scattering diffractometers

require better and therefore, more complex collimation, which to some

extent overlaps with the monochromatization described below, but otherwise

will not be considered in this

book.2

Divergence slits reduce angular dispersion of the incident x-ray beam in

the plane perpendicular to the goniometer axis, in other words they are used

to control the so-called in-plane divergence, which is further discussed in

Chapter

Angular divergence in the direction parallel to the goniometer

axis is known as the axial divergence and it is controlled by using the so-

called Soller slits. Soller slits are usually manufactured from a set of parallel,

equally spaced thin metal plates as shown in

Figure

2.10,

top.

Each pair of the neighboring plates works similar to a regular divergence

slit. The major differences in the design of Soller slits, when compared to

For a line focus, a typical width of the cathode projection

mm;

a typical take-off

angle

6'.

Hence, a typical size of the source in powder diffraction is

0.1

nun.

Advanced collimation and monochromatization techniques are described in: D.K. Bowen

and B.K. Tanner, High resolution x-ray diffractometry and topography, Taylor

Francis,

London/Bristol, PA

(1998).

118

Chapter

divergence slits, is in the requirement to minimally affect the in-plane size of

the beam, regardless of the distance between the plates. This is done to

maximize the intensity of the incident beam and thus, to maintain the high

quality of powder diffraction patterns. Soller slits, installed on both the

primary and diffracted beam paths, substantially reduce axial divergence of

the x-ray beam and asymmetry (see section 2.9.2) of Bragg peaks. The axial

divergence of the beam collimated by Soller slits can be estimated in the

same way as for a single divergence slit:

where

is the distance between the parallel plates in

and

is the plate

length (also in mm) as shown at the bottom of

Figure

2.10.

Thus, the axial

divergence can be reduced or increased by varying either or both the length

of the plates

(I)

and the distance between them

(6).

Divergence

Slit

Soller

Slits

Figure

2.10.

The schematic showing how the x-ray beam is collimated by using both the

divergence and Soller slits (top). The beam, collimated in-plane by the divergence slit, is

further collimated axially by the Soller slits. The coordinates in the middle of the drawing

indicate the corresponding directions. The bottom part of the figure illustrates the analogy of

Eq.

2.6

with

Eq.

2.5.

Fundamentals of diffraction

2.3.2

Monochromatization

addition to collimation, the x-ray beam should be monochromatized by

reducing the intensity of white radiation and by eliminating the undesirable

characteristic wavelengths from the x-ray spectrum, leaving only a single

usable wavelength. As noted above, when conventional x-ray sources are

employed, the two-wavelengths beams

(Kal plus Ka2) are acceptable

because the complete removal of the Ka2 component substantially reduces

the intensity of the incident beam and therefore, increases the time of the

experiment needed to obtain high quality x-ray diffraction data. When

synchrotron sources are used, the monochromatization process selects a

single wavelength from a continuous x-ray spectrum. The most common

methods utilized in the instrumental monochromatization of x-ray beams are

as follows:

Using a

p-filter (conventional x-ray sources only).

Using diffraction from a crystal monochromator (any source, including

neutrons).

Pulse height selection using a proportional counter (x-rays).

Energy resolution using a solid-state detector (x-rays).

2.3.2.1

Absorption of x-rays and p-filters

The simplest monochromatization of x-rays produced in a conventional

source can be performed by means of filtering, which takes advantage of the

peculiar variation of the x-ray absorption properties of chemical elements.

When x-rays penetrate into the matter, they are partially transmitted and

partially absorbed. Thus, when an x-ray beam travels the infinitesimal

distance,

dx,

its intensity is reduced by the infinitesimal fraction dI/I (Figure

2.11,

left), which can be defined using the following differential equation

where

is the proportionality coefficient expressed in the units of the

inverse distance, usually in cm-'. This coefficient is also known as the linear

absorption coefficient of a material.

After integrating

Eq.

2.7, the transmitted intensity (I,) is easily calculated

in terms of a fraction of the initial intensity of the beam (Io):

120

Chapter

Wavelength,

Figure

2.1

The schematic explaining the phenomenon of absorption of x-rays by the matter

(left) and the typical behavior of the linear absorption coefficient of any chemical element as a

function of the wavelength of the x-rays (right). The gradual reduction of the transmitted

intensity (left) is represented by the gradual change of the shade of grey of the arrow

indicating the x-ray beam.

The linear absorption coefficient of any chemical element is a function of

the wavelength, and a typical p(h) dependence is shown in

Figure

2.11,

right.

the range of wavelengths, which is of interest to filtering

applications, the p(h) function consists of two continuous branches separated

by an abrupt change in the absorption properties at a certain photon energy.

The point, at which the discontinuity of the absorption coefficient occurs, is

called the

absorption edge, and the linear absorption coefficient (p)

changes its value by a factor of

The continuous change of the liner absorption along each of the two

branches is approximately defined as p

kZ3h3, where Z is the atomic

number of the chemical element and

is a constant, specific for each of the

two continuous parts of the absorption function. The continuous branches

correspond to the absorption occurring due to random scattering of photons

by electrons, which is observed in all directions, thus reducing the number of

photons in the transmitted beam in the direction of the propagation vector.

The appearance of the discontinuity

(i.e. the presence of the

absorption

edge) is known as the true absorption and it can be understood by

considering

Eq.

2.3.

As the wavelength decreases, the energy of the x-ray

photons increases and at a certain h it becomes sufficient to excite

electrons.' This not only causes a rapid increase in the number of the

absorbed photons but also results in the transitions of upper level electrons to

At higher wavelengths (lower energies), L- and M-absorption edges also appear due to L-

and M-electrons being excited from their ground states.