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

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Fundamentals of diffraction

121

vacant K levels in the atoms of the absorber

a photoelectric effect, during

which a fluorescent x-ray photon can be emitted

any direction. Both

scattered and true absorption result in the exponential reduction of the

transmitted intensity, as defined by

Eq.

2.8.

The monochromatization using a p-filter employs the presence of the K

absorption edge to selectively absorb Kp radiation and transmit the Kal and

Ka2 parts of the x-ray spectrum, as shown schematically in

Figure

2.12.

Thus, a properly selected p-filter material has its K absorption edge below

the wavelength of the Ka, characteristic line and just above the wavelength

of the KP line.'

The general rule for choosing the p-filter is to use a material, which is

rich in a chemical element, one atomic number less than the anode material

in the periodic table. This assures proper location of the K absorption edge,

i.e. between the Kal and Kp lines. For heavy anode materials (e.g. Mo), this

rule can be extended to two atomic numbers below the element of the anode.

list of P-filter elements, suitable for the most commonly used anode

materials, is found in

Table

2.1.

Thus, for

Cu anode, a foil made from

will work best as the p-filter, while for Mo radiation both

and Zr are

good p-filters. The former material (Zr) is more often used in practice

because its K absorption edge is closer to the wavelength of the Kal line.

Wavelength,

Figure

2.12.

Left

the schematic of the x-ray emission spectrum shown as the solid line

overlapped with the schematic of the

y(h)

function of the properly selected P-filter material

(dotted line). Right

the resultant distribution of intensity after filtering as a function of the

wavelength.

Better monochromatization can be achieved when using the so-called balanced filters. The

first filter material is chosen to suppress

and other short wavelengths. The second filter

material is selected such that its

absorption edge is located above the wavelength of the

KaZ

component and the thickness of each material is selected to equally suppress the

intensity. As a result, the balanced filter reduces the intensity of both short and long

wavelengths. Balanced filters are seldom used today because the resulting intensity loss is

usually higher when compared with a well aligned crystal-monochromator.

122 Chapter 2

The major disadvantages of p-filters are: i)

they are incapable of

complete elimination of the KP intensity, and ii)

they leave a considerable

amount of white radiation after filtering. Furthermore, any p-filter also

reduces all intensities in the spectrum at wavelengths higher than the

corresponding absorption edge (see Figure 2.12). This results in the

reduction of the intensity of the

KCX,,~ components, although obviously the

latter are reduced by a much smaller factor than the intensity of the KP line.

Fundamentally, the use of a p-filter improves

IKa/IKP and IKa/Iwhite ratios and

this improvement is proportional to the thickness of the filter. For example,

assume that the ratio of the intensities of the Kal and KP spectral lines in the

unfiltered spectrum is approximately

After passing through a properly

designed P-filter, it becomes -100

500 to 1.

Considering Eq. 2.8 and the as-produced ratio of intensities between Kal

and Kp lines (5:1), the filtered IKa,/IKP intensity ratio can be expressed as

follows:

where

~l,

and pp are the linear absorption coefficients of the filter material

for hKa and hKP, respectively, and t is the thickness of the p-filter. This

equation can also be used to calculate the filter thickness, t, which is

necessary to obtain the desired &/IKP ratio. Thus, depending on the

particular need and the application, a compromise is made between the

purity of the spectrum and the intensity of the Ka radiation.

Absorption coefficients for all chemical elements are usually tabulated

(see Table

2.3)

in the form of mass absorption coefficients,'

pip

(the units

are cm2/g), instead of the linear absorption coefficients, p.

The linear absorption coefficient of any material (solid, liquid or gas) is

then calculated as

where

is the mass fraction of the chemical element in the material, (p.1~)~

is elemental mass absorption coefficient, and p, is the density of the

material. For example, the liner absorption coefficient of the stoichiometric

This is reasonable because absorption of x-rays is proportional to the probability of a

photon to encounter an atom when passing through matter. This probability

directly

proportional to the number of atoms in the unit volume, i.e. to the density of the material.

Fundamentals of diffraction

123

mixture of gaseous hydrogen and oxygen (2 moles of

per 1 mole of

02)

only -111200 of the linear absorption coefficient of water because the

density of water is just over 1200 times greater than the density of the

mixture of the two gases at atmospheric pressure and room temperature.

Table

2.3. Mass absorption coefficients (in cm2Jg) of selected chemical elements for the

commonly used anode materials.' The mass absorption coefficients of the best @-filter

elements for the corresponding anode are shaded.

Fe Cu Mo

Ka KP Ka KP Ka K@ Ka K@

0.412 0.405 0.400 0.396 0.391 0.388 0.373 0.370

He 0.498 0.425 0.381 0.335 0.292 0.268 0.202 0.197

. .

C 15.0 11.2 8.99 6.68 4.51 3.33 0.576 0.458

24.7 18.6 14.9 11.0 7.44 5.48 0.845 0.645

37.8 28.4 22.8 17.0 11.5 8.42 1.22 0.908

...

Sc 516 403 332 256 180 137 20.8 14.9

Ti 358 277 200 152 23.4 16.8

399 309 219 166 26 18.7

Cr 247 185 29.9 21.5

Mn 270 207 33.1 23.8

Fe 302 232 37.6 27.1

Co 124 96.0 78.5 60.0 41.0 29.6

144 112 91.3 69.8 46.9 34.0

Cu 153 118 96.8 74.0 49.1 35.7

. . .

Sr 328 256 210

161 113 85.9 90.6 67.2

358 279 229

176 124 94.0

Zr 386 300 247 191 139 101

416

325 267 205 145

110

Mo 442 345 284

219 154 117 18.8 13.8

2.3.2.2

Crystal monochromators

different and more complex but much improved monochromatization

approach takes advantage of diffraction from a high-quality single crystal,

properly positioned with respect to the propagation vector of x-rays. The

examples of commonly used crystal monochromator materials include

pyrolitic graphite, Si, Ge, and

LiC1.

nearly perfect single crystal is placed at a specific angle

(eM)

with

respect to the primary or the diffracted x-ray beams and, according to the

Braggs' law

(Eq.

2.1 only discrete wavelengths can be transmitted at this

Taken from the International Tables for Crystallography, vol.

Second edition, A.J.C.

Wilson and

Prince, Eds., vol. C, Kluwer Academic Publishers, Boston/Dordrecht/

London (1999).

Also, see section 2.6.1.

124

Chapter

angle. Assuming that

1, the single transmitted wavelength, A,, is a

function of the corresponding interplanar distance of the crystal, dMhkl, and

eM.

reality, even the best crystal monochromators always allow some

wavelength dispersion in the transmitted beam because of the unavoidable

imperfections.

2d:[

sineM

nh,

(2.11)

The general principle of operation of a single crystal monochromator is

illustrated in

Figure

2.13. The x-rays in the divergent beam that reach the

monochromator form slightly different angles (ranging from 8, to 02) with

the crystal. They are reflected from the set of crystallographic planes,

hkl,

which are parallel to the crystal surface.

According to Braggs' law (Eq. 2.1 I), both the incident and diffracted

beams with identical wavelengths should form the same angle with the

surface of the crystal. Hence, each wavelength,

A,, is diffracted at a particular

angle 0,

82 which yields an uneven spatial distribution of wavelengths

in the beam reflected by the crystal.

effect, the shorter wavelengths are

grouped at the low Bragg angles (they are darker in

Figure

2.13) and the

longer wavelengths are observed at the high Bragg angles (they are lighter in

Figure

2.13). For example, the KP line falls into the low Bragg angle range

and the Kal/Ka2 doublet falls into the high angle range. The former or the

latter can be easily selected by a slit properly installed in the path of x-rays

reflected by a crystal monochromator.

Even though the diffracted beam is not perfectly monochromatic at any

specific angle due to various imperfections (defects, distortions, stresses,

etc.) present in the crystal monochromator, the separation of

and

wavelengths is large enough so that the elimination of the KP and nearly all

Monochromator

slit

Monochromator

Figure

2.13.

The schematic explaining the principle of monochromatization using a single

crystal monochromator. Generally

(e2

0,)/2.

The directions of the propagation vectors

are indicated

arrows.

Fundamentals of diffraction

125

white x-rays during the monochromatization is usually not a problem.

However, the separation of

Kal

and

Ka2

wavelengths requires more than

just a simple arrangement shown in

Figure

2.13. The wavelength (photon

energy) resolution can be improved by using diffraction from two or more

monochromators placed in sequence. We note that the diffraction from the

sample, in the first approximation, works as a preliminary monochromator.

Thus, the distribution of the wavelengths may be assumed uniform in the

whole range of incident angles (from

€4

Figure

2.13) but this is no

longer the case in the beam reflected by the first monochromator or by the

sample.

The following two configurations (parallel and angular), which are

shown in

Figure

2.14, are in common use to improve the

monochromatization of the beam. Both diffracting plates can be crystal

monochromators, or one can be the sample while another is the

monochromator. The latter arrangement is especially common in powder

diffractometry.

Two parallel plates

(Figure

2.14, left) remove the

line completely.

The

Ka,

and

Ka2

doublet can be separated at specific conditions when the

first plate is used as a monochromator but at a cost of the reduced beam

intensity. When the second plate is used as monochromator, the resultant

intensity is higher but the

KallKa2

doublet is nearly impossible to separate.

This layout also improves the collimation of the beam. The second

arrangement

(Figure

2.14, right) is more efficient. Angular orientation of the

two crystals separates

Kal

and

Ka2

lines quite well due to the larger spatial

difference between the two wavelengths after diffraction from the second

plate. Yet again, this is usually done at a considerable intensity loss penalty.

The three most common geometries of a crystal monochromator and a

sample, used in powder diffraction, are illustrated in

Figure

2.15 and their

characteristics are compared in

Table

2.4. Diffracted beam monochromators

(Figure

2.15a, b) have a relatively high intensity output (or in other words

have low intensity losses) but do not separate the

Kal

and

Ka2

doublet. The

removal of both the

component and white radiation is excellent. The

Monochromator

or sample

Figure

2.14.

Parallel (left) and angular (right) arrangements of two crystal monochromators

or the sample and the crystal monochromator commonly used to improve the

monochromatization of the resultant x-ray beam.

126

Chapter 2

latter is especially important when the sample is strongly fluorescent. The

diffracted beam monochromators, by far, are more commonly used than the

primary beam monochromators. The advantage of the angular configuration

(Figure

2.15b)

is in the lower torque exerted on the detector arm when

compared with the linear configuration, but the crystal surface should be

curved to match the radius of the monochromator focusing circle for best

results.

Figure

2.15.

The three different monochromator/sample geometries used in powder

diffraction: a) flat

diffracted

beam monochromator, parallel arrangement;

curved

dzffracted

beam monochromator, angular arrangement, and c) flat

primary

beam monochromator,

parallel arrangement.

focus of the x-ray source,

sample,

crystal monochromator,

detector,

radius of the monochromator focusing circle,

radius of the

goniometer focusing circle.

Fundamentals

diffraction

Table

2.4.

Comparison of the three common monochromator1sample configurations used in

powder diffractometry.

Property

Position in the beam Diffracted Diffracted Primary

Orientation relatively the sample Parallel Angular parallel

Shape of the crystal

FlatICurved Curved FlatICurved

Intensity loss Low Low LowIHigh

Kal

and

Kaz

separation

No No NoIComplete

Alignment Simple Simple Simple/Complex

Additional torque on the detector arm High Low None

Use with area detectors Impossible Impossible Possible

Labeling of the columns corresponds to

Figure

2.15.

The most important advantage of the primary beam monochromator

(Figure 2.152) is the possibility to separate KallKa2 doublets, which is

especially important when working with complex

diffraction

patterns where

the maximum resolution of Bragg peaks is critical. A primary beam

monochromator is the only type of geometry that can be used with area

detectors (see section 2.4.4). The main disadvantages are high intensity

losses, the need for precise alignment, and potential for high fluorescent

background in some combinations of materials and photon energies.

2.3.2.3

Pulse height selection

and

energy resolution

Monochromatization by pulse height selection using a proportional

detector is based on the fact that the signal generated in the detector is

proportional to the energy of the absorbed x-ray photon and therefore,

inversely proportional to its wavelength. This approach is discussed in the

following section, where the proportional detector is described.

Monochromatization by pulse height selection is far from ideal and it does

not even result in the complete elimination of the KP intensity. Nevertheless,

its use substantially improves the quality of powder diffraction data by

reducing the background noise, especially when combined with a P-filter.

Relatively recently developed cooled solid-state detectors find more and

more use as both detectors and monochromators because of their extremely

high sensitivity to the energy of absorbed photons, which enables precise

energy discrimination. Thus, the detector can be tuned to register

wavelengths only within certain limits, e.g. Ka, and Ka2 energies, or in

some of the most recent applications, only

KP component. The latter is one

of the ways to obtain single wavelength x-rays without using crystal

monochromators and, therefore, without substantial loss of intensity except

for the fact that the intensity of the KP line is only -115 of the intensity of

the

Kal line. Another advantage of this approach is a clean registered

diffraction pattern because the majority of white radiation is also eliminated.

The disadvantages of this monochromatization technique arise from the

128

Chapter

limitations intrinsic to solid-state detection and they will be briefly discussed

below.

2.4

Detection

x-rays

The detector is an integral part of any diffraction analysis system and its

major role is to measure the intensity and, sometimes, the direction of the

scattered beam. The detection is based on the ability of x-rays to interact

with matter and to produce certain effects or signals, for example to generate

particles, waves, electrical current, etc., which can be easily registered.

other words, each photon entering the detector generates a specific event,

better yet, a series of events that can be recognized and fiom which the total

photon count (intensity) can be determined. Obviously, the detector must be

sensitive to x-rays (or in general to the radiation being detected) and should

have an extended dynamic range and low background noise.'

2.4.1

Detector efficiency, linearity, proportionality and resolution

important characteristic of any detector is how efficiently it collects

x-ray photons and then converts them into a measurable signal. Detector

efficiency is determined by first, a fraction of x-ray photons that pass

through the detector window (the higher, the better) and second, a fraction of

photons that are absorbed by the detector and thus result in a series of

detectable events (again, the higher, the better). The product of the two

fractions, which is known as the absorption or quantum efficiency, should

usually be between

0.5

and

The efficiency of modern detectors is quite high, in contrast to the x-ray

film, which requires multiple photons to activate a single grain of photon-

sensitive silver halide. It is important to keep in mind that the efficiency

depends on the type of the detector and it normally varies with the

wavelength for the same type of the detector. The need for high efficiency is

difficult to overestimate since every missed (i.e. not absorbed by the

detector) photon is simply a lost photon. It is nearly impossible to account

for the lost photons by any amplification method, no matter how far the

amplification algorithm has been advanced.

The linearity of the detector is critical in obtaining correct intensity

measurements (photon count). The detector is considered linear when there

For the purpose of this consideration, the dynamic range is the ability of the detector to

count photons at both the low and high fluxes with the same effectiveness, and by the

background noise we mean the events similar to those generated

the absorbed photons,

but occurring randomly and spontaneously in the detector without photons entering the

detector.

Fundamentals of diffvaction

129

is a linear dependence between the photon flux (the number of photons

entering through the detector window in one second) and the rate of signals

generated by the detector (usually the number of voltage pulses) per second.

any detector, it takes some time to absorb a photon, convert it into a

voltage pulse, register the pulse, and reset the detector to the initial state, i.e.

make it ready for the next operation. This time is usually known as the dead

time of the detector

the time during which the detector remains inactive

after it has just registered a photon.

The presence of the dead time always decreases the registered intensity.

This effect, however, becomes substantial only at high photon fluxes. When

the detector is incapable of counting every photon due to the dead time, then

some of them could be absorbed by the detector but remain unaccounted,

i.e.

become lost photons. It is said that the detector becomes non-linear under

these conditions. Thus the linearity of the detector can be expressed as: i)

the maximum flux in photons per second that can be reliably counted (the

higher the better); ii)

the dead time (the shorter, the better), or iii)

the

percentage of the loss of linearity at certain high photon flux (the lower

percentage, the better). The latter is compared for several different types of

detectors in

Table

2.5

along with other characteristics.

The proportionality of the detector determines how the size of the

generated voltage pulse is related to the energy of the x-ray photon. Since x-

ray photons produce a certain amount of events (ion pairs, photons of visible

light, etc.) and each event requires certain energy, the number of events is

generally proportional to the energy of the x-ray photon and therefore, to the

inverse of its wavelength. The amplitude of the generated signal is normally

proportional to the number of these events and thus, it is proportional to the

x-ray photon energy, which could be used in pulse-height discrimination.

Usually, the high proportionality of the detector enables one to achieve

additional monochromatization of the x-ray beam in a straightforward

fashion: during the registration, the signals that are too high or too low and

thus correspond to photons with exceedingly high or exceedingly low

energies, respectively, are simply not counted.

Table

2.5.

Selected characteristics of the most common detectors using Cu

radiation.

Linearity loss Proportionality Resolution Energy per No. of

at 40,000 cpsa for Cu

event (eV) eventsb

Scintillation <1%

Very good 45% 350 23

Proportional

15%

Good, but fails at 14% 26 310

high photon flux

Solid-state Up to 50% Pileup in mid-range 2% 3.7 2200

cps

counts per second.

Approximate number of ion pairs or visible light photons resulting from a single x-ray

photon assuming Cu

radiation with photon energy of about

keV.

130 Chapter

Finally, the resolution of the detector characterizes its ability to resolve x-

ray photons of different energy and wavelength. The resolution

(R)

defined as follows:

where

is the average height of the voltage pulse and

is the spread of

voltage pulses. The latter is also defined as the full width at half-maximum

of the pulse height distribution in Volts. The resolution for Cu

radiation

for the main types of detectors is listed in Table

2.5.

Thus, the resolution is a

function of both the number of the events generated by a single photon and

the energy required to generate the event, and it is critically dependent on

how small is the spread in the number of events generated by different

photons with identical energy.

other words, high resolution is only viable

when every photon is absorbed completely, which is difficult to achieve

when the absorbing medium is gaseous, but is nearly ideal in the solid state

simply due to the difference in their densities (see section 2.3.2.1).

2.4.2

Classification

detectors

Historically, the first and the oldest detector of x-rays, which was in use

for many decades, is the photographic film. Just as the visible light, the x-ray

photons excite fine particles of silver halide when the film is exposed to x-

rays. During the development, the exposed halide particles are converted

into black metallic silver grains. Only the activated silver halide particles,

i.e. those that absorbed several x-ray photons (usually at least

photons), turn into metallic silver.

This type of detector is simple but is no longer in common use due to its

low proportionality range, and limited spatial and energy resolution.

Furthermore, the film development process introduces certain

inconveniences and is time consuming. Finally, the information stored on the

developed photographic film is difficult to digitize.

modern detectors the signal, which is usually an electric current, is

easily digitized and transferred to a computer for further processing and

analysis.

general, detectors could be broadly divided into two categories:

ratemeters and true counters.

a ratemeter, the readout is performed after

hardware integration, which results, for example, in the electrical current or

a voltage signal that is proportional to the flux of photons entering the

detector. True counters, on the other hand, count individual photons entering

through the detector window and being absorbed by the detector.